The Modelling and Optimisation of High Performance Internal Combustion Engines

by Julian Ross Panting

A thesis submitted for the degree of Doctor of Philosophy of the Faculty of Engineering, University of London, and the Diploma of Imperial College

July 1993

Department of Mechanical Engineering Imperial College of Science, Technology and Medicine University of London

1 Abstract

The project described addresses the numerical modelling of two types of small, novel internal combustion engines of unusually high power/weight ratio in order to obtain data upon their suitability for automotive application. The particular data sought are reliable estimates of the part and full load thermal efficiencies and the maximum specific power output level. The engine types studied are an unusual form of two shaft gas turbine with and without a heat exchanger, and a compound highly turbocharged, spark ignition, high speed, four stroke piston engine. The latter may be considered a hybrid of gas turbine and spark ignition piston engines. Maximum specific power output projections are based upon the assumption of high turbine entry temperatures of the order of 1600K. This in turn would require future use of ceramics technology if an uncooled turbine were to be used. It is assumed the engine is coupled with a continuously variable transmission, which permits greater flexibility in engine operation and hence enhanced performance levels. The numerical models used are of well known types - iterative matching calculations based on turbomachinery maps for the gas turbine simulations and the filling and emptying model type for the compound turbocharged engine calculations. A feature of the studies as applied to the latter engine type is the use of a sophisticated multi - dimensional optimisation algorithm to maximise simulated engine performance. The particular algorithm used is of the well known conjugate - gradient type. With this algorithm, the value of a number of independent input parameters can each be chosen to obtain the best engine performance, with this being defined, for the purposes of this work, as the specific power output.

2 Acknowledgements

My thanks go to the following people. To my supervisor, Dr. N.Baines for providing support and encouragement and for arranging a special grant which enabled my third year studies to continue. To my parents and sister for also providing support and encouragement and to my parents for providing considerable financial assistance. To Rev. W.Raines for arranging my student accommodation. To Dr. S.Etemad for permitting me to use his section's computing facilities. Finally to the Imperial College Computer Centre help desk for providing much helpful advice.

3 CONTENTS

Chapter (1) Considerations for the Optimum Design of a High Specific Output, High Thermal Efficiency Powertrain (1.1) Overview 23 (1.2) Continuously Variable Transmission Design 27 (1.3) Hybrid Vehicles- 31 (1.4) Compact, High Efficiency Engines - the Gas Turbine verses the Compound Highly Turbocharged Engine 34 (1.5) Simultaneously Optimising Many Design Parameters and the use of Continuously Variable Mechanisms 36 (1.6) Emissions Considerations 39 (1.7) Turbine Entry Temperature Limits 40 (1.8) The Structure of this Thesis 41 (1.9) References 43

4 Chapter (2) The Determination of Fluid Properties (2.1) Overview 57 (2.2) The Calculation of Fluid Properties for 'Low' Temperatures 59 (2.3) The Calculation of Fluid Properties for 'High' Temperatures 62 (2.3.1) Burnt Fuel/Air Mixture Properties 62 (2.3.2) Concerning the Calculation of the Specific Gas Constant 66 (2.3.3) Unburnt Fuel Vapour Properties 67 (2.3.4) Combining Burnt and Unburnt Gas Mixture Properties 68 (2.3.5) The Fuel Heat Content Value 69 (2.4) Summary 70 (2.5) References 71

5 Chapter (3) Automotive Gas Turbine Studies (3.1) Overview 90 (3.2) Basic Gas Turbine Performance Analysis 92 (3.3) The use of Turbomachinery Characteristics 97 (3.4) The General Calculation Procedure Invoking the use of Turbomachinery Characteristics and Variable Properties 99 (3.5) Modelling a Single Shaft, Constant Speed Gas Turbine without Variable Geometry 102 (3.6) Modelling the Twin Shaft Differential Gas Turbine without Variable Geometry 105 (3.7) A Note Concerning the Differential Gas Turbine Operating Schedule 107 (3.8) Alternative Gas Turbine Concepts which were not Modelled 107 (3.9) Summary 109 (3.10)References 110

6 Chapter (4) The Compound Highly Turbocharged Engine (4.1) Overview 129 (4.2) The Theoretical Advantages of the Compound Turbocharged Engine 131 (4.2.1) The Efficiency of the Compound Turbocharged Engine in a Limiting Case 131 (4.2.2) Consideration of the Optimum Pressure Ratio for Efficiency for the Gas Turbine 133 (4.2.3) The Compound Turbocharged Engine Considered as a High Pressure Ratio Unit 135 (4.2.4) The Specific Power Outputs of Various Engine Types Compared 139 (4.2.5) Further Considerations 141 (4.3) A Brief Review of Past Compound Turbocharged Engine Designs 142 (4.4) Justifying Further Research into the Compound Turbocharged Engine 143 (4.5) The Proposed Design Layout 144 (4.5.1) Four Cylinder Layout 145 (4.5.2) Four Stroke, Spark Ignition, High Speed Cycle 145 (4.5.3) Hybrid Sleeve/Aspin Valves 145 (4.5.4) Swash Plate Crank Mechanism 147 (4.5.5) Variable Cylinder Compression Ratio 148 (4.5.6) Throttleless Concept 149 (4.5.7) High TET Operation 150 (4.5.8) The Overall Concept 150 (4.6) The Turbine Entry Temperature Limit 152 (4.7) Transient Response 154 (4.8) A Simple Quasi-Steady Compound Turbocharged Engine Model 154 (4.9) Summary 157 (4.10)References 158

7 Chapter (5) Time - Marching Numerical Models of Naturally Aspirated and Compound Turbocharged Spark Ignition Piston Engines (5.1) Overview 175 (5.2) A Comparison of Modelling Strategies 176 (5.3) The Filling and Emptying Model 178 (5.3.1) The Governing Ordinary Differential Equations 178 (5.3.2) The Mass Flow Differential Equation 180 (5.3.2.1) The Basic Mass Flow Rate Equation 180 (5.3.2.2) Determination of the Flow Coefficient and Port Area 182 (5.3.3) The Energy Flow Rate Differential Equation 184 (5.3.3.1) Combustion Simulation 184 (5.3.3.2) The Heat Transfer Rate 187 (5.3.3.3) Modelling Direct Fuel Injection 188 (5.3.3.4) Cylinder Volume Calculation 189 (5.3.3.5) Frictional Loss Calculations 190 (5.3.4) Convergence of the Solution 191 (5.3.5) The Order of Integration 192 (5.4) Modifications to Produce a Compound Turbocharged Filling and Emptying Model 192 (5.4.1) The use of Turbomachinery Characteristics 192 (5.4.2) Modelling an Aftercooler 197 (5.4.3) The use of a Variable Cylinder Volumetric Compression Ratio 198 (5.4.4) Estimating the Turbine Entry Temperature 199 (5.5) Determining the Power Output and Brake Thermal Efficiency 199 (5.6) Summary 200 (5.7) References 201

8 Chapter (6) Ensuring the Stability of the Integration of the Mass Balance O.D.E. (6.1) Overview 215 (6.2) An Example of the Mass Flow Rate Calculation Instability Problem 216 (6.3) Analytic Stability of Integration of an O.D.E. 217 (6.4) The Numerical Stability of the First Order Euler Scheme 221 (6.5) Applying the Stability Condition 222 (6.6) Numerical Integration and Ill - Conditioned Problems 223 (6.7) Controlling the Ill - Conditioned Problem 226 (6.8) The Complete Algorithm 229 (6.9) Summary 229 (6.10)References 231

9 Chapter (7) Minimisation of a Function of a Vector - and the Application of this Technique to Powertrain Design Optimisation (7.1) Overview 236 (7.2) Multi - Dimensional Optimisation Methods Employing Line Minimisations 239 (7.2.1) The Orthogonal Directions Method 240 (7.2.2) The Method of Steepest Descent 241 (7.2.3) The use of Conjugate Directions 241 (7.2.3.1) The Conjugate Directions Method 244 (7.2.3.2) The Conjugate Gradient Method 245 (7.2.4) Quasi - Newton or Variable Metric Methods 246 (7.3) Multi - Dimensional Optimisation Employing the 'Monte-Carlo' Technique 247 (7.4) The Choice of Multi - Dimensional Optimisation Method 248 (7.5) Bracketing the Minimum and Employing a Line Minimisation Technique 249 (7.6) Calculating the Distances to the Input Vector Boundaries 253 (7.7) Evaluating the Gradient Vector 256 (7.8) Convergence of the Solution 257 (7.9) Applying the Multi - Dimensional Minimisation Technique to Powertrain Design Optimisation 257 (7.10)Summary 259 (7.11)References 260

10 Chapter (8) Numerical Results (8.1) The Gas Turbine Models 262 (8.2) Validation of the Piston Engine Models 265 (8.3) Optimisation Results 267 (8.3.1) Employing Test Mathematical Functions 267 (8.3.2) Employing the Quasi-Steady Piston Engine Models 267 (8.3.3) Employing the Filling and Emptying Piston Engine Models 269 (8.3.4) Part Validation of the Optimisation Results 272 (8.4) Further Aspects of the Filling and Emptying Piston Engine Modelling 274 (8.4.1) Increasing the Bore/Stroke Ratio of the Naturally Aspirated Engine 274 (8.4.2) Some Adjustments to the Compound Turbocharged Engine Model Input Parameters 274 (8.4.3) Part Load Efficiency Contours 276 (8.4.4) Further Data from the Compound Turbocharged Engine Model 278 (8.5) Comparing the Gas Turbine and Piston Engines' Specific Power Output Figures 278 (8.6) References 280

11 Chapter (9) Conclusions and Suggestions for Further Work (9.1) Conclusions 320 (9.1.1) The Gas Turbine Simulations 320 (9.1.2) The Compound Turbocharged Engine Simulations 321 (9.1.3) Comparisons Between the Gas Turbine and Compound Turbocharged Engine Simulations 323 (9.1.4) The Optimisation Procedure 325 (9.2) Suggestions for Further Work 326 (9.2.1) Modifications to the Filling and Emptying Method 326 (9.2.2) Empirical Measurements 327 (9.2.3) The Detonation Limit 327 (9.2.4) Increasing the Specific Power Output 327 (9.2.5) The Navier-Stokes Equations 329 (9.2.6) Development of the Unconstrained Optimisation Algorithm 330 (9.2.7) Constrained Optimisation 330 (9.2.8) Optimising Additional Parameters 331 (9.3) References 332

12 Appendices

Appendix Appendix Number Title

(A1) The Entropy Function 336 (A2) Second Order Interpolation from Tabulated Property Values 338 (A3) The Calculation of Gas Turbine Performance Assuming Constant Fluid Properties 342 (A4) The Calculation of Gas Turbine Performance Invoking the use of Variable Properties 346 (A5) A Simplified Analysis of the Compound Turbocharged Engine Cycle 349 (A6) The Analysis of the Engine Balancing of an Engine Fitted with a Swashplate Crank Mechanism 356 (A7) The Compound Turbocharged Engine Layout 359 (A8) The Quasi-steady Piston Engine Model 361 (A9) The Numerical Integration Scheme Employed for the Filling and Emptying Model 366 (A10)The Mass Flow Rate as a Function of Pressure Ratio 373 (All) A Method to Calculate the Cylinder Wall Temperature 377 (Al2) Evaluating the Stability Parameter 378 (A13)The Step Size Control Method used for the Filling and Emptying Model 382 (A14)The Gradient Vector of a Quadratic Function 386 (A15)Proof that Conjugate Directions give Rise to Exact Minimisation of a Quadratic Function in Exactly n Line Minimisations 388 (A16)Derivation of the Conjugate Gradient Algorithm 390 (A17)Comparisons of Estimated Relative Program Execution Time, for the Two 'Conjugate' Optimisation Methods 395 (A18)The Constrained Optimisation Problem 397

13 List of Figures

Figure Number Figure Title

(1.1) Viscous Coupling 46 (1.2) Variable Pulleys 47 (1.3) Some CVTs of Sliding Mechanical Contact Form 48 (1.4) Electrical CVT 49 (1.5) Hybrid Mechanical/Electrical CVT 50 (1.6) Hybrid Mechanical/Hydraulic CVT 51 (1.7) Various Driving Modes of the Hybrid Vehicle 52 (1.8) Regenerative Braking 53 (1.9) Two Automobile Gas Turbine Configurations 54 (1.10) The Compound Turbocharged Engine Concept 55 (1.11) Turbine Life as a Function of TET 56

(2.1a) Specific Heat Constant at Constant Pressure versus Temperature for Air at Low Temps 73 (2.1b) Ratio of Specific Heats versus Temperature for Air at Low Temperatures 74 (2.1c) Specific Enthalpy versus Temperature for Air at Low Temperatures 75 (2.1d) Non-Dimensional Entropy Function versus Temperature for Air at Low Temperatures 76 (2.2) Second Order Interpolation Method used to Obtain Property Values 77 (2.3a) Specific Heat Constant at Constant Pressure versus Temperature for Equilibrium Gas 78 (2.3b) Ratio of Specific Heats versus Temperature for an Equilibrium Gas 79 (2.3c) Specific Enthalpy versus Temperature for an Equilibrium Mixture of Burnt Gas or Air 80 (2.3d) Non-Dimensional Entropy Function versus Temperature for an Equilibrium Gas 81 (2.3e) Specific Gas Constant versus Temperature for an Equilibrium Gas 82

14 Figure Number Figure Title

(2.4a) Specific Heat Constant at Constant Pressure versus Temperature for Octane Vapour 83 (2.4b) Ratio of Specific Heats versus Temperature for Octane Vapour 84 (2.4c) Specific Enthalpy versus Temperature for Octane Vapour 85 (2.5) Empirical Combustion Efficiency vs. Equivalence Ratio Curve (Cubic Spline Curve Fit) 86

(3.1) The Simple Gas Turbine without Heat Exchanger 112 (3.2) Thermal Efficiency versus Cycle Pressure Ratio for a Simple Gas Turbine 113 (3.3) The Gas Turbine Plus Heat Exchanger 114 (3.4) Thermal Efficiency versus Cycle Pressure Ratio for a Gas Turbine Plus Heat Exchanger Cycle Temperature Ratio = 1200K / 288K 115 (3.5) Thermal Efficiency versus Cycle Pressure Ratio for a Gas Turbine Plus Heat Exchanger Cycle Temperature Ratio = 1600K / 288K 116 (3.6a) Pressure Ratio versus Non-Dimensional Mass Flow Function for a Centrifugal Compressor 117 (3.6b) Isentropic Efficiency v Non-Dimensional Mass Flow Function for a Centrifugal Compressor 118 (3.7a) Pressure Ratio versus Non-Dimensional Mass Flow Function for a Radial Turbine 119 (3.7b) Isentropic Efficiency v Non-Dimensional Mass Flow Function for a Radial Turbine 120 (3.8) The Construction of a Speed Line 121 (3.9) The Iterative Solution Process for the Single Shaft Gas Turbine 122 (3.10) The Differential Gas Turbine with Mechanical Epicyclic Transmission 124

15 Figure Number Figure Title

(3.11) The Iterative Solution Process for the Differential Gas Turbine 125 (3.12) Various Gas Turbine Design Concepts 127

(4.1) The High Pressure Ratio, Low Heat Release, Compound Turbocharged Engine Cycle 160 (4.2) The Sleeve Valve 161 (4.3) The Aspin Valve 162 (4.4) The Hybrid Sleeve/Aspin Valve 163 (4.5) The Swashplate Mechanism 164 (4.6) A Particular Type of Swashplate Mechanism 165 (4.7) Variation of the Cylinder Volumetric Compression Ratio through Alteration of the Piston Stroke 166 (4.8) Variation of the Cylinder Volumetric Compression Ratio through Alteration of the Clearance Volume 167 (4.9) Compound Turbocharged Engine Concept with Two Stages of Intercooling and Auxiliary Combustion 168 (4.10) Simplified Pressure-Volume Diagram for the Compound Turbocharged Engine 169

(5.1) The Filling and Emptying Method Schematically Illustrated for a Four

Cylinder Engine 203 (5.2) The Filling and Emptying Model Plus an Unsteady Dynamic Flow Model for the

Manifolds 204 (5.3) Mass Flow Characteristic versus Pressure

Ratio 205 (5.4) Non-Dimensional Poppet Valve Lift versus

Non-Dimensional Crank Angle 206

16 Figure Number Figure Title

(5.5) Max Valve Lift to Diameter Ratio versus Valve Opening Duration in Crank Angle

Degrees 207 (5.6) Valve Flow Coefficient versus Valve

Lift/Diameter Ratio 208 (5.7) Mass Fraction Burnt Curve versus

Non-Dimensional Crank Angle 209 (5.8) Combustion Duration in Crank Angle Degrees

versus Equivalence Ratio 210 (5.9) The Modified Square Form Fuel Injection

Characteristic 211 (5.10) Compressor Pressure Ratio versus Mass Flow

Rate 212 (5.11) Compressor Isentropic Efficiency versus Mass

Flow Rate 213 (5.12) Turbine Isentropic Efficiency versus

Pressure Ratio 214

(6.1) Implicit Integration of the Mass Flow Equation at 2500 rpm (step size = 1/4 degree

CA) 232 (6.2) Variation of Mass Flow Rate Curves with

Engine Speed 233 (6.3) Comparison of True and Approximate Mass Flow

Functions 234 (6.4) Error Between True and Approximate Mass Flow

Functions 235

(7.1) The Use of Line Minimisations to Optimise a Function 261

(8.1a) Part Load Thermal Efficiency versus Specific Power Output for Single Shaft Gas Turbine

Max TET is 1200K 281

17 Figure Number Figure Title

(8.1b) Part Load Thermal Efficiency versus Specific Power Output for Single Shaft Gas Turbine Max TET is 1600K 282 (8.2) TET versus Specific Power Output for the Single Shaft Gas Turbine 283 (8.3) Compressor Pressure Ratio versus Specific Power Output for Single Shaft Gas Turbine 284 (8.4a) Part Load Thermal Efficiency versus Specific Power Output for the DGT Max TET is 1200K 285 (8.4b) Part Load Thermal Efficiency versus Specific Power Output for the DGT Max TET is 1600K 286 (8.5) TET versus Specific Power Output for the Differential Gas Turbine 287 (8.6) Compressor Pressure Ratio versus Specific Power Output for Differential Gas Turbine 288 (8.7a) Non-Dimensional Turbine Speed versus Non-Dimensional Compressor Speed for the DGT Max TET is 1200K 289 (8.7b) Non-Dimensional Turbine Speed versus Non-Dimensional Compressor Speed for the DGT Max TET is 1600K 290 (8.8a) Non Dimensional Turbine Speed versus Specific Power Output for the DGT Max TET is 1200K 291 (8.8b) Non Dimensional Turbine Speed versus Specific Power Output for the DGT Max TET is 1600K 292 (8.9a) Non-Dimensional Gross Turbine Torque versus Specific Power Output for the DGT Max TET is 1200K 293 (8. 9b) Non-Dimensional Gross Turbine Torque versus Specific Power Output for the DGT Max TET is 1600K 294

18 Figure Number Figure Title

(8.10) The use of a Three Dimensional Orthogonal Grid to Validate a Three Dimensional Optimisation 295 (8.11) Specific Power Output versus Piston Engine Speed for a Naturally Aspirated Engine 296 (8.12a) Thermal Efficiency versus Piston Engine Speed for a Naturally Aspirated Engine 297 (8.12b) Thermal Efficiency versus Specific Power Output for a Naturally Aspirated Engine 298 (8.13a) Thermal Efficiency versus Piston Engine Speed for a Compound Turbocharged Engine 299 (8.13b) Thermal Efficiency versus Specific Power Output for a Compound Turbocharged Engine 300 (8.14) Cylinder Pressure versus Non-Dimensional Cylinder Volume 301 (8.15a) Cylinder One Inlet Mass Flow Rate versus Crank Angle 302 (8.15b) Cylinder One Exhaust Mass Flow Rate versus Crank Angle 303 (8.16a) Compressor Mass Flow Rate versus Cylinder One Crank Angle 304 (8.16b) Turbine Mass Flow Rate versus Cylinder One Crank Angle 305 (8.17) Specific Power Output versus Piston Engine Speed for a Compound Turbocharged Engine 306

(9.1) Isothermal Compression Work Divided by Isentropic Compression Work versus Pressure Ratio 333 (9.2) Two Stage Compression Work Divided by One Stage Compression Work versus Pressure Ratio 334

19 Figure Number Figure Title

(A2.1) Second Order Interpolation 341 (A5.1) The Pressure-Volume Diagram for the Simplified Compound Turbocharged Engine

Cycle 355

20 List of Tables

Table Number Table Title

(2.1) The Constants used for the Calculation of Air Properties at Low Temperatures 87 (2.2) Base Values of the Specific Gas Constant 88 (2.3) The Constants used for the Calculation of Unburnt Fuel Vapour (Octane) Properties 89

(3.1) The Optimum Thermal Efficiencies of the Simple Gas Turbine and the Corresponding

Optimum Pressure Ratios 128

(4.1) Optimum Efficiency Levels of a Simple Gas Turbine for a given Cycle Temperature Ratio, and the Corresponding Compressor Pressure Ratio 170 (4.2) Approximate Compressive Pressure Ratios for a Compound Turbocharged Engine at Two Different Altitudes 171 (4.3) Specific Power Outputs of Various Engine Types 172 (4.4) Napier Nomad Engine Design Details 173 (4.5) Wallace Differential Compound Engine Design Details 174

(8.1) Comparisons Between the Specific Power Output of a 4.5 Litre V8, Naturally Aspirated Nissan Engine and the Equivalent

Simulated Values 307 (8.2) Comparisons Between the Performance of a 2.2 Litre Conventionally Turbocharged Lotus Four Cylinder Engine, and the Equivalent

Simulated Values 308

21 Table Number Table Title

(8.3) Comparisons Between the Analytically Calculated Minimum of a Quadratic Function and the Equivalent Calculated Numerical Values 309 (8.4) The Fixed Input Parameters used for the Quasi-Steady Simulation Model Optimisations 310 (8.5) The Optimised Input Vectors and Number of Line Minimisations for the Quasi-Steady Model Optimisations 311. (8.6) Output Parameters from the Quasi-Steady Model Optimisations 312 (8.7) The Fixed Input Parameters used for the Filling and Emptying Model Optimisations 313 (8.8) The Optimised Input Vectors and Corresponding Number of Line Minimisations for the Filling and Emptying Model 314 (8.9) Simulated Output Parameters for the Filling and Emptying Model Optimisations 315 (8.10) Maximum Detected Errors in the Filling and Emptying Model Optimisations 316 (8.11) The Maximum Specific Power Output for a Simulated Naturally Aspirated Engine for Three Different Bore/Stroke Ratios 317 (8.12) Adjusted Figures for the Compound Turbocharged Engine Filling and Emptying Model 318 (8.13) Comparisons of the Maximum Specific Power Outputs of Various Engine Types 319

Illustrations

Engineering Drawings of the Proposed Compound Turbocharged Engine Layout Placed in a Pocket at the end of the Thesis, Side View and Plan View

22 Chapter (1)

Considerations for the Optimum Design of a High Specific Output, High Thermal Efficiency Powertrain

(1.1) Overview

Small, high performance powertrains enable an overall reduction in vehicle weight to be achieved, through the reduction in weight of the powertrain itself and also because this in turn permits a reduction in absolute stiffness and therefore a reduction in the weight of the vehicle structure. The reduction in weight enables a less powerful engine to be used (considering only vehicle acceleration requirements) which in turn leads to an even smaller, lighter engine. Thus a favourable feedback loop is set in place, and provided powertrain efficiency can be maintained at a good level (e.g. an overall efficiency level of 20+ %) over a wide range of power outputs, then a vehicle of high overall efficiency should result.

Continuing the theme of attaining high efficiency levels, if the powertrain contains some form of continuously variable transmission (henceforth referred to as a CVT), which enables the transmission ratio between drive and load to be continuously adjusted between upper and lower limits, the efficiency can be further increased. This is because the transmission permits the engine to adopt operating parameters (such as fuel/air ratio, ignition timing, engine speed and variable valve timing events if available) which optimise the engine efficiency for a given load, rather than the load/engine speed curve being dictated by a fixed transmission ratio.

A feature of this work is the simulated coupling of a light, efficient internal combustion engine to an electrical CVT. Such an electrical CVT has the following possible benefits

23 in addition to the above:

(i) if the electric motors of the transmission are used 'in reverse' as alternators during braking, then battery units can be recharged, (ii) the internal combustion engine can be by-passed for periods, with power directly transferred from battery units to the electric motors of the transmission - this could reduce local emissions levels in built up areas by providing a part time electric vehicle.

The first point suggests that an increased overall vehicular efficiency is possible in 'stop-start' driving, though it should be noted that a given number of batteries can only absorb a certain level of electric current before damage occurs. Thus only a certain level of regenerative braking can be provided.

Though an electrical CVT is of particular interest, the theoretical studies described in this thesis have not modelled an electrical CVT in particular, but rather have assumed any form of CVT with an idealised transmission efficiency of 100%. The justification for this is that it has much simplified the initial studies. Furthermore, estimation of the transmission efficiency and torque characteristics of an electrical transmission would require fairly detailed knowledge of such a transmission's design, and such design studies were considered beyond the scope of this thesis.

References (1), (2) and (3) describe research and development work which studies high speed electrical transmissions, designed with small high speed gas turbines in mind. Such devices are of the permanent magnet rotor type, to eliminate the need for high speed slip rings. The use of a new type of permanent magnet material, neodymium iron boron, has made such machine types more feasible. It

24 should be noted that the transmission efficiency of such units is not as high as that of mechanical transmissions. Depending on the load and speed, the efficiency of a high speed electrical alternator alone is in the mid 80's to low 90's as a percentage, whereas the efficiency of a mechanical transmission will generally be of the order 95+ %. Note, however that there are possibilities for increasing an electrical transmission's efficiency, refer to section (1.2).

Much research work is being performed into the design of hybrid internal combustion engine / electric motor vehicles, henceforth described as hybrid vehicles. Refer for example to (4), (5) and (6). Such vehicles are a means of reducing urban pollution levels where electric motor propulsion would be employed, while maintaining a good range and performance outside of urban areas by switching to internal combustion engine propulsion. The incorporation of adequate numbers of batteries into such a hybrid vehicle makes the design of a suitable lightweight internal combustion engine particularly important because of the weight of the requisite batteries.

The design of a lightweight powerplant which displays both high power to weight ratio together with good thermal efficiency across a broad power spectrum is a considerable challenge. Small gas turbine engines can be made lightweight, though achieving good efficiency levels at all power outputs has been a problem for experimental units (7). (Typical thermal efficiency levels are 20% at the design point without a heat exchanger, 40% with a heat exchanger, though generally much reduced at part load in both cases). In contrast, small diesel piston engines demonstrate good efficiency levels, but are relatively heavy. (Typical thermal efficiency levels are around 40% for a small diesel over a wide power spectrum). Spark ignition piston engines represent something of a compromise between the two, with typical peak efficiency figures in the high 20's as a

25 percentage, and power to weight ratios also less than a small gas turbine but more than a diesel. Turbocharging of course increases the power to weight ratio of a piston engine.

In order to obtain the best fuel economy and emissions over a wide range of power outputs, continuously variable features are increasingly being developed for internal combustion engines. These include variable valve timing, ignition timing, and fuel injection rate for piston engines, and for gas turbines secondary air admission to the combustor. All these are in addition to the CVT previously mentioned. The more variables there are, the more difficult it becomes to optimise the powertrain, and this is particularly true for the novel engines considered here, for which operating experience is limited or non-existent. The process of optimisation therefore forms an important part of this project. Optimisation procedures, in conjunction with variable controlling mechanisms, enable the performance of an engine to be maximised. Performance can here mean maximum power output or maximum thermal efficiency for a given power output, or minimum emissions levels.

Optimisation can be done either theoretically or through maximising the measured performance of a prototype powertrain on a testbed in conjunction with a specialised form of computer control for the powertrain. In this project, the optimisation approach is entirely theoretical, as work has not yet progressed to the building of prototypes. In any event, the theoretical optimisation approach is a first step that must be employed in the design process, prior to optimisation of a real engine which would be designed around the initial theoretical optimisation studies. This is partly because it is not possible to vary some parameters in real prototype engines, though such parameters can easily be varied computationally, e.g. piston engine bore/stroke ratio.

26 In this project, two possible alternative approaches have been adopted to a lightweight, efficient powerplant - a differential two shaft gas turbine with a heat exchanger and a small compound highly turbocharged spark ignition piston engine.

The compound turbocharged engine concept was thought to represent the greater design potential, and also had a greater number of variable input parameters, and so it was to this concept that the mathematical optimisation procedures were applied. In addition, it would have been difficult to apply the basic unconstrained optimisation method which was developed to the gas turbine studies due to the need to control the turbine entry temperature. This could be achieved in the compound turbocharged piston engine simulations through limitation of the compressor pressure ratio and hence speed. The two shaft differential gas turbine has therefore been crudely optimised, by operating the compressor so that it runs on the curve of maximum isentropic efficiency, and by operating the turbine so that the turbine entry temperature is held at a maximum value.

All studies are entirely theoretical, employing computational models. To maintain simplicity, the models have not addressed emissions considerations, nor transient engine behaviour. These are topics that would ultimately require further study if any of the design concepts were to become a working reality.

(1.2) Continuously Variable Transmission Design

A CVT is a power transmission system that permits a continuous variation of the speed ratio between the input and output power shafts, the ratio being bounded by upper and lower limits. The advantage of employing a CVT is that the internal combustion engine which provides the power

27 input can be held continuously at optimum operating points, to maximise the engine efficiency, engine power output or minimise emissions. Any real transmission operates with an efficiency of less than 100%, and this complicates the choice of the optimum transmission ratio, because the transmission efficiency will be a function of this ratio. This latter complication is not accounted for in these studies.

Various types of CVT are available. They are:

(i) viscous coupling (ii) variable size pulley(s) and expandable belt (iii)mechanical sliding contact (iv) electrical transmission (v) conventional gear train plus hydraulic pump and motor (vi) conventional gear train plus electrical transmission

For an illustration of transmission type (i) see fig (1.1). This does not give a directly controllable transmission ratio because the device is of a passive nature and adapts its speed ratio according to the torque/speed characteristics of the drive and the load. This type of unit also imposes large viscous losses, especially when the slip of the viscous coupling is high. This type of unit is therefore not considered suitable for the application of interest.

For illustrations of transmission types (ii) and (iii) see fig (1.2) and fig (1.3) respectively. There are many possible variations on design type (iii). These options permit directly controllable transmission ratios, and the transmission efficiency of such units is high. Essentially, then, such units should be suitable for the application of interest, option (iii) being preferred due to its potentially greater compactness. However, the high rotational speeds of small turbomachines produces high

28 surface speeds which may be a problem for both options. For a given design of turbomachine the rotational speed is inversely proportional to the diameter, so the problem remains approximately the same whatever the size of engine. This is seen by considering that the average rotodynamic tip blade speed is limited by material constraints' to be an approximate constant. The blade speed is generally made as high as possible, subject to material strength constraints, to maximise the power transfer of the turbomachine. As this blade speed is directly related to the transmission surface th.it tha speed, it is seen A.surface speed is also an approximate constant for all sizes of turbomachine. However, as smaller turbomachinery components have a higher rotational frequency (for a given form of design), fatigue limitations of the transmission in a smaller device could also become important. This is because of the higher number of rotational cycles in a given operating time span.

The electrical CVT (option (iv)) is illustrated in fig (1.4). This does not suffer from direct limitations to the surface speed of the component, because there is no requirement for surface contact. However there is still a limitation on the diameter in order to control centrifugal stresses. References (1), (2) and (3) describe practical design studies into high speed electrical alternators and motors. It is probable that the overall transmission efficiency of option (iv) would be lower than for options (ii) and (iii). However, research into high speed electrical alternators is at an early stage, and further work may bring further benefits in transmission efficiency. It would be possible to boost the transmission efficiency of electrical transmissions by one or all of the following,

(iv)(a) reduce the current density (at the expense of increased component size), (iv)(b) improve the stator design to minimise copper loss, (iv)(c) 2-educe component speed at low power transfer rates

29

(which reduces the relative size of viscous drag losses relative to the power being transferred), (iv)(d) reduce component diameter (at the expense of component length), (iv)(e) perform design optimisation to produce the right balance between the cooling requirement and windage loss, (iv)(f) use a low viscosity, low density gas in the alternator/electric motor interior, e.g. hydrogen.

Options (iv)(a) and (b) reduce the heat losses through electrical resistance, while options (iv)(c)-(f) reduce the losses due to viscous drag. Option (iv)(f) brings problems of extra complexity and sealing difficulties, and may not be a practical option.

The continuously variable high speed electrical transmission is preferred for the hybrid vehicle application. Some form of electrical transmission must be provided in such vehicles, and it appears opportune to make such a transmission the only component of the power transfer mechanism, rather than employ a mechanical CVT plus electrical transmission design. The latter 'hybrid transmission' (as opposed to hybrid vehicle) is illustrated in fig (1.5). Some combination of mechanical CVT plus electrical transmission could however become a necessity because of possible (torque / speed) and / or (efficiency / speed) limitations of the electrical components. This is due to the facts that at low power outputs the efficiency of the constant speed high speed alternator falls, while the high speed electrical motor is limited to a maximum torque output at low rotational speeds. (Regarding the last point, the motor torque at low speeds is limited by the capability for heat dissipation, therefore it is possible to over-run substantially for short periods e.g. for acceleration.)

30 Option (v) is illustrated in fig (1.6). The principle of this type of mechanism is that a variable displacement hydraulic pump takes power from one shaft of an epicyclic geartrain and feeds it to another, via an hydraulic motor. By varying the displacement of the hydraulic pump it is found the overall transmission ratio can be varied. Reference (8) describes a design study into such a mechanism, which demonstrates a good overall transmission efficiency over a broad power range. The hybrid mechanical gear/hydraulic pump and motor CVT could also be designed as an hybrid mechanical gear/electric alternator and motor CVT, and this in turn has parallels with the concept illustrated in fig (1.5). This transmission type is itemised as type (vi). By mounting the main alternator on a high speed shaft, the alternator size can be kept relatively small.

The studies presented in this thesis do not include the detailed design of the power transmission mechanism. Efforts have been concentrated upon the determination of the thermodynamic possibilities of potential power units. It is important to note, however, that these design studies do assume some form of CVT, with the specific design of this mechanism yet to be decided.

(1.3) Hybrid Vehicles

Hybrid vehicles, which at present only exist in prototype form for automotive usage, are designed to contain a hybrid of internal combustion engine and battery driven electric motor propulsion. In urban areas, power would be provided by battery units directly driving electric motors, eliminating automotive urban emissions. On the open road, power would be provided by the internal combustion engine, which could also recharge the batteries. Conceptual illustrations of hybrid vehicles are shown in fig (1.7).

A rapidly growing world vehicle fleet is producing an

31 increasing global exhaust emissions problem, (9), (10). A number of alternative technologies have been studied to address this challenge. Among these are electric vehicles, hybrid vehicles, alternatively fueled vehicles and hybrid fueled vehicles. These are reviewed extensively in (10). Assessing the relative environmental benefits of these various options is complex. In this context, the case for the hybrid vehicle relative to the electric vehicle is discussed.

There is concern that pure electric vehicles would not necessarily reduce overall emissions levels, as fossil fuelled power stations would provide the bulk of the energy necessary to recharge the batteries of such vehicles. In turn, these power stations would produce emissions of their own. There have even been suggestions that, in the long term, a large number of electric vehicles would increase overall emissions levels - though they would reduce urban emissions levels. The environmental benefit of the hybrid vehicle relative to the electric vehicle is critically dependent upon such parameters as power stations emissions levels, power station generation and transmission efficiency, the hybrid vehicle's emissions levels and the overall vehicular efficiency of the hybrid and electric automobiles being considered. Thus, for example, inefficient, high emissions power stations and low emissions hybrid vehicles favour consideration of hybrid vehicles relative to electric vehicles and vice versa. Future technology, in the form of cleaner, more efficient power stations will therefore favour the electric vehicle. An increased number of power stations would be necessary to provide recharging capacity for a large number of electric vehicles, though due to excess power grid capacity the required percentage increase would not be as large as simple calculations suggest. It is open to question whether an increased number of nuclear power stations which could be required to recharge electric vehicle's batteries would be

32 acceptable. Though such power stations do not produce atmospheric pollution directly, they do of course produce radioactive waste. Renewable power sources, such as hydro-electric power, do not produce waste but their number cannot be determined arbitrarily - i.e. there have to be suitable environmental conditions to enable them to be built.

Putting aside the discussion of the relative environmental impact of hybrid and electric vehicles, there is another powerful argument in favour of the hybrid vehicle. Purely electrical vehicles suffer from a severe power output and range limitation problem. This in turn is due to the limitations of present battery technology. Hybrid vehicles can switch to internal combustion engine driving mode outside of urban areas, and their potential range and power output is therefore much extended. Reference (4) describes such a hybrid vehicle concept, employing a small gas turbine, while (5) is a description of the same vehicle in a non-technical journal. Reference (6) also describes a gas turbine driven hybrid vehicle.

Additional benefits accrue from hybrid vehicle designs. The electrical transmission of such vehicles can be used as a CVT giving benefits in vehicular efficiency as this permits the internal combustion A to be always operated at some optimum condition. A further benefit is that an electrical transmission can be operated 'in reverse', hence providing braking at the driven wheels if the electric motors are used as alternators, which in turn produce electric current to recharge battery units. This is regenerative braking, with some of the kinetic energy being reclaimed as chemical energy within the batteries. This reclaimed energy can then be used on demand. Only some of the energy is reclaimed due to inefficiencies within the overall system. The process is illustrated in fig (1.8).

33 (1.4) Compact, High Efficiency Engines - the Gas Turbine verses the Compound Highly Turbocharged Engine

As explained in section (1.1), lightweight powerplants provide overall vehicular efficiency benefits. These benefits become particularly important in hybrid vehicles, as such designs require a number of battery units in order to facilitate the electric vehicle part requirement of such designs. These battery units in turn entail a weight addition to the vehicle, which can be offset with a lighter primary powerplant. It is important, however, that the search for a lighter powerplant does not entail too large a compromise in the thermodynamic efficiency of that powerplant, for this would be counter productive in terms of producing a vehicle of high overall efficiency. The immediately obvious candidate for a lightweight internal combustion engine is the gas turbine. Small gas turbines generally employ heat exchangers to enable sufficiently high thermal efficiencies to be attained. However, despite this, the thermal efficiency of such units is often poor, particularly at part load, (7). In addition, the transient response of gas turbines is usually poor. This is due to the fact that gas turbines are high speed, low torque devices.

The compound highly turbocharged engine is an alternative to a high pressure ratio gas turbine without a heat exchanger. A high pressure ratio gas turbine will generally display good thermal efficiency levels, even without a heat exchanger. However, the pressure ratio of a small gas turbine is limited by the following considerations:

(i) with a small, high pressure ratio gas turbine, the flow passage areas at the high pressure ends of the compressor and turbine become very small, much reducing the isentropic efficiencies of these components due to boundary layer effects and disproportionately large clearance areas.

34 (ii) high pressure ratio compressors have very narrow regions of stable operation, and measures to combat this result in extra expense and complexity, e.g. variable geometry, multiple low pressure ratio compressors in series or a combination of both. (iii)to keep the design simple, small gas turbines use single centrifugal compressors and there is a design limit to the pressure ratio that can be achieved by such a single centrifugal compressor design. Thus, the higher the pressure ratio, the larger the number of compressor and turbine stages, which increases the weight and complexity of the engine. Modest pressure ratios (up to about 6:1) can be achieved with a single stage centrifugal compressor with reasonable flow range. (iv) high pressure ratios imply large pressure forces on the engine structure and casings, which thus require heavier components.

The author has studied two gas turbine concepts. These are:

(i) a simple, single shaft, constant speed design, (ii) a more sophisticated twin shaft design in which the turbine and compressor are physically separate and are able to rotate at variable, and different, speeds.

Concept (ii) is known as a differential gas turbine, and gives more freedom to optimise the thermodynamic potential of the engine. Fig (1.9) illustrates these two concepts. Concept (ii) is made more feasible by the advent of the high speed electrical transmission discussed in sections (1.1) and (1.2).

The compound turbocharged engine differs from the gas turbine, in that there is a two stage compression and expansion process. Air is compressed, generally in a centrifugal compressor, from where it passes to a compact

35 piston engine. Here the air, mixed with fuel in a spark ignition engine, is further compressed, burnt, and passes through the first stage of expansion. After the first stage of expansion, the exhaust gases are passed from the piston engine to the turbine, where the second stage of expansion occurs. Both the turbine and the piston engine produce power, while the compressor absorbs power. These units must therefore couple their power outputs and inputs. The overall concept is illustrated in fig (1.10). The bulk of the project has been concerned with demonstrating, theoretically, the potential of the compound turbocharged unit. Automotive compound turbocharged engines have never been commercially produced. The latter point is considered to be for two main reasons,

(i) at modest turbine entry temperatures (-1200K) the power output and thermal efficiency gains of the compound turbocharged engine over the conventionally turbocharged engine are relatively modest, (ii)this is offset by the mechanical complexity of providing a CVT mechanism between the piston engine and turbine and compressor.

In the scheme considered in this project, TETs are higher (up to 1500K) which in turn increases the relative advantages of the compound turbocharged engine. Further, it is considered that the advent of high speed electrical transmissions makes for a more feasible overall concept.

(1.5) Simultaneously Optimising Many Design Parameters, and the use of Continuously Variable Mechanisms

A feature of the studies as applied to spark ignition piston engine simulations has been the use of a multi - variable optimising algorithm. There are many parameters which influence the performance of a spark ignition engine. Engine performance is here measured as either overall thermal

36 efficiency or overall power output. These two output values cannot in general be optimised simultaneously. Examples of parameters that can be optimised are ignition timing, equivalence ratio, valve events timing, and others. In order to provide a more complete engine simulation/design package, a multi - variable optimisation procedure has been developed to predict the optimum values of these parameters.

It was intended that such an optimisation procedure would be used for the design of continuously variable mechanisms, such as,

(i) variable valve timing (ii)CVTs.

Mechanisms of type (i) enable the optimum valve timing to be adopted, whatever the engine speed. There is increasing interest in variable valve timing mechanisms, such technology being reviewed in (11). Such mechanisms incorporate variations on the cam/spring/poppet valve theme, or use more sophisticated actuation devices, for example an electro - pneumatic system. Though the optimisation procedure is ideally suited to use in the design of variable valve mechanisms, limitations on available computer time have not enabled the process of optimising a continuously variable valve train to be adopted.

CVTs enable an engine to be held in an optimum power, or optimum efficiency mode. For example, in optimum efficiency mode where all the variable parameters are set to optimise efficiency, there is a unique power versus engine speed curve. The engine speed can be selected to provide the demand power output according to this curve, with the CVT adjusting the engine speed versus road speed ratio as appropriate. When the engine is set to optimise power levels there is an additional power verses engine speed curve, and this curve can also be used to select a transmission ratio

37 in a similar fashion.

The concept, then, is to operate the engine in an 'efficiency' mode up to an engine speed near that at which the maximum optimised power level occurs. There is then a transition region in which a weighted optimisation of power and efficiency occurs, passing gradually from entirely optimising efficiency to entirely optimising power output. The maximum, and optimised, power output is then achieved. This can be described in more detail as follows.

If N1 < Nh

where N1 is a piston engine speed at which the thermal efficiency is optimised, and Nh is the engine speed at which maximum power is generated, then, if,

N - N a = (1.2) N - N h 1 the parameter (3 is maximised over the transition region, where,

= (1 - al 7/(N) a w(N) 77(N1 ) w(Nh) (1.3)

and 77 is the thermal efficiency, w is the power output.

Thus, the process of optimising a powertrain for either overall thermal efficiency or power output, independently of an assumed load verses engine speed curve, is equivalent to invoking the use of a CVT.

38 (1.6) Emissions Considerations

The emissions output of the simulated engine types has not been specifically calculated in this project, the aim instead being to find projected power output and efficiency levels. Nevertheless comments are here made on the likely emissions levels.

Gas turbines operate at lower peak cycle temperatures than piston engines and consequently, because the levels of burnt gas mixture chemical dissociation are lower, the emissions levels of gas turbines are lower than for piston engines. Overall, the gas turbine can be considered a very 'clean' engine, and this is an important advantage over the piston engine.

There is no particular reason to suppose the basic compound turbocharged engine produces especially higher or lower emissions levels than for the equivalent naturally aspirated piston engine. There is, however, a possible derivative of the compound turbocharged engine which is more of a hybrid between the gas turbine engine and the former. Here, additional combustion occurs after the burnt fuel/air mixture (which must be fuel lean) exits the piston engine and before it enters either the turbine or a LP turbine. The additional combustion occurs at relatively low temperature, and is therefore likely to reduce overall emissions levels by reducing the amount of chemical dissociation. This measure would reduce overall thermal efficiency, though may increase specific power output. This configuration has not been modelled.

It would be possible to develop an optimisation procedure to optimise power output or efficiency subject to the constraints that various emissions levels are at, or below, a constraining upper level. This is then constrained optimisation. Here, an output variable may be optimised

39 subject to the constraints that one or more constrained variables are either less than or greater than preset values. A constrained optimisation procedure was not developed within this project, though is a consideration for future work.

(1.7) Turbine Entry Temperature Limits

The turbine entry temperature, or TET for short, is limited by the material properties of the turbine used - in the small engines envisaged by these studies the complexities inherent in providing turbine cooling are not considered appropriate. Figure (1.11), taken from (12) gives the life of a nickel alloy turbine as a function of TET. It is seen (for the stress levels considered) the life of a nickel alloy turbine at a TET of 1200K has a best value of 107 hours or -1000 years and therefore this value is conservative for an uncooled nickel alloy turbine. It is likely that in the near term future ceramic turbines will become a production reality. Reference (13) describes the future potential use of non-metallic components in aeroengines, while reference (14) estimates that the future service temperature for ceramic composites may be in excess of 1400° C (= 1700K).

From the above considerations, the gas turbine simulations were performed with two turbine entry temperature limits. The first was for 1200K, appropriate to an uncooled nickel alloy turbine. The second was for 1600K, which is considered appropriate for future technology uncooled ceramic turbines. The compound turbocharged simulations were performed over a range of TETs, varying from 1390K to 1500K. The higher TET figures are therefore appropriate to future technology uncooled ceramic turbines.

40 (1.8) The Structure of this Thesis

The structure of this thesis is now described. Chapter (2), 'The Determination of Fluid Properties', describes the algorithms that were used to calculate fluid properties, these algorithms being used in the engine simulation programs.

Chapter (3), 'Automotive Gas Turbine Studies', describes the design of two computer simulation programs used to simulate two types of automotive gas turbine, with the particular aim of producing a net thermal efficiency verses power output contour for each type of engine.

Chapter (4), 'The Compound Highly Turbocharged Engine', describes the design concept for this type of engine, along with a simple form of simulation program.

Chapter (5), 'Time - Marching Numerical Models of Naturally Aspirated and Compound Turbocharged Spark Ignition Piston Engines', describes time - marching simulation models used to estimate the performance of naturally aspirated and compound turbocharged spark ignition piston engines.

Chapter (6), 'Ensuring the Stability of the Integration of the Mass Balance O.D.E.', describes work done to overcome instability problems encountered in the development of the algorithm described in chapter (5).

Chapter (7), 'Minimisation of a Function of a Vector - and the Application of this Technique to Powertrain Design Optimisation', describes the design of an unconstrained numerical optimisation procedure applied to the computer simulations described in chapters (4) and (5).

Chapter (8), 'Numerical Results' presents the collective numerical results.

41 Chapter (9), 'Conclusions and Suggestions for Further Work' presents the collective conclusions and also provides suggestions for further work.

The references, figures and tables for each chapter are at the end of the relevant chapter. There are, in addition, a number of appendices.

42 (1.9) References

(1) Fenocchi, A. The Design and Development of a Small High Speed Generator PhD thesis, 1991 Imperial College of Science, Technology and Medicine Univ. of London

(2) Fenocchi, A. ;Etemad, M.R.S. ;Besant, C.B. ;Ristic, M. Baines, N.C. Turbo-Electric Transmission: a New Concept for Vehicles ISATA 1991 Conference

(3) Chudi, P. ;Malmquist, A. Development of a Small Gas Turbine Driven High-Speed Permanent Magnet Generator The Royal Institute of Technology S-100844 Stockholm, Sweden 1989

(4) The Volvo Environmental Concept Car 1992 Volvo Car Corporation Environmental Report No. 29

(5) Howard, K. Volvo Shapes Up Autocar and Motor Haymarket Motoring Publications Ltd. Vol 194, No.5, 4996 28 October 1992

-(6) Chevis, R.W. ; Walsh, P.P. ; Coulson, M. ; Everston, J. Hybrid Electrical Concepts using Gas Turbines C427/20/206, Electric Vechicles Seminar, I. Mech. E. Autotech 1991

43 (7) Weaving, J.H. Small Gas Turbines Nominated Lecture, Automobile Division Proceedings, Proc. I. Mech. E., 1961-62, no. 6, pp221

(8) Grainger, A. ; Luff, J. ; Mead, T. ; Myers, D. Design of a Continuously Variable Transmission Undergraduate Design Project Private Communication Imperial College of Science, Technology and Medecine Univ. of London 1992

(9) Walsh, W.P. Transport and the Environment; Local, Regional and Global Challenges C389/404, 925001, FISITA 92 conference, I. Mech. E. The Vehicle and the Environment, Vol. 1

(10)Cragg, C. Cleaning up Motor Car Pollution: New Fuels and Technology A Financial Times Management Report published and distributed by Financial Times Business Information 1991

(11)Ahmed, T. ; Theobald, M.A. A Survey of Variable-Valve Actuation Technology SAE Trans 1989 Section 3, 891674

(12)Baines, N.C. ; Panting, J.R. ; Etemad, M.R. Besant, C. A Gas Turbine-Electric Vehicle Concept I. Mech. E. paper C389/031 FISITA conference, Total Vehicle Dynamics, vol. 2 1992

44 (13)Jeal, R.H. Meeting the High Temperature Challenge - the Non-Metallic Aeroengine Metals and Materials vol 4 No 9 sept 1988

(14)Lewis, M.H. ; Leng-Ward, G. Advanced Engineering Ceramics Metals and Materials vol 7 No 6 June 1991

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56 Chapter (2) The Determination of Fluid Properties

(2.1) Overview

In order to predict the power outputs and thermal efficiencies of internal combustion engines using numerical algorithms, it is necessary to calculate the thermodynamic properties of the working fluid. These properties are functions of composition, temperature and pressure. For the numerical algorithms described in this thesis changes with respect to pressure were, however, neglected. Ignoring property changes with pressure was justified because,

(i) in small gas turbines, the pressure changes are small, resulting in negligible additional property changes. In the studies undertaken, the maximum pressure was only 5 bar in such engines. (ii) in spark ignition engines, whether they be turbocharged or not, the peak pressures are higher, of the order of 100 bar. Thus the theoretical corrections to property values are larger than for case (i). However, such corrections were considered to be small in relation to the errors inherent in the assumptions leading to construction of the numerical algorithms. Such assumptions were that there was bulk homogeneity in a control volume, changes in dissociation energies were insignificant and that various physical processes could be modelled through the use of empirical formulae. Thus, correcting the property values for changes in pressure was considered inappropriate. (iii)It may be appropriate to make pressure corrections to the property values when modelling compression ignition engines. Such engines generally employ higher peak pressures than spark ignition engines. However, compression ignition engines were not modelled in this

57 project.

The following sections describe the algorithms that were used to calculate fluid property values. Different algorithms were used for spark ignition engine modelling as compared with gas turbine modelling. This was because,

(i) peak temperatures in the spark ignition models were expected to be higher than was allowable for the algorithm used to calculate fluid properties for the gas turbine modelling, (ii) in the spark ignition engine models, allowance was made for the effect of unburnt fuel vapour on the calculated property values.

In retrospect, it was discovered that peak modelled temperatures in the spark ignition engine were lower than had been anticipated. A typical maximum value was = 2300K with a volumetric compression ratio of 10:1, while values of = 3000K had been expected. In the case of the spark ignition engine modelling, temperatures of 3000K would have resulted in the polynomial curve fits, used for the gas turbine fluid property modelling, being inaccurate. The gas property calculations used in the gas turbine modelling in any case assume no dissociation, a poor assumption at temperatures of 3000K.

The gas turbine models that were used did not specifically make use of a fuel energy content value, as they simply calculated temperature changes without additionally calculating the fuel/air ratio and fuel flow rate. However, the piston engine models did require a value for the fuel energy content.

58 (2.2) The Calculation of Fluid Properties for 'Low' Temperatures

For the gas turbine models, property values were calculated according to the work of (1). This work involved fitting fourth order polynomial curves to theoretically calculated property values. The theoretical calculations did not allow for the effect of dissociation, i.e. the gas is assumed to have a fixed composition. Two ranges of temperature were used for the fourth order polynomial curve fits. The first range was 200K - 800K, while the second range was 800K - 2200K. Though corrections to the property values according to fuel/air ratio were available, these were not used. This was because,

(i) gas turbines run at lean overall fuel/air ratios, in order to control the turbine entry temperature. Thus the corrections for property values with respect to fuel/air ratio are small compared with the property values calculated for air, i.e. a fuel/air ratio of zero. (ii) Invoking the corrections for property values with respect to fuel/air ratio required extra iterative procedures in the simulation programs to calculate the fuel/air ratio. This extra complexity was not considered worthwhile.

Property value calculations centred around an empirical definition for the value of the specific heat constant at constant pressure,

2 3 4 c (T) = c + c T + C T + C T + C T o 1 2 3 4 (2.1) where c is the specific heat constant at constant pressure (kJ kg-1 K-1), c, are the empirical constants, and T is the temperature (K).

59 The ratio of specific heats, y(T), was then found from the following equation,

c (T) c (T) P (T) - cv(T) cp(T) - R (2.2) noting that the specific gas constant, R, is assumed constant, taking a value of 0.2871 kJ kg-1 K_1. The latter part of equation (2.2) is only valid when R is constant with temperature, the validity of this assumption being ensured by reference (1) assuming a theoretical fuel which gives stoichiometric products of the same molar mass as air. This in turn assumes zero dissociation of the burnt fuel/air products. The assumption of constant specific gas constant is therefore good over the temperature range employed (200K - 2200K).

Now, the specific enthalpy, h(T), is defined by,

h(T) = f c (T)dT (2.3) 0 P

(assuming the specific enthalpy is a function of temperature only) which then implies,

1 2 3 1 4 5 h(T) = c T + T + c T +— T + c T + C (2.4) o 2— c1 1 2 4 c3 5 4 h where ch is a suitable constant of integration and h(T) is in units of kJ kg-1.

A quantity that was required for turbomachinery calculations was the entropy function, defined by,

c (T) (T) = P dT (2.5) T ref

A non - dimensional entropy function may be defined,

60

0(T) - (2.6)

where the specific gas constant, R, was again assumed constant. Appendix (A1) describes the use of the non - dimensional entropy function.

Thus, the non - dimensional entropy function was calculated by the following equation,

„ 1 ,0 (T) = — c ln(T) + C 1T + C 2 T2 + C + C T4 R o 2 3 3 4 4

+ cf (2.7)

where cf is a suitable constant of integration.

The coefficients cn are given in table (2.1) for the lower and upper temperature ranges, adapted from (1) to give SI rather than imperial units.

The empirically calculated property values are plotted against temperature, for the temperature range 200K - 2200K, as follows, PROPERTY DESCRIPTION SYMBOL UNITS FIGURE

specific heat constant at constant pressure c (T) kJ kg-1 K 1 (2.1a)

ratio of specific heats 7(T) (2.1b)

specific enthalpy h(T) kJ kg-1 (2.1c) 0o(T) non - dimensional (2.1d) entropy function

61 (2.3) The Calculation of Fluid Properties for 'High' Temperatures

(2.3.1) Burnt Fuel/Air Mixture Properties

A different form of algorithm was used for the calculation of burnt fuel/air mixture properties for a temperature range which extended to temperatures greater than 2200K. The empirical formulations described in section (2.2) were only valid to this temperature, and it was considered that peak simulated in - cylinder temperatures in a simulated spark ignition piston engine would be considerably greater than this. In the event, it was discovered that peak simulated in - cylinder temperatures were typically only slightly greater than 2200K. Nevertheless, the alternative method for calculating burnt fuel/air mixture property values, which is described in this section, was retained for the piston engine simulations.

The property values used were taken from tabulations provided in reference (2). These tabulations allowed for the effects of dissociation energies upon property values. Therefore, gas composition is assumed to be that of an equilibrium mixture at the given temperature and pressure. Dissociation effects are only significant at higher temperatures, e.g. T > 2000K. The property values given in (2) assume a fuel chemical formula of CnH2n i.e. the fuel contains twice as many hydrogen as carbon atoms. Tabulations were made for every 100K temperature increment from temperatures of 200K to 4000K inclusive. In addition, values were recorded for three different equivalence ratios, 0.0, 0.6 and 1.0. The equivalence ratio is in turn defined as the ratio of the burnt fuel/air ratio to the stoichiometric burnt fuel/air ratio. The tabulations were recorded for a pressure of 1 bar.

A two dimensional second order interpolation method was used

62

to calculate a particular property value from nine particular tabulated property values. The calculated property value was thus a function of temperature and equivalence ratio. The assumption of the chemical form of the fuel fixes the stoichiometric air/fuel ratio as = 14.79 which in turn enables the equivalence ratio to be calculated. The interpolation process is illustrated in fig (2.2) and described in Appendix (A2).

Two property values were required that are not tabulated in (2). These were the particular gas constant, R, which may not be assumed constant at higher temperatures, and the non - dimensional entropy function, 0°. The entropy function was only required for turbomachinery calculations and was therefore only needed in turbocharged engine simulations.

To calculate the specific gas constant as a function of temperature, we start with,

6h = Su + 6(RT) (2.8)

where 6h is a small change in specific enthalpy, Su is a small change in internal energy, and equation (2.8) is derived through the definition of specific enthalpy.

Then,

c 6T = c ST + 6 (RT) (2.9)

assuming that, at constant equivalence ratio, the specific heat values c and c are functions of temperature only, i.e. an ideal gas.

Then,

6 (RT) = (cp cv)ST (2.10)

63

T d(RT) = A(RT) = T (cPv)dT - c (2.11) f .1 ref Ti-ef

T A(RT) = I c (11 - ldT (2.12) lJ ref

If R is constant with temperature, then equation (2.10) reduces to the perfect gas relationship,

R = c - c (2.13) p v

Using a second order numerical integration procedure, A(RT) was thus found from equation (2.12) for different temperatures and the three equivalence ratios of 0.0, 0.6 and 1.0. In each integration process, the equivalence ratio was defined as constant. The existing tabulations for cp and were employed to evaluate the integrals.

Then,

1 R(T) = [ A(RT) + R(Tref )Tref (2.14)

by the definition of A(RT). The reference temperature was taken as 288K, with the values of R(Tref) used for the three different equivalence ratios given in table (2.2) and taken from values given in (3). For the interpolation from the values given in (3), the chemical formula assumed for the fuel was of the form CnH2n. Reference (3) gives the molar mass of the burnt fuel/air mixture, rather than the particular gas constant. However, the particular gas constant (J K-1 kg-1 ) is equal to Boltzman's constant (J K-1 ) divided by the molecular mass of the gas (kg), (4). Then,

k A R - (2.15) At

64 where k is Boltzman's constant, At is the molar mass of the gas and A is Avogadro's number. Equation (2.15) was therefore used to provide the base values of R at a given temperature of 288K, while equation (2.14) was used to calculate the values of R over a range of temperatures using these base values.

The non - dimensional entropy function was found by a similar second order numerical integration process, from equations (2.5) and (2.6). The reference temperature was again taken as 288K at which point 00(T) was defined as zero. Note that equation (2.6) is only valid over the temperature range for which, at constant equivalence ratio, the particular gas constant, R, can be assumed approximately constant with variation in temperature. Thus, the non - dimensional entropy function was only defined to a maximum temperature of 2300K.

The values obtained for R(T,O) and 00(T,0), where 0 is the equivalence ratio, were then added to the tabulations provided by (2). The resulting calculated property values are plotted against temperature as follows,

PROPERTY DESCRIPTION SYMBOL UNITS FIGURE specific heat constant c (T,0) kJ k41 K-1 (2.3a) at constant pressure ratio of specific heats T(T,0) (2.3b) specific enthalpy h(T,O) kJ kg-1 (2.3c) non - dimensional e(T,O) (2.3d) entropy function particular gas constant R(T,I) kJ kg-1 K-1 (2.3e)

65

(2.3.2) Concerning the Calculation of the Specific Gas Constant

It was later discovered that reference (2) tabulates the molecular mass of burnt fuel/air mixture, enabling the specific gas constant, R(T,4) to be directly calculated from equation (2.15). The already tabulated values of R were then checked against such calculations. It was found there was a small error at sufficiently low temperatures (0.24% for air at 2500K), the error becoming significant at higher temperatures (8.1% for air at 4000K). However, since the piston engine simulations typically never gave temperatures greater than 2500K, the errors are not deemed significant. The error is explained by noting that the specific heats values must be replaced by their corresponding effective values when dissociation is allowed for. Thus,

E vi tif

a = C (2.16) P eff aT

E vi hf

°f c V = c + (2.17) eff aT

cP eff Jeff C (2.18) v eff where i is the specii number, v are the specii volumetric concentrations for an equilibrium mixture, hi, are the specii formation enthalpies in kJ mot-1 and M is the equilibrium mixture molar mass in kg. The effective values for the specific heat constant at constant pressure (c ) and the pelf ratio of specific heats (7,17) must then be substituted in place of c and 7 into equation (2.12) to yield the correct

66 change in the specific gas constant, R.

(2.3.3) Unburnt Fuel Vapour Properties

In the piston engine simulations, allowance was made for the effect of unburnt fuel vapour on overall thermodynamic properties. The properties of the unburnt fuel vapour were assumed to be similar to those of Octane, as suggested by (5). In a similar manner to that described in section (2.2) and equation (2.1), polynomial curve fits were used to approximate the value of the specific heat constant at constant pressure. For temperatures ranging from 298K to 1500K, data from reference (6) was used, with a third order polynomial curve fit. For temperatures ranging from 1500K to 1922K data from reference (7) was used. This data employed tabulated values, and a second order polynomial curve fit was produced by the author to approximate the tabulated values over this temperature range. At temperatures below 298K and above 1922K, a linear extrapolation of the empirical curves was employed.

The ratio of specific heats was found via equation (2.2), in which it is assumed that the particular gas constant is constant, taking a value of 72.8 J kg-1 K-1 in this case. The latter value was taken from (8), with the chemical formula for the fuel assumed to be that of Octane, C8H18. (Reference (8) actually gave the molar mass for Octane, the specific gas constant then being calculated from equation (2.15)). The assumption of constant particular gas constant is equivalent to the assumption of constant molar mass of the fuel.

The specific enthalpy values were calculated in a similar fashion to that described by equation (2.4). In the turbomachinery calculations, the unburnt fuel fraction was assumed to be zero, and therefore entropy function values were not required for the unburnt fuel vapour. The

67 coefficients cn for the unburnt fuel vapour properties are defined in table (2.3) in an analogous manner to table (2.1).

The resulting property values obtained are plotted against temperature as follows,

PROPERTY DESCRIPTION SYMBOL UNITS FIGURE specific heat constant cp(T) kJ kg-1 K-1 (2.4a) at constant pressure ratio of specific heats 7(T) - (2.4b) specific enthalpy h(T) kJ kg-1 (2.4c)

(2.3.4) Combining Burnt and Unburnt Gas Mixture Properties

In general, the combined property values were simply mass weighted according to the value of the unburnt fuel fraction, A. This is defined as,

m - ubf (2.19) where rflubf is the mass of the unburnt fuel, and m is the total mass, both for a control volume. Then, for example,

R = Rbf ( 1 — A ) Rut&A (2.20) where the subscript bf stands for burnt fuel/air mixture properties, while the subscript ubf is the equivalent for unburnt fuel. The specific enthalpy was calculated in an analogous fashion, while the ratio of specific heats is the ratio of two mass weighted values,

68 c (1-A) + c A P P ubf bf (2.21) - r Cp (1—A) + r

bf L ubf x

(2.3.5) The Fuel Heat Content Value

The value of the heat of reaction for the complete forwards reaction of gasoline burning with air was required for use in the piston engine simulation algorithms. It is given by,

v h ir — V h I r f f (2.22) AH - V ((fuel) (fuel) where h are the molar specific formation enthalpies for each component of product and reactant mixtures, v are the volumetric concentrations of the mixture components, i and j denote individual species, r denotes reactants, p denotes .41 is the molar mass of the fuel in kg and AH products, (fuel) is the heat of reaction related to the amount of fuel burnt. Equation (2.22) enables the overall chemical energy release in a complete forwards reaction to be evaluated.

The heat of reaction is identically equal to the calorific value of the fuel. Reference (5) gives a value of 43.9 MJ kg-I as a typical calorific value of gasoline fuels, which is assumed to be the lower, rather than higher, calorific value. This figure was erroneously adjusted to a value of 44.1 MJ kg-1, as it was originally thought that the calorific value was not identically equal to the heat of reaction. The latter figure was therefore used in the piston engine simulations, though the error is small. This is especially true considering that the original figure given by (5) is approximate, as gasoline fuels vary slightly in composition and therefore calorific value.

69 Equation (2.22) will always give the energy released by a chemical reaction. However this value will not always be constant, because the initial and final chemical compositions will not be constant, i.e. the values v r and vJp. will vary. This is in turn due the phenomenon of dissociation, that is at higher temperatures and richer fuel/air ratios the reaction will not be complete.

The latter observation can be allowed for through direct evaluation of equation (2.22). A simpler method, however, is to use an empirical combustion efficiency, such that,

AH = an . AH (2.23) actual comb complete where the first term is the approximate actual heat of reaction, the second term is the combustion efficiency and the third term is the heat of reaction for complete combustion, i.e. reaction with no dissociation occurring.

According to data provided by (9), the combustion efficiency was assumed to be a function of equivalence ratio. A spline curve fit was provided for the data, the details of this procedure being given in (10). Assuming that the combustion efficiency is a function of equivalence ratio only is slightly erroneous, as the amount of dissociation occurring will also depend upon the temperature. The curve used for the assumed form of combustion efficiency verses equivalence ratio is shown in fig (2.5).

(2.4) Summary

Simple numerical algorithms were developed to enable fluid thermodynamic properties to be calculated. Such algorithms are essential to enable the accurate performance predictions of engine simulation programs.

70 (2.5) References

(1) Fielding, D ; Topps, J.E.C. Thermodynamic Data for the Calculation of Gas Turbine Performance Aero. Res. Coun. R. & M., no. 3099, 1959

(2) Banes, B. ; McIntyre, R.W. ; Sims, J.A. Properties of Air and Combustion Products with Kerosine and Hydrogen Fuels Published by Bristol Siddely Engines Ltd. on behalf of A.G.A.R.D. 1967

(3) Kaye, J. ; Keenan, J.H. Gas Tables, Thermodynamic Properties of Air, Products of Combustion and Component Gases Compressible Flow Functions John Wiley and Sons, 1948

(4) Bird, G.A. Molecular Gas Dynamics Oxford Engineering Science Series, Clarendon Press, 1976 p.12

(5) Goodger, E. Hydrocarbon Fuels; Production, Properties and Performance of Liquids and Gases Macmillan 1975

(6) Patterson, D.J. ; Van Wylen, G.J. Empirical Heat Capacity Equations for Ideal Gases J. Heat Transfer, vol. 85, 1963, pp281-2

71 (7) Mott Souders, J.R. ; Matthews, C.S. ; Hurd, C.O. Relationship of Thermodynamic Properties to Molecular Structure. Heat Capacities and Heat Content of Hydrocarbon Vapours Ind. Eng. Chem., vol. 41, 1949, pp1037-48

(8) Rossini, F.O. (ed) Selected Values of Properties of Hydrocarbons and Related Compounds Am. Petrol Inst. Report 44, 1953

(9) Matthews, R.D. Relationship of Brake Power to Various Energy Efficiencies and Other Engine Parameters: The Efficiency Rule Int. J. of Vehicle Design, vol. 4, no. 5, 1983, pp491-500

(10)Press, W.H. ; Flannery, B.P. ; Teukolsky, S.A. ; Vetterling, W.T. Numerical Recipes, the Art of Scientific Computing Fortran Edition Cambridge Univ. Press, 1989

72 Fig 12.10 Specific Heat Constant at Constant

K) Pressure versus Temperature for Air at Low Temps /

kg 1.3 kJ/ (

1.2 - t Pressure, Cons t a t t Cons Hea ic if 1.0 ec 200 460 660 860 1000 12'00 400 160018002000 2200 Sp Temperature, (K)

Fig (2.1bj Ratio of Specific Heats versus Temperature for Air at Low Temperatures 1.45

1.40

4= a)o 1.35 - CL

1.30 -

1.25 200 460 660 860 10'00 1200 1400 1600 1800 2000 2200 Temperature, (K) Fig (2lc) Specific Enthalpy versus Temperature for Air at Low Temperatures

2600

2200 -

) kg

kJ/ 1800 - [ lpy,

ha 1400 - t En ic if ec Sp

200 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Temperature, (K) Fig (2.1d) Non - Dimensional Entropy Function versus Temperature for Air at Low Temperatures 8

(-)

ion, 6 t Func

y 4 trop

l En 2 iona

imens 0 D - Non LL Li 200 460 660 860 1000 1200 1400 1600 1800 2000 2200 Temperature, (K) FIRST 2ND olecER iNTEAPNA-TroNS (OF ivdito-i -nicrea NEE, •TroasE) toBTAirs, fbikir'S A,B1C

gEcewb 214o oece.2 iN-raeeborriets.) , Frivt, PartM 0 ) IoM Poiv15 A, BAG, f () is -nit playsieri vitimE of 1.4-r6ecrr

Ft Gi (2.2) SECOND oeDF-t2 r)-)E—rmoiD

UE../:) TO 03,90v PROP6eVy VALVES

77 Fig (2.3a) Specific Heat Constant at Constant

K] Pressure versus Temperature for Equilibrium Gas /

kg 7 0=1.0 kJ/ [

e, 6 ur ess 5 0=0.5 Pr t 4 Cons t 0=0.0 a t s 3 Con t 2 Hea ic if ec 1000 2000 3000 4000 Sp Temperature, (K)

Fig (2.3b) Ratio of Specific Heats versus Temperature for an Equilibrium Gas 1.45

1.40 — ..___..• r 1.35 — =8 u 1.30 — 4=. D a) 0 - •4 in 1.25 0 1.20 — -__, 00- c:K 1.15 — 0=1.0 1.10 0 lobo 2000 3000 4000 Temperature, (11 Fig (2.3c) Specific Enthalpy versus Temperature for an Equilibrium Mixture of Burnt Gas or Air

12000 11000 - 10000 -

) 9000 Ti kg J/

k 8000 - (

y, 7000 Ti lp 6000 Ti tha

En 5000 - ic if 4000 - ec

Sp 3000 - 2000 - 1000 0 0 10'00 I 2000 T 3000 4000 Temperature, (K) Non-Dimensional Entropy Function, (-) -2.0 10.0 -1.0 7.0 0.0 8.0 3.0 4.0 5.0 9.0 - 6.0 2.0 1.0 T

0

Fig (2.3d)Non-DimensionalEntropyFunction versus TemperatureforanEquilibriumGas 560

Temperature, (11 1000

15b0

2d00

2500 Fig (2.3e) Specific Gas Constant versus Temperature for an Equilibrium Gas 0.5

0=1.0 k.) 0.4 - _._;• a Ill 0=0.5 c 0 U 0=0.0 U3a c.o 0.3 - U i'l= F3 a) 0.. CJ1

0.2 Ii 1000 2000 I 3000 4000 Temperature, (K) Specific Heat Const at Const Pressure, (kJ/kg/K) 1 200

Pressure versusTemperatureforOctaneVapour Fig (2.4a)SpecificHeatConstantat 660

1000

Temperature, (K) 1400

1800

2200

2600

3000

Fig (2.4b) Ratio of Specific Heats versus Temperature for Octane Vapour 1.06

m 1.05 VT -0 o a) = 1.04 - U

C.) N CL 03 cn 1.03 - .1, 4- o 0 :475 0 f: 1.02 -

1.01 1 200 600 ' 10001 I 1400' I 1800' 1 2200 1 2600 3000 Temperature, (K) Specific Enthalpy, (kJ/kg) 10000 -,_ 12000 _ 11000 -_. 7000 -_ 4000 -._ 5000 - 3000 -_ 8000 2000 - 9000 - 6000 -. 1000 - o 200 -

_ .

1

Fig (2.4c)SpecificEnthalpyversusTemperature 600 i

I

1000 for OctaneVapour I 1400 Temperature, (K)

1

1800

I

2200 i

1

2600 i

1 3000 Fig (2.5) Empirical Combustion Efficiency vs. Equivalence Ratio Curve (Cubic Spline Curve Fit) 1.0

,] ❑ ❑ ❑ (-)

ncy, 0.9 - ie ic f

Ef 0.8 - ion t bus

m 0.7 - d Co te 0.6 - ima t Es 0.5 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Equivalence Ratio, (-) c T is in Kelvins o n s t 200 -s T < 800 800 ..5 T :5 2000

-1 +1.0189134 +7.9872959x10 co

-4 -4 c -1.3783636x10 +5.3392159x10 1

-7 c +1.9843397x10-7 -2.2881694x10 2

-11 c +4.2399242x10-10 +3.7420857x10 3

-13 c -3.7632489x10 0 4

1 c -1.7034633 +4.7320631x10 h

C -1.9997658x10+1 -1.6171711x10+1 f

Table (2.1) The Constants used for the Calculation of Air Properties at Low Temperatures

87 equivalence gas ratio constant 0 R (J kg-1 K 1)

0.0 287.044

0.6 287.409

1.0 287.632

Table (2.2) Base Values of the Specific Gas Constant

88 c T is in Kelvins 0 n s t 298.16 ts T < 1500 1500 5-- T 1922

-1.4126484 +2.0832190x10+3 co

-1 +1.8397817x10 +2.2526489 c1

-5 -4 -9.9985303x10 -4.6529x10 C2

-8 +2.1072621x10 0 C3

base constant for enthalpy = h(298.16) = 3.156x10" J kg 1 calculations

Table (2.3) The Constants used for the Calculation of Unburnt Fuel Vapour (Octane) Properties

89 Chapter (3) Automotive Gas Turbine Studies

(3.1) Overview

The small gas turbine has for many years been considered as an alternative powerplant for automobiles (1). Such a unit has the attractions of an intrinsically high power to weight ratio, smoothness in operation and low exhaust emissions characteristics. The low exhaust emissions are a consequence of the need to limit peak combustion temperatures, as there are material limitations of the combustion chamber and turbine. Piston engines can tolerate higher peak combustion temperatures because of the non - continuous nature of their operating cycle. Thus, in the latter case, the time averaged cycle temperature is of more importance than the peak cycle temperature with regard to material limitations. Gas turbines, by contrast, operate on a continuous cycle principle. The lower peak combustion temperatures of the gas turbine then, in turn, reduce the amount of chemical dissociation which reduces pollution levels.

Despite the advantages of the gas turbine and many experimental studies, there has yet to be a production automotive unit of this type. This is for two main reasons,

(i) small gas turbines have been found to demonstrate poor part load thermal efficiency levels, (ii)gas turbines demonstrate poor transient response.

At the beginning of the work described by this thesis, a simple constant speed single shaft gas turbine, employing an electrical transmission, was being studied, (2). One advantage of the constant speed gas turbine is that it eliminates problem (ii). In addition, the design of the electrical system is simplified, due to the constant frequency of the output current from the alternator.

90 In order to improve their thermal efficiency levels, small gas turbines designed for automotive use are generally fitted with a heat exchanger. The constant speed, single shaft unit was no exception. However, the theoretically predicted part load thermal efficiency was still poor, i.e. problem (i) had not been overcome.

As an alternative concept to the above, a form of twin shaft gas turbine was considered. The aim was to produce a design concept that was predicted to attain good levels of thermal efficiency across a broad power output range. In addition, three other alternative gas turbine concepts are described by this chapter, though none of these concepts were theoretically modelled.

Basic predictions of gas turbine design point performance can be made by assuming constant compressor and turbine isentropic efficiencies and constant fluid properties. This approach is described and illustrated in this chapter, and used to highlight the reasons that small gas turbines generally employ a heat exchanger.

More realistic performance predictions require the assumption of variable fluid properties and variable compressor and turbine isentropic efficiencies. This approach was adopted to produce predictions of the alternative twin shaft gas turbine, these predictions then being compared with predictions of the performance of the constant speed, single shaft gas turbine. The latter calculations were performed by making comparable assumptions to the twin shaft case.

The results of this work are presented in a combined results chapter, chapter (8).

91 (3.2) Basic Gas Turbine Performance Analysis

The layout of the basic gas turbine without heat exchanger is illustrated in fig (3.1). Adiabatic compression of the working fluid is followed by combustion which may be assumed in a simplified analysis to occur at constant pressure. This is then followed by adiabatic expansion back to atmospheric pressure. The compression stage absorbs power, while the expansion stage creates power. There will thus be a net power output if the power produced by the turbine exceeds the power absorbed by the compressor.

In this section, constant fluid properties and compressor and turbine isentropic efficiencies are assumed in the analysis of gas turbine performance. In addition, combustion inefficiencies and total pressure drops in the combustion chamber and heat exchanger, if fitted, are neglected. Furthermore, the mechanical efficiency is assumed to be unity. The last three assumptions will result in small additional errors.

The net thermal efficiency of a gas turbine without a heat exchanger is then given, according to Appendix (A3), by,

( .71 t t c) (c - 1) Ti (3.1) th c[a, (t - 1) + 1 - c] where,

T MAX t T (3.2) AMB I ENT is the cycle temperature ratio, and

r-11 [ c = r (3.3)

92 is the isentropic temperature ratio, with

AMAX Y - (3.4) P PAMB I ENT being the cycle total pressure ratio. In addition, 17 is the compressor isentropic efficiency, nt is the turbine isentropic efficiency, and T is the ratio of specific heats.

From equation (3.1), the thermal efficiency is seen to be a function of cycle pressure ratio, for a constant cycle temperature ratio and other parameters. There will then be an optimum cycle pressure ratio for maximum thermal efficiency. This optimum efficiency level will in turn increase with increasing cycle temperature ratio, as will the pressure ratio at which it occurs.

In fig (3.2), the thermal efficiency of a gas turbine as given by equation (3.1) is plotted against cycle pressure ratio for two different cycle temperature ratios. The two different cycle temperature ratios correspond with values of (TmAx/T AMBIENT ) of (1200K/288K) and (1600K/288K). The value of T corresponds with the ASA International Standard AMBI ENT Atmosphere figure for sea level altitude, to three significant figures. The lower value for TmAx corresponds with a typical turbine entry temperature for an uncooled metal turbine, while the upper value corresponds with a typical turbine entry temperature for a cooled metal or uncooled ceramic turbine. The turbine and compressor isentropic efficiencies are taken to be equal to 0.8, while the ratio of specific heats, r, is taken to be 1.35.

In table (3.1), the corresponding peak efficiency levels, along with the pressure ratios at which they occur, are itemised. The tabulated values are for three different values of z: 1.3, 1.35 and 1.4. The tabulated values were

93 generated using equation (3.1) and a simple line minimisation technique. This latter technique is described in section (7.5).

The assumption of constant properties leads to rather inaccurate predictions of thermal efficiency as given by equation (3.1). Nevertheless, the figures are accurate enough to indicate that the cycle pressure ratio required for optimum gas turbine efficiency will generally be higher than is practicably achievable for a small unit. The reasons for the limitation on the achievable cycle pressure ratio are itemised in section (1.4).

It can also be demonstrated that, although the design point efficiency of a gas turbine may be acceptable, if any or all of the parameters roc, rat, r or t fall in value significantly then so does the thermal efficiency level. This in turn usually leads to problems in achieving acceptable part load efficiency levels for the simple gas turbine.

The partial solution to these problems invoked for a gas turbine of modest pressure ratio is to employ a heat exchanger. A gas turbine plus heat exchanger is found to have a high efficiency level at a very low cycle pressure ratio.

The gas turbine plus heat exchanger cycle is illustrated in fig (3.3). Here, heat energy exhausted from the turbine is used to pre-heat air exiting the compressor via a heat exchanger. This is prior to the combustion of the air with fuel. Thus for a given turbine entry temperature the amount of heat energy required to be added by combustion is reduced, but only if the turbine exhaust temperature is greater than the compressor exit temperature.

A heat exchanger effectiveness, c, is defined as,

94 - actual temperature rise acheived in heat exchanger theoretical max temperature rise in heat exchanger

(3.5)

The effectiveness varies between zero and unity, the value of zero corresponding with the case of the simple gas turbine. The theoretical efficiency level, making the same assumptions as those leading to equation (3.1), is then given by,

ri t 1 c t (1 - — ) - (c - 1) nth (1-c) + c t - c t c .t - (c-l+ -ric )} - (c-1+ Tic )

(3.6)

which is again derived in Appendix (A3).

The peak theoretical efficiency levels given by equation (3.6) increase with increasing values of c and t. However, a plot of thermal efficiency versus pressure ratio according to this equation shows a quite different form from that of the simple gas turbine. Peak efficiency generally occurs at a very modest pressure ratio, while at higher pressure ratios the efficiency falls with increasing pressure ratio. The latter effect is accounted for by noting that at a high cycle pressure ratio the compressor exit temperature will be higher than the turbine exhaust temperature. The heat exchanger will then have an increasingly adverse effect as the cycle pressure ratio is increased, i.e. an increasing amount of pre - cooling of air exiting the compressor will be occurring.

Graphs of theoretical efficiency levels verses pressure ratio for the gas turbine plus heat exchanger are shown in

95 fig (3.4) and fig (3.5). Figure (3.4) is for a value of t=1200/288, while fig (3.5) is for a value of t=1600/288. Curves are shown for different values of the heat exchanger effectiveness; c = 0, 0.3, 0.5, 0.7 and 0.9, the case of c = 0 of course corresponding with that of the simple gas turbine.

The thermal efficiency of a gas turbine plus heat exchanger will still fall if the parameters lit or t fall in value. A reduction in the value of the parameter r will only cause a reduction in efficiency if the current pressure ratio is less than or equal to the pressure ratio for peak efficiency. Thus, the problem of attaining acceptable part load efficiency levels is reduced in severity. Therefore the reason for automotive gas turbines generally employing a heat exchanger to boost efficiency levels is seen.

Note that there is also a cycle pressure ratio corresponding with peak power output. The power output level is given by,

c T p AMBIENT w - (Ti n t - c) (c - 1) (3.7) TI C c t C

where w is the work output per unit mass of working fluid, and c is the specific heat constant at constant pressure. This equation is derived in Appendix (A3). The equation is, according to the assumptions made, applicable to a gas turbine with or without an heat exchanger, i.e. the presence of an heat exchanger primarily affects the amount of heat input during combustion, rather than the work output.

Equation (3.7) again results in an optimal pressure ratio, this time for power output, for a given value of cycle temperature ratio, t, and other parameters. It is noted that designing a gas turbine which operates at a low pressure ratio, and hence which makes best use of a heat exchanger, will generally result in a rather poorer power output level as given by equation (3.7).

96 (3.3) The Use of Turbomachinery Characteristics

For more accurate estimation of gas turbine performance, the consideration of variable compressor and turbine isentropic efficiency levels and flow matching of the compressor and turbine must be invoked. In order to achieve this, turbomachinery characteristics, or maps, are generally employed.

Turbomachinery maps make use of non - dimensional data groups, refer for example to (3). As applied to gas turbines, the data is generally simplified to the use of pseudo non - dimensional parameters. This is due to the fact that the gas constant, R, can be considered constant over the temperature range employed and thus removed from the truely non - dimensional parameters.

There are then four key non - dimensional or pseudo non - dimensional data groups. These are,

(i) The isentropic efficiency TI (ii) The total pressure ratio

(iii) The pseudo non - dimensional N = NP;T (3.8a) speed function

(iv) The pseudo non - dimensional M' 1///T (3.8b) mass flow function m Ap r where m' is the mass flow rate, A is the characteristic turbomachinery component area, p is the total pressure at entry to the component, T is the total temperature at entry to the component, N is the physical rotational speed of the component, and ', as before, is the ratio of specific heats at entry to the component.

97 Nc, Ni M and N Truely non - dimensional forms of t c it were used, these parameters being measured relative to a design value. The suffix c is used to describe the fact that a quantity relating to the compressor is being employed, while suffix t relates likewise to the turbine. The corresponding truely non - dimensional are then N, Nt, Nic and Nit and given by,

= / N (3.9a) cdesign

1\lt / Nt (3.9b) design

Mc / (3.9c) design

Mt = Mt/ Mt (3.9d) design where the suffix 'design' describes the fact that data at a reference design point is employed.

The four parameters 17, r , N and M were stored in graphical form, with two different graphs each for the compressor and turbine. Data for a centrifugal compressor was taken from (4), while that for a radial turbine was taken from (5). Both sets of data required some degree of extrapolation for use in the simulations described in the following sections, particularly for the turbine.

The data is presented in figs (3.6a), (3.6b), (3.7a) and (3.7b). Figures (3.6) are for the compressor, while figs (3.7) are for the turbine. Figures (a) are plots of pressure ratio versus non - dimensional mass flow function, while figs (b) are plots of isentropic efficiency versus non - dimensional mass flow function. Each curve is for one value of non - dimensional speed parameter.

In fig (3.6a), if the slope of r verses NI. becomes P c

98 positive, then compressor operation becomes unstable. This is known as compressor surge, and the point at which surge occurs is given by a characteristic joining the points of zero slope in fig (3.6a). Points within the surge region do not constitute a valid solution for the pressure ratio at the given value of M.

An interpolation procedure was required to enable parameter values to be obtained from these maps. The procedure used was a simple first order type. For a given value of N , a speed line was constructed, as schematically indicated in fig (3.8). Then a simple 'ordinate to abcissa' or 'abcissa to ordinate' interpolation procedure was invoked from this constructed speed line.

(3.4) The General Calculation Procedure Invoking the Use of Turbomachinery Characteristics and Variable Properties

When the use of turbomachinery maps and variable properties is invoked, gas turbine performance can not be calculated using simple equations. Instead, computational iterative procedures must be employed.

Performance levels, in particular efficiency levels, for the two gas turbine concepts described in the subsequent two sections were predicted according to the following main assumptions:

(i) property values were a function of temperature only, (ii) turbine entry temperatures and component isentropic efficiency levels were calculated according to the use of turbomachinery maps, (iii) a heat exchanger, if fitted, was simulated according to an effectiveness parameter.

Assumption (i) is discussed in chapter (2), while the use of maps itemised as (ii) is discussed in section (3.3) and

99 sections following this one. The use of a heat exchanger, itemised as (iii), is discussed in section (3.2).

Additional assumptions were,

(iv) combustion efficiency was 100%, (v) there was a 2% total pressure drop in the combustion chamber, (vi) there was no total pressure drop in the heat exchanger, (vii) mechanical efficiency was 100%, (viii)transmission efficiency was 100%, (ix) the mass flow rates through compressor and turbine were equal, (x) the maximum compressor pressure ratio was 5:1.

Assumption (iv) is good given the lean overall fuel/air ratios at which gas turbines generally operate. The figure given in assumption (v) is a little arbitrary, though is a typical order of magnitude figure for practical gas turbines. Assumption (vi) results in some error, though was used due to the difficulty in calculating an actual total pressure drop through an heat exchanger. Assumption (vii) is good for gas turbines, while assumption (viii) is discussed in chapter (1). Assumption (ix) is equivalent to that that the mass flow rate of fuel is small compared with the mass flow rate of incoming air. This is a good assumption: for stoichiometric combustion the mass flow rate of fuel is approximately 1/15 that of the incoming air. Furthermore, gas turbines generally operate at very lean overall fuel/air ratios, (a typical overall equivalence ratio for a gas turbine being 20%) reducing the relative mass flow rate of fuel still further. Assumption (x) sets an attainable upper limit to the cycle pressure ratio for small, single stage, gas turbines.

The essential procedure was then to calculate a series of

100

temperature increments, along with the compressor and turbine isentropic efficiencies. This then enabled the gas turbine thermal efficiency levels to be calculated, for different values of the heat exchanger effectiveness.

Consider the gas turbine cycle with heat exchange schematically illustrated in fig (3.3). Using the nomenclature of fig (3.3),

- W 11(1'4) ] — [h (T2) — NT) wont in [h(T3) 'nth q,. h(T3) - h(T'2) (3.10)

where h(T) is the specific enthalpy, and w and q are specific works and specific heat input. The calculation of h(T) was as according to section (2.2).

Now, the maximum temperature rise on the cold side of the heat exchanger is given by,

— 2-2' = T T (3.11) AT2-2 max 4 2

hence,

AT2-2 = c.AT2-z , = c(T4 - T2) (3.12)

thus,

T2 = T2 + c(T4 — T2) (3.13)

The procedure was then as follows,

(1) select an ambient temperature, T1, (ii) select a compressor operating point on the compressor map: from this r , NI,.• and N.• are known, P c

101 ■ (iii) with the known values of Mc and Nc use the compressor map to calculate 71c, n (iv) with the known values of , rp and T1, calculate T2, (v) calculate r = 0.98.r pt pc (vi) iteratively calculate T3, Mt and N using matching equations between the compressor and turbine, where T3 is the TET, (vii) from the known values of Mt and N use the turbine map to obtain 71t, (viii)from the known values of rat, r and T3, calculate p t T4, (ix) for various values of heat exchanger effectiveness, calculate T'2 from equation (3.13), (x) calculate the overall thermal efficiencies for various values of heat exchanger effectiveness from equation (3.10).

The procedure was repeated for steps (ii) - (x), for a number of compressor operating points. The value employed for the ambient temperature, T1, was 288K. The overall calculation procedure is described in more mathematical detail in Appendix (A4).

(3.5) Modelling a Single Shaft, Constant Speed Gas Turbine without Variable Geometry

As a comparative basis for a more complex twin shaft gas turbine concept, a simulation procedure for a single shaft, constant speed gas turbine without variable geometry was designed. Such a design concept was already being considered by the author's research group, (2).

As outlined in the previous section, the key part of the algorithm was to calculate iteratively the turbine entry temperature and turbine isentropic efficiency, which in turn entails calculation of the variables N and Mt. Physically,

102

this corresponds with matching of the turbine and compressor mass flow rates and speeds, according to the flow characteristics of these components.

Equations (3.8a), (3.9a) and (3.9b) may be combined to give,

1 • t. ) 1 2 Nt 1 (At/Ac) (r,/rt) (Tc/T 1 Nt/Nc [(At/Ac) (zcirt.) (Tc/Tt) 'design [Nt/Nc 3 design }

(3.14) while equations (3.8b), (3.9c) and (3.9d) may be combined to give,

1 M. )(7c/7t) 1 7 1 (Ac/Ad (pc/pd t _ { (let /mic )(Tt/Tc [(ret /Wc )(Tt/Tc) (7cPa't) ]dsnI 1 [ (Ac/At.) (Pc/Pt) J dsni iviMc

(3.15) where the subscript dsn denotes design conditions. Equations (3.14) and (3.15) can in turn be simplified. The ratio of the (actual turbine to compressor area ratio) to the (design turbine to compressor area ratio) is a constant, which will be denoted by A. The ratio of turbine and compressor physical speeds is a constant. The constant is in this case unity, but would be altered if a step - up or step - down gearbox was introduced between the compressor and turbine. The compressor inlet temperature is assumed constant, which implies that 7c is constant. The mass flow rates through the compressor and turbine are assumed equal. Thus, simplifying equations (3.14) and (3.15),

1 1 1 * 2 I N: 2 2 * ) (7tTt ) 4 (ztT t design (3.16) N:

103 and,

12 * r T P T t t t Mt 4 (3.17) { • [r 3 • p design design 7t Mc t. where,

• = 7/7c (3.18) the value for 7 being the corresponding value at a temperature of 288K.

Equations (3.16) and (3.17) give an implicit solution for the turbine entry temperature, Tt. The equations must be satisfied simultaneously for the appropriate values of N:, Mt and T. For given design points of the compressor and turbine, and a given compressor operating point, the input variables which alter the solution to these equations are N: and M.

The method of the solution to the equations has to be iterative. The simple bisection technique was used, this being described in detail in reference (6). With this technique, the solution of an equation is first bracketed between upper and lower solution limits. The bracketing interval size is then repeatedly halved, always ensuring the solution remains bracketed. The process continues until the bracketing interval becomes small enough to permit sufficient solution accuracy. Because two equations have to be solved, a two dimensional version of this technique is employed. Details of the iterative technique are shown schematically in fig (3.9).

In the simulations performed, A was set to unity for a maximum TET of 1200K and to a value of 1.16 for a maximum TET of 1600K. The compressor was operated on successive

104 points on the 100%, N speed line.

(3.6) Modelling The Twin Shaft Differential Gas Turbine without Variable Geometry

The twin shaft differential gas turbine comprises a compressor and turbine which are not physically connected, along with a combustion chamber and heat exchanger. Thus there is freedom in choosing the relative speeds of turbine and compressor. The concept was illustrated in fig (1.9), is described in reference (2) and reviewed in reference (7).

The differential gas turbine employing a mechanical linkage in the form of an epicyclic transmission is illustrated in fig (3.10). This differs from the scheme considered in that a CVT is not employed. For such a mechanical system, (8) states that there are advantages in part load efficiency and transient response. Anderson (9) and Tso (10) theoretically study such a layout, but come to differing conclusions regarding its potential benefits over a more conventional automotive gas turbine. The layout considered here is quite different, because a CVT is employed between the compressor and turbine. The author is unaware of previous studies into such a layout.

The assumed non - constant speed of operation, plus the freedom in selecting turbine and compressor relative speeds, results in the following,

(i) the compressor can be operated at maximum isentropic efficiency for a given pressure ratio, i.e. along the maximum efficiency line. (ii) the turbine speed can be adjusted to maintain the turbine entry temperature at a high value, maximising overall thermal efficiency. (iii)the operating schedule maintains Tic, TIt and t at high values, while allowing r to fall at low power Pc

105

outputs. Thus, maximum use is made of an heat exchanger, if fitted.

Point (ii) requires further explanation. It is well known that increasing the turbine entry temperature of a gas turbine results in an increase in overall thermal efficiency, and this can also be theoretically demonstrated. Adjusting the rotational speed of a radial turbine allows the turbine entry temperature to be adjusted because the corresponding mass flow parameter, Ni•t, is a function of speed. This can be demonstrated as follows: consider equation (3.17) which is here repeated,

1 2 p T t Mt t t

M rp design design :to t

• It is seen that raising the value of Mt raises the value of T , as 7 drops slightly with increasing temperature. Note that equation (3.14) is not required, as there is no need for speed compatibility between the compressor and turbine. Equation (3.17) can be solved directly for Mt , once a value for T t is selected. However, this calculated mass flow parameter must lie between upper and lower limits, defined by the turbine map. If it does not, the selected value of Tt cannot be attained, i.e. the turbine entry temperature must either lie above or below the preferred value. A value for N must still be found, to enable a calculation of the turbine isentropic efficiency to be made via map interpolation. The required speed parameter was calculated iteratively, by means of one dimensional bisection. This was done by finding the value of Nt which gives the required value of N•, from the turbine map, for the given value of r . The overall method of calculation is outlined in pt schematic form in fig (3.11).

Note that the use of an axial turbine would not permit the

106 same operating principle. This is because the mass flow parameter of an axial turbine is largely independent of the speed parameter.

The variable A was adjusted to give the maximum range over which the turbine entry temperature took its preset maximum value. These values for A were 0.94 for a maximum TET of 1200K and 1.085 for a maximum TET of 1600K. For a small part of the simulated unit's operating range, it was necessary that the turbine entry temperature value be less than the desired maximum. This occurred at low pressure ratios, and was due to the small variation in turbine non-dimensional mass flow parameter at such pressure ratios. The turbine speed parameter was defined with an arbitrary minimum value of 40%. This parameter can not be allowed to fall to zero in value, otherwise the turbine can produce no power output.

(3.7) A Note Concerning the Differential Gas Turbine Operating Schedule

In section (3.6), an operating principle for the differential gas turbine was described which was based on maximising the turbine entry temperature. However, the theoretically ideal operating principle is to choose a turbine speed parameter such that an optimal combination of turbine entry temperature and turbine isentropic efficiency is produced. Such an operating schedule could be realised, in combination with optimising compressor speed, by an optimisation procedure of the type described in chapter (7).

(3.8) Alternative Gas Turbine Concepts which were not Modelled

The twin shaft differential gas turbine operates on the principle that variation of the turbine mass flow parameter enables the turbine entry temperature to be maintained at a high value. The primary method of varying power output is to

107 adjust the compressor pressure ratio. Figures (3.12a) to (3.12c) illustrate alternative gas turbine concepts that in principle could achieve a similar operating schedule. None of these alternative concepts were numerically modelled. In the figures, an electrical transmission is assumed.

Figure (3.12a) illustrates a a non - constant speed single shaft gas turbine with variable geometry for the turbine but not the compressor. With this design, an axial turbine could be employed, as turbine entry temperature variation is not through turbine speed variation, but rather through variable geometry for the turbine.

Figures (3.12b) and (3.12c) illustrate designs that are similar to each other. Both designs are of twin shaft nature, with the turbine split into an HP and LP unit. The compressor is attached to the LP unit, and adjustment of the compressor pressure ratio is through adjustment of the speed of this LP shaft. The HP turbine is free to rotate at a separate speed from the LP shaft, and thus if the HP turbine is radial this enables adjustment of the turbine mass flow parameter. The LP turbine could be radial or axial, though an axial turbine behind the radial HP turbine makes for a more elegant design.

Where the two latter designs differ is that in the first the LP turbine has an excess of power over that required to drive the compressor, while in the second it has a deficiency. Thus, in the first design, an alternator is attached to the LP shaft, while in the second an electrical motor is attached (driven by the HP shaft alternator). It would also be possible to produce a hybrid of these two designs, with such a unit behaving as one type or the other according to the operating point.

In principle, all three of the designs illustrated in figs (3.12a) to (3.12c) should demonstrate similar operating

108 characteristics to that of the differential gas turbine. One problem with the twin shaft differential gas turbine is that the gross power of the turbine must be absorbed by a suitable power transmission mechanism. In conventional gas turbines only the net power (turbine output minus compressor input) need be absorbed. If an electrical transmission is fitted, this translates to the requirement for a large alternator. The three designs outlined in this section circumnavigate this problem.

(3.9) Summary

Small gas turbines require heat exchangers in order to achieve adequate efficiency levels. A twin shaft differential gas turbine plus heat exchanger was theoretically modelled, while a constant speed single shaft gas turbine plus heat exchanger was used to provide reference data. Thermal efficiency verses power output curves were obtained for various values of heat exchanger effectiveness. The detailed results of this work are presented in chapter (8).

109 (3.10) References

(1) Penny, N. Rover Case History of Small Gas Turbines SAE paper 660361, 1966

(2) Baines, N.C. ; Panting, J.R. ; Etemad, M.R. ; Besant. C. A Gas Turbine-Electric Vehicle Concept I. Mech. E. Paper C389/031 FISITA conference, Total Vechicle Dynamics, vol. 2 1992

(3) Cohen, H. ; Rogers, G.F.C. ; Saravanamutoo, H.I.H. Gas Turbine Theory Longman Scientific and Technical, 1989

(4) Private Communication Rolls-Royce Plc, Ansty, England 1988

(5) Large, G.D. ; Meyer, L.J. Cooled Variable-Area Radial Turbine Technology Program Final Report (Garret Turbine Engine Co.) NASA CR-165408 January 1982

(6) Press, W.H. ; Flannery, B.P. ; Teukolsky, S.A. ; Vetterling, W.T. Numerical Recipes, the Art of Scientific Computing Fortran Edition Cambridge Univ. Press, 1989

(7) Car Design and Technology Brackland Publishing Plc August 1992

110 (8) Judge, A.W. Small Gas Turbines and Free Piston Engines Chapman and Hall Ltd., 1960

(9) Anderson, D.G.R. The Differential Gas Turbine for Automotive Applications: a Feasibility Study MSc/D.I.C. thesis, Imperial College of Science, Technology and Medicine, Univ. of London, Sept. 1977

(10) Tso, C.W. Investigation of Differential Gas Turbine for Automotive Applications MSc/D.I.C. thesis, Imperial College of Science, Technology and Medicine, Univ. of London, Sept. 1976

111 (oxigurbloN EYfrioluST inn.er Cr*Ameeg GAS Afe.

QIN

co meeessat Tu RGINE

FIG (3.1) THE SIMPLE GrAS

TueeINE t A 11" 7 )--1 owr &err EXCHANCtER.

112 Fig (3.2) Thermal Efficiency versus Cycle Pressure Ratio for a Simple Gas Turbine 0.35

0.30 - t=1600/288 H 0.25 - iency,

ic 0.20 - Eff l 0.15 t=1200/288 Therma

0.10 le c

Cy 0.05

0.00 0 lb h 310 40 550 60 710 80 910 160 1i0 1L0 Cycle Pressure Ratio, (--) Tt)Rani e 2 WIN 3

V 4- I-16AT Eicroictr4Ne4 GoMemS-rpoN CI-14.1•V eq.

2!

GI (3.3) 11-rE. G, A Tu as siva S I-1 AT 1-1 AN G ER

114 Cycle Thermal Efficiency, (-) 0.45 0.40 - 0.00 0.05 0.30- 0.35 - 0.20 0.25 0.10 0.15

0

Fig (3.4)ThermalEfficiencyversusCyclePressure Ratio foraGasTurbinePlusHeatExchanger n J.

Cycle Temperature Ratio=1200/288K Cycle Pressure Ratio,(- 10

1S

20

25

30 Cycle Thermal Efficiency, H 0.4 0.3 0.5 0.2 0.0 0.1 Fig (3.5)ThermalEfficiencyversusCyclePressure Ratio foraGasTurbinePlusHeatExchanger Cycle Temperature Ratio=1600K/288K I

Cycle Pressure Ratio,(-) 1 1 0

1~5

20

25

30 Compressor Pressure Ratio, [-) 1 0.0 I 0.1 Fig (3.6a)PressureRatioversusNon-Dimensional Mass FlowFunctionforaCentrifugalCompressor I 0.2 Non -Dimensional MassFlowFunction,(-) I 0.3 I 0.4 I 0.5 I 0.6 1 0.7 I 0.8 I 0.9 I 1.0 I

1.1 Fig (3.6b) Isentropic Efficiency v Non-Dimensional Mass Flow Function for a Centrifugal Compressor 0.9

._U 0_ L._0 0.7 - -4-1 = 1-1 CD I-. cn 03 - L- 0 Cri rno 0.6 -

E 0 CD 0.5 I I I 1 I I I I 1 1 I I 1 IIIIIIII 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Non - Dimensional Mass Flow Function, (-) Fig (3.7a) Pressure Ratio versus Non-Dimensional Mass Flow Function for a Radial Turbine 7

6

5 H io, t

Ra 4

3 Pressure Nt=120%

t=40% 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 11 Non-Dimensional Mass Flow Function, I-) Turbine Relative Speeds Increase in 20% Increments Fig (3.7b) Isentropic Efficiency v Non-Dimensional Mass Flow Function for a Radial Turbine 0.9 0.8 0.7 H 0.6 iency,

ic 0.5 ff E 0.4 ic

trop 0.3

Isen 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Non-Dimensional Mass Flow Function, (-) x.

•Xel = XI +- LX1-6 — Xt

+ 9 (x-m-xx

FIG (3.8) 17-1 E CoNS-T0c4C.TroNI OF- P SPC-4r) LI NE

121 select r , r M. , Nc Pc P t * bracketing speeds N t L N* Ri with N > Nt t Li Ri

find TETLI by iteratively inverting (a ) equation ( 3 . 16) and employing the value N Li BOX (A) find from turbine map employing Nt M*t Li Li and r P t find TET by iteratively inverting Li ( b ) equation ( 3 . 17) and employing the value M Li

1

• TET —* TET repeat BOX (A) for Ri(a)1 Nt t , Ri, Ri

IS (HTET >TET ) AND (TET TET ))) Ll(a) Li(b) RI(a) RI(b)

IYE S SOLUTION NOT BRACKETED

Nt = Nt = Nt L Li R Ri

1

N t = (Nt + Nt )/ 2 try L R 1

N , TET repeat BOX (A) for , , M (b) TET(a) t try t try

IS - TET TET I TET(a) (b) 1/ (b) < c IYE S

122 solns. are N t try TET=(TET + TET(b) )/2 * (a) Mt try

IF TET TET (a)< ( b ) IF TET < TET Li(a) Li (b) * Nt = Nt L try ELSE

NtR= N*t try ENDIF ELSE IF TET > TET Li(a) Li(b) * Nt = Nt L try ELSE

Nt H=N t try ENDIF ENDIF I

Fig (3.9) The Iterative Solution Process for the Single Shaft Gas Turbine

123 Co ract.6-ri Or-.1 crol-maek

OUT T PS..06ct SHAT. CoNNEC-T60 To Pc NET crecr CARJEls-tt C .r-

Co mta2ESasa— Co A.,N$ -Toga. - GnNNECtto To -Sur,* -re 0.4s G(E-g.

FIG (3. to) T4-1E pi F TImergi NE- LJP-7'H M eci-o NI CAL Ert Y CLi C -T2.ANiSre1 lo 1\-)

124 select r , r M / TETMAX P N., C t • m* bracketing speeds Nt ' "t • • with Nt > N find Mt non- iteratively from eqn. (3.17) at TET=TETMAX

from turbine map and N , r find m•t •t 4. • IS M t > t L NO 'YES

NO SOLUTION find M• from turbine map and N• , r t t P t

IS M t NO TYES

N N turbine mas flow t t param. = Mt N t try 2 TET found from Mt 1 and eqn. (3.17) Fi from turbine speed find Mt try param. N turbine map and = • N r t try P t 1

IS IMt - Mt / Mt < try NO IYES

125

IF M t > Nit solutions are, try * * TETMAX, Nit, Nt try Nit = N *t II try ELSE

Nt = N *t L try END I F 1

Fig (3.11) The Iterative Solution Process for the Differential Gas Turbine

126 vAiziAfece. Gr60 /.1(-lay

Co p4FreC:Skie -rufeatNE

CoP.481.4Sir 0-s4,41116Z

COMPRESSOR LP ALTEe4AIDeS 11.43.6VE

CoMeASIoN elorPiStit

(c) Kra He &mom "nposskie Le ComAUSSA MaToe -rOtrainic ALTEAvA-roR

FIG (3.12) V41.210145 GAS 7ora3in)E-1

PES i6INI CO kJ cep-T-5

N. Q . 1-1 CAT EXCI-ANGIE:RS Nar Si-lawx3)

127 ratio of specific heats = 7 = 1.40 ; 1.35 ; 1.30 component isentrpic efficiency = 80%

cycle optimum pressure ratios case temp for the different 7 opt ratio effcy r t Popt %

200 (1) 1288 8.6 ; 10.8 ; 14.4 22

1600 (2) 288 17.3 ; 23.3 ; 34.1 30

16 00 (3) 223 50.4 ; 75.2 ; 128.6 45

Table (3.1)

The Optimum Thermal Efficiencies of the Simple Gas Turbine and the Corresponding Optimum Pressure Ratios

The calculations are based upon the assumption of constant fluid properties

128 Chapter (4) The Compound Highly Turbocharged Engine

(4.1) Overview

The compound highly turbocharged engine was investigated as a possible alternative to the small gas turbine plus heat exchanger as an automotive powerplant. The concept is a means to achieve high overall pressure ratios at all power outputs, and to this end may in fact be seen as a counterpart to the high pressure ratio gas turbine without heat exchanger.

As described in section (1.4), a compound turbocharged engine has a two stage compression and expansion process. Initial compression is in a rotodynamic or positive displacement compressor. From here, the compressed inlet air is admitted to the piston engine, in some cases passing through an aftercooler on the way. In the piston engine, a fuel/air mixture is further compressed and ignited, and the the first stage of expansion follows. After the first stage of expansion, the burnt fuel/air mixture is exhausted from the piston engine, to pass through the exhaust turbine which may be radial or axial in design. In the turbine, the second stage of expansion occurs, and the working fluid is exhausted from the engine. The concept was illustrated in fig (1.10).

The compressor absorbs power, the turbine produces power, and the piston engine can either absorb or produce power depending upon the degree of frictional losses and the enthalpy gain of the exhaust fluid relative to the inlet air. The power inputs and outputs of these three units must be coupled through some form of transmission. Here lies the key difference between a compound turbocharged engine and a conventionally turbocharged engine. In the latter type, the turbine simply powers the compressor, and there is no direct

129 power transfer between those mechanisms producing the first and second stages of compression and expansion. In the former type, the turbine power output directly contributes to overall power output through an interconnecting transmission system.

There are a number of choices concerning the design of a compound turbocharged engine, such as whether it should be four - stroke or two - stroke, compression ignition or spark ignition type, or include an aftercooler. There was insufficient time to investigate all of these options, and the four stroke spark ignition type with and without aftercooler was chosen for study through numerical modelling. The initial sponsors of the project were also more interested in the spark ignition four stroke type of engine. The compression ignition engine does however possess theoretical advantages, which are discussed in more detail in section (4.2.3).

A closely related concept to the compound turbocharged engine is the free - piston engine, in which the piston engine produces no net power output. In such an engine, the gross power output of the piston engine is balanced by frictional losses and by the enthalpy gain of the exhaust fluid relative to the inlet air. The free piston engine may be seen as a special case of the compound turbocharged engine and as such was not expected to offer any relative advantage over the latter. In particular, the cylinder volumetric compression ratio is constrained to take a particular value for given operating conditions, such as fuel/air ratio, compromising design freedom. Free piston engines therefore have a self adjusting cylinder volumetric compression ratio. Further, the free piston engine relies for its effectiveness upon the turbine inlet pressure being higher than the compressor outlet pressure. It was found that at high piston engine speeds and with optimal valve timings in the simulated compound turbocharged engine the

130 reverse occurred. This in turn implies that a high speed free piston engine would offer no advantage over a gas turbine engine. Reference (1) discusses the free piston engine in more detail. Because the free piston engine was not expected to offer any advantage over the compound turbocharged engine it was not investigated in this project.

This chapter describes the theoretical advantages of the compound turbocharged engine concept and gives a brief review of past compound turbocharged engine designs. There is a description of the proposed design layout that was numerically modelled and an illustration of the constraints on turbine entry temperature in such an engine. The transient response of such a unit is briefly discussed. The reasons for numerical investigation of the compound turbocharged engine are expounded. A simple quasi - steady model of this engine type that was used to give an initial estimate of performance levels is described. (The following chapter describes a more complex time - marching model).

(4.2) The Theoretical Advantages of the Compound Turbocharged Engine

(4.2.1) The Efficiency of the Compound Turbocharged Engine in a Limiting Case

A simplified analysis of the compound turbocharged engine cycle is first considered. It is based on the following assumptions,

(i) isentropic expansion and compression, (ii) no heat loss from the cylinders of the piston engine, (iii)no frictional losses, (iv) constant volume combustion at 'top dead centre', (v) fluid properties are constant, (vi) no mass addition during combustion.

131

The pressure - volume and temperature - entropy diagrams for this type of cycle are schematically illustrated in fig (4.1). Referring to fig (4.1), it is noted that,

T + ST 2 23 as ----> 1, thermal carnot (4.1) T2

Likewise,

w aa c (4.2) carnot n carnot v6T23

where T2 is the temperature after compression, ST23 is the additional temperature change produced by combustion, 71 denotes efficiency, w is the specific work output, Sclin is the specific heat input produced by combustion and c is the specific heat constant at constant volume.

It is thus seen, according to an idealised model, that such a cycle produces very high efficiencies, in fact tending towards the maximum achievable Carnot efficiency. If the temperature after compression (T2) is large, then a small relative temperature change during combustion (8T23) can still produce a high work output per unit mass of working fluid (of order cvST23). High temperature changes after compression would in turn result from high overall pressure ratios. The arguments are developed more fully in Appendix (A5).

The same arguments apply to the Joule cycle which is used to provide a simplified analysis of gas turbine performance, i.e. conditions (4.1) and (4.2) still hold. However, it is more difficult to achieve large pressure ratios in gas turbines, for the reasons noted in section (1.4). A compound turbocharged engine can more easily attain high pressure ratios, due to the two stage compression and expansion processes, one of which is of the positive displacement type. Thus, for a given overall pressure ratio, the compressor and turbine pressure ratios will be lower. Also,

132 conditions (4.1) and (4.2) hold for the Otto cycle, typically used to provide a simplified analysis of the performance of a naturally aspirated spark ignition engine. However, naturally aspirated engines cannot achieve the same specific outputs for a given cubic capacity of piston engine, due to the fact that they are not highly supercharged.

Cycle irreversibilities result in the fact that in practice the Carnot efficiency cannot be approached. For a given upper temperature limit on the cycle, this then produces an optimum theoretical pressure ratio for efficiency, which may in itself be unachievable. If component inefficiencies (degree of heat loss, frictional loss, compression and expansion irreversibilities and the total pressure losses across valve ports) can be maintained at a low level, this theoretical optimum pressure ratio will be high, as will the efficiency level and specific output that is achieved.

(4.2.2) Consideration of the Optimum Pressure Ratio for Efficiency for the Gas Turbine

It is now illustrated that a high pressure ratio engine is an advantage, and hence that the compound turbocharged engine represents a suitable means to achieve such high pressure ratios. This is achieved by considering the optimum pressure ratio for efficiency for a gas turbine without a heat exchanger. For this, the following assumptions for the gas turbine analysis were made,

(i) compressor and turbine isentropic efficiencies are constant, (ii) fluid properties are a function of temperature only, (iii)there is a 2% total pressure loss in the combustion chamber, (iv) mechanical and transmission efficiencies are equal to unity,

133 (v) combustion efficiency is equal to unity, (vi) there is no mass addition during combustion.

The performance of such a cycle, for given ambient temperature, turbine entry temperature and compressor pressure ratio may then be calculated. This then enables the optimum pressure ratio for maximum cycle thermal efficiency to be calculated. The fluid properties algorithm described in section (2.2) was employed, and compressor and turbine isentropic efficiencies of 80% were assumed. The calculation of gas turbine efficiency then proceeds as given by Appendix (A4).

The resulting calculations of thermal efficiency were used to find the optimum value of this parameter at the optimum compressor pressure ratio, for given values of ambient temperature, TET and compressor and turbine isentropic efficiency. This was done using a simple one dimensional line minimisation algorithm, as is described in section (7.5). This analysis then has parallels with that of section (3.2), but it is more accurate because consideration of variable properties is invoked.

The data from this optimisation procedure is presented in table (4.1). The values of (T3/Ti) used are: (1200K/288K), (1600K/288K), (2000K/288K) and (2000K/223K). The upper and lower values of T1 correspond with the ambient temperatures at sea level altitude and 10 km altitude respectively, both according to the ASA International Standard Atmosphere. The three values of T3 correspond approximately, in increasing order, with the use of an uncooled metal turbine, the use of a cooled metal or uncooled ceramic turbine, and possible future use of a cooled ceramic turbine.

The lower values of (T3/T1) are applicable to smaller engines, while the larger values are more applicable to large engines, particularly high by - pass ratio aircraft

134 turbofans. It can be seen that the optimum compressor pressure ratio for best efficiency is high. For example, case (2) indicates an optimum compressor pressure ratio of approximately 22:1, at which point the efficiency level is approximately 30%. This pressure ratio is much too high to be achieved with small gas turbines. Likewise, the optimum pressure ratio for the future technology aero - engine of case (4) is very high at approximately 76:1, at which the thermal efficiency level is at a high value of approximately 41%. This high pressure ratio includes pressure recovery in the engine intake, and may therefore represent an achievable figure for a large engine fitted to a high speed aircraft.

A number of further points need to be made about the procedure. Firstly, there will also be an optimum pressure ratio for maximum specific power output, which will generally be lower than the optimum pressure ratio for maximum efficiency. Secondly, the slope of 'nth versus r is shallow near the maximum, so that a relatively large percentage drop in the pressure ratio from the optimum will initially result in a small percentage drop in the efficiency value. Thirdly, the provision of any bleed air for turbine cooling will reduce the achievable efficiency levels slightly.

(4.2.3) The Compound Turbocharged Engine Considered as a High Pressure Ratio Unit

The compound turbocharged engine is a potential solution to the problem of attaining the high pressure ratios in small engines for good thermal efficiency. Such an engine is suited to the development of high overall pressure ratios because of its two stage compression and expansion process, one of which is of the positive displacement type.

The basic thermodynamic analysis of such an engine is similar to that of the gas turbine, though with an assumed

135 constant volume rather than constant pressure combustion phase. The calculation of the optimum pressure ratio for efficiency for the compound turbocharged engine is more difficult than for a gas turbine, as there are now three constraints. These are the turbine entry temperature, the mean in - cylinder temperature and the temperature prior to combustion. The latter figure need only be considered in spark ignition engines, where it is important to prevent the phenomenon of detonation.

If heat losses, frictional losses, turbomachinery irreversibilities and the total pressure losses across valve ports can all be kept at a low level, the optimum pressure ratio for efficiency for this unit will be high, as will the efficiency level that is attained. Specific outputs, measured in power output for a given cylinder volume of piston engine, will be high if the degree of supercharging is high.

In addition, when the compressor is operating at low speed and hence pressure ratio, an adjustable cylinder volumetric compression ratio can maintain the overall pressure ratio, as described in section (4.5.5). This compares with the case of the simple gas turbine without heat exchanger, where a drop in rotational speed implies a fall in overall pressure ratio and hence thermal efficiency.

There are, however, practical limitations to achieving high overall cycle pressure ratios in compound turbocharged engines. In spark ignition engines, the volumetric compression ratio must be limited to control detonation. Compression ignition engines have the advantage that higher pressure ratios can be employed, subject to constraints on engine weight. If the maximum pressures in such a unit are high, then the unit must be made sufficiently strong to contain these pressures.

136

Though compression ignition engines have the advantage that they can tolerate higher pressure ratios, they have the disadvantage that the maximum rotational speeds of such engines is rather lower. This reduces specific power output. For example, racing spark ignition engines can achieve speeds of 14,000 rpm (2), while the maximum speeds of small high speed compression ignition engines appears to be limited to no more than approximately 6000 rpm.

In considering the pressure ratios that can be achieved with spark ignition compound turbocharged engines, it has been found that the overall volumetric compression ratio must typically be limited to a maximum of approximately 10:1, at sea level altitude and with poppet valves, to control detonation. Section (4.5.3) discusses the case when non-poppet valves are fitted.

Using this figure for the maximum volumetric compression ratio, the pressure ratio prior to combustion may be found. For this, an alternative form of the non - dimensional entropy function was defined,

1 c dT x°(T) R J T (4.3) Tref

since,

cV = cp - R (4.4)

for constant gas constant, R, then,

xo ) = vio(T) 1n(4- ) (4.5) ref

by the definition of the standard non - dimensional entropy function, te(T).

Then, during compression,

137 T v 2 As 2 c dT f P v RdTv - Ax°(T) + ln( 4)\ (4.6) f T v 1 1 1

Then, if the overall volumetric compression ratio is known to be r and the compression is isentropic,

V 2 AX°(T) In( = ln(r,) (4.7) vi

Equation (4.7) can be inverted iteratively, the method of bisection being used, (3), to find the temperature at the end of compression. Then, the pressure ratio may be found from the ideal gas equation as, T ) r = rv T2 (4.8)

where T2 is the temperature after compression found from equation (4.7), and T1 is the initial temperature.

Equations (4.7) and (4.8) were solved for two conditions, the fluid properties algorithm described in section (2.2) being used. The two conditions were for sea level altitude and an altitude of 10 km. Suitable atmospheric conditions were taken from the ASA International Standard Atmosphere. In the latter case it is more difficult to estimate a suitable overall volumetric compression ratio which avoids detonation. Initial compression to one bar was assumed, followed by the volumetric compression ratio of 10:1 (this is not necessarily correct). For the initial stage of compression, isentropic conditions were assumed, employing the non - dimensional entropy function te(T) to calculate the temperature after this first stage. For this calculation refer to Appendix (A4).

The results of this analysis are given in table (4.2). It is

138 seen that at sea level altitude, the pressure ratio prior to combustion is approximately 24:1, while at 10 km altitude it is approximately 161:1. In the latter case the temperature is high, and thus it might be necessary to reduce this compression ratio to prevent detonation.

The pressure ratios obtained are higher than those of an equivalent gas turbine. A small gas turbine will not generally operate at pressure ratios as high as 24:1, while a large turbofan will not generally operate at a pressure ratio as high as 161:1. Note also that the overall expansion ratio will exceed these figures, due to the pressure rise during combustion.

(4.2.4) The Specific Power Outputs of Various Engine Types Compared

The specific power outputs of various engine types are now considered. Comparisons are made between,

(i) a naturally aspirated spark ignition racing formula one piston engine, (ii) a conventionally turbocharged spark ignition racing formula one piston engine, (iii)an experimental automotive gas turbine, (iv) a turboprop gas turbine engine, (v) a large high by - bass ratio turbofan gas turbine engine, to obtain a specific maximum power output figure, in kW kg-1.

Data for case (i) was taken from (2), for present day formula one engines. For case (ii) data was taken from (4). The latter data is for the case when the maximum compressor boost pressure was restricted by regulations to 4 bar. Data is not available for the unrestricted case. Data for case

139 (iii) was taken from (5) for a gas turbine of 3:1 pressure ratio and heat exchanger effectiveness of 85% and unlike the other comparative figures includes the mass of part of the transmission in the form of the high speed alternator. The engine mass figure was therefore corrected with data from (6) to subtract the weight of the alternator. For cases (iv) and (v) data was taken from (7). The power output figure for this last engine was estimated according to the data given and certain assumptions. The main assumptions were that the compressor and turbine isentropic efficiencies were 80%, the total pressure drop in the combustion chamber was 2%, the TET was 1600K and the ambient temperature was 288K. The compressor pressure ratio was given by (7) as 35:1. Calculations of the specific work output (in J kg-1) then proceeded as given by Appendix (A4). The engine total mass flow rate, by-pass ratio and engine mass then enable the specific work output (kW kg 1) to be calculated from,

-3 W - m'w (4.9) (B r + 1)m x 10

where W is the resulting specific output (kW kg-1), m' is the total engine mass flow rate (kg s-1 ), Br is the bypass ratio, w the specific work output calculated from Appendix (A4), (J kg-1), and m the total engine mass (kg).

The data for the five cases is presented in table (4.3). The resulting figures show values of 3.88 kW kg 1 for the naturally aspirated racing engine and 5.14 kW kg-1 for the -1 turbocharged racing engine, 1.05 kW kg for the experimental automotive gas turbine and 4.31 kW kg 1 for the turboprop. This last figure is assumed to be for sea level altitude, the altitude figure not being quoted in (7). The figure for the turbofan is 7.96 kW kg-1.

Thus it is seen that the specific power output of highly (non - compound) turbocharged racing engines is comparable with that of large aero - engines. This in turn suggests a

140 favourable outlook for the compound highly turbocharged engine, which has the potential to generate even higher specific power outputs. However, a point to be borne in mind is that formula one engines are only required to last a few hours before a complete rebuild, whereas an aero - engine is designed to operate for many hours with the minimum of maintenance. In particular, the fact that turbocharged racing engines need only last a few hours permits a very short turbine life and hence a higher turbine entry temperature. This in turn permits a higher degree of supercharging (refer to section (4.6)).

(4.2.5) Further Considerations

A first additional point is that compound turbocharged engines should theoretically demonstrate higher thermal efficiencies than a corresponding naturally aspirated unit. In a naturally aspirated engine, the working fluid is exhausted at relatively high pressure. However, in the compound turbocharged engine, this pressure energy is recovered with expansion of the working fluid down to atmospheric pressure in an exhaust turbine.

A second additional point is that with rotodynamic compressors, higher boost pressures will occur at higher mass flow rates and hence higher piston engine speeds. Thus the IMEP of the piston engine rises with speed, to some extent counteracting the rise in FMEP to produce a more constant piston engine mechanical efficiency, with the mechanical efficiency given by,

BMEP IMEP - FMEP FMEP - 1 (4.10) 77mech IMEP IMEP IMEP

141 (4.3) A Brief Review of Past Compound Turbocharged Engine Designs

A brief review of past compound turbocharged engine designs was undertaken. Only three examples of such engine designs were found. These were,

(i) The Wright Turbo Compound Engine, (ii) The Napier Nomad Engine, (iii)The Wallace Differential Compound Engine.

Engine (i) is described in (8), engine (ii) in (9) and engine (iii) in (10).

The Wright Turbo Compound Engine is the only known production engine of the compound turbocharged type. Known details of this engine are few. The engine was for aeronautical use, and was of four stroke gasoline type with eighteen radially arranged cylinders. The cylinders were divided into three groups of six, each group feeding a separate turbine geared into the crankshaft system. It is believed a centrifugal supercharger, geared to the crankshaft, was employed. Reference (8) quotes a peak brake thermal efficiency of 36%, which is high for a spark ignition engine.

The Napier Nomad was a complex experimental aero - engine designed circa 1950. The engine was a two stroke compression ignition type, with twelve cylinders arranged into two horizontally opposed banks of six. Underneath this piston engine was set a twelve stage axial compressor and a three stage axial turbine. The compressor and the turbine were mounted on the same shaft. This was geared into the crankshaft of the piston engine via an epicyclic gear set and a complex infinitely variable gear of the Beier type (sliding mechanical contact form of mechanism). The propeller shaft was also driven from the epicyclic gear set.

142 A typical air/fuel ratio for this engine was 40:1 (approximately 0.4 equivalence ratio) and performance could therefore be augmented by additional combustion after the piston engine and prior to the turbine. Further performance gains are achieved through the use of water injection downstream of the compressor (which increases maximum permissible boost for a given turbine entry temperature) and through consideration of the small amount of jet thrust produced at the turbine exhaust.

Specific data for this engine is presented in table (4.4). Attention is drawn to the maximum figure of 1.77 ekW kg 1 (ekW stands for estimated kilowatts, and allows for the small amount of jet thrust produced). Note is also made of the peak brake thermal efficiency of 42%. The specific output figure can be compared with the figures in table (4.3).

The Wallace Differential Compound Engine is an experimental unit only, designed with road haulage in mind. The piston engine of opposed two stroke compression ignition form drives the annulus of an epicyclic gear train. The sunwheel drives the positive displacement compressor. The planet carrier drives the output shaft into which are geared both the main power turbine and the auxiliary (or stall) turbine. The main power turbine is of the inwards radial flow type. The specific data for this engine is presented in table (4.5).

(4.4) Justifying Further Research into the Compound Turbocharged Engine

The foregoing section demonstrates that the compound turbocharged engine has been found to be a viable prime mover, although not one which has yet found commercial acceptance. However, the present proposal differs from the three described in the previous section in several ways. The

143 differences in the intended concept are itemised as follows,

(i) the engine is of small, high speed, rather than large, low speed type, (ii) the engine has a variable cylinder volumetric compression ratio, (iii)the engine is intended to be fitted with hybrid sleeve/Aspin valve gear, (iv) the engine is throttleless in concept, (v) the engine has a high turbine entry temperature.

The design concept is described in more detail in the following section. In the event, time has precluded simulation of item (iii). Nevertheless, items (i),(ii),(iv) and (v) represent an unique combination of design elements. Thus, further investigation of this type of engine is justified on the basis that no other unit of this type is known to have previously existed.

(4.5) The Proposed Design Layout

The proposed design layout has the following features,

(i) four cylinders, (ii) four stroke, spark ignition, high speed unit, (iii) hybrid sleeve/Aspin valves, (iv) swash plate type crank, (v) variable cylinder volumetric compression ratio, (vi) throttleless operation, (vii) high nominal turbine entry temperature of 1600K.

These items are discussed in order.

144 (4.5.1) Four Cylinder Layout

Four cylinders were chosen as with the crank mechanism and layout employed, four cylinders are the minimum required for complete dynamic balancing.

(4.5.2) Four Stroke, Spark Ignition, High Speed Cycle

A spark ignition engine type was chosen because such engines can operate at high speeds and this later proved to be an important advantage. The four stroke cycle was chosen for initial study, there being insufficient time to investigate the two stroke cycle option.

(4.5.3) Hybrid Sleeve/Aspin Valves

The sleeve valve is a thin sleeve of metal sandwiched between the piston and cylinder wall, fig (4.2). By an appropriate movement of this sleeve, ports in the cylinder wall are uncovered by holes in the sleeve. During compression, combustion and expansion, the part of the sleeve containing its ports is retracted to lie above a Junk Ring in the Junk Head, refer to fig (4.2). This produces better sealing at high pressures. For a four stroke engine cycle, a simple sleeve mechanism cannot operate in a linear direction only and must therefore provide some rotational movement.

The Aspin valve is of conical type, with its axis lying along the cylinder centre line, fig (4.3). Rotation of this valve at half engine speed opens and closes ports in the cylinder wall. In practice, such valves suffered from lubrication problems on the conical face, which is held at pressure against its seat to provide a gas seal. The Aspin valve is described in more detail in (8).

It is proposed that a hybrid sleeve/Aspin valve be used, as

145 illustrated in fig (4.4). In this mechanism, sealing of the cylinder at high pressures is provided by the sleeve valve, which moves in the linear sense only. Thus the problem of lubricating an Aspin valve held at pressure against its seat is removed. When the sleeve is 'down', the rotating Aspin valve controls whether the inlet or exhaust port is open. When the sleeve is 'up' the ports are closed.

The hybrid sleeve/Aspin valve is considered to offer a number of advantages. These are,

large port areas and flow coefficients, resulting in high volumetric efficiency, higher volumetric compression ratio can be employed, there is no engine speed limitation from valve 'bounce', mechanical driving power for this type of mechanism may be less than for a poppet valve train, (iii)(e) the engine may have a lower lean limit, reducing emissions.

The Aspin valve demonstrates high volumetric efficiency, (8). Both the Aspin valve and the sleeve valve appear to permit higher volumetric compression ratios, (8), (11). Ricardo, (11) was able to increase the volumetric compression ratio on a four valve per cylinder engine from 4.8:1 to 5.8:1 on an equivalent sleeve valve engine, representing a 21% increase. The maximum compression ratio was in both cases defined by the detonation limit, the values for this parameter being low because of the use of a low octane fuel. It appears that the removal of a hot exhaust poppet valve from the combustion chamber delays the onset of detonation. Thus, higher compression ratios can be employed.

146 Poppet valves actuated by a cam mechanism generally require the use of return springs. At high engine speeds, resonances in the springs cause the valves to bounce, destroying the design valve timings and stressing the valves. In an attempt to overcome this problem, some modern racing engines employ gas springs, which are less subject to resonance, (12). Another approach is to use desmodromic cams, in which a second cam is used to define the return motion of the valve. The hybrid sleeve/Aspin valve does not use a cam mechanisms and therefore does not suffer from speed limiting valve bounce problems. This is an important consideration given that the optimum speeds for the maximum power output of compound turbocharged engines were calculated to be high, refer to chapter (8).

In reference (11), it was found that the power required to drive a sleeve valve train was considerably lower than for an equivalent poppet valve train. It was found that Aspin valved spark ignition engines would run at very lean fuel/air mixtures, a lean limit quoted to occur at an equivalence ratio of approximately 0.6 in (8). This is believed to be due to centrifugal forces in the combustion chamber creating a stratified charge effect.

As already noted, limitations on project time have dictated that the time marching engine simulation algorithms described in chapter (5) have initially proceeded assuming the use of four poppet valves per cylinder, i.e. two inlet and two exhaust valves.

(4.5.4) Swash Plate Crank Mechanism

To obtain a compact layout with perfect dynamic balancing, a four piston unit with swashplate type crank mechanism is to be used, fig (4.5). The four cylinders are arranged around a common axis, their respective centrelines being parallel to the crank centreline. Such a mechanism results in no

147 harmonics higher than the primary because the motion is pure simple harmonic. The primary forces are balanced if one swashplate is used and the pistons are all 90° out of phase with each other. The primary torques are then not balanced, and two counter balancing secondary swash plates must be used to counteract these torques. The counter balances can double as flywheels. One counter balance is needed to provide a counter torque against the torque produced by the swash plate itself. The second counter balance, which must counter rotate at engine speed, balances the torques produced by the accelerations of the pistons. Such a unit will then be perfectly dynamically balanced. This is an important consideration for a potentially high speed unit. An analysis of the dynamic balancing of this engine configuration is given in Appendix (A6).

Various engine layouts with the piston axis placed parallel to the crank centre line have been attempted experimentally (8), (13). Reference (13) describes the 'Dyna-Cam' engine that uses a cam form of mechanism to describe the piston motion, rather than a swash plate. This engine is type certified in the U.S.A. for aircraft use - though it is not a production engine.

The particular type of crank mechanism that might be employed is based upon a simple harmonic motion machine designed as part of TRIDAC - the Royal Aircraft Establishment's three dimensional analogue computer, employed between c. 1955 to 1966, (14). By analogue means, the mechanism provided sine and cosine values as a function of input angle. The mechanism is illustrated in schematic form in fig (4.6).

(4.5.5) Variable Cylinder Compression Ratio

At high boost pressures it is necessary to limit the cylinder volumetric compression ratio, so that the overall

148 compression ratio is sufficiently low to avoid the onset of detonation. At lower piston engine speeds, the mass flow rate demands of the piston engine are lower, and thus the mass flow rate through the compressor must be lower. The characteristics of rotodynamic compressors are such that a substantially lower mass flow rate results in a lower pressure ratio across the compressor. At such lower boost pressures, a fixed cylinder volumetric compression ratio then results in a low overall pressure ratio. This in turn leads to a low thermal efficiency for the engine.

The solution to this problem is a variable cylinder volumetric compression ratio. At first, it was anticipated a variable swash plate angle would be employed, thus varying piston stroke. This concept is illustrated in fig (4.7). This type of mechanism has the disadvantages that it is both mechanically complex and reduces the swept capacity at high boost pressures. The latter effect is to the detriment of maximum specific output.

Therefore, the engine simulations are based upon the idea that the entire swash plate mechanism can be translated back and forth slightly, reducing or increasing the clearance volume at top dead centre, refer to fig (4.8). The swept volume then remains constant.

(4.5.6) Throttleless Concept

A throttleless concept implies greater part load efficiency levels. A conventional butterfly throttle used on conventional spark ignition engines produces a large total pressure drop in the inlet manifold at small throttle openings. This reduces the efficiency of the engine, by imposing a larger pumping loop in the engine cylinders. Elimination of this butterfly valve removes this source of engine inefficiency. Section (4.7) discusses the transient response of such an engine concept in more detail.

149 (4.5.7) High TET Operation

Higher turbine entry temperatures permit higher compressor pressure ratios to be used and result in higher specific outputs, as discussed in section (4.6). For a smaller engine, the high turbine entry temperature limit could be achieved through the use of a ceramic turbine, such devices presently being in the experimental stage. For a larger engine, the high turbine entry temperature could be achieved by the same means, or through the use of a cooled nickel alloy turbine. Cooling is impractical for small turbines.

(4.5.8) The Overall Concept

The overall concept design layout is presented in Appendix (A7) with a centrifugal compressor and an axial turbine placed in line with the main engine centre line. The valve gear drives would run parallel with the main centreline of the engine. A separate alternator would be provided for the piston engine and turbine/compressor shaft. Due to the relatively high maximum speeds of the piston engine, the alternator on this output shaft can be kept relatively small. Additional features which can be considered, are,

(ix) an additional combustion chamber downstream of the piston engine and upstream of either the main turbine, or between an HP and a LP turbine, (x) intercooling or aftercooling of the compressed charge.

It is possible that item (ix), with additional combustion occurring at a relatively low temperature, may assist in reducing pollutant levels. This is because the amount of chemical dissociation is reduced. In addition, unburnt hydrocarbons leaving the piston engine would be burnt in the

150 secondary combustion chamber. It is not clear whether secondary combustion would increase or reduce the overall power output. It is likely to reduce unit thermal efficiency, due to the low secondary expansion ratio.

Intercooling and aftercooling are here defined as separate concepts. Intercooling means a series of compressions, each followed by charge cooling in a heat exchanger. Thus, adiabatic compression is replaced by a process that approximates isothermal compression (prior to the charge being admitted into the piston engine). Aftercooling means one compression stage in an adiabatic compressor, followed by one stage of cooling in a heat exchanger, and then the inlet charge is admitted to the engine. Another option which was considered is water injection, where water injected into the inlet manifold cools the inlet charge by virtue of its evaporation. Calculations showed, however, that the quantity of water required for sustained operation at maximum load was excessive.

Intercooling and aftercooling permit the following observations,

(x)(a) the increased inlet charge density increases the piston engine mass flow rate for a given piston engine speed, (x)(b) the reduced inlet charge temperature permits a higher cylinder compression ratio before the detonation limit is reached, (x)(c) the reduced inlet charge temperature permits a higher boost pressure before the turbine entry temperature limit is reached, (x)(d) with intercooling, the compressor work input is reduced.

Items (x)(a) - (c) all increase the indicated mean effective pressure of the piston engine, with the consequence that

151 power output is increased. By increasing the indicated mean effective pressure, the friction mean effective pressure becomes a smaller fraction of the former, and the two effects combined result in higher unit thermal efficiency and higher piston engine mechanical efficiency. Item (x)(d) also boosts overall power output level in an intercooled engine. An aftercooler was modelled, but an intercooler and additional combustion chamber were not. An overall engine concept, with two stages of intercooling and secondary combustion is illustrated in fig (4.9).

(4.6) The Turbine Entry Temperature Limit

The turbine entry temperature must be maintained at or below a value commensurate with the material limitations of the turbine. The primary factors affecting the turbine entry temperature are boost pressure, and the maximum temperature during combustion. This is seen through consideration of a simplified pressure volume diagram for the compound turbocharged engine, fig (4.10). Referring to fig (4.10), point A is at the end of the isentropic compression of the working fluid in the compressor (no intercooler or aftercooler is considered). This point occurs at approximately the same specific volume as for the turbine inlet, at point E. Point E in turn occurs after isentropic compression of the working fluid after exiting the compressor, followed by constant volume combustion, followed by isentropic expansion. Raising the boost pressure so that the compressor exit point moves to B will cause the turbine inlet point to move to D, which is at a higher temperature than point E. Also, raising the maximum temperature at point C will also cause the turbine entry temperature to rise. It has also been found that raising the overall compression ratio causes the TET to fall, other parameters being constant.

The assumed TET limit of 1600K, can be achieved, either with

152 an uncooled nickel alloy turbine or an uncooled ceramic turbine. A key question, however, is the life of the turbine at the given operating temperature and applied stress level. For example, the specific outputs of well over 750 kW dm-3 achieved by formula one racing engines of some years ago (4) are likely to have been achieved with turbine entry temperatures of the order of 1500K. This can be ascertained by the fact that rule changes were eventually enforced to limit the compressor pressure ratio to 4:1. At compressor pressure ratios somewhat greater than this (e.g. 6:1), a simple cycle analysis of the type described in section (4.8) shows that the TET is of the order of 1500K for an aftercooled engine, of overall volumetric compression ratio 20:1, at stoichiometric fuel/air ratios. It is believed nickel alloy turbines were used in these engines, the high turbine entry temperatures being able to be withstood simply because racing engines are only required to last for a few hours. This is further demonstrated by the fact that racing engines for qualifying, which only have to last a few minutes, demonstrated a further power increase. rhus, the design life of the turbine is an important design parameter,

( along with material properties. Refer, for example, to fig (1.11). It is possible to envisage an engine design in which the turbine can be replaced at servicing intervals, the shorter design life of the turbine then permitting a higher operational TET.

It is noted that an engine running at a leaner overall fuel/air ratio can operate at an higher boost level, while still maintaining the turbine entry temperature at a sufficiently low level. Therefore, the maximum power output might not occur with a slightly fuel rich mixture, as is usual for naturally aspirated spark ignition engines.

153 (4.7) Transient Response

The primary method of altering power output is considered to be through increasing or reducing piston engine speed, a CVT permitting a continuous variation in the speed of the engine relative to the load. Therefore, some means must be provided for accelerating or decelerating the engine. In a hybrid vehicle, this can be achieved through adjustment of the alternator load. For example, reducing the alternator load allows the engine to accelerate, the temporary power deficiency being provided by batteries and electric motors.

Other means for adjusting the transient power level include adjusting the cylinder volumetric compression ratio, adjusting the boost pressure and altering the inlet valve timing if variable valve events are available.

(4.8) A Simple Quasi - Steady Compound Turbocharged Engine Model

A simple quasi - steady model of the compound turbocharged spark ignition engine was produced to provide an initial estimation of engine performance. Firstly, a quasi - steadl, model of a naturally aspirated spark ignition engine was produced. This was based on the following assumptions,

(i) fluid properties were a function of temperature and equivalence ratio (but not unburnt fuel fraction), (ii) the fuel heat of reaction was 44.1 MJ kg 1, with an empirical combustion efficiency curve defined as a function of equivalence ratio, (iii)combustion was at constant volume, (iv) there was mass addition during combustion, (v) in - cylinder compression and expansion was defined by an instantaneous polytropic index, (vi) intake and exhaust pressures were constant, (vii)friction mean effective pressure was calculated

154

according to an empirical curve, as a function of piston engine speed.

Items (i) and (ii) are as described in sections (2.3.1) and (2.3.5) respectively. Item (iii) is a suitable simplified approach. Item (iv) allows for the mass addition of fuel. Item (v) allows the in - cylinder heat loss to be calculated. By setting heat loss as a constant fraction of compression or expansion work, an effective instantaneous polytropic index can be calculated as a function of the ratio of specific heats. That is,

(1 + QL).5w ST - (4.11) exp c and,

Sw (1 - QL) STcomp C (4.12) where ST is a small change in temperature, QL is the heat loss factor, Sw a small amount of specific work output, c the specific heat constant at constant volume and the subscripts exp and comp denote expansion and compression respectively. By setting QL at 30%, the overall heat loss was typically calculated to be approximately 25% of overall net heat input. This was judged to be a suitable fraction according to data provided by (15).

Item (vi) is a suitable simplifying assumption, though there are two choices for the exhaust pressure. The first is to set the exhaust pressure as that at the start of the exhaust stroke, the second is to set the exhaust pressure at atmospheric pressure. The second choice appeared to produce more realistic simulation predictions. Item (vii) is taken from an empirical relationship given in (15). An identical relationship was used in the time marching models described in the next chapter. A more detailed description of this

155 empirical formulation is therefore given in section (5.3.3.5).

Further development of the algorithm produced a compound turbocharged engine model (with or without aftercooler), with the following additional assumptions,

(viii)the compressor and turbine had polytropic efficiencies of 80%, (ix) the pressures at the compressor inlet, compressor exhaust/piston engine intake, piston engine exhaust/turbine inlet and turbine exhaust were all constant, (x) the aftercooler temperature drop was modelled according to an 'effectiveness' parameter, (xi) the cylinder volumetric compression ratio was calculated according to the compressor volumetric compression ratio.

The detailed calculations for these two schemes are presented in Appendix (A8). A first order Euler integration scheme was used. The calculations were performed for one cycle only, with the in - cylinder equivalence ratio at the start of the cycle assumed to be zero.

An energy cross check was employed to provide partial validation of the schemes. If,

e = E 8Q in E 6Qout E Min - E Mout (4.13) E 8141.- E Mout - E SU

where Q represents the total heat transfer, H the total enthalpy transfer, W the total work transfer and U the total internal energy of the fluid within the cylinder, then e should theoretically be zero when summed over one cycle. However, because finite difference algorithms are inexact the variable e takes a small non - zero value in practice.

156 Typically the magnitude of this error was found to be 0.2% of the net heat input, which was judged to be suitably small figure.

(4.9) Summary

The theoretical advantages of the compound turbocharged engine were discussed. A brief review of past engine designs of this type was undertaken. The justification for further research into the small, high speed spark ignition compound turbocharged engine with variable cylinder volumetric compression ratio and high TET was given. The proposed design layout was described. A simple numerical model of the compound turbocharged engine was implemented, prior to a more complex time marching model.

157 (4.10) References

(1) Judge, A.W. Small Gas Turbines and Free Piston Engines Chapman and Hall Ltd., 1960

(2) Henry, A. Autocourse 1991-92 Hazelton Publishing, p.30

(3) Press, W.H. ; Flannery, B.P. ; Teukolsky, S.A. ; Vetterling, W.T. Numerical Recipes, the Art of Scientific Computing Fortran Edition Cambridge Univ. Press, 1989

(4) Otobe, Y. ; Osamu, G. ; Miyano, H. ; Kawamoto, M. Aoki, A. ; Ogawa, T. Honda Formula One Turbo-charged V-6 1.5L Engine SAE Trans, 890877 1989

(5) Mackay, R. Gas Turbine Generator Sets for Hybrid Vehicles SAE Tech Pap Ser, International Congress and Exposition, Detroit, USA, Feb 24-28 1992, 920441 (SP-915)

(6) Pullen, K.R. A Case for the Gas Turbine Series Hybrid Vehicle Battery, Electric and Hybrid Vehicles, I.Mech.E. conference proceedings, 10-11 December 1992

(7) Jane's All the World's Aircraft 1990-91 Jane's Information Group Ltd.

158 (8) Setright, L.J.K. Some Unusual Engines Mechanical Engineering Publications, 1975 Published for the I.Mech.E.

(9) Chatterton, E. ; Sammons, H. Napier Nomad Aircraft Diesel Engine SAE Trans., vol. 63, 1955, p.107

(10)Wallace, I.J. The Differential Compound Engine SAE Trans., vol. 76, 670110 1967

(11)Ricardo, H.R. The High-Speed Internal Combustion-Engine Blackie and Son Ltd., 4th. ed., 1953

(12)Car Design and Technology Yearbook 1992 Brackland Publishing Plc

(13)Pilot Pilot Publishing Co. Ltd. Jan. 1982

(14)D.R.A. Museum, Farnborough, Hampshire, England

(15)Heywood, J.B. Internal Combustion Engine Fundamentals McGraw Hill, 1988

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169 case cycle optimum optimised number temp compressor efficiency ratio pressure 96 ratio for efficiency t r pc nth opt opt

1 1200/288 10.2 21.8

2 1600/288 21.6 29.7

3 2000/288 41.6 35.3

4 2000/223 75.7 41.3

Table (4.1) Optimum Efficiency Levels of a Simple Gas Turbine for a given Cycle Temperature Ratio, and the Corresponding Compressor Pressure Ratio

170 altitude compressive temperature km pressure after ratio compression r T (K) P 2

0 24.44 703.9

10 160.9 1519.

Table (4.2) Approximate Compressive Pressure Ratios for a Compound Turbocharged Engine at Two Different Altitudes

171 case engine power engine specific no. type output weight output kW kg kW/kg

(i) Honda RA121E 582 150 3.88 naturally aspirated SI piston engine (ii) Honda RA167E 750 146 5.14 turbocharged SI piston engine (iii) experimental 30 28.5 1.05 automotive gas turbine (iv) Pratt and Whitney 2073 481 4.31 PW126A turboprop * (v) Rolls-Royce Trent 34585 4345 7.96 turbofan

Table (4.3) Specific Power Outputs of Various Engine Types

-1 total airflow 870 kg s , bypass ratio 5:1,

compressor pressure ratio 35:1

172 Item Item Value

Engine Capacity 40.9 dm3

Maximum Power at 3057 ekW Take Off

Peak Brake Thermal 42% Efficiency

Peak Specific Power 1.77 ekW/kg Output 74.7 ekW/dm3

Max Compressor 8.25:1 Pressure Ratio Peak Compressor 87.5% Isentropic Efficiency Pressure Ratio Across 4.6:1 Turbine at Sea Level Peak Turbine Isentropic 86% Efficiency Typical TET 934K BMEP at Maximum Boost 14 bar

Table (4.4) Napier Nomad Engine Design Details

173 Item Item Value

Estimated Engine 3.25 dm3 Capacity Maximum Power 231 kW Output Peak Brake Thermal 43% Efficiency

Peak Specific 71.1 kW/dm3 Power Output

Typical Compressor 4:1 Pressure Ratio Maximum TET 1020K Maximum BMEP 17.7 bar Maximum Piston 2,500 rpm Engine Speed

Table (4.5) Wallace Differential Compound Engine Design Details

174 Chapter (5)

Time - Marching Numerical Models of Naturally Aspirated and Compound Turbocharged Spark Ignition Piston Engines

(5.1) Overview

To investigate the potential of the small, high speed compound turbocharged engine, a numerical algorithm that is more sophisticated than the one described in the last chapter (the quasi - steady model) was developed. At an interim stage, this entailed the development of a naturally aspirated engine model, which was then modified to produce the full compound turbocharged engine model. Given the relatively extensive modifications then required, it was decided to develop the model from first principles rather than modify an existing computer code.

This chapter describes modelling strategies of the time - marching type. It was important to limit program execution time to enable the optimisation procedure described in chapter (7) to be employed in conjunction with the basic simulation procedure. Such a consideration then yielded the choice of method. This is of the well known 'filling and emptying' model type. Though a brief overview of the method is given, it is more the intention of this chapter to describe the modelling aspects particular to this application. These aspects are described for the naturally aspirated model, followed by a description of the modifications required to produce a compound turbocharged engine model.

A major problem in the development of the algorithm has been that of ensuring stability of the numerical integration scheme. This problem is briefly mentioned in this chapter and expounded on in chapter (6).

175

(5.2) A Comparison of Modelling Strategies

Three basic methods were considered. These were,

(i) a full three dimensional model, (ii) the filling and emptying model, (iii)the filling and emptying model plus a one dimensional compressible unsteady flow model for the inlet and exhaust manifolds.

Method (i) employs a full three dimensional grid for the engine cylinders and inlet and exhaust manifolds. Pressure and temperature are stored for each grid point, with the method of solution of the governing partial differential equations either by the finite element or finite difference techniques.

Method (ii) is considerably simpler, and effectively employs one grid point for each of the cylinder volumes and inlet and exhaust manifold volumes. Each grid point then stores an average fluid temperature and the total mass for the given control volume, with the composition of each control volume assumed to be homogeneous. The control volume temperature and mass are incremented by a finite difference technique. For example, a first order scheme is given by,

dT ST - dt St am . ( dm dm ) (at dt St

where T is the control volume temperature, m is the control volume mass, the subscript i denotes influx, the subscript e denotes efflux and t is time. The total volume is known, and the control volume pressure may therefore be found via the ideal gas equation,

mRT P - V (5.2)

176 where R is the specific gas constant. A schematic illustration of the method for a four cylinder engine is given in fig (5.1).

Method (iii) is a modification of the basic filling and emptying technique. Cylinder volumes are considered in the same manner as for method (ii), but the inlet and exhaust manifold control volumes are replaced by dynamic gas flow models that employ a one dimensional grid in space. Such models become important when the consideration of pressure waves in the manifolds is significant. This in turn becomes apparent when,

a ?: 0(1) (5.3) where,

a - LN (5.4) l7RT and L is the length of the manifold, N is the piston engine rotational frequency, and the averaged term is the mean speed of sound in the gas in the manifold. Equation (5.4) relates the time taken for one engine cycle to the time taken for a small amplitude pressure pulse to travel to one end of the manifold and be reflected back to the first end again.

A typical method of solution for the gas flow dynamic model is the method of characteristics, which is a first order finite difference technique, (1). A schematic illustration of the scheme is given in fig (5.2). Because the treatment of gas flow modelling is more sophisticated in method (iii), rather than (ii), a more reliable estimation of volumetric efficiency results. This is particularly so when condition

177 (5.3) holds.

For this project it was important to select a relatively simple modelling scheme which entailed the minimum of program execution time. This was to enable the use of the optimisation technique described in chapter (7). Therefore, method (ii) was chosen, and this is the method henceforth described in this chapter.

(5.3) The Filling and Emptying Model

(5.3.1) The Governing Ordinary Differential Equations

Two basic governing ordinary differential equations are employed. These equations must be applied to each control volume, there being six such volumes in total in a four cylinder model (the inlet manifold, the four cylinder volumes and the exhaust manifold). The relevant equations are the mass flow equation, given by,

dm _ dm dm (5.5) dt dt dt e and the energy flow equation, given by,

dU dQ dQ dW dH dH (5.6) dt dt dt dt dt dte where U is control volume total internal energy, Q is the control volume total heat transfer, We is the total work transfer out of the control volume and H is the total enthalpy transfer across the control volume boundaries. For the inlet and exhaust manifolds, We and Q are taken to be zero.

Equation (5.5) must be expanded as follows,

178

dm dm dm _ a a (5.7a) dt dt 1 dte dm dm dm bf bf bf (5.7b) dt • dt dte

dm dm dm ubf ubf ubf (5.7c) dt • dt dte

where the subscript a denotes air prior to combustion, subscript bf denotes burnt fuel, and subscript ubf denotes unburnt fuel. The composition parameters are then determined from,

= bf A in (5.8a) a

ubf = — mM (5.8b)

where 0 is the equivalence ratio, 4 is the stoichiometric air/fuel ratio and A is the unburnt fuel fraction. Fluid properties are then taken to be a function of T, 0 and A as described in section (2.3).

Now,

dUdu dm dt - m dt + u dt ( au d0 + au dA, dT ) = r dt + c --- ao dt ax v dt + u dtdm (5.9)

where u is the specific internal energy and cv the specific heat constant at constant volume. Also,

dH dm dt - h dt (5.10) dWdV mRT dV dt - P (5.11) e dt V dt

179 Therefore, substituting equations (5.9), (5.10) and (5.11) into equation (5.6) and rearranging,

dT 1 RT dV dt c L V dt V 1 dQ dQdm dm dm ) h - h m ( dt dte u dt dt e dt e au d0 au dA 80 dt ax dt (5.12)

Equations (5.5), (5.7), (5.8) and (5.12) then constitute the basic scheme. It is noted that equation (5.12) is simplified by neglecting consideration of dissociation. Equations (5.5), (5.7) and (5.12) must be replaced by equivalent finite difference equations to enable practical solution to be made. A hybrid third/fourth order method was employed, as described in Appendix (A9).

(5.3.2) The Mass Flow Differential Equation

(5.3.2.1) The Basic Mass Flow Rate Equation

The basic mass flow rate equation describes the flow rate through a port and is based on the following assumptions,

(i) there is an isentropic static pressure drop through the port up to the throat or vena contracta, the ratio of the area of the vena contracta to the port area being defined by the flow coefficient, cf , (ii) the static pressure remains constant downstream of the vena contracta for subsonic or sonic flow, with a static pressure fall occurring in supersonic flow, (iii)the ratio of specific heats, 7, remains constant during the isentropic pressure drop, (iv) the total pressure drop produced by a conventional butterfly throttle was not modelled.

180 Items (i)-(iii) are standard simplifying modelling assumptions, and are further described in (2). Item (iv) produces a full-throttle engine simulation for the naturally aspirated engine, though the method of throttle control for the compound turbocharged engine is different. The mass flow rate is then given by,

_ pAc dm f (5.13) dt a where pi is the upstream total pressure, A is the port area, a is the upstream speed of sound at the upstream stagnation temperature and C is a function of pressure ratio defined below. Forwards flow is defined by the total pressure ratio across the port, r , exceeding unity, where,

Pi r = -- (5.14) P

and p is the downstream static pressure (equal to the downstream total pressure in a filling and emptying model). If r is less than unity, the designation of the upstream and downstream control volumes is reversed. If r is equal to unity, no flow occurs.

The non - dimensional parameter C is given by,

1 - (5.15)

unless,

r r (5.16) Pchoke

where,

181

7 7 1 1 7-1 ) (5.17) choke 2 in which case the flow is choked and,

7+1 2(7-1) 2 Cchoke (5.18)

These equations are derived in Appendix (A10). For a value of 7 of 1.4 the resulting form of C verses rp is plotted in fig (5.3). This analytic form for C was later replaced by a more approximate form, based on the analytic form, which gave greater algorithm numerical stability. This is described in chapter (6).

(5.3.2.2) Determination of the Flow Coefficient and Port Area

The flow across cylinder valve ports is first considered. Data for a four poppet valve per cylinder engine was taken from (3). According to this data, total inlet port area for the two inlet valves was taken as 23.36% of the cylinder head area. The total exhaust port area was taken as 70% of this value, i.e. 16.35% of cylinder head area. The flow coefficients for the inlet and exhaust valves are then taken as a function of non - dimensional valve lift. The parameter * . L is defined as,

L L - /D (5.19) (L/D) max where L is a non - dimensional valve lift, L is the dimensional valve lift, and D is the dimensional valve throat diameter for one valve. This is plotted as a function of non - dimensional crank angle, A* , in fig (5.4). The parameter, A* ,is given by,

182

* * - *o A* - (5.20)

where * is the crank angle, *0 is the crank angle at which the valve opens, and A* is total crank angle duration for which the valve is open. Figure (5.4) is in turn defined by the cam profile.

The parameter (L/D)max is defined as a function of A* by a characteristic consisting of two adjoining straight line portions. There is a sloped portion of the characteristic, defined by the cam contact stress limit. This is given by,

1.053 x 10-4 = (A* - A*0) (5.21a)

A*o = 158.690 (5.21b)

where t is in units of degrees, and D is in units of metres. An upper limit to (L/D),,,,,, is given by the cam eccentricity not exceeding valve tappet radius. This limit is taken to be at a value of (L/D) max of 40%. The two resulting straight line portions of the non - dimensional maximum valve lift verses total valve opening duration in degrees is plotted in fig (5.5). The characteristic is for a cylinder size of 125cc, with a bore/stroke ratio of unity. The two straight line portions of the characteristic are joined together by a small curve of arbitrary radius in order to give a continuous derivative of maximum valve lift with respect to crank angle valve lift duration.

The resulting flow coefficient is then taken from,

0 -s A* 1 ; cf = f(L = f(L/D)

otherwise; cf = 0 (5.22)

The characteristic defined by the function in equation (5.22) is plotted in fig (5.6). The same characteristic was used for the inlet and exhaust valves.

183

In the case of the naturally aspirated engine, the port area for the inlet and exhaust manifolds was taken to be three times the respective flow area for one cylinder. The flow coefficient, cf, was taken to be unity. For the compound turbocharged engine, the mass flow rate into and out of the inlet and exhaust manifolds was calculated in a different manner described in section (5.4.1).

(5.3.3) The Energy Flow Rate Differential Equation

Requirements specific to the in - cylinder control volume energy flow rate differential equation (equation (5.12)) are now considered.

(5.3.3.1) Combustion Simulation

The rate of heat release during combustion must be determined. This is done using an empirical rate of fuel burning relationship coupled with an empirical combustion duration expression. The rate of fuel burning is defined by a cosine law,

* 1 * f(1, ) = (1 - cos[rr.* ]) (5.23)

* where ? is a non-dimensional crank angle defined in an analogous fashion to equation (5.20). The amount of fuel burnt in any given crank angle step is then given by,

am = - [f(15 + 819-) - )3m ubf ubf (5.24) 0

with,

ambf = -Smubf (5.25)

where M is the unburnt fuel mass immediately prior to ubf0 combustion. The function f(19-) is entirely empirical and

184 reflects the fact that the rate of combustion in spark ignition engines is experimentally observed to start slowly, speed up during mid combustion and then reduce in rate towards the end of combustion. This function is plotted against ** in fig (5.7).

The combustion duration period, A*, must then be determined in order to calculate ** as a function of *. For this, data from (4) was employed. Comprehensive single - cylinder engine data for three combustion chamber types were analyzed statistically to quantify the effects of engine operating conditions and chamber geometric variables on combustion characteristics (and of combustion on engine performance stability). The tests were only performed at low engine speeds (a maximum speed of 2000rpm was employed). Operating condition variables of interest were air - fuel ratio, residual fraction, spark timing, engine speed and fueling level (trapped fuel per cycle).

Some of these corrections were considered inappropriate as applied here. For example, the correction for spark timing requires previous knowledge of the spark timing for maximum torque, which is not generally available. The correction for engine speed gave too large a value for A* when applied at high engine speeds, e.g. if the engine speed was greater than 10,000rpm this correction would give a value for A* of order 180°. The correction for fueling level was considered to be small. Thus the resulting expression considered here corrects for air - fuel ratio and residual fraction. Adapted from (4), the expression took the form,

A* = 35° + 20.384(4/0)(1/[2ry ]) (5.26) cyl where ti is in degrees, A is the stoichiometric air - fuel ratio, 0 is the equivalence ratio, and the last round bracketed term was found to be a suitable estimate of the residual fraction. The term r is the cylinder volumetric cyl

185 compression ratio. The residual fraction is defined as the ratio of the mass of fluid trapped in the cylinder at the end of the exhaust stroke to the mass of fluid ingested during the intake stroke. Expression (5.26) is plotted against 0, for various values of r , in fig (5.8). For cyl later runs of the simulation programs equation (5.26) was considered to have little physical justification and was replaced by a constant value of At of 60°, this being a typical order of magnitude figure for the combustion duration period as given by (5).

The resulting figure for the combustion duration period, A*, must be regarded as very approximate, representing an order of magnitude estimation rather than a precise value. The estimation of the fuel burning rate is a particular difficulty in the filling and emptying method. Reference (6) describes two other empirical methods, the use of the Wiebe function or alternatively a two zone combustion model. The Wiebe function is entirely empirical and still requires specification of the combustion duration period, A*. The basic equations for the two zone combustion model, which assumes the presence of burnt mixture and unburnt mixture zones in the combustion chamber divided by a discontinuity, are unclosed and must be closed by specification of the fuel burning rate. This can be done by employing the cosine law given by equation (5.23), by employing the Wiebe function or through estimation of the turbulent burning velocity. Specification of the turbulent burning velocity in turn requires an empirical expression for the turbulence intensity and therefore does not remove the empiricism.

The resulting heat release during a finite crank angle step is then given by,

6Qi = Ali 8m Tlcomb complete bf (5.27) with the terms 7)comb and AHcomplete being defined in section

186

(2.3.5).

(5.3.3.2) The Heat Transfer Rate

The heat transfer rate through the cylinder walls is empirically determined. In an extensive review of empirical methods, Marzouk (7) selects Woschni's method (8) as the most accurate. This was the method used in this application. The empirical formulation then derived is,

0. 8 -O. 2 p 8 -.0 53 VT h = 127.93 B * T 2NS + c „1 (p - po) 1 2 p v 1 2 (5.28)

2 where h is the heat transfer coefficient [J m K-1sec-1], B is the cylinder bore [m], p is the cylinder pressure [bar], T is the in cylinder temperature [K], V is the cylinder volume, N is the engine rotational frequency [sec 1]and S is the piston stroke [m]. The parameters cl and c2 are empirical constants, while the subscript 1 on the thermodynamic parameters denotes conditions at the start of combustion and the subscript 0 denotes motored conditions.

Equation (5.28) is derived through a dimensional analysis, with the square bracketed term being an estimate of the in-cylinder turbulence velocity. The empirical constants have the following values,

during admission of inlet air c1=6.18 c20.0 and expulsion of exhaust gas

during compression c1 =2.28 c2'=0.0 -3 during combustion and prior c1=2.28 c =3.24x10 to the exhaust valve opening 2'

the coefficients c1 and c2 being determined empirically by Woschni (8) for a compression ignition engine. The motored pressure po is not known directly. To obtain an estimate of

187 its value, a parallel numerical integration process for the parameter /00 is undertaken, assuming isentropic compression and expansion from the pressure value at the start of combustion (p1 ). If pc, exceeds p at any point, the parameter c2 is set to zero.

The overall heat transfer rate is then given by,

dQ dt - h[Ahead (T - Thead ) e + A (T T + A (T T ) wal 1 wall piston piston (5.29) where A denotes area (m2), 'head' denotes cylinder head, 'wall' denotes cylinder wall and 'piston' denotes piston. Based upon computational data in (9), the boundary wall temperatures were taken as fixed in value as follows,

Thead = 473K Twall = 423K T piston= 523K

This assumption of fixed boundary wall temperatures is rather approximate and could be improved upon in a more sophisticated model. Refer, for example, to Appendix (All).

(5.3.3.3) Modelling Direct Fuel Injection

Fuel was modelled as being injected directly into the cylinder during the intake stroke based upon an assumed modified square form for the fuel injection rate. Injection occurred late in the intake stroke, to reduce the amount of unburnt fuel lost through any back flow. Injection started at 60° after top dead centre and finished 170° after top dead centre. The fuel was assumed to vapourise instantaneously upon injection and to have zero initial enthalpy. The fuel injection process was defined by three

188 stages,

* dm 5- * ubf 16 (5.30a) 0 4 I * d* 3 * dm 1 3 ubf 4 — < — ; * (5.30b) 4 4 d* 3 * dm 3 * ubf 16 * 1; = (1 - ) (5.30c) 4 d* 3 with ** defined in an analogous fashion to equation (5.20) and,

* dmm dm ubf . = m * ubf (5.31) d* o d** with M being the total amount of fuel injected. The ubf 0 relationships defined by equations (5.30) are plotted in fig (5.9). The total area underneath this characteristic is unity.

The resulting mass of fuel injected in any crank angle step is then,

* * 1, + 511 • dirt ubfA, 3 = m --- ul/ m ubf ubf * (5.32) 0 * d* * which may be determined exactly from equations (5.30).

(5.3.3.4) Cylinder Volume Calculation

The cylinder volume is a function of crank angle,

V V v(*) = 25 + vc — —cos2 (19-) (5.33)

189 where V(*) is the cylinder volume, V, is the swept volume and V the clearance volume. This is function has a C sinusoidal form, which is correct if the motion of the pistons is pure simple harmonic, as is the case for an engine with a swashplate crank mechanism. For an engine with a conventional crank mechanism, there is a small error in the definition of cylinder volume particularly if the ratio of the connecting rod length to piston stroke is relatively small.

The clearance volume is calculated as a function of swept volume and cylinder volumetric compression ratio as follows,

V V 1 (5.34) c r V - c y 1

The energy flow rate differential equation requires the time derivative of the cylinder volume. This is given by,

dV(*) dV(i) d* Vs d* dt di dt 2 sin(*) dt (5.35)

(5.3.3.5) Frictional Loss Calculations

The frictional loss calculations were based on an empirical formulation for the friction mean effective pressure. The formulation is based upon data provided in (9). It takes the form,

2 FMEP = 0.97 + 0.15r + 0.05r (5.36) where FMEP is in units of bars and r is the piston engine speed in thousands of revolutions per minute. The mechanical efficiency of the piston engine is then given by,

BMEP IMEP - FMEP - 1 FMEP (5.37) 7/mech IMEP IMEP IMEP

190 where BMEP is the brake mean effective pressure and IMEP is the indicated mean effective pressure.

Equation (5.36) is only accurate up to modest engine speeds, e.g. 6,000rpm, and is also only accurate for a conventional crank mechanism. In the simulations performed, the use of this empirical formulation has, in the case of the compound turbocharged engine, been extrapolated to application at high engine speeds, e.g. 20,000rpm, and to an unconventional swash plate type crank mechanism.

(5.3.4) Convergence of the Solution

Integration is initiated with starting values of control volume temperature, air mass, unburnt fuel mass and burnt fuel mass. Based upon the input equivalence ratio and an estimate of the volumetric efficiency, an initial value for the quantity of fuel injected per cylinder per cycle is estimated. The numerical integration then proceeds, with the quantity of fuel injected per cylinder per cycle adjusted according to the iterative relationship,

macs k, i Amulpf Amubf [(0 - 0k, 1 (5.38) k+l,i k,1

Ara where ubf is the mass of fuel injected per cylinder per cycle, k is the iteration number, i is the cylinder number, m acs is the mass of in - cylinder air immediately prior to combustion, A is the stoichiometric air/fuel ratio, 0 k, i is the actual equivalence ratio immediately after combustion on the k'th iteration for the i'th cylinder and 0 is the desired equivalence ratio.

Convergence to the desired accuracy is typically achieved in less than twenty simulated engine cycles, though this figure increases as manifold volumes are made large relative to the

191 maximum cylinder volume. A further six loops are then performed, with the output parameters being averaged over these six loops.

(5.3.5) The Order of Integration

An Euler first order scheme was the first method employed for the numerical integration of the two governing ordinary differential equations. The resulting first order finite difference algorithm is given by equations (5.1). This was found to be insufficiently accurate and to lead to numerical instability of the mass flow rate differential equation at low simulated engine speeds. For example, an energy cross check method analogous to equation (4.13) gave an error that was typically 2% of the net heat input in magnitude. The method eventually employed was a hybrid third/fourth order type. The method is described in Appendix (A9). With this method, typical energy cross check errors are of the order -3 -2 10 to 10 %, while the stability of the algorithm is much improved.

(5.4) Modifications to Produce a Compound Turbocharged Filling and Emptying Model

The modifications required to the above algorithm necessary to produce a compound turbocharged engine model are now described.

(5.4.1) The use of Turbomachinery Characteristics

Turbomachinery characteristics, for a centrifugal compressor and an undefined turbine type, were used to define the upstream inlet manifold and downstream exhaust manifold boundary conditions. The assumption is made of quasi - steady flow, where the steady flow turbomachinery characteristics are used to define the dynamic flow. For the compressor, the inputs are,

192 (i) the upstream temperature, ambT ,

(ii) the upstream pressure, oamb (iii)the instantaneous pressure ratio across the compressor, r , (iv) the compressor area ratio, Ac, (v) the ratio of the compressor speed to the design compressor speed, N .

From these inputs, the outputs are,

(i) the instantaneous compressor mass flow rate, m',

(ii) the compressor exit enthalpy, h(Tce (iii)the instantaneous compressor work input, W.

The mass flow parameter for the compressor is defined by,

cV am b Mc A (5.39) c pwrib

where A is an adjustment for the area of the compressor relative to the design area. Thus,

• Pam b = M cA c (5.40) T v am b

where no allowance is made for the variation of the compressor characteristics with the ratio of specific heats, T. Now,

Mc = f (N*c, ) (5.41)

* For a given simulation N will be fixed (it is a program input) this in turn reflecting the fact that compressor/turbine speed is controlled through the use of a CVT. Thus,

193 Pam b m' = e(rp )A c (5.42) amb where e() is a generalized function. The compressor mass flow rate characteristics for an ambient temperature of 288K, an ambient pressure of 1 bar and a compressor area ratio of unity are plotted in fig (5.10). They are based upon the same data as the compressor characteristics described in chapter (3). The source of data for these characteristics is (10). This compressor offers higher isentropic efficiencies than is normal for turbochargers, at the expense of reduced map width. There is some inaccuracy in the extrapolation of these characteristics to relative compressor speeds greater than 100% and less than 60%. There has also been a slight error in the extrapolation of the characteristics to a relative compressor speed of 120%, as will be seen from fig (5.10). However, the error is not large, and is not expected to have significantly affected the simulation predictions. Note that, in the surge region the characteristics do not reflect reality in that they are here defined with a small negative slope. This permits the integration process to remain stable even in the surge region though the results then obtained are physically meaningless. The surge margin is provided as a program output, where,

- Ms - m' x 100% (5.43) des 1 gn is the surge margin, m' is the mass flow rate at which surge occurs for the given compressor speed, and Trldi esign is the design mass flow rate of the compressor. Thus, it is possible to check manually whether the solution has converged to a point in the surge region. It should be noted that defining the surge region as an area of positive slope on the compressor mass flow characteristics is not exact,

194 but rather an approximate guide. That is, compressors can sometimes operate stably in regions of small positive slope.

The compressor exit enthalpy is calculated as in Appendix (A4). The instantaneous compressor work input is then defined by,

W = m'[h(T ) - h(T )) (5.44) c e ce amb

where h(T ce) is the compressor specific exit enthalpy (0 and A are assumed to be zero). Referring to Appendix (A4), an efficient iterative solution for the compressor exit enthalpy and temperature was employed. This is a combination of the Newton - Raphson and bisection techniques, as described in (11). With the compressor characteristics employed, the compressor pressure ratio can theoretically fall below unity at very low compressor speeds. If this occurs, the enthalpy change across the compressor is set to zero, which then defines a zero compressor work input. This is unlikely to be realistic, though in practice the condition never occurred.

The solution for the compressor exit enthalpy requires the use of the compressor isentropic efficiency value,Tic. The estimation of this value was again based on the compressor characteristics of (10), albeit with modifications to give a non - zero compressor efficiency at low mass flow rates. This is because the modifications to the mass flow rate characteristics in the surge region give pressure ratios greater than unity at zero mass flow rate. This would then result in infinite compressor exit temperatures and enthalpies if the compressor isentropic efficiency approached zero in this region (refer to Appendix (A4)). Since such operating points represent a 'dummy' operating condition, the modifications are not of significance. The resulting efficiency characteristics are plotted in fig (5.11), with,

195

71C = j (MC ) = 1 (r ) (5.45)

where j() and l() are generalized functions. Thus, for a fixed compressor speed, the instantaneous compressor efficiency is a function of the instantaneous pressure ratio only.

The estimation of the instantaneous turbine mass flow rate and work output proceeds in a broadly similar fashion. The turbine characteristic is assumed to be independent of turbine speed, and the mass flow rate is estimated by the same function as was illustrated in fig (5.3), i.e. the turbine mass flow rate is taken to be the same as for an isentropic nozzle of the same pressure ratio. Then,

P t t m' = A a C (rp 7 ) (5.46) t

where m' is the turbine mass flow rate, pi is the total pressure upstream of the turbine, At is the turbine choke area, a1 is the stagnation speed of sound upstream of the turbine, r is the turbine pressure ratio and t is the P t ratio of specific heats upstream of the turbine. The non - dimensional function C() was defined in section (5.3.2.1). This function was then modified for low pressure ratios to give a more stable integration scheme. This is mentioned in section (5.3.2.1) and described in detail in chapter (6). Since the turbine usually operates at a pressure above the point at which these modifications take effect, these modifications are not generally important.

The estimation of turbine exit enthalpy proceeds as for Appendix (A4) with the turbine pressure ratio defined as the ratio of the exhaust manifold pressure to the ambient pressure. This requires the use of a turbine isentropic efficiency value. This was estimated from a speed

196 independent characteristic illustrated in fig (5.12). This is in turn based on typical graphical data given in (12). For typical turbine pressure ratios achieved in the simulation, this gave an isentropic efficiency in excess of 80%. The instantaneous power output of the turbine is then given by,

W = m'[h(T ) - h(T )) (5.47) t t te where TU is the TET, Tte is the turbine exit temperature and the value of Wt is set to zero in the case of reverse flow.

(5.4.2) Modelling an Aftercooler

As described in chapter (4), the incorporation of an aftercooler enables the specific power output to be boosted considerably. An aftercooler was modelled in the following simple fashion. As in Appendix (A4), the compressor exit enthalpy, h(Tce), was found. This value was iteratively inverted to find the compressor exit temperature, Tce . This was again done using a combination of the Newton - Raphson and bisection techniques, (11). The inlet manifold inlet temperature and enthalpy influx rate were then found from,

T = T C (T - T ) (5.48) i ce ce amb

H' = reh(T ) (5.49) . 1 where T is the resulting inlet manifold inlet temperature, c is an effectiveness parameter analogous to that defined for heat exchangers for gas turbines (chapter (3)), T amb is the ambient temperature and H; is the rate of enthalpy influx to the inlet manifold. The effectiveness value, e, was assumed to be 70% when an aftercooler was modelled.

197 (5.4.3) The use of a Variable Cylinder Volumetric Compression Ratio

As described in chapter (4), the compound turbocharged engine requires the use of a variable cylinder volumetric compression ratio, in order to maintain the nominal overall volumetric compression ratio at a value of r of 10:1. max Thus,

r = r /r (5.50) V V V cy 1 max c where r is the cylinder volumetric compression ratio and cyl r is the estimated compressor volumetric compression ratio, based upon the compressor exit temperature prior to any aftercooling. The value of r should ideally be based upon a mean value of compressor exit pressure and temperature, once the integration procedure has converged. This then entails an iterative calculation of r . V However, this extra iterative procedure would significantly increase program execution time. Therefore, r, is estimated at the beginning of the program run based upon the compressor operating at its maximum efficiency condition at the given compressor speed. The clearance volume, Ve, was then calculated from equations (5.50) and (5.34). The resulting algorithm will overestimate the overall pressure ratio when the compressor pressure ratio is greater than the pressure ratio for maximum isentropic efficiency at the given compressor speed and vice versa. However, since optimum power outputs and efficiencies will occur when the compressor is operating close to its curve of maximum isentropic efficiency the error was not deemed significant.

198 (5.4.4) Estimating the Turbine Entry Temperature

The turbine entry temperature is an important limiting operating parameter. Its mean value was estimated using the mass weighted average,

mt IT tidt T (5.51) t i fl Teldt 0 where T is the instantaneous turbine entry temperature and T is the time for one engine period, equal to the time for two piston engine revolutions.

(5.5) Determining the Power Output and Brake Thermal Efficiency

The overall power output and brake thermal efficiency figures are the key program outputs. For the piston engine, the power output is given by,

W n [3(TID (FMEP)V] (5.52) P = T dt ‘-"" where W is the mean piston engine power output [Watts), n is the number of cylinders [-], T is the time for one engine period [sec), P is the instantaneous in - cylinder pressure [Nm 2], V is the instantaneous cylinder volume [m3], t is time [secs], FMEP is the friction mean effective pressure [Nm-2] and V, is the swept cylinder volume [m3]. In this application n was equal to four. The averaging is performed once the iterations have converged.

For the compound turbocharged engine, the time weighted averages of equations (5.44) and (5.47) are,

199 T W f W'dt (5.53) c T 0

T Wt = T t dt (5.54)

The overall work output is then,

W = W p W W t (5.55) with the last two terms of equation (5.55) being zero in a naturally aspirated engine. The brake thermal efficiency is then defined by,

WT nth (5.56) n Am AH f complete where Am is the mass of fuel injected per cylinder per cycle and AHcomplete is the calorific value of the fuel.

(5.6) Summary

An accurate numerical integration scheme to estimate the performance of the compound turbocharged engine was required. Various types of algorithm were considered, the chosen algorithm representing a good compromise between accuracy and speed of execution. This algorithm was of the well known 'filling and emptying' type. The algorithm was initially developed for a naturally aspirated engine, and the modelling assumptions particular to this application were described. The further modifications to produce a compound turbocharged engine model were also described.

200 (5.7) References

(1) Benson, R.S ; Garg, R.D. ; Woollatt, D. A Numerical Solution of Unsteady Flow Problems Int. J. Mech. Sci., vol. 6, pp117-144, 1963

(2) Benson, R.S. The Thermodynamics and Gas Dynamics of Internal Combustion Engines Vol. 1, Oxford Science Publications, 1982

(3) Private Communication Lotus Engineering, Hethel, Norfolk, England 1991

(4) Young, M.B. Cyclic Dispersion - Some Quantitative Cause and Effect Relationships SAE miscallaneous paper 800459 (1980)

(5) LoRosso, J.A. ; Tabaczynski, R.J. Combustion and Emissions Characteristics of Methanol, Methonal-Water and Gasoline-Methonal Blends in a Spark Ignition Engine Paper no. 769019, Proceedings of the 11th. Intersociety Energy Conversion Engineering Conference, Lake Tahoe, Nevada. pp122-132, Sept. 1976

(6) Ramos, J.I. Internal Combustion Engine Modeling Hemisphere Publishing Corp., 1989

(7) Marzouk, A.M. Simulation of Turbocharged Diesel Engines under Transient Conditions PhD thesis, Imperial College of Science, Technology and Medicine, Univ. of London, 1976

201 (8) Woschni, G. A Universally Applicable Equation for the Instantaneous Heat Transfer Coefficient in the Internal Combustion Engine SAE Trans. 670931, 1967

(9) Heywood, J.B. Internal Combustion Engine Fundamentals McGraw Hill, 1988

(10)Private Communication Rolls-Royce Plc, Ansty, England 1988

(11)Press, W.H. ; Flannery, B.P. ; Teukolsky, S.A. Vetterling, W.T. Numerical Recipes, the Art of Scientific Computing Fortran Edition Cambridge Univ. Press, 1989

(12)Cohen, H. ; Rogers, G.F.C. ; Saravanamuttoo, H.I.H. Gas Turbine Theory Longman Scientific and Technical, 1989

202 Gys-iNcek 04E.

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0.8 -_,

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0.1

Fig (5.4)Non-DimensionalPoppetValveLift 0.2 versus Non-DimensionalCrankAngle

Non-Dimensional CrankAngle,(-) 0.3

0.4

0.5

0.6

0.7

0.8

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Fig (5.5)MaxValveLifttoDiameterRatioversus 160 Valve OpeningDurationinCrankAngleDegrees

260 Valve Opening Duration, (Degrees) I 2k240

260

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360 Fig (5.6) Valve Flow Coefficient versus Valve Lift/Diameter Ratio 0.8 0.7 -

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211

Fig (5.10) Compressor Pressure Ratio versus Mass Flow Rate 9.0 8.5

(-1 8.0 - 7.5 - contrived region 6 7.0 - surge 120% 14:5 6.5 - Tine 6.0 - 5.5 - 'cn 5.0 - 100% °' 4.5 - 4.0 - 90% N N 3.5 3.0 80% L. 2.5 - 70% 0 2.0 - 60% C3 1.5 - 1.0 Nc=0% 0.5 0.0 0.1 0.2 0.3 0.4 0.5 06 Mass Flow Rate, (kg/sec) Isentropic Efficiency, 1-) 0.4 - 0.2 0.3 - 0.5 0.6 0.7 0.8 - 0.9 0.0

1 Fig (5.11)CompressorIsentropicEfficiencyversus

0.1

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1

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' is1 1 7 '1 1 8 '1'9 20 Chapter (6) Ensuring the Stability of the Integration of the Mass Balance O.D.E.

(6.1) Overview

The numerical integration of ordinary differential equations can result in instability if the step size of the independent variable is too large. Such instability results in large oscillatory errors in the dependant variable, these errors then swamping the true solution.

In the filling and emptying engine model, two ordinary differential equations are integrated over time. These are the mass balance differential equation and the energy balance differential equation. In practice, with a time step size corresponding to an one degree crank angle increment, the numerical integration of the latter differential equation proved always to be stable. However, if the simulated engine speed was low, the simulated cylinder volume was small or the simulated bore/stroke ratio was high then the integration of the mass balance differential equation became highly oscillatory. The values of mass flow rate obtained through the numerical integration process were then swamped by numerical errors. The problem was particularly acute in this application because engines were simulated as having small cylinder sizes, large relative port areas and high bore/stroke ratios. Measures to control this error were therefore required.

With a sufficiently small time step size any numerical integration scheme will become stable provided that it is not unconditionally unstable. With the filling and emptying method considered here the integration is not generally unconditionally unstable, but rather conditionally stable. Thus, a sufficiently small time step size, equating to a sufficiently small crank angle step size, will ensure

215 stability of the filling and emptying scheme. However, a small crank angle step size carries with it a penalty of much increased program execution time. This is not generally a problem if the simulation is to be run in isolation. However, within the context of this project the simulation procedure was to be used within an optimisation shell. The optimisation procedure invokes the simulation procedure of the order of a thousand times per optimisation. Thus, limitation of simulation execution time to no more than a few minutes was an important consideration. The analysis described by this chapter was therefore applied in an attempt to guarantee program stability while at the same time maintaining a reasonably large step size. The measures described have only been partially successful as there are still combinations of simulation input parameters that will cause instability of the algorithm to occur.

(6.2) An Example of the Mass Flow Rate Calculation Instability Problem

At an earlier stage of algorithm development, the numerical integration processes employed were of first order. The scheme here employed was of the implicit, rather than explicit, type. The explicit first order scheme, more generally known as an Euler scheme, is defined by,

1T1 = 111 1+1 (6.1) where m is a control volume fluid mass, i is the step number and T is the time step size. Note that here only one inflow to the control volume is considered.

The implicit first order scheme is however defined by,

2 m = m + m IT1' M" 1+1 1+1 1 1 i i 1 1 (6.2) the second derivative being estimated by a finite difference

216 method applied to previous values of the first derivative m'. An implicit scheme was employed as reference (1) indicates that such a scheme has better stability properties than the alternative explicit scheme.

The resulting numerical integration solution, obtained at a simulated engine speed of 2500rpm, a time step size corresponding with a 0.25° crank angle step size, a cylinder size of 125cc and a bore/stroke ratio of unity is shown in fig L6.1). This is seen to be highly oscillatory particularly for the inlet mass flow rate. The graphical data presented corresponded with a solution obtained for one cylinder of a four cylinder simulation, on the first loop (first cycle) of the integration. Reference (2) also notes problems with the stability of integration of the mass balance O.D.E., for a second order filling and emptying method calculation.

Evidently, the form of solution obtained by an implicit first order method is unsatisfactory. Further, the oscillatory problem becomes worse as the simulated engine speed is reduced. In some cases this could lead to failure of the algorithm, as the apparent instantaneous temperature in a control volume could drop below the lower limit of temperature defined for the gas properties calculation algorithm (presently this lower limit is 100K).

(6.3) Analytic Stability of Integration of an O.D.E.

Before considering the numerical stability of the integration of an O.D.E. it is first necessary to consider the analytic stability of such an integration process.

Whether an integration process is analytically stable depends upon the answer to the question: Does the numerical integration process always become stable as the step size of the independent variable becomes infinitesimally small? The

217

question is equivalent to another: Does the effective physical process being modelled remain stable as the independent variable advances in value?

Consider the analytic integration of the mass flow balance equation, taking into account one inflow only, as follows:

t mt - mo = m' (t)dt (6.3) 0 where t is time, and the superscript indicates the exact solution for the final mass value, mt.

Now suppose a small error, co(m) is introduced into the value of the mass mo at time t=0. Then, noting that the time step size is here assumed constant in size,

mo - mo = -co (m) (6.4)

and,

n=tiat am' mt - m = -c (m) II 1 + (t)St I ,as St ---4 0 t o n= 0 am (6.5)

because,

* am m (St) - m(St) ---4 —Co(M)I 1 + am (0)St ,as St ----4 0 (6.6)

and,

218 * am' ---4 -el (m) 1 + m (2St) - m(2St) am (St) St I

am ' am' (m) [ 1 + (o) St I [ 1 + ----> c0 am am (St)St

as St ---4 0 (6.7)

Now, the natural number, e, may be defined,

N 1 e ---> ( 1 + --N ) ,as N ---4 + co (6.8) then,

ami are a. mat am (t)St --> e , as St --> 0 (6.9)

(provided .71Tam' (t) remains finite)

Thus,

n=t/6 * am'(t)St) rat - int --> - co(m) n e as St ----4 0 n=0 (6.10)

8., (t)dt. * f 7.7 int - mt = -co (m) e ,lim St ----4 0 (6.11)

Then the initial error will decay in size, or remain the same size, if

am' am (t) < 0 and is finite, for all t (6.12)

219 This is the condition that guarantees the integration of the mass balance O.D.E. is analytically stable. The term in equation (6.12), henceforth termed the stability parameter, is derived in Appendix (Al2). It is seen there that the term is generally finite and zero or negative, and larger in modulus when the port area to cylinder fluid mass ratio is high. The first exception to condition (6.12) is that a positive slope plotted on a compressor map will result in a positive value for the stability parameter. This then results in a system that is physically unstable, and this is of course equivalent to the phenomenon of compressor surge. A second exception is found by noting that,

a am' a.S r ---4 1, ----4 W 4 ----4 co (6.13) p Or OM

where r is the pressure ratio across two control volumes, and C is the nondimensional mass flow parameter, see Appendix (Al2). The negative infinite value of the stability parameter at a pressure ratio of unity results in a local failure of the derivation leading to equation (6.11), the derivation assuming that the stability parameter is always finite. It is not clear, therefore, whether the integration procedure is analytically stable at a pressure ratio of unity. However, a pressure ratio of precisely unity will generally exist over only an infinitesimal time span, and therefore even if the integration process is analytically unstable at this point, the net effect upon the stability of the overall solution will be negligible.

The overall conclusion is that the filling and emptying model is always analytically stable (considering the mass balance O.D.E. only), provided any compressor employed does not operate in the surge region.

220 (6.4) The Numerical Stability of the First Order Euler Scheme

Analysis of the algorithm's stability reverted to consideration of an explicit, rather than implicit, scheme, the implicit scheme having been unsuccessful in giving any greater margin of stability. The numerical stability of the first order Euler scheme was therefore considered. The derivation in section (6.3) leading to condition (6.12) shows that for a sufficiently small time step size, the numerical integration of the mass balance O.D.E. will always be stable, provided any compressor that is simulated is not operating in a surge region. The next question is: How large can the numerical time step size be made while still ensuring numerical stability?

For the first order Euler scheme, the increase of control volume mass is defined by,

n TrI — mo = E n 1-1 St1-1 (6.14) 1=1

noting that the time step size is now non - constant. Imagine now a small error co(m) is introduced at time t=0, in an analogous fashion to the derivation of the stability condition in section (6.3). Then the additional error c (m) at time step i=2 is given by,

am' cl(m) = co(m)( 1 + -Rn— (0)6t0) (6.15) and, in a similar fashion the additional error c2(m) at time step i=3 is given by,

e2 (M) = cl (M) I 1 am (6t0)6t1) (6.16)

aM' c2 (TO = co (m) ( 1 + (0)St ) ( 1 + (8 t am o am/ o)45t ) (6.17)

221

The general condition for the growth of the error over subsequent time steps of the initial error co (m) is therefore given by,

n -1 am/ (E at )at ) (m) = c on) n ( i + (6.18) En o ,=, am J.. , ,_,

The error cn(m) will therefore not be larger in modulus than the initial error co(m) provided that the product of terms on the right hand side of equation (6.18) is less than or equal to unity in modulus. This is guaranteed if,

1 + —am '(t i)att or,

am' 5 (t ) 6 t -s 0 -2 arl i i (6.20) or,

2 Sts- I i (6.21) —aaln-T: (ti )

This is the condition for numerical stability of an Euler integration scheme of the mass balance O.D.E..

(6.5) Applying the Stability Condition

The stability condition (6.21) was applied to an Euler integration scheme of the mass balance O.D.E.. A safety factor, S , was introduced such that,

222

2S at = min I atmax (6.22) 4(:1 (t i)1

where S=0.75 and Stmax corresponds with a crank angle increment of 1°.

This stability criterion was applied to a normally aspirated engine simulation at a moderate engine speed of 2500rpm. The condition, however, was found to be completely inadequate for ensuring stability, with large oscillations in the calculated mass flow rates still occurring. Condition (6.22) resulted in a time step size corresponding to Stmax being selected for most of the duration of the simulation, with small regions where the time step size was slightly less than this.

(6.6) Numerical Integration and Ill Conditioned Problems

The conclusions resulting from the work of section (6.5) are:

either (i) the stability analysis as applied is incorrect, or (ii) the numerical integration procedure at moderate simulated engine speed is not numerically unstable - instead some other form of error is occurring.

Assuming conclusion (i) is not true, conclusion (ii) leads to the consideration of ill - conditioned problems.

An ill - conditioned problem is defined as one where the output value or values of a calculation are extremely sensitive to errors in some or all of the input values. Thus an ill - conditioned problem is subtly different to an inherently unstable problem. In an unstable problem, small errors grow exponentially, while in an ill - conditioned

223 problem the size of the errors in the output value(s) are themselves very large.

In the problem of interest, the input value is the pressure ratio across a port, r , while the output value is the calculated mass flow increment,

Am1 =mi+1 - (6.23)

Now, for an Euler scheme,

• Am = = (kg) T (6.24) I i i I where the superscript denotes the exact solution obtained with no error in the calculated pressure ratio (but still subject to the inexact solution of a numerical integration scheme), while k is a dimensional scaling factor and C is the non - dimensional mass flow rate parameter. Equation (6.23) holds for one inflow into the control volume only. Note that errors in the calculated pressure ratio, which in turn affect the value of C, result primarily from errors in the calculated control volume mass, m. This is because, from the ideal gas law,

mRT P v (6.25) using standard thermodynamic notation, and,

C = f(r (6.26) where 7 is the ratio of specific heats. For the full derivation of equation (6.26), see Appendix (A10). Then,

o c (Am) = -(Am - Am) - a(kam c (m) ti (6.27) where c (m) is the error in the calculated control volume

224

mass at the i'th step. Using the results of Appendix (Al2),

a c i(m) c (Am) = -k r c— ---T (6.28) 1 1 p ar m i i p i i

Then, rearranging equation (6.28) slightly,

c (Am) r i ac 1 — k r — — (6.29) c (m) I p ar Ill i i p i 1

The key point to note from equation (6.29) is, that for a finite step size,

a C c i(Am) as r --, 1 ; —> + CO 4 P ar c (m) P i (6.30)

Fig (5.3) illustrates a graph of C verses r . The method P thus becomes severely ill - conditioned as the simulated pressure ratio across a port approaches unity, if ki remains non - zero.

The error c i(m) results from truncation error (due to the use of an incomplete Taylor series for the numerical integration) and, to a lesser extent, inexact arithmetic (due to roundoff errors that occur with any computer).

At lower simulated engine speeds, the simulation process will operate for a larger percentage of the time at lower pressure ratios with a large value of the dimensional parameter, k. This is due to the fact that more time is available for mass transfer between control volumes for any given crank angle step. In addition, at lower simulated engine speeds, the mass truncation error is likely to be larger. This is in turn due to the larger volumetric

225 efficiency, resulting in larger higher derivatives for the mass flow, refer to fig (6.2).

Note that at very low simulated engine speeds (e.g. 500rpm) it is possible that a crank angle step of one degree results in a problem which is both ill - conditioned and inherently unstable.

From the foregoing analysis, it is seen that the integration of the mass balance O.D.E. is inherently ill - conditioned, and that measures to limit the ill - conditioning problem are required.

(6.7) Controlling the Ill - Conditioned Problem

Two steps were taken to limit the magnitude of the ill - conditioning problem that occurred with the numerical integration of the mass balance O.D.E.. These were:

(i) retain a finite value for the stability parameter for all values of r , (ii)reduce the error size c (m).

Method (i) was achieved by modifying the C verses r characteristic from its true analytic form. A finite slope of semi - arbitrary value 6.6 was used for low values of r , while at higher values of r the approximate characteristic has a curve of fixed radius, this curve joining a straight portion of the approximate characteristic at the choke point. A comparison of the approximate and the analytically correct characteristics is shown in fig (6.3), for a value of the specific heats, 7, of 1.4.

The relative error between the true and approximate mass flow characteristics is given by,

E - f(rp,7) (6.31)

226

where E is the relative error, C is the true mass flow characteristic value, and C the approximate value. The relative error, E, is plotted in fig (6.4) against pressure ratio, for a value of the specific heats, 7, of 1.4. The error is zero for a pressure ratio corresponding to the choke value and upwards, becoming +100% as rp ---4 1. The latter fact is seen by invoking L'Hospital's rule, (3), which is,

df(x) f(x) dx g(x) dg(x) as f(x),g(x) ----4 0 (6.32) dx then,

. ac — ac ar ar P P __.) + co E ---> ,, ---4 +1 ,as r ---4 1 ac + co a r (6.33)

Though the relative error is large at a pressure ratio of unity, it decreases rapidly with increasing pressure ratio. The relative error is seen from fig (6.4) to be approximately +20% at a pressure ratio of 1.05, +10% at a pressure ratio of 1.225, and close to zero at a pressure ratio of 1.6.

Method (ii) was to reduce the error c (m). For computer calculations performed in 32-bit mode, which is the norm for Fortran programs, the relative roundoff error = 5 x 10-8. Nothing could be done to reduce this roundoff error, other than invoking double precision type parameters to store the mass value. It was, however, considered that the roundoff error was generally smaller than the truncation error.

The truncation error can be reduced by employing a higher order integration method, in addition to a step control

227 algorithm. The technique eventually employed was a third order integration method for the mass balance O.D.E. described in detail in Appendix (A9), the step control method being described in Appendix (A13). The step control method is given by,

I Ti I ti such that c (6.34) max where Ti is the estimated local truncation error. A suitable was found to be 10;7 an exceedingly small value for cma x value which highlights the severity of the ill - conditioning problem. However, it should be noted that the total error is cumulative, i.e.

(m) = E T (6.35) .1=1

Therefore, the cumulative truncation error can become several orders of magnitude greater than the local truncation error over many integration steps.

The simulated engine speed was limited to a minimum value of 2000rpm for a bore/stroke ratio of unity and cylinder size of 125cc. The number of integration steps required per loop became very large at lower simulated speeds, e.g. = 5000 steps at 2000rpm, increasing to = 14000 steps at 500rpm for the same engine parameters. In addition, at the lower simulated engine speed, the solution once more became slightly oscillatory. This is believed to be due to the larger time step sizes at lower simulated engine speeds. Equation (6.29) shows that for a given error in the control volume mass value, c(m), the larger time step sizes, z, associated with lower simulated engine speeds for a given crank angle step size, will give rise to increased errors in the estimation of the mass increment, c(Am). The integration process also became highly unstable at large bore/stroke ratios and small cylinder sizes, e.g. a bore/stroke ratio of

228 7:1, a cylinder size of 40cc and a simulated piston engine speed of 20,000 rpm.

The measures described by this section were not completely satisfactory as they compromised accuracy. However, they did generally permit a stable algorithm with a reasonable number of integration steps per loop.

(6.8) The Complete Algorithm

For completeness, an algorithm that guaranteed numerical stability, as opposed to well - conditioning, of the third order integration method was invoked. This is a straightforward development of condition (6.21), and is described in detail in Appendix (A13), along with the step control method for controlling the truncation error. However, this section of the program code appeared to have no effect on step size for the range of simulated engine speeds employed, i.e. L- 2500rpm.

(6.9) Summary

Numerical integration of the mass balance O.D.E. frequently gave rise to large oscillations in the calculated mass flow rate as time steps advanced. Calculations showed the method was generally analytically stable. For a first order Euler scheme, a technique to ensure numerical stability was employed, but this was unsuccessful in solving the oscillation problem. The resulting conclusion was that, at moderate simulated engine speed, the integration process was ill - conditioned rather than inherently unstable. Measures were taken to reduce the magnitude of the ill - conditioning problem, and these were partially successful in removing the oscillations in the calculated mass flow rate at the expense of reduced accuracy. At very low simulated engine speeds, it is possible that the integration process is both ill - conditioned and inherently unstable, for a typical crank

229 angle step size of one degree. For this reason, an algorithm to ensure the numerical stability of integration of the mass balance O.D.E. is retained, in addition to the algorithm that ensures well - conditioning of the problem.

230 (6.10) References (1) Press, W.H. ; Flannery, B.P. ; Teukolsky, S.A. ; Vetterling, W.T. Numerical Recipes , the Art of Scientific Computing Fortran Edition Cambridge Univ. Press, 1989

(2) Carden, C.M. Gas Dynamics in Exhaust Systems of Turbocharged Medium-Speed Diesel Engines PhD thesis, Imperial College of Science, Technology and Medicine, Univ. of London, 1989

(3) Peterson, T.S. Calculus with Analytic Geometry Harper and Brothers, 1960

231 Fig (6.1) Implicit Integration of the Mass Flow Equation at 2500 rpm (step size = 1/4 degree CA) 0.06 10--1

C13U1 0.05 -II' 0.04 exhaust mass 60 0.03 flow rate 0 c 0.02 inlet mass flow rate 0.01

En L.)" 0.00

-0.01 D -0.02 - "a) -o .E -0.03 - -0.04 0 160 360 540 720 Cylinder One Crank Angle, (degrees) MASS Wm/ ttv_i

LAP") Srte0 CUalle (LAeC4Gr. V uWilton) 62aues

r-ri Art S FEED Oth2v6 C Sevoc-64. TRONcfrilon) 6a2c425)

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233 Mass Flow Function, (-) 0.4 0.8 0.3 0.5 0.6 0.7 0.9 0.2 0.0 0.1 Fig (6.3)ComparisonofTrueandApproximate 1.5 Mass FlowFunctions Pressure Ratio, (-)

2.0 Relative Error, (%) 100 -10 50- 70- 40- 80- 60- go 1.0 -

Fig (6.4)ErrorBetweenTrueandApproximateMass I

1.5 I

Pressure Ratio, (-)

Flow Functions I

I 2.0 2.5 Chapter (7) Minimisation of a Function of a Vector - and the Application of this Technique to Powertrain Design Optimisation

(7.1) Overview

Mathematical optimisation can be used to maximise the potential of a new design concept or, alternatively, an existing design. Such a method allows the use of genuinely optimised key design parameters, rather than for these to be chosen semi - arbitarily or through protracted trial and error methods.

A note is firstly made that multi-dimensional mapping of a function, in order to determine the optimal value, is impractical. This can be illustrated as follows. To resolve one input variable to within one per cent of the optimum value over a given range of the input variable requires 102 function evaluations, where each function evaluation produces the output value which is to be optimised. Therefore, to resolve the optimum of n input values to the same accuracy of one per cent requires 102n function evaluations. Thus, even for a moderate value of n of 5, then 10 10 function evaluations are required. Even if each function call can be evaluated in only one second, this translates to a total time of over 300 years. Clearly, then, an optimisation procedure which is considerably more efficient than multi-dimensional mapping must be applied.

The mathematical requirement is to maximise or minimise an output function which represents the output of interest (e.g. specific power output) from a mathematical model for the problem of concern. But for the presence of a minus sign, maximisation and minimisation are mathematically equivalent. To avoid verbosity only minimisation will be mentioned henceforth. The input to the function is a vector

236 which contains the variables for which the function is to be optimised. Elements of the input vector must, in the methods considered here, be mutually independent. However, problems where the input vector elements are not independent can be restated in a equivalent form with independent vector elements. For example, a four dimensional input vector with two interdependent vector directions can be rewritten as a three dimensional vector with independent elements. That is, if,

T X = (X ,X ,X ,X ) X =j (X ) 1 2 3 4 with 4 3 (7.1) where j() is a generalized function and x is the four dimensional input vector with two interdependent elements. Then x may be replaced by y where,

T y = (x1 1x2 ,x3 ) (7.2)

In powertrain design optimisation, the overall method is complicated slightly by the fact that each element of the input vector has a lower and an upper limit. This then corresponds with physical limitations to the problem. For example, if one element of the input vector is the equivalence ratio, this has a minimum value of zero while it is pointless increasing this parameter to a value above approximately 1.3 due to the fall in combustion efficiency at high equivalence ratios. Thus, the mathematical strategy for the overall problem is,

find x such that f(x )s f(x) for all x subject to xi s xi s xi , for i=1 to n min max (7.3)

where x is the input vector, x is the input vector at which the minimum lies, f(x) is the function to be minimised and n is the number of dimensions of the input vector.

237 A general problem with minimisation routines is that they are usually designed to find a local minimum where the gradient vector is zero in modulus, or, in a numerical method, approximately zero. Such a local minimum may or may not be the true global minimum. Reference (1) describes a method that uses an upper bound on the second derivatives of the function f(x) to establish the true global minimum. However, this technique is liable to be slow in multi-dimensions and establishing suitable upper bounds on the second derivatives is also a problem. A more usual approach is to use a random number generator to select the initial starting vector for the minimisation routine. Thus, several executions of the minimisation routine will see the procedure invoked from different starting vectors. If the same minimum is found several times from different starting vectors, this is likely to be the true global minimum. Conversely, if there are several local minima this strategy should see they are all found. In practice, however, the heavy demands of the optimisation technique on computing time when applied to the filling and emptying method described in chapter (5) precluded the use of this repeated minimising strategy. Thus, it cannot be known for certain whether the results obtained by the minimisation method applied to the filling and emptying type model are for the true global minimum. It should also be noted that one minimisation technique described in this chapter (the 'Monte-Carlo' technique) does not require repeated execution in this way.

An extension of the basic minimisation technique is to that of constrained minimisation. This can be stated mathematically as follows,

find x such that f(x )s f(x) for all x subject to xi s xi s xi , for i=1 to n min max and subject to c (x) s C , for j=1 to m (7.4)

238 where c (x) are the constraining functions of the input J vector, C are the constraints and m is the number of constraints. This then enables an optimisation to be performed subject to certain physical limitations, such as,

(i) TET TET max (ii) Ti n to give compatibility th th [ -Tith Tith min min with equations (7.4)) (iii) [NO] s [NO]max where TET is the turbine entry temperature, T1th is the brake thermal efficiency and [NO] the concentration of nitrogen oxide exiting the simulated engine. Constrained optimisation presents a very powerful mathematical tool, but within the time constraints of this project it was not possible to develop such a method.

This chapter describes five minimisation methods that employ repeated line minimisations (i.e. minimising along a given vector direction) and one non line minimisation method. The choice of method is then given. A line minimising technique is described, as is the method for calculating the vector distance to the boundaries defined by the upper and lower limits of each element of the input vector. The chosen method requires the partial first derivatives of the function f(x) to be calculated, and the method for achieving this is described. The criteria for algorithm convergence are given. The manner in which the chosen optimisation technique was applied to powertrain design optimisation is described. Finally, there is a summary.

(7.2) Multi-Dimensional Optimisation Methods Employing Line Minimisations

The methods considered in this section all employ a succession of line minimisations. The technique is to choose

239 a starting vector x 1 and to minimise from this starting vector in a vector direction p 1 to find a minimum in this direction. The vector position at which this minimum lies is ascribed to be a new point x 2 and line minimisation then proceeds in a new vector direction -L 2 (which must be a different vector direction from direction _p_ 1 ). The process is repeated until a minimum is found. The technique is itemised as follows,

(i) k=1, set x k (ii) choose a line minimisation direction p k * (iii)find a such that f(x + a p k) s f(x k + for --k °-P-k ) all permissible a (iv) setx 10.1 =x k +ap* k (v) if convergence achieved, STOP (vi) set k = k + 1 (vii)go to (ii)

The only difference between the different line minimisation techniques lies at step (ii), i.e. the method used to select a new vector direction for line minimisation. This has an important bearing on algorithm efficiency.

Several line minimisation techniques are now described.

(7.2.1) The Orthogonal Directions Method

The orthogonal method is a very simple technique which employs mutually perpendicular vector directions in which to perform the line minimisations. These can be chosen to be the base vector directions. This method employs no information about the function of interest when choosing the line minimisation directions.

240 (7.2.2) The Method of Steepest Descent

The method of steepest descent employs function derivatives information and may be considered to be a first order method in that it does not consider function curvature. The vector direction chosen for the line minimisation to proceed in is given simply by,

P k = -g (x k) (7.5) which is the 'most downhill' direction. Each element of the gradient vector g (x k) is given by,

af(x) gi ax , for i=1 to n (7.6)

If the gradient vector cannot be calculated directly then it can be found by finite difference methods (refer to section (7.7)).

(7.2.3) The use of Conjugate Directions

Two methods are considered that employ conjugate directions. Such methods implicitly consider both the function first derivatives and function curvature and may therefore be considered second order methods.

A general multi-dimensional function with zero third derivatives is defined by the quadratic equation,

1 T T f(x) = x -b . X C (7.7) where A is the square n x n matrix with elements fixed in value, b is a constant vector and c is a constant. All continuous functions will approximate to a quadratic function over a small region. Methods employing conjugate directions make the slightly questionable assumption that the function being minimised approximates to a quadratic

241

function non-locally.

Each element of the matrix A at the i'th row and j'th column is defined by,

a2f(x) a - (7.8) U ax ax j 1

i.e. matrix A contains the function second derivatives, and is also seen to be symmetric. The first point is seen by noting that the gradient vector g (x) can be expressed as,

g (x) = A. x - b (7.9)

which is derived in Appendix (A14). Hence,

af(x) inXn ] ax - [aux1 + a12x2 + ...+ a - b (7.10)

and the partial differential of equation (7.10) with respect to x gives equation (7.8). Equation (7.9) shows that a stationary point of the function f(x) is given by,

A x* =b or x* = A l b (7.11)

This is a minimum if

aii > 0 for i=1 to n (7.12)

Now, mutually conjugate directions are defined by,

T p I.A.p j = 0 ,j # i, i=1 to n, j=1 to n (7.13)

and mutually conjugate directions are used for the line minimisations. Note, however, that there are an infinite set of mutually conjugate directions. The conjugate directions algorithms described in this section will choose a particular set of conjugate directions, though the

242 convergence properties of the algorithms are independent of which set of directions are chosen.

That conjugate directions are 'good' directions to minimise in is seen as follows. Consider a general problem, where the first minimum is found at x 2 by moving in direction2 1, fig (7.1). It is now desired to find a new minimum by moving in a direction p 2. By definition of the minimum in direction p 1,

gT(X 2). p = 0 (7.14)

A condition that movement in the direction p 2 does not 'interfere' with the first minimisation is therefore,

g (X 2+ P 2)* P = (7.15) where a is the distance moved in direction p 2. It will be shown that the conditions for equation (7.15) to hold are,

T either a= 0 or p 1 .A.-- p 2 = 0 (7.16) the first condition in equation (7.16) is the trivial case that zero movement distance from the first minimum will not interfere with the first minimisation. The second condition, however, is precisely that for conjugate directions. This will be shown as follows. From equation (7.9),

T T p 1.A.[X 2 + ap 2 )-p——1.b =0 (7.17)

Hence,

T T a p 1 .A.-- p 2 + p 1 .(A. --x 2 - —b ) = 0

a PCA* 12- 2 + -12- 1 (-31- 2) = which, from equation (7.14) gives conditions (7.16).

243 Appendix (A15) proves that a set of n mutually conjugate directions will lead to the exact minimum being found in exactly n line minimisations as applied to a n dimensional quadratic function, in exact arithmetic.

Two methods that generate conjugate directions for the line minimisations to proceed in are now considered. The first method does not evaluate the gradient vector while the second method does.

(7.2.3.1) The Conjugate Directions Method

The conjugate directions method employs the use of conjugate directions without evaluating the gradient vector. The method proceeds as follows,

(i) initially set the minimisation direction vectors p 1...p n to the base vector directions i 1...i n (ii) perform an initial line minimisation in direction p , hence set —1x (iii) for k=1 to n find a such that f(x k + a p k) f(x . + a p k) for all permissible a, " setx k+1 =x k +ap k, if convergence achieved, STOP (iv) for j=1 to n-i set p j = p (v) define p=n x n+1- —x 1 (vi) set x==x —n+1 n+1 (vii)go to (iii)

Reference (1) shows that the directions generated at step (v) will be mutually conjugate when the algorithm is applied to a quadratic function. Because the gradient vector is not evaluated, it takes some while to build up these conjugate directions, one new conjugate direction being generated every n line minimisations. Therefore n sets of n line minimisations generate the required n conjugate directions. There is an initial line minimisation required at step (ii),

244 plus the final n line minimisations required with the full set of conjugate directions. Thus (n(n+l) + 1) line minimisations will see the exact minimum of a quadratic function being found, in exact arithmetic.

The basic method as given has been found to be unreliable, and references (1) and (2) give modifications to the basic algorithm which are designed to improve its reliability.

(7.2.3.2) The Conjugate Gradient Method

The conjugate gradient method employs the gradient vector to find the conjugate directions used for the line minimisations. Because this additional information is available, the conjugate directions are generated more quickly by this method than by the method described in the last section. Against this, if the gradient vector is estimated by finite difference methods, then additional function evaluations will be required. The basic method is,

(i) set x 1 (ii) set p = -g (x 1) (iii) perform an initial line minimisation in direction p 1, hence set21 2, if convergence achieved, STOP (iv) for k=2 to n

g ( X k ) .1g (X k ) (a)set p k = -g (x k) + T -P k-1 g (X k -1 ) .g (X k-1 ) (b)minimise in direction p k , hence set x k."1 (c)if convergence achieved, STOP (v) set x = x — 1 — n+1 (vi) go to (ii)

Appendix (A16) shows that, applied to a quadratic function, each direction generated at step (iv)(a) will be mutually conjugate. Thus minimisation of a quadratic function is achieved in exactly n line minimisations, in exact arithmetic.

245 It was found by the author that the method was sensitive to errors in the gradient vector, g (x), which could lead to failure of the algorithm. For example, if performing a two dimensional optimisation of equivalence ratio and piston engine speed in rpm, the degree of variation of the equivalence ratio is 0(1), while that of the engine speed is 0(104). This implies gradient values are typically 104 times less in the direction of the engine speed vector than in the direction of the equivalence ratio vector. Therefore a relative error of only 10-4 in the evaluation of the gradient component in the direction of the equivalence ratio vector is liable to lead to failure of the algorithm. Therefore, the basic unconstrained minimisation problem, equation (7.3), was modified to,

_ find x such that f(x) f(X) for all R subject to 0 :5 X L5 1, for i=1 to n (7.20)

i.e. the input vector is normalised with each element lying between zero and unity. This ensures that the gradient components in each of the base vector directions are of a similar order of magnitude. The optimisation algorithm calculations are then based on the normalised vector X . When the simulation routine is invoked, this normalised vector is converted to the 'true' vector, x . The i'th component of the true vector is calculated from the i'th component of the normalised vector as follows,

xi = xi (xi — x ) + xi (7.21) max imin min

(7.2.4) Quasi-Newton or Variable Metric Methods

As for methods employing conjugate directions, quasi-newton (or variable metric) methods assume the function to be minimised approximates to a quadratic function. Then,

246 equations (7.9) and (7.11) combined give,

* x - x = -A 1.(g (x k) - g (x )) = -A 1.g (x ) (7.22) — -k k where x is the minimum. Thus, a direction defined as,

k.1 = g (x k) (7.23) is a 'good' direction to minimise in. Quasi-newton methods iteratively approximate the inverse Hessian matrix, A 1, by a matrix Hk, such that,

(X k ) k+1 = (7.24) and

Hk Hk-1 + Uk-1 (7.25) where Uk is the update matrix. As for methods employing conjugate directions, quasi-newton methods may be considered to be second order methods. Different quasi-newton methods differ in the manner in which the update matrix is calculated. Two popular methods are the Davidon - Fletcher - Powell (DFP) and Broyden - Fletcher - Goldfarb - Shanno (BFGS) algorithms, which both lead to exact minimisation of a function in n dimensions of quadratic form in n line minimisations, (3).

(7.3) Multi-dimensional Optimisation Employing the 'Monte-Carlo' Technique

The 'Monte-Carlo' technique is a colloquial name for a general statistical type method relying upon the use of a random number generator. Applied to multi-dimensional optimisation, the method proceeds as follows,

247 (1) set the best minimum figure, fmin, to a large positive figure significantly greater in modulus than the estimated function minimum value (ii) normalise the input vector space so that each element of the input vector, k, lies between zero and unity (iii) for i=1 to n (a) generate a random number, r, between 0 and 1 (b) set X = r (iv) evaluate f(X) (v) if f(R) < fmin (a) set fmin = f(X) (b) store X (vi) if sufficient loops performed, STOP (vii) go to (iii)

The technique is seen to consist of performing a large number of trial function evaluations, storing the minimum value found. It is an extremely robust, simple technique. However, a little analysis shows that such a method is likely to be inefficient in practice. This is seen by noting that if each element of the numerical solution input vector is within 0.1% of the true solution over the maximum base vector span of unity, then 103n function evaluations will generally be required. This assumes a perfect random number generator, and is a reflection of the ratio of the total vector space to the approximate solution vector space. Thus, even for moderately large n (e.g. n = 5) a large number of function evaluations are required to approach the solution.

(7.4) The Choice of Multi-Dimensional Optimisation Method

As suspected, the orthogonal directions and Monte-Carlo methods proved inefficient when applied to the minimisation of trial mathematical functions. They were therefore rejected on this basis. The method of steepest descent, whilst providing a serviceable algorithm, does not possess

248 the property that exact minimisation of a quadratic function is achieved in a finite number of iteration steps. Being a first order method, the technique is liable to be inefficient when function curvature is large. The method was therefore rejected in favour of one of the conjugate directions, conjugate gradient or quasi-newton techniques. Appendix (A17) shows that the conjugate gradient and quasi-newton techniques are more efficient when minimising a quadratic function than the conjugate directions technique. This is even allowing for the extra number of function evaluations required to evaluate the gradient vector for the conjugate gradient and quasi-newton techniques. Thus the choice is between the conjugate gradient and one of the quasi-newton techniques. Applied to a quadratic function each of these techniques are identically efficient and thus either method would be a good choice. The conjugate gradient method was chosen purely on the basis that its algorithm is simpler.

(7.5) Bracketing the Minimum and Employing a Line Minimisation Technique

The chosen algorithm requires a line minimisation technique. The 'Golden Section' Search method was used. Initially, a minimum must be bracketed between upper and lower limits. This proceeds as follows,

(i) set NOTBRKT = FALSE (ii) establish initial vector x ki line minimisation direction vector p k (iii) establish amin, amax as defined by input vector space boundaries (iv) set Aa = (a - a )/10 max max (v) set aa = amin (vi) set x =x +a p - a - k a k (vii) set f = ) , set f a f (x a 1 = fa (viii) set a = a + b mln Aa

249 (ix) setx b =x k + ab p k (x) set fb = f(x b) (xi) set a = a+ 20a c mi n set x =x +ap (xii) —c k c — k (xiii) set f = f(x— c ) (xiv) set i=1 (xv) if (fb < fa) and (fb < f )

then minimum bracketed, store aa, ab, ate , fb RETURN to main optimisation routine (xvi) if i=9 then minimum not bracketed set NOTBRKT = TRUE f = f 2 c if f 1 f2 min = f store ab = a , fb 1 else store ab = max fb = f2 endif RETURN to main optimisation routine endif (xvii) as =ab,fa =fb, ab = ac, fb = f c , ac = ab + Oa (xviii)f = f(x + a p ) k c k (xix) i=i+1 (xx) go to (xv)

Such a process ensures either,

(a) at step (xv) a value fb at vector position 2E.b lying between vector positions x and -x-c is less than both f and f c , with I x c a -2—C a I ( (Xmax amin) 5 (b) at step (xvi) the minimum is not bracketed, and the minimum end value lies at either ( x + a p k) or - - k min + a p ) at ( --x k max — k

If (a) is true a minimum must lie between x a and x c, and the subsequent line minimisation procedure can continue. Conversely, if (b) is true, no interval which can be

250

guaranteed to contain a minimum has been found. Thus the line minimisation procedure is not subsequently invoked. Further, the conjugate gradient algorithm must be reinitiated, as this procedure relies on a bracketed minimum being found at each iteration step, k, to ensure its convergence properties.

If (a) is true, the line minimisation procedure continues as follows,

(i) pass fb, aa ab, a as parameters from bracketing routine

(ii) set 'golden' constant, G = 2 + (iii) set RIGHT = FALSE (iv) if NOT(RIGHT) then

a = aa + G(ac - a a ) else

a = ac - G(ac - aa) endif

(v) f=f(x k +a P k ) (vi) if (a > a ) then if (f s fb) then a = a a b a = a b fb = f else a = a endif else if (f s fb) then a = a ab = a fb = f else aa a endif endif

251 cta )/(ct a (vii) if (((a - max — min )) < c ) then store ab RETURN to main algorithm (viii) )/(ac - aa)) > if (((ab - aa 2 ) then RIGHT = TRUE else RIGHT = FALSE endif (ix) go to (iv)

Such a procedure ensures that at each subsequent step, the minimum is bracketed to lie in a an interval G (=0.618) times the previous bracketing interval, (3). Thus if a fractional tolerance for c of 10-3 is used, as was the case in the optimisations performed, then

m N - ln(5 x 10-3) G = 5c or ln(G) (7.26)

where N is the number of line minimisation iterations. The figure of five appears in equation (7.26) because the minimum is already bracketed to lie within a region 20% the size of the maximum interval size. Therefore, 12 iterations are required to give the desired tolerance convergence accuracy. Added to the average number of function evaluations required for the bracketing routine which takes a value of 7, this gives an average of 19 function evaluations per line minimisation, excluding the gradient vector evaluation.

The Golden Section Search algorithm is a simple, reliable technique. Reference (3) describes an alternative method that uses a combination of the Golden Section Search and quadratic interpolation methods, which it is claimed is more efficient. In practice, such a method was found to be similarly efficient to the current method when applied to a quadratic function. This is surprising given that, once a minimum has been bracketed, quadratic interpolation should

252

see the minimum of a quadratic function being found in exactly one iteration.

(7.6) Calculating the Distances to the Input Vector Boundaries

The optimisation procedure, with a constrained region for the input vector space, requires the calculation of the maximum and minimum allowable movement distance from the vector k. A line minimisation proceeds along a vector direction given by,

x =x +ap (7.27) — -k -k

where x is a general point in the minimisation direction p k . Consider a four dimensional vector space. One boundary is defined by the superplane (actually a three dimensional surface) given by,

S = y0 + b1 al 4- b2 a2 + b4 a4 (7.28)

where bi are variables and the vectors yo and ai are constant, the latter in this case being defined as unit vectors. The superplane considered has no component in the direction given by the third element of a general vector, i.e.

T S = yo + [b1 , b2, 0, b4] (7.29)

Thus at the point where the line meets the superplane,

x + a p k = y +b +b +b a (7.30) -- k o 1 a1 2 a2 14 4

or,

- b2 a b = y - L.: (7.31) cc p k - bl al 2 - 4 a4 o - - k

253 Equation (7.31) may be written,

al a 2 p k a4 -- -- yo 1 i 1111 _ - _ _ (7.32) 1 0 - -b O Pkl 1 Yol 0 1 0 -b Pk2 2 1102 0 0 pko 0 a = 1 - --kX 0 n 1 -b - 0 - 'k4 - - 4 - -Y04-

The vector y0 is unimportant in determining a, apart from the value of unity which appears in this vector and which defines the position of the superplane. The matrix in equation (7.32) can be inverted to determine the solution vector z , though only the value a is required. The form of this system of equations may be generalised to n dimensional vector space and for all superplanes defining the constrained vector space.

Equation (7.32) was inverted using the LU decomposition technique, (3). To protect against numerical overflows the solution for a was obtained using Cramer's rule. For the above system of equations this is given by,

la ,a ,y -x ,a I D3 i 1 2 0 — k 4 I a = D (7.33) , a2 , p k , a4 where D stands for determinant. This method of solution can be generalised to n dimensional vector space and for all superplanes defining the vector space boundaries. LU decomposition enables the determinants given by equation (7.33) to be efficiently found. Numerical overflows, which are in danger of occurring when p k is nearly parallel to S , are protected against by storing a determinant as a sign and log value, i.e.

254 log (d ) 10 L D = d S 10 (7.34) from which equation (7.33) is solved through,

[log (d )-log (d )) 10 3L 10 L a = d 3sd s 10 (7.35)

If the exponent in equation (7.35) is large in value the expression is not evaluated, a instead being set to a large positive or negative value as appropriate. The same occurs if the system of equations given by expression (7.32) is found to be singular.

The final values for a are then given, in a four dimensional problem, by,

a = min (a, a , a , a (7.36a) max +2 +3 +4 )

amin = max (a-1 , a-2, a-3, a-4) (7.36b) where the values in parenthesis are those obtained for each of eight constraining superplanes (at vector positions 0 and 1 in each of the base vector directions), '+' denoting a positive value and '-' denoting a negative value.

It was later realised that solving the complete system of equations as given by expression (7.32) is unnecessary. For example, the third equation in this system of equations may be solved directly to give,

1 — X k3 a - (7.37) Pk 3 though the result obtained is identical to that of evaluating equation (7.33).

255

(7.7) Evaluating the Gradient Vector

The chosen optimisation method requires evaluation of the gradient vector. This is achieved through a finite difference method as follows,

f(x + Axi ) - f(x - Axi) (7.38) g, 21 Axi

where gi is the approximate i'th component of the gradient vector and,

Ax = (0,...,0,Ax,0,...,0)T (7.39)

ith component

This is a second order finite difference gradient evaluation which therefore has the property of maintaining exact convergence of the chosen method in n line minimisations, applied to a quadratic function in exact arithmetic. Care has to be taken when the vector x is close to a boundary, to avoid the function evaluations required for gradient evaluation overlapping the boundary. For example, if x is near to the i'th lower boundary, define,

f(x + 2Axi ) - f(x) (7.40) g, 21 Ax 1

Equation (7.40) is a first order gradient evaluation which therefore represents a slight error in the algorithm, the intended method being second order.

The value used for Ax was,

Ax = 0.02(xmx - Xmin ) = 0.02(1 - 0) = 0.02 (7.41)

256

thus Ax represents a relatively large interval size. This was to smooth out statistical fluctuations in the output function, f(x).

(7.8) Convergence of the Solution

Convergence is deemed to have been achieved, when,

k-1 c, for i=1 to n (7.42) 1 -

and

Xk X -1 k-21 se for i=1 to n (7.43) I

i.e. in the last two successive iteration steps, each component of the normalised input vector has changed by an amount less than, or equal to, c. A value of 10-3 was generally used for c, though in practice the available computation time did not always permit full convergence to be achieved.

(7.9) Applying the Multi-dimensional Minimisation Technique to Powertrain Design Optimisation

The specific power output, w(X), (in kW/dm3) was maximised for the quasi-steady naturally aspirated and compound turbocharged engine models by setting,

f(5) = -w(k) (7.44)

The variables optimised were,

x1 = fuel/air equivalence ratio x2 = piston engine speed

and, additionally in one case,

257 x3 = the overall compression ratio the compressor pressure ratio and heat exchanger effectiveness being fixed in the case of the compound turbocharged engine.

In the naturally aspirated filling and emptying engine model, equation (7.44) was again applied to maximise the specific power output and the variables optimised were, x1 = fuel/air equivalence ratio x2 = ignition timing x3 = inlet valve opening angle x4 = inlet valve opening duration xs = exhaust valve opening angle x6 = exhaust valve opening duration x7 = piston engine speed

The compound turbocharged filling and emptying model was additionally optimised for the following variables,

X = compressor area ratio relative to design area 8 x9 = turbine choke area the heat exchanger effectiveness being fixed in value, while the compressor speed was fixed in value to give the desired compressor pressure ratio. Additional optimisations were performed with, x1 = bore/stroke ratio the fuel/air equivalence ratio being fixed in value at 0.98 as this was the best value given by previous optimisations. Time precluded the optimisation of three further variables as follows,

258 x10 = inlet manifold volume x11 = exhaust manifold volume = the overall compression ratio x12

(7.10) Summary

Mathematical optimisation of a multi-dimensional function was employed to enable 'best' design parameters for an engine concept to be chosen, rather than for these to be selected semi-arbitrarily. An optimisation method that is efficient when applied to a quadratic function was chosen. This method was applied to naturally aspirated and compound turbocharged piston engine simulations, to maximise simulated specific power output.

259 (7.11) References (1) Brent, R.P. Minimization without Derivatives Prentice-Hall, 1973

(2) Powell, M.J.D. An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives Comp. J., no. 7, pp155-162, 1964

(3) Press, W.H. ; Flannery, B.P. ; Teukolsky, S.A. ; Vetterling, W.T. Numerical Recipes, the Art of Scientific Computing Fortran Edition Cambridge Univ. Press, 1989

260 LI NES 0 p C0 nt CTANT -i..".1G11ON -\ VALUE PI tZEC110to 42i ...... ------..

STA/2-n ni Ct VELTO le

-A cm (i.1) -The use oc- umE mitvirvmsernomS

To OPTIMISE A -FuNCTIOA-3

261 Chapter (8) Numerical Results

(8.1) The Gas Turbine Models

Results from the numerical modelling of gas turbines of maximum compressor pressure ratio of 5:1 and two different maximum TETs of 1200K and 1600K are now presented. The case of the constant speed, single shaft gas turbine is first considered. Numerical data for this engine type is presented as follows,

FIGURE ILLUSTRATED FEATURE (8.la,b) part load thermal efficiency (-) versus specific power output (kW cm-2) (8.2) TET(K) versus specific power output (kW cm-2) (8.3) compressor pressure ratio (-) versus specific power output (kW cm-2)

Figure (8.1a) is for the case when the maximum TET is 1200K, and figure (8.1b) is for the case when the maximum TET is 1600K.

The specific power output figure is given in terms of the net power output for a given cross sectional area, and is based upon the maximum cross sectional area of the compressor impeller. In this manner, the quoted specific power output is directly proportional to the actual power output but independent of machine size. The alternative of providing a specific power output figure in units of energy produced per unit mass of inlet air gives a figure that is not directly proportional to the actual power output, as it does not consider the compressor mass flow rate. From these figures it is seen,

(i) the thermal efficiency rises fairly linearly with power output, increasing with increased heat exchanger

262 effectiveness and TET. However, the thermal efficiency at very low relative power outputs is poor even with high heat exchanger effectiveness and TET. (ii) The peak thermal efficiency with high effectiveness heat exchanger and high TET is high. (iii)The power output reduces fairly linearly with reducing TET. (iv) The compressor pressure ratio only varies by approximately 20% over the entire power output range, being lower at lower power outputs. (v) The maximum specific power output of the gas turbine operating with a TET of 1600K is approximately double that of the gas turbine operating with a TET of 1200K.

In the single shaft, constant speed gas turbine the net torque output is directly proportional to the power output. However, since the engine is constant speed the transient response of this unit is not a relevant consideration. Thus, the torque/speed characteristic is not important.

The case of the two shaft differential gas turbine with continuously variable interconnecting transmission is now considered. Numerical data for this engine type is presented as follows,

FIGURE ILLUSTRATED FEATURE (8.4a,b) Part load thermal efficiency (-) versus specific power output (kW cm-2) (8.5) TET (K) versus specific power output (kW cm-2) (8.6) Compressor pressure ratio (-) versus specific power output (kW cm-2) (8.7a,b) Non-dimensional physical compressor speed (-) versus non-dimensional physical turbine speed (-) (8.8a,b) non-dimensional physical turbine speed (-) versus specific power output (kW cm-2) (8.9a,b) non-dimensional gross turbine torque (-) versus specific power output (kW cm-2)

263 where figures (a) are for the case when the maximum TET is 1200K and figures (b) are for the case when the maximum TET is 1600K. The conclusions drawn from this data are as follows,

(i) with a low to moderately effective heat exchanger, e.g. 50%, the part load efficiency characteristics of the differential gas turbine are similar to the corresponding case of the single shaft, constant speed gas turbine. (ii) The peak and particularly part load efficiencies with a high effectiveness heat exchanger are high. With such a high effectiveness heat exchanger, the differential gas turbine is able to maintain a high thermal efficiency at low relative power outputs. This is because the differential gas turbine is operating in a low pressure ratio, high TET regime at low power outputs, making the maximum use of a high effectiveness heat exchanger. (iii)There is a significant fall in compressor pressure ratio at low relative power outputs. (iv) The TET can be maintained at its maximum permitted value down to very low relative outputs. (v) The speed relationship between the compressor and turbine is complex, making a continuously variable interconnecting transmission essential. (vi) The gross turbine torque falls with power output, suggesting poor transient response. (vii)The maximum specific power output of the differential gas turbine operating with a TET of 1600K is approximately double that of the differential gas turbine operating with a TET of 1200K.

Thus, the complexity incurred by adopting the continuously variable transmission, differential gas turbine is only worthwhile if a heat exchanger of high effectiveness is

264 fitted. In this case, there is a considerable increase in part load efficiency.

(8.2) Validation of the Piston Engine Models

The piston engines were validated against test data. Reference (1) gives data for a naturally aspirated eight cylinder spark ignition engine, while (2) gives data for a turbocharged, four cylinder spark ignition engine. Table (8.1) presents specific data for the naturally aspirated engine. The equivalence ratio was estimated as that given for maximum power output in the optimisation procedures, namely an equivalence ratio of 0.98, this value being discussed below. In the table, comparisons between the 3 measured maximum specific power output (kW dm ) and the predictions of this same parameter in the quasi-steady and filling and emptying models are given. The measured value is 46 kW dm 3, which compares with estimates of 51 kW dm 3 and 40 kW dm-3 for the quasi-steady and filling and emptying models respectively. The quasi-steady model therefore shows an overestimate of 11%, while the filling and emptying model shows an underestimate of 13%. In the case of the former, the overestimate is considered to be caused by the assumptions of constant volume combustion and a volumetric efficiency of 100%. In the case of the latter, the shortfall is thought to be due to the fact that the empirical combustion efficiency characteristic is inaccurate. The optimisation results described in the next subsection consistently predicted that the maximum simulated power output occurred at an equivalence ratio of -0.98, whereas it is well known that in reality the equivalence ratio for maximum power output is of the order of 1.1. Reference (3) presents data which shows that the change from an equivalence ratio of 0.98 to that for maximum power output (- 1.16 for the fuel used) produces approximately 10% gain in power output, accounting for much of the discrepancy in the filling and emptying naturally aspirated engine model.

265 Table (8.2) presents similar comparative data for a spark ignition turbocharged engine. The valve port areas, ignition timing, valve timings and equivalence ratio were not known and had to be estimated. The equivalence ratio was again set to a value of 0.98, the predicted simulated optimum value, while the ignition timing and valve timings were also set to predicted optimum values for the naturally aspirated engine. The valve port areas were based upon data provided by (4). When the compound work is subtracted from the simulated results, the comparisons are as follows: actual engine specific output, 91 kW dm-3; quasi-steady model simulated output, 73 kW dm-3; filling and emptying model simulated output, 82 kW dm-3. The quasi-steady model therefore gives a shortfall of 20%, while the filling and emptying model gives a shortfall of 11%. The shortfall in the former is considered to be due to the assumption of constant exhaust stroke pressure equal to the pressure at the end of the piston engine expansion stroke. This gives a much bigger pumping loop, reducing simulated piston engine power output (though increasing simulated turbine power output). The quasi-steady naturally aspirated model did not make this assumption, instead the exhaust pressure was assumed to fall instantaneously to atmospheric pressure at the start of the exhaust stroke. As far as the compound turbocharged quasi-steady engine simulations are concerned, the assumption of this high piston engine exhaust pressure is considered to give an inaccurate estimation of the power split between the piston engine and rotodynamic components, but a reasonable estimate of overall specific power output. As before, a large fraction of the discrepancy between actual specific power outputs and the filling and emptying model estimate of the specific power output is considered to be due to the inaccurate combustion efficiency characteristic. Table (8.2) also gives comparisons between the actual and estimated thermal efficiencies.

266 In summary, the quasi-steady model is relatively simple and can only be expected to give approximate estimates of power output and thermal efficiency. The filling and emptying model has more potential to give reliable estimates of these parameters , but currently requires further development to improve its accuracy. Nevertheless, it is able, at present, to provide order of magnitude estimates of specific power output and thermal efficiency.

(8.3) Optimisation Results

(8.3.1) Employing Test Mathematical Functions

To test the optimisation algorithm, the method was applied to quadratic functions for which the position of the minimum, x*, and the minimum value, f(x ), are known. Applied to such functions, the algorithm consistently produced answers which were very close to the exact solutions, and also produced these results in the theoretically predicted number of line minimisations. Table (8.3) shows the results for one case, which was a seven dimensional problem. The solution was achieved in 9 line minimisations, which is equal to the theoretical value of 7 plus 2 additional line minimisations which are required to test that convergence has been achieved.

(8.3.2) Employing the Quasi-Steady Piston Engine Models

The quasi-steady piston engine models were used to provide initial test usage for the optimisation algorithm, and also to provide order of magnitude cross checks with the optimisation results produced by using the filling and emptying simulations. The quasi-steady models were not expected to provide particularly accurate simulation results. In all cases the net specific power output (kW dm-3) was optimised. The fixed input parameters used for the optimisations are shown in table (8.4). The optimised

267 input vectors (either two or three dimensional) along with the number of line minimisations required to reach a solution are shown in table (8.5). Table (8.6) shows the optimised output parameters. Case (I-1) is for a naturally aspirated engine, while all the other cases are for a compound turbocharged engine with or without aftercooling. The overall volumetric compression ratio excludes density changes in the aftercooler. The results produced show that,

(i) the optimised specific power output and piston engine speed increase with increasing boost pressure and level of aftercooling. (ii) The overall thermal efficiency is of the order of 20% in all but case (VI-1). In the latter case, the thermal efficiency is approximately 10%. (iii)The rotodynamic power output is simulated as being considerably higher than the piston engine power output in the compound turbocharged engine model. The difference increases with increasing boost pressure. The high simulated rotodynamic power output is believed to be a consequence of the assumption of constant piston engine exhaust pressure equal to the pressure at the end of the piston engine expansion stroke. (iv) At high boost pressures, the simulated TET is high, as is the simulated specific power output. (v) The number of line minimisations required to reach a solution is generally greater than the theoretical number for a quadratic function. The two extra line minimisations required to test for convergence have been subtracted from the figures given in table (8.5). (vi) An attempt to predict the optimum overall volumetric compression ratio (case (VI-1)) has resulted in a value of this parameter of 5.5:1 being selected by the optimisation algorithm. This was the lower limit value. At this condition, the net thermal efficiency is unsuprisingly very low while the TET is high. Only

268 a limited amount of confidence is placed in this prediction.

In cases (V-1) and (VI-1) the net rotodynamic power output is a negative fraction of the piston engine power output. This is because the piston engine is simulated as absorbing power. In all cases the optimised equivalence ratio is close to 0.98. This is considered to be numerically correct, but reflects an error in the empirical combustion efficiency profile as discussed in section (8.2).

(8.3.3) Employing the Filling and Emptying Piston Engine Models

A number of different parameters were optimised in the filling and emptying engine models. Table (8.7) gives those parameters that were fixed in value, for the six different optimisations. Table (8.8) gives the values of the optimised parameters along with the number of line minimisations required to achieve a solution, while table (8.9) gives output simulation values at the optimised points. In all cases, the optimisations were performed to maximise specific power output. The results arising from this work may be summarised as follows,

(i) the optimised specific power output and piston engine speed increase with increasing boost pressure and level of aftercooling. (ii) The overall thermal efficiency is between 20% and 25% in all cases. (iii) The rotodynamic power fraction increases with increasing boost level. (iv) At high boost pressures the simulated TET is high as is the simulated specific power output. (v) The optimum valve timings at high boost pressure and consequently high piston engine speed have a very large degree of overlap. In cases (III-2), (IV-2) and

269 (V-2) the exhaust valve closes after fuel injection into the cylinder has begun (at 60° ATDC). This will adversely affect the thermal efficiency as fuel is being lost through the exhaust valve, but will have a minimal effect upon the simulated specific power output. The thermal efficiency figures are therefore corrected in section (8.4.2). (vi) The optimum valve timings for the compound turbocharged engine provide, by virtue of their large degree of overlap, a substantial amount of excess air. This has the benefit, through dilution, of lowering the TET. (vii) For the compound turbocharged engine, the net energy production per unit mass of inlet air is substantially the same whatever the level of boost pressure, at - 600 kJ kg-1. A comparable figure for a gas turbine of pressure ratio 9:1 and TET of 1600K is - 300 kJ kg 1. (viii)The number of line minimisations required to converge to a solution is rather greater than can be expected for a quadratic function. The extra line minimisations required to test for convergence have been subtracted from the figures given in table (8.8). (ix) It was also found that at high piston engine speeds, the heat loss as a fraction of gross heat input was small, e.g. 3% at 20,000 rpm.

Thus, as with the quasi-steady model, the high speed, highly supercharged, compound turbocharged engine is confirmed as having very high specific power outputs. Tables (8.6) and (8.9) may be compared, and show reasonable agreement in the predicted specific power outputs and thermal efficiencies. However, the predicted power split between the piston engine and rotodynamic components differs markedly. As discussed previously, this is believed to be due to the assumption, in the quasi-steady model, of a constant piston engine exhaust

270 pressure equal to the pressure at the end of the piston engine expansion stroke. In the filling and emptying model, the piston engine exhaust pressure drops rapidly as the exhaust valve is opened, increasing the piston engine work but reducing the turbine work. As with the quasi-steady model optimisations, the filling and emptying model optimisations consistently predicted an equivalence ratio of approximately 0.98 as the best for maximum power output. This is considered to reflect an error in the simulation procedure, as described in section (8.2).

In two cases ((II-2) and (V-2)) the bore/stroke ratio was optimised. In case (II-2) the optimisation algorithm has converged at the defined upper limit in the bore/stroke ratio of 7:1. It was not possible to increase the bore/stroke ratio to values higher than this due to the danger of algorithm instability. Case (V-2) has also converged to a high bore/stroke ratio value of 4.33:1. High bore/stroke ratios allow larger valve areas for a given swept capacity, but the validity of extrapolating the simulation model to such high bore/stroke ratios is questionable. The validity of the friction mean effective pressure calculations is also questionable at simulated engine speeds of 20,000 rpm. The given definition of the maximum valve lift as dictated by the cam contact stress limit is also likely to be invalid at these high piston engine speeds. However, because the optimum valve opening duration is so large at these high piston engine speeds, the valves will be operating in a regime where their maximum allowable lift is dictated by the tappet radius, rather than the cam contact stress limit. It is noted that in cases (II-2), (III-2) and (IV-2) the optimisation has converged such that the compressor is operating in the surge region. These figures were therefore corrected by adjusting the compressor size as described in section (8.4.2).

271 (8.3.4) Part Validation of the Optimisation Results

It is not possible to validate absolutely the multi-dimensional optimisation technique. To achieve such absolute validation would require the construction of a multi-dimensional 'map' of the solution, which would be prohibitive in computing time requirements. Indeed, the optimisation procedure is designed to eliminate the need for such maps. A number of points can be made about the likely validity of the optimisation procedure, however, as follows,

(i) the optimisation procedure has been tested against quadratic functions, for which the optimum point is known, with complete success. Thus, provided the application of the optimisation procedure to engine simulation models is not ill-conditioned, it can be expected to work with such models as well. (ii) The gradient vector was calculated at the supposed optimum point and in all but one case the components of the gradient vector were found to be small. Typically, the gradient vector components were such that there would be between a 1% and 5% change in the optimised output variable if a straight line of gradient equal to the given gradient component was extrapolated over the complete interval. (iii)Because the optimisation procedure took a considerable time to converge, it was designed to be stopped and restarted after a given number of line minimisations. When the optimisation routine was employed in this manner, the optimised variable could be observed to systematically increase.

In case (V-2), when one component of the gradient vector was found to be large (giving a 100% change in the output variable if a straight line equal to the gradient component was extrapolated over the entire interval) at the supposed optimum, this might be explained if the optimum lay on a

272 sharp peak in the the output function, so that a small displacement from the real optimum value would give a large gradient value. The vector component of the concern was the compressor area ratio, which affects the compressor pressure ratio. The optimisation results obtained in case (V-2) must, however, be regarded as slightly suspect. A further note is that the gradient values calculated through finite difference methods can be subject to errors, particularly with the large interval size used here.

To give extra confidence in the optimisation procedure, four of the filling and emptying model optimisations were partially validated as follows. An orthogonal grid was constructed in the input vector space with its apex lying at the supposed optimum position. Figure (8.10) illustrates this for a three dimensional case. Each line of the grid lies along a base vector direction. Along each such direction, the output function value was evaluated and the value a was found, where,

a = 0 negative value, large in modulus compared with f a = max(a k-1 , f - f (5 k ) ) for k = 1 to m (8.1) where k is the point number, m is the number of points, f is the supposed optimum value and x k is the test point. Noting that f(x) is the negative of the specific power output, a will be positive if the 'best' test point found is 'better' than the supposed optimum value and negative if the converse is true. The final parameter is expressed as a percentage,

a a — R1 x 100 % (8.2) If

The results from this work are shown in table (8.10). In

273 case (I-2i), m was equal to 61, while in the other cases m was equal to 16. It can be seen from the table that in the latter three cases the best test point found lies at a slightly greater specific power output value than the supposed optimum value, while in the first case the supposed optimum was the best value. Overall, this work confirms beyond reasonable doubt that the supposed optimum values found lie very close to a true local optimum in specific power output. Unfortunately, there is no easy way to test whether this is the global optimum.

(8.4) Further Aspects of the Filling and Emptying Piston Engine Modelling

(8.4.1) Increasing the Bore/Stroke Ratio of the Naturally Aspirated Engine

Time precluded the optimisation of the bore/stroke ratio for the naturally aspirated engine. To give some indication of the importance of this parameter, curves of specific power output versus piston engine speed were generated for three bore/stroke ratios: 1:1, 2:1 and 3:1 at a cylinder size of 125cc. These curves are illustrated in fig (8.11). The valve and ignition timings used were as optimised in case (I-2ii). It is seen that the maximum specific power output increases with increasing bore/stroke ratio, though the increment in specific power output when the bore/stroke ratio is increased from 2:1 to 3:1 is small. This suggests that a bore/stroke ratio of 3:1 at the given cylinder size is close to the optimum. The results at the maximum specific output points are summarized in table (8.11).

(8.4.2) Some Adjustments to the Compound Turbocharged Engine Model Input Parameters

As noted in section (8.3.3), most of the optimisations of the filling and emptying model of the compound turbocharged

274 engine resulted in a slightly negative surge margin. An iterative process was therefore used to correct to a positive surge margin, varying the compressor size but keeping all other parameters constant. In addition, the fuel injection timings were revised to ensure that no fuel was lost through the exhaust valves. The adjusted figures are shown in table (8.12). The specific power output figures are slightly reduced from the relevant optimised case, which confirms that the optimisation procedure is functioning correctly.

Numerical experiments have confirmed that operating an engine at a high degree of supercharge but at reduced piston engine speed increases the thermal efficiency at the expense of a reduced specific power output. For example, for case (V-2) the piston engine speed was reduced to 11,500 rpm, the bore/stroke ratio reduced to 2:1 and the compressor area ratio was adjusted to maintain a similar compressor pressure ratio. The simulated specific power output fell from 1274 kW dm-3 to 686 kW dm 3 (a 46% reduction) but the thermal efficiency figure rose from 24.7% to 28.6% (a 16% increase). By increasing the ratio of the indicated mean effective pressure to the friction mean effective pressure and hence increasing the piston engine mechanical efficiency, a high degree of supercharge at low piston engine speeds, e.g. 3000 rpm, is likely to give net thermal efficiency figures in excess of 30%. However, in this case, the maximum specific power output will be much reduced.

The nominal overall volumetric compression ratio figure of 10:1 (excluding the density change in the aftercooler), which was used in the compound turbocharged filling and emptying model engine simulations, is considered to be very conservative. This is for two reasons,

275 aftercooling reduces the temperature at the end of the compression process, the projected valve design is likely to give a more detonation resistant combustion chamber than when poppet valves are used.

For example, the turbocharged road car engine described in (2) uses an equivalent compression ratio of 12.5:1. The highly turbocharged racing engines described in (3) use an equivalent compression ratio of 20:1, though this is with a special high octane fuel. Ricardo (5) notes that the use of sleeve valves, as opposed to poppet valves, permits a 20% increase in volumetric compression ratio. Experiments to determine the effect of increasing the compression ratio proved inconclusive. For case (V-2), increasing the nominal overall compression ratio to 20:1 gave a slight reduction in simulated specific power output and a slight gain in thermal efficiency.

(8.4.3) Part Load Efficiency Contours

Part load efficiency contours were generated for the filling and emptying models of the naturally aspirated and compound turbocharged engines. The efficiency characteristics for the simulated naturally aspirated engines are shown in figs (8.12a) and (8.12b) for three different bore/stroke ratios of 1:1, 2:1 and 3:1. The valve and ignition timings are as optimised in case (I-2ii). For the compound turbocharged engine the relevant data is shown in figs (8.13a) and (8.13b). This simulated engine is as for case (V-2), but the piston engine speeds are reduced and the compressor speed adjusted accordingly to give a positive surge margin. This then results in a reduction of compressor pressure ratio at reduced piston engine speed. The efficiency contours for the naturally aspirated engine are seen to rise with reduced power output and piston engine speed, as is to be expected from the reduction in mechanical losses. The peak thermal

276 efficiency at low engine speed and full throttle opening is of the order of 30%. The compound turbocharged engine simulations had been expected to give a similar rise in thermal efficiency at part load. Instead, the efficiency characteristic shown gives a reduced efficiency at low power outputs, e.g. 16% at 4% load or 5,000 rpm verses 25% at 100% load or 20,000rpm. This was found to be due to the fact that at low piston engine speeds the optimised valve timings for high piston engine speeds gave an extremely large amount of excess air, e.g. 480% excess air at a piston engine speed of 5,000 rpm. The excess air in turn entails a reduction in thermal efficiency, there being a loss associated with pumping this air through the engine. This is confirmed by the fact that at low simulated piston engine speeds the rotodynamic power fraction took a large negative value. Tests have shown that more conventional valve timings with reduced overlap enable the thermal efficiency to approach 30% at low piston engine speeds. To enable a full part load efficiency contour to be generated using variable valve timings would entail the use of the optimisation algorithm to maximise thermal efficiency at several different piston engine speeds. Time has precluded this approach. Nevertheless, it can be inferred that variable valve timing would be a significant advantage for the compound turbocharged engine, allowing maximum specific power outputs at high piston engine speeds and maximum thermal efficiency at low piston engine speeds.

277 (8.4.4) Further Data from the Compound Turbocharged Engine Model

For case (V-2), additional graphical output is provided as follows,

FIGURE ILLUSTRATED FEATURE (8.14) Pressure volume diagram for cylinder one (8.15a) Cylinder one inlet mass flow rate versus crank angle (8.15b) Cylinder one exhaust mass flow rate versus crank angle (8.16a) Compressor mass flow rate versus cylinder one crank angle (8.16b) Turbine mass flow rate versus cylinder one crank crank angle (8.17) Specific power output versus piston engine speed

Figure (8.14) shows that the thermodynamic cycle approximates well to one of constant volume combustion, while the cylinder pressure drops rapidly upon opening of the exhaust valve to give a very small pumping loop. Figures (8.15) show that the optimised valve timings give a very small amount of reverse flow through the inlet valve and zero reverse flow through the exhaust valve. It can be seen from figure (8.16a) that the compressor mass flow rate is relatively constant, while the turbine mass flow rate shown in fig (8.16b) has more variation. From fig (8.17) it is seen the power output drops rapidly when the piston engine speed is reduced from the optimum, with the rate of power reduction lessening as the piston engine speed is further reduced.

(8.5) Comparing the Gas Turbine and Piston Engines' Specific Power Output Figures

The specific power outputs, in units of power output per unit engine mass (kW kg-1), of the various engine types simulated were compared. The results are given in table

278 (8.13), though it should be borne in mind that these results are very approximate. All figures exclude the transmission weight. The specific power output figures used for the compound turbocharged engines are those given in table (8.12), while the figure used for the naturally aspirated engine is based on the maximum simulated specific power output for an engine of bore/stroke ratio 3:1, as given in table (8.11). Compressor and turbine mass are estimated from data in (6). The combined compressor and turbine mass are scaled according to the compressor area, giving a combined mass of 10kg for a peak mass flow rate of 0.4 kg sec 1, at which the compressor area ratio has been defined as unity. The physical compressor area of a compressor of area ratio of unity is 80.1 cm 2, allowing the maximum specific power output values given in figs (8.1a), (8.1b), (8.4a) and (8.4b) to be converted from kW cm 2 to kW kg-1. The heat exchanger masses for the gas turbine are taken from (7), which gives masses of 5kg and 30kg for heat exchanger effectiveness values of 0.5 and 0.95 respectively, for a compressor area ratio of unity. For the piston engine mass calculations, data from (2) was used to give a specific engine weight of 82 kg dm 3. To this were added the relevant compressor and turbine masses according to the projected compressor area ratio at a swept engine capacity scaled to -3 1 dm .

The data shows that the minimum specific power output figure is given by the naturally aspirated piston engine, and the maximum by the compound turbocharged engine with high compressor pressure ratio. The various gas turbine engine configurations and moderately supercharged compound turbocharged engines lie between these two extremes.

279 (8.6) References

(1) Kinoshita, M. ; Shiga, S. ; Hirai, T. ; Sugihara, K. ; Fukuhara, T. Development of a New-Genration High Performance 4.5 Liter V8 Nissan Engine SAE Trans., Section 3, 900651 (SP-823), 1990

(2) Bloomfield, J.H. ; Wood, S.P. Clean Power - Lotus 2.2Lt Charge Cooled Engine SAE Trans., Section 3, 900269, 1990

(3) Otobe, Y. ; Goto, 0. ; Miyano, H. ; Kawamoto M. Aoki, A. ; Ogawa, T. Honda Formula One Turbo-charged V-6 1.5L Engine SAE Trans., 890877, 1989

(4) Private Communication Lotus Engineering, Hethel, Norfolk, England 1991

(5) Ricardo, H.R. The High-Speed Internal-Combustion Engine Blackie and Son Ltd., 4th ed., 1953

(6) Pullen, K.R. A Case for the Gas Turbine Series Hybrid Vehicle Battery, Electric and Hybrid Vehicles, I. Mech. E. Conference Proceedings, 10-11 December 1992

(7) Baines, N.C. ; Panting, J.R. ; Etemad, M.R. ; Besant, C. A Gas Turbine-electric Vehicle Concept I. Mech. E. paper C389/031, FISITA conference, Total Vehicle Dynamics, vol. 2, pp27-32 1992

280 Fig (8.10 Part Load Thermal Efficiency versus Specific Power Output for Single Shaft Gas Turbine 0.4

(-) ncy, ie ic ff l E rma he T

0.0- I I I I I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Specific Power Output, (kW/[square cm] Max TET is 1200K (Fig 8.1b) Part load Thermal Efficiency versus Specific Power Output for Single Shaft Gas Turbine 0.5

) (- iency, ic l Eff Therma

0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Specific Power Output, (kW/[square cm] ) Max TET is 1600K (Fig 8.2) TET versus Specific Power Output for the Single Shaft Gas Turbine 1600

%...... '—' 1_7 1200 — 1 co" I-- (4 1100 —

1000 —

900 —

800 1 I I 0.0 01.1 I 0.2 I 0.3 I 0.41 I 0.5 I 0.6II I 0.7 I 0.8It1 I 0.9 I 1.0 I 1.1 1.2 Specific Power Output, (kW/[square cm] )

(Fig 8.3) Compressor Pressure Ratio versus Specific Power Output for Single Shaft Gas Turbine 5.0

t/3 CDC13 Ot 4.5

C/9 4.4 4.3 E 4.2 4.1 0.0 0.1 01.2 01.3 01.4 01.5 01.6 01.7 01.8 0.91 I 1.01 1.11 1.2 Specific Power Output, (kW/[square cm] ) Fig (8.4a) Part Load Thermal Efficiency versus Specific Power Output for the DGT 0.4

0.0 0.0 0.1 0.2 0.3 I 0.4 I 0.5 I 0.6 0.7 Specific Power Output, (kW/[square cm] j Max TET is 1200K Fig (8.4b) Part Load Thermal Efficiency versus Specific Power Output for the DGT 0.5

-) ( cy, ien ic Eff l Therma

0.0i iI 11111111i 0.0 0.1 0.21 0'.3 01.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 12 Specific Power Output, (kW/[square cm] Max TAT is 1600K Turbine Entry Temperature, (K) 1400 - 1000 - 1200 1700 1300 - 1500 - 1600 - 1100 - 900 0.0

Fig (8.5)TETversusSpecificPowerOutputfor 0.2 0.30.4O Specific Power Output,(kW/[squarecm] the DifferentialGasTurbine I .5 O I .6 O Max. TET1200K I .7 0.80 Max. TET1600K 1 .9 1.0i11.2

Compressor Pressure Ratio, H 5 4 3 2 1 00 0.10.20.30.40.5 0.6 0.70.80.91.01.112

Specific PowerOutputforDifferentialGasTurbine Fig (8.6)CompressorPressureRatioversus Specific Power Output,(kW/[squarecm]

Fig (8.7a) Non-Dimensional Turbine Speed versus Non-Dimensional Compressor Speed for the DGT 1.0

-) 0.9 - ( d, e e 0.8 - Sp ine

b 0.7 - Tur l 0.6 - iona 0.5 - imens D - 0.4 Non

0.4 0.5 0.6 0.7 0.8 0.9 10 Non-Dimensional Compressor Speed, (-) Max TET is 1200K Fig 03.7b) Non-Dimensional Turbine Speed versus Non-Dimensional Compressor Speed for the DGT 1.0

)

(- 0.9 - d, ee

Sp 0.8 - ine b

Tur 0.7 - l iona 0.6 - 00 ens 0 im D

- 0.5 - Non 0.4 I " I I I I I I I I I I I I 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Non-Dimensional Compressor Speed, (-) Max TET is 1600K

Fig (8.8a) Non Dimensional Turbine Speed versus Specific Power Output for the DGT 1.0

0.9 - H d, ee 0.8 - Sp ine b 0.7 - Tur l 0.6 - iona s 0.5 - imen

D 0.4 Non

0.3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Specific Power Output, (kW/[square cm] Max TET is 1200K Fig (8.8b) Non Dimensional Turbine Speed versus Specific Power Output for the DGT 1.0

H 0.9 - d, ee Sp

0.8 - e in b

Tur 0.7 - l

iona 0.6 - ens im D

0.5 - Non

0.4 i ii 0.0 0.1 0.2 0.31 0.51 0.61 0.91 1.0 1.1 1.2 Specific Power Output, (kW/[square cm] Max TET is 1600K Fig (8.90 Non-Dimensional Gross Turbine Torque versus Specific Power Output for the DGT

1.0

(-) 0.9 - ue, rq 0.8 - To e 0.7 - in b r 0.6 - Tu 0.5

Gross 0.4- l 0.3- iona s 0.2- imen

D 0.1-i -

Non 0.0 IIIIIIIIIIIII 0.0 0.1 0.2 0.3 0.4 0.5 0.6 07 Specific Power Output, (kW/[square cm] j Max TET is 1200K

Fig (8.9b) Non-Dimensional Gross Turbine Torque versus Specific Power Output for the DGT

10-1 1.0

a) 0.9

Cr

0 0.8

0) 0.7 0.6

C/3 0.5 0 0.4 a 0.3 a) 0.2 E 0.1 - 0 0.0 0.0 0.1 IIOI.2 OI.3 OI.4 OI.5 O I.6 0.71 OI.8 OI.9 1.0I 1.1 1.2 Specific Power Output, (kW/[square cm] ) Max TET is 1600K tr.ctu-tArrE -res-rpoikoS /NJ

F(6 ($.1o) 1i-4e LASE- of- A THeee oini&r\gioNAL

02T+-loGroNAL G alio To V At.- i GATE A

"rfrl fer56. 0 Irn61,4Stotugrt- 0 PTI rn Sprr so 1,J

295 Fig (8.11) Specific Power Output versus Piston Engine Speed for a Naturally Aspirated Engine 60 E 50 — -CD= bore/stroke=1 40 4-7 ci_ t.) 30

0 20

0 cr) 10 2000 4000 6000 81100 10600 12000 Piston Engine Speed, (rpm) Fig (8.12a) Thermal Efficiency versus Piston Engine Speed for a Naturally Aspirated Engine 0.35

0.30

-)

( 0.25 iency,

ic 0.20

Eff bore/stroke= l

rma 0.15 - The 0.10 -

0.05 2000 40'00 60'00 80'00 r 10600 12000 Piston Engine Speed, (rpm) Fig (8.12b) Thermal Efficiency versus Specific Power Output for a Naturally Aspirated Engine 0.35 -

0.30 -

›: 0.25 - Li 9a.

0.20 -

0 3 -6 0.15 -

0.10 -

0.05 10 210 k 40 0 60 Specific Power Output, (kW/[cubic dm] Fig (8.13a) Thermal Efficiency versus Piston Engine Speed for a Compound Turbocharged Engine 0.25

E

0.15 5 12 13 116 117 18 19 20 21 Piston Engine Speed, (krpm) Fig (8.13b) Thermal Efficiency versus Specific Power Output for a Compound Turbocharged Engine 0.25

E

0.15 1 i 0 200 I 460 I 660 I 860 I 1000 I 1200 1 1400 Specific Power Output, (kW/[cubic dm] j Fig (8.14) Cylinder Pressure versus Non-Dimensional Cylinder Volume 120

100 -

) r ba

( 80 -

60 - Pressure,

der 40 - in l Cy 20 -

0 0.3 I 0.4 ' 0.5 I 0.6 I 0.7 I 0.8 I 0.9 10 Non-Dimensional Cylinder Volume, (-) Fig (8.15a) Cylinder One Inlet Mass Flow Rate versus Crank Angle 0.4

) 0.3 /sec kg

( 0.2 te, Ra 0.1 - Flow

Mass 0.0 -

-0.1 Ii 0 160 1 360 1 540 7k Crank Angle, (degrees) Fig (8.15b) Cylinder One Exhaust Mass Flow Rate versus Crank Angle 0.6

0.5 -

)

/sec 0.4 - kg (

te, 0.3 - Ra

0.2 Flow s Mas 0.1

0.0 0 io 1 360 1 540 720 Crank Angle, (degrees) Fig (8.16a) Compressor Mass Flow Rate versus Cylinder One Crank Angle 0.355

-153 0.350

0 0 U- U) U)

0.345 0 160 360 540 720 Crank Angle, (degrees) Fig (8.16b) Turbine Mass Flow Rate versus Cylinder One Crank Angle 0.45

) 0.40 /sec kg (

te, 0.35 Ra Flow 0.30 - Mass

0.25 160 1 360 1 540 0 Crank Angle, (degrees) Specific Power Output, (kW/[cubic dm 1000 - 1200 1300 1100 - 500 - 300 - 200 - 400 - 700 - 600 - 800 - 900 - 100 - 0

5

a Engine SpeedforaCompoundTurbocharged Fig (8.17)SpecificPowerOutputversusPiston -

e 6 -io-i-i121314-616 Piston EngineSpeed, (krpmJ 1 1 7 lb192021 item item description value cylinder size, c.c. 561.77 bore/stroke ratio 1.125 volumetric compression ratio 10.2:1 inlet valve area/ cylinder head area, I; 33.39 exhaust valve area/ cylinder head area, 96. 25.18 equivalence ratio 0.98 (e) ignition timing B.T.D.C.,° 21.9 (e) valve timings inlet valve opens B.T.D.C., ° 0 inlet valve closes A.B.D.C.,° 68 exhaust valve opens B.B.D.C.,° 60 exhaust valve closes A.T.D.C.,° 8 specific power output, kW dm-3 @ 6000rpm 46.2 comparable quasi-steady model specific power output, kW dm-3 @ 6000rpm 51.0 comparable filling and emptying model specific power output, kW dm-3 @ 6000rpm 40.3

Table (8.1) Comparisons Between the Specific Power Output of a 4.5 Litre V8, Naturally Aspirated Nissan Engine and the Equivalent Simulated Values

(e) - estimated value

307

item item description value

cylinder size, c.c. 543.5 bore/stroke ratio 1.25:1 cylinder volumetric compression ratio 8:1 compressor pressure ratio ©6500rpm 1.9:1 charge cooler effectiveness, %, @6500rpm 69 inlet valve area/ cylinder head area, 96 22.4 (e) exhaust valve area/ cylinder head area, % 16.4 (e) equivalence ratio 0.98 (e) ignition timing, B.T.D.C.,° 28.6 (e) valve timings inlet valve opens B.T.D.C., ° 12.7 (e) inlet valve closes A.B.D.C.,° 34.1 (e) exhaust valve opens B.B.D.C.,° 43.5 (e) exhaust valve closes A.T.D.C.,° 9.3 (e)

specific power output/ thermal efficiency / turbine entry temperature,all @ 6500rpm -3 kW dm / 96' / K 91/22/1253 comparable quasi-steady model figures (minus compound work) 73/19/1200 comparable filling and emptying model figures (minus compound work) 82/26/1314

Table (8.2) Comparisons Between the Performance of a 2.2 Litre Conventionally Turbocharged Lotus Four Cylinder Engine, and the Equivalent Simulated Values

(e) - estimated value

308 analytical numerical solution results

minimum * value, f(x ) -14.0000000 -14.0000000

vector at * minimum, x * 1.0009 x1 1.00000 * 1.00000 0.99998 x2 * 1.00000 0.99946 x3 * x4 1.00000 1.0001 * X 1.00000 0.99994 5 * 1.00000 0.99996 x6 * x7 1.00000 1.0001

Table (8.3) Comparisons Between the Analytically Calculated Minimum of a Quadratic Function and the Equivalent Calculated Numerical Values

309 case no. compressor aftercooler overall pressure effectiveness compressive ratio 96 volumetric compression ratio

I-1 - - 10:1 II-1 1.9 70 10:1 III-1 4.0 0 10:1 IV-1 4.0 70 10:1 V-1 7.5 70 10:1 VI-1 7.5 70 -

In all cases, the ambient pressure was 1.0132 bar, the ambient temperature was 288K, and in cases II-1 to VI-1 the compressor and turbine polytropic efficiencies were 8096

Table (8.4) The Fixed Input Parameters used for the Quasi-steady Simulation Model Optimisations

310 case equivalence piston nominal no. of no. ratio engine overall line speed compressive mins. rpm volumetric compression ratio

I-1 0.98161 8310.4 - 6 11-1 0.98182 12238.0 - 4 111-1 0.98182 14589.0 - 6 IV-1 0.98122 17775.0 - 5 V-1 0.98211 24150.0 - 4 VI-1 1.0051 24625.0 5.5 2

Table (8.5) The Optimised Input Vectors and Number of Line Minimisations for the Quasi-Steady Simulation Model Optimisations

311 output case no. parameter I-1 II-1 III-1 IV-1 V-1 VI-1

specific power output, kW dm-3 56.6 169.3 286.2 502.9 1232 1360 TET, K - 1310 1657 1591 1856 2189

t o r 0

4. I-

0 E

r 1-- r

1:1 ct ( P- i

r Fi o 0 - 186 785 724 -1305 -382 thermal effcy., % 20.5 22.3 20.3 21.3 18.8 10.4 BMEP, bar 8.2 5.8 2.7 4.1 -5.1 -23.5 piston engine mech. effcy., % 59.0 36.1 16.1 17.5 -18.2 -205

Table (8.6) Output Parameters from the Quasi-Steady Model Optimisations

312

case no. I-2i I-2ii 11-2 111-2 IV-2 V-2 input parameter

equivalence - - 0.98 - - 0.98 ratio valve timings IVO, BTDC,° 15 - - - - - IVC, ABDC,° 50 - - - - - EVO, BBDC,° 55 - - - - - EVC, ATDC,° 10 - - - - - piston engine speed, rpm 6500 - - - - - cylinder capacity, cc 125 125 125 125 125 40 manifold sizes, cc 1000 1000 1000 1000 1000 300 bore/stroke ratio 1:1 - 1:1 1:1 1:1 - compressor speed, % - - 57 90 90 120 aftercooler effective- ness, % - - 70 0 70 70 no. of dimensions of variable input vector 2 7 9 9 9 9

In all cases, the ambient pressure was 1.0132 bar, the ambient temperature was 288K and the nominal overall volumetric compression ratio was 10:1

Table (8.7) The Fixed Input Parameters used for the Filling and Emptying Model Optimisations

313 case no.

I-2ic I-2iia II-2a III-2a IV-2 V-2 optimised input parameter

equivalence 0.9804 0.9867 - 0.9860 0.9832 - ratio ignition timing BTDC,° 20.4 21.9 25.0b 31.0 32.6 28.6b valve timings IVO, BTDC,° - 12.7 66.0 62.8 68.6 76.2 IVC, ABDC,° - 30.9 28.5 42.8 50.0 33.1 EVO, BBDC,° - 43.3 18.1 78.2 82.9 70.2 EVC, ATDC,° - 10.5 33.6 111.3 97.3 116.1 piston engine speed, rpm - 7314 12124 11176 11556 20040 bore/stroke ratio - - 7:1 - - 4.33:1 compressor area ratio - - 1.7 0.74 0.99 0.66 turbine 2 area, cm - - 7.09 6.13 7.80 6.39 total no. of line mins. 26 15 16 40 32 17

Table (8.8) The Optimised Input Vectors and Corresponding Number of Line Minimisations for the Filling and Emptying Model

(a) not a fully convergent solution (b) fixed combustion duration period of 60° employed (c) small relative convergence tolerance of 10-5 employed

314 output case no. parameter I-2i I-2ii 11-2 111-2 IV-2 V-2 specific power output, kW dm-3 43.9 47.2 169.6 239.3 327.0 1274 TET, K - - 1294 1508 1471 1509 roto- dymamic power/ piston engine power, % - - 21.4 54.8 44.5 139 thermal effcy., % 24.5 22.5 22.2 22.7 23.7 24.7 compressor pressure ratio - - 2.0 4.2 4.2 7.1 turbine pressure ratio - - 2.0 3.4 3.4 5.7 BMEP, bar 8.1 7.8 13.8 16.6 23.5 31.9 volumetric effcy., % 92.2 95.3 136 137 125 182 piston engine mech. effcy., % 66.6 62.7 57.7 65.1 71.5 57.0 surge margin, % - - -6.0 -1.8 -4.6 9.0 mean piston speed, ms-1 11.7 13.2 6.0 20.1 20.8 9.4 specific work output, J kg-1 717 664 562 603 643 582 compr. area/ swept volume, - 1 cm - - 0.27 0.12 0.16 0.33 cylinder head area/swept vol. - 1 cm 0.185 0.185 0.676 0.185 0.185 0.716 turbine choke area/swept vol. - 1 cm - - 0.014 0.012 0.016 0.040 Table (8.9) Simulated Output Parameters for the Filling and Emptying Model Optimisations

315 max. error case * no. ( f - max f 1 s6,, f* )

I-2i (2D) -0.034 I-2ii (7D) +0.020 111-2 (9D) +0.028 IV-2 (9D) +0.036

Table (8.10) Maximum Detected Errors in the Filling and Emptying Model Optimisations

316 simulated bore/stroke ratio output parameter 1:1 2:1 3:1

specific power output, kW dm-3 47.2 51.0 52.0 piston engine speed, rpm 7314 8000 8000 thermal effcy.,% 22.5 21.8 22.1 mech. effcy.,96. 62.1 58.7 59.2

volumetric effcy., 96 95.3 97.3 98.0

Table (8.11) The Maximum Specific Power Output for a Simulated Naturally Aspirated Engine for Three Different Bore/Stroke Ratios

317 case no. adjusted adjusted adjusted adjusted compressor specific thermal surge area power effcy., margin, ratio output, % kW dm-3

11-2 1.18 162.1 21.9 2.9 111-2 0.654 232.1 22.7 5.7 IV-2 0.831 315.1 23.6 5.7 V-2 - 1263 24.9 -

Table (8.12) Adjusted Figures for the Compound Turbocharged Engine Filling and Emptying Model

318 estimated engine specific type power output, kW kg-1

gas turbine max TET=1200K, c=0.95 1.3 max TET=1600K, c=0.95 2.4 max TET=1200K, c=0.5 3.3 max TET=1600K, c=0.5 6.3

naturally aspirated piston engine 0.6 compound turbocharged piston engine r = 2.0:1, c=0.7 1.5 P c r = 4.2:1, c=0.0 2.5 P c r = 4.2:1, c=0.7 3.2 P c r = 7.1:1, c=0.7 10.4 P c

Table (8.13) Comparisons of the Maximum Specific Power Outputs of Various Engine Types

c = heat exchanger/ aftercooler effectiveness r = compressor pressure ratio P c

319 Chapter (9) Conclusions and Suggestions for Further Work

(9.1) Conclusions (9.1.1) The Gas Turbine Simulations

Iterative turbine and compressor matching procedures demonstrate the potential of the two shaft differential gas turbine plus heat exchanger. Provided that a heat exchanger of high effectiveness is fitted, the simulated net thermal efficiency verses power output characteristic of this unit is excellent. For example, with a heat exchanger effectiveness of 90% and TET of 1600K, the brake thermal efficiency of this unit is greater than or equal to 40% from 20% to 100% of maximum power output. The peak brake thermal efficiency is 47%. This is against the accepted doctrine that small gas turbines always demonstrate poor part load efficiency levels. Some allowance would have to be made, however, for the incorporation of a realistic level of transmission efficiency, currently assumed to be 100%. The estimated specific output of this engine is 3.3 kW kg-1, including the high effectiveness heat exchanger but excluding the transmission. This compares with a figure of approximately 0.6 kW kg-1 for a highly rated naturally aspirated spark ignition engine, excluding transmission. With a low to moderately effective heat exchanger (e.g. with an effectiveness of 0.5), the differential gas turbine showed little advantage over the single shaft, constant speed gas turbine. The transient response of the differential gas turbine is likely to be poor, the gross torque verses speed characteristic of the turbine showing an almost linear reduction with turbine speed. Overall, the differential gas turbine is an attractive low weight, high thermal efficiency, low emissions unit, subject to reservations concerning transient response.

320 (9.1.2) The Compound Turbocharged Engine Simulations

The compound turbocharged spark ignition engine, time - marching, filling and emptying model also demonstrates that this unit is a potential low weight device with good thermal efficiency levels across a broad power output range. For example, with a turbine entry temperature of 1500K, the estimated maximum specific output of this engine with an aftercooler of effectiveness 70% currently stands at 1263 kW dm 3 or = 10.4 kW kg 1 excluding transmission. The latter figure is subject to errors in the estimation of the specific engine weight in units of kg dm\-3. Brake thermal efficiency is = 25% at maximum power output, falling to 16% at low load with fixed valve timings, with a corresponding figure of approximately 30% with variable valve timings, assuming a transmission efficiency of 100%.

The high turbine entry temperatures employed in the simulations are essential in order to achieve the high specific outputs given in the last paragraph. Reducing the turbine entry temperature to = 1300K reduces the optimised specific power output to = 170 kW dm-3 or 1.5 kw kg-1. Thus, for a small engine, the use of a ceramic turbine would be required to realise the high specific outputs. For a large engine, e.g. of several thousand kilowatts;, a cooled nickel alloy turbine could be used. It should be noted, however, that the use of volumetric compression ratios higher than 10:1 has been found to reduce the TET for a given compressor pressure ratio and fuel/air ratio.

The value of compounding coupled with high turbine entry temperatures is also seen, the turbine/compressor unit adding = 140% to the overall power output for the aftercooled engine at the turbine entry temperature of 1500K. Thus, the maximum specific output of the piston engine in isolation is 526 kW dm-3, and its brake thermal

321 efficiency = 10%. This compares with specific outputs of conventionally turbocharged formula one racing engines of approximately 750 kW dm-3, with a compressor pressure ratio limited to 4:1.

A major problem with such an engine type is likely to be that of ensuring sufficient engine longevity. As noted in the previous paragraph, the specific output of the piston engine itself is simulated as being less than that of conventionally turbocharged formula one engines of some years ago. This suggests that the overall projected specific outputs are practically achievable. However, racing engines only have to last for a few hours before a complete rebuild. It is not known whether an engine with such high specific outputs could be made to last, for example, for tens of thousands of hours between rebuilds. Some compromise on specific output might be necessary in order to achieve sufficient engine longevity. However, the factor limiting engine life is likely to be fatigue. Therefore the use of heavier and stronger components may ensure sufficient engine longevity.

There are several features which presently limit the accuracy of the filling and emptying model. These are the calculation of the in - cylinder heat release rate during combustion, the calculation of the rate of heat transfer from the cylinder walls, the estimation of the allowable cylinder volumetric compression ratio as dictated by the detonation limit, the estimation of the valve port mass flow rate and the calculation of the friction mean effective pressure. These errors are particularly large in this application as the empirical models have been extrapolated to simulations at high piston engine speeds and large bore/stroke ratios for which the validity of these models is questionable. These calculations could be improved in accuracy with further development of the empirical models. Alternatively, the more complex approach of solving the

322 Navier - Stokes partial differential equations may be the route forward for improving simulation accuracy. Thus, the performance figures presently given should be seen as order of magnitude estimates rather than precise values.

The compound highly turbocharged unit is unlikely to suffer from such severe transient response problems as the differential gas turbine, but against this emissions levels of the basic unit will be higher. However, it is supposed that auxiliary combustion at low temperatures may provide emissions levels that are competitive with the gas turbine. Alternatively, it may be possible to run the engine fuel lean with the hybrid Aspin/sleeve valve considered suitable. For further discussion of the use of an auxiliary combustion chamber and the use of hybrid sleeve/Aspin valves refer to chapter (4).

The compound highly turbocharged engine is thus demonstrated as an attractive proposition when high power to weight ratio is a prime requirement. Though initially considered as a potential automotive powerplant, it is also believed this type of unit could have applications in the aeronautical field, as an alternative to small to medium sized turboprops.

(9.1.3) Comparisons Between the Gas Turbine and Compound Turbocharged Engine Simulations

In all cases, higher TETs gave higher specific power outputs, and in the case of the gas turbine, gave higher thermal efficiency. It will be assumed here that TETs of the order of 1600K for uncooled turbines will become technically feasible in the medium term future. In this case, the overall simulation results may be summarised as follows,

(i) if high thermal efficiency at both full and part load is a priority, the differential gas turbine with high

323 effectiveness heat exchanger is the best choice and gives a maximum specific power output figure of the order of 3.3 kW kg-1 at a pressure ratio of 5:1 and TET of 1600K, 5.5 times as great as that of a conventional spark ignition, naturally aspirated piston engine. (ii) If thermal efficiency at full load is a priority, but the part load efficiency is of less significance, then the single shaft, constant speed gas turbine with high effectiveness heat exchanger is the best choice. For the same operating parameters, this engine has a virtually identical maximum specific power output as for case (i). (iii)If maximum specific power output is the main priority, the compound highly turbocharged engine is the best choice. This engine is predicted to give a maximum specific power output figure of the order of 10.4 kW kg-1 with a compressor pressure ratio of 7.1:1 and a TET of approximately 1500K. This specific power output figure is in turn 17.3 times as large as that of a conventional spark ignition, naturally aspirated piston engine.

As far as point (iii) is concerned, the single shaft gas turbine and differential gas turbine, both with a moderately effective heat exchanger of effectiveness 0.5, give very similar part load efficiency characteristics to the compound highly turbocharged engine with fixed valve timings optimised to give maximum specific output. However, the gas turbines, with this moderately effective heat exchanger, have an inferior maximum specific power output of 6.3 kW kg-1. Further, if the compound turbocharged engine can be fitted with variable valve timings then its part load efficiency characteristic will be superior. Against this, the single shaft gas turbine is a much simpler machine than the compound turbocharged engine.

324 The transmission weight of the compound turbocharged engine will be higher if one alternator is directly geared to the piston engine. This is because the rotational speed of this alternator will be approximately that of a comparable gas turbine alternator. The ideal solution would be to gear up an alternator, from the piston engine, to run at high speed, e.g. 100,000 rpm. It is not known whether this is technically feasible.

(9.1.4) The Optimisation Procedure

The multi - dimensional optimisation routine was demonstrated as an effective engine design tool. For example, applied to the unaftercooled compound turbocharged engine, the simulated power output rose from an initial estimate of 149 kW dm 3 to a final optimised value of 3 239 kW dm at a compressor pressure ratio of 4:1. This value would rise further if the volumetric compression ratio, bore/stroke ratio and the manifold sizes were also optimised. The optimisation procedure is thus extremely useful in determining suitable engine operating parameters, rather than the values of these being estimated semi - arbitarily. Such an algorithm could be particularly useful when applied to real prototype engines, as opposed to simulated engines.

One problem with the optimisation algorithm is that of lengthy overall execution time. Basic simulation time for the function being evaluated must be limited to no more than a few minutes for the optimisation algorithm to be practically applied. Even so, a seven dimensional optimisation, for example, will take of the order of 48 hours when applied to a function that takes 5 minutes to execute.

325 (9.2) Suggestions for Further Work

(9.2.1) Modifications to the Filling and Emptying Method

Other than modifications to the empirical methods employed, described in the next subsection, there are considered to be two main modifications required for the filling and emptying models. These are,

(i) the use of an improved step control method, (ii) a more accurate estimation of the combustion efficiency.

Item (i) is intended to give an algorithm that can guarantee numerical stability under all simulated operating conditions, and may allow a less approximate mass flow rate verses pressure ratio characteristic to be used. The present characteristic was modified from the analytic form to give a more stable algorithm.The use of a better step control method would also permit the simulation to operate more satisfactorily at low piston engine speeds, small cylinder sizes and large bore/stroke ratios. At present, it is not entirely clear the form that the improved step control algorithm would take.

As far as item (ii) is concerned, it has been found that the empirical combustion efficiency characteristic employed is inaccurate. It is proposed that the combustion efficiency calculations could in future be based upon the direct calculation of the change in overall formation enthalpy assuming an equilibrium gas. In this manner, dissociation energies are accounted for. The rate of change of overall formation enthalpy with temperature and equivalence ratio are then additional terms which must be included into the energy flow ordinary differential equation numerical integration scheme.

326 (9.2.2) Empirical Measurements

The filling and emptying model relies upon a number of empirical calculations. These are,

(i) the mass fraction burnt verses crank angle characteristic, (ii) the heat transfer coefficient calculation, (iii)the friction mean effective pressure verses piston engine speed characteristic.

It is suggested that empirical measurements of these variables be made for a test engine operating at high piston engine speeds, large bore/stroke ratio, with hybrid sleeve/Aspin valves and a swashplate type crank mechanism. Using the subsequent data as inputs for the numerical model would then considerably increase simulation accuracy.

(9.2.3) The Detonation Limit

The detonation limit, as given by the maximum permitted compression ratio, is currently very crudely determined. It is proposed that a more sophisticated approach be adopted in the determination of this limit, but it is currently not clear how this could be achieved.

(9.2.4) Increasing the Specific Power Output

Calculations have suggested that the optimum pressure ratio for maximum specific power output for a gas turbine does not give much increase over the specific power output that can be achieved by a gas turbine of 5:1 pressure ratio and a TET of 1600K. For example, the specific power output of a gas turbine with a TET of 1600K, a heat exchanger effectiveness of 0.5 and a compressor pressure ratio of 7:1 is approximately calculated to increase from 6.3 kW kg to 6.9 kW kg-1. The corresponding figure at a compressor

327 pressure ratio of 9:1 is 7.0 kW kg-1. The last small incremental increase suggests that a pressure ratio of 9:1 is close to the optimum for maximum specific power output at the given TET.

However, there are considerable possibilities for increasing the specific power output of the compound highly turbocharged engine. These are,

higher compressor pressure ratios, with a two stage centrifugal compressor and charge cooling between each stage and after the second stage, (ii) an optimised overall compression ratio, (iii)the use of hybrid sleeve/Aspin valves as opposed to poppet valves, (iv) as a possible option, to use two stroke engines.

Higher compressor pressure ratios are likely to increase the specific power output of the engine, measured in kW kg 1. A suitable compressor pressure ratio might be 12:1. Intercooling, as opposed to aftercooling, reduces the total compression work allowing increases in both the specific power output and the thermal efficiency. Figure (9.1) shows the ratio of isothermal compression work to isentropic compression work versus pressure ratio, assuming constant properties and a ratio of specific heats, 7, of 1.4. At a pressure ratio of 12:1, isothermal compression work is approximately 68% of isentropic compression work. A more realistic assessment of the work reduction is however given by considering the work input of two stages of non-isentropic compression with a non-ideal charge cooler between each stage and comparing this with the work input of one stage of non-isentropic compression. Thus, fig (9.2) shows the work ratio of a two stage compression process with the isentropic efficiency of each stage 80% and a charge cooler effectiveness of 70%, compared with a single stage compression at isentropic efficiency 80%. The pressure

328 ratios of each stage of the two stage process are equal (this having been found to be the optimum split to reduce the work input to a minimum) while properties are again assumed constant with 7 = 1.4. According to this data, the compression work is reduced to 89% of that of the single stage compression work at a pressure ratio of 12:1. On the basis of this comparison, two stage intercooling may not give sufficient power and efficiency gains to be worth the extra complexity. It should be noted, however, that two stage intercooling will give a lower final charge temperature, allowing additional specific power output gains and increased overall compression ratios.

Higher overall compression ratios than presently modelled would give higher thermal efficiencies, lower TETs and possibly increased specific power outputs. Alternatively, a lower compression ratio may be the optimum for maximum specific power output. Hybrid sleeve/Aspin valves are considered to be suitable for the development of high volumetric efficiency, high overall compression ratios and for use at high piston engine speeds. Two stroke engines allow higher specific power outputs and a simpler design.

One option that was considered was to place a heat exchanger duct around the turbine exhaust and use this waste heat to power a small gas turbine or steam turbine. For example, in case (V-2) the turbine exit temperature is approximately 1100K. However, when non-ideal heat exchangers are taken into account, approximate calculations showed the extra power gain was small, and not worth the extra complexity.

(9.2.5) The Navier-Stokes Equations

The Navier-Stokes equations applied to piston engine modelling allow increases in algorithm accuracy at the expense of increased complexity and program execution time. The most accurate algorithms assume non-equilibrium gas

329 kinetics. References (1) and (2) describe the Navier-Stokes equations. Such models eliminate the need for the empirical mass fraction burnt verses crank angle curve and the empirical heat transfer coefficient calculations. They also eliminate the rather approximate description of the mass flow rate across valve ports verses pressure ratio, which in turn removes a source of algorithm instability. The development of a Navier-Stokes model of the compound turbocharged engine is a consideration for future work. However, it is uncertain whether such algorithms can be designed to run sufficiently quickly to be used in conjunction with the optimisation routine.

(9.2.6) Development of the Unconstrained Optimisation Algorithm

A brief description of a version of the conjugate gradient algorithm which is claimed to be more efficient than the basic conjugate gradient algorithm for non-quadratic functions is given by (3). This algorithm is known as the Polak-Ribiere conjugate gradient method, and is considered as a possible development of the basic optimisation algorithm. It is described in more detail in Appendix (A16).

(9.2.7) Constrained Optimisation

Constrained optimisation is an extension of the basic unconstrained optimisation technique, as described in chapter (7). Constrained optimisation would permit the following,

(i) optimisation of the compound turbocharged engine specific power output, subject to the TET being at or below an upper limit, (ii) optimisation of the compound turbocharged engine specific power output, subject to the thermal efficiency being at or above a lower limit,

330 (iii)both of the above, (iv) optimisation of the differential gas turbine compressor and turbine speeds, subject to the TET being at or below an upper limit.

It is suggested that a constrained optimisation technique be developed. An outline of a method to achieve this is given in Appendix (A18).

(9.2.8) Optimising Additional Parameters

Three parameters that have not been optimised in the filling and emptying model of the compound turbocharged engine are the overall compression ratio, the inlet manifold volume and the exhaust manifold volume. Optimising such additional parameters would involve no major modification to method or code.

331 (9.3) References

(1) White, F.M. Viscous Fluid Flow McGraw-Hill, 1974

(2) Heywood, J.B. Internal Combustion Engine Fundamentals McGraw-Hill, 1988

(3) Ciarlet, P.G. Introduction to Numerical Linear Algebra and Optimisation Cambridge Univ. Press (English Translation), 1991 p.320

332 Fig (9.1) Isothermal Compression Work Divided by Isentropic Compression Work versus Pressure Ratio 100

90 -

So°Th 80 - 16 0 70 0

60 -

50 0 110"210 30 421 50 Pressure Ratio, (-) Fig (9.2) Two Stage Compression Work Divided by One Stage Compression Work versus Pressure Ratio 100

95 -

0 90 -

L.- 0

85 -

80 0 lb 210 k 4!0 50 Pressure Ratio, (-) Appendices

335 Appendix (A1) The Entropy Function

The entropy function is used to calculate the temperature change for isentropic, adiabatic compression or expansion. For a fluid, a small change of entropy is defined by,

S u S v Ss - T + P T (A1.1)

where s is the specific enthalpy, u the specific internal energy, T the temperature, P the pressure and v the specific volume. This gives,

Sh SP Ss = T v T (A1.2)

by the definition of the specific enthalpy where h is the specific enthalpy, or,

S S P Ss = c p--- TT R P (A1.3)

where R is the specific gas constant and the use of the ideal gas law has been invoked. The specific gas constant is assumed to be constant. The non-dimensional entropy function is defined as,

1 T e(TiO) = j- I c (Tp '0) dT (A1.4) R T ref

where 0 is the equivalence ratio assumed constant during compression or expansion. Then the change of specific entropy divided by the specific gas constant is given by,

As(T,O) - Ate(T,0) - Aln(P) R (A1.5)

336 Thus for isentropic compression and expansion respectively,

Ate(T,O) = ln(r_ ) C

Ate(T,O) = -1n(r ) Pt where r is the pressure ratio during compression and r P c Pt is the expansion pressure ratio. The non-dimensional entropy function is known as a property of temperature and equivalence ratio and therefore equations (A1.6) may be inverted iteratively to find the temperature after isentropic compression or expansion.

337

Appendix (A2) Second Order Interpolation from Tabulated Property Values

The desired property value *(x) is to be found from the three known tabulated values *(xL0), *(xmD) and *(xm) where *() is the property value of interest, x lies between 0, or the XLO and xM and x is either the equivalence ratio, temperature, T. For a diagrammatic interpretation refer to fig (A2.1). The values h, k and j are defined by,

h = x - XLO (A2.1a)

X X HID (A2.1b) k X X L()

X X HI J X X (A2.1c) LO

A second order Taylor's series is used (i.e. third and higher derivatives are neglected) so that,

! *(xL0) = *(x) - *1(x)h + *"(x) 2 (A2.2a) 2h 2 1.7(X),up ) = k *(x) + *1 (x)kh + *"(x) (A2.2b)

*(xm) = *(x) + *1(x)jh + *"(x) 32h22 (A2.2c)

where one prime indicates the first derivative of 6(x) with respect to x, and two primes indicates the second derivative. Equations (2.2) may be written in matrix form,

2 h 2 -h 1 k2h2 kh 1 *(x ) (A2.3) 2 MID j2h2 jh 1 2 *(xHI )

The desired value is i3(x) as a function of k, j, *(xLD),

338 1.7(x MID) and 13-(xM). This will now be obtained from equations (A4.3) by Gaussian elimination. The first step is to subtract k2[row 1] from [row 2] and to subtract j2[row 1] from [row 3] which obtains - - 2 h *" ( x) ) 2 -h 1 t/(xLc

0 kh(1+k) 1-k2 (x) *(xmID ) -k219. (XLO ) 0 jh(1+j) 1-j2 *(x) .2 'a ()CHI ) — 3 'a(xi.o ) (A2.4)

The second step is to subtract [(j(l+j))/(k(1+k))][row 2] from [row 3] in equations (A2.4) which yields,

h M -h 1 15" (x) 0 kh(l+k) 1-k2 (x) _j2) -j(1+j)(1_k2) 0 0 (1 k(l+ k) 1Y(x)

1.1(xL0)

*(xmD ) - k 2flx1,0)

▪ ( jk(l+ j2)*(xL0) k(1+k) 15(xMD) 1+kj) (A2.5)

The third row of equations (A2.5) yields the required value for 15(x). With a little algebra one obtains,

j+1 k (k— j) *(x) — j 1 —*(x1 ) , (x ) 15(x ) ) ( 1+j) (k—j) 3 HI n+1 MD k+1 Lc' (A2.6)

which is the desired result. A final point is that the denominators of equations (A2.lb) and (A2.lc) approach zero

339 as x approaches xL0. Therefore, when x is very close, or at, xLO , a first order interpolation method is used. Since the interpolation distance, h, defined in equation (A2.la), is then very small the use of a first order interpolation over this region represents negligible inaccuracy compared with the use of a second order method. It was realised after the equations had been formulated that the problems of the denominators in equation (A2.lb) and (A2.lc) approaching zero under certain circumstances could be eliminated by expressing the equations in a different form. However, time has not permitted this modification.

340 0Gen-ter r, T-uNg--rioNI, 04'

Aix)

1 NPLL7 X L° X X ii 7 PAf2.AMETh

i Sl utt '. (A 2_. I)

SE conic) o. RDEFL 1 N.)-T 2.e oc-fiTs csr4

341 Appendix (A3) The Calculation of Gas Turbine Performance Assuming Constant Fluid Properties

Making the assumption of constant fluid properties, the thermal efficiency and specific work output of the simple gas turbine and gas turbine plus heat exchanger are calculated. Figure (3.3) gives a schematic illustration of the thermal cycle. Further assumptions are zero total pressure loss in the combustion chamber and that the mechanical and combustion efficiencies are both equal to unity. Making these assumptions it is possible to derive algebraic equations that describe gas turbine efficiency and specific work output.

In this appendix, T denotes the temperature, w denotes the specific work, q denotes the specific heat input, 17 denotes the efficiency, t denotes the cycle temperature ratio and c denotes the isentropic temperature ratio for the given pressure ratio. The subscripts c, t, th, 1, 2, 2', 3, and 4 denote the compressor, the turbine, thermal, the compressor inlet, the compressor outlet, the heat exchanger cold side outlet, the turbine inlet and the turbine outlet respectively.

The simple gas turbine without heat exchanger is first considered. The isentropic cycle temperature ratio is defined as,

7-1

c = r (A3.1) where r is the cycle pressure ratio and 7 the ratio of specific heats. The cycle temperature ratio is defined as,

T t = T3 (A3.2)

342

Temperatures around the cycle are defined in terms of the ambient temperature, T1, as follows,

T2 = T1 1 + — (c - 1)] (A3.3a) C l C

T3 = T1t (A3.3b)

T4 = T1 t [ 1 — Tit ( 1 — --1c- ) ] (A3.3c) where equations (A3.3a) and (A3.3c) invoke the definition of the compressor and turbine isentropic efficiencies respectively. Because the specific heat constant at constant pressure is assumed constant, the thermal efficiency calculation is based on temperature changes only. Thus,

wt —TAT [T3 - T4] - [T2 - Ti] 71 _ c (A3.4) th q T3 — T2 hence,

71 t (1 - —1 ) - --111 (c - 1) t c C n (A3.5) th 1 t — [1 + (C — 1 ) 1 ] c which gives,

(c - 1)(11c7Itt - c) n th — C[Ti(t — 1) + 1 — c] (A3.6)

simple gas turbine

For the calculation of the efficiency of the gas turbine plus heat exchanger, the compressor specific work input and turbine specific work output are unchanged for given operating parameters. However, the combustion chamber specific heat input is altered. The combustion chamber specific heat input is proportional to the turbine entry

343 temperature minus the heat exchanger cold side exhaust temperature, i.e.,

q oc T T' a T — [ + — )) 3 2 3 T2 c(T4 T2 (A3.7) where the heat exchanger cold side exit temperature, T12, has been defined in terms of a heat exchanger effectiveness, c, the heat exchanger hot side inlet temperature, T4, and the heat exchanger cold side inlet temperature, T2. Thus substituting equations (A3.3) into equation (A3.7),

q a T [ t - (1 + [c - 1] 1 ) (1 - c) - ct(1 - N[l - 1 71 - ])] (A3.8)

Applying a constant of proportionality of unity to equation (A3.8) and substituting this expression as the denominator of equation (A3.4) one obtains,

t (1 - (c -1) c t 71 = th r7t(1-c) + 71 C t — Cf c c t — (c-1+ TIC )} — (C-1+11C)

gas turbine plus heat exchanger (A3.9)

For both the simple gas turbine and the gas turbine plus heat exchanger, the specific work output is given by,

w = w ) — ( - )] t - w = cP [(T3 - T4 T2 T1 (A3.10) where c is the specific heat constant at constant pressure. Substituting equations (A3.3) into equation (A3.10) one obtains,

344

c pT w = ..n c1 [-ricritt - ci [c - 1] (A3.11) 'IC

simple gas turbine and gas turbine plus heat exchanger

345

Appendix (A4) The Calculation of Gas Turbine Performance Invoking the use of Variable Properties

The gas turbine cycle plus heat exchanger is considered, as illustrated in fig (3.3). To simulate a simple gas turbine without heat exchanger the heat exchanger effectiveness at step (VI) is set to zero. The calculation of the thermal efficiency and specific work output for such a cycle, assuming fluid properties are a function of temperature only and that mechanical and combustion efficiencies are equal to unity, proceeds as follows,

(I) For the given ambient temperature, T1, calculate the compressor isentropic exit temperature, T2s , where,

1,6° (T2s ) = (T1 ) (A4.1) C

(too)-1 loo T2s T 2s (A4.2)

equation (A4.1) is derived in Appendix (A1), r is the compressor pressure ratio and the inverse equation (A4.2) is solved iteratively.

Calculate the compressor exit enthalpy, h(T2), and compressor exit temperature, T2 , where,

h(T2s ) - h(T1 ) h(T2 ) - + h(T1) (A4.3) TIC

by the definition of the compressor isentropic efficiency, where n is the compressor isentropic efficiency. The temperature T2 is then calculated iteratively from the enthalpy h(T2). This temperature value is only needed when a heat exchanger is being simulated.

346 Set the turbine entry temperature, T3, and calculate the turbine entry enthalpy, h(T3).

(IV) Calculate the turbine isentropic exit temperature, T , where,

(T4s ) = (T3 ) In (r._) A t ( T3 ) ln( [1 "IP ]r ) P Pc (A4.4)

(01-1 (tpo ( T (A4.5) 4s

r is the turbine pressure ratio, p is the Pt compressor exit total pressure, Ap is the combustion chamber total pressure drop, p is the compressor exit total pressure, equation (A4.4) is derived in Appendix (A1) and equation (A4.5) is solved iteratively.

(V) Calculate the turbine exit enthalpy, h(T4), and the turbine exit temperature, T4, from,

h(T4 ) = h ( T3 ) — rit[h(T3) - h(T4s ) (A4.6)

by the definition of the turbine isentropic efficiency, where -qt is the turbine isentropic efficiency. The temperature T4 is then calculated iteratively from the enthalpy h(T4). This temperature value is only needed when a heat exchanger is being simulated.

(VI) Calculate the heat exchanger exit temperature, T2, where,

347 - T2 = T2 + C (T4 T2) (A4.7)

by the definition of the heat exchanger effectiveness, E.

(VII) Calculate the specific work output, w, where,

w = wt - Ian = [h (T3) - h(T4)] - [h (T2) - h(Ti ) ] (A4.8)

and wt is the specific turbine work output and wc is the specific compressor work input.

Calculate the overall thermal efficiency, where, (VIII) 11th'.

W - w - (A4.9) nth gin h(T3) - h(T12 )

where gin is the specific heat input.

348 Appendix (A5) A Simplified Analysis of the Compound Turbocharged Engine Cycle

A simplified analysis of the compound turbocharged engine cycle is applied making the following assumptions,

(i) the combustion and mechanical efficiencies are equal to unity, (ii) the compression and expansion processes are adiabatic and isentropic, (iii)fluid properties are constant, (iv) combustion occurs adiabatically and at constant volume, (v) the mass addition due to the combustion of fuel is assumed to be negligible, (vi) no aftercooler or intercooler is considered.

The pressure-volume diagram for this type of cycle is shown in fig (A5.1). Isentropic compression occurs in a rotodynamic compressor from points 1 to 2', followed by isentropic compression in the piston engine from points 2' to 2. Constant volume, adiabatic combustion occurs in the piston engine from points 2 to 3. There are then two stages of isentropic expansion, from points 3 to 3' in the piston engine and from points 3' to 4 in the rotodynamic turbine. Point 3' occurs at the same specific volume as for point 2'. There is in addition a 'pumping loop' 2'-5-6-3'. In the analysis that is applied, T is the temperature, cp the specific heat constant at constant pressure, cv the specific heat constant at constant volume, r the ratio of the specific heats, V the volume, v the specific volume, P the pressure, w the specific work and q the specific heat input.

The specific work output and net thermal efficiency are now calculated. The specific work output of the piston engine is given by,

349

W = (loss of internal energy on expansion stroke) piston - (gain of internal energy on compression stroke) - (pumping work)

which gives,

M ( cyl C (T - T')-(T - Ti d (P' - 1)(--1- ) piston v AM )( 3 3 2 2 3 P2 Am (A5.1)

where the mass specific quantities are related to the mass admitted and exhausted per cylinder stroke, Am, rather than the mass in the cylinder, The cylinder swept volume is V . The initial temperature in the cylinder is assumed to be equal to T. The ratio of the in-cylinder mass to the mass admitted per piston stroke is defined by,

M V +V (V + V r, cyl s c c - Rv = Am V r - 1 s[ (V S + VC) - V-, ]/ c v (A5.2)

where V is the clearance volume and r is the cylinder volumetric compression ratio. The pumping work may be found as follows,

( V ) P' P' 3 2 w = - R(T; - T'2) pump (P'3 P'2 ) 1 -Ers = p3 = (Cp Cv) (T' - T2) (A5.3)

where the fluid density is assumed constant during the inlet and exhaust strokes and where the ideal gas law has been used. Thus, equation (A5.1) becomes,

= R c i(T - T') - (T2 - T2' )) - (cp - cv ) (T'3 - T'2 ) wpiston v v 3 3 (A5.4)

350

The work output of the rotodynamic components is given by,

= (enthalpy drop thru turbine) Wrotor - (enthalpy gain thru compressor)

or,

= c (T' -) - c (T' - T1 ) (A5.5) Wrotor P 3 p 2

The net work output is then given by,

W W + w net piston rotor = CvRv(T3 — T2) — v - 1)(T; - T'2) - cp(T4 Tnet (R - Tl) (A5.6)

and this may be seen to comprise of the heat input minus the internal energy gain of the fluid trapped in the cylinders at the end of the exhaust stroke minus the enthalpy gain of the exhaust fluid relative to the inlet air. The net thermal efficiency is given by,

Wnet cvRv(T2 - T2) - (Rv- (T13 r2) cp(T4 Ti) nth (T3 — T2) (A5.7)

The case is now considered where the cylinder volumetric compression ratio, rv, is large so that R = 1. Then equation (A5.7) simplifies to,

.1 [T4/ T1 ] - 1 T1 th 1 — [T2/ T3 ) I T3 (A5.8)

The ratio of the exhaust temperature to the inlet temperature may be re-expressed,

351

r 1 T V V4 V2 7 4 4 exp - R (A5.9) V T V 3 1 r 1 1 comp where the results used are that the exhaust pressure and inlet pressure are equal, the specific volume during combustion is constant and the compression and expansion processes are isentropic. The subscript exp denotes the expansion process, the subscript comp denotes the compression process, while R is the rise in pressure ratio P associated with combustion, i.e.,

P 3 (A5.10) P2

Similarly, the inverse of the temperature ratio occurring during combustion may be re-expressed as,

T 2 2 T = R (A5.11) 3 3

because the specific volume during combustion is assumed constant. Substituting equations (A5.9) and (A5.11) into equation (A5.8) one obtains,

1 R - 1 T p 1 = 1 — 7 nth (A5.12) [ 1 - X 1 T3

The case is now considered where R is only just greater than unity by a small amount, 8. Then,

R = 1 + a —17 (A5.13a)

R 1 1 - (A5.13b) so that,

352

(1/ T n = 1 - 7 - 1 1) = 1 - (A5.14) th ( 1 [ T T - ]) 3 3 with the result that the overall thermal efficiency is approximately equal to the Carnot efficiency and equal to the Carnot efficiency in the limiting case when R is equal to unity. The overall work output is then given by,

7-1 W 1111 C — = 1111 C (r - 1 + c) (A5.15) TiCarnot v(T T ) vT 45 y 3 2 1 comp where m' is the air intake mass flow rate. Thus, to maintain both the Carnot efficiency at a high level and to give high work outputs the compressive volumetric compression ratio must be high. A high mass flow rate will also give higher power outputs.

The value of R for maximum specific output is now found. The specific heat input is given by,

q = (T3 - T2) = c,T3 (1 - Rp1 ) (A5.16)

Multiplying equation (A5.12) by equation (A5.16) gives,

1 W = C (T (1 — /9 1) - T T(R 7 - 1)) (A5.17) v 3 1 p

The maximum value of w for given values of Tl and T3 is found by taking the partial differential of w with respect to R and setting this equal to zero. Thus,

1-7 aW 2 (T R T R aR Cv 3 p 1 (A5.18)

Setting this equal to zero gives,

353 T3 R = (A5.19) P T [ 1 and thus the value of R for maximum specific output differs P from that for maximum thermal efficiency.

354 1- kt SSUW

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ll htE przESSUgE VOL-tim 111 ‘41-tt.1 -FoK

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ENGIN E. CYCL E.

355 Appendix (A6) The Analysis of the Engine Balancing of an Engine Fitted with a Swashplate Crank Mechanism

An analysis of the engine balancing of an engine fitted with a swashplate type crank mechanism is applied. The engine is assumed to have four cylinders arranged with their axis parallel to the centreline of the output shaft, operated by a swashplate type crank mechanism. Appendix (A7) presents approximate engineering drawings of such a layout.

The displacement, L1(*), of the first piston is defined by the sinusoidal displacement of the piston. Thus,

Li(*) = [ 2s + LL2s cos(*) (A6.1) where i is the angular displacement of the swashplate, defined to be zero when the first piston is at top dead centre, L is the height of the clearance volume, and L is the piston stroke. The velocity and acceleration of the first piston may then be found from,

L S L ' (*) = 2 sin 05)15' (A6.2) L L L S 2 s 2 L " (* ) = cos(*)191 + s sin(*)*" = cos Mt,' 1 2 2 2 (A6.3) the second term on the right hand side of equation (A6.3) being set to zero for constant rotational speed of the swashplate. The piston accelerations for the remaining three 3 cylinders are found by adding n and 2n to 1g in equation (A6.3) respectively. Thus,

356

L L " = L "13 + TT— ) = - sin(*)*'2 (A6.4) 2 1 2 2

L L "(i + n) = - cos(*)*12 (A6.5) L3" 1 2S

3n L4" L"(191 + 2 ) = 2 sin(*)*'2 (A6.6)

Thus, the resultant primary force produced by the four pistons is,

L3" F = L1 " (*) + L2"09 + (19) + L4"(1,) mL */ 2 2s [cos(*) - sin(*) - cos(*) + sin(*)] = o (A6.7) where m is the piston mass. That is, the resultant primary force is identically equal to zero. As there are no harmonics higher than the primary, all forces for this type of layout are in balance.

The primary torques will now be analysed. Consider fig (A6.1) below. The figure is from an end on view of the cylinders with the torque planes for cylinders one and two shown. The parameter R is the radial distance from the cylinder centreline to the power output shaft centreline.

cyl 1 2 0 mRL 19' 2 T - 2 mRL 19' 2s cos(*) 0 cyl 2 mRL 1Y 2 T2 — 2 sin(*)

Fig (A6.1) The resultant torque plane for cylinders one and two

It will be seen there is a resultant torque plane of

357 magnitude Tr which counter rotates relative to the direction of engine rotation. There is a second torque from cylinders three and four which is in the same plane and is of the same magnitude. Thus, the overall resultant primary torque is given by,

T = ml2L 1Y2 in a plane rotating at angular velocity - (A6.8)

In addition to the resultant primary torque produced by the accelerations of the pistons, there will also be a primary torque produced by the swashplate itself the plane of which rotates at engine speed. There are no torques higher than the primary.

358 Appendix (A7) The Compound Turbocharged Engine Layout

The proposed layout for the compound turbocharged engine with fixed valve timing is presented in two engineering drawings, contained in a pocket at the back of the thesis. The engine has four cylinders, arranged co-axially with the piston engine power output shaft centreline. The pistons are attached to the power output shaft by a swashplate type crank mechanism, adjacent pistons being 90° out of phase with each other. There are two main counter balancing swashplates, one rotating with the power output shaft to balance the torque produced by the primary swashplate and one driven by an epicyclic geartrain and counter rotating at engine speed to cancel out the torque produced by the accelerations of the pistons. The valve gear type is of hybrid sleeve/Aspin form, as described in chapter (4). There

1 is an outer sleeve driven at 2 engine speed for each cylinder, which directly drives the Aspin valve. The outer sleeves are attached to individual swashplates and counter balancing swashplates which drive each individual inner sleeve and each individual counter balancing sleeve, one of both for each cylinder. The inner sleeves move in an axial sense only. There are two counter balances for each outer sleeve to cancel out the force and torque produced by the rotating Aspin valve. There are two alternators, one driven directly from the piston engine and one driven from the compressor and turbine shaft, which runs co-axially with the piston engine power output shaft. The compressor is of single stage centrifugal type, while the turbine is of two stage axial type. Some engine parameters as optimised in the four poppet valve per cylinder filling and emptying model are,

359 Total swept capacity 160cc Bore/stroke ratio 4.33:1 Max. piston engine speed 20,040 rpm Compressor/turbine speed - 120,000rpm Power output at max. speed 1263 kW

360 Appendix (A8) The Quasi-Steady Piston Engine Model

The equations that describe a simple quasi-steady engine model are developed. The temperatures and pressures around the engine cycle are calculated to enable calculation of the specific work output and thermal efficiency. Fluid properties are assumed to be a function of temperature and equivalence ratio (but not unburnt fuel fraction), while the numerical integration procedure used is of first order type. Firstly, two definitions are made,

V + V rv s c cy 1 R v - 1 (A8.1) V rv cyl

is the ratio of the mass in the cylinder (after cylinder filling and before combustion) to the mass ingested by the cylinder intake stroke, where V, is the swept cylinder volume, V is the cylinder clearance volume and r is the cylinder volumetric compression ratio. Specific work outputs are related to a unit mass of air ingested by the cylinder during the intake stroke. Also,

m + m a f max max f ax — 1 + — 1 + (A8.2) R - M Ill m a a

is the ratio of the in-cylinder mass at the end of combustion to the in-cylinder mass at the start of combustion, where ma is the mass of air at the start of

combustion, mf is the total mass of fuel injected during max

combustion, 0max is the equivalence ratio at the end of combustion and A is the stoichiometric air/fuel ratio.

361 (A8.1) Calculating the Temperature Increments

(i) Stage 1 to 2, adiabatic compression from ambient pressure to final compressor exit pressure with polytropic efficiency "laic (not applicable to naturally aspirated engine). By the definition of the polytropic efficiency, the temperature increment over a small pressure increment is given by,

ST - T 7[T,0] - 1 8p (A8.3) 71m 7[T,0] which is summed over the entire pressure increment.

(ii)Stage 2 to 2', modelling an aftercooler of effectiveness c (not applicable to naturally aspirated engine model). The aftercooler exit temperature is given by,

T' = T2 — C(T2 Tl) (A8.4) by the definition of the aftercooler effectiveness. Then,

'2/T 1 v'2 = v T (A8.5) 1 P2/Pi from the ideal gas equation. For the naturally aspirated engine, set v'2 = vl, v2 = vl = Tl and p2 = pi.

(iii)Calculation of the volumetric compression ratio. The cylinder compression ratio is based upon the overall volumetric compression ratio (excluding density changes in the after cooler) being equal to r , where the latter is max a program input. Thus,

r max racyl v / V (A8.6) l 2 and,

362

V v - 2 (A8.7) 3 r cy 1

(iv) Stage 2' to 3, the piston engine compression process. A heat loss factor, QL , is introduced such that the heat loss from the piston engine is a constant fraction of the work input. Then, a small temperature increment is given by,

R[T,O)T av 6T = - (1 - Q) cv[T,0 v (A8.8)

which is summed over the entire specific volume change, av being negative.

(v) Stage 3 to 4, the combustion process. Constant volume combustion is assumed. The total internal energy change is given by,

+ au 60) (A8.9) 6U = uam + mau = uam + m(c 6T ao

from which,

— (6U - u6m) au 60 6T - (A8. 10) V

Also,

6U = TI Ati 6 m (A8.11) comb f f

wherecomb is the combustion efficiency, Ah is the fuel calorific value and the subscript f denotes fuel. Further,

6m am f f 60 6m (A8.12) + ICI 4 + 0 f

where the subscript a denotes air and 0 is the equivalence ratio. Substituting equations (A8.11) and (A8.12) into

363

equation (A8.10) yields,

Ah - u[T,0] ) comb f aU [T,04 50 4 + 80 { (A8.13) 6T - cv [T,0] which is summed over the entire equivalence ratio change.

(vi)Stage 4 to 5, piston engine expansion stoke. By analogy with section (iv),

R[TrOmax]T 6v 6T = - ( (A8.14) 1 QL) c[T,Omax)v where the heat loss is a constant fraction of the work output and equation (A8.14) is summed for the entire volume increment and,

V = V /R 4 3 m (A8.15a)

V = v'/R (A8. 15b) S 2 m

(vii)Stage 5 to 6, turbine expansion (not applicable to the naturally aspirated engine). By analogy with section (i),

7[T,Omax] - 1 6T = 71 tT 6p (A8.16) 0o 7 [T.O.a.] p

which is summed for the total pressure increment, defined as the ambient pressure minus the pressure at the end of the piston engine expansion stroke.

364

(A8.2) The Work Transfers and Overall Thermal Efficiency

The specific works relative to the mass of air ingested during the intake stroke may now be calculated as follows,

w1-2 = h[T2,0] - h[T ,0] (A8.17a)

= (1 + QL)Rv(u[T3,0] - u[T2,0]) (A8.17b)

combAh f max q3-4 .s4 Rv (A8.17c)

O W4-5 = ( 1 - QL)RvRm(u[T4 (1)max ] u[T max (A8.17d)

wP = R (p v 5 p2) (v 2 v3) (A8.17e)

TAT 1/ ) fric = RFMEP[Np ](v'2 - 3 (A8.17f)

W5-6 = Rm(h[T5 , 4)max ] h[T6 , max ] (A8.17g) where w1-2 ,Wpump and w5-6 are set to zero for the naturally aspirated engine. The specific work output, in units of kW dm 3, is given by,

W = (14 W - w w 4-5 5-6 w1-2 we -3 pump Eric -6 2 2P ) x 10 (A8.18) where N is the piston engine rotational speed in rev sec-1. The overall thermal efficiency is given by,

W + W - - - 4-5 5-6 141-2 W2,-3 pump Inci th comb g3-4 71 (A8.19)

365

Appendix (A9) The Numerical Integration Scheme Employed for the Filling and Emptying Method

In the numerical integration scheme employed for the filling and emptying method, most terms are evaluated to third order accuracy. However, some terms can be integrated exactly (discounting roundoff errors), while one term is evaluated to fourth order accuracy. Derivatives higher than the the first are generally evaluated using a finite difference scheme, using the present and two previous values of the first derivative. Using previous values, rather than future values, saves on computation time and simplifies the algorithm.

(A9.1) Evaluating the Higher Derivatives by Finite Difference Means

Using Taylor's series, the following relationships may be written,

1 2 1 IV 3 f' = f' - f" St + — f"St - — f 8t (A9.1a) 1 2 2 12 2 12 6 12

1 2 1 IV 3 f' = f' - + — f"St - f 0 2 f"2 St02 2 02 St02 (A9.1b)

where f', f', f'2 are the penultimate, the last and the present values of the first derivative of the parameter of interest, f"2 is the present second derivative, f ' is the IV 2 present third derivative and f2 is the present fourth derivative. The values St and St 12 02 are the time increments between the last and present steps and the penultimate and present steps respectively. Manipulation of equations (A9.1) yields,

(f' - f')Stz - (f' - fi)8t2 f t 2 1 02 2 0 12 1 IV + f6t St St 2 at1 2 6t02 (atO 2 - at1 2 ) 6 2 12 02 (A9.2a)

366 2[(f'2 - f; )6t02 - (f21 - f'0)6t12) f/ It 2 6t126t02(6t02 - 6t12 ) 1 IV —f (6t ) 3 2 12 + St02 (A9.2b)

The first term on the right hand side of each of the above equations is the desired finite difference relationship, while the second is the estimate of the fourth order truncation error.

(A9.2) Evaluating the Higher Derivatives of the Pressure-Volume Integral

An integral of the form,

:45t r RT dV d* (A9.3) I J t V d* dt dt must be evaluated in the general scheme of equations, where R is the specific gas constant, T is the temperature, V is the volume, 2 is the crank angle and t is time. Setting,

dw RT dV d* dt V d* dt (A9.4) enables the integral I to be approximately evaluated as follows,

1 2 1 3 1 IV 4 I = Wi St — 2 w"St + 6 ww6t + 24 w St (A9.5)

The second derivative in equation (A9.5) can be evaluated by the finite difference method given in section (A9.1). However, it can also be evaluated exactly, giving greater accuracy. This is done as follows,

w' d ( RT dV d* dt ( V d* dt (A9.6)

367

or,

w' ( T dR dV R dT dV RT r dV 12 RT d2V) V dt di V dt d* V2 L di V d*2

dt (A9.7)

where,

dR aR dT 8R do aR dA dt aT dt + — dt— + ax dt (A9.8)

and 0 is the equivalence ratio, X is the unburnt fuel fraction, and the partial derivatives in equation (A9.8) are estimated by finite difference means based on the fluid properties algorithm. The ordinary differentials on the right hand side of equation (A9.8) are known. The following relationships are also used,

1 dV (r - 1)sin* V d* (r + 1) - (r - 1) cos* (A9.9a)

2V (r - 1) cos* 1 d (A9.9b) Vd*2 (r + 1) - (r - 1)cos*

which are derived from,

VV V = 2s + V ) - 2S cost (A9.10)

and,

V + V s c r - (A9.11) y V

where V is the swept cylinder volume, V is the cylinder clearance volume and r is the cylinder volumetric compression ratio. The third and fourth derivatives in equation (A9.5) are evaluated by substituting le2', w;' and

w"0 in place of f',2 f'1 and f'0 in equations (A9.2). Thus

368

expression (A9.3) is evaluated to fourth, rather than third, order.

(A9.3) The Overall Scheme

The basic integration scheme is given by equations (5.5), (5.7), (5.8) and (5.12) in chapter (5). These equations are applied to each control volume, the numerical formulations of these equations being designed to increment a control volume's mixture component masses, total mass and temperature.

(A9.3.1) Calculating the Control Volume's Mass Increment

The overall mass increment is given by,

t.4.6t 6m = f dm dm dt t dt i dte

= 6m - Sin 6m 1 e ubf inj

„ 1 (mil me r t3 - re)St + '' ) 1 —2 (M1 - Mel ) 6t2 66 a + 6mubf (A9.12) inj

where the first derivatives in equation (A9.12) are evaluated directly from equations (5.13), (5.40) or (5.46) as appropriate, the higher derivatives are evaluated from equations (A9.2), and ra is the unburnt fuel injected ubf inj into a cylinder volume which may be found exactly from equation (5.32). Note also that,

6m = 8111 + am + am + 6m (A9.13a) e i t i iman cyll cy12 cyl 3 cy14

SM = 6111 + 6m + 6m + 6m (A9.13b) e e e e eman cy11 cy12 cy13 cy14

The component mass increments may be found as follows,

369

- Sma — Auth [Sin{ ousa, I I1 — A.)] e (A9.14a)

[8 mi A cP 4.J 4 ) (1 - 6m 0 I ) (1 - Xu )] bf 111 u i [^ Pl u + ambf (A9.14b) comb

- [S (A9.14c) amubf = ( 8rn A ) in Atde+ Arnubt 8ubfin ti i comb where u denotes the upstream conditions, A denotes the stoichiometric air/fuel ratio, and comb denotes a term arising from combustion evaluated exactly from equations (5.23), (5.24) and (5.25). Equations (A9.14) can be written alternatively as expressions for the first derivatives of m , m , m m and m with respect to a bf Tnubf ubf bf ubf in j comb comb time. Then one may write,

, dm dm dm dm a ( b f _ bf) a 1 dO = — 0 1 a - (A9.15a) dt ma dti dte ma( dti dte ) dm dm dA _-1 ( ub f ubf) A dm - dm m (A9.15b) dt m dt i dte ( dt i dte these expressions being used to evaluate equation (A9.8). The new mass values at the next step are found by adding the mass increments given by equations (A9.14) to the old mass values. It is then necessary to find the control volume temperature increment.

370

(A9.3.2) Calculating the Control Volume's Temperature Increment

Equation (5.12) of chapter (5) shows that the temperature increment is given by,

t+St ST = RT dV [ V dt

+ 1 i dQ dQ u dm + h dm - h dm ) m t dt 1 dte dt i dt i e dt e j _ au dO au dA ] ac dt ax dt dt (A9.16)

The volume change term and heat flux terms are zero for the inlet and exhaust manifolds. Chapter (5) describes in more detail how the heat flux terms are calculated. Equation (A9.16) may be written,

t+St AQ 1 ti-St ST = --1 RT dV I + f cadY dt f V dt at -I- 2- m cV t c V cV t 1 [11(T, 0 + SO, A + 8A) - u(T, (A9.17)

where the first term on the right hand side of equation (A9.17) is evaluated as in section (A9.2) and the second term can be evaluated exactly. The third term in equation (A9.17) is given by,

dY = 1 _ dQ u dmdm+ h - h dm (A9.18) dt m ( dte dti e dte

so that,

t+S I tdY dt = Y'St + —Y"St2 + 1—Y'"at 3 (A9.19) dt 2 6 j t

where the higher derivatives are evaluated as in section (A9.1). The average specific heat constant at constant

371 volume is evaluated by a predictor-corrector method as follows,

0 60, cV = f(T, + A + 6A) (A9.20a) 1

ST - Su (A9.20b) 1 c V 1

0 = f(T + ST , 0 + 60, A + SA) (A9.20c) 1 ,12

C = (C + CV V2 v v (A9.20d) 1 v2

ST - Su (A9.20e) C V

and thus the final temperature increment is found. The piston engine work increment is found by taking the negative of the first integral on the right hand side of equation (A9.17) and multiplying it by the control volume mass.

372 Appendix (A10) The Mass Flow Rate as a Function of Pressure Ratio

The mass flow rate, as a function of pressure ratio, is derived. The assumptions are,

(i) constant fluid properties, (ii) isentropic expansion to a throat, with constant pressure downstream of the throat if the pressure ratio across the port is less than or equal to the choking pressure ratio and a pressure reduction otherwise, (iii)a quasi-steady model is used.

The resulting relationship for the mass flow rate is used for the filling and emptying model for the following,

(a) the inlet port of the inlet manifold for a naturally aspirated engine, (b) the cylinder inlet ports, (c) the cylinder exhaust ports, (d) the exhaust port of the exhaust manifold for both a naturally aspirated and compound turbocharged engine.

(A10.1) Unchoked Flow

The upstream stagnation conditions are designated '01' while the throat conditions are designated '2'. The effective throat area is given by,

F =2 C A (A10.1) 2 f 2 where A2 is the actual throat area and c is a flow coefficient which lies between zero and unity in value. The continuity equation is written,

(A10.2) In' = p2u2F2

373

where m' is the mass flow rate, p is the fluid density and u is the fluid velocity. The energy conservation equation for steady flow is,

1 2 C T = C T + —U (A10.3) p 01 p 2 2 2 where c is the specific heat constant at constant pressure. P The speed of sound is given by,

a = 1/?;;--' (A10.4) where 7 is the ratio of specific heats and R is the specific gas constant. Substitution of equation (A10.4) into equation (A10.3) yields,

2 2 7 - 1 2 = + U a01 a2 2 2 (A10.5)

The isentropic conditions which exist between the upstream stagnation point and the throat gives,

7 P2 p2 (A10.6) Pot = [ pot thus,

[ p:2 11 P2 (A10.7) 01 1

Substituting equations (A10.5) and (A10.7) into equation (A10.2) gives,

2 1 a 2 II 2 nil - p01 F2 2'12 P 2 a01 { 7 - 1 11 p012 ) (A10.8) a 01

374 The isentropic conditions which exist between the upstream stagnation point and the throat gives,

a 2 n 7 2 T2 a (A10.9) 01 Tot P01

Thus the resulting equations for unchoked flow are,

m'=kC p01cf A2 k - a 2 7-1 1 } 2 PO1F2 22'2 P2 I 7 a01 - 1 Poi

r s r choke

(A10.10)

The choked pressure ratio, r being defined in the next Pchoke section.

(A10.2) Choked Flow

Up to the pressure ratio at which choked flow occurs, there is an increasing mass flow rate with increasing pressure ratio as given by equations (A10.10). When the pressure ratio is such that sonic flow occurs at the throat, the mass flow rate is a maximum. Thus, for pressure ratios greater than this critical pressure ratio, the mass flow rate remains constant at the maximum value. An increasing pressure ratio then results in an increasing pressure reduction downstream of the throat. When the flow at the throat is sonic, equation (A10.5) becomes,

375

2 2 7 - 1 I a = 1 + 01 a2 2 (A10.11)

or

01 7 + 1 2 (A10.12) [ aa2 2 —

then, from equation (A10.9),

7 p + 1 ) 7-1 2 = 1 7 1 (A10.13) 2 r p01 P choke

Substituting equation (A10.13) into equations (A10.10) yields,

MI = kE (A10.14) choke choke

01cf A2 k — p a01 7+1

= 71 2 choke 7+1

r -›: r P P choke

376

Appendix (A11) A Method to Calculate the Cylinder Wall Temperature

An averaged heat flux is used to calculate the cylinder wall temperature. The averaged heat flux equation is,

q = k(T - ya)Ax = he (T - T ) (A11. 1) wl c w2 c where q is the mean heat flow rate out of the cylinder per cycle per unit area, k is the thermal conductivity of the cylinder wall, Tw1 is the average inner wall temperature, Tw2 is the average outer wall temperature, Ax is the thickness of a thin cylinder wall, h is the heat transfer coefficient for the coolant and T is the coolant temperature, the last two values both assumed constant. Equation (A11.1) may be rewritten as two simultaneous equations,

kAx T - kAx Tw2 = q

hcTw2 =q+hTc

Using these equations to eliminate Tw2 one obtains,

1 T = a + + T (A11.4) -[ kAx hc which is the required result for the inner wall temperature. This parameter would then be calculated iteratively, a new value for T being estimated based on the immediately previous value for q.

377

Appendix (Al2) Evaluating the Stability Parameter

The stability parameter for the mass balance O.D.E. is defined in chapter (6) as,

stability parameter = am'am (Al2.1) where m' is the mass flow rate and m is the control volume mass. This parameter has an important bearing on stability. For mass flow rate through a compressor, the stability parameter is evaluated by finite difference means, otherwise it is evaluated analytically.

(Al2.1) Evaluating the Stability Parameter for Mass Flow through a Compressor

For mass flow through a compressor, the stability parameter is found through finite difference means. From the ideal gas equation and the definition of the compressor pressure ratio, r ,

r - mRT/ V (Al2.2) p c p u mRT/u u u V u where u denotes upstream conditions. Thus,

ar P c am in (Al2.3)

A first order finite difference expression for the stability parameter is therefore given by,

Sr m'(r + Sr ) - mi(r ) r P P am' am' c c c am ar am Sr

(Al2.4)

378

where the compressor mass flow rate is a function of pressure ratio only. In the finite difference evaluation, ar was set to a value of 0.01. The stability parameter will only be zero or negative (a requirement for stability) provided that,

am' 0 (Al2.5) ar

(Al2.2) Evaluating the Stability Parameter for Mass Flow which is not Occurring through a Compressor

For mass flow which is not occurring though a compressor the mass flow rate is given by,

m' = kE (Al2.6)

where the negative sign occurs for outflow from a control volume and the positive sign occurs to inflow into a control volume. This equation is derived in Appendix (A10). The two cases considered in deriving the stability parameter are for inflow into, and outflow out of, the control volume.

(Al2.2.1) Inflow into a Control Volume

For inflow into a control volume, the stability parameter is given by, ak u ar ar am' ac — k ac aC ac am am ku am ar am a7 am =ku ar am (Al2.7)

u denoting that k applies to the upstream control volume, while the partial differentials of k with respect to m, and 7 with respect to m are zero. Now, the pressure ratio is defined by,

379

Pu mRT /V mRT/ V (Al2.8) which implies that,

ar = P (Al2.9) art

Thus,

k r am' _ u p ae (Al2.10) am m Or

inflow into a control volume

(Al2.2.2) Outflow from a Control Volume

For outflow from a control volume, equations (Al2.6) and (Al2.7) become,

am' ak ac ar Om am k ar am (Al2.11)

The partial differential of k with respect to m being non-zero for reverse flow, as k then applies to the control volume parameters. Now, the parameter k is defined by,

pc A mRTt:f A k - af - aV (Al2.12) refer to Appendix (A10). This then implies,

ak _ k Om (Al2.13)

The pressure ratio is defined by,

r - p mRT/ V (Al2.14) R T i V P Pdd d d d

380 d denoting downstream conditions. This then implies that,

Or P (Al2.15) am

Substituting equations (Al2.13) and (Al2.15) into equation (Al2.11) yields,

am' k 1c ac am r p ar (Al2.16)

outflow from a control volume

(Al2.3) The Sign of the Stability Parameter

It is seen that the stability parameter derived in section (Al2.2) is always zero or negative, while that derived in section (Al2.1) is conditionally zero or negative. A non-positive stability parameter is required to ensure numerical stability.

381

Appendix (A13) The Step Size Control Method used for the Filling and Emptying Model

As described in chapter (6), the step control method consists of two separate stages,

to control the truncation error in the mass balance O.D.E., to ensure numerical stability of the mass balance O.D.E..

These two separate stages are now described.

(A13.1) Controlling the Truncation Error (A13.1.1) Estimating the Fourth Derivative

The third order method employed for numerical integration of the mass balance O.D.E. requires evaluation of the fourth derivative of mass with respect to time to enable estimation of the truncation error. This requires knowledge of the first derivative of mass with respect to time at the present and three previous points. Then, using Taylor's series,

1 2 1 IV 3 f' = f' f"St + f"'St + -6- f St (A13.1a) 3 0 0 03 2 0 03 0 03

1 2 1 IV 3 f' = f' + + —f"'St + f St 2 0 f"0 St 02 2 0 02 0 02 (A13.1b)

1 2 1 IV 3 f' = f' + f" St + — f"'St f St (A13.1c) 1 0 0 01 2 0 01 6 0 01

where StM is the time between the O'th and 1st step, St02 the time between the O'th and 2nd step and St03 the time between the O'th and 3rd step. Elimination of f" and f'" yields,

382 036t03 - 6% 026t02 - TERM1 = (6t 36%1) / (6t 6 t026 t01 ) 2 - S TERM2 = 6t02(6t 02 tot NUM = 6(f; - f'(3. + (f' - f01 ) ((St02TERM1 - St03)/St ()) - TERM1(f'2 - f')) 3 2 DENOM = St03 - 6t03 St01 - TERM1.TERM2 fIV = o NUM/DENOM (A13.2) the fourth derivative being constant over the interval (0-3) in a fourth order analysis.

(A13.1.2) Using the Estimated Fourth Derivative to Control the Truncation Error

The numerical integration error is approximately given by,

1 2 1 3 + 1 IV 4 cf --• 2— c(f")6t + 6 c(f"')St 24 f St (A13.3) where c(f") and c(f1 ") are the errors inherent in estimating the second and third derivatives respectively, St is the next time step and c is a relative error. Appendix (A9), equations (A9.2) shows that,

1 IV - St 6 t c(f") = 6 f 12 O2 (A13.4a) 1 e (v " ) = 7 fiv (6t12 + 8t02) (A13.4b)

Thus,

2 3 IV + 4 4.. 24cf = f [26t12St02ot 3 (St12 + St02 )6t + St ] (A13.5) setting,

383

4 (A13.6a) = —3 (8t12 8t02)

(A13.6b) g = 2at12 St 02

7 = 24minpL-1,fp/ over all control volumes (A13.6c)

and setting the approximate equality to an equality, then the following polynomial in St is obtained,

4 3 2 at °tat pst — CT = 0 (A13.7)

which is solved iteratively to find at. An error in the calculation of (3 was found in the numerical algorithms, though this will generally result in over estimation rather than under estimation of the truncation error. Note also that,

fIV = (fIV - fIV) (A13.8) e n

where n is the control volume number, i denotes influx and e denotes efflux, i.e. the fourth derivative used is the sum of the influx and efflux components for each volume.

(A13.2) Ensuring Numerical Stability

For a first order scheme, stability is ensured by,

am' - 1 < 1 + < 1 (A13.9)

as derived in chapter (6). The equivalent relationship for a third order scheme is,

1 am" 2 1 am' 3 — < 1 + am ' St + , St + St < 1 am 2 0111 6 am (A13.10)

Since the stability parameter (.5Fam' ) is always negative for a conditionally stable problem, equation (A13.10) may be

384

written,

2 1 3 t + ) — "--amL11' (at + 2 am' 6 am' at 2= 0 (A13.11)

which is polynomial in St which can be solved iteratively, given that from Appendix (A9), equations (A9.2),

+ am" at02 St12 15t (A13.12a) am' St12 02

am '' ' 2 am' St (A13.12b) at12 02

the stability parameter being calculated as given by Appendix (Al2). Equation (A13.11) is applied to each control volume, with the total stability parameter value for each of the control volumes being the sum of the influx and efflux components. The time step value obtained is then the minimum of the time steps obtained for each control volume.

(A13.3) The Final Time Step Calculation

The final time step obtained is the minimum of,

(i) that corresponding with a crank angle increment of 0 30 (ii) that given by controlling the truncation error, (iii)that given by ensuring numerical stability.

385 Appendix (A14) The Gradient Vector of a Quadratic Function

A quadratic function is defined by,

f(x) = 2 ir.A. x - bT. x + c (A14.1) where x is a general vector with n elements, A is the square, symmetric Hessian matrix of size n x n with constant coefficients, b is a constant vector with n elements and c is a constant. Each element of the gradient vector, for which an expression in terms of A, b and x is required, is given by,

af(x) i=1 to n (A14.2) gi ax

For simplicity, consider a function where n=3. Equation (A14.1) may be expanded to give,

f (x) = x ( a x + a X + a X - b1) 1 2 11 1 2 12 2 13 3 1 1 1 + x( a x + a X + -a X - b2 ) 2 2 21 1 2 22 2 2 23 3 1 1 1 + x( a x+ a x + - a x b3) + C 3 2 31 1 .2 32 2 2 33 3 (A14.3) where aij is the element of A lying on the i'th row and the j'th column and ri is the i'th element of r . The Hessian matrix is symmetric, thus,

a = a i it (A14.4) which implies that,

386

1 2 1 2 1 2 —a x + — a x + —a x f(x) = 2 11 1 2 1 2 2 2 13 3

+a xx +a xx +a XX 12 1 2 13 1 3 23 2 3 - b x - b X - b X c l l 2 2 3 3 (A14.5)

thus,

of {E x ) - b (A14.6) gi ax 1 j i .1 =1 a

Since equation (A14.6) is true for every row of the gradient vector, this then implies that,

g (x) = A x - b (A14.7)

which is the required result. Equation (A14.7) may be generalized to any number of elements of the input vector, n.

387

Appendix (A15) Proof that Conjugate Directions give Rise to Exact Minimisation of a Quadratic Function in Exactly n Line Minimisations

A quadratic function is defined by,

T f (x) = 1 xT Ax -b x + c (A15.1) 2 — —

where A is the constant Hessian matrix, b is a constant vector and c is a constant. Mutually conjugate directions p k are defined by,

T p I A p j = 0, i=1 to n, j=1 to n, i*j (A15.2)

where p andp J are independent vector directions. Suppose 2 k is reached in k steps, where the descent directions are defined by equation (A15.2). Then,

T p s. g (x k) = 0, s=1,2,...,k-1 (A15.3)

where g (x k) is the gradient vector at point k.

Proof

from Appendix (A14),

k g (x ) =Ax k - b (A15.4)

Repeated line minimisations are performed in independent vector directions, so that equation (A15.4) may be written,

k-1 g (x k) = A(x s„.1 +E Ap) - b , 0< s s k-1 j + 1 (A15.5)

or,

388 k-1 g (x k ) g (x s„.1) + E Aj Ap j , 0< s k-1 j=s+1 (A15.6) so that,

k-1 T T p s g (X k) = s g (x ) + E pTsA p -- — s+1 j=s+1 0 < s k -1 (A15.7)

The first term on the right hand side of equation (A15.7) is zero by the definition of a line minimum. The second term is zero by the definition of conjugate directions. Therefore condition (A15.3) is proved to be true. Thus,

T T p . g (x ) -Er g (x 11+1) = P 2' g (x nfl) = n - n1.1 (A15.8)

If p s are independent vectors, the only way this can be true is if,

I .1 (X "1) I = (A15.9) thus the minimum has been found in exactly n line minimisations, and the proof is complete.

389 Appendix (A16) Derivation of the Conjugate Gradient Algorithm

The proof proceeds by induction as follows,

(i) initially it is assumed that for s=1 to k-1 P s directions are mutually conjugate. A recursive relationship to generate a k'th mutually conjugate direction is assumed. (ii) It is then shown that assumptions (i) lead to the k'th direction being mutually conjugate. (iii)It is shown that, with suitable initial direction p 1 the first two directions are mutually conjugate with the algorithm employed. Thus, the proof by induction is complete.

Finally, a variation on the basic conjugate gradient scheme, known as the Polak-Ribiere conjugate gradient algorithm, is described.

Some general definitions are now made. The k'th position vector 21 k is reached after a line minimisation in direction k-1 apart from x which is an arbitrary initial point. The gradient vector g (x k) applies at point while A is the symmetric Hessian matrix with constant k coefficients. The basic proof now proceeds.

(A16.1) Proof of the Basic Algorithm

(i) assume,

T p i A p = 0, i=1 to k-i, j=1 to k-1, i*j (A16.1) then from Appendix (A15), equation (A15.3),

T p s . g (x i) = 0, 0< s i-1, 1< i< k (A16.2)

390

A recursive relationship of the form,

Pk = g (x k) ak-i (A16.3)

is assumed.

(ii)(a) It is now proved that,

pT Ap = 0, 0< s k-2 (A16.4) — s — k

assuming that,

T k = 0 k -1A 1D- (A16.5)

Condition (A16.4) is proved as follows; from equation (A16.3),

T T T sA p k sA g (x ) .. (A16.6) p -p k k P k-1 P k

which from condition (A16.5) gives,

T —kpT As — p = - ( x s+1 - x s) A g (x k ) (A16.7)

but, from Appendix (A14),

g (x k) =Ax k - b (A16.8)

thus,

g (x 2) - g (x 1) = A( x 2 — x 1) (A16.9)

which gives,

T ) TAT = 1)) - ( x - x )TA ( 2) - g (x -2 — 1 ( —x 2- —x 1 (A16.10)

as A is symmetric. Thus equation (A16.7) becomes,

391

T T p sA p i = - ( g (x si.1) - g (x s) ) g (x i ) (A16.11)

From equations (A16.2) and (A16.3),

T (-g (x s) + 7, p s_i) . g (2L. I) = 0, 0< s 5 i-1, 1< i 5 k (A16.12) or,

T g (X s ) . g (x 1) = 0, 0< s 5 i-1, 1< i 5 k (A16.13)

thus equation (A16.11) becomes,

T p sA p k = 0, 0 < S 5 k-2 (A16.14)

(ii)(b) It is now shown that with suitable choice of 7k, condition (A16.5) will hold. From equation (A16.3),

T T T A g (x ) +7 p A p l ic-iAl-k = - —P k-1 — — k k — k-1 — k-1 (A16.15)

setting this equal to zero,

T p k_IA g (x k) (A16.16) k - A 1::)T k-1” -:1:13- k - 1

Thus, from the definition of a line minimum and equation (A16.3),

T (x x A g (x ) k — k - 1) — — k Tk T (x - x ) A(- g (x k_i) + 7 p ) --k -- k -1 k-1 — k-2 (A16.17)

From equation (A16.10) this gives,

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T (x g k ).(— g (x— k ) - g (x k-1 )) ?I k T I g (X k ) - g (X k- 1 ) ) . (- --9- (-2C-- k -1) + 7k-1 -2:)- k-2) (A16.18)

From equations (A16.2) and (A16.13) this may be simplified to,

g T (x k) . g (x k)

wk T (A16.19) g (X k -1) . g (X k-1 )

which will ensure that the k'th direction is mutually conjugate to the preceding k-1 mutually conjugate directions.

(iii) It is now shown that with suitable initial minimisation direction, p 1, the two initial minimisation directions will be conjugate. First, set,

P 1 = ---9-- (x 1) (A16.20) then, from the preceding definition of the algorithm,

g T (x 2) . g (x 2) p . - g (x (A16.21) — 2 — — 2 ) T g (x 1) g (X 1) . -9- (x 1) then,

T p iA p 2 = T T -9- (x 2) • g (x 2) - x ) A ( x 2 — 1 [ - g (x 2) T g (x 1 )1 g (X 1) • g (X 1)

(A16.22)

393 The condition for a line minimum gives that,

T T g (X 2 ). p 1 = - g (X 2) g (x 1) = 0 (A16.23)

Thus, substituting equation (A16.23) and (A16.10) into equation (A16.22) gives,

p 2 = 0 (A16.24)

Thus, the first two directions are conjugate, and the proof is complete.

(A16.2) An Alternative Formulation of the Conjugate Gradient Algorithm

Reference to equations (A16.18) and (A16.19) shows that the conjugate gradient algorithm may alternatively be written with,

T g (X k).( g (x k) g (x k..1)) (A16.25) k T g ) . g (x ) - (x- k-1 - k-1

This alternative formulation is identical with that of (A16.19) when applied to a quadratic function. However, this alternative formulation, known as the Polak-Ribiere conjugate gradient method, has generally been found to be more efficient when applied to a non-quadratic function.

394 Appendix (A17)

Comparisons of Estimated Relative Program Execution Time, for the Two 'Conjugate' Optimisation Methods

The number of function evaluations required for exact minimisation of a quadratic function in exact arithmetic, is compared for 2 different numerical techniques. The 2 techniques are the 'conjugate directions' and 'conjugate gradient' methods.

A value for the average number of function evaluations required for each line minimisation is needed in the calculation. A value of 19 is taken, which is correct for a fractional convergence tolerance of 10-3 with the 'golden search' method, including the evaluations necessary for bracketing the minimum. A better line minimisation routine would give a lower value for this figure.

The results of the calculations follow.

N = total number of required function evaluations n = number of dimensions

CASE (i) = conjugate directions method; N = 19.[ n(n + 1) + 1]

CASE (ii) = conjugate gradient method, 2nd order gradient calculation; N = n(19 + 2n)

395 N n CASE (i) CASE (ii)

2 133 46 5 589 145 9 1729 333 co 19n2 2n2

Thus, according to this analysis, the conjugate gradient method is superior to the conjugate directions method.

396 Appendix (A18) The Constrained Optimisation Problem

The general constrained optimisation problem for a function f(x), constrained by constraining functions cJ (x) over a limited input vector space, is given by,

* 4. find x such that f(x )s f(x) for all x subject to xi s xi s xi , for i=1 to n min max i and subject to c (x) s Ci , for j=1 to m (A18.1) where n is the number of input vector components, m the number of constraints and C are constants. This problem may be translated to an equivalent unconstrained optimisation of the function s(x), where s(x) is equal to f(x) plus a penalty function, a(x). That is,

s(x) = f(x) + a(x) where a(x) = (3 if any of cj(x) > Cj, where (3 > 0, a(x) = 0 otherwise (A18.2)

The penalty value, 6, must be a suitably large value. Using this approach, the constrained optimisation problem is turned into a much simpler unconstrained optimisation problem. Care must be taken, however, if the function partial derivatives are evaluated by finite difference methods in order to determine line minimisation directions. Such derivatives information must be based upon the function values f(x) rather than s(x), as the latter function is discontinuous at the penalty boundary. If the basic method employed is of the conjugate gradient type, then the iterative procedure must be reinitiated if the last vector point found lies on the penalty boundary. This is because the function gradient in the line minimisation direction will then not generally be zero, and the conjugate gradient

397 algorithm relies on such a zero gradient value to ensure its convergence properties.

398