A simplified simulation of gas turbine engine operation

John Olsen & John Page School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, N.S.W., 2052. [email protected] & [email protected]

Abstract. This work outlines the analysis used in simulating the operation of a gas turbine engine. Its starting point is the analysis of Cumpsty (2006). Cumpsty’s work suggests that it is possible to simplify the behavior of gas turbine engines enormously by taking into account aspects that are common to all turbine maps, i.e., the collapse of curves representing differing constant non-dimensional spool rotational speed. As a result, the relationship between the pressure ratio and the non-dimensional mass flow rate as well as the pressure ratio and the isentropic efficiency are essentially independent of the rotational speed of the spool. Following Cumpsty, we develop equations to enable us to calculate both pressure ratios and normalised mass flow rates through compressors so that we can plot the behavior of these engines on compressor maps.

numbering regime adopted in this work, i.e., station 2 at 1. INTRODUCTION the inlet to the compressor, station 3 at the inlet to the combustion chamber, station 4 at the inlet to the turbine For straight and level flight, aircraft require a constant and station 5 at the inlet to the propelling nozzle. throughput of energy to maintain altitude and speed. In many aircraft applications, gas turbine engines are utilised to provide forward thrust. It can be shown that the net thrust Tn is:

n & ( j −= VVmT ) (1)

where m& is the mass flow rate of air through the engine (we ignore the mass flow rate of fuel through the engine), Vj is the magnitude of the velocity of the jet issuing from the propelling nozzle with respect to the aircraft and V is the magnitude of the velocity of the incoming airflow with respect to the aircraft. To maximise propulsive efficiency, Vj must be kept low meaning that the mass flow rate must be kept high. For an aircraft powered by gas turbine engines, the rate at which the flow issuing from the jet nozzles can transfer energy to the aircraft is equal to the product of the thrust and velocity of the incoming airflow with respect to the aircraft. If we increase rate at which the jet transfers energy to the aircraft so that it exceeds the energy required for straight and level unaccelerated flight, the aircraft will climb. Similarly, if we decrease the rate at which the jet transfers energy to the aircraft to a level below that required for straight and level unaccelerated flight, the aircraft will descend. Changing the pressure ratio across the engines propelling nozzles

controls the rate at which the jet transfers energy to the aircraft. This is the job of the gas turbine engine. Figure 1. Basic engine configurations a) single-spool turbojet, b) two-spool turbospool and c) two-spool 1.1. Basic operation of gas turbines turbofan, given with the numbering regime that is used in this work. The numbering regime is the same as that To understand the simulation, we need to offer a brief used by Cumpsty (2006). explanation of how gas turbine engines work. For further information we refer to reader to Gunston In a single spool engine, all compressor and turbine (2006) and Hünecke (2003). Figure 1 shows diagrams stages connect to the same shaft which is commonly of the basic engine configurations for the single-spool called the spool. We distinguish between engine turbojet, the two-spool turbojet and the more common compressor operation at design and off design two-spool turbofan engine. The figure also shows the conditions. The design condition for a commercial

airliner typically corresponds to engine operation when k the aircraft cruises at a pressure height of around ⎡ ⎤ ()−1k ⎛ − 1k ⎞ 2 (3) 35,000ft and an airspeed specified as a Mach 0 ⎢1PP += ⎜ ⎟ M ⎥ . number ⎣ ⎝ 2 ⎠ ⎦ of 0.85. At this design condition, the rotational velocity of the spool and the velocity of the incoming For most gas turbine engine operating conditions, the air to the engine, are such that the inlet air impinges on flow from the propelling nozzles is choked. This is the compressor blades at zero degrees of incidence. At emphasised by Cumpsty (2006) and supported by El- this condition, the isentropic efficiency of the Sayed (2008) and Crane (2005). This means that the compressor is maximised. This ideal flow arrangement velocity at the nozzle throat has reached the local speed is not met at other engine operating conditions such as of sound. Changing the stagnation pressure and take off and landing, where there is a significant temperature through it can only alter the flow from a reduction in compressor efficiency. The practical need fixed convergent nozzle. Therefore the purpose of the to operate engines at off design conditions limits the gas generator (i.e. the compressor, combustor and the pressure ratio that a multiple stage compressor can turbine) is to provide the propelling nozzle with a flow attain if connected to a single-spool. of gas sufficient to choke the propelling nozzles. If we control the stagnation pressure and/or temperature of As the thermodynamic efficiency of an engine is a the flow through the propelling nozzles, we can vary function of pressure ratio, a second and sometimes third the energy transfer to the aircraft. spool is required to drive compressors. This minimises the pressure that the compressor fitted to a single spool It is normal practice to convert turbine and compressor needs to achieve. Modern multi-spool compressors can isentropic efficiencies into polytropic efficiencies for achieve pressure ratios of around 45:1. reasons that need not concern us here (see for example, Cumpsty (2006), Cumpsty (2004) or Dixon (2008)). In this work, we consider two-spool rather three-spool This polytropic efficiency will also remain constant engines. Turbofan engines require a minimum of two- throughout the turbine or compressor. Unlike the spools. Typically, the low-pressure spool drives the fan. isentropic efficiency, it is included in the relationship The role of the fan is two fold. Firstly, it acts as a between the temperature ratio and the pressure ratio compressor pressurising air prior to entering the engine across turbine and compressors. core where a multiple stage compressor driven by the high-pressure spool further compresses the air. 2. ANALYSIS USED IN THE PRESENT Secondly, the fan accelerates a much larger mass flow SIMULATION BASED ON CUMPSTY”S of air, which improves the propulsive efficiency of the REASONING engine including reducing energy lost do to noise.

The intention of our present work is to simulate the 1.2. A note on some of the terminology used in this behaviour of different configurations of gas turbine work engines as an aid to teaching aerospace propulsion to

It will be noted that in this paper that we specify undergraduate students at our university. Our approach temperatures and pressures as stagnation quantities T differs from the industry codes in that it takes advantage 0 of the simplifications suggested in the analysis of gas and P0 rather than as the more common static quantities T and P. For example, we specify the stagnation turbines by Cumpsty (2006). temperature of air at the inlet to the compressor as T02. The stagnation temperature is the temperature at a gas 2.1. The simplifications can achieve if bought to rest without loss of energy These simplifications arise from aspects common to all through transfers of either heat or work in either a turbine maps. Turbine maps are basically graphs which reversible or irreversible manner. Atkins (2007) depict the relationship between the pressure ratio across provides a good introduction to the concepts of heat, a turbine with either the non-dimensional mass flow work and reversibilility. The airflow through a gas rate through the turbine or the isentropic efficiency of turbine engine is always compressible and so we the turbine. specify its velocity as a Mach number M. The Mach number is a dimensionless number, which expresses the The first simplification arises from the turbine maps ratio of the flow velocity to the local speed of sound. which show the relationship between pressure ratio and Stagnation temperature is therefore expressed as: non-dimensional mass flow rate per unit area. Cumpsty argues that curves plotted at differing constant non- ⎡ ⎛ − 1k ⎞ 2 ⎤ dimensional spool rotational speed, practically collapse 0 ⎢1TT += ⎜ ⎟ M ⎥ (2) ⎣ ⎝ 2 ⎠ ⎦ on top of one another. This means that the relationship between the pressure ratio and the non-dimensional where k is the ratio of specific heat capacities. The mass flow rate is essentially independent of the stagnation pressure relies on the assumption that the rotational speed of the spool. Although properly flow is bought to rest in a reversible manner and is designed turbines and nozzle guide vanes do not expressed as: operate in the choked condition, we simplify the

analysis greatly if we accept that they behave as though and they are effectively choked. Therefore the non- 2 η p ()−1k dimensional mass flow rate per unit area m is taken to T ⎛ A ⎞ 2k η p ()−− 1k 05 = ⎜ 4 ⎟ . (8) be approximately constant, since for choked flow, the ⎜ ⎟ T04 ⎝ A9 ⎠ normalised mass flow rate per unit area is dependent only on the isentropic index of the gas k = ( cp / cv ) As a result, we can express both the stagnation pressure where c and c are the specific heat capacities of the p v ratio (P04 /P05) and the stagnation temperature ratio gas at constant pressure and volume (see Çengel & (T04 /T05) across the turbines as functions of the ratio of Boles, 2006 for a review of these thermodynamic the turbine inlet area to the nozzle exit area A4 /A9 . terms). We see this dependence in equation 4 below: P /P and T /T become constants once A /A is ( +− 1k ) 04 05 04 05 4 9 Tcm fixed and are therefore independent of the rotational & 04p k ⎛ + 1k ⎞ ()−1k2 (4) m4 = = ⎜ ⎟ . speed of the spool. Engineers check the effect of A /A PA 1k 2 4 9 044 − ⎝ ⎠ through experiment. Its effect on the position of the engine operating curve with respect to the compressor Similarly, the second simplification comes from the surge or stall curve on the compressor map is one thing turbine maps which show the relationship between we intend demonstrating to students through our pressure ratio and isentropic efficiency. Again, simulation. Cumpsty argues that curves plotted at differing constant non-dimensional spool rotational speed, practically The stagnation pressure ratio across the compressor (P03 collapse on top of one another. This means that the /P02) is dependent on the rotational speed of the spool. relationship between the pressure ratio and the The requirement that the work output of the turbine isentropic efficiency is essentially independent of the matches the work absorbed by the compressor provides rotational speed of the spool. Following on from a further constraint, i.e.: comments made in section 1.2, we find that: −=− TTcTTc . a ( )(e 0504p0203p ) (9) η p ()−1k

⎛ ⎞ k T05 P05 (5) where c is the specific heat capacity of the exhaust gas = ⎜ ⎟ . pe T04 ⎝ P04 ⎠ and cpa is the specific heat capacity of the air passing through the compressor. Once the pressure ratio across These simplifications hold true for practically all engine the turbine is fixed, the pressure ratio across the operating conditions. compressor is also fixed. We can express the term in brackets on the right hand side of equation 9 in terms of 2.2. Analysis of single-spool turbojet engine the area ratio A4 /A9 which in the real engine is fixed and therefore is a constant. Making use of the Following Cumpsty’s approach which assumes that the relationship between the stagnation temperature ratio propelling nozzle is choked and that the nozzle guide and stagnation pressure ratio for a compressor, i.e.: vanes upstream of the turbine are effectively choked, it follows that: ()−1k T ⎛ P ⎞ η pk 03 ⎜ 03 ⎟ (10) ( +− 1k ) = ⎜ ⎟ . T02 ⎝ P02 ⎠ k ⎛ + 1k ⎞ ()−1k2 (6) mm 94 == ⎜ ⎟ . − 1k 2 ⎝ ⎠ it is possible to write an expression for the stagnation pressure ratio across the compressor as: Generally, for air, we take k to be 1.4 whilst for exhaust gases; we take k to be 1.3. This results in the normalised η p k mass flow rate per unit area becoming a number P ⎛ ⎛ T ⎞⎞ ()−1k depending on the isentropic index of the gas. In this 03 ⎜1 += β ⎜ 04 ⎟⎟ (11) P ⎜ ⎜ T ⎟⎟ work as we give the equation, we can easily 02 ⎝ ⎝ 02 ⎠⎠ accommodate changes in k in the simulation. Making use of equation (5) and assuming that the stagnation ⎛ 2 η p ( −1k ) ⎞ ⎜ ⎟ c ⎛ A ⎞ 2k η p ()−− 1k conditions downstream of the propelling nozzle are the where the constant β pe ⎜1 −= ⎜ 4 ⎟ ⎟ . same as those upstream of the propelling nozzle, it c ⎜ ⎜ A ⎟ ⎟ pa ⎜ ⎝ 9 ⎠ ⎟ follows that the pressure ratio and the temperature ratio ⎝ ⎠ across the turbine can be expressed as functions of the area ratio ( A4 / A9 ). the polytropic efficiency and the It can be shown that the normalised mass flow rate isentropic index of the gas, namely: through the compressor is equal to the following:

-2k

⎛ ⎞ η p ()2k-1-k P05 A4 (7) = ⎜ ⎟ P04 ⎝ A9 ⎠

( +− 1k ) −1 ( −1k ) k + 1k ()−1k2 ⎛ T ⎞ η k ⎛ ⎞ 04 T023 ⎛ P023 ⎞ p m2 = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ (18) 2 ⎜ T ⎟ ⎜ ⎟ − 1k ⎝ ⎠ ⎝ 02 ⎠ T02 ⎝ P02 ⎠ 1 (12) and 2 ⎛ c ⎞ ⎛ A ⎞⎛ P ⎞ ( −1k ) × ⎜ p ⎟ ⎜ 4 ⎟⎜ 03 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎛ PT ⎞ η pk c p ⎝ A2 ⎠⎝ P02 ⎠ 03 03 (19) ⎝ e ⎠ = ⎜ ⎟ . 023 ⎝ PT 023 ⎠ Knowledge of the normalised mass flow rate per unit area and the pressure ratio enable us to plot the It is possible to write an expression for the stagnation operating point on a compressor map. Note that the pressure ratio across the high-pressure compressor as: pressure ratio across the compressor is expressed in η p k terms of the ratio of inlet temperatures (T04 / T02 ). The P ⎛ ⎛ T ⎞⎞ ()−1k altitude and velocity of the incoming airflow with 03 = ⎜1 + β ⎜ 04 ⎟⎟ (20) ⎜ HPS ⎜ ⎟⎟ respect to the aircraft at which the aircraft operates P023 ⎝ ⎝ T02 ⎠⎠ determines T02. The quantity of energy released in the combustion chamber through the burning of fuel ⎛ 2 η p ()−1k ⎞ determines T . T is usually limited to around 1700K c ⎜ 2k η ()−− 1k ⎟ 04 04 pe ⎛ A4 ⎞ p due to material considerations. where the constant βHPS ⎜1 −= ⎜ ⎟ ⎟ . c ⎜ ⎜ A ⎟ ⎟ pa ⎜ ⎝ 45 ⎠ ⎟ ⎝ ⎠ 2.3. Analysis of two-spool turbojet engine The HPS subscript stands for high-pressure spool. For Figure 1b) shows the number regime used in the the low-pressure compressor, we express the pressure analysis. In the two-spool engine, a turbine drives each ratio as: spool. The flow is effectively choked at the entry to both turbines, i.e., at station 4 and 45. Therefore, ηpk P ⎛ ⎛ T ⎞⎞ ()−1k ()+− 1k 023 ⎜ 04 ⎟ (21) = 1 + βLPS ⎜ ⎟ ()−1k2 ⎜ ⎜ ⎟⎟ k ⎛ + 1k ⎞ P02 ⎝ ⎝ T02 ⎠⎠ mmm 9454 === ⎜ ⎟ − 1k ⎝ 2 ⎠ (13) 2 η ()−1k . p c p ⎛ A ⎞ 2k η p ()−− 1k where the constant β = e ⎜ 4 ⎟ LPS c ⎜ A ⎟ Again, applying the same type of analysis as for the pa ⎝ 45 ⎠ single spool engine, it follows that the pressure ratio ⎛ 2 η p ()−1k ⎞ ⎜ ⎟ across the two turbines are: ⎛ A ⎞ 2k η p ()−− 1k ⎜1 −× ⎜ 45 ⎟ ⎟ . ⎜ ⎜ A ⎟ ⎟ -2k ⎜ ⎝ 9 ⎠ ⎟ P ⎛ A ⎞ η p ()2k-1-k ⎝ ⎠ 045 = ⎜ 4 ⎟ (14) P ⎜ A ⎟ 04 45 ⎠⎝ The LPS subscript stands for low-pressure spool. and Equation 12 again determines the normalised mass flow -2k rate through the low-pressure compressor. The ⎛ ⎞ η p ()2k-1-k P05 A45 (15) normalised mass flow rate through the high-pressure = ⎜ ⎟ , P045 A9 ⎠⎝ compressor is:

( +− 1k ) While the temperature ratios across the two-spools are: −1 k ⎛ + 1k ⎞ ()−1k2 ⎛ T ⎞ ⎜ 04 ⎟ m23 = ⎜ ⎟ ⎜ ⎟ 2 η p ()−1k − 1k ⎝ 2 ⎠ ⎝ T023 ⎠ T ⎛ A ⎞ 2k η p ()−− 1k (22) 045 = ⎜ 4 ⎟ (16) 1 T ⎜ A ⎟ ⎛ c ⎞ 2 ⎛ A ⎞⎛ P ⎞ 04 45 ⎠⎝ × ⎜ p ⎟ ⎜ 4 ⎟⎜ 03 ⎟ and ⎜ c ⎟ ⎜ A ⎟⎜ P ⎟ ⎝ pe ⎠ ⎝ 23 ⎠⎝ 023 ⎠ 2 η ()−1k p 2k η ()−− 1k T05 ⎛ A45 ⎞ p (17) = ⎜ ⎟ . Again, knowledge of the normalised mass flow rate per T045 A9 ⎠⎝ unit area and the pressure ratios enable us to plot the operating point on compressor maps. Again, making use of the relationship between the stagnation temperature ratio and stagnation pressure 2.4. Analysis of two-spool turbofan engine ratio for each compressor, i.e.: Figure 1c) shows the number regime used in the analysis. Again the flow is effectively choked at the entry to both turbines, i.e., at station 4 and 45. Therefore, equation 13 is still valid as it applies to the

core flow. The flow in the by-pass duct at station 19 is The term in brackets represents the temperature ratio also choked. for most operating conditions. The across the fan, i.e., T013 / T02. To find this temperature normalised mass flow rate of air per unit area th rough ratio, the term in brackets representing the temperature the fan will be: ratio and equation 25 must be solved in an iterative solution. Note that in the above analysis, the only time ( +− 1k ) we use an iterative solution is for the fan and never for & Tcm 013pbp k ⎛ + 1k ⎞ ()−1k2 mbp = = ⎜ ⎟ (23) the compressors. The non-dimensional mass flow rate PA 01319 − 1k ⎝ 2 ⎠ through the fan is given in equation 23. . 3. SOME REMARKS ABOUT THE SIMULATION Note that k for air is different than for the exhaust gas that passes through the turbine and propelling nozzles. We are currently in the process of developing an in The mass flow rate through the by-pass duct can house simulator based on the above simplifications. We therefore be written as: of course considered the adoption of existing simulation programs, such as GSP by NLR and GasTurb 11 ()+− 1k ⎛ ⎞ (Joachim Kurzke). GSP uses a comprehensive k ⎛ + 1k ⎞ ()−1k2 ⎜ 19 ⎟ PA 013 m& bp = ⎜ ⎟ combustion model but uses only generic component − 1k ⎝ 2 ⎠ ⎜ c ⎟ T (24) ⎝ p ⎠ 013 maps for compressors and turbines. We have no , intention of modeling combustion, as the user to examine a range of gas turbine operation will vary T04. Which Cumpsty approximates as: GasTurb 11 is coded in the Delphi programming environment. It also models the combustion process but 2.65 uses an iterative method to match the performance of m& bp ≈ α T013 , (25) the compressor to the turbine. We intend to use ‘Map Collection 2’ in our simulation which are available from where α is a constant. The mass flow rate through the Joachim Kurkze. As we use Cumpsty’s text in our core will be: aerospace propulsion course, our aim is to develop a

()+− 1k package that will clearly demonstrate the basic ⎛ ⎞ k ⎛ + 1k ⎞ ()−1k2 ⎜ 4 ⎟ PA 04 principals that we teach to our students. At the same m& core = ⎜ ⎟ − 1k ⎝ 2 ⎠ ⎜ c ⎟ T (26) time our program will allow key parameters such as k pe 04 ⎝ ⎠ and ηp to be easily changed. , We have taken a decision to program the simulator in where k is the isentropic index for air. The by-pass ratio Python. Python along with its scientific extensions has for a turbofan engine is defined as: the advantage of being free and open source. Also, Python is suited to object orientated programming. We m& bp therefore create components like compressors and BPR = turbines as objects within the Python environment. m& core Object orientated programming enables the Cum psty also suggests that the ratio of the temperature minimisation of program flow control. increase of t he air through that passes to the core to the to the temperature increase of the air that passes 4. ACKNOWLEDGEMENTS through the fan to the by-pass duct remains constant The authors would like to thank Mr. G. A. Fernando for regardless of whether the engine operates at design or his work in developing this simula tor. off-design conditions. Therefore,

− TT 02023 5. REFERENCES ξ = . (27) − TT 02013 N. Cumpsty; 2006, Jet propulsion. A simple guide to the aerodynamic and thermodynamic design and For the high and low pressure compressors, the pressure performance of jet engines, 2nd Edition, Cambridge ratios and the mass flow rates will be the same as for University Press. the two-spool engine (see equations 12, 20, 21 and 22). The difference is that for the turbofan engine, we need B. Gunston; 2006, The development of jet and turbine to know the pressure ratio across the fan. By applying aero engines, 4th Edition, Patrick Stephens Limited. the 1st law of thermodynamics for open systems to the low-pressure spool, it can be shown that: K. Hünecke; 2003, Jet engines, fundamentals of theory, design and operation, Airlife Publishing Limited, η p k England. P ⎪⎧⎡⎛ β ⎞⎛ T ⎞⎤ ⎪⎫()−1k 013 = ⎨⎢⎜ LPS ⎟⎜ 04 ⎟⎥ + 1⎬ . (28) Atkins, P; 2007, Four laws that drive the universe, P BPR + ξ ⎜ T ⎟ 02 ⎩⎪⎣⎢⎝ ⎠⎝ 02 ⎠⎦⎥ ⎭⎪ Oxford University Press.

A. F. El-Sayed; 2008, Aircraft propulsion and gas turbine engines, CRC Press, Taylor Francis Group, p- 508.

D. Crane; 2005, Aircraft maintenance technician series: Powerplant, 2nd Edition, Aviation Supplies and Academics, Inc., Newcastle, Washington, p-509.

N. A. Cumpsty; 2004, Compressor aerodynamics, Krieger Publishing Company, Malabar, Florida.

S. L. Dixon; 2005, Fluid mechanics, thermodynamics of turbomachinery, 5th Edition, Elsevier-Butterworth- Heinemann, Amsterdam, Boston.

Y. A. Çengel & M. A. Boles; 2006, Thermodynamics, an engineering approach, 5th Edition, McGraw Hill Higher Education. http://www.gspteam.com/main/main.shtml.

Joachim Kurkze; 2007, http://www.gasturb.de/Products/ GasTurb_11/gasturb_11.html.