NUMERICAL STUDY OF THE HEAT TRANSFER IN A MINIATURE JOULE-THOMSON COOLER
TEO HWEE YEAN
NATIONAL UNIVERSITY OF SINGAPORE
2004 NUMERICAL STUDY OF THE HEAT TRANSFER IN A MINIATURE JOULE-THOMSON COOLER
TEO HWEE YEAN (B.Tech Mech. Engrg (Hons.), NUS)
A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2004 ACKNOWLEDGEMENTS
Acknowledgements
There are many friends, colleagues and lecturers as well as institutions to whom I would like to express my thanks for their contribution and helpful information.
I would like to express my thanks to Prof. Ng Kim Choon for his valuable comments and useful assistance regarding the topics in heat transfer of fluids and thermodynamics.
I would also like to mention thanks for the kind foreword and the ideas and discussions from Assistant Prof. Chua Hui Tong and Dr Wang Xiaolin.
Last but not least let me express my warmest thanks to the National University of Singapore and A*STAR for giving me the opportunity and full support, without which this project could not have been completed.
Thank you.
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Filename:TeoHY.pdf TABLE OF CONTENTS
Table of Contents
PAGE
Acknowledgements i
Table of Contents ii-v
Summary vi
Nomenclature vii-x
List of Figures xi-xiii
List of Tables xiv
Chapter 1 Introduction 1
1.1 Background 1
1.1.1 Recuperative Heat Exchanger 1
1.1.2 Regenerative Heat Exchanger 4
1.2 Present Trend 9
1.2.1 Open Cycle Cooling Systems 9
1.2.2 Inefficiencies & Parasitic Losses in Real Cryocooler 10
1.3 Objectives and Scopes 12
Chapter 2 Joule-Thomson Cooler Fundamentals 16
2.1 Parameters & Characteristics 19
2.1.1 The Flows 19
2.1.2 Capillary Tubes 23
2.1.3 J-T Coefficients & Throttle Valves 25
2.2 Refrigeration Cycle 29
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2.2.1 Stage 5 to 1 31
2.2.2 Stage 1 to 2 31
2.2.3 Stage 2 to 3 31
2.2.4 Stage 3 to 4 32
2.2.5 Stage 4 to 5 32
2.3 Hampson-Type J-T Cryostat 33
2.4 Experimental Model 37
Chapter 3 Governing Differential Equations 40
3.1 Geometry Model 40
3.1.1 Helical Coil Capillary Tube 40
3.1.2 Helical Coil Fins 41
3.2 High Pressure Cryogen in the Helical Coil Capillary Tube 48
3.3 Helical Coil Capillary Tube 50
3.4 Helical Coil Fins 50
3.5 Shield 51
3.6 External Return Cryogen 51
3.7 Spacers 53
3.8 Entropy Generation for Internal Cryogen 54
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Chapter 4 Numerical Prediction 55
4.1 Computational Fluid Dynamics 55
4.2 Dimensionless Governing Differential Equations 57
4.2.1 High Pressure Cryogen (Single Phase Flow) 58
4.2.2 High Pressure Cryogen (Two Phase Homogenous Flow)
58
4.2.3 Helical Coil Capillary Tube 58
4.2.4 Helical Coil Fins 58
4.2.5 Shield 58
4.2.6 External Return Cryogen 59
4.2.7 Entropy Generation 59
4.3 Properties and Areas 60
4.3.1 Fanning Friction Factors 60
4.3.2 Convective Heat Transfer Coefficients 61
4.3.3 Thermodynamic and Transport Properties of Argon 61
4.3.4 Thermal Conductivities of Materials 68
4.3.5 Heat Transfer Areas 69
4.4 Boiling Heat Transfer 70
4.4.1 Nucleate Pool Boiling 71
4.4.2 Pool Film Boiling 74
4.4.3 Jet Impingement Boiling 75
Chapter 5 Results & Discussion 76
5.1 Temperature-Entropy (T-s) Diagram 77
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5.2 Cooling Capacity 79
5.3 Coefficient of Performance and Figure of Merit 83
5.4 Effectiveness and Liquefied Yield Fractions 84
5.5 Temperature and Pressure Distributions 86
Chapter 6 Conclusions & Recommendations 88
6.1 Conclusions 88
6.2 Recommendations 90
References R-1
Appendix A – Operation Manual for Simulation Program A-1
Appendix B – Fortran 90 Source Code – Main Program B-1
Appendix C – Fortran 90 Source Code – IMSL Subroutine (DBVPFD) C-1
Appendix D – Fortran 90 Source Code – IMSL Subroutine (FDJAC) D-1
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Summary
The miniature Joule-Thomson (J-T) cooler is widely used in the electronic industry for the thermal management of power intensive electronic components because of special features of having a short cool-down time, simple configuration and having no moving parts.
In this thesis, the sophisticated geometry of the Hampson-type J-T cooler is analyzed and incorporated into the simulation, so that the model can be used as a design tool. The governing equations of the cryogen, helical tube and fins, and shield are coupled and solved numerically under the steady state conditions, and yield agreements with the published experiments to within 3%.
The characteristics of flow within the capillary tube and external return gas are accurately predicted. The temperature versus entropy, cooling capacity versus load temperature, and cooling capacity versus input pressure charts are plotted and discussed. The conventional way of simulating a Hampson-type J-
T cooler, which is accompanied by a host of empirical correction factors, especially vis-à-vis the heat exchanger geometry could now be superseded.
The effort and time spent in designing a Hampson-type J-T cryocooler could be greatly reduced. By avoiding the use of empirical geometric correction factors, the model produces the real behavior during simulation.
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Filename:TeoHY.pdf NOMENCLATURE
Nomenclature
A Areas of contact m2 cp Isobaric specific heat J/(kg.K)
Coef Heat Transfer Coefficient W/(m2.K) cv Isochoric specific heat J/(kg.K)
D,d Diameter of tubes m ds Grid length along s-axis m d s Dimensionless grid length along s-axis - f Fanning friction factor - f(T,P) f is a function of T and P -
G Mass velocity kg/(m2.s) h Specific enthalpy J/kg k Thermal conductivity W/(m.K)
Ls Total length of capillary tube m m& Mass flow rate kg/s
M Molecular Weight g/mol
Mv Volumetric flow rate SLPM p Perimeter of heat transfer area m
P Pressure Pa or N/m2
Pitchm Pitch of capillary tube m
Pitchfin Pitch of fins m
C µ Pr Prandtl number = p - k
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q Heat transfer per unit mass W/kg
Q& Heat transfer W
Ro Universal gas constant J/(kg.K)
ρUD Re Reynolds number = H - µ
S& Specific entropy J/kg.s
T Temperature K u Average velocity of fluid m/s x Quality of fluid - y Liqufied yield fraction -
Greek Letters
α Helical angle
β Helical angle
γ Non-linear coefficient
λ Dimensionless conduction parameter
µ Fluid dynamic viscosity
µJ-T Joule-Thomson coefficient
σ Stefan-Boltzmann constant
ρ Fluid density
ε Emissitivity
θ Dimensionless temperature
Φ Dimensionless pressure for hot fluid
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Filename:TeoHY.pdf NOMENCLATURE
ψ Dimensionless pressure for returned fluid
Superscipts & Subscripts
0 Initial state
1,2,3,4,5 State points amb Ambient or room temperature and pressure conditions f High pressure incoming fluid fa High pressure vapor state in two-phase condition fl High pressure liquid state in two-phase condition finm Contact between capillary tube & fins (Area) fin Capillary fins fm Contact between high temperature fluid and capillary tube (Area) g Saturated fluid in gas state
H Hydraulic hel Helical in Inlet l Low pressure returned fluid m Capillary tube man Mandrel min Minimum ml Contact between capillary tube and returned fluid (Area) out Outlet pc Polycarbonate
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r Radiation ref Refrigeration s Shield sh Shield si Shield inside so Shield outside ss Stainless steel
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Filename:TeoHY.pdf LIST OF FIGURES
List of Figures
Page
Figure 1.1 Classification of Recuperative Cycles Heat Exchangers 2
Figure 1.2 Regenerative Cycles Heat Exchanger 4
Figure 2.1 Joule-Thomson Cycle and Temperature-Entropy Diagram 19
Figure 2.2 Contours of velocity head non-dimensionalized with ½ρU2 21
Figure 2.3 Development of the axial velocity fields at Re=104 & Pr=7 21
Figure 2.4 Development of secondary velocity fields at Re=104 & Pr=7 22
Extracted from [21]
Figure 2.5 Mean axial velocity distribution and vectors of means 22
secondary flows in curved and helically coil pipes
Extracted from [20]
Figure 2.6 The CFD models for the ordinary helix centerline 23
Extracted from [24]
Figure 2.7 The streamline patterns near the top of the arch. 23
Extracted from [24]
Figure 2.8 Typical J-T Cryostat Nozzle Schematic Diagram 26
Figure 2.9 Schematic of J-T Inversion Curve 28
Figure 2.10 Basic Joule-Thomson Cycle 30
Figure 2.11 A Real Hampson-type Joule-Thomson Cryocooler 35
Figure 2.12 Schematic of Hampson-Type Joule-Thomson Cryocooler 36
Figure 2.13 Schematic of Experimental Apparatus 37
Extracted from [12]
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Figure 2.14 Photograph of Experimental Apparatus Extracted from [12] 39
Figure 3.1 Helical Coil Notations – Capillary Tube 43
Figure 3.2 Helical Coil Notations – Fins 44
Figure 3.3 Elevation View of Helical Coil Capillary Tube and Fin 45
Figure 3.4 Plan View of Helical Coil Capillary Tube and Fin 46
Figure 3.5 Cross-Sectional View of Helical Coil Capillary Tube and Fin 47
Figure 3.6 Tf, Tl, Tm and Tfin Relations 50
Figure 4.1 Variation of Argon Density against Temperature 63
Figure 4.2 Variation of Argon Specific Heat against Temperature 63
Figure 4.3 Variation of Argon Entropy against Temperature 64
Figure 4.4 Variation of Argon Enthalpy against Temperature 64
Figure 4.5 Variation of Argon Viscosity against Temperature 65
Figure 4.6 Variation of Argon Thermal Conductivities against 65
Temperature
Figure 4.7 Temperature-Entropy Charts for Argon 67
Figure 5.1 Simulated T-s Diagram (CASE 1) 78
Figure 5.2 Simulated T-s Diagram (CASE 5) 78
Figure 5.3 Effect of the Load Temperature on the Cooling Capacity 79
Figure 5.4 Effect of the Input Pressure on the Cooling Capacity 80
Figure 5.5a Effect of the Normalised Volumetric Flowrate on the 81
Cooling Capacity
Figure 5.5b Effect of the Volumetric Flowrate on the Cooling Capacity 82
Figure 5.5c Effect of the Volumetric Flowrate on the Cooling Capacity 82
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Figure 5.6 Coefficient of Performance and Figure of Merit under 84
Different Inlet Pressure
Figure 5.7 Variation of Effectiveness & Liquefied Yield Fraction under 86
Different Inlet Pressure
Figure 5.8 Temperature and Pressure Distribution along the Finned 87
Heat Exchanger
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Filename:TeoHY.pdf LIST OF TABLES
List of Tables
Page
Table 1.1 Performance of Regenerative Cryocoolers 8
Table 1.2 Advantages & Disadvantages of Store Expendable 10
Cryogen
Table 2.1 Maximum Inversion Temperature 28
Table 2.2 Approximate inversion line locus for Argon (Perry, 1984) 29
Table 2.3 Dimensions of J-T Cryostat 37
Table 2.4 Experimental Data and Measured Results of T1 39
Table 4.1 Specifications of Dimensionless Parameters 59
Table 4.2 Thermal Conductivities of Materials 69
Table 4.3 Heat Transfer Specifications and Areas 69
Table 5.1 A Comparison between Experimental Data & Simulated 77
Results
Table 5.2 Variations of Effectiveness, Liquefied Yield Fraction and 85
COP under Different Input Pressure
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Filename:TeoHY.pdf CHAPTER 1 INTRODUCTION
Chapter 1 Introduction
This chapter presents a brief introduction of the different types of cryocoolers.
Heat exchangers based on different types of cooling cycles, namely recuperative and regenerative, were briefly discussed. The objectives and scope of the project are discussed at the end of this chapter.
1.1 Background
1.1.1 Recuperative Heat Exchangers
The recuperative cryocooler is analogous to a DC electrical device in the sense that the refrigerant flows steadily in a direction. This one-directional flow is often an advantage because they can transport the refrigerant over fairly large distances to do spot cooling at several locations. The recuperative heat exchangers have two separate flow passages and the streams continuously exchange heat with each other. Such heat exchangers are relatively inexpensive to manufacture.
There are three basic types of regenerative heat exchangers. These are characterized by their thermodynamic cycles of operation and names of original investigators, namely Linde-Hampson, Claude, and Joule-Brayton.
The configuration details are shown in Figure 1.1 below.
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COMPRESSOR RECUPERATIVE HEAT EXCHAGER
COOLING
J-T VALVE AFTER COOLING a) LINDE-HAMPSON TYPE HEAT EXCHANGER
COMPRESSOR RECUPERATIVE HEAT EXCHAGER
COOLING
J-T VALVE AFTER COOLING b) CLAUDE TYPE HEAT EXCHANGER COLD EXPANSION ENGINE
COMPRESSOR RECUPERATIVE HEAT EXCHAGER
COOLING
COLD EXPANSION AFTER ENGINE COOLING c) JOULE-BRAYTON TYPE HE AT EXCHANGER
Figure 1.1 Classifications of Recuperative Cycles Heat Exchangers
i. Linde-Hampson and Claude Type Heat Exchangers
The Joule-Thomson (J-T) cryocooler device is very similar to the
vapour-compression cycle used in household refrigerators except for
the use of a non-CFC refrigerant to reach cryogenic temperatures and
the need for a very effective heat exchanger to span such a large
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temperature difference. In a domestic refrigerator, oil from the oil-
lubricated compressor dissolves in the CFC refrigerants and remains in
solution even at the cold end.
The irreversible expansion that occurs at the J-T valve leads to cooling
only for non-ideal gases below the inversion temperatures. Nitrogen
and Argon gases are typically used for refrigeration at 77 K & 84 K
respectively, but the input pressure is usually about 200 bar in order to
achieve reasonable efficiencies. Hydrogen gas, pre-cooled by a
nitrogen stage, is used for refrigeration at 20 K, and a helium stage is
used to achieve 4 K. More often a 4 K J-T system is pre-cooled to 15 ~
20 K with a regenerative refrigerator.
Single-stage J-T coolers that use nitrogen or argon with miniature
finned-tube heat exchangers have been used in large quantities for
rapid (a few seconds) cool-down of infrared sensors. These open
systems use high pressure gas from a small storage cylinder.
ii. Joule-Brayton Type Heat Exchangers
Another common recuperative cryocooler is the Brayton cycle
refrigerator. An ideal gas such as helium or a helium-neon mixture can
be used on this cryocooler because of the reversible expansion that
occurs in either the reciprocating or turbo-expanders. As a result, only
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one fluid is required for all temperatures and much lower pressure
ratios are needed.
This cycle is commonly used in large liquefaction systems (with a final
J-T stage) and it has a high reliability due to the use of gas bearings on
the turbo-expanders. This cycle is generally not practical or efficient for
refrigeration powers less than 10 W at 80 K because of the machining
problems encountered with such small turbo-expanders. As a result, its
application to the cooling of superconducting electronics is rather
limited.
Wc Wc Wc Reservoir
Orifice Qc,Tc Q ,T Q ,T c c Qc,Tc h h
Regenerator Pulse Regenerator Displacer Regenerator Tube Displacer
Q ,T Q ,T QE,TE E E E E d) STIRLING TYPE e) PULSE TUBE TYPE d) GIFFORD-MCHAHON TYPE HEAT EXCHANGER HEAT EXCHANGER HEAT EXCHANGER
Figure 1.2 Regenerative Cycles Heat Exchanger
1.1.2 Regenerative Heat Exchangers
The primary heat exchanger is known as a regenerator or a regenerative heat exchanger. It consists of some form of porous material with high heat capacity, through which the working fluid flows in an oscillating manner. Heat is transferred from the fluid to a porous matrix (stacked screens or packed
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spheres) during the hot blow (fluid flowing from the warm end) and returned to the fluid from the matrix during the cold blow (fluid flowing from the cold end).
Because of the single flow channel, regenerators are very simple to construct.
The rapid decrease in heat capacity of most matrix materials at low temperatures causes a rapid decrease in regenerator performance below about 10 ~ 15 K.
As a result, all regenerative refrigerators are usually limited to temperatures above 8 ~ 10 K. The cryogen in nearly all regenerative systems uses helium gas. Temperatures down to about 50 K are usually achieved with single-stage cold heads, whereas two or more stages are used to achieve lower temperatures. From a thermodynamic stance, more stages lead to higher efficiencies, but the additional manufacturing complexity shall be considered in any practical device.
Typical frequencies of these cryogcoolers vary from about 2 Hz to 60 Hz. An oscillating displacer causes the working fluid to be compressed when it is at the warm end and to be expanded when it is at the cold end. There are four basic types of mechanical cryocooler which incorporates regenerative heat exchangers. These are generally classified by the thermodynamic cycle on which they operate, specifically:
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i. Stirling;
The Stirling refrigerator, which has the highest efficiency among all of
the regenerative cryocoolers, is the oldest and most common of the
regenerative systems. The Stirling cycle was invented for use as a
power system in 1816 and first commercialized as a cryocooler in 1954.
The schematic diagram of Stirling heat exchanger is shown in Figure
1.1 (c) above.
ii. Pulse Tube;
The pulse-tube refrigerator is a recent variation of the Stirling
refrigerator. The moving displacer is replaced by an orifice and
reservoir volume. The original version of the pulse-tube refrigerator was
developed in the mid-1960s, but a more powerful orifice version was
introduced in the 1980s. The pressure oscillation is most commonly
provided by a Stirling cycle compressor but a Gifford-McMahon
compressor and valves are sometimes used with a sacrifice in
efficiency.
In the pulse-tube refrigerator, the compressed, hot cryogen flows from
the pulse tube through the warm heat exchanger and the orifice. The
expanded cold cryogen in the pulse tube flows past the cold heat
exchanger when the cryogen from the reservoir returns to the pulse
tube. These systems are analogous to AC electrical systems.
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Except for the Gifford-McMahon refrigerator, the compressor or
pressure wave generator in the regenerative system has no inlet and
outlet valves. As a result, it produces an oscillating pressure in the
system, and void volumes must be minimized to prevent a reduction in
the pressure amplitude.
A thermo acoustic driver was used to drive a pulse-tube refrigerator in a
joint project between National Institute of Standards and Technology
(NIST) and Los Alamos National Laboratory in 1989. It achieved 90 K
and became the first cryocooler with no moving parts. The schematic
diagram of a pulse tube cycle heat exchanger was shown in Figure 1.1
(d) above.
iii. Gifford-McMahon;
Gifford-McMahon refrigerator was developed in the mid-1950s using
the same type of cold head as the Stirling crycooler. However, the
pressure oscillation is generated by using valves switch between the
high and low pressure sides of an air conditioning compressor modified
for use with helium gas. Oil in the high pressure gas is removed by
extensive filters and adsorbers before the gas enters into the cold head.
The use of valves to provide the pressure oscillation greatly reduces the
system efficiency compared with the Stirling cryocooler, but it allows the
use of inexpensive oil-lubricated compressors. These Gifford-McMahon
refrigerators, now manufactured by the thousands from cryopumps,
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magnetic resonance imaging shield cooling, ground-based satellite
communications systems, and research applications, are available in
both one and two stage units. The schematic diagram of Gifford-
McMahon heat exchanger is shown in Figure 1.1 (e) above.
iv. Vuilleumier.
Vuilleumier cryocooler uses an input of thermal energy at high
temperature to generate cyclic pressure fluctuations of the cryogen
contained in the closed volume of the unit. The pressure variations
were produced by the action of a reciprocating displacer shuttling the
working fluid periodically from an ambient temperature space to a high
temperature space through a regenerator. Extremely low temperature
up to 0.1 K can be produced by this approach.
The performance of regenerative cryocoolers is summarized in Table 1.1 below:
Table 1.1 Performance of Regenerative Cryocoolers Cooler Temperature Cooling Advantages Disadvantages Range Power Stirling 300 → 50 K 100 mW / 5 W Simple Poor efficiency Limited Compact autonomy (one shot), No moving parts Susceptibility to gas purity Pulse May replace Coolers in the Compact Efficiency may be slightly Tube G-M and near future Robust lower than Stirling Stirling No moving parts Reliable Gifford- 300 → 2.5 K 5 W / 200 W Simple Poor efficiency McMahon 1 W / 20 W Robust Induced vibrations Reliable Vuilleumier 100 → 0.1 K µ W / few W Compact Limited autonomy No moving parts Poor efficiency “Unlimited” lifetime Fully passive
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1.2 Present Trend
For the J-T cryocooler, significant improvement in efficiency has been made in the last few years by replacing pure nitrogen or argon with a mixture of nitrogen, methane, ethane, and propane. Temperatures of 80 K can be easily achieved with four or five times the efficiency of a nitrogen system with a lower pressure on the compressor output. An efficient, long-life compressor for a J-T refrigerator is still needed, but till to-date, no one has produced a comprehensive and accurate engineering model that predicts and analyses the behavior and flow of the cryogen in the cryocooler.
1.2.1 Open Cycle Cooling Systems
A widely used method for low capacity cryogenic refrigeration cycle involves the use of a stored, cold, expendable cryogen which eventually vaporizes and is vented to the atmosphere. The principal method is a solid, liquid or gas vaporizes and escapes from the storage dewar. The dewar may be opened to the atmosphere or sealed with a vent valve so that it is operated under pressure.
The advantages and disadvantages of the stored expendable cryogen are tabulated below:
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Table 1.2 Advantages & Disadvantages of Store Expendable cryogen Advantages Disadvantages Reliable and absolute guaranteed Inevitable loss of the cryogen due to cooling for a predictable period heat leaks and loses Low cost Periodic replenishment Uncomplicated storage dewars The continued loss of cryogen from stored cryogen requires the provision of relative large volume and mass storage requirements for extended operations. Easily obtainable Some cryogen such as Helium are not widely available in the market Quiet & no electromechanical effects Not subject to mechanical breakdown or unscheduled interruption of power supply
1.2.2 Inefficiencies and Parasitic Losses in Real Cryocoolers
The real cryocooler operates practically in a markedly different way from the ideal situations. The characteristics are briefly discussed below:
i. Compressor
In reality, the movement of the piston in a compressor is quasi-
sinusoidal. The expansion piston leads the compression piston by a
phase angle generally about 90°. This results in the overlapping in the
motion of the compression and expansion pistons thus inducing a
deformation of the ideal work diagram and a loss in efficiency.
ii. Dead Volumes
The cryogen in an ideal regenerator is usually assumed to be totally
expelled from the cold volume and the generator when it undergoes
compression. In real practice, the existing dead volumes “waste” part of
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the compression work. The reduction of void volume decreases the
efficiency in the heat transfer process.
iii. Pressure Drop
Pressure drops in the regenerator matrix and heat exchangers induces
a reduction of the amplitude of pressure variation in the expansion
space compared with the pressure variation in the compression space.
This results in a decrease of the specific refrigeration effect and a
relative increase of the compression work.
iv. Non-isothermal Operation
In an ideal regenerative cycle, reversible isothermal compression and
expansion processes are assumed. In a real machine, large variation of
the gas temperature is observed either in the compression or in the
expansion volume owing to the limited heat transfer surface area. This
results a significant loss in efficiency. When it is technically possible,
heat exchangers are introduced on both sides of the regenerator. The
heat transfer between the cycle working gas and the ambient or cold
heat sinks will be improved.
v. Regenerator or Counterflow Heat Exchangers Inefficiency
Thermal efficiency, ε, of a regenerator is defined as:
AmbientTemp − MeanTemp ε = AmbientTemp −ColdEndTemp
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In all cases, the efficiency is always greater than zero but lower than 1.
The efficiency is always dependent on the design of the heat
exchanger. Extensive effort should be devoted to theoretical and
numerical simulation of the heat exchanger in order to produce an
optimum heat exchanger.
vi. Thermal Losses
Thermal conduction along the walls of the heat exchanger reduces the
net cooling power of a practical cryocooler. High strength and low
thermal conductivity material is recommended for the application.
Limited experimental and theoretical works are reported in the literature due to its complexity of geometries, variable physical properties of compressible cryogen. This thesis presents the mathematical models as well as the complete governing equations of the flow in the J-T cryostat.
Nitrogen and Argon are typically used for refrigeration at 77K and 84 K, respectively. In this thesis, Argon has been selected as the cryogen due to its ease of availability, low cost and being able to achieve relatively low cryogenic temperature with no moving parts.
1.3 Objectives and Scopes
The objectives of this research are as follows:
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i. To establish a theoretical model to perform a numerical simulation to
predict the characteristics of the cryogen, argon along the helical coil
capillary tube on a miniature J-T cryocooler;
ii. To validate the simulation results against the experimental data
obtained from previous research;
iii. Improve the design of miniature J-T cryocooler using the computational
simulation instead of a “trial and error” approach, so that the optimum
design of Hampson-type miniature J-T cryocooler can be accurately
modeled and predicted.
iv. To eliminate the use of empirical correction factors, especially vis-à-vis
the heat exchanger geometry.
The boundary conditions are based on the data measured from the previous experiment. Thus the main assumption for this project is the data obtained from the previous experiments are valid and accurately measured.
This thesis consists of six chapters:
This chapter presents an introduction of the type of heat exchangers under difference working cycles, namely recuperative and regenerative, the present trend and the comparison of real versus ideal cryocooler. The objectives of this research project are also listed down.
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Chapter 2 discusses the fundamentals of the miniature J-T cryocooler. The refrigeration cycle, previous researches, theoretical and the experimental models were also discussed in this chapter.
Chapter 3 presents the mathematical model of helical capillary and fins, and the governing differential equations to the cryogenic flow inside the helical coil capillary tube. The continuity, momentum and energy equations for the steady- state flow inside the capillary tubes as well as the return fluid were discussed.
The conduction equations of capillary tube, the fins, the mandrel and the shield along the flow direction, are also coupled to form a complete engineering fundamental and were solved numerically.
Chapter 4 presents the governing differential equations which are cast in dimensionless form. The thermo-physical properties of the cryogen, Argon, which is used for the calculations, are also discussed. The thermo-physical properties were obtained from NIST [2]. The computational errors due to the inaccuracy of thermo-physical properties are minimized to the lowest possibility. The heat transfer areas are derived from first principle and mathematical models derived in Chapter 3. Nucleate boiling of jet impingement is incorporated in the calculation of the cooling load and performance of the J-T cryostat. However, the effect on jet impingement is relatively small compared to the two-phase cooling.
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Chapter 5 shows the results obtained from the numerical simulations and are compared with the experimental values. The Coefficient of Performance
(COP) and effectiveness of the heat exchanger were examined and calculated respectively. The Temperature-Entropy (T-s) diagram with the incorporation of the J-T inversion curve is presented. The trend is similar to a typical T-s chart in the literature. Effects of the load temperature and input pressure on the cooling capacity are also plotted.
Chapter 6 gives the conclusions and recommendations.
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Chapter 2 Joule-Thomson Cooler Fundamentals
The miniature Joule-Thomson (J-T) cryocooler has been a popular device in the electronic industry. It is widely used for rapid cooling of infrared sensors and electronics devices due to its special features of having a short cool-down time, simple configuration and no moving parts (Aubon [3], Joo et al. [4],
Levenduski et al. [5]). The cooling power of the Joule-Thomson cryocooler is generated by the isenthalpic expansion of a high pressure gas through a throttling (capillary) device, i.e. the Joule-Thomson (J-T) effect. The performance of this cooler is amplified and improved by using the recuperative effect of the expanded gas to pre-cool the incoming stream inside the capillary tube in a counter-flow heat exchanger arrangement.
Numerical studies on the J-T coolers have hitherto been focusing on the prediction of cool-down rates albeit with an extensive use of empirical correction factors for the heat exchanging geometry.
There are limited experimental and theoretical works reported on the prediction of the flow characteristics for the Hampson-type Joule-Thomson (J-
T) cooler. Maytal [6] analyzed the performance of an ideal flow regulated
Hampson-type Joule-Thomson (J-T) cooler. The prediction was not realistic because the heat-and-mass transfers among the cryogen, tube wall, Dewar and mandrel were not considered. Chou et al. ([7], [8]) reported
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CHAPTER 2 JOULE-THOMSON COOLER FUNDAMENTALS
experimental results and preliminary numerical predictions on the transient characteristics of a Hampson-type Joule-Thomson (J-T) cooler.
A one-dimensional model incorporating momentum and energy transport equations was presented. However, secondary flow, torsion effect caused by the helical capillary tube and fins, and the choking of flow were not considered.
Constant idealized heat transfer coefficients of the tube wall, Dewar and mandrel were used in the simulation although they actually vary with the temperature.
Chien et al. [9] simulated the transient characteristics of a self-regulating
Hampson-type Joule-Thomson (J-T) cooler. However, this paper concentrated primarily on the development of the self-regulating Hampson-type Joule-
Thomson (J-T) cooler by bellows control mechanism. The simulation approach was similar to that of Chou et al. [8]
Recently, Ng et al. ([10], [11], [12], [13]) simulated the performance of a
Hampson-type Joule-Thomson (J-T) cooler on its effectiveness, flow characteristics, heat conduction and liquefied yield fraction. Again, the torsion, secondary flow effect, and the choking of flow were not considered. Straight tube and straight fins were used in the simulation.
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In this thesis, the sophisticated geometry of the Hampson-type heat exchanger is analyzed and incorporated into the simulation model. The characteristics of high pressure gas, return gas, the mandrel, capillary tubes and fins are numerically simulated. The choking of flow in the capillary tube is also considered. The performance of the Hampson-type Joule-Thomson (J-T) cooler in steady state condition is accurately predicted. The conventional way of simulating the Hampson-type Joule-Thomson cooler, which is accompanied by a host of empirical correction factors, especially vis-à-vis the heat exchanger geometry could now be superseded. The effort and time spent in designing a Hampson-type Joule-Thomson (J-T) cooler could be greatly reduced. Since we have totally avoided the use of empirical geometric correction factors, the model is a very helpful design tool.
This thesis concentrates on the Linde-Hampson type miniature J-T cryocooler.
The stainless steel capillary tubes are finned with copper ribbon and wound in a helical annular space between two co-axial cylinders (White [14] and Barron
[15]). The schematic diagram of a typical process is shown in Figure 2.1 below:
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Compressor 300 1 5 1 5
Heat 270
Exchanger
240
HP BP HP BP
210
180
Heat
T (K) Exchanger 150
2
120
h
2 4 90
J-T Valve
3 60 3 4
2.5 3 3.5 4 4.5 5 5.5 6 6.5 S (kJ/kg.K) Qc
Figure 2.1 Joule-Thomson Cycle and Temperature-Entropy Diagram
2.1 Parameters & Characteristics
2.1.1 The Flows
Both the laminar and turbulent flows in helical coil pipes are subject to present research although some works have revealed the main characteristics of the flows. The curvature shape creates secondary motions, causes the difference in axial momentum between fluid particles in the core and wall regions. The core fluid encounters a higher centrifugal force than the fluid near to the outer wall which is pushed towards the inner wall. Eustice (1911) was the first person to present the concept of the secondary flow in helical coil pipes and
Taylor (1929) subsequently presented the secondary flow by injecting ink into the water, flowing through a coil pipe.
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Dean (1927) derived a solution for the low Reynolds number flow in helical coil pipes which exhibited a typical secondary flow pattern with two vortices. Dean number, De, was then introduced for characterizing the magnitude and shape of secondary motion of flow through a torus.
Adler (1934) presented experimental results of laminar and turbulent flow and
Wang (1981), used the perturbation method to solve the laminar helical coil problems with small curvature and torsion based on a non-orthogonal helical co-ordinate system. Patankar ([16], [17]) predicted the development of turbulent flow in curved pipes by a finite-difference approach. The details of the velocity contours are presented in Figure 2.2 as shown below.
Germano ([18], [19]) proposed the Germano number, Gn, which is used to describe the torsion effect on the flow in a helical coil pipe. Hϋttl [20] elaborated further on the influence of curvature and torsion of turbulent flow in a helically coil pipe. They performed several DNS on fully developed flow through toroidal and helical coil pipes and showed the turbulence structures appearing in instantaneous velocity fields. Lin and Ebadian ([21], [22]) investigated the effect of inlet turbulence level on the development of 3-D turbulent flow and the heat transfer in the entrance region of a helically coil pipe by means of fully elliptic numerical study. The results are shown in
Figures 2.3 and 2.4.
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0.5 0.7
0.8 0.8
0.9 0.9 0.7 Outside Outside 0.5 of Bend of Bend 0.8 0.9 0.5 0.7
(a) Angular Position along Bend = 0o (a) Angular Position along Bend = 45o
Figure 2.2 Contours of velocity head non-dimensionalized with ½ρU2
1.07 1.12 1.01 1.07
(a) Outer Outer
1.06 1.06 1.01 1.01
Outer Outer (b)
s/dh=8 s/dh=4
4 Development of the axial velocity fields at Re=10 and Pr=7: (a) I=2%; (b) I=40% Figure 2.3 Development of the axial velocity fields at Re=104 & Pr=7
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s/dn = 4 s/dn = 8 s/dn = 16
Figure 2.4 Development of secondary velocity fields at Re=104 & Pr=7 Extracted from [21]
Recently, Thomas J. Hüttl [20] computed the influence of curvature and torsion on turbulent flows in curved and helically coiled pipes. The details of mean axial velocity distribution and vectors of mean secondary flows in curved and helically coiled pipes were shown in Figure 2.5.
DT DT DT
DXXH DXXH DXXH
Mean axial velocity Vectors at the mean Turbulent kinetic component for toroidal secondary flow in energy for toroidal pipe (DT) and helical toroidal (DT) and helical flow (DT) and helical (DXXH) pipe flow (DXXH) pipes pipe flow (DXXH)
Figure 2.5 : Mean axial velocity distribution and vectors of means secondary flows in curved and helically coiled pipes Extracted from [20]
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Mori, Liu and Yamaguchi [24] presented the effect of the 3-D distortion on flow in the ordinary helix circular tube model of the aortic arch by Computational
Fluid Dynamic (CFD) solutions as shown in Figures 2.6 and 2.7.
The CFD models for the ordinary helix centerline
Figure 2.6 : The CFD models for the ordinary helix centerline Extracted from [24]
α=15o α=30o α=45o
Figure 2.7 : The streamline patterns near the top of the arch. Extracted from [24]
2.1.2 Capillary Tubes
Capillary tubes are commonly used as expansion and refrigerant controlling devices in small vapour compression refrigeration systems. They usually come in two types:
i. Adiabatic where refrigerant expands from high pressure to low
pressure;
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ii. Non-adiabatic capillary tubes where the refrigerant expands to low
pressure but the capillary tube is set up to form a heat exchange
relationship with the suction line.
Bolstad and Jordan [25] proposed an analytical solution for adiabatic capillary tubes based on the homogeneous flow and constant friction factor. The flow equations were solved based on the conservation of mass, momentum and energy using a simplified method. Marcy [26] developed the approach further based on viscosity for the calculation of two-phase Reynolds number. Hopkins
[27] made a graphical presentation based on the Bolstad, Jordan and Marcy equations. Cooper et al. [28] developed rating curves based on Hopkins’ work for capillary tube selection. Rezk and Awn [29] improved the charts further.
The analysis was later coupled with Whitesel [30] and ASHRAE [31] charts for capillary tube selection were produced.
Maczek and Krolicki (1981) used variable frictional factors and developed a model for the adiabatic capillary tubes. However, there were unexplained trends in the deviations between the calculated model and experimental data presented by him. Bansal and Rupasinghe [32] presented a simple empirical model for sizing both the adiabatic and non-adiabatic capillary tubes using
HFC-134a and suggested that the methodology can be extended to other refrigerants. Chien [9] and Chou [8] conducted the transient characteristics study of a self-regulating J-T cryocooler which predicted the transient behaviors and cool down times numerically. The calculations of the heat
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transfer areas were not discussed and the authors did not explain clearly how the equations were solved.
Ng, Xue and Wang [12] evaluated the performance of J-T cryocooler on effectiveness, flow and various heat conduction losses and liquefied yield fraction. This thesis is the continuation of the current research.
2.1.3 J-T Coefficients & Throttle Valves
The J-T expansion is an irreversible thermodynamic process resulting in a change in fluid temperature due to the change in fluid pressure at constant enthalpy, or namely the isenthalpic process. The flow of high pressure fluid is restricted by the smaller diameter of orifice or smaller cross-sectional area of the J-T nozzle. Flow pressure is reduced drastically at the nozzle tip where
“choked” or “shock” flow is usually occurred. The process is isenthalpic with no work done, nor heat or internal energy being generated.
If the temperature at the fluid before expansion is below the inversion temperature, it will be reduced during the isenthalpic process. If the fluid before expansion is above the inversion temperature, the temperature will be increased during the isenthalpic process. In addition, if the initial fluid temperature is sufficiently low and near the saturated temperature, the fluid will undergo a phase change during the expansion process and two-phase fluid will be generated.
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The development of the fixed orifice cryostat started since late 1950’s. The flow rate control was improved in 1960’s with the advent of the demand flow of
J-T cryostat. Brian and Sidney [33] carried out an experiment which used argon as working fluid to produce a two-phase jet which impinged horizontally onto a heated aluminium surface. The results showed that nucleate boiling phenomenon is independent of the mass flow rate of the jet, whereas the peak nucleate and film boiling heat flux valves were found to be mass flow dependent.
A common problem encountered with J-T nozzles is blockage of the flow orifice by condensed contaminants in the expanding fluid due to its tiny size. A typical J-T cryostat nozzle is shown in the Figure 2.8 below.
J-T Cooler Fin Stealing threat Valve seat
Glass Dewar Mandrel High pressure tube
Figure 2.8 Typical J-T Cryostat Nozzle Schematic Diagram
The basic expression for the enthalpy variation is defined as follows:
⎡ d ⎤ ⎛ ν ⎞ (2.1) dh = c p dT + ⎢ν − T⎜ ⎟ ⎥dp ⎣⎢ ⎝ dT ⎠ p ⎦⎥
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The ratio of the temperature variation (∂T ) to the pressure drop ()∂p can be determined during a J-T expansion. This ratio is known as the Joule-Thomson effect coefficient µ J − T and defined as:
⎛ ∂T ⎞ 1 ∂h 1 ⎡ ⎛ ∂υ ⎞ ⎤ µ = ⎜ ⎟ = ( ) = ⎢ν −T⎜ ⎟ ⎥ (2.2) J −T ⎜ ∂p ⎟ c ∂p T c ∂T ⎝ ⎠h p p ⎣⎢ ⎝ ⎠ p ⎦⎥
From the above relation, the J-T coefficient of a real working fluid is defined as the temperature drop of the fluid divided by its pressure drop under isenthalpic conditions.
There is no temperature variation associated with an isenthalpic expansion for an ideal gas. At sufficiently low temperature and pressure, µJ-T is usually positive for a typical real gas. At sufficiently high temperature and pressure,
µJ-T is usually negative. The temperature versus pressure graph can be plotted for a typical real gas under an isenthalpic process. The locus of zero-slope points is plotted and it is called inversion curve. Temperature increases with increasing pressures for an isenthalpic process, reaches a maximum point and then starts to decrease with increasing pressure. The temperature corresponding to the maximum point is called the “inversion temperature”. This is shown in the Figure 2.9 below. At zero pressure, the maximum inversion temperature has practical importance whereas the minimum is often omitted.
The maximum inversion temperatures of most commonly used fluids are shown in Table 2.1 below:
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Table 2.1 Maximum Inversion Temperature Fluid Max. Inversion Temperature (K) Oxygen 761 Argon 722 Nitrogen 622 Air 603 Neon 250 Hydrogen 202 Helium 40
From the inversion curve, the locus of points for which µJ-T = 0 is called the maximum inversion temperature. This curve is divided into two regions. For the zone within the inversion curve where the adiabatic J-T effect is positive, the decrease of pressure leads to decrease of temperature. For the zone within outside the inversion curve where the adiabatic J-T effect is negative, the decrease in pressure leads to an increase in temperature.
Temperature (K) 900
800 Max. Inversion Temp Tmax Isenthalpic Line
700
600
Inversion Line Heating Region U < 0 500 J-T
Cooling Region U > 0 Isenthalpic Line 400 J-T Max. Inversion Pressure
300 Critical Point Isenthalpic Line 200
T 100min Boiling Point Isenthalpic Line 0 Po Pma x 0 200 400 600 800 1000 Pressure (bar)
Figure 2.9 Schematic of J-T inversion line
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Francis and Luckhurst (1962) investigated the J-T coefficient and claimed that the law of “corresponding states” fits the isothermal J-T coefficient but not the adiabatic J-T coefficient because of the specific heat and this conforms to pure gases as well as their mixtures. Gunn et al. (1966) developed a generalized inversion curve to fit the experimental data of real gases that have a small acentric factor. Miller (1970) further correlated the experimental data of a number of real gases with a generalized inversion curve and concluded that the inversion curve could be described reasonably well by the two-parameter corresponding states theory for all gases but mentioned that this is not valid for
H2, He and Ne.
Table 2.2 Approximate inversion line locus for Argon (Perry, 1984) P, bar 0 25 50 75 100 125 150 175 200 225 Tmax, K 765 755 744 736 726 716 708 694 683 671 Tmin, K 94 97 101 105 109 113 118 123 128 134
P, bar 250 275 300 325 350 375 400 425 450 475 Tmax, K 657 643 627 610 591 569 544 515 478 375 Tmin, K 141 148 158 170 183 201 222 248 288 375
2.2 Refrigeration Cycle
The J-T cycle is a commonly used and well-understood refrigeration cycle.
Variations of this cycle are used in many applications such as home refrigerators and automotive air-conditioning because of its simplicity, reliability and efficiency. Different working fluids are used to achieve a wide range of cooling temperatures.
A typical refrigeration cycle is shown in Figure 2.10. A compressor is used to compress the working fluid from stage 5 to stage 1. Then the fluid is allowed to
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pass through a counter-flow heat exchanger from stage 1 to stage 2 which will be cooled by low-pressure return fluid. The high pressure fluid is expanded isenthalpically from stage 2 through a throttle valve of which produces two phases fluid to stage 3. The fluid is used to adsorb the heat load at constant temperature and pressure to stage 4. The low-pressure fluid will be channeled and returned to stage 5 and forms a complete refrigeration cycle. The details of the operation cycles are discussed below:
Temperature, T Isothermal compression 1 5
5’
Heat Exchange Q
2
Isenthalpic throttling
Tcr
Two phase region 3 4 Cooling Process
Entropy, S
Figure 2.10 Basic Joule-Thomson Cycle
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2.2.1 Stage 5 to 1
In this process, the gas is compressed to high pressure. This is an isothermal process and no work is done in the process. The work input required by the compressor in the isothermal process as follows:
P5 W&input = m& RT ln( ) (2.3) P1 where P5 & P1 are the pressure of the fluid at inlet and outlet, respectively
2.2.2 Stage 1 to 2
Pre-cool process by the low-pressure return fluid and is also known as isobaric process. The high pressure drops along the flow direction in capillary tube.
This is due to the conservation of energy through conduction and convection process and the conservation of momentum through frictional losses. The pressure and temperature drop in the process is usually considered as part of total throttling process.
Q&1~2 − Q& 4~5 = 0 (2.4)
2.2.3 Stage 2 to 3
This is an isenthalpic process, which is also known as the throttling process.
The high pressure fluid is throttled through the J-T throttle valve. The governing equation for this throttling process as follows:
h2 = h3 (2.5)
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where h2 and h3 denotes enthalpies at states 2 and 3, respectively. Change of entropy and the entropy generation have to be monitored closely in the process. “Choked” or “shocked” will be occurred when the change of entropy is equal to zero which leads to undesirable results obtained in the process. If the change of entropy is less than zero, the fluid obtains in stage 4 will be completely liquid. Care is taken in the design of throttle valves so that this phenomenon is avoided.
2.2.4 Stage 3 to 4
This is an isothermal saturated evaporation process. The liquefied fluid generates the refrigeration capacity and is used to cool the external heat load.
The governing equation for this isothermal process as follows:
Q& = m& f hfg = m& f ( hg − hf ) (2.6) where hfg is the latent heat of vaporization of the liquefied fluid.
2.2.5 State 4 to 5
This is a constant pressure heating process. The low-pressure return fluid is used to pre-cool the incoming high-pressure fluid. The low-pressure fluid is warmed up to a higher temperature before input to the compressor in stage 1.
It is to be noted that the low-pressure real fluid can only be warmed up to temperature at stage 5’ instead of ideal stage 5. Both this process and the isobaric pre-cool process (stage 1 to 2) are usually combined into a
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recuperative heat exchanger. The governing equation for this heating process is as follows:
(u 2 − u 2 ) Q& −W& = ∆h + ∆KE + ∆PE = h − h + out in + g( Z − Z )(2.7) out in 2 out in where Q& is heat transfer rate
W& is work input
∆h changes of enthalpy
∆K.E. changes of Kinetic Energy
∆P.E. changes of Potential Energy
For adiabatic and zero work process, the above equation reduces to
∆h = 0 ⇒ hout = hin (2.8)
2.3 Hampson-Type J-T Cryostat
A typical miniature Hampson-type J-T cryostat operating in an open cycle, single expansion device is used in this research project. The assembly details of the J-T cryostat are shown in Figure 2.11. This is a self-regulating counter- flow heat exchanger where high pressure fluid flows inside the helical coil capillary tube and low pressure return fluid flows within the helical coil capillary tube and the mandrel. Pressure drops along the helical coil capillary tube and throttles at the expansion valve and produces the cryogenic temperature at about 80 K. The saturated fluid then is used to pre-cool the high pressure fluid
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in the counter-flow heat exchanger before channels to the outlet of the J-T cryostat. The overall length and width of the J-T cryostat is about 100 mm and
5 mm in size, respectively. Its typical dimensions are shown in Table 2.3.
The Hampson-type miniature Joule-Thomson cryocooler has been gaining in popularity and is commonly used for the thermal management of power- intensive electronic devices due to its compactness, simple configuration and having no moving parts. Figure 2.12 shows the schematic of a typical
Hampson-type Joule-Thomson cryocooler used in the simulation. The main advantage this cryocooler is they can transport the refrigerant over a fairly large distance for spot cooling at several locations.
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Figure 2.11 A Real Hampson-type Joule-Thomson Cryocooler
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Incoming High Returning Low Pressure Gas Pressure Gas ÿØÿà
Mandrel
Nylon Strings (Spacers) Cross- Sectional Hampson- View type J-T Helical Coil Heat Capillary Tube Exchanger
Helical Coil Fins
Shield
Return Flow Path
Expansion Nozzle Cavity Electronics Chip
Figure 2.12 Schematic of Hampson-Type Joule-Thomson Cryocooler
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Table 2.3 Dimensions of J-T Cryostat Items Internal Diameter, Outer Diameter, mm mm Capillary Tube, helical pitch = 1.0 0.3 0.5 mm Mandrel 2.3 2.5 Shield 4.5 4.8 Fins, height = 0.5 mm, thickness = 1.0 mm, secondary pitch = 0.25 mm Length of heat exchanger = 50.0 mm
2.4 Experimental Model
An experimental model had been set up and performed by Xue H., Ng K.C. and Wang J.B. [12] in the year 2001. This section is extracted and modified from the previous research with consent from the writers. The schematic diagram for the experimental apparatus is shown in Figure 2.13.
Figure 2.13 Schematic of Experimental Apparatus Extracted from [12]
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Pure argon gas at 120-180 bars was obtained from a commercial argon tank.
The built-in gas regulator and gas filter helped to regulate and filter the argon gas to minimize the chances of choking due to impurities. The high pressure argon was expanded in the helical coil fined capillary tube where the J-T effect takes place. The argon gas was chocked at the nozzle tip at the end of the capillary tubes reached the vapor liquid equilibrium state corresponding to its saturation pressure and temperature. An external DC thick film resistant heater was incorporated to measure the cooling capacity up to 4.5 W.
Temperatures were measured by 0.5mm diameter Type K thermocouples with mineral-insulated stainless steel sheathed probes. The thermocouples were calibrated with an accuracy of ±1 °C. Three pressure transducers with signal conditioners and temperature compensated functions were utilized to record pressures. The pressure transducers were calibrated with a relative accuracy of 1%. A microbridge mass airflow sensor AWM 5104VA with the linearity error of ±3% reading, was used to record the heat transfer due to the airflow directed towards the surface. All sensors were connected to a HP 34970A data logger. The photograph of the setup is shown in Figure 2.6 below.
P2, T2, P5, T5, Mv and T1 were monitored in the experiment. The highest inlet pressure of P2=179.12 bar was used and minimum temperature of 108.70 K was achieved in the experiment. Five sets of different inlet pressure were tested and measured. The experimental values were recorded and tabulated in Table 2.4.
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Table 2.4 Experimental Data and Measured Results of T1 Extracted from [12]
P2 (bar) P5 (bar) Mv (SLPM) T2 (K) T5 (K) T’1 (K) 179.12 1.7272 13.927 291.49 110.36 282.57 169.86 1.7460 13.102 291.40 110.42 283.73 160.10 1.6362 11.943 292.25 109.90 284.77 149.66 1.4713 10.948 292.14 109.28 284.90 140.47 1.3426 10.145 291.94 108.70 284.98
Figure 2.14 Photograph of Experimental Apparatus Extracted from [12]
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Chapter 3 Governing Differential Equations
The performance of a Hampson-type Joule-Thomson (J-T) cooler could be determined if the amount of energy transfer among the high pressure gas, tube wall, fins and return gas at any instant is evaluated. This chapter presented the geometry model of the helical coil capillary tube and fins, and the governing differential equations to the cryogen flow inside and outside the helical coil capillary tube respectively. The governing equations for the energy balance, conservation of mass, the thermal conduction and radiation are discretized, coupled and simulated numerically. These governing equations are listed in the following sections.
3.1 Geometry Model
3.1.1 Helical Coil Capillary Tube
The geometry models of a helical tube and fins are derived and presented below. The notations are shown in Figures 3.1 and 3.2, respectively.
Let r = a cosθ ˆi + a sinθ ˆj+ bθ kˆ (3.1)
1 ˆr = ()acosθ ˆi + a sinθ ˆj + bθ kˆ (3.2) a2 + ( bθ )2
1 Tˆ = ()− a sinθ ˆi + acosθ ˆj + b kˆ (3.3) a2 + b2
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1 nˆ = [cosθ−(1+ b ) sinθ] ˆi + [sinθ+ (1+ b )cosθ] ˆj− []a(1+ 1 b 2 ) kˆ 1+(1+ b )2 + a 2 (1+ 1 b )2 (3.4)
ˆr, Tˆ and nˆ forms an orthogonal system.
3.1.2 Helical Coil Fins
ˆer( = cosβ ˆr + sinβ nˆ (3.5)
ˆeβ = − sinβ ˆr + cosβ nˆ (3.6)
ˆ ( ˆ eT = T (3.7)
ˆ ˆ ˆ ( where er( , eβ and eT form an orthogonal system
ˆ Therefore, at any T , the location of the helical fin, rfin is defined as follows:
rfin = r cosβ ˆr + r sinβ nˆ (3.8) and its absolute location, r2nd is described by
r2nd = r + rfin (3.9) or alternatively,
ˆ ˆ ˆ r2nd = f x i + f y j + f z k (3.10) where
⎡ b+a2 b ⎛ b+a2 b ⎞ ⎤ r(sin β) − cos θ −⎜b− ⎟sin θ ⎢ 2 2 ⎜ 2 2 ⎟ ⎥ rcos β 2+a b ⎝ 2+a b ⎠ f = acos θ+ (cos θ−sin θ )+ ⎣ ⎦ x 2 2 2 2 2 2+a b ⎛ b+a2 b ⎞ ⎛ b+a2 b ⎞ ⎡ 1 b+a2 b ⎤ ⎜ ⎟ +⎜b− ⎟ +a2 1− ( ) ⎜ 2 2 ⎟ ⎜ 2 2 ⎟ ⎢ 2 2 ⎥ ⎝2+a b ⎠ ⎝ 2+a b ⎠ ⎣ b 2+a b ⎦ (3.11)
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⎡ b+a2 b ⎛ b+a2 b ⎞ ⎤ rsin β − sin θ+⎜b− ⎟cos θ ⎢ 2 2 ⎜ 2 2 ⎟ ⎥ rcos β 2+a b ⎝ 2+a b ⎠ f = asin θ+ (cos θ+sin θ)+ ⎣ ⎦ y 2 2 2 2 2 2+a b ⎛ b+a2 b ⎞ ⎛ b+a2 b ⎞ ⎡ 1 b+a2 b ⎤ ⎜ ⎟ +⎜b− ⎟ +a2 1− ( ) ⎜ 2 2 ⎟ ⎜ 2 2 ⎟ ⎢ 2 2 ⎥ ⎝2+a b ⎠ ⎝ 2+a b ⎠ ⎣ b 2+a b ⎦
(3.12)
⎡ a ⎛ b + a 2 b ⎞⎤ r sin β a − ⎜ ⎟ ⎢ ⎜ 2 2 ⎟⎥ a r cos β ⎣⎢ b ⎝ 2 + a b ⎠⎦⎥ f z = bθ − − b 2 2 2 2 2 2 2 2 2 + a b ⎛ b + a b ⎞ ⎛ b + a b ⎞ ⎡ 1 b + a b ⎤ ⎜ ⎟ + ⎜b − ⎟ + a 2 1− ( ) ⎜ 2 2 ⎟ ⎜ 2 2 ⎟ ⎢ 2 2 ⎥ ⎝ 2 + a b ⎠ ⎝ 2 + a b ⎠ ⎣ b 2 + a b ⎦ (3.13)
⎡∂( f x ) ∂( f y ) ∂( f y ) ∂( f x )⎤ where the Jacobian, Jm = ⎢ ⋅ − ⋅ ⎥ , and f z is a constant ⎣ ∂β ∂r ∂β ∂r ⎦
(3.14)
The relationship between θ and β is defined as
2 2 θ −θ0 2π ⋅(θ −θ0 )⋅ a + b β − β0 = ⋅ 2π = (3.15) ∆θ Pitchfin
2 2 where Pitchfin = a + b ⋅ ∆θ and β0 = 0 when θ0 = 0 (3.16)
The surface of the helical fins is described by:
ˆ rsurf = rfin cosβ ˆr + rfin sinβ nˆ + ( rfin − r )tanα T (3.17)
where r ≤ rfin ≤ Wfin cosα+ r (3.18)
Hence its absolute description is given by
abs ˆ ˆ ˆ rsurf = acosθ i + a sinθ j + bθ k + rsurf (3.19)
The elevation, plan and cross-sectional views of the helical coil capillary tube and fin are presented in Figures 3.3, 3.4 and 3.5, respectively.
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Z (kˆ )
Helical Coil Fins
Helical Coil Capillary Tube r
θ a s Y (ˆj) X (ˆi )
Figure 3.1 Helical Coil Notations – Capillary Tube
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Details of Section “A”
Figure 3.2 Helical Coil Notations - Fins
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Helical Coil Capillary Tube Helical Coil Fins Pitchfin
Pitchm b
rfino rmi
rmo=rfini
a
Figure 3.3 Elevation View of Helical Coil Capillary Tube and Fin
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Figure 3.4 Plan View of Helical Coil Capillary Tube and Fin
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Mandrel Core
Shield
Spacer 2 Spacer 1 Capillary Fins Tube
Figure 3.5 Cross-Sectional View of Helical Coil Capillary Tube and Fin
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3.2 High Pressure Cryogen in the Helical Coil Capillary Tube
Continuity Equation
Since the capillary tube diameter is much smaller compared to the capillary tube length, approximately 1 to 1840, one-dimension steady state flow is assumed. The conservation of mass of the high pressure cryogen inside the helical capillary tube could be expressed as,
dm& f = 0 (3.22) ds where m& f = G f A f (3.23)
For compressible fluid, the continuity equation may be expressed as,
∂u ∂ρ ρ f + u f = 0 (3.24) f ∂s f ∂s where s refers to the natural helical axis
Momentum Equation
The pressure inside the capillary tube drops rapidly due to the high velocity of the flow and viscosity of the cryogen. The one-dimensional pressure drop along the natural helical direction (or the s-direction) of the capillary tube is given by,
dp 2 f ρ u 2 d( ρ u 2 ) f = − f f f − f f (3.25) ds Dmi ds or alternatively, it is expanded as,
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2 2 dp f 2 f f G f G f ⎡∂ρ f dT f ∂ρ f dp f ⎤ = − + ⎢ + ⎥ (3.26) ds ρ D 2 ∂T ds ∂p ds f mi ρ f ⎣⎢ f f ⎦⎥
where G f = u f ρ f (3.27) and for compressible fluid,
dρ (T , p ) ∂ρ dT ∂ρ dp f f f = f f + f f (3.28) ds ∂T f ds ∂p f ds
Energy Equations
The temperature of the high pressure cryogen varies along the capillary tube due to the drop of pressure, frictional loss and heat transfer between the gas and the tube wall. It could be expressed as,
⎡ dT ⎛ ∂ν ⎞ dp d( u2 2 )⎤ f ⎜ f ⎟ f f Coe f (Tm −T f )πDmi = G f Af ⎢c pf + ν f −T f + ⎥ ⎢ ds ⎜ ∂T ⎟ ds ds ⎥ ⎣ ⎝ f ⎠ ⎦
(3.29) or alternatively,
⎡ dT T ∂ρ dp G2 ∂ρ dT ∂ρ dp ⎤ ⎢ f 1 f f f f f f f f ⎥ Coe f (Tm −T f )πDmi = G f Af c pf + ( + ) − ( + ) ⎢ ds ρ ρ2 ∂T ds ρ3 ∂T ds ∂p ds ⎥ ⎣ f f f f f f ⎦ (3.30)
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Figure 3.6 Tf, Tl, Tm and Tfin Relations
3.3 Helical Coil Capillary Tube
The conductive energy balance equation in the helical coil capillary tube is,
2 d T Coe f (Tm −T f )( Afm ds ) Coe (T −T )( A ds ) 2kT (Tm −T fin )( Afinm ds ) m = − − l m l ml − 2 ds Amkm Amkm Amkm (3.31)
km ⋅ k fin where kT = (3.32) k fin ⋅W fin + km ⋅ H fin
3.4 Helical Coil Fins
The conductive energy balance equation for the helical coil fins wound around the helical coil capillary tube can be expressed as,
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2 d T fin Coel (T fin −Tl )( Afinl ds ) 2kT (T fin −Tm )( Afinm ds ) = − − (3.33) 2 ds Afink fin Afink fin
3.5 Shield
The conductive energy balance equation for the shield could be written as,
d 2T Coe (T −T )πD Coe πD (T 4 −T 4 ) sh = − l sh l si − r si sh amb (3.34) 2 dz Asiksh Asiksh or it could be expressed in terms of natural coordinates (or the s-direction) as,
d 2T Coe (T −T )πD ( L ds ) Coe πD (T 4 −T 4 ) sh = − l sh l si s − r si sh amb (3.35) 2 2 2 ds Asiksh( ds dz ) Asiksh( ds dz ) where the conversion factor between ds (or the s-direction) and dz (or the z- direction) is given by,
2 ds ( Pitch 2π ) + R 2 = m curve (3.36) dz Pitchm / 2π
3.6 External Return Cryogen
Continuity Equation
The conservation of mass is expressed as,
dm & l = 0 (3.37) dz where m& l = Gl Al (3.38)
For compressible fluid, the continuity equation is defined as,
∂u ∂ρ ρ l + u l = 0 (3.39) l ∂z l ∂z
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Momentum Equation
The cryogen pressure outside the return path drops gradually due to the pressure difference between the outlet and expansion cavity is small. The one- dimensional pressure drop along the primary direction (or the z-direction) of the heat exchanger is presented as,
dp 2 f ρ u 2 d( ρ u 2 ) − l = l l l + l l (3.40) dz DHl dz
where Gl = ul ρl (3.41)
For compressible fluid, the temperature and pressure-dependent density could be written as, dρ (T , p ) ∂ρ dT ∂ρ dp l l l = l l + l l (3.42) dz ∂Tl dz ∂pl dz or alternatively, the momentum equation is expanded as,
2 2 dpl 2 flGl Gl ⎡∂ρl dTl ∂ρl dpl ⎤ − = − + 2 ⎢ + ⎥ (3.43) dz ρl DHl ρl ⎣ ∂Tl dz ∂pl dz ⎦
The equation could be expressed in natural coordinates (or the s-direction) as dp 2 f G 2 G 2 ⎡∂ρ dT ∂ρ dp ⎤ l l l l l l l l (3.44) = − 2 ⎢ + ⎥ ds ρ l DHl ( ds dz ) ρ l ⎣∂Tl ds ∂pl ds ⎦ where the conversion factor between ds (or the s-direction) and dz (or the z- direction) is given by,
2 ds ( Pitch 2π ) + R 2 = m curve (3.36) dz Pitchm / 2π
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Energy Equation
The temperature of low pressure cryogen varies along the return path due to the pressure drop. The energy equation along the primary axial direction (or the z-direction) is written as,
⎡ 2 ⎤ Aml Afinl dTl ⎛ dνl ⎞dpl d(ul 2) Coe l(Tl −Tm ) +hl(Tl −Tfin ) +hl(Tl −Tsh )πDshi =Gl Al ⎢cpl +⎜νl −Tl ⎟ + ⎥ dz dz ⎣⎢ dz ⎝ dTl ⎠ dz dz ⎦⎥ (3.45)
Alternatively, the energy equation could be expressed in the natural direction
(or the s-direction) as,
Aml Afinl 1 Coel (Tl − Tm ) + hl (Tl − Tfin ) + []hl (Tl − Tsh )πDshi = ds ds ( ds dz ) 2 ⎡ dTl 1 Tl ∂ρl dpl Gl ∂ρl dTl ∂ρl dpl ⎤ Gl Al ⎢cpl + ( + 2 ) − 3 ( + )⎥ ⎣ ds ρl ρl ∂Tl ds ρ ∂Tl ds ∂pl ds ⎦
(3.46) where the conversion factor between ds (or the s-direction) and dz (or the z- direction) is given by,
2 ds ( Pitch 2π ) + R 2 = m curve (3.36) dz Pitchm / 2π
3.7 Spacers
Nylon strings, which possess an extremely low thermal conductivity, is used to wind round the helical capillary tube and fins, so as to limit the cross-sectional area available to the returning low pressure gas and thereby enhancing its contact with the fins and the primary helical capillary tube. In our model, we assume that the spacers play the sole role of limiting the heat exchange cross- sectional area.
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3.8 Entropy Generation for Internal Cryogen
The entropy generation equation is used to assess the choking position of the high pressure cryogen in the capillary tube and is expressed as, dS& ⎧ ⎡ dT ⎛ T dρ ⎞ dp ⎤ dp ⎫ ⎛ T −T ⎞ A gen ⎪ 1 f ⎜ 1 f f ⎟ f 1 f ⎪ m f fm (3.47) = m& f ⎨ ⎢cpf + + ⎥ − ⎬ − hf ⎜ ⎟⋅ ds T ds ⎜ ρ ρ 2 dT ⎟ ds ρ T ds ⎜ T ⎟ ds ⎩⎪ f ⎣⎢ ⎝ f f f ⎠ ⎦⎥ f f ⎭⎪ ⎝ m ⎠ or it is simplified as, dS& G A ⎡ dT T dρ dp ⎤ ⎛ T −T ⎞ A gen = f f c f + f f f − h ⎜ m f ⎟ ⋅ fm (3.48) ⎢ pf 2 ⎥ f ⎜ ⎟ ds Tf ⎣⎢ ds ρ f dTf ds ⎦⎥ ⎝ Tm ⎠ ds
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Chapter 4 Numerical Prediction
This chapter presented the governing differential equations which cast in dimensionless form. The thermo-physical properties of the cryogen, Argon, which is used for the calculations, were also discussed. The thermo-physical properties were obtained from the NIST [2]. The computational errors due to the inaccuracy of thermo-physical properties were minimized to the lowest possibility. The heat transfer equations are derived from first principle and the mathematical models derived in Chapter 3. Nucleate boiling of jet impingement was incorporated in the calculation of the cooling load and performance of the
J-T cryostat.
4.1 Computational Fluid Dynamics
The first attempt FLUENT [34] with the formulae stated in Chapter 3, was used to analyze and solve the flow characteristics.
The preprocessor, Gambit [35] was used as a tool to develop the sophisticated geometry of the counter-flow recuperative heat exchanger. The helical coil shape capillary fins and tubes were subsequently decomposed to hex meshing with control over clustering.
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See Figures 3.3, 3.4 and 3.5 of the helical coil capillary tube and fin developed by using Gambit [35].
However, only one circular out of 50 turns could be produced due to the limitation of the computational power and the complexity of the helical coil shaped. The torsion effect of fins was clearly shown in the geometry. The
“secondary helical coil fins” do not repeat by itself after second turn.
The geometry was first developed with Super Computer HP J6700 powered with 2 PA RISC 8600 750 MHz processors and a HP Visualize-fx10 Pro graphics accelerator, with 4 GBytes of physical memory, by Graphic User
Interface (GUI) method. However, only one quarter of a complete circle could be created due to low physical memory.
Then SGI Origin 2000 Super Computer powered by 16 MIPS R10000 CPUs in a cache-coherent non-uniform memory access (cc-NUMA) architecture with
16-CPU of total performance of 7120 Mflops with 6 GBytes of memory was used to developed the geometry by batch processing method.
Another Super Computer, the Compaq GS320 alphaserver (code named
Wildfire) configured with 22 EV67 731 MHz Alpha 21264 CPUs and 11 GBytes of physical memory, 22-CPU of total performance of 32000 Mflops was also used to develop the helical coil capillary tube and fins’ shape by a batch processing method.
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It took more than 3 weeks processing times to develop only a complete circular shape as shown in Figures 3.2, 3.3 and 3.4. The development process could not be continued further due to low physical memory encountered.
Therefore, it can be concluded that the performance of the Hampson-type J-T cryocooler could not be numerical simulated and evaluated by the
Computational Fluid Dynamic software.
Nevertheless, this approach gives a better understanding of the geometries of counter-flow heat exchanger. It was further developed by using the Microsoft
® Excel 2002, Copyright © Microsoft Corporation 1985-2001 spreadsheet and the cross-sectional view was produced as shown in Figure 3.5 above.
4.2 Dimensionless Governing Differential Equations
The governing differential equations were discretized and solved with boundary conditions at two points, using a variable order, variable step size finite difference method with deferred corrections in Fortran 90 programming language (Microsoft Developer Studio, Copyright © 1994-95 Microsoft
Corporation). The Microsoft IMSL Math Library, DBVPFD and DFDJAC subroutines were used in the computations. To ease the computational efforts, the governing differential equations were normalized and presented in dimensionless forms. Natural coordinate of s-axis was introduced in the calculations. The dimensionless governing differential equations are as follows:
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4.2.1 High Pressure Cryogen
Momentum Equation
2 2 P dΦ f 2 f f G f G f ⎡∂ρ f ∆T dθ f ∂ρ f P dΦ f ⎤ 0 0 (4.1) = − + 2 ⎢ + ⎥ Ls d s ρ f Dmi ρ f ⎣⎢∂T f Ls d s ∂p f Ls d s ⎦⎥
Energy Equation
Coef ∆T(θm −θ f )πDmi = ⎡ 2 ⎤ ∆T dθ f 1 ( ∆Tθ f +Tco ) ∂ρ f P0 dΦ f Gf ∂ρ f ∆T dθ f ∂ρ f P0 dΦ f Gf Af ⎢cpf +( + 2 ) − 3 ( + )⎥ ⎣⎢ Ls ds ρ f ρ f ∂Tf Ls ds ρ f ∂Tf Ls ds ∂pf Ls ds ⎦⎥ (4.2)
4.2.2 Helical Coil Capillary Tube
2 ∆T d θ m h f ∆T(θ m −θ f )( A fm ds ) hl ∆T(θ m −θ l )( Aml ds ) 2kT ∆T(θ m −θ fin )( A finm ds ) 2 = − − − Ls d s Am km Am km Am km (4.3)
4.2.3 Helical Coil Fins
2 ∆T d θ fin hl ∆T(θ fin −θl )( Afinl ds ) 2kT ∆T(θ fin −θ m )( Afinm ds ) 2 = − − (4.4) Ls d s Afin k fin Afin k fin
4.2.4 Shield
2 4 4 ∆T d θ sh hl ∆T(θ sh −θl )πDsi ( Ls ds ) hrπDsi [( ∆Tθ sh + Tco ) − ( ∆Tθ amb + Tco ) ] 2 = − 2 − 2 Ls d s Asi ksh ( ds dz ) Asi ksh ( ds dz ) (4.5)
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4.2.5 External Return Cryogen
Momentum Equation
P dΨ 2 f G 2 G 2 ⎡∂ρ ∆T dθ ∂ρ P dΨ ⎤ 0 l l l l l l l 0 l (4.6) = − 2 ⎢ + ⎥ Ls d s ρl DHl ( ds dz ) ρl ⎣ ∂Tl Ls d s ∂pl Ls d s ⎦
Energy Equation
⎡ Aml Afinl ⎤ 1 ⎢Coel∆T(θl −θm ) + hl∆T(θl −θ fin ) + hl∆T(θl −θsh )πDshi ⎥ = ⎣ ds ds ⎦ ( ds dz ) 2 ⎡ ∆T dθl 1 ( ∆Tθl + Tco ) ∂ρl P0 dΨl Gl ∂ρl ∆T dθl ∂ρl P0 dΨl ⎤ Gl Al ⎢cpl + ( + 2 ) − 3 ( + )⎥ ⎣ Ls ds ρl ρl ∂Tl Ls ds ρ ∂Tl Ls ds ∂pl Ls ds ⎦ (4.7)
4.2.6 Entropy Generation
S dE G f A f ⎡ ∆T dθ f ( ∆Tθ f + Tco ) dρ f P dΦ f ⎤ Tm − T f A fm 0 n = ⎢c + 0 ⎥ − h L d s ( ∆Tθ + T ) pf L d s 2 dT L d s f ( ∆Tθ + T ) ds s f co ⎣⎢ s ρ f f s ⎦⎥ m co (4.8)
Table 4.1 Specifications of Dimensionless Parameters
Dimensionless Longitudinal Length : s s = Ls Dimensionless Pressures :
Pf Pl Φ = Ψl = f P P0 0
Dimensionless Temperatures : ∆T = Tamb −Tco
T f −Tco Tm −Tco Tsh −Tco θ f = θ m = θ sh = Tamb −Tco Tamb −Tco Tamb −Tco
Tl −Tco T fin −Tco Tamb −Tco θ l = θ fin = θ amb = Tamb −Tco Tamb −Tco Tamb −Tco
Entropy : S gen En = S0
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4.3 Properties and Areas
4.3.1 Fanning Friction Factors
The high pressure cryogen flowing through the helical coil encounters centrifugal force across the tube. This results in the formation of a pressure gradient between a maximum pressure at the outer wall and a minimum pressure at the inner wall. A secondary flow is superimposed on the main flow and the point of maximum velocity is moved towards the outer wall due to torsion. The frictional loss is much higher compared to the flow in a straight tube.
Timmerhaus and Flynn ([36], [37]) suggested an empirical expression of
Fanning friction factor for flow in a helical coil as follows:
Dmi −0.2 f f ( p f ,T f ) = 0.184(1.0 + 3.5 ) Re( p f ,T f ) (4.9) DHx
For cryogen flowing inside a circular duct, the Reynolds number is defined as:
ρ f u f DHf G f DHf Re( p f ,Tf ) = = (4.10) µ f ( p f ,Tf ) µ f ( p f ,Tf )
For cylindrical tube, the conventional definition for hydraulic diameter is:
Cross _ Sectional _ Area D = 4⋅ (4.11) Hx Wetted _ Perimeter
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For return cryogen, the Fanning friction factor is defined as,
−0.2 f l ( pl ,Tl ) = 0.184 Re( pl ,Tl ) (4.12)
4.3.2 Convective Heat Transfer Coefficients
Timmershaus and Flynn ([36], [37]) proposed an empirical equation to estimate the convective heat transfer for cryogenic process in a helical circular as follows:
High Pressure Cryogen
−0.2 −2 3 Dmo Coe f = 0.023c p G f Re Pr (1.0 + 3.5 ) (4.13) DHf
Low Pressure Cryogen
−0.4 −2 3 Coel = 0.26c p Gl Re Pr (4.14)
Radiative heat transfer coefficient between the ambient and the shield
σ Coer = (4.15) 1 ε s + ( Ash Ar )(1 ε r − 1) where es and er are emissitivities of the shield and ambient and σ is Stefan-
Boltzmann constant.
4.3.3 Thermodynamic and Transport Properties of Argon
Liquid argon is a clear, colourless fluid with properties similar to those of liquid nitrogen. At 1 atmospheric pressure liquid argon boils at 87.3 K and freezes at
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83.9 K. Saturated liquid argon has a larger molecular weight and it is denser than oxygen at 1 atmospheric.
Argon is present in atmospheric air in a concentration of 0.934 % by volume or
1.25 % by weight. Since the boiling point of argon lies between that of liquid oxygen and that of liquid nitrogen (slightly closer to liquid oxygen), a crude grade of argon (90 to 95 % purity) can be obtained by adding a small auxiliary argon recovery column in an air separation plant.
The thermo-physical properties of the real gas, Argon, were used for the simulations. The thermodynamic and transport properties were obtained from
NIST [2]. The thermodynamic properties of argon were determined with a
Helmhotz energy equation (FEQ), a modified Benedict-Webb-Rubin equation
(mBWR), and an extended corresponding states model (ECS). The viscosity and thermal conductivity values were determined with a fluid specific model and a variation of the ECS method.
The variation of density of Argon against temperature was plotted as shown in
Figure 4.1.
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1600
1400
1200
1000 3
/m 800 g K
, 200 bar y it s 600 160 bar Den
100 bar 400
60 bar 200
1 bar 0
-200 90 120 150 180 210 240 270 300 Temperature, K
Figure 4.1 Variation of Argon Density against Temperature
The specific heat of argon is a function of temperature and density. The variation of specific heat of Argon against temperature was plotted as shown in
Figure 4.2 below.
10
9
8
7 .K) g 6 /(k J
at, k 5 60 bar He
ific 4 Spec 3 100 bar
2 160 bar
1 200 bar 1 bar
0 90 120 150 180 210 240 270 300 Temperature, K
Figure 4.2 Variation of Argon Specific Heat against Temperature
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The variation of entropy against temperature was plotted as shown in Figure
4.3.
4
1 bar
3.5
3 gK) k ( 60 bar
kJ/ 100 bar
, 2.5 y 160 bar p
o 200 bar tr En
2
1.5
1 90 120 150 180 210 240 270 300 Temperature, K
Figure 4.3 Variation of Argon Entropy against Temperature
The variation of enthalpy against temperature was plotted as shown in Figure
4.4.
200
150
1 bar 100
60 bar g
/k 50 160 bar 200 bar , kJ 100 bar py l
tha 0 n
E 90 120 150 180 210 240 270 300
-50
-100
-150 Temperature, K
Figure 4.4 Variation of Argon Enthalpy against Temperature
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The variation of viscosity against temperature was plotted as shown in Figure
4.5.
300
250
200 , uPa.s
y 150 scosit Vi
100
160 bar 200 bar 50 60 bar 100 bar 1 bar
0 90 120 150 180 210 240 270 300 Temperature, K
Figure 4.5 Variation of Argon Viscosity against Temperature
The variation of thermal conductivity against temperature was plotted as shown in Figure 4.6.
0.16
0.14
0.12 K) m
W/( 0.1 ty, i tiv 0.08 duc l Con
a 0.06 m er
h 200 bar T 0.04 160 bar 60 bar 100 bar
0.02 1 bar
0 90 120 150 180 210 240 270 300 Temperature, K
Figure 4.6 Variation of Argon Thermal Conductivity against Temperature
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The uses of argon result largely from its property of inertness in the presence of reactive substances but also from its low thermal conductivity, low ionization potential, and good electrical conductivity.
i. Fabricated Metal Products – Argon is used in electric arc welding as an
inert gas shield, alone or mixed with other gases. It is used both with
consumable electrodes, most often similar in composition to the material
being welded, or with non consumable tungsten electrodes in order to
protect the weld from the atmosphere.
ii. Iron and Steel Production – Argon is used to mixed and degas carbon
steel and steel castings, to purge molds, and to provide neutral
atmospheres in heat treating furnaces.
iii. Nonferrous Metals Production – The major use of argon is nonferrous
metals production is to furnish an inert atmosphere in vessels in which
titanium tetrachloride is reduced to metal sponge by the Kroll process.
iv. Chemicals – Argon is used for blanketing and atmosphere control in the
production of chemicals. It prevents reaction with elements in the air and
used as a dilutent, reduces in a controlled fashion the rate of reactions
involving gases.
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900 500 300 200 700 300 100 1100 kg/m3 150 290 Const. 280 Density 140 270
260 Isobar 50 130 250 800
240 600 120 230 400
220 300 110 ) 210 200
200 150 100 30 20 190 Isenthalpy 100 Line mperature (K e 180
T 90 80 170 60 80 160 50
150 40 10 30 70 140 Saturated 20 130 Liquid 60 15 Line 30 120 10 10 6 110 -10 4
-30 2 100 -50 -70 0.1 kJ/kg 1 bar -90 90 -110 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 Specific Entropy (J/kgK)
Figure 4.7 Temperature-Entropy Charts for Argon
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v. Electronic Components – Argon atmospheres are used in integrated
circuit manufacturing, in single crystal growing furnaces, and as a carrier
gas in constructing the circuits.
vi. Other Uses – About 1 % of the demand for argon results from its use in
lamps. Infrared and photoflood lamps are filled with a mixture of about
88% argon and12% nitrogen. The high density and low reactivity of argon
help retard the rate of evaporation of the tungsten filament, thus
prolonging the life of the lamp. The nitrogen prevents arcing.
4.3.4 Thermal Conductivities of Materials
Temperature-dependent thermal conductivities of Copper, Monel, Stainless
Steel and Polycarbonate were used in the simulation. Copper was used as fins wound along the stainless steel’s capillary tube. The assembly was inserted into the shield, which is made by Monel alloy, and insulated with
Polycarbonate.
The relevant correlations were summarized as shown in Table 4.2 (Flynn [37] and Perry et al., [38])
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Table 4.2 Thermal Conductivities of Materials
Materials Correlations Relative k - W/(mk) Errors T - K (Perry [38]) Copper 0.2413T 2 − 47.775T + 2848,(60K ≤ T ≤ 100K ) (Fins) < 1.5% k fin = { 0.028T 2 − 1.525T + 608,(100K ≤ T ≤ 300K ) Monel Alloy k = 6.5169lnT −14.76,( 40K ≤ T ≤ 400K ) < 1.0% (Mandrel) ma Stainless Steel k = 5.0353lnT − 13.797,( 40K ≤ T ≤ 400K ) < 1.0% (Capillary Tube) m Polycarbonate k = 0.22,( 40K ≤ T ≤ 40K ) - (Shield) sh
4.3.5 Heat Transfer Areas
The heat transfer specifications and areas were computed and tabulated in
Table 4.3 as shown below:
Table 4.3 Heat Transfer Specifications and Areas
Descriptions Expression Length ds ds ds = ⋅ Pitch dz m
Length dfin Pitch D dfin = 74π ⋅ ( fin )2 + ( mo )2 2π 2
Total length of capillary tube Ls = 50 ⋅ ds
Total length of fins Lfin = 50 ⋅ dfin Total cross-sectional areas of π( D2 − D2 ) A = si Mandrel shield excluding mandrel core Tot 4 in dz direction Cross-sectional areas of rmi 2π Am = ∫∫[J m ]dβdr where a=1.5 mm capillary tube in dz direction 0 0 Cross-sectional areas of rspc1 2π Aspc1 = ∫∫[Jm ]dβdr where a=2.14 mm spacer 1 in dz direction 0 0 Cross-sectional areas of rspc 2 2π Aspc2 = ∫∫[Jm ]dβdr where a=1.375 mm spacer 2 in dz direction 0 0 Cross-sectional areas of fins in rfino 2π Afin = ∫∫[Jm ]dβdr where a=1.5 mm dz direction rfini 0
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Cross-sectional areas of return Al = ATot − Am − Aspc1 − Aspc2 − Afin fluid flow in dz direction
Al Hydraulic diameter of return fluid, DHx = 4 ⋅ π( Dsi + DMandrel ) + Lspc1 + Lspc2 + Lm + Lfin
2π 2 2 where Lspc1 = ∫0 f x + f y dβ , a=2.14 mm, f z =constant and r= rspc1
2π 2 2 Lspc2 = ∫0 f x + f y dβ , a=1.375 mm, f z =constant and r= rspc2
2π 2 2 Lm = ∫0 f x + f y dβ , a=1.5 mm, f z =constant and r= rm
β 2 2 2 L = f + f dβ , a=1.5 mm, f =constant, r= rfin , 0 ≤ β ≤ 2π & β1 > β2 fin ∫β1 x y z
Contact areas between fluid Afm = πDmi ⋅ ds and capillary tube per ds
Contact areas between fins Afinm = dfin ⋅Wfin and capillary tube per ds
Contact areas between return Aml = πDmo ⋅ ds − dfin ⋅Wfin fluid and capillary tube ⎡ Pitch D ⎤ Pitch D Contact areas between fins A = ⎢74π ⋅ ( fin )2 + ( mo + H )2 + dfin⎥H +74π ⋅W ⋅ ( fin )2 + ( mo )2 finl 2π 2 fin fin fin 2π 2 and return fluid per ds ⎣⎢ ⎦⎥ Areas of fluid flow in capillary πD2 A = mi tube along ds direction f 4 Cross-sectional areas of π( D2 − D2 ) A = mo mi capillary tube along ds m 4 direction
Cross-sectional areas of fins Afin = Wfin ⋅ H fin along ds direction Total areas of shield π( D2 − D2 ) A = so si si 4
4.4 Boiling Heat Transfer
Boiling heat transfer occurs in a convective heat transfer of fluid that encounters phase change. If the liquid is sub-cooled, there may be no net vapor generation and thus evaporation occurs.
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The two main categories of boiling are:
i. Pool boiling;
Pool boiling is similar to natural convection for single phase flow. It
occurs when a heated surface is immersed in a “pool” of liquid. Four
regions occur: Free convection evaporation, nucleate boiling,
transition boiling or unstable film boiling, and stable film boiling.
ii. Forced convection boiling;
This occurs when the fluid boils as it flows within a flow passage.
A saturated liquid not in contact with its own vapor does not boil, even if the heating surface is slightly warmer than the saturation temperature. This phenomenon arises because surface tension produces an additional force that maintains the vapor in bubbles at a pressure higher than the pressure of the liquid.
4.4.1 Nucleate Pool Boiling
Nucleate pool boiling correlation is incorporated to calculate the cooling capacity of the J-T cryocooler.
Rohsenow et al. [39] introduced the first correlation of nucleate boiling as follows:
1 3 C (T − T ) ⎡ σ ⎤ pl co sat Q A fl = Csf ⎢ ⎥ (4.16) C (Pr )n µ C ⎢ g( ρ − ρ ) ⎥ fg l ll fg ⎣ f l f g ⎦
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where Csf = 0.013 for cryogens except helium
Csf = 0.169 for helium
n = 1.0 for water and 1.7 for all other fluids
The total cooling capacity is calculated as follows:
( Q A )Total = ( Q A )Nucleate + ( Q A )Load (4.17) or
−1 3 µ C (T − T ) ⎡ σ ⎤ ( Q A ) = ll pl co sat ⎢ l ⎥ + h (T − T ) (4.18) Total C (Pr )n g( ρ − ρ ) l l co sf l ⎣⎢ ll la ⎦⎥
1 Q C (T − T ) ⎡ σ ⎤ 3 p f sat A f l = Csf ⎢ ⎥ (4.19) C (Pr )n µ C ⎢ g( ρ − ρ ) ⎥ Latent fl fl Latent ⎣ fl f g ⎦ where Csf = 0.013 for cryogens except helium
n = 1.0 for water and 1.7 for all other fluids
Another pool boiling correlation is that of Kutateladze (Brentari and Smith
1965; Kutateladze 1952), which has been verified as follows for cryogenic fluids including N2, O2, H2, and He:
0.3 Q ⎡ ⎤ σ ρ f Ja A fl g 0.7 = 0.0007 ⎢ ⎥ ( K p ) (4.20) (Pr)0.65 µ C ⎢ g( ρ − ρ ) ⎥ ρ fl Latent ⎣ fl f g ⎦ fl
P where K = sat (4.21) p gσ ( ρ − ρ ) fl fl f g
C (T − T ) and Jacob number, Ja = p f sat (4.22) CLatent
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Tong and Tang (1997) had carried out the prediction of pool boiling heat transfer. They observed that the nature of the heating surface (rough, smooth, etcetera) may have significant effect on the heat flux. In general, the scatter of pool boiling heat transfer data is somewhat greater than that of single-phase forced convection heat transfer.
The critical heat flux or the heat flux at the peak nucleate boiling point is an important point on the boiling point. For highest heat transfer rate in boiling, operation of thermal systems (evaporations, etcetera) near the critical heat flux is desirable. If the heat flux is increased slightly above the critical heat flux, the boiling pattern may change suddenly to film boiling, leads to undesirable high surface temperature.
The form of the relationship for the peak nucleate boiling heat flux was developed by Kutateladze (1948) from dimensional analysis and experimental data on boiling on tube surfaces. Zuber (1958) developed a similar expression with a slightly different numerical coefficient. Lienhard, Dhir, and Riherd (1973) developed the following relationship for larger flat-plate heaters:
1 / 4 ⎡ g( ρ − ρ )σ ⎤ fl f g fl ( Q / A )Max = 0.1492ρ f C Latent ⎢ ⎥ (4.23) g ⎢ ρ 2 ⎥ ⎣ f g ⎦
All fluid properties are evaluated at the fluid saturation temperature, Tsat. This expression is valid for (Bo)½ > 2.7, where the Bond number is defined as:
g( ρ − ρ )L2 Bond Number, Bo = fl f g (4.24) ρ f l
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The characteristic length, L, is the heater width.
For horizontal cylindrical heaters, the critical heat flux was correlated as follows by Sun and Lienhard (1970):
1 / 4 ⎡ g( ρ − ρ )σ ⎤ fl f g fl ( Q / A )Max = 0.1492KD ρ f CLatent ⎢ ⎥ (4.25) g ⎢ ρ 2 ⎥ ⎣ f g ⎦
The geometrical factor, KD, is as follows:
¼ KD = 0.781 + 1.993 exp[-2.432 Bo ] (4.26)
The characteristic dimension in the Bond number is the cylinder diameter.
4.4.2 Pool Film Boiling
Film boiling is an inefficient mode of heat transfer. It often happens in many cryogenic systems. During the initial cool-down immediately after the J-T valve, the surface temperature is sufficiently high that film boiling is almost always achieved.
For film boiling on a horizontal tube, the convective contribution to the heat transfer may be determined as follows from Westwater and Breen (1962):
¼ Nu = 0.62 KD (Rab/JaG) (4.27)
-½ KD = 1.0 for Bo >= 13.8 or KD = (0.6 + 0.442 Bo) for Bo < 13.8 (4.28) where the “film boiling Rayleigh number”, Rab is defined as:
gρ ( ρ − ρ )D3 Pr Ra = f g fl f g G (4.29) b µ 2 f g
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The Jacob number, JaG is defined as:
CG (Tf − Tsat ) JaG = (4.30) ( CLatent )e where CG = 64 for laminar flow, vapour (Re < 2000)
CG = 0.316 for turbulent flow, vapour (3,000 CG = 0.184 for vapour Re>50,000 ( C Latent )e = C Latent + 0.34CG (T f − Tsat ) (4.31) Nu.K .∆T ( Q / A ) = G (4.32) D where KG = Conductivity 4.4.3 Jet Impingement Boiling The jet impingement boiling correlation on the heated surface proposed by Brian [33] is used to estimate the heat flux at the load. Q A = 181.1463 ()∆T 1.218 (4.33) where ∆T is the temperature difference between the surface and the measured bulk fluid. National University of Singapore 75 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION Chapter 5 Results and Discussion This chapter presents the results from numerical simulation and discusses the deviation of the results obtained compared to the experimental results. The current simulation is applied to a Hampson type J-T cooler which is shown in the Figure 2.12. The simulation results are compared with the experimental data as shown in Table 5.1 . Only the relative errors of the outlet temperatures of the return cryogen are compared with the simulated results as suitable sensors were not available to measure the actual pressure and temperature in the capillary tube. It is noted that the measured return cryogen outlet temperature is slightly higher than the simulated results. This could be due to: i. the use of the polycarbonate instead of a Dewar flask, which inevitably results in the increased heat gain during the experiment. ii. readings of the temperature sensor at the return cryogen outlet are affected by the ambient conditions, on account of the minuteness of the exit port. However, it is observed that the relative errors between the simulations and experiments all fell within 3%. National University of Singapore 76 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION Table 5.1 A Comparison between Experimental Data & Simulated Results Case Pressure (bar) Temperature (K) Mv Temperature (Toutlet, K) Relative p1 p3 Tinlet Tsat=f(p3) (SLPM) Experiment Simulation Error, % 1 179.12 1.7272 291.49 92.68 13.927 282.57 276.93 2.00 2 169.86 1.7460 291.40 92.80 13.102 283.73 277.12 2.33 3 160.10 1.6362 292.25 92.11 12.060* 284.77 278.53 2.19 4 149.66 1.4713 292.14 90.99 10.948 284.90 279.20 2.00 5 140.47 1.3426 291.94 90.06 10.145 284.98 279.34 1.98 *The originally reported experimental value (11.943 SLPM) was erroneous. Since the experimental mass flow rates behave essentially linearly with the input pressure, the present value is obtained by linear interpolation. 5.1 Temperature-Entropy (T-s) Diagram Figure 5.1 and 5.2 show the simulated Temperature-Entropy diagram of the cryogen in the Hampson-type Joule-Thomson (J-T) cooler. The trend is similar to a typical T-S chart in the literature. However, it is contrary to the commonly held view of insignificant pressure drop in the initial expansion path, the pressure drops much more rapidly due to the higher frictional loss and expansion process in the high pressure gas channel than that in the low pressure channel. This in turn increases the cooling capacity of the Hampson- type Joule-Thomson (J-T) cryocooler and demonstrates the efficiency of the recuperative method in improving the performance of the Hampson-type Joule-Thomson (J-T) cryocooler substantially. The Joule-Thomson (J-T) inversion curve has also been plotted into the chart. To the right of the chart is where the cooling process occurs. National University of Singapore 77 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION 320 Input Pressure : 179.12 bar Isobar (1.7272 bar) Input Temperature : 291.49 K 1 280 5' JT_InversionCurve Isobar (1.55 bar) 240 Regeneration re (K) 200 Isobar (179.12 bar) eratu 2 Temp 160 Isobar (81.40 bar) 120 3 4 Two-Phase Cooling 80 0 1000 2000 3000 4000 5000 Specific Entropy (J/kgK) Figure 5.1 Simulated T-s Diagram (CASE 1) 320 Input Pressure : 140.47 bar Isobar (1.3426 bar) Input Temperature : 291.94 K 1 280 5' JT_InversionCurve Isobar (1.212 bar) 240 Regeneration re (K) 200 eratu p Isobar (140.47 bar) 2 Tem 160 Isobar (72.72 bar) 120 Isenthalpic Process 3 4 80 Two-Phase Cooling 0 1000 2000 3000 4000 5000 Specific Entropy (J/kgK) Figure 5.2 Simulated T-s Diagram (CASE 5) National University of Singapore 78 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION 5.2 Cooling Capacity Figure 5.3 presents the effect of the load temperature on the cooling capacity. With an increase in the cooling load temperature, the cooling capacity increases greatly. This corroborates with the basic theory commonly used in traditional air-conditioning and refrigerant systems. It is noted that the cooling capacity is linearly proportional to the load temperature. 9.00 8.00 Case 1 W 7.00 , y 6.00 g Capacit n oli Case 5 Co 5.00 4.00 3.00 85 90 95 100 105 110 115 120 125 Load Temperature, K Figure 5.3 Effect of the Load Temperature on the Cooling Capacity National University of Singapore 79 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION The higher the input pressure, the higher is the cooling capacity that could be achieved. This is evident from the simulation results as shown in Figure 5.4. Within the simulated range, the cooling capacity increases as the input pressure increases. It is observed from the chart that the cooling capacity increases gently at the lower range of pressures while it increases more rapidly at the higher range. 8.0 7.5 7.0 Cooling Load Temperature : 110 K 6.5 6.0 5.5 ooling Capacity, W C 5.0 Cooling Load Temperature : 95 K 4.5 4.0 140 150 160 170 180 Input Pressure, bar Figure 5.4 Effect of the Input Pressure on the Cooling Capacity National University of Singapore 80 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION Figures 5.5(a), (b) and (c) show the effect of the flow rate on the cooling capacity. For a valid range, the higher the flow rate supplies to the heat exchanger, the higher the cooling capacity could be achieved. The choked point happens when the entropy generation is equal to zero. 8.20 Case 1: P=179.12 bar, T=291.49 K, Mv=13.927 SLPM Case 4: P=149.66 bar, T=292.14 K, Mv=10.948 SLPM 7.80 Case 1 Choked Point 7.40 (W) 7.00 6.60 No J-T Invalid Capacity 6.20 Cooling Range oling Choked Point Co 5.80 Case 4 5.40 5.00 4.60 0.80 0.90 1.00 Normalised Volumetric Flow Rate Figure 5.5(a) Effect of the Normalised Volumetric Flowrate on the Cooling Capacity National University of Singapore 81 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION 8.1 Case 1: P=179.12 bar, T=291.49 K 8 7.9 Choked Point ) 7.8 W ( y t i c 7.7 pa Ca 7.6 g n i ol 7.5 Co 7.4 7.3 No J-T Cooling Invalid Range 7.2 11.5 12 12.5 13 13.5 14 14.5 Volumetric Flow Rate (SLPM) Figure 5.5(b) Effect of the Volumetric Flowrate on the Cooling Capacity 5.3 Case 4: P=149.66 bar, T=292.14 K 5.1 Choked ) Point W ( y t aci 4.9 ng Cap i Cool 4.7 No J-T Invalid Cooling Range 4.5 9.25 9.75 10.25 10.75 11.25 Volumetric Flow Rate (SLPM) Figure 5.5(c) Effect of the Volumetric Flowrate on the Cooling Capacity National University of Singapore 82 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION 5.3 Coefficient of Performance and Figure of Merit The Coefficient of Performance (COP) of the heat exchanger is examined. For an ideal cryocooler, it is defined as, Heat Absorbed from Low - Temp source h5 − h1 COPi = = (5.1) Net Work Input T5(s5 − s1) − (h5 − h1) where as for the practical cryocooler, it is shown as, Heat Absorbed from Low - Temp source h − h COP = = 5' 1 (5.2) Net Work Input T5'(s5' − s1) − (h5' − h1) The Figure of Merit is given by, COP FOM = (5.3) COPi The Coefficient of Performance and Figure of Merit of the Hampson-type J-T cryocooler under different inlet pressure conditions are plotted as shown in Figure 5.6. It is shown that the higher the input pressure, the higher the Coefficient of Performance and the Figure of Merit of the Hampson-type J-T cryocooler. National University of Singapore 83 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION 0.14 0.81 0.13 0.80 Figure of Merit ce 0.12 an 0.79 rm 0.11 Perfo f 0.10 0.78 o Carnot Cycle ent gure of Merit 0.09 i ci 0.77 F effi 0.08 Real J-T Cryocooler Co 0.76 0.07 0.06 0.75 140 145 150 155 160 165 170 175 180 Input Pressure (bar) Figure 5.6 Coefficient of Performance and Figure of Merit under Different Inlet Pressure 5.4 Effectiveness and Liquefied Yield Fractions The effectiveness of the heat exchanger Flynn [37] is defined as, h − h ε = 5' 1 (5.4) h5' − h f The liquefied yield fraction of the heat exchanger Flynn [37] is given by, (h − h ) − (1− ε)(h − h ) h − h y = 5 1 5 4 = 5' 1 (5.5) (h5 − h f ) − (1− ε)(h5 − h5 ) h5' − h f National University of Singapore 84 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION The variations of effectiveness and liquefied yield fractions of the Hampson- type J-T cryocooler under different input conditions are shown in Table 5.2. Table 5.2 Variations of Effectiveness, Liquefied Yield Fraction and COP under Different Input Pressure Input Conditions Liquefied Coefficient of Yield Performance Case Pressure Temp. Effectiveness Fraction Carnot Real J- (bar) (K) (%) (%) Cycle T Cycle (%) (%) 1 179.12 291.49 92.82 9.50 11.28 9.08 2 169.86 291.40 92.95 9.03 10.98 8.77 3 160.10 292.25 93.27 8.46 10.34 8.20 4 149.66 292.14 93.67 7.95 9.69 7.66 5 140.47 291.94 93.86 7.45 9.12 7.14 Figure 5.7 shows the variations of effectiveness and liquefied yield fractions of the Hampson-type J-T cryocooler under different inlet pressure conditions. It is shown from the chart that: i. The Hampson-type J-T cryocooler has relatively low yields, only about 10% of the gas circulated becomes liquid; ii. The higher the input pressure, the higher the liquefied yield fraction could be achieved; iii. The effectiveness of the Hampson-type J-T cryocooler decreases with the increase of input pressure; National University of Singapore 85 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION iv. The liquefied yield is sensitive to the effectiveness of the Hampson- type heat exchanger. 1.00 0.100 0.98 0.095 n o i t 0.96 0.090 c a r ess n d F e el v 0.94 0.085 i Y ed ffecti i f E 0.92 0.080 que i L 0.90 0.075 0.88 0.070 140 145 150 155 160 165 170 175 180 Input Pressure (bar) Figure 5.7 Variation of Effectiveness & Liquefied Yield Fraction under Different Inlet Pressure 5.5 Temperature and Pressure Distributions The expansion process of argon gas is successfully simulated by the computer code. The temperature and pressure distributions along the capillary tube and its fins are shown in Figure 5.8. The key features of the fluid expansion are: (i) The temperature and pressure drop are linear as the gas expands along the capillary tube; National University of Singapore 86 Filename:TeoHY.pdf CHAPTER 5 RESULTS AND DISCUSSION (ii) The throttling effect occurs at the nozzle exit; and (iii) The pressure drops of the returning gas is insignificant, but the temperature profile through the recuperative heat exchanger is non- linear. 300 200 280 282.57 K 180 260 Tf Flow Inside Tubing Flow Over Outside Tubing 240 160 220 140 ) 200 Tl K 180 120 ar) b re ( u 160 re ( at 100 r u e 140 Pf 120 80 Press Temp 100 110.36 K 60 80 ThThrottlingrottling in in 60 exexpansionpansion ca vicavityty 40 40 Experimental Data Input Pressure : 179.12 bar 20 20 Input Temperature : 291.49 K Pl 0 0 0.2L 0.4L 0.6L 0.8L L 0.8L 0.6L 0.4L 0.2L Length of Heat Exchanger Fig. 5.8 Temperature and Pressure Distributions along the Finned Heat Exchanger National University of Singapore 87 Filename:TeoHY.pdf CHAPTER 6 CONCLUSIONS & RECOMMENDATIONS Chapter 6 Conclusions & Recommendations 6.1 Conclusions A practical design tool for the Hampson-type Joule-Thomson (J-T) cooler has been successfully developed that accounts for the correct geometrical areas of the helical capillary tube and fins. Such a code captures accurately the performance characteristics of the working fluid where the predicted temperature and pressure of the gas at outlet of the cryocooler agrees to within 3% of the experimental data. In addition, the code provides tracking of the entropy changes which permits the accurate sketch of the T-s diagram; and has pedagogical value for evaluating the heat transfer under the process path. Significant improvements are observed in the predictions and these are attributed to three major factors such as: i. the correct estimation of heat exchange areas of Hampson-type J-T device; ii. the improved routines for fluid properties; and iii. the distributed computational procedure. National University of Singapore 88 Filename:TeoHY.pdf CHAPTER 6 CONCLUSIONS & RECOMMENDATIONS This simulation is useful for the prediction, estimation and evaluation of the Hampson-type J-T cooler. It provides a realistic design solution for the manufacturers of J-T cryocoolers and eliminates the use of empirical geometric correction factors thus avoiding the “trial and error” procedures commonly adopted. 6.2 Recommendations To seek further influencing factors in a Hampson-type J-T cryocooler, the followings suggestion are recommended: i. Two-phase flow subroutines Two-phase homogenous flow of high pressure cryogen might occur near the end of expansion path prior to the isenthalpic process at the nozzle. Two-phase homogenous governing equations should be incorporated to monitor the quality of the cryogen. ii. Better analysis of argon fluid properties In Figure 4.2, there is a high surge of specific heat value at 60 bars and 156 K. The simulations are closed to this region. This is not desirable and could be due to the numerical disorder of the argon properties’ subroutines; more thorough search on literature for better analysis of argon fluid properties should be performed. National University of Singapore 89 Filename:TeoHY.pdf CHAPTER 6 CONCLUSIONS & RECOMMENDATIONS iii. Truncation errors The equations are solved by the Personal Computer, which make use of 32-bit micro-processor’s technology. The simulated results could be further improved if 64-bit micro-processor’s technology is used. National University of Singapore 90 Filename:TeoHY.pdf REFERENCES References [1] Walker G. and Bingham E.R., Low-Capacity Cryogenic Refrigeration (1994) [2] Lemmon E.W., Peskin A.P., McLinden M.O., Friend D.G. Thermodynamic and Transport Properties of Pure Fluids. NIST Standard Reference Database 12, Version 5.0 (2000) [3] Aubon C.R., Joule-Thomson Cooling – An Overview. SPIE. Vol. 915 (1998), pp. 32-33 [4] Joo Y., Dieu K., Kim C., Fabrication of Monolithic Microchannels for IC Chip Cooling, IEEE. (1995), pp. 362-367 [5] Levenduski R., Scarlotti R., Joule-Thomson Cryocooler for Space Applications. Cryogenics. Vol. 36 (1996), pp. 859-866 [6] Maytal B. Z., Performance of Ideal Flow Regulated Joule-Thomson Cryocooler. Cryogenics, Vol. 34 (1994), pp. 723-726 National University of Singapore R-1 Filename:TeoHY.pdf REFERENCES [7] Chou, F.C., Wu, S.M. and Pai, C.F. Prediction of Final Temperature following Joule-Thomson Expansion of Nitrogen Gas. Cryogenics Vol. 33 (1993), pp. 857-862 [8] Chou F.C., Pai C.F., Chien S.B. and Chen J.S., Preliminary Experimental and Numerical Study of Transient Characteristics for a Joule-Thomson Cryocooler. Cryogenics Vol. 35 (1995), pp. 311-316 [9] Chien S.B., Chen L.T. and Chou F.C., A Study on the Transient Characteristics of a Self-Regulating Joule-Thomson Cryocooler. Cryogenics, Vol. 36 (1996), pp. 979-984 [10] Ng Kim Choon and Wang Jinbao, Xue Hong., Modeling of Recuperative Heat Exchanger in a Joule-Thomson Cooler, IMECE2001/HTD-24278, ASME (2001) [11] Ng K.C., Xue H., Wang J.B., Experimental and Numerical Study on a Miniature Joule-Thomson Cooler for Steady-state Characteristics. Int. J. Heat & Mass Trans. Vol. 45 (2002), pp 609-618 National University of Singapore R-2 Filename:TeoHY.pdf REFERENCES [12] Xue H., Ng K.C., Wang J.B., Performance Evaluation of the Recuperative Heat Exchanger in a Miniature Joule-Thomson Cooler. Applied Thermal Engrg, Vol. 21 (2001), pp. 1829-1844 [13] Wang JinBao., Experimental and Numerical Study on A Miniature Joule- Thomson Cooler, National University of Singapore (2000) [14] White, Guy K., Experimental Techniques in Low-Temperature Physics, Oxford: Clarendon Press (1987), pp. 59-73 [15] Barron, Randall F. Cryogenic Heat Transfer, Taylor & Francis (1999), pp. 265-368 [16] Patankar S.V., Pratap V.S. and Spalding D.B., Prediction of Laminar Flow and Heat Transfer in Helically Coiled Pipes. J. Fluid Mech. Vol. 62, Part 3 (1973), pp. 539-551 [17] Patankar S.V., Pratap V.S. and Spalding D.B., Prediction of Turbulent Flow in Curved Pipes. J. Fluid Mech, Vol. 67 (1974), pp. 583-595 [18] Germano M., On the Effect of Torsion on a Helical Pipe Flow J. Fluid Mech, Vol. 125 (1982), pp.1-8 National University of Singapore R-3 Filename:TeoHY.pdf REFERENCES [19] Germano M., The Dean Equations Extended to a Helical Pipe Flow J. Fluid Mech., Vol. 203 (1989), pp. 289-305 [20] Thomas J. Hϋttl. Influence of Curvature and Torsion on Turbulent Flow in Curved and Helically Coiled Pipes. Lehrstuhi fϋr Fluidmechanik Technische Universität Mϋnchen Boltzmannsir. 15, 85748 Garching, Germany International Journal of Heat and Fluid Flow, Vol. 21 (2000), pp. 345-353 [21] Lin C.X. and Ebadian M.A., The Effects of Inlet Turbulence on the Development of Fluid Flow and Heat Transfer in a Helically Coiled Pipe. International Journal of Heat and Mass Transfer, Vol. 42 (1999), pp. 739- 751 [22] Yang G. and Ebadian M.A., Soliman H.M., An Experimental and Theoretical Study of Laminar Fluid Flow and Heat Transfer in Helical Coils. 8-IC-23, pp. 321-326 [23] Zheng B., Lin C.X., and Ebadian M.A., Combined Laminar Forced Convection and Thermal Radiation in a Helical Pipe. International Journal of Heat and Mass Transfer, Vol. 43 (2000), pp. 1067-1078 National University of Singapore R-4 Filename:TeoHY.pdf REFERENCES [24] Mori D., Liu H., Yamaguchi T., Effect of the 3D Distortion on Flow in the Ordinary Helix Circular Tube Model of the Aortic Arch. 12th Conference of the European Society of Biomechanics, Dublin (2000) [25] Bolstad M.M. and Jordan R.C., Theory and Used of Capillary Tube Expansion Device. Part II, Non-adiabatic flow. Refrigerating Engineering, Vol. 57 (1949), pp. 572-583 [26] Marcy G.P., Pressure Drop with Change of Phase in a Capillary Tube. Refrigerating Engineering, Vol. 57 (1949), pp. 53-57 [27] Hopkins N.E., Rating the Restrictor Tube. Refrigerating Engineering, Vol. 58 (1950), pp. 1087-1095 [28] Cooper L., Chu C.K. and Brisken W. R., Simple Selection Method for Capillaries Derived from Phusical Flow Conditions. Refrigerating Engineering, Vol. 65 (1957), pp. 37-46 [29] Rezk A. and Awn A., Investigation on Flow of R12 through Capillary Tubes. In Proceedings of the 15th International Congress of Refrigeration, Commission, B2. Vol. 2 (1979), pp. 443-452 National University of Singapore R-5 Filename:TeoHY.pdf REFERENCES [30] Whitesel H.A., Capillary Two-Phase Flow, part II. Refrigerating Engineering, Vol. 65(9) (1957), pp. 35-47 [31] ASHRAE. ASHRAE Handbook – Equipment, Ch. 19. American Society of Heating, Refrigerating, and Air Conditioning Engineers, Inc. [32] Bansal P.K. and Rupasinghe A.S., An Homogeneous Model for Adiabatic Capillary Tubes, Applied Thermal Engrg, Vol. 18 (1998), pp. 207-219 [33] Brian A. Sherman, Sidney H. Schwartz. Jet Impingement Boiling Using a JT Cryostat. HTD, ASME Vol. 167 (1991), pp. 11-17 [34] FLUENT, A Computational Flow Dynamics (CFD) software, Copyrighted © 1995-2003 by Fluent Incorporated [35] GAMBIT, A Geometry Development Software, Copyright © 1995-2003 by Fluent Incorporated [36] Timmerhaus, K.D. and Flynn, T.M. Cryogenic Process Engineering. Plenum Press, New York, USA. (1989) National University of Singapore R-6 Filename:TeoHY.pdf REFERENCES [37] Flynn M.T., Cryogenic Engineering, Marcel Dekker Inc., NewYork (1997) [38] Perry, Robert H. Green, Don W. and Maloney, James O. Perry’s Chemical Engineers’ Handbook, New York McGraw-Hill (1997), pp. 2-318 to 2-337 [39] Rohsennow, Warren M., Hartnett, James P. and Cho Young I. Handbook of Heat Transfer., 3rd Edition, MCGRAW-HILL (1998), pp. 85-125 National University of Singapore R-7 Filename:TeoHY.pdf APPENDIX A OPERATION MANUAL FOR SIMULATION PROGRAM A-1 Appendix A Operation Manual for Simulation Program Step Procedures 1 Set up a new project workspace namely, “JT_Cooler”, in Fortran 90 2 A sub-directory of “JT_Cooler” will be automatically created (C:\MSDEV\Projects\JT_Cooler) 3 Copy all working files into the sub-directory by using Window Explorer. The files are: Ar01 Mod-Gas Data.F90 - C1 Mod-Double Precision.F90 C2 Sub-Heat Transfer Area.F90 C3 Sub-Radiation Coeff.F90 C4 Sub-Thermal Cond of Solid Materials.F90 Core_bwr.for Core_cpp.for Core_de.for Core_ecs.for Core_feq.for Core_mlt.for Core_ph0.for Core_stn.for Flash2.for Flash_sub.for Flash_pas.for Heat_Transfer_Areas.F90 Idealgas.for Mix_hmx.for Prop_sub.for R_Fluid_Prop.F90 R_Liquid_Prop.F90 Realgas.for ReturnFluid.F90 ReturnLiquid.F90 Sat_sub.for Setup.for A-2 Setup2.for Thesis_Report.F90 Trns_ecs.for Trns_tcx.for Trns_vis.for Trnsp.for Two_Phase_Critical_T.F90 Two_Phase_Properties.F90 Utility.for X_sub.for 4 NIST [2] program shall be installed in the computer system prior to the compilation and execution of the project. This is to allow the registry files being created in the computer system so that the subroutines could be used in the Fortran program. The properties of cryogen are obtained from the NIST [2] subroutines. 5 Under the Fortran 90, select “Inset-Files into Projects…”. A menu is pop-up and all files in the C:\MSDEV\Projects\JT_Cooler” shall be selected and click “OK”. 6 Under Fortran 90, select “File-Open Workspace” from the pull-down menu and choose for “Thesis_Report.mdp”. 7 Under the Fortran 90, select “Build-Settings” in the pull-down menu. Then choose “Link” from the pop-up window. “maths.lib” and “mathd.lib” shall be keying in into the “Object/library modules”. This is allowed the program to use single and double-precision programming approach from the Fortran 90 library. Then hit “enter”. 8 Under Fortran 90, select “File-Open…” from the pull-down menu and select file’s name “Thesis_Report.F90”. Main program is opened. You are ready to compile the program. A-3 9 Set TIMES=1 and FineTune=1 (line 4). Press “Alt+F8” to compile the source code, or alternatively, you can select “Build-Rebuild All…” from the pull-down menu to compile the source code. 10 Press “Ctrl+F5” to execute the compiled program, or alternatively, you can select “Build-Execute Thesis_Report.exe” to run the compiled program. 11 The program will be iterated in the IMSL sub-routine (ie. DBVPFD) until either its stop due to memory full or the program is diverged as shown in the working MSDOS pop-up window. Please do not panic in this situation. 12 To click on the “open” icon to open the following files: “A_Temp.dat” – This file captured all convergence values each iteration from DBVPFD. The smaller value the closer to the solutions. Look for the lowest convergence value. If the Max. abs. residual is not less than .4000000E-04, then record the lowest computed value and its Newton-Iteration’s number. This Newton-Iteration number should be an integer and having a value lower than 15. If the Max. abs. residual is less than .4000000E-04 then proceed to step (16). Let the Newton-Iteration number to be X. 13 Proceed to the main program. Set the FineTune (line 4) to 2. Then proceed to line 96. Amend the statements as follows: Case(2) If (K.EQ.51*X) I=0 If ((K.GE.51*X+1).AND.(K.LE.51*X+50)) THEN 14 Repeat step (9) to (13). Then open the file “A_Cp_Eqn_1.dat” and choose all data, copy into the file “Ini_Data.F90”. Care shall be taken to replace the respective data of CASE(TIMES) conditions. Reset the FineTune to 1. A-4 15 Then back to the main program and reset the FineTune back to 1 and choose the next CASE(TIMES). 16 If the Max. abs. residual is less than .4000000E-04, the properties of the cryogen at each grid points (i.e 51 as set in the program) can be obtained from the followings files: High Pressure Cryogen A_Pf.dat : Pressure A_Tf.dat : Temperature A_Sf.dat : Entropy Return Low Pressure Cryogen A_Pl.dat : Pressure A_Tl.dat : Temperature A_Sl.dat : Entropy The temperature of capillary tube, fins and shield at each grid points can also be obtained in the following files. A_Results.dat 17 The following files will be created during the execution of the program. A_Ini.dat A_FcnEqn.dat A_Jac.dat A_Bc.dat A_Result.dat A_Temp.dat A_Fcn.dat A_Cp_Eqn_1.dat A_Pf.dat A_Pl.dat A_Tf.dat A_Tl.dat A_Sf.dat A_Sl.dat A-5 APPENDIX B FORTRAN 90 SOURCE CODE MAIN PROGRAM B-1 MAIN PROGRAM : THESIS_REPORT.F90 THESIS_REPORT.F90 MODULE NTU USE DP IMPLICIT NONE INTEGER(n):: BC=1,TIMES=1,FineTune=1 !Times means for the different initial conditions REAL(q)::P0,Mv,Tco,Thi,Tlexp,E0,Psat,Tamb,Plexp REAL(q)::G_f,G_l INTEGER(n)::NTurns !NTurns, No. of coils in capillary tube, it is always 1 no. less than NINIT !Let Tamb = 298K CONTAINS SUBROUTINE INPUT SELECT CASE(times) CASE(1) P0=17.912D6;Plexp=0.17272D6;Mv=13.927D0;Thi=291.49D0;Tco=92.6 84307407D0;Tlexp=282.57d0;Tamb=298.0d0;E0=2.907328353d3;Psat=6.81 0626742d5 CASE(2) P0=16.986D6;Plexp=0.17460D6;Mv=13.102D0;Thi=291.40D0;Tco=92.8 00867976D0;Tlexp=283.73d0;Tamb=298.0d0;E0=2.906650814d3;Psat=6.83 72246d5 CASE(3) P0=16.010D6;Plexp=0.16362D6;Mv=11.943D0;Thi=292.25D0;Tco=92.1 06152225D0;Tlexp=284.77d0;Tamb=298.0d0;E0=2.912536960d3;Psat=6.60 91893d5 CASE(4) P0=14.966D6;Plexp=0.14713D6;Mv=10.948D0;Thi=292.14D0;Tco=90.9 93293822D0;Tlexp=284.90d0;Tamb=298.0d0;E0=2.919598042d3;Psat=6.34 45564d5 B2 MAIN PROGRAM : THESIS_REPORT.F90 CASE(5) P0=14.047D6;Plexp=0.13426D6;Mv=10.145D0;Thi=291.94D0;Tco=90.0 56936107D0;Tlexp=284.98d0;Tamb=298.0d0;E0=2.926247786d3;Psat=6.10 40286d5 END SELECT END SUBROUTINE INPUT END MODULE NTU ! !####### ####### SUBROUTINE OF ODEs SYSTEM ############## SUBROUTINE FCNEQN (NEQNS, T, Y, P, DYDT) USE DP USE NTU USE SIZE_JT IMPLICIT NONE ! Specifications for arguments INTEGER(n),INTENT(IN)::NEQNS REAL(q),INTENT(IN)::T,P REAL(q),DIMENSION(NEQNS),INTENT(IN)::Y REAL(q),DIMENSION(NEQNS),INTENT(OUT)::DYDT REAL(q)::Pf,Tf REAL(q)::d_f,h_f,s_f,Cv_f,Cp_f,hjt_f,x_f,DpDrho_f,eta_f,U_Energy_f,sigma_f, tcx_f,DrhoDT_f,DpT_f,Re_f REAL(q)::Pl,Tl REAL(q)::h_l,s_l,Cv_l,Cp_l,hjt_l,x_l,DpDrho_l,eta_l,U_Energy_l,sigma_l,tcx_l ,DrhoDT_l,DpT_l,d_l REAL(q)::Tm,Tfin REAL(q)::Kt,Km,Km_T,Kfin,Kfin_T,Kman,Kman_T INTEGER(n)::K,I,j REAL(q)::Coe_f,Coe_l,f_l,Re_l,f_f,En,En1,En2 REAL(q)::Tsh,Ksh,Hr,Dhr_T,Ksh_T K=K+1 B3 MAIN PROGRAM : THESIS_REPORT.F90 CALL INPUT CALL AREA Tf=Y(1)*(Tamb-Tco)+Tco Tl=Y(2)*(Tamb-Tco)+Tco Tm=Y(3)*(Tamb-Tco)+Tco Tsh=Y(4)*(Tamb-Tco)+Tco Tfin=Y(5)*(Tamb-Tco)+Tco !dTm/ds = Y(6) !dTfin/ds = Y(7) !dTsh/ds = Y(8) Pf=Y(9)*P0 !phi Pl=Y(10)*P0 !For monitoring purpose print*,'FCNEQN',K print*,' Pf ',Pf,' Tf ',Tf print*, ' Pl ',Pl,' Tl ',Tl print*, ' Tm ',Tm,' Tfin ',Tfin print*, ' Tsh',Tsh !For Tf CALL R_Fluid_Prop (Pf,Tf,d_f,h_f,s_f,Cv_f,Cp_f,hjt_f,x_f,DpDrho_f,eta_f,U_Energy_f,& & sigma_f,tcx_f,DrhoDT_f,DpT_f) CALL ReturnFluid (Re_f,f_f,Coe_f,Cp_f,eta_f,tcx_f) !For Tl CALL R_Liquid_Prop (Pl,Tl,d_l,h_l,s_l,Cv_l,Cp_l,hjt_l,x_l,DpDrho_l,eta_l,U_Energy_l,& & sigma_l,tcx_l,DrhoDT_l,DpT_l) CALL ReturnLiquid (Re_l,f_l,Coe_l,Cp_l,eta_l,tcx_l) B4 MAIN PROGRAM : THESIS_REPORT.F90 !For storing into files Select Case(FineTune) Case(1) write(9,*) k,' Pf ',Pf,' Tf ',Tf write(9,*) ' Pl ',Pl,' Tl ',Tl write(9,*) ' Tm ',Tm,' Tfin ',Tfin write(9,*) ' Tsh ',Tsh write(9,*) Case(2) If (K.EQ.1071) I=0 If ((K.GE.1172).AND.(K.LE.1172)) THEN I = I+1 write(99,*) 'Ypoints(1,',I,')=',Y(1) write(99,*) 'Ypoints(2,',I,')=',Y(2) write(99,*) 'Ypoints(3,',I,')=',Y(3) write(99,*) 'Ypoints(4,',I,')=',Y(4) write(99,*) 'Ypoints(5,',I,')=',Y(5) write(99,*) 'Ypoints(6,',I,')=',Y(6) write(99,*) 'Ypoints(7,',I,')=',Y(7) write(99,*) 'Ypoints(8,',I,')=',Y(8) write(99,*) 'Ypoints(9,',I,')=',Y(9) write(99,*) 'Ypoints(10,',I,')=',Y(10) Write(98,*) Pf Write(97,*) Pl Write(96,*) Tf Write(95,*) Tl Write(94,*) s_f Write(93,*) s_l EndIf B5 MAIN PROGRAM : THESIS_REPORT.F90 Case(3) If ((K.GE.1).AND.(K.LE.54)) THEN I=I+1 Write(98,*) Pf Write(97,*) Pl Write(96,*) Tf Write(95,*) Tl Write(94,*) s_f Write(93,*) s_l EndIf End SELECT If (I.ge.(NTurns+1)) I=I+1 If (I.eq.70) stop !Calculates conductivity of Capillary Tube, Fins, Mandrel & Spacer CALL k_Copper(Tm,Km,Km_T) CALL k_SSteel(Tfin,Kfin,Kfin_T) CALL k_Monel(Tl,Kman,Kman_T) CALL k_PC(Ksh,Ksh_T) CALL RAD(Tsh,hr,Dhr_T) Kt=Km*Kfin/(Kfin*Lm+Km*Hfin) !Units of Kt = W/(m2K) !Single Phase Internal Fluid Flow, Tf DYDT(1)= -((Ls*(2.0D0*A_fluid*f_f*G_f**3*(d_f**2*DpDrho_f - G_f**2 + d_f*DpDrho_f*DrhoDT_f*(Tco - Tco*Y(1) + Y(1)*Tamb)) + Coe_f*d_f**2*Dmi**2*(d_f**2*DpDrho_f - G_f**2)*Pi*(Tco - Tamb)*(Y(1) - Y(3))))/ (A_fluid*Dmi*G_f*(Tco - Tamb)*(Cp_f*(d_f**4*DpDrho_f - d_f**2*G_f**2) + DpDrho_f*DrhoDT_f**2*G_f**2*(Tco - Tco*Y(1) + Y(1)*Tamb)))) !External Return Fluid, Tl DYDT(2) = -((Ls*(Afinl*Coe_l*D_Fluid_Flow*d_l**2*(d_l**2*DpDrho_l + G_l**2)*(Tco - Tamb)*(Y(5) - Y(2)) - Aml*Coe_l*D_Fluid_Flow*d_l**2*(d_l**2*DpDrho_l + G_l**2)*(Tco - Tamb)*(Y(2) - Y(3)) + (1.5d0*ds/dsdz)*(- B6 MAIN PROGRAM : THESIS_REPORT.F90 (Coe_l*D_Fluid_Flow*d_l**2*Dshi*(d_l**2*DpDrho_l + G_l**2)*Pi*(Tco - Tamb)*(Y(2) - Y(4))) + 2.0D0*f_l*G_l**3*(-(d_l**2*DpDrho_l) + G_l**2 + d_l*DpDrho_l*DrhoDT_l*(Tco*(-1.0D0 + Y(2)) - Tamb*Y(2)))*X_Area_Liquid_Flow)))/ (D_Fluid_Flow*(1.5d0*ds/dsdz)*dsdz*G_l*(Tco - Tamb)*(Cp_l*d_l**2*(d_l**2*DpDrho_l + G_l**2) + DpDrho_l*DrhoDT_l*G_l**2*(-2.0D0*d_l - DrhoDT_l*(Tco - Tco*Y(2) + Tamb*Y(2))))*X_Area_Liquid_Flow)) !Eqn (C) : Tube Conduction,Tm DYDT(3) = Y(6) DYDT(6) = -Coe_f*Ls**2*Afm/ds/A_CapillaryTube/Km*(Y(3)-Y(1)) - Coe_l*Ls**2*Aml/ds/A_CapillaryTube/Km*(Y(3)-Y(2)) - 2.0D0*Kt*Ls**2*Afinm/ds/A_CapillaryTube/Km*(Y(3)-Y(5)) !Eqn (D) : Fin Conduction,Tfin DYDT(5) = Y(7) DYDT(7) = -Coe_l*Ls**2*Afinl/ds/X_AreaFin/Kfin*(Y(5)-Y(2)) - 2.0D0*Kt*Ls**2*Afm/ds/X_AreaFin/Kfin*(Y(5)-Y(3)) !Eqn (F) : Shield Energy,Tsh DYDT(4) = Y(8) DYDT(8) = -Coe_l*Pi*Dshi/dsdz**2*Ls**2/Ksh/Ash*(Y(4)-Y(2)) - hr*Pi*Dshi*Ls**2/Ksh/Ash/dsdz**2*(((Tamb-Tco)*Y(4)+Tco)**4- Tamb**4)/(Tamb-Tco) !Single Phase Internal Fluid Flow, Pf DYDT(9)= (DpDrho_f*G_f*Ls*(A_fluid*(-2.0D0*Cp_f*d_f**3*f_f*G_f + 2.0D0*DrhoDT_f*f_f*G_f**3) + Coe_f*d_f**2*Dmi**2*DrhoDT_f*Pi*(Tco - Tamb)*(Y(1) - Y(3))))/ (A_fluid*Dmi*P0*(Cp_f*(d_f**4*DpDrho_f - d_f**2*G_f**2) + DpDrho_f*DrhoDT_f**2*G_f**2*(Tco - Tco*Y(1) + Y(1)*Tamb))) !External Return Fluid, Pl DYDT(10)= (DpDrho_l*G_l*Ls*(- (Afinl*Coe_l*D_Fluid_Flow*d_l**2*DrhoDT_l*(Tco - Tamb)*(Y(5) - Y(2))) + Aml*Coe_l*D_Fluid_Flow*d_l**2*DrhoDT_l*(Tco - Tamb)*(Y(2) - Y(3)) + (1.5d0*ds/dsdz)*(Coe_l*D_Fluid_Flow*d_l**2*DrhoDT_l*Dshi*Pi*(Tco - Tamb)*(Y(2) - Y(4)) +2.0D0*f_l*G_l*(Cp_l*d_l**3 - B7 MAIN PROGRAM : THESIS_REPORT.F90 DrhoDT_l*G_l**2)*X_Area_Liquid_Flow)))/ (D_Fluid_Flow*(1.5d0*ds/dsdz)*dsdz*P0*(Cp_l*d_l**2*(d_l**2*DpDrho_l + G_l**2) + DpDrho_l*DrhoDT_l*G_l**2*(-2.0D0*d_l - DrhoDT_l*(Tco - Tco*Y(2) + Tamb*Y(2))))*X_Area_Liquid_Flow) !Entropy Check, En=En1 - En2 !En>0, Single gas phase !En=0, Shocked or chocked occurs, the quality of working fluid to be monitored. 2 phase equations to be used !En<0, Working fluid becomes completely liquid state, Liquid phase equation to be used. En1 = G_f*A_fluid/((Tamb-Tco)*Y(1)+Tco)*(Cp_f*(Tamb-Tco)/Ls*DYDT(1) + (1.0d0/d_f+((Tamb-Tco)*Y(1)+Tco)/d_f**2*DrhoDT_f)*P0/Ls*DYDT(9)- 1.0D0/d_f*P0/Ls*DYDT(9)) En2 = Coe_f*(Tamb-Tco)*(Y(3)-Y(1))/((Tamb-Tco)*Y(3)+Tco)*Afm/ds En = En1 - En2 write(8,*) K write(8,*) En1,En2,En write(2,*) k,DYDT write(2,*) END SUBROUTINE FCNEQN ! !################ SUBROUTINE OF JACOBIANS ################## SUBROUTINE FCNJAC(NEQNS, T, Y, P, DYPDY) USE DP USE NTU USE SIZE_JT IMPLICIT NONE ! Specifications for arguments INTEGER(n),INTENT(IN)::NEQNS REAL(q),INTENT(IN)::T,P REAL(q),DIMENSION(NEQNS),INTENT(IN)::Y B8 MAIN PROGRAM : THESIS_REPORT.F90 REAL(q),DIMENSION(NEQNS,NEQNS),INTENT(OUT)::DYPDY INTEGER(n)::MM,NN,LDFJAC PARAMETER (MM=10,NN=10,LDFJAC=10) INTEGER(n)::NOUT1 REAL(q)::FJAC(LDFJAC,NN), XSCALE(NN), XC(NN), FC(MM), EPSFCN INTEGER(n)::i,j,k EXTERNAL UMACH,FCN, DFDJAC DATA XSCALE/10*2.0D0/ j=j+1 EPSFCN = 0.01D0 XC = Y ! XSCALE = Y CALL FCN (MM, NN, XC, FC) CALL DFDJAC (FCN, MM, NN, XC, XSCALE, FC, EPSFCN, FJAC, LDFJAC) DYPDY = FJAC CALL UMACH (2, NOUT1) write(3,*) j Do k = 1,10 write(3,*) ' k I DYPDY(k,I)' Do I = 1,10 write(3,*) k,I,DYPDY(k,I) END Do write(3,*) END Do END SUBROUTINE FCNJAC B9 MAIN PROGRAM : THESIS_REPORT.F90 ! !########### SUBROUTINE OF BOUNDARY CONDITIONS ########### SUBROUTINE FCNBC(NEQNS, YLEFT, YRIGHT, P, F) USE DP USE NTU,ONLY:BC,Psat,P0,Tlexp,Tamb,Tco,Plexp,Thi IMPLICIT NONE INTEGER(n),INTENT(IN)::NEQNS REAL(q),INTENT(IN)::P REAL(q),DIMENSION(NEQNS),INTENT(IN)::YLEFT,YRIGHT REAL(q),DIMENSION(NEQNS),INTENT(OUT)::F INTEGER(n)::k ! Define boundary conditions SELECT CASE(BC) CASE(1) F(1)= YLEFT(1)-((Thi-Tco)/(Tamb-Tco)) !At s=0, Tf=Thi F(2)= YLEFT(6) !At s=0, dTm/ds=0 F(3)= YLEFT(7) !At s=0, dTfin/ds=0 F(4)= YLEFT(8) !At s=0, dTsh/ds=0 F(5)= YLEFT(9)-1.0D0 !At s=0, Pf=P0 F(6)= YRIGHT(2) !At s=Ls, Tl=Tco F(7)= YRIGHT(6) !At s=Ls, dTm/ds=0 F(8)= YRIGHT(7) !At s=Ls, dTfin/ds=0 F(9)= YRIGHT(8) !At s=Ls, dTsh/ds=0 F(10)= YRIGHT(10)-Plexp/P0 !At s=Ls, Pl=Plexp END SELECT k=k+1 write(4,*) k,F write(4,*) END SUBROUTINE FCNBC B10 MAIN PROGRAM : THESIS_REPORT.F90 !############ SUBROUTINE OF FCN (EST.JACOBIAN) ############# SUBROUTINE FCN (MM, NN, Y, Ffcn) USE DP USE NTU USE SIZE_JT IMPLICIT NONE INTEGER(n),INTENT(IN)::MM,NN REAL(q),DIMENSION(NN),INTENT(IN)::Y REAL(q),DIMENSION(MM),INTENT(OUT)::Ffcn REAL(q)::Pf,Tf REAL(q)::d_f,h_f,s_f,Cv_f,Cp_f,hjt_f,x_f,DpDrho_f,eta_f,U_Energy_f,sigma_f, tcx_f,DrhoDT_f,DpT_f,Re_f REAL(q)::Pl,Tl REAL(q)::h_l,s_l,Cv_l,Cp_l,hjt_l,x_l,DpDrho_l,eta_l,U_Energy_l,sigma_l,tcx_l ,DrhoDT_l,DpT_l,d_l REAL(q)::Tm,Tfin REAL(q)::Kt,Km,Km_T,Kfin,Kfin_T,Kman,Kman_T INTEGER(n)::K,J REAL(q)::Coe_f,Coe_l,f_l,Re_l,f_f,En REAL(q)::Tsh,Ksh,Hr,Dhr_T,Ksh_T K=K+1 CALL INPUT CALL AREA PRINT*,'FCN ',K Tf=Y(1)*(Tamb-Tco)+Tco Tl=Y(2)*(Tamb-Tco)+Tco Tm=Y(3)*(Tamb-Tco)+Tco Tsh=Y(4)*(Tamb-Tco)+Tco Tfin=Y(5)*(Tamb-Tco)+Tco !dTm/ds = Y(6) !dTfin/ds = Y(7) B11 MAIN PROGRAM : THESIS_REPORT.F90 !dTsh/ds = Y(8) Pf=Y(9)*P0 !phi Pl=Y(10)*P0 !The subroutine will use lower value to estimate the jacobians, this step is to prevent the NIST properties subroutines cannot be used. The result is compromised but negligible If (Tl.le.83.9d0) Then Print*,'Tl is less than 83.9 K',Tl Tl=83.90d0 ! Pause endif !For Tf CALL R_Fluid_Prop (Pf,Tf,d_f,h_f,s_f,Cv_f,Cp_f,hjt_f,x_f,DpDrho_f,eta_f,U_Energy_f, sigma_f,tcx_f,DrhoDT_f,DpT_f) CALL ReturnFluid (Re_f,f_f,Coe_f,Cp_f,eta_f,tcx_f) !For Tl CALL R_Liquid_Prop (Pl,Tl,d_l,h_l,s_l,Cv_l,Cp_l,hjt_l,x_l,DpDrho_l,eta_l,U_Energy_l, sigma_l,tcx_l,DrhoDT_l,DpT_l) CALL ReturnLiquid (Re_l,f_l,Coe_l,Cp_l,eta_l,tcx_l) !Calculates conductivity of Capillary Tube, Fins, Mandrel & Spacer CALL k_Copper(Tm,Km,Km_T) CALL k_SSteel(Tfin,Kfin,Kfin_T) CALL k_Monel(Tl,Kman,Kman_T) CALL k_PC(Ksh,Ksh_T) CALL RAD(Tsh,hr,Dhr_T) Kt=Km*Kfin/(Kfin*Lm+Km*Hfin) !Units of Kt = W/(m2K) CALL ReturnFluid (Re_f,f_f,Coe_f,Cp_f,eta_f,tcx_f) CALL ReturnLiquid (Re_l,f_l,Coe_l,Cp_l,eta_l,tcx_l) B12 MAIN PROGRAM : THESIS_REPORT.F90 !Single Phase Internal Fluid Flow, Tf Ffcn(1)= -((Ls*(2.0D0*A_fluid*f_f*G_f**3*(d_f**2*DpDrho_f - G_f**2 + d_f*DpDrho_f*DrhoDT_f*(Tco - Tco*Y(1) + Y(1)*Tamb)) + Coe_f*d_f**2*Dmi**2*(d_f**2*DpDrho_f - G_f**2)*Pi*(Tco - Tamb)*(Y(1) - Y(3))))/ (A_fluid*Dmi*G_f*(Tco - Tamb)*(Cp_f*(d_f**4*DpDrho_f - d_f**2*G_f**2) + DpDrho_f*DrhoDT_f**2*G_f**2*(Tco - Tco*Y(1) + Y(1)*Tamb)))) !External Return Fluid, Tl Ffcn(2) = -((Ls*(Afinl*Coe_l*D_Fluid_Flow*d_l**2*(d_l**2*DpDrho_l + G_l**2)*(Tco - Tamb)*(Y(5) - Y(2)) - Aml*Coe_l*D_Fluid_Flow*d_l**2*(d_l**2*DpDrho_l + G_l**2)*(Tco - Tamb)*(Y(2) - Y(3)) + (1.5d0*ds/dsdz)*(- (Coe_l*D_Fluid_Flow*d_l**2*Dshi*(d_l**2*DpDrho_l + G_l**2)*Pi*(Tco - Tamb)*(Y(2) - Y(4))) + 2.0D0*f_l*G_l**3*(-(d_l**2*DpDrho_l) + G_l**2 + d_l*DpDrho_l*DrhoDT_l*(Tco*(-1.0D0 + Y(2)) - Tamb*Y(2)))*X_Area_Liquid_Flow)))/ (D_Fluid_Flow*(1.5d0*ds/dsdz)*dsdz*G_l*(Tco - Tamb)*(Cp_l*d_l**2*(d_l**2*DpDrho_l + G_l**2) + DpDrho_l*DrhoDT_l*G_l**2*(-2.0D0*d_l - DrhoDT_l*(Tco - Tco*Y(2) + Tamb*Y(2))))*X_Area_Liquid_Flow)) !Eqn (C) : Tube Conduction,Tm Ffcn(3) = Y(6) Ffcn(6) = -Coe_f*Ls**2*Afm/ds/A_CapillaryTube/Km*(Y(3)-Y(1)) - Coe_l*Ls**2*Aml/ds/A_CapillaryTube/Km*(Y(3)-Y(2)) - 2.0D0*Kt*Ls**2*Afinm/ds/A_CapillaryTube/Km*(Y(3)-Y(5)) !Eqn (D) : Fin Conduction,Tfin Ffcn(5) = Y(7) Ffcn(7) = -Coe_l*Ls**2*Afinl/ds/X_AreaFin/Kfin*(Y(5)-Y(2)) - 2.0D0*Kt*Ls**2*Afm/ds/X_AreaFin/Kfin*(Y(5)-Y(3)) !Eqn (F) : Shield Energy,Tsh Ffcn(4) = Y(8) Ffcn(8) = -Coe_l*Pi*Dshi/dsdz**2*Ls**2/Ksh/Ash*(Y(4)-Y(2)) - hr*Pi*Dshi*Ls**2/Ksh/Ash/dsdz**2*(((Tamb-Tco)*Y(4)+Tco)**4- Tamb**4)/(Tamb-Tco) B13 MAIN PROGRAM : THESIS_REPORT.F90 !Single Phase Internal Fluid Flow, Pf Ffcn(9)= (DpDrho_f*G_f*Ls*(A_fluid*(-2.0D0*Cp_f*d_f**3*f_f*G_f + 2.0D0*DrhoDT_f*f_f*G_f**3) + Coe_f*d_f**2*Dmi**2*DrhoDT_f*Pi*(Tco - Tamb)*(Y(1) - Y(3))))/ (A_fluid*Dmi*P0*(Cp_f*(d_f**4*DpDrho_f - d_f**2*G_f**2) + DpDrho_f*DrhoDT_f**2*G_f**2*(Tco - Tco*Y(1) + Y(1)*Tamb))) !External Return Fluid, Pl Ffcn(10) = (DpDrho_l*G_l*Ls*(- (Afinl*Coe_l*D_Fluid_Flow*d_l**2*DrhoDT_l*(Tco - Tamb)*(Y(5) - Y(2))) + Aml*Coe_l*D_Fluid_Flow*d_l**2*DrhoDT_l*(Tco - Tamb)*(Y(2) - Y(3)) + (1.5d0*ds/dsdz)*(Coe_l*D_Fluid_Flow*d_l**2*DrhoDT_l*Dshi*Pi*(Tco - Tamb)*(Y(2) - Y(4)) + 2.0D0*f_l*G_l*(Cp_l*d_l**3 - DrhoDT_l*G_l**2)*X_Area_Liquid_Flow)))/ (D_Fluid_Flow*(1.5d0*ds/dsdz)*dsdz*P0*(Cp_l*d_l**2*(d_l**2*DpDrho_l + G_l**2) + DpDrho_l*DrhoDT_l*G_l**2*(-2.0D0*d_l - DrhoDT_l*(Tco - Tco*Y(2) + Tamb*Y(2))))*X_Area_Liquid_Flow) write(7,*) k,Ffcn write(7,*) RETURN END SUBROUTINE FCN !############### MAIN PRAGRAM ######################## PROGRAM THESIS_REPORT USE DP USE Ar_DATA,ONLY:R USE SIZE_JT USE NTU !NINIT indicates the no. of grid points to be discretized !NTurns is always 1 no. less than NINIT IMPLICIT NONE INTEGER(n),PARAMETER::MXGRID=20000,NEQNS=10,NINIT=51,LDYFIN =NEQNS,LDYINI=NEQNS ! SPECIFICATIONS FOR LOCAL VARIABLES INTEGER(n)::NCUPBC,NFINAL,NLEFT,NOUT B14 MAIN PROGRAM : THESIS_REPORT.F90 REAL(q)::PISTEP,TLEFT,TOL,TRIGHT REAL(q),DIMENSION(NEQNS)::ERREST REAL(q),DIMENSION(MXGRID)::TFINAL REAL(q),DIMENSION(NINIT)::TINIT REAL(q),DIMENSION(LDYFIN,MXGRID)::YFINAL REAL(q),DIMENSION(LDYINI,NINIT)::YINIT REAL(q)::d_f,h_f,s_f,Cv_f,Cp_f,hjt_f,x_f,DpDrho_f,eta_f,U_Energy_f,sigma_f, tcx_f,DrhoDT_f,DpT_f REAL(q)::U,Uf,Ul LOGICAL::LINEAR, PRINTING INTEGER(n)::I,J REAL(q)::Q_Load,A_Load,T_Load ! SPECIFICATIONS FOR SUBROUTINES EXTERNAL DBVPFD,UMACH,FCN,FDJAC ! SPECIFICATIONS FOR FUNCTIONS EXTERNAL FCNBC,FCNEQN,FCNJAC OPEN(Unit=1,File='A_Ini.dat',Status='unknown') OPEN(Unit=2,File='A_FcnEqn.dat',Status='unknown') OPEN(Unit=3,File='A_Jac.dat',Status='unknown') OPEN(Unit=4,File='A_Bc.dat',Status='unknown') OPEN(Unit=5,File='A_Result.dat',Status='unknown') OPEN(Unit=6,File='A_Temp.dat',Status='unknown') OPEN(Unit=7,File='A_Fcn.dat',Status='unknown') OPEN(Unit=8,File='A_Entropy.dat',Status='unknown') OPEN(Unit=99,File='A_Cp_Eqn_1.dat',Status='unknown') OPEN(Unit=98,File='A_Pf.dat',Status='unknown') OPEN(Unit=97,File='A_Pl.dat',Status='unknown') OPEN(Unit=96,File='A_Tf.dat',Status='unknown') OPEN(Unit=95,File='A_Tl.dat',Status='unknown') OPEN(Unit=94,File='A_Sf.dat',Status='unknown') OPEN(Unit=93,File='A_Sl.dat',Status='unknown') OPEN(unit=9,file="A_R-bc-con.dat") B15 MAIN PROGRAM : THESIS_REPORT.F90 !Initialisation NLEFT =5 NCUPBC=0 TOL =1.0D-12 TLEFT =0.0D0 TRIGHT=1.0D0 PISTEP= 0.0D0 PRINTING = .TRUE. LINEAR= .FALSE. !NTurns: Number of coils for capillary tube is always 1 no. less than grid points NTurns = NINIT - 1 Call AREA CALL INPUT !To get the boundary conditions of gas !These statements to find out G, hf, f_f, hl CALL MaxU(Uf,P0,Thi,Mv) CALL R_Fluid_Prop (P0,Thi,d_f,h_f,s_f,Cv_f,Cp_f,hjt_f,x_f,DpDrho_f,eta_f,U_Energy_f,& & sigma_f,tcx_f,DrhoDT_f,DpT_f) G_f=d_f*Uf ! Conservation of mass flow rate = G_f*A_f = G_l*A_l G_l=G_f*A_fluid/X_Area_Liquid_Flow ! Define TINIT and YINIT DO I=1,NINIT TINIT(I)=TLEFT+(I-1)*(TRIGHT-TLEFT)/REAL(NINIT-1) END DO CALL IniData(YINIT,LDYFIN,NINIT,Times) B16 MAIN PROGRAM : THESIS_REPORT.F90 CALL DBVPFD (FCNEQN,FCNJAC,FCNBC,FCNEQN,FCNBC,NEQNS,NLEFT,NCUPBC,& & TLEFT,TRIGHT,PISTEP,TOL,NINIT,TINIT,YINIT,LDYINI,LINEAR,PRINTING ,& & MXGRID,NFINAL,TFINAL,YFINAL,LDYFIN,ERREST) !For NTurns greater than 50 rounds If (NINIT.gt.51) Then DO I=52,NINIT DO J=1,5 !NEQNS YINIT(J,I)=0.8 END DO YINIT(6,I)=1.0 DO J=7,8 YINIT(J,I)=-0.2 END DO YINIT(9,I)=0.8 YINIT(10,I)=0.1 END DO EndIf ! Solve problem print*,'Congratulations! The program Converged' ! Print results CALL UMACH (2, NOUT) !To retrieve statements from filename:NewAr_Printing.f90 CLOSE(1,Status='keep') CLOSE(2,Status='keep') B17 MAIN PROGRAM : THESIS_REPORT.F90 CLOSE(3,Status='keep') CLOSE(4,Status='keep') CLOSE(5,Status='keep') CLOSE(6,Status='keep') CLOSE(7,Status='keep') CLOSE(8,Status='keep') CLOSE(9,Status='keep') CLOSE(99,Status='keep') CLOSE(98,Status='keep') CLOSE(97,Status='keep') CLOSE(96,Status='keep') CLOSE(95,Status='keep') CLOSE(94,Status='keep') CLOSE(93,Status='keep') CLOSE(92,Status='keep') END PROGRAM JT_STEADY !########################## THE END ##################### B18 APPENDIX C FORTRAN SOURCE CODE IMSL SUBROUTINE (DBVPFD) C-1 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved BVPFD/DBVPFD (Single/Double precision) Solve a (parameterized) system of differential equations with boundary conditions at two points, using a variable order, variable step size finite difference method with deferred corrections. Usage CALL BVPFD (FCNEQN, FCNJAC, FCNBC, FCNPEQ, FCNPBC, N, NLEFT, NCUPBC, TLEFT, TRIGHT, PISTEP, TOL, NINIT, TINIT, YINIT, LDYINI, LINEAR, PRINT, MXGRID, NFINAL, TFINAL, YFINAL, LDYFIN, ERREST) Arguments FCNEQN — User-supplied SUBROUTINE to evaluate derivatives. The usage is CALL FCNEQN (N, T, Y, P, DYDT), where N — Number of differential equations. (Input) T — Independent variable, t. (Input) Y — Array of size N containing the dependent variable values, y(t). (Input) P — Continuation parameter, p. (Input) See Comment 3. DYDT — Array of size N containing the derivatives y’(t). (Output) The name FCNEQN must be declared EXTERNAL in the calling program. FCNJAC — User-supplied SUBROUTINE to evaluate the Jacobian. The usage is CALL FCNJAC (N, T, Y, P, DYPDY), where N — Number of differential equations. (Input) T — Independent variable, t. (Input) Y — Array of size N containing the dependent variable values. (Input) P — Continuation parameter, p. (Input) See Comments 3. DYPDY — N by N array containing the partial derivatives ai,j = ∂ fi / ∂ yj evaluated at (t, y). The values ai,j are returned in DYPDY(i, j). (Output) The name FCNJAC must be declared EXTERNAL in the calling program. FCNBC — User-supplied SUBROUTINE to evaluate the boundary conditions. The usage is CALL FCNBC (N, YLEFT, YRIGHT, P, H), where N — Number of differential equations. (Input) YLEFT — Array of size N containing the values of the dependent variable at the left endpoint. (Input) YRIGHT — Array of size N containing the values of the dependent variable at the right endpoint. (Input) P — Continuation parameter, p. (Input) See Comment 3. IMSL Math Library FPS 4.0 Books Online C2 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved H — Array of size N containing the boundary condition residuals. (Output) The boundary conditions are defined by hi = 0; for i = 1, . . . , N. The left endpoint conditions must be defined first, then, the conditions involving both endpoints, and finally the right endpoint conditions. The name FCNBC must be declared EXTERNAL in the calling program. FCNPEQ — User-supplied SUBROUTINE to evaluate the partial derivative of y’ with respect to the parameter p. The usage is CALL FCNPEQ (N, T, Y, P, DYPDP), where N — Number of differential equations. (Input) T — Dependent variable, t. (Input) Y — Array of size N containing the dependent variable values. (Input) P — Continuation parameter, p. (Input) See Comment 3. DYPDP — Array of size N containing the partial derivatives ai,j = ¶fi /¶yj evaluated at (t, y). The values ai,j are returned in DYPDY(i, j). (Output) The name FCNPEQ must be declared EXTERNAL in the calling program. FCNPBC — User-supplied SUBROUTINE to evaluate the derivative of the boundary conditions with respect to the parameter p. The usage is CALL FCNPBC (N, YLEFT, YRIGHT, P, H), where N — Number of differential equations. (Input) YLEFT — Array of size N containing the values of the dependent variable at the left endpoint. (Input) YRIGHT — Array of size N containing the values of the dependent variable at the right endpoint. (Input) P — Continuation parameter, p. (Input) See Comment 3. H — Array of size N containing the derivative of fi with respect to p. (Output) The name FCNPBC must be declared EXTERNAL in the calling program. N — Number of differential equations. (Input) NLEFT — Number of initial conditions. (Input) The value NLEFT must be greater than or equal to zero and less than N. NCUPBC — Number of coupled boundary conditions. (Input) The value NLEFT + NCUPBC must be greater than zero and less than or equal to N. TLEFT — The left endpoint. (Input) TRIGHT — The right endpoint. (Input) PISTEP — Initial increment size for p. (Input) If this value is zero, continuation will not be used in this problem. The routines FCNPEQ and FCNPBC will not be called. IMSL Math Library FPS 4.0 Books Online C3 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved TOL — Relative error control parameter. (Input) The computations stop when ABS(ERROR(J, I))/MAX(ABS(Y(J, I)), 1.0).LT.TOL for all J = 1, . . . , N and I = 1, . . . , NGRID. Here, ERROR(J, I) is the estimated error in Y(J, I). NINIT — Number of initial grid points, including the endpoints. (Input) It must be at least 4. TINIT — Array of size NINIT containing the initial grid points. (Input) YINIT — Array of size N by NINIT containing an initial guess for the values of Y at the points in TINIT. (Input) LDYINI — Leading dimension of YINIT exactly as specified in the dimension statement of the calling program. (Input) LINEAR — Logical .TRUE. if the differential equations and the boundary conditions are linear. (Input) PRINT — Logical .TRUE. if intermediate output is to be printed. (Input) MXGRID — Maximum number of grid points allowed. (Input) NFINAL — Number of final grid points, including the endpoints. (Output) TFINAL — Array of size MXGRID containing the final grid points. (Output) Only the first NFINAL points are significant. YFINAL — Array of size N by MXGRID containing the values of Y at the points in TFINAL. (Output) LDYFIN — Leading dimension of YFINAL exactly as specified in the dimension statement of the calling program. (Input) ERREST — Array of size N. (Output) ERREST(J) is the estimated error in Y(J). Comments 1. Automatic workspace usage is BVPFD N(3N * MXGRID + 4N + 1) + MXGRID * (7N + 2) + 2N * MXGRID + N + MXGRID DBVPFD 2N(3N * MXGRID + 4N + 1) + 2 * MXGRID(7N + 2) + 2N * MXGRID + N + MXGRID Workspace may be explicitly provided, if desired, by use of B2PFD/DB2PFD. The reference is CALL B2PFD (FCNEQN, FCNJAC, FCNBC, FCNPEQ, FCNPBC, N, NLEFT, NCUPBC, TLEFT, TRIGHT, PISTEP, TOL, NINIT, TINIT, YINIT, LDYINI, LINEAR, PRINT, MXGRID, NFINAL, TFINAL, YFINAL, LDYFIN, ERREST, RWORK, IWORK) The additional arguments are as follows: IMSL Math Library FPS 4.0 Books Online C4 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved RWORK — Floating-point work array of size N(3N * MXGRID + 4N + 1) + MXGRID * (7N + 2). IWORK — Integer work array of size 2N * MXGRID + N + MXGRID. 2. Informational errors Type Code 4 1 More than MXGRID grid points are needed to solve the problem. 4 2 Newton's method diverged. 3 3 Newton's method reached roundoff error level. 3. If the value of PISTEP is greater than zero, then the routine BVPFD assumes that the user has embedded the problem into a one-parameter family of problems: y’ = y’(t, y, p) h(ytleft, ytright, p) = 0 such that for p = 0 the problem is simple. For p = 1, the original problem is recovered. The routine BVPFD automatically attempts to increment from p = 0 to p = 1. The value PISTEP is the beginning increment used in this continuation. The increment will usually be changed by routine BVPFD, but an arbitrary minimum of 0.01 is imposed. 4. The vectors TINIT and TFINAL may be the same. 5. The arrays YINIT and YFINAL may be the same. Algorithm The routine BVPFD is based on the subprogram PASVA3 by M. Lentini and V. Pereyra (see Pereyra 1978). The basic discretization is the trapezoidal rule over a nonuniform mesh. This mesh is chosen adaptively, to make the local error approximately the same size everywhere. Higher-order discretizations are obtained by deferred corrections. Global error estimates are produced to control the computation. The resulting nonlinear algebraic system is solved by Newton's method with step control. The linearized system of equations is solved by a special form of Gauss elimination that preserves the sparseness. Example 1 This example solves the third-order linear equation subject to the boundary conditions y(0) = y(2p) and y’(0) = y’(2p) = 1. (Its solution is y = sin t.) To use BVPFD, the problem is reduced to a system of first-order equations by defining y1 = y, y2 = y’ and y3 = y². The resulting system is IMSL Math Library FPS 4.0 Books Online C5 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved Note that there is one boundary condition at the left endpoint t = 0 and one boundary condition coupling the left and right endpoints. The final boundary condition is at the right endpoint. The total number of boundary conditions must be the same as the number of equations (in this case 3). Note that since the parameter p is not used in the call to BVPFD, the routines FCNPEQ and FCNPBC are not needed. Therefore, in the call to BVPFD, FCNEQN and FCNBC were used in place of FCNPEQ and FCNPBC. C SPECIFICATIONS FOR PARAMETERS INTEGER LDYFIN, LDYINI, MXGRID, NEQNS, NINIT PARAMETER (MXGRID=45, NEQNS=3, NINIT=10, LDYFIN=NEQNS, & LDYINI=NEQNS) C SPECIFICATIONS FOR LOCAL VARIABLES INTEGER I, J, NCUPBC, NFINAL, NLEFT, NOUT REAL ERREST(NEQNS), PISTEP, TFINAL(MXGRID), TINIT(NINIT), & TLEFT, TOL, TRIGHT, YFINAL(LDYFIN,MXGRID), & YINIT(LDYINI,NINIT) LOGICAL LINEAR, PRINT C SPECIFICATIONS FOR INTRINSICS INTRINSIC FLOAT REAL FLOAT C SPECIFICATIONS FOR SUBROUTINES EXTERNAL BVPFD, SSET, UMACH C SPECIFICATIONS FOR FUNCTIONS EXTERNAL CONST, FCNBC, FCNEQN, FCNJAC REAL CONST, FCNBC, FCNEQN, FCNJAC C Set parameters NLEFT = 1 NCUPBC = 1 TOL = .001 TLEFT = 0.0 TRIGHT = 2.0*CONST('PI') PISTEP = 0.0 PRINT = .FALSE. LINEAR = .TRUE. C Define TINIT DO 10 I=1, NINIT TINIT(I) = TLEFT + (I-1)*(TRIGHT-TLEFT)/FLOAT(NINIT-1) 10 CONTINUE C Set YINIT to zero DO 20 I=1, NINIT CALL SSET (NEQNS, 0.0, YINIT(1,I), 1) 20 CONTINUE C Solve problem CALL BVPFD (FCNEQN, FCNJAC, FCNBC, FCNEQN, FCNBC, NEQNS, NLEFT, & NCUPBC, TLEFT, TRIGHT, PISTEP, TOL, NINIT, TINIT, & YINIT, LDYINI, LINEAR, PRINT, MXGRID, NFINAL, & TFINAL, YFINAL, LDYFIN, ERREST) C Print results CALL UMACH (2, NOUT) IMSL Math Library FPS 4.0 Books Online C6 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved WRITE (NOUT,99997) WRITE (NOUT,99998) (I,TFINAL(I),(YFINAL(J,I),J=1,NEQNS),I=1, & NFINAL) WRITE (NOUT,99999) (ERREST(J),J=1,NEQNS) 99997 FORMAT (4X, 'I', 7X, 'T', 14X, 'Y1', 13X, 'Y2', 13X, 'Y3') 99998 FORMAT (I5, 1P4E15.6) 99999 FORMAT (' Error estimates', 4X, 1P3E15.6) END SUBROUTINE FCNEQN (NEQNS, T, Y, P, DYDX) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL T, P, Y(NEQNS), DYDX(NEQNS) C SPECIFICATIONS FOR INTRINSICS INTRINSIC SIN REAL SIN C Define PDE DYDX(1) = Y(2) DYDX(2) = Y(3) DYDX(3) = 2.0*Y(3) - Y(2) + Y(1) + SIN(T) RETURN END SUBROUTINE FCNJAC (NEQNS, T, Y, P, DYPDY) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL T, P, Y(NEQNS), DYPDY(NEQNS,NEQNS) C Define d(DYDX)/dY DYPDY(1,1) = 0.0 DYPDY(1,2) = 1.0 DYPDY(1,3) = 0.0 DYPDY(2,1) = 0.0 DYPDY(2,2) = 0.0 DYPDY(2,3) = 1.0 DYPDY(3,1) = 1.0 DYPDY(3,2) = -1.0 DYPDY(3,3) = 2.0 RETURN END SUBROUTINE FCNBC (NEQNS, YLEFT, YRIGHT, P, F) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL P, YLEFT(NEQNS), YRIGHT(NEQNS), F(NEQNS) C Define boundary conditions F(1) = YLEFT(2) - 1.0 F(2) = YLEFT(1) - YRIGHT(1) F(3) = YRIGHT(2) - 1.0 RETURN END Output I T Y1 Y2 Y3 1 0.000000E+00 -1.123191E-04 1.000000E+00 6.242319E05 2 3.490659E-01 3.419107E-01 9.397087E-01 -3.419580E01 3 6.981317E-01 6.426908E-01 7.660918E-01 -6.427230E-01 4 1.396263E+00 9.847531E-01 1.737333E-01 -9.847453E-01 5 2.094395E+00 8.660529E-01 -4.998747E-01 -8.660057E-01 6 2.792527E+00 3.421830E-01 -9.395474E-01 -3.420648E-01 7 3.490659E+00 -3.417234E-01 -9.396111E-01 3.418948E-01 IMSL Math Library FPS 4.0 Books Online C7 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved 8 4.188790E+00 -8.656880E-01 -5.000588E-01 8.658733E-01 9 4.886922E+00 -9.845794E-01 1.734571E-01 9.847518E-01 10 5.585054E+00 -6.427721E-01 7.658258E-01 6.429526E-01 11 5.934120E+00 -3.420819E-01 9.395434E-01 3.423986E-01 12 6.283185E+00 -1.123186E-04 1.000000E+00 6.743190E-04 Error estimates 2.840430E-04 1.792939E-04 5.588399E-04 Example 2 In this example, the following nonlinear problem is solved: y² - y3 + (1 + sin2 t) sin t = 0 with y(0) = y(p) = 0. Its solution is y = sin t. As in Example 1, this equation is reduced to a system of first-order differential equations by defining y1 = y and y2 = y’. The resulting system is In this problem, there is one boundary condition at the left endpoint and one at the right endpoint; there are no coupled boundary conditions. Note that since the parameter p is not used, in the call to BVPFD the routines FCNPEQ and FCNPBC are not needed. Therefore, in the call to BVPFD, FCNEQN and FCNBC were used in place of FCNPEQ and FCNPBC. C SPECIFICATIONS FOR PARAMETERS INTEGER LDYFIN, LDYINI, MXGRID, NEQNS, NINIT PARAMETER (MXGRID=45, NEQNS=2, NINIT=12, LDYFIN=NEQNS, & LDYINI=NEQNS) C SPECIFICATIONS FOR LOCAL VARIABLES INTEGER I, J, NCUPBC, NFINAL, NLEFT, NOUT REAL ERREST(NEQNS), PISTEP, TFINAL(MXGRID), TINIT(NINIT), & TLEFT, TOL, TRIGHT, YFINAL(LDYFIN,MXGRID), & YINIT(LDYINI,NINIT) LOGICAL LINEAR, PRINT C SPECIFICATIONS FOR INTRINSICS INTRINSIC FLOAT REAL FLOAT C SPECIFICATIONS FOR SUBROUTINES EXTERNAL BVPFD, UMACH C SPECIFICATIONS FOR FUNCTIONS EXTERNAL CONST, FCNBC, FCNEQN, FCNJAC REAL CONST C Set parameters NLEFT = 1 NCUPBC = 0 TOL = .001 TLEFT = 0.0 TRIGHT = CONST('PI') PISTEP = 0.0 PRINT = .FALSE. LINEAR = .FALSE. C Define TINIT and YINIT DO 10 I=1, NINIT TINIT(I) = TLEFT + (I-1)*(TRIGHT-TLEFT)/FLOAT(NINIT-1) IMSL Math Library FPS 4.0 Books Online C8 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved YINIT(1,I) = 0.4*(TINIT(I)-TLEFT)*(TRIGHT-TINIT(I)) YINIT(2,I) = 0.4*(TLEFT-TINIT(I)+TRIGHT-TINIT(I)) 10 CONTINUE C Solve problem CALL BVPFD (FCNEQN, FCNJAC, FCNBC, FCNEQN, FCNBC, NEQNS, NLEFT, & NCUPBC, TLEFT, TRIGHT, PISTEP, TOL, NINIT, TINIT, & YINIT, LDYINI, LINEAR, PRINT, MXGRID, NFINAL, & TFINAL, YFINAL, LDYFIN, ERREST) C Print results CALL UMACH (2, NOUT) WRITE (NOUT,99997) WRITE (NOUT,99998) (I,TFINAL(I),(YFINAL(J,I),J=1,NEQNS),I=1, & NFINAL) WRITE (NOUT,99999) (ERREST(J),J=1,NEQNS) 99997 FORMAT (4X, 'I', 7X, 'T', 14X, 'Y1', 13X, 'Y2') 99998 FORMAT (I5, 1P3E15.6) 99999 FORMAT (' Error estimates', 4X, 1P2E15.6) END SUBROUTINE FCNEQN (NEQNS, T, Y, P, DYDT) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL T, P, Y(NEQNS), DYDT(NEQNS) C SPECIFICATIONS FOR INTRINSICS INTRINSIC SIN REAL SIN C Define PDE DYDT(1) = Y(2) DYDT(2) = Y(1)**3 - SIN(T)*(1.0+SIN(T)**2) RETURN END SUBROUTINE FCNJAC (NEQNS, T, Y, P, DYPDY) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL T, P, Y(NEQNS), DYPDY(NEQNS,NEQNS) C Define d(DYDT)/dY DYPDY(1,1) = 0.0 DYPDY(1,2) = 1.0 DYPDY(2,1) = 3.0*Y(1)**2 DYPDY(2,2) = 0.0 RETURN END SUBROUTINE FCNBC (NEQNS, YLEFT, YRIGHT, P, F) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL P, YLEFT(NEQNS), YRIGHT(NEQNS), F(NEQNS) C Define boundary conditions F(1) = YLEFT(1) F(2) = YRIGHT(1) RETURN END Output I T Y1 Y2 1 0.000000E+00 0.000000E+00 9.999277E-01 2 2.855994E-01 2.817682E-01 9.594315E-01 3 5.711987E-01 5.406458E-01 8.412407E-01 IMSL Math Library FPS 4.0 Books Online C9 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved 4 8.567980E-01 7.557380E-01 6.548904E-01 5 1.142397E+00 9.096186E-01 4.154530E-01 6 1.427997E+00 9.898143E-01 1.423307E-01 7 1.713596E+00 9.898143E-01 -1.423307E-01 8 1.999195E+00 9.096185E-01 -4.154530E-01 9 2.284795E+00 7.557380E-01 -6.548903E-01 10 2.570394E+00 5.406460E-01 -8.412405E-01 11 2.855994E+00 2.817683E-01 -9.594313E-01 12 3.141593E+00 0.000000E+00 -9.999274E-01 Error estimates 3.906105E-05 7.124186E-05 Example 3 In this example, the following nonlinear problem is solved: with y(0) = y(1) = p/2. As in the previous examples, this equation is reduced to a system of first-order differential equations by defining y1 = y and y2 = y’. The resulting system is The problem is embedded in a family of problems by introducing the parameter p and by changing the second differential equation to At p = 0, the problem is linear; and at p = 1, the original problem is recovered. The derivatives ¶y’/¶p must now be specified in the subroutine FCNPEQ. The derivatives ¶f/¶p are zero in FCNPBC. C SPECIFICATIONS FOR PARAMETERS INTEGER LDYFIN, LDYINI, MXGRID, NEQNS, NINIT PARAMETER (MXGRID=45, NEQNS=2, NINIT=5, LDYFIN=NEQNS, & LDYINI=NEQNS) C SPECIFICATIONS FOR LOCAL VARIABLES INTEGER NCUPBC, NFINAL, NLEFT, NOUT REAL ERREST(NEQNS), PISTEP, TFINAL(MXGRID), TLEFT, TOL, & XRIGHT, YFINAL(LDYFIN,MXGRID) LOGICAL LINEAR, PRINT C SPECIFICATIONS FOR SAVE VARIABLES INTEGER I, J REAL TINIT(NINIT), YINIT(LDYINI,NINIT) SAVE I, J, TINIT, YINIT C SPECIFICATIONS FOR SUBROUTINES EXTERNAL BVPFD, UMACH C SPECIFICATIONS FOR FUNCTIONS EXTERNAL FCNBC, FCNEQN, FCNJAC, FCNPBC, FCNPEQ C DATA TINIT/0.0, 0.4, 0.5, 0.6, 1.0/ DATA ((YINIT(I,J),J=1,NINIT),I=1,NEQNS)/0.15749, 0.00215, 0.0, IMSL Math Library FPS 4.0 Books Online C10 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved & 0.00215, 0.15749, -0.83995, -0.05745, 0.0, 0.05745, 0.83995/ C Set parameters NLEFT = 1 NCUPBC = 0 TOL = .001 TLEFT = 0.0 XRIGHT = 1.0 PISTEP = 0.1 PRINT = .FALSE. LINEAR = .FALSE. C CALL BVPFD (FCNEQN, FCNJAC, FCNBC, FCNPEQ, FCNPBC, NEQNS, NLEFT, & NCUPBC, TLEFT, XRIGHT, PISTEP, TOL, NINIT, TINIT, & YINIT, LDYINI, LINEAR, PRINT, MXGRID, NFINAL, & TFINAL, YFINAL, LDYFIN, ERREST) C Print results CALL UMACH (2, NOUT) WRITE (NOUT,99997) WRITE (NOUT,99998) (I,TFINAL(I),(YFINAL(J,I),J=1,NEQNS),I=1, & NFINAL) WRITE (NOUT,99999) (ERREST(J),J=1,NEQNS) 99997 FORMAT (4X, 'I', 7X, 'T', 14X, 'Y1', 13X, 'Y2') 99998 FORMAT (I5, 1P3E15.6) 99999 FORMAT (' Error estimates', 4X, 1P2E15.6) END SUBROUTINE FCNEQN (NEQNS, T, Y, P, DYDT) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL T, P, Y(NEQNS), DYDT(NEQNS) C Define PDE DYDT(1) = Y(2) DYDT(2) = P*Y(1)**3 + 40./9.*((T-0.5)**2)**(1./3.) - (T-0.5)**8 RETURN END SUBROUTINE FCNJAC (NEQNS, T, Y, P, DYPDY) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL T, P, Y(NEQNS), DYPDY(NEQNS,NEQNS) C Define d(DYDT)/dY DYPDY(1,1) = 0.0 DYPDY(1,2) = 1.0 DYPDY(2,1) = P*3.*Y(1)**2 DYPDY(2,2) = 0.0 RETURN END SUBROUTINE FCNBC (NEQNS, YLEFT, YRIGHT, P, F) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL P, YLEFT(NEQNS), YRIGHT(NEQNS), F(NEQNS) C SPECIFICATIONS FOR LOCAL VARIABLES REAL PI C SPECIFICATIONS FOR FUNCTIONS EXTERNAL CONST REAL CONST C Define boundary conditions PI = CONST('PI') F(1) = YLEFT(1) - PI/2.0 IMSL Math Library FPS 4.0 Books Online C11 BVPFD/DBVPFD (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved F(2) = YRIGHT(1) - PI/2.0 RETURN END SUBROUTINE FCNPEQ (NEQNS, T, Y, P, DYPDP) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL T, P, Y(NEQNS), DYPDP(NEQNS) C Define d(DYDT)/dP DYPDP(1) = 0.0 DYPDP(2) = Y(1)**3 RETURN END SUBROUTINE FCNPBC (NEQNS, YLEFT, YRIGHT, P, DFDP) C SPECIFICATIONS FOR ARGUMENTS INTEGER NEQNS REAL P, YLEFT(NEQNS), YRIGHT(NEQNS), DFDP(NEQNS) C SPECIFICATIONS FOR SUBROUTINES EXTERNAL SSET C Define dF/dP CALL SSET (NEQNS, 0.0, DFDP, 1) RETURN END Output I T Y1 Y2 1 0.000000E+00 1.570796E+00 -1.949336E+00 2 4.444445E-02 1.490495E+00 -1.669567E+00 3 8.888889E-02 1.421951E+00 -1.419465E+00 4 1.333333E-01 1.363953E+00 -1.194307E+00 5 2.000000E-01 1.294526E+00 -8.958461E-01 6 2.666667E-01 1.243628E+00 -6.373191E-01 7 3.333334E-01 1.208785E+00 -4.135206E-01 8 4.000000E-01 1.187783E+00 -2.219351E-01 9 4.250000E-01 1.183038E+00 -1.584200E-01 10 4.500000E-01 1.179822E+00 -9.973146E-02 11 4.625000E-01 1.178748E+00 -7.233893E-02 12 4.750000E-01 1.178007E+00 -4.638248E-02 13 4.812500E-01 1.177756E+00 -3.399763E-02 14 4.875000E-01 1.177582E+00 -2.205547E-02 15 4.937500E-01 1.177480E+00 -1.061177E-02 16 5.000000E-01 1.177447E+00 -1.479182E-07 17 5.062500E-01 1.177480E+00 1.061153E-02 18 5.125000E-01 1.177582E+00 2.205518E-02 19 5.187500E-01 1.177756E+00 3.399727E-02 20 5.250000E-01 1.178007E+00 4.638219E-02 21 5.375000E-01 1.178748E+00 7.233876E-02 22 5.500000E-01 1.179822E+00 9.973124E-02 23 5.750000E-01 1.183038E+00 1.584199E-01 24 6.000000E-01 1.187783E+00 2.219350E-01 25 6.666667E-01 1.208786E+00 4.135205E-01 26 7.333333E-01 1.243628E+00 6.373190E-01 27 8.000000E-01 1.294526E+00 8.958461E-01 28 8.666667E-01 1.363953E+00 1.194307E+00 29 9.111111E-01 1.421951E+00 1.419465E+00 30 9.555556E-01 1.490495E+00 1.669566E+00 31 1.000000E+00 1.570796E+00 1.949336E+00 Error estimates 3.448358E-06 5.549869E-05 IMSL Math Library FPS 4.0 Books Online C12 APPENDIX D FORTRAN SOURCE CODE IMSL SUBROUTINE (DFDJAC) D-1 FDJAC/DFDJAC (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved FDJAC/DFDJAC (Single/Double precision) Approximate the Jacobian of M functions in N unknowns using forward differences. Usage CALL FDJAC (FCN, M, N, XC, XSCALE, FC, EPSFCN, FJAC, LDFJAC) Arguments FCN — User-supplied SUBROUTINE to evaluate the function to be minimized. The usage is CALL FCN (M, N, X, F), where M — Length of F. (Input) N — Length of X. (Input) X — The point at which the function is evaluated. (Input) X should not be changed by FCN. F — The computed function at the point X. (Output) FCN must be declared EXTERNAL in the calling program. M — The number of functions. (Input) N — The number of variables. (Input) XC — Vector of length N containing the point at which the gradient is to be estimated. (Input) XSCALE — Vector of length N containing the diagonal scaling matrix for the variables. (Input) In the absence of other information, set all entries to 1.0. FC — Vector of length M containing the function values at XC. (Input) EPSFCN — Estimate for the relative noise in the function. (Input) EPSFCN must be less than or equal to 0.1. In the absence of other information, set EPSFCN to 0.0. FJAC — M by N matrix containing the estimated Jacobian at XC. (Output) LDFJAC — Leading dimension of FJAC exactly as specified in the dimension statement of the calling program. (Input) IMSL Math Library FPS 4.0 Books Online D2 FDJAC/DFDJAC (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved Comments 1. Automatic workspace usage is FDJAC M units, or DFDJAC 2 * M units. Workspace may be explicitly provided, if desired, by use of F2JAC/DF2JAC. The reference is CALL F2JAC (FCN, M, N, XC, XSCALE, FC, EPSFCN, FJAC, LDFJAC, WK) The additional argument is WK — Work vector of length M. 2. This is Algorithm A5.4.1, Dennis and Schnabel , 1983, page 314. Algorithm The routine FDJAC uses the following finite-difference formula to estimate the Jacobian matrix of function f at x: where ej is the j-th unit vector, hj = 1/2 max{|xj|, 1/sj} sign(xj), is the machine epsilon, and sj is the scaling factor of the j-th variable. For more details, see Dennis and Schnabel (1983). Since the finite-difference method has truncation error, cancellation error, and rounding error, users should be aware of possible poor performance. When possible, high precision arithmetic is recommended. Example In this example, the Jacobian matrix of is estimated by the finite-difference method at the point (1.0, 1.0). C Declaration of variables INTEGER N, M, LDFJAC, NOUT PARAMETER (N=2, M=2, LDFJAC=2) REAL FJAC(LDFJAC,N), XSCALE(N), XC(N), FC(M), EPSFCN EXTERNAL FCN, FDJAC, UMACH C DATA XSCALE/2*1.0E0/, XC/2*1.0E0/ C Set function noise EPSFCN = 0.01 IMSL Math Library FPS 4.0 Books Online D3 FDJAC/DFDJAC (Single/Double precision) © 1990-1995 Microsoft Corporation. All rights reserved C Evaluate the function at the C current point CALL FCN (M, N, XC, FC) C Get Jacobian forward-difference C approximation CALL FDJAC (FCN, M, N, XC, XSCALE, FC, EPSFCN, FJAC, LDFJAC) C Print results CALL UMACH (2, NOUT) WRITE (NOUT,99999) ((FJAC(I,J),J=1,N),I=1,M) 99999 FORMAT (' The Jacobian is', /, 2(5X,2F10.2,/),/) C END C SUBROUTINE FCN (M, N, X, F) C SPECIFICATIONS FOR ARGUMENTS INTEGER M, N REAL X(N), F(M) C F(1) = X(1)*X(2) - 2.0E0 F(2) = X(1) - X(1)*X(2) + 1.0E0 C RETURN END Output The Jacobian is 1.00 1.00 0.00 -1.00 IMSL Math Library FPS 4.0 Books Online D4