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SOLUTION OF THE IDEAL ADIABATIC STIRLING

MOD EL WITH COUPLED FIRST ORDER D IFFERENTIALEQUATIONS BY THE PASIC METHOD

A Thesis Presented t o

The Faculty of the

Fritz J. And Dolores H. Russ

College of Engineering and Technology

Ohio U niversity

In Partial Fulfillment

of the Requirements for the D egree

Master of Science

by

Eric Thomas Malroy

June, 1998



ACKNOWLEDGMENTS

I would first like to thank my Branch Chief,Mr. Derrick Cheston of NASA Lewis

Research Center, for allowing me to take time off to complete this thesis.

I would also like to thank Dr. Lloyd Herman, Dr. Hajrudin Pasic and Dr. Israel

Urieli for being on the defense committee. Also, I am grateful that D r. Pasic took the

time to explain his method of solving ODES. He was able to take a complex subject and

make it easy to understand. His diligence, example and scholarship were an inspiration .

Particularly, I am gratefbl that Dr. Urieli was my advisor at Ohio University. It

has been a real honor to have one of the experts in S tirling analysis teach me. His

diligence in making me see the importance of having well documented programs will

always be usefbl. His time spent pouring over my programs and thesis will not be

forgotten . Also, his excellence in scholarship has pushed me fbrther. I am gratefbl.

I would also like to state that I still owe a paper to D r. Pasic and Dr. U rieli, which

I have promised to research and write - hough it may take me some time to complete. I

will not forget.

My parents and two brothers have also been a source of encouragement to me

throughout the time I have studied at Ohio University. Their encouragement has kept me

going during the times I felt overwhelmed.

Finally, I would like to thank my Lord and Savior Jesus Christ who is the grand

architect of this magnificent universe who gives me the hope and reason for living.



Table of Contents

.... . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .cknowledgements.. iii

. . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. .able of Conten ts.. iv

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ist of Tables. .vi.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ist of Figures. .vii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ist of Sym bols.. x

1 Introduction

1 . 1 Background. . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..2 The Stirling Engine. . 5

. . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . ..3 Literature Review of Stirling Analysis.. .12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..4 Thesis Background.. -22

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..5 Thesis Outline. .2 4

2 The Ideal Adiabatic Model

. . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . ..1 Introduction.. .27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..2 The Basic Model. .28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..3 Development of Equations.. . 31

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..4 Volume V ariations. .42

. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . ...5 Method of Solution.. .48

. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . ..6 Summary 50

3 The P asic Method

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..1 Introduction.. . 5

. . . . . . . . . . . . . . . . . . . . . . . . . . ..2 Picard's Iterations by Successive Approximations.. .52



. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . ..3 Collocation Method 53

.......................................................................4 Shooting Method 54

............................5 Synthesis of Methods to Formulate the Pasic Method 56

3.6 Pasic Method for Higher Order Differential Equations ........................60

4 Application of the Pasic Method to the Ideal Adiabatic Stirling Model

4.1 Introduction...........................................................................63

...................................................................2 Problem Description 64

4.3 Complications in the Application of the Pasic Method ..........................66

4.4 Program Structure .................................................................... 70

4.5 Results . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .76

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..6 Conclusion 82

5 Conclusions and Future Research

..............................................................................1 Conclusion 83

. . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . ..2 Future Research 84

...........................................................................................eferences 88

Appendix A

Drive Mechanisms and Pictures of Stirling Engine .................................... 91

Appendix B

......................................................................esults of Analysis 100

Appendix C

........................................................................rogram Modules 107

Appendix D

...................................................................utput from Programs 120

Abstract



List of Tables

Table 1.1 Basic Assumptions Made by Schmidt ............................................ 13

Table 1.2. Classification of Simulation Approaches as Described by Urieli . . . . . . . . . . . . . . .17

Table 1.3: Relationship between Type of Analytical Model and Numerical Integration

Scheme as Presented by Organ .......................................................... 17

Table 1.4: Engineering Thermodynamic and Gas Dynamic

.........................................................................onservation Laws 20

...................................................able 1 .5 . Symbols and Associated Meaning 21

.....................................................able 1.6 . Questions that Thesis Addresses 25

Table 2.1: Ideal Adiabatic Stirling Machine Differential and Algebraic

....................................................quations as Presented by Urieli 41

........able 2.2. Numerical Scheme of the Classical Fourth-order Runge-Kutta Method 49

...able 4.1 Essential Equations Required to Solve Temperature Differential Equations 65

Table 4.2. Summary of Results ..................................................................7



vii

List of Figures

Figure 1 . 1 Finke lstein's Conception of what Stirling's 2 Hp Engine of 18 18 May Have

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ooked Like 2

Figure 1.2. Ideal Isothermal Stirling Cycle . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

Figure 1.3. Parasitic Losses Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Figure 2.1. Urieli's Ideal Adiabatic Stirling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0

Figure 2.2. Generic Control Volumes Used to Formulate Equations . . . . . . . . . . . . . . . . . . . . . . . .33

Figure 2.3 Regenerator Linear Tem perature Profile as given by Urieli . . . . . . . . . . . . . . . . . . . . . 9

.igure 2.4. Cross Section of the N V Philips 1-98 Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Figure 2.5:Schematic Showing the FordIPhilips Four Cylinder Double-Acting Eng ine

. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . .onfiguration 44

Figure 2.6:Swash Plate Drive Mechanism Show ing Two of Four Cylinders and R elated

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .inusoidal Equations

45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .igure 2.7.The Ross D-90 Engine 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .igure 2.8.Inverted Ross Yo ke with Corresponding Equations 47

Figure 3.1 The Shooting Method Changes a Boundary Value Problem into an Initual-

. . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . .alue Problem 55

Figure 4.1 Graph Showing Fixed-point Iterations and Collocation Points for

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .emperature T, Over Same Sub-domain as Figure 4.2 67

Figure 4.2: Graph Showing Fixed-point Iterations and Collocation Points for

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .emperature T, Over Same Sub-domain as Figure 4.1 68

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .igure 4.3 Adiabatic Module Algorithm 71



...Vlll

Figure 4.4 . Stirlingl Module Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 4.5. Effects of Number of Sub-domains on Temperature (Ross-90) . . . . . . . . . . . . . . .78

Figure4.6.

Effects of Num ber of Sub-domains on Error (Q,). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Figure 4.7 . Effects of Number of Sub-domains on Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0

Figure A 1 Rhombic Drive and Equation Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

Figure A.2. Ross Yoke and Equation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .igure A.3 : Ross-90 Stirling Engine 94

. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . .igure A.4. Ross-90 Foil Regenerato r and Cooler 95

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .igure A.5 . Ross-90 Engine Partially Disassembled 96

. . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . .igure A.6. V-Configuration Drive Mechanism 97

Figure A.7: Ford-Philips 4-215 Schmidt Analysis Equations for Sinusoidal Volume

. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . .pproximations 98

Figure A.8. Maximum Linear Displacement of Inverted Ross Yoke Drive . . . . . . . . . . . . . . . . . 9

Figure B .1 Ross-90 Engine: Comparison of Temperature Results between Runge-Kutta

and Pasic Method . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . .01

Figure B.2: Ross-90 Engine: Comparison of Heat Results between Runge-K utta and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .asic Method 102

Figure B.3 : Ross-90 Engine: Com parison of Work Results between Runge-K utta and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .asic Method 103

Figure B.4 : Effects of Number of Sub-domains on T, and T, T emperatures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Ross-90) 104



ix

Figure B.5: Ford-Philips 4-215 Engine: Comparison of Work Results between Runge-

Kutta and Pasic Me thod.. . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . . . . . . . ..105

Figure B.6: Effects of Number of Sub-domains on T, and T, Temperatures

(Ford-Philips 4-215). .. . . . . . . . . . .. . . . .. . .. . . . . . . .. .. . . . . . . . . . . 106



List of Symbols

The subscripts are primarily used to identify the various com ponents in the Stirling

engine. For example, V, refers to the compression space volume while V' refers to the0

expansion space volume. The variable gAm refers to the interface mass flow between the

regenerator and cooler. The positive mass flow is arbitrarily defined as positive going in

the direction of com pression to expansion space.

Cv, Cv(V

coefficien ts for the two-dimensional third-order polynomial shape

function; piston area [m2]

area, cross sectional area [m2]; coefficien ts for the P asic m ethod

vector specifying the coefficien ts in the Pasic method

effective length (used with drive mechanisms) [m]

height of crank [m]

midsection o f horizontal to edge of crank [m]

n by n matrix specifying the collocation points for the Pasic method

initial conditions for the Pasic method.

specific heat with constant pressure; specific heat w ith con stant pressure in

relation to temperature [Jkg-K ]

specific heat with constant volume; specific heat with constant volume in

relation to temperature [Jkg -K ]

piston diameter [m]

eccen tricity of compression piston on V-configuration drive [m]



P o r p

eccentricity of expansion piston on V-configuration drive [m]

specific total energy [J kg ]

refers to the function or value of the function with one or more

independent variables

vector of values o f the ODE at the collocation points

total energy [J]; error (numerical analysis)

force [N]

mass flux [kg/s-m2]

gravitational acceleration [mls2]

mass flow [kgls]

small sub-domain used with the Pasic method or Runge-Kutta method

[radians, degrees or other units related to independent variable]

enthalpy [Jk g]

coefficients for the Runge-Kutta method; index to collocation points

length of regenerator [m]

length of hypotenuse of Ross yoke [m]

mass [kg]

total mass of work ing fluid in Stirling engine [kg]

pressure during cycle [Pa]

power per unit mass [Wlkg]

total heat [J1



radius of crank [m]

gas constant [kJ/kg-K]

dum my variable for integration (num erical analysis)

total entropy [J/K]

time [s]; indepen dent variable

temperature; temperature with respect to x over the regenerator ["C or K]

specific internal energy [Jlkg]

velocity [m/s]

volume [m3]

swept volume of compression space [m3]

swept volum e of expansion space [m3]

total work [J]

independ ent variable; refers to distance between crank pivot point and

midsection of yoke base [m]; arbitrary distance within regenerator [m].

th e independ ent variable for the collocation points; the "i" is the index the

individual points

depend ent variable; length of displacement of the piston [m]

th ey value one step ahead ofyi

th e dependent variable values derived from th e collocation points in the

Pasic method; the "i" is the index referring t o the individu al collocation

points

height [m]; dependent variable for shape fbnction



Greek

CL

P C

Y

rl

8

Bmax

P

4

Subscripts

C

ck

clc

cle

e

h

Hhe

hr

phase advance ang le [radians or degrees]

angle of V-configuration drive [radians or degrees]

ratio of specific heats [none]

efficiency [none]

crank angle [radians or degrees]

crank angle at maximum linear displacement [radians or degrees]

density [kg/mA3]

angle related to crank angle on drive mechanisms [radians or degrees]

compression space

compression-cooler interface

compression space clearance

expansion space clearance

expansion space

heater

high temperature heat reservoir

heater-expansion space interface

heater-regenerator interface

inlet; index to collocation points



k

krL

max

min

n

0

X

cooler

cooler -regenerator interface

low tem perature heat reservoir

maximum value o f variable

minimum value of variable (i.e . y,;, = minimum linear displacement)

index to variable; index referring to the fixed-point iterations

outlet (Stirling engines); initial value (numerical analysis)

in the vector direction of x

Superscripts

m refers to the order of the derivative (numerical analysis)

-

(bar) the a pproxim ate value (numerical analysis)

Other Symbols

D derivative of a variable with respect crank angle 8 (i.e. DT = dTIde)

d partial differential symbol

A incremen tal step

C summ ation notation

d ordinary differential symbol

f o means fbnction of one or more independent variables

I derivative symbol (i.e.y' = dyldx ,where x is some arbitrary independen t

variable)



integral symbol

natural logarithm



Chapter 1

Introduction

1.1 Background

The Rev. Robert Stirling, a Scottish Presbyterian minister patented the Stirling

engine, in 1816. This engine operates on a closed cycle, which means that it encloses a

constant mass of fluid and has no mass flow in or out of the engine. Tw o reciprocating

pistons shuttle the working fluid back and forth through a process of compression,

heating, expansion and cooling, resulting in a positive net production of work. The

engine operates fiom any external heat source. Additionally, the engine will provide

cooling if work is put into the machine as predicted fiom basic thermodynam ics. Figure

1.1 shows one of Stirling's first engines.

Stirling's invention was extra-ordinary for his time because the laws of

therm odynam ics were not yet formulated. Additionally, he used a device for storing

energy, called a regenerator , between the compression and expansion spaces . This device

collects the heat fiom the hot fluid when it flows from the expansion space to the

compression space and returns the heat during the reversal of the flow. Stirling's keen

insight made him aware of the value of the regenerator in storing energy, which many

people afterwards failed to hlly comprehend [I] . Most of the early Stirling engines,

following the Rev. Stirling, did not have regenerators. These engines are commonly

referred to as air engines. Stirling engines with regenerators and air engines excelled as



Figure 1.1 Finkelstein's Conception of what Stirling's 2 Hp Engine of 1818MayHave Looked Like [2

prime movers for "water pumps and fans and for small engine applications such as

sewing machines, laboratory centrifbges and mixers, organ air pumps, gramophones, and

window display turntables" [3]. The internal combustion engine (Otto cycle) later took

over as the predominant engine by the end of the century, due primarily to the infancy of

the field o f materials during this time [3, 41. A "renaissance" of the Stirling engine began

in 1937 at the Philips Research Laboratories in Eindhoven, The Netherlands, where small

generators used Stirling engines as prime movers [5]. By 1952 the research team at

Philips produced their small "102C" engine that achieved a specific power output 30

times greater than that of earlier hot air engines. Further progress continued, and in 1973



3

the Ford-Philips 4-215 experimental automotive Stirling engine attained a 300 times

increase in specific power compared with earlier Stirling engines [3]. These

technological improvements coupled with the energy crisis in the 1970s accelerated

interest in Stirling engines [ 5 ] . Many viewed the Stirling engine as a potential solution to

the energy crisis, since it could minimize fuel consumption with its potentially high

efficiency. Additionally, the Stirling engine has a multi-fuel capability [ l ] that could

reduce the dependence on fossil fuels. Any heat source will run the engine, such as

burning rice husks, methane gas, or hydrogen. Many researchers proposed developing

large-scale manufacturing of solar concentrators to generate electricity as a solution to the

crisis related to fossil he1 depletion. In 1978 a claim was made that these

collector/engine/generators could save over 100 million barrels of oil by 2020. To

accomplish this, they need to start a large-scale production of the engine in 2000 [5].

Geothermal, nuclear and coal are some other potential energy sources that could power

Stirling engines. Stirling developers also suggested many other environmentally friendly

applications of the engine. The oil crisis exhausted itself as the OPEC nations

fragmented and additional oil reserves were found in Mexico, the North Sea and

elsewhere. Government support was also trimmed in the 1980's as other economic

concerns took priority over the long-term energy shortages associated with fossil fuels.

Although potential applications existed for the use of Stirling engines, market

conditions were unfavorable for the proposed large-scale manufacture of these engines.

Energy costs were relatively low, and industry seemed to have little interest in new and

unproven methods of reducing fossil he1 consumption. Another barrier to the production



4

of the Stirling engine was the economic costs associated with the required tooling

changes. Fo r example, highly efficient Stirling refrigerators were designed and built with

substantial energy savings. The industry resisted the large-scale manufacture o f these

refrigerators, due primarily to the tooling costs and the major changes that it would bring

upon the industry. These refrigerators w ould eliminate the CFC s that traditional

refrigerators use.

Another factor in the resistance o f industry to developing the Stirling engine is the

lack of consistency between the analysis and the actual engine performance [2, 61.

Twenty-pe rcent design margins are typically used in Stirling design. Thes e wide margins

are a hindrance in the marketplace that needs superior performance [7]. This

inconsistency puts an additional cost on the Stirling engine, so many companies back

away from the development of the engine because o f the added risk.

Environmental concerns still abound as third world nations become more

industrialized and the population of countries continue to grow. The potential high

efficiency, the multi-fbel energy source capability (including solar energy), mechanical

simplicity, and potentially low emission of pollution are some advantages of this engine.

When properly designed, the engine is quiet in operation, has low vibration and can

potentially obtain the highest specific work output of any closed regenerative cycle.

Also, it is possible to hermetically seal the engine, thus achieving potentially high

reliability [I] . All these favorable characteristics continue to arous e interest among

Stirling developers and others interested in the efficient use of energy sources, possibly

without using foss il fbels.



1.2 The Stirling Engine

The Stirling engine is a machine that operates on a closed regenerative

thermod ynam ic cycle. Typically, the engine has five main sub-compo nents: the

expansion space, comp ression space, cooler, regenerator and heater. Th e engine also has

a drive mechanism that controls the volume variations during the cycle and transfers the

linear alternating motion of the pistons into the angular velocity of the drive shaft. The

workin g fluid is shuttled back and forth during the cycle between the hot expansion space

and the cold compression space. When a larger percentage of the working fluid is in the

expansion space, a larger percentage of the total mass o f fluid is at a higher temperature.

This increases the pressure of the fluid. Typically, the pressure is nearly sinusoidal over

the cycle. The drive mechanism is designed such that the total volum e increases at

approxim ately the sam e cycle interval, during which time the high pressure occ urs. This

results in a net production of wo rk over the cycle. The interval wh en th e working fluid is

at higher pressure and the volume is expanding occurs over a significant portion of the

cycle. This produces the work. Th e regenerator stores energy from the working fluid as

the gas flows from the hot expansion space to the cold comp ression space. Upo n reversal

of the flow, the cold fluid is heated as it recaptures the energy stored in the regenerator.

This storage and retrieval of heat occurs once for each cycle and results in a decrease in

heat loss, thus causing a greater efficiency of the engine. The Stirling cycle can also be

reversed t o produce cooling by designing the drive mechanism such that the expansion o f

fluid occurs over the same cycle interval, during which the pressure is lowest. This



thermodynamic process requires work to be applied to the system during the cycle to

produce the cooling.

Any reversible heat engine, such as an ideal Stirling engine, has the efficiency of

a Carn ot engine 181. This efficiency is defined in the following equa tion:

The temperatures TL and T' stand respectively for the low and high temperature of the

two heat reservoirs. The temperatures in the two reservoirs determ ine the efficiency,

which is the highest possible that an engine can have. The limiting factor is the

tem peratu re that the materials within the engine can withstand. Typically, the m aximum

tem peratu re is about 1000 K, which gives an ideal efficiency o f 70%, if the low

temperature is 300 K . As Urieli points out, the typical ideal efficiencies of the O tto and

Diesel engines are 60% and 63% respectively for the same temperatures. The Stirling

engine is therefore only marginally better; however, if the maximum temperature is

increased to 1600 K, by using improved m aterials, an efficiency of 81% is possible [I].

Figure 1.2 shows the Ideal Isothermal Stirling model, which has been the "ideal

model" used to describe the Stirling engine. The purpose of the "ideal model" is to

provide the upper limit work o f the cycle and the upper limit efficiency of the engine. It

is a benchmark against which designers can compare their engines. The Ideal Isothermal

Stirling model is beneficial in that it does provide a good estimate of the work of the

cycle, but it fails to accurately model the efficiency of the cycle.



Compression Expansion Constant Volume

Space Space Displacement

+

Constant Volume

! . ....A,.<

..,:. ;L... , ? .

.., $. ?.. P

Regenerator CompressionIDisplacement

A Th= Constant hot

s~ n k emperature

4

2 fTk = Constant 1

Cold Sink Tempb

1Expansion

4

+

ConstantVolume

Displacement

Figure 1.2 Ideal Isothermal Stirling Cycle

The cylinder device that has an expansion space, compression space and

regenerator is used to explain the Ideal Isothermal Stirling Model in Figure 1.2. Also, the

P-V and T-S diagrams are given for the Stirling cycle. Four engine diagram s describe the

four different piston positions corresponding to points 1 through 4 on the P-V and T-S

diagrams. Starting at position 1, the working fluid is in the cold side, called the



8

compression space. Notice that the pressure is lowest at this location. The piston moves

and compresses the gas in process 1 to 2, which reduces the volume to a m inimum. Note

that work is done on the working fluid for this process, since the work is the integral of

the P-V diagram from 1 to 2. The temperature is isothermal during this process and the

entropy decreases, due to the loss of heat. Next, the cold fluid is heated as it passes

throu gh the hot regenerator in process 2 to 3. Notice that the working fluid cannot

expand, so the pressure rises to a maximum , since the fluid is in the hot side or expansion

space. Process 3-4 is the expansion of the gas to the maximum volume. Mo st the work

occurs during this process, since the wo rk is the integral of the P-V diagram from 3 to 4.

Th e entropy is greatest at point 4, due to the added heat. Heat is removed and the volume

is constant for process 4 to 1, which completes the cycle. The volume is at a maximum

during this part of the cycle as the working fluid is pushed through the regenerator and

cooled. N o energy is converted to work from 4 to 1, given that the volum e is constant

during this process, similar to process 2 to 3. The heat is removed from th e hot fluid and

stored back in the regenerator.

Several points are worth noting about the Ideal Isothermal model. No heater or

cooler is found in Figure 1.2. The model assumes that the expansion and compression

spaces are isothermal, which inherently implies having unrestricted heat flow in the

compression and expansion spaces. The isothermal assumption causes the heater and

cooler to be redundant, which contradicts reality where the heat transfer characteristics of

the compression and expansion spaces are very close to being adiabatic. The heater and



9

cooler are therefore needed. The Isothermal assumption is used to simplify the analysis

even though this assumption introduces contradictions into the model.

The analysis of Stirling engines is a complex subject. The various parameters are

highly interdependent and require optimization. Designers have to balance the

"competing" variables by optimization, which often requires sophisticated software. We

will next examine some of the losses or irreversibilities of the Stirling engine. The

purpose of this discussion is to better understand the need for software tools that enable

the optimization of the Stirling engine.

The second law of thermodynamics implies that all real engines deviate from their

theoretical models due to irreversibilities. The same is true for Stirling engines where the

engine experiences a number of losses caused by the irreversibilities. The two main

groups of losses are the flow dissipation and parasitic losses.

The flow dissipation is caused by internal heat generation, due to frictional drag,

which causes a pressure drop. The viscous dissipation occurs in the main components of

the Stirling engine, but primarily occurs In the regenerator. Also, it can be substantial in

the heater and cooler. The flow dissipation causes a loss of power, due to the pressure

drop [I].

There are a number of parasitic losses. The first is seal leakage, sometimes

called blow-by. This results when gas leaks through the rings or through the close fitting

clearance seal during high pressure. Normally, this occurs when the fluid is on the hot

side (expansion space) and the pressure is near maximum. The loss of mass results in a

loss of pressure because the hot gas transports energy. The next loss is the gas spring



10

hysteresis loss. This loss is due to the imperfect reversibility of gas springs. Some of the

energy of the spring is converted to kinetic and thermal energy in the fluid, which is lost

to the surround ings resulting in a loss of energy. Crank-type engines usually have small

hysteresis loss whereas free piston engines usually have significant hysteresis loss [I].

Another parasitic loss is the conduction loss associated with the engine . Heat flows

through the walls and through the pistons from the hot side to the cold side. This loss can

be large given that it is usual to operate the engine at a maximum temperature difference

between the expansion and compression spaces to increase the efficiency. Another loss is

the convection heat transfer loss within the hollow piston because of its motion.

Normally, this loss is ignored.

The append ix gap losses are another group of parasitic losses. The piston is made

long to isolate the seal from the hot side. Seals cannot tolerate the high temperatures, so

a long piston helps alleviate this problem. There is a gap associated with the long piston,

which causes three main losses. The first is the shu ttle loss that results from the motion

of the piston. When the wall of the cylinder is the same temperature as the wall of the

piston, say at the midstroke time, then no heat transfer occurs. When the piston moves

such that the volume is smallest in the expansion space, the colder walls of the piston

receive heat from the hotter cylinder. The piston moves back to midstroke and continues

moving until maximum volume is found in the expansion space. The wall temperature of

the cylinder is less than the wall temperature of the piston at this point, so heat is

transferred to the cylinder. This shuttling of heat continues for each cycle. The next

append ix gap loss is the enthalpy loss associated with the working gas flow in and out of



the appendix gap and the corresponding enthalpy loss. The last appendix gap loss is the

hysteresis heat lost to the cylinder walls, due to the temperature variations induced by the

pressure variations in the expansion space. Oftentimes, hysteresis heat loss is neglected,

since it is normally insignificant [I ]. Figure 1. 3 shows a concise picture of the parasitic

losses and the corresponding typical losses for a helium-charged Stirling engine. It

should be noted that Figure 1.3 shows typical values for a fiee piston engine.

Figure 1.3 Parasitic Losses Diagram [I]

It should be obvious by now that the simplicity of the Stirling engine is

misleading. The mechanical mechanisms may be simple, but the thermodynamic

processes are not. The magnitude of the fluid velocity and heat transfer rates both

oscillate during the cycle. Each sub-com ponent is interrelated with all the others. For

example, the shape and size of the regenerator will affect the pressure loss and heat



12

transfer rates. The heat transfer rates of the regenerator affect both the heating and

cooling of the fluid in the heater and cooler. The pressure drop in the regenerator, caused

by the viscous losses, effects the working fluid velocities, which also alters the heat

transfer rates in the cooler and heater. Additionally, the heating and coo ling affect the

pressure, though very minutely, which in turn impacts the appendix gap losses and seal

losses. The analysis of Stirling engines is complex making it necessary for the engineer

to ma ke idealized assumptions. Also, the complexity requires the designer to use

numerical optim ization tools.

1.3 Literature Review of Stirling Analysis

The evolution of the analysis of Stirling engines begins with Schmidt, who first

applied the thermodynam ic m odeling approach to Stirling engines. H e assumed that the

gas in the expansion and compression spaces wer e isothermal. This assumption

alleviates the comp lexity associated with the temperature variations in the expansion and

compression spaces, by holding these temperatures constant. Seven main assumptions

are made using th e Schm idt analysis, as shown in Table 1.1. One o f the key assumptions

is that the volume variation in the expansion and compression spac es are sinusoidal. This

important assumption enables the formulation of closed form solutions - ones that solve

algebraically. Bef ore the advent of computers, it was prerequisite to simplify the analysis

to prevent having to perform intricate and lengthy calculations with many chances of

error. The com plexity of the analysis is one reason why little improvem ent in the

Schm idt analysis wa s realized for nearly a century. The S chmidt analysis is beneficial in

that the work calculated during the cycle can be found fairly accurately. Th e analysis



13

fails, however, to accurately determine the heat flows during the cycle, which has an

adverse impact on the efficiency. The Isothermal assumption causes the calculated

efficiency to be the Carnot efficiency [4], which is normally two or three times the

efficiency of actual Stirling engines [4]. Another value of the analysis is that many

designers use the initial calculations of the Schmidt analysis to size a Stirling engine 191.

Table 1.1 Basic Assumptions Made by Schmidt

1. The temperatures in the compression and expansion spacesare isothermal.

2. The mass of the working fluid is constant, which im pliesthat no leakage occurs.

3 . The equations of state of a perfect gas apply.

4. The speed of the machine is constant.

5 . Cyclic steady state is established.

6. The kinetic and potential energies of the gas are neglec ted.

7. The volume variations are sinusoidal.

Finkelstein modified the Schmidt analysis by allowing non-isothermal conditions

for the cylinders [9]. His method constrains the thermodynamic processes of the heat

exchang ers to be ideal, while allowing for non-isothermal (including adiabatic) cylinders.

The solu tion requires numerical analysis, since no direct algebraic equations are obtained.

The Finkelstein model is significant, since it was the first major modification of the

Schmidt analysis that allowed for non-isothermal conditions, including adiabatic

conditions in the exp ansion and compression spaces. Finkelstein was also the first to use

the upstream temperature of the fluid to describe the fluid temperature entering or exiting



14

the expansion and compression spaces. Finkelstein set a new standard in Stirling

analysis.

Two main approaches grew out of the work of Schmidt and Finkelstein, thus

setting the fou ndations for Stirling analysis. The first approach is to decouple the

analysis. This means that the irreversibilities are treated separately from the basic ideal

cycle by m odifying the basic ideal relations. For example, the losses due to pressure d rop

and non-sinusoidal motion of the moving parts, can be subtracted from the Schmidt

closed-form work equation giving a better estimate of work per cycle[9].

A more

accurate efficiency is then found from the calculated work. The advan tage of this method

is that a detailed accounting of the losses is produced which is usehl in design

modifications. Creswick, Qvale, Rios and Smith were some pioneers using this

approach. Additionally, Martini derived a simplified decoupled ana lysis that gave a

reasonably good indication of the performance and operation of a S tirling engine [9]. He

also wrote a manual on Stirling engines which has closed-form equations, but some

discrimination is recommended by Walker in the use of his so called "second order

design m ethods" [4].

The coupled approach in the analysis of Stirling engines uses a numerical

simulation of the com plete cycle withou t any attempt to decoup le the analysis. This

approach is generally more complex, because the conservation laws (two or three) are

used along with the equation of state to develop a set of differential equations.

Additionally, the Stirling engine is divided into various control volumes, and the

differential equations are derived for each. These equations are solved numerically with



15

the aid of a computer by time stepping integration. The term "coupled" is somewhat

misleading, because some decoupling occurs by the use of the thermodynamic model.

For example, the thermodynamic approach neglects flow patterns which effects heat

transfer. Additionally, assumptions are sometimes made that the flow is at steady state

whe n in reality it oscillates. This assumption causes the time dependent effects, which

couple between subsystems, to be neglected. The decoupling has relatively insignificant

effects (usually), so we use the term "coupled". Kirkly's [9] coupled analysis was the

first to account for non-isothermal conditions in the expansion and com pression spaces as

well as the pressure drop losses and imperfect regeneration. Finkelstein's later analysis

had the same capability as Kirkly's, but also included imperfect heat transfer.

Furthermore, Finkelstein based his work on the standard integral forms of the energy and

continuity equations [9]. Urieli extended the Finkelstein analysis t o include the

conservation of momentum and kinetic energy effects. This work significantly advanced

the analysis of Stirling engines [2].

Both the coupled and decoupled approaches have value. The decoupled approach

is generally highly idealized since the relationships between the various subsystems are

neglected. The use of this method is common in the design of Stirling engines and is

used by Walker [4], Senfi [3], West [ lo ] and others. The purists (such as Organ [2]) are

highly critical of the decoupled analysis used in the design of Stirling engines. On the

other hand, West claims that the simpler decoupled analysis is not established to be

inferior to th e more complex "third order" or coupled analysis [lo ]. Both arguments have

validity and depend on the individual performing the analysis or design. The decoupling



16

does produce highly idealized results, but the em pirical data can substantiate the results.

The coupled analysis is more accurate in many respects, but it also makes significant

assumptions. For example, Organ uses the gas dynamics approach, but his method

cannot be used with more than one dimension . Does neglecting the other dimensions

produce any worse results than the decoupled analysis that is substantiated by em pirical

data? Further research will have to answer this question. The "em pirical" designer often

gravitates to the simpler decoupled analysis, which can be validated by empirical data.

The "theoretical" designer often gravitates to the more sophisticated software and

modeling techniques required by the coupled analysis.

Martini first proposed the classification of simu lation approaches as first, second,

or third orde r analysis based on increased difficulty. This led to am biguity, since the

second order classification was not well defined and some inconsistencies in terms w ere

encountered. Furthermore, the third order analysis includes nodal analysis with two and

three conservation of energy laws. The added momentum law is significantly more

involved. This lack of distinction in the complexity by this classification schem e lead

Organ to propose a more hndamental approach based on the number of conservation

laws used in the analysis [l11. The three conservation laws are the conservation of mass.

mom entum, and energy. Urieli first established the code description as found in Tab le

1.2. Later O rgan adapted it slightly (specifically for a gas dynamic approach) to specify

if certain terms were left out by including a minus sign as a superscript. The essential

features are defined as well as the integration scheme. Table 1.3 shows Organ's

classification scheme.



Tab le 1.2 Classification of Simulation Approaches as Described by Urieli [l l]

Table 1.3 Relationship between Type of Analytical Model and Num ericalIntegration S cheme as Presented by Organ [2]

Type

1m2mF

2mE

3mEF

3rnEFa

Features

Conserva tion of mass - Ideal Isothermal Model, includes the Schmidt analysisConserva tion of mass and momentum (represented by F for 'force'). Typicallywould include an Isothermal model in which the pressure drop e ffects due toflow fiiction have been included. This would normally be considered a Quasi-Steady-Flow model in that the acceleration of the working gas is ignored.

Conserva tion of mass and energy. Typically would include the Ideal Adiabaticmodel having ideal heat exchangers.

Conservation of mass, energy and mom entum, in the quasi-steady-flow case.Most of the nodal analysis techniques fall into this class.

Conserva tion of mass, energy, and mom entum, in the Non-Steady-Flow case.The "a" does not represent a further conservation law, but rather signifies the

inclusion of the gas acceleration term. This is the highest level ofsophistication available in Stirling cycle machine analysis and requires the fullgas dynamic treatment.

Analysis Type

CL- 1

CL-2MF - (i.e.,without term

in a(p+')lat

CL-2MF

CL-3ME

CL-3MFE

Essential Features

Solution forp, F-,T not affected by sound

wave propagation or temperature gradientdiscontinuities. Integration intervalsindependen t and open to fiee choice.

Governing equations suggest diffusion-typeflow

Pressure information in numerical modelmust propaga te at more or less the sam espeed as in the real gas circuit.

Heat transfer and friction act on individualfluid particles giving rise to fluid propertygradient discontinuities.

Discon tinuities in gradients of all properties.Full gas dynamics treatment.

Integration Schem e

Fixed grid method - Ax,

At chosen forcomputational .convenience.

Ax, At chosen to allowappropriate diffusionrate

Method ofCharacteristics or fixed

grid scheme - Ax, At setby reference to local

sound speed.Langrange coordinates

Method ofCharacteristics



18

Th e recent metho ds of analysis can be broken do wn in to five main types, as Urieli

does [ l l ] . No significant changes have occurred in the methods of analysis since 1983;

however, h rt h e r research has continued since then. These five analysis methods are the

following: linearization methods; adiabatic analysis; nodal analysis; finite element

analysis and th e method of characteristics.

The first method of analysis is linearization methods, which includes three types

of analysis. The first type of analysis is based upon the Metho d of Perturbations, which

adds an error term (or perturbation). An approximate value is required to start the

analysis, which makes the method ideally suitable for Stirling analysis since the Schmidt

results can provide the initial guess. Organ and Rix have presented solutions based on

this method [12, 131. The second type of analysis is phasor ana lysis in which vectors are

used to describe the engine. This method is similar to approximating the solution a s the

first harmonic whe re all terms of the series are truncated but one. The advantage o f the

method is the extreme speed and relative ease of implementation. Isshiki uses this

method [14]. The third type of analysis is based upon harmon ic analysis with one o r

more higher order terms. The solution is assumed to be harmonic and o ne can truncate

the series to any number desired. The pioneers of this method were R ios and S mith with

others continuing the work such as Rauch, Chen, Griffen, and West 121. The software

HF AS T developed by Mechanical Technology Inc. is based upon harmonic analysis [15].

Additionally, the Philips Research Laboratories Stirling program may have used this

method [2].



19

The second method of analysis is adiabatic analysis. This model is a more

accurate representation of the cycle than the Schmidt analysis. The heater and the cooler

both serve a us e h l purpose with this analysis, unlike the Ideal Isothermal analysis. Also,

fudge factors can take into account the pressure drop and heat transfer losses as Lee

demonstrated [ll]. Urieli was the pioneer in this method of analysis [ I , 91. A more

comprehensive discussion of this method is given in Ch apter 2.

The third method of analysis is nodal analysis (3rnEF). This method utilizes the

three conservation laws and has been the mainstay in Stirling analysis. Non e of these

methods include the working gas acceleration terms, due to the added complexity and

comp uter solve time associated with this added term [ l l ] . Finkelstein, Vanderburg,

Urieli, Kirkley, Heames, Tew, and G edeon are some of the main contributors to the nodal

analysis method [lo].

Th e fourth method of analysis is the finite element analysis (3mEF). This m ethod

is well developed and has been used in stress analysis, heat transfer and fluid flow

problems. One possible disadvantage to this method is that shape hn cti on s are

formulated that are based upon a continuous domain, which is not true for the Stirling

engin e [2]. This is a minor problem though, since most the other methods neglect the

flow discontinuities. Datta and Larson have formulated the method by using the Galerkin

weighted residual sformat [l l ] . One advantage of this system is that it cannot only be

formulated for one dim ension, but fo r two and three dimensions.

The last method of analysis is the method of characteristics, which is used

commonly in gas dynamics. The gas dynamics approach is hnda m ent ally different from



20

the thermodynamic approach, upon which the nodal analysis is based. Table 1.4 shows

the conservative laws stated both in a thermodynamic form and a gas dynamics (fluid

flow) form. The hnda m en tal difference is that the nodal formulations treat the

incrementals as total derivatives in equations (1.2) through (1.4) of Table 1.4. The gas

dynamics form transforms the partial differentials prior to descretization, so that fluid

properties are justifiable in total differential form. The thermodynamic form causes the

diffusion rates and pressure propagation to be dependent on the descretization form at (i.e.

At and Ax ratios - Courant Num bers) [2]. Notice that the symbols are found in Table 1.5.

Table 1.4 Engineering Thermodynamics and Gas Dynamics Conservation Law s [2]

Conservation o f mass:

C @ - z g ~ = ~ m (1.2a)in our

ap a-+ - (pF) = O (1.2b)

d t axConservation of momentum:

CF C ~ A K C g ~ e D(mlL,) (1.3a)x in out

av aF 1 ap- + P - + - - + F = O (1.3b)at p a x

Conservation o f energy:

1 1D Q - D W + C ~ A ( ~ + - F ~g , z ) - z g ~ ( ' h + - ~ ' g , z ) = DE

in 2 ou t 2

(1.4a)

a 1 d P I- [ P ( ~ v ~ + - ~ 2 ) ] - q * p = - - [ ~ P ( c ~ T + - + - P ~ ) ] (1.4b)at 2 ax ~2



Table 1.5 Symbols and Associated Meaning

Symbol Meaning [example units]

Cv Specific heat at constant volume [J/kg-K]

E Total energy [J1

F Force [N]

gA Mass flow [kgls]

gr Gravitational constant [kg/m2]

k Enthalpy [Jlkg]

m Mass [kg]

P Pressure [Pa]

4' Power per unit mass [Wlkg]

Q Total heat [J]

t Time [s]

T Temperature [K]

V Velocity [mls]

W Work [J]

x Distance [m]

z Height [m]

P Density [kg/m3]

Other Symbols

D Derivative of dependent variable with respect to independent

variable (i.e. DE = dElde where d is a total derivative operator)

a Partial derivative operator

The method of characteristics solves fluid problems that have discontinuous wave

propagation. This method is ideally suited to analyze the Stirling engine where both

wave propagation and discontinuities are present. Organ presents the analysis using this



22

method and argues for its use in Stirling analysis [Z]: Larson and Taylor also

demonstrate the method of characteristics in their analysis of Stirling engines [lo].

Calandrelli and Rispoli use the h method, which is also based upon gas dynamics and is a

variation of the method of characteristics [16]. One disadvan tage of this method is that it

cannot be used w ith mo re than one dimension, which is a severe limitation.

1.4 ThesisBackground

Many people today are unfamiliar with the Stirling Engine and the potential

environm ental benefits that the engine can bring. Few Stirling engines are found in

common use and its application remains predominantly in research and specialized

applications. The analysis of the Stirling engine is relatively new comp ared to the

internal combustion engine. Any new analysis tools can assist in bringing the engine

from the lab to ordinary use. The objective of this thesis is to improve the numerical

analysis that solves the Ideal Adiabatic model. These improvem ents in th e numerical

analysis should assist in the design and optimization of Stirling engines, thus assisting in

the transition of the engine to everyday use.

Th e Ideal Adiabatic model is well known in the "Stirling world". Dr . Urieli of

Ohio University developed the Ideal Adiabatic model and wrote software utilizing the

Classical Fourth Order Run ge Kutta method [I]. The Ideal Isothermal model incorrectly

implies that heat transfer occurs in the expansion and compression spaces rather than in

the heat exchanger spaces. The Ideal Adiabatic model corrects this by assuming that an

adiabatic process occurs in both the expansion and compression spaces, thus making the



23

model more refined. Many Stirling designers have used this method in the analysis and

design of Stirling engines. Urieli and Berkowitz have suggested that this model be used

as the "standard Ideal model because it is a more accurate representation of the of the

Stirling engine [I]. The model is made by descretizing the engine, typically, into five

basic parts: the heater, cooler, regenerator, expansion space, and compression space. The

conservation laws along with the ideal gas law from thermodynamics are used to

formulate differential equations for each cell. The Isothermal analysis has the luxury of

allowing the integration of the differential equations so that closed-form equations are

found, while the Ideal Adiabatic model requires numerical analysis, due to the added

complexity of the temperature variations. Dr. Urieli uses the Classical Runge-Kutta

method for solving the differential equations, but this method is numerically expensive in

that many repetitions are required to solve the differential equations for each step.

Designers often have to perform significant numbers of iterations of the cycle in the

optimization of the Stirling engine. Typically, they examine the many effects that were

previously explained, such as appendix gap losses or viscous dissipation. The solve-time

for such operations can be lengthy and expensive in terms of the CPU time spent.

There are several difficulties in solving the model. Particularly, the differential

equations are boundary-value problems and not initial-value problems. The boundary

conditions are such that the temperatures of the gas in the expansion and compression

spaces at the start of the cycle should equal their respective values at the end of the cycle.

In order to solve this problem, a guess is made at the starting temperatures for the

compression and expansion spaces. After the end of the cycle, the newly calculated



values are used for the next guess of the initial temperatures. Convergence is obtained

after several repetitions. One other difficulty of the problem is that the fluid, between

control volumes, takes on the upstream temperature during the analysis, which causes the

differential equations to be nonlinear. This does not allow closed-form solutions to be

formulated from the differential equations. Finkelstein was the first to set the gas

temperatures of the fluid flowing between control volumes to the ups tream temperature.

Dr. Pasic, also of Ohio University has recently developed a new method of

solving differential equations. The method is extremely versatile and adaptable and can

be used in solving boundary-value problems. The number of times solving the

differential equations is reduced because a larger sub-domain is used. His method is

based on a fixed-point iteration scheme. The Runge-Kutta method is based on the Taylor

series where many calculations of the derivatives are required for each small step.

The objective of this thesis is to apply the Pasic method to the Ideal Adiabatic

model to solve the differential equations. Table 1.6 lists the questions that the thesis

addresses. Program modules were written in C language to solve the Ideal Adiabatic

model using the Pasic method. Also, programs that Urieli developed were adapted for

this investigation. The two central concerns of the thesis were the speed of the

calculations and the magnitude of the error.

1.5 Thesis Outline

The analysis of Stirling engines is complex when the model includes non-

isothermal working spaces. Closed form equations are impossible to formulate due to the



25

nonlinearity of the differential equations. Chapter 2 presents the ideal adiabatic model,

which has been the ideal baseline analysis, which better reflects the Stirling performance

as compared with the Schmidt analysis. The description of the model is presented, and

the set of ordinary differential equations (ODE S) is formulated along w ith the other

associated analysis variables. Chapter 3 presents the Pasic method for solving ordinary

differential equations. The Ideal Adiabatic Stirling model and more complex nodal

Table 1.6 Questions that Thesis Addresses

1. Can the Pasic m ethod be applied to the Ideal Adiabatic model?

2. Will the correct results be obtained?

3 . Will the fixed-point iterations used by the Pasic m ethod (Picard type iterations)converge?

4. Will the conditional statemen ts adversely affect the Pasic method , since a larger sub-domain is used?

5. Will the larger sub-domains (steps) work in general?

6. How fast is the m ethod?

7 . Is there an optimal step size?

8.

How much error will result?



26

analysis both require numerical analysis to solve the resulting set of ODEs. A typical

method, which solves the ODEs, is the classical fourth order Runge-Kutta method. This

method is computationally intense resulting in long solve times when multi-cycle

program runs are invoked during the design process. Chap ter 3 presents Pasic's new

method for solving ODEs which may reduce substantially the CPU solve time for a

Stirling engine cycle. This chapter also describes the underlying mathematical schemes

associated with the Pasic method. The formulation of the Pasic method for first, second,

third, and higher order OD Es is briefly considered. Chapter 4 applies the Pasic method in

solving the coupled first order boundary-value ODEs formulated from the ideal adiabatic

model. This chapter presents the two main program algorithms, which describe the Pasic

metho d when applied to the ideal adiabatic model. Furthermore, the results compare the

fourth order Runge-Kutta method and the Pasic method (fourth degree) when applied to

two different Stirling engines: the Ford-Ph ilips 4-215 and the Ross-90 engine. Chapter 5

concludes the thesis and presents the h tu r e research possibilities.



Chapter 2

The Ideal Adiabatic Model

2.1 Introduction

Dr. Israel Urieli developed the Ideal Adiabatic model while pursuing his doctoral

work at the University of the Witwatersrand, Johannesburg. This model is similar to the

Ideal Isothermal model developed by Schm idt. The main exception is that the working

gas in the cylinders is assumed to undergo an adiabatic process rather than an isothermal

process. An Ideal model should represent the highest efficiency that the modeled engine is

capable of producing and closely follow the real cycle. The Otto and Diesel cycle models

adhere to these requirements and prove to be Ideal models in the true sense. The

Isothermal model does not closely follow the real cycle because the compression and

expansion processes of the gas are closer to adiabatic. This model is used m ore because

of the analytic tractability of the closed-form solutions. Finkelstein first applied adiabatic

conditions to the working spaces and showed that the ideal efficiency was not equal to the

Carnot efficiency [9], which the isothermal process in the cylinders would p redict. Urieli,

therefore, developed the Ideal Adiabatic model and recommends this model in the analysis

of Stirling engines [171.

The first section presents the basic Ideal Adiabatic model with its relevant

nomenclature and basic assumptions as first presented by Urieli [I]. Next the derivation

of the equa tion set is presented along with the table of equations. The set of equations



28

was taken from class notes given by Urieli. The derivations were taken from Urieli 111

with slight modifications to add hr th er clarity to the development of the equations. Some

of the more commonly used drive mechanisms are presented in the next section . The last

section outlines the method of solution as first developed by Urieli, where the classical

Runge-Kutta method is presented along with the solution approach to obtain the boundary

conditions.

2.2 The Basic Model

Schmidt was the first to perform an analysis of the Stirling engine where he

assumed that the expansion and compression spaces were isothermal. This isothermal

assumption creates a paradox because it predicts the heater and cooler to be unnecessary.

In reality, the convection heat transfer in the expansion and compression spaces is very

small and nearly an adiabatic process. The assumed isothermal conditions incorrectly

models the heating and cooling to occur in the expansion and compression spaces, thus

causing the redundancy of the heater and cooler. Finkelstein was one of the first to adapt

his Stirling model so that non-isothermal conditions exist. His model used a convection

coefficient that could be forced to simulate adiabatic conditions. Furthermore, Finkelstein

used the upstream tem perature to find the enthalpy of the flowing fluid. This essentially is

a step hnction at the point in the cycle where the fluid changes directions, which causes

the non-linearity in the formulated differential equations.

Urieli created his adiabatic model by descretizing, similar to Schmidt, with five

control volumes (see F igure 2.1). He formulated his equation set based upon an adiabatic



process in the compression and expansion spaces. Also, he used the upstream

temperatures to represent the temperature of the fluid moving between control volumes

similar to the approach used by Finkelstein. The models made by Finkelstein and Urieli

were more refined; however, there is more complexity with the adiabatic compression and

expansion spaces. The closed-form equations are not obtained as in the Isothermal model.

The differential equations cannot be solved by direct integration because the derived

differential equations are nonlinear and no direct integration is possible. This is no

problem with the advent of computers since numerical analysis can be used to solve the

differential equations.

Figure 2.1 show s the Ideal Adiabatic model. Notice that there a re five control

volumes, which are the compression space, expansion space, heater, cooler, and

regenerator. Furthermore, the graph shows the temperatures of the gas in each control

volume. The heater and cooler gas temperatures are constant over the cycle and are

respectively, Th and T,. The regenerator gas temperature is assumed to be linear, varying

from Th to T,. The mean effective temperature of the regenerator gas is found from the

assumption that the temperature of the gas varies linearly over the length of the

regenerator. This gas temperature is derived later in the chap ter in equations 2.39 to 2.44

as originally given by Urieli [11.Figure 2.1 also shows the four boundaries between the five control volumes.

Enthalpy is transported between control volumes by the mass flow rate gA, where g is the

mass flux and A is the cross sectional area at the interface between control volumes. The

subscripts ck, kr, rh, and he respectively describe the interfaces between the following



control volumes: compression spacelcooler, coolerlregenerator, regeneratorlheater and

heaterlexpansion space. For example, the mass flow gA,k represents the mass flow

between the com pression space and cooler. Notice that the tempera ture of the mass flow

between the cooler and regenerator is Tk independent of the direction of flow, as is the

temperature between the regenerator and the heater Th. The compression space/cooler

and expansion sp aceh ea ter interface, however, have discontinuities in temp erature as is

Compression

space Regenerator

Expansion

c Cooler r space

k .Heater e

7

Figure 2.1 Urieli's Ideal Ad iabatic Stirling Model [I]



3 1

shown on the temperature graph in Figure 2.1. These are the discontinuities in

temperature first described by Finkelstein that require the following conditional

statements:

if gAck> 0 then Tckt , otherwise Tckt k (2.1)

if gAhe> 0 then Thet i otherwise Thet e (2.2)

The Ideal Adiabatic model also assumes that the control volumes all have the same

pressu re at any given point in the cycle. In other words, no pressure drops occur in the

five contro l volumes due to flow losses . Additionally, it is assumed that there is no loss of

mass around the seals of the piston to the environment, so that the total mass M of the

working fluid stays constant over the cycle.

2.3 Development of Equations

Urieli first developed the set of equations for the Ideal Adiabatic model and based

the solution on solving the pressure differential Dp and the mass derivative Dm, [I] .

Later , Urieli modified the solution in terms of the temperature differentials Tc and Te. The

equations are again presented here since they are central to this thesis and have not been

published in the open literature in this form . The equations were form ulated by Urieli [ I] ,

however, some modifications were made to add clarity.

This section first presents some basic relationships that a re used to derive the set of

equations for the Ideal Adiabatic model. Two main equations are subsequently developed:

the differential pressure and pressu re. Finally, the other equations that describe the Ideal



Adiabatic model are derived: mass flow rates; the differential temperatures; the mean

effective temperature of the regenerator; and the differential energy equa tions.

The Ideal Adiabatic model is based upon the conservation of mass principle,

conservation of energy principle, and the equation of state. Figure 2 .2 shows a control

volume for a pistodcylinder device and a heat exchanger that has an inlet and outlet mass

flow. The following equations are based upon these control volumes, however, initially

we start with the basic thermodynamic equations that have an inlet and outlet port as well

as heat flow. No tice that the control volumes are designated by the dotted line in Figure

2 .2 . The conservation o f mass principle is

%Ai - %A, = Dm

and the conservation of energy principle is

DQ DW +CgA; A;+ g2/2+ g,zi) - Cg A, (k , + ~ 2 1 2 g,2,) = DE ( 2 . 4 )

where the subscript i and o stand for the inlet and outlet ports. Additionally, the equationof state is the following:

where R is the gas constant. The internal energy and enthalpy equations in differential

form, for an ideal gas, are the following:

du = C, (T ) d T ( 2 . 6 )

d k = C, (T) d T ( 2 . 7 )

The symbol u represen ts the internal energy in this case where the symbol k represents the

enthalpy both per unit ma.ss. The symbol C, is the specific heat with constant pressure



3 3

where y is the ratio of specific heat capacities. Additionally, the following relations hold

for an ideal gas:

Cp=C,+R (2.8)

y=Cp/C, (2 .9)

C, =R y/ (y - 1) (2.10)

C,,=RI(y- 1) (2.11)

1-~-.1-'-.- '-~

Control

Volumes

a) Adiabatic Piston Cylinder Device b) Isothermal Heat Exchanger

Figure 2.2 Generic Control Volumes Used to Formulate Equations

The first objective is to formulate an equation for the differential pressure. Next,

an equation for the pressure is derived. Both these equations need to be in a useful form.

We begin to accomplish the first objective by noticing that the term DE in the energy

equation (2.4) is

DE = D (me) (2.12a)

where the total energy per unit mass is



34

e = u + v 2 / 2 + & Z (2.12b)

Also, assuming that the specific heat capacity is constant with temperature, the internal

energy and enthalpy are

u = G T (2.13)

k = CpT (2.14)

Rewriting the energy equation by substituting in equations 2.12, 2.13, and 2.14, and

neglecting the kinetic and potential energy of the entering, exiting, and control volume

fluids, gives the following:

D Q - D W +CpT,gAi-CpTogAo=C,D(mT) (2.15)

The differential form of the equation of state is obtained by taking the log of equation

(2.5) and differentiating:

Dplp + DVIV = Dmlm + DTIT (2.16)

Figure 2.1 shows the five control volumes within the Stirling engine. Since the total fluid

in the Stirling engine is self-contained, the total mass of the system for any given crank

angle is the following:

m c + m k + m r + m h + m e = M (2.17)

Differentiating this equation gives

Dm, + Dmk + Dm, + Dmh+ Dm, = 0 (2.18)

Since the volume and temperatures are constant for the three heat exchangers, the

differential equation of state reduces to

Dmlm = Dplp



35

Equation (2.19) as applied to the three heat exchangers is next substituted into equation

(2.18):

Dmc+Dme+Dp(mklp+mdp + mNlpl=

0 (2.20)

Substituting for the mlp terms for the equation of state gives the following:

Dmc+Dm, + (DpIR)(Y,ITk+ VJT,+ VhlTd= 0 (2.21)

The objective is to solve for the differential pressure Dp by eliminating the tw o term s Dm,

and Dm,. Applying the conservation of energy equation to the expansion and com pression

spaces will give the tw o differential mass terms. For the compression space, equation

(2.15) is

D Q, - C P T c k g 4 , D W c + C , D ( m c T c ) (2.22)

The compression space is adiabatic, thus DQc = 0. Furthermore, the work is defined by

DW, = p DV, , and the rate of accumulation of gas, Dmc, is -gAck. Thus equations 2.22

reduce to

C pTckDm, = p D V , + C , D ( m , T , ) (2.23)

Substituting the ideal gas relations, (2.8), (2.10), and (2.1I), as well as the equation of

state (2.5) into (2.23) reduces to the following:

Dm, = ( P D V , + Vc D P / y ) / ( R T , k ) (2.24)

A similar relation is found fo r the mass accum ulation in the expansion space:

Dme = ( P DVc +Ve DP ' ) (RTbe) (2.25)

Substituting equation (2.24) and (2.25) into (2.21) and simplifjing results in the

differential pressure:



Notice that the volume variations V,, Ve and their derivatives formulated in section 2.3.

The masses are determined fiom the equation of state applied to each control

volume. The equations for the masses are the following:

mc = PVC / ( R T c )

The pressure is found by substituting equations, (2.27) through (2.31) into the mass terms

of equation (2.17). Rearranging and solving for pressure gives the following:

p = M R / ( V c I T , + Vk / T , + V , / T , + Vh T h + V , / T e )

whereM refers to the total mass of the gas.

The other equations that describe the Ideal Adiabatic model are presented next.

The mass flow in a control volume is found from the conservation of mass principle of

equation (2.3), which says that the mass accumulation of a control volume is equal to the

difference in the inlet and outlet mass flow rate. Applying the conservation of mass

principle to each control volume results in the following:

gAck= -Dm,



gAhe=Dm, =gA, -Dm,

The differential temperatures for the expansion and compression spaces are formulated

from the differential form of the equation of state. The temperatures of these spaces are

DT , = T , ( D p / p +DV, /Vc - D m , / m , ) (2.37)

DT, = T e ( D p / p + D V e V , -D m e / m e ) (2 .38)

The differential temperature equations are the most important, due to the fact that these

equations are solved to obtain the solution. Prior to solving, however, the analysis

variables Dp, DV and Dm for both the compression and expansion spaces, need to be

derived in relation to some function f(Tc, TJ in order to solve the coupled differential

temperature equations DT , and DT,. Any method that solves ordinary differential

equations can then be used t o obtain the solution.

Another important equation is the mean effective temperature of the gas in the

regenerator. Urieli has already presented this derivation in Appendix A.2 of [I], but for

the sake of completeness it is restated. The mean effective temperature for the regenerator

is found by integrating the linear temperature of Figure 2 . 3 over the length L, of the

regenerator. The temperature at any distance x is

T ( x )= (T , - T, )x / L, +T,

By definition, this temperature in terms of the ideal gas equa tion of state is



The total mass of gas, m,,n the regenerator is found by integrating the following:

Since dV, =A , d x and the density relates to the equation of state by p = pRT, the total

mass of the regenerator is the following:

Integrating equation (2.42) and simpliQing results in

Comparing Equation (2.40) and (2.43)gives the relationship for T,, which is

T , = (Th - Tk) ln(Th / Tk) (2.44)

The derivation of the energy relations is also important. The equations for the

compression and expansion spaces that describe the differential work are

DW c = p D V c (2.45)

D W, = pD V e (2.46)

where the differential total work is

D W = D W , + D W ,

The differential heats for the heater, cooler, and regenerator are based on equation (2.22).

Substituting the equation of state (2.5) and the specific heat relation (2.8) into equation



39

(2.22) and applying to each individual heat exchanger we obtain the three differential heat

equations

DQk =Vk DPC , / R - C , (Tk gA, -T ' g A h ) (2.48)

DQr =Vr DP C, / R - C p (Th gAh -T* gA*) (2.49)

DQh = v h D P C , 1 R - q (T* 8 4 -The @he) (2.50)

Since the heat exchangers are isothermal and the regenerator temperature is linear in

relation to x (see Figure 2.I), the following temperature relations are given by d e w t i o n :

Th =T, (2.51 )

T, =Th (2.52)

Figure 2.3 Regen erator Linear Temperature Profile as given by Urieli [I]



40

The differential temperature equations (2.37) and (2.38) have not been solved to

obtain closed-form solutions, due t o the non-linearity o f the differential equations. Table

2.1 presents a summary of the fbndamental differential and algebraic equations that define

the Ideal Adiabatic Stirling model. Numerical methods are required t o solve the

differential equations. Chapter 4 presents the solution o f the equations utilizing the Pasic

method.



Table 2.1 Ideal Ad iabatic Stirling M achine Differential and A lgebraic

Equations as Presented by Urieli [18]

p = M R / ( V c T c +Vk / T k +V, / T , +Vh /T h +V, /T , ) Pressure

Dp =- P (DVc Tck + Dye / The )

/Vc/Tck+Y (Vk/Rk +V r/T, +Vh/Th)+Ve/Thel

me = pV e / ( R T , ) Masses

mk = p V k / ( R T k )

m r = pV r ' ( R T , )

mh = P V ~ t R T h )

m e = p V e ' ( R T e )

Dmc = ( p D V c + V c D p / y ) / ( R T C k ) Mass AccumulationsDm e = (p D V C D ~ / ~ ) / ( R T h e )

D m , = m k D p / p

Dmr = m r D p / p

Dmh = m , D p / p

gAck= -Dmc Mass Flows

gA , = gAck -Dm,

gAhe= Dm, = gA , -Dmh

gA* =gAhe + Dm,

i fgAck> 0 th en Tckt c else Tckt k Conditional

i fg A he > O h en T h e t T h else T h e t T e Temperatures

DTc = T c ( D p / p + D V c V , - D mc / m c ) Temperatures

DT, = T , ( D p / p + D t / t -D m ,/m ,)

DQk = Vk DpCv / R - C p (Tck gAck - TbgAkr) Energy

DQr =Y, D p C v / R - C p (T , gA , -T , gA ,)

DQh =Vh DpCV / R - C p ( T, gA* -The gAhe)

DWc = pDVc

D We= pDV,

D W = D W c +D W e

w=wc w e



2.4 Volume Variations

The volume variations within the compression and expansion spaces are an

important aspect of the S tirling engine and affect all the key variables. The mass flow

rates, heat transfer rates, pressure, yo rk , etc., are all dependen t on the volume variations

during the cycle. The device that regulates the relationship of the volumes within the

expansion and compression spaces is the drive mechanism. Many different drive

mechanisms exist and have been used with Stirling engines. Figure 2 .4 show s a Rhom bic

drive mechanism, which was invented by M eijer and was used extensively by N. V. Philips

in Holland. Figure A-1 shows ,the equations for this type of engine. Tw o o ther drive

mechanisms are the Ross Yoke (Figure A-2), and V-configuration drive (Figure A-6).

Additionally, Figures 2.6 and 2.8 show the swash plate and the inverted R oss yoke drive

mechanisms. All the drive mechanisms serve the purpose of converting the linearly

alternating motions of the compression and expansion piston into the usefbl motion o f a

rotating shaft.

We will focus on the Ford-Philips 4-21 5 and the Ross-90 drive mechanisms, due to

the fact that we analyzed these two engines. The Ford-Philips has a swash plate drive

mechanism. This experimental automobile engine was developed under a joint project

between N. V. Philips, of Holland, and Ford Motor Com pany in 1972 . It is a double-

acting engine with four cylinders where one cylinder is connected to the adjacent cylinder

as shown in Figure 2.5 . The swash plate drive mechanism is sinusoidal with each piston



Exhnurt artkt L L L

8umr-oir inlet

Cylinder

Conpnsston sporeO~rplacer od

PlSton

Piston rod

Rston pk e

Rsto n cm ect n g rod

O~spbcerm m c t n g r od

olrpbca p k e

Figure 2.4 Cross Section of the N. V. Philips 1-98 Engine [5]

90 degrees ou t of phase with the adjacent piston. Figure 2.6 shows the swash plate drive

and gives the related equa tions. Figure A.7 shows the sinusoidal equations that were used

to m odel the Ford-Philips 4-215 engine volume variations.

The Ross-90 is an engine developed by Andy Ross from Columbus Ohio, which

uses air as the working fluid. The engine is used in a lab for undergraduate students at



44

Ohio University. The Ross-90 has an inverted Ross yoke drive mechanism. Notice the

size of the Ross-90 engine shown in Figure 2.7. The equ ations that describe the volum e

are show n in Figure2 .8 .

FigureA.8

shows the calculation of the maxirnumy displacement

for both the expansion and compression spaces. Other pictures of the engine and

com ponen ts are shown in Appendix A (Figure A.3 through A.5).

Figure 2.5 Schematic Showing the Ford-Philip's Four-Cylinder Double-ActingEngine Configuration 15)



Figure 2.6 Swash Plate Drive Mechanism Showing Tw o of Four

Cylinders and Related Sinusoidal Equations



46

The inverted Ross yoke drive mechanism is a variation of the Ross yoke drive,

where the triangular plate is inverted or turned upside down [I]. Notice how the

triangular plate is repositioned to conserve space in the cylinder block for the drive

mechanism.

Figure 2.7 TheRossD-90 Engine (191



Displacements

Volume variationsv. = vCle + Ae (ymU - Y e )

VC = VclC + Ac ( ~ m a x - Yc )

d V e / d 8 = A e r [ c o s 8 - ( b 2 / b , ) s i n 8 + ( r / b , ) s i n 8 c o s 8 ]

d V c / d 8 = A , r [ c o s e + ( b 2 / b l ) s i n 8 + ( r / b , ) s i n 8 c o s 8 ]

Figure 2.8 Inverted Ross Yoke with Corresponding Equations



2.5 Method of Solution

The equations of Table 2 .1 define the Ideal Adiabatic model. The objective is to

solve the temperature differential equations found in equations [2.37] and [2.38] for both

the expansion and compression spaces. These differential equations are coupled in that

the gas temperature in the compression space influences the gas temperature in the

expansion space and vice versa (DT, =fi(T,, TJ and DT, =fi(T,,T J). The problem is a

boundary-value problem given that the initial temperature of the gas in the respective

expansiodcompression space must equal the temperature of the gas in the

expansiodcompression space at the end of the cycle. Urieli has solved this problem by

treating the boundary-value problem as an initial-value problem and guessing at the initial

cond itions for both the compression and expansion spaces. Normally, the initial condition

of the compression space is set to the temperature of the cooler while the expansion space

is set to the temperature of the heater. Urieli had a keen understanding of the

thermodynamics of the engine for he realized that by taking the end solution of the first

cycle and using it as the initial condition in the next cycle that eventually cyclic stability

would prevail. In other words, the end temperature would match the initial temperature of

a cycle. As in a regular S tirling engine, the engine initially starts off in a transient state

where the cycle temperatures change from cycle to cycle. Eventually, cyclic stability is

obtained where the temperatures are the same from cycle to cycle for the respective

compression and expansion spaces, but this usually takes many cycles due to the thermal

mass of the engine. Urieli was perceptive in realizing that the Adiabatic model should0

follow in a similar manner and obtain cyclic stability in significantly fewer cycles due to the



absence of the thermal mass in the Adiabatic model. H e solved this problem using the

classical fourth-order Runge-Kutta method and showed that by using the respective

temperatures of the expansion and compression spaces as the initial conditions for the next

cycle, that cyclic stability is normally obtained in three t o five cycles. This is significant

given that root-finding techniques for tw o independent variables normally require solving

the cycle about the same number of times to obtain the solution, but with much more

complexity. The test that Urieli used for convergence wa s to see if the combined

temperature differences (between beginning and end of cycle) of the expansion and

compression spaces was within one degree Celsius. This is the test case for heat engines

but for cryogenic coolers the acceptable error may be much smaller.

The method for solving the classical fourth-order R unge-Kutta method is shown in

Table 2.2 . Notic e that each differential equation is solved four times for each step value h

Table 2.2 Numerical Scheme of the Classical Fourth-order Runge-K utta

Method [20]

y , + ,= y i + [ + ( k , + 2k, + 2k, + k , ) ] h

k, = f ( x , , y , )

k , = f ( x , + t h , y , + + h k , )

k , = f ( x i + i h , y i + + h k 2 )

k , = f ( x i +h,y i +h k , )



50

to obtain the kl through k4 values. Also, the solution of the ki values is consecutively

solved starting at kl and ending with k4. This numerical process is expensive in the use of

CPU time requiring many steps over the domain because eight derivatives are solved for

each small step, h, given that we have tw o coupled differential equations.

2.6 Summary

This chapter presen ts the Ideal Adiabatic model as developed by Urieli. The

equations for the model are formulated and Table 2.1 shows the relevant equations. The

method that Urieli used to solve the Ideal Adiabatic model is presented. The classical

fourth-order Runge-Kutta method is given and shown to be numerically expensive.

Another numerical method could enhance the Ideal Adiabatic model by decreasing the

number of numerical iterations. This leads into the next topic about the Pasic method,

which is presented in Chapter 3 .



Chapter 3

The Pasic Method

3.1 Introduction

Dr. Pasic of Ohio University in Athens has recently developed a new method in

numerical analysis for solving ordinary differential equations (ODEs). This method

shows much promise in solving nonlinear differential equations and has been used to

solve problems related to mechanical engineering in such fields as dynamics, controls

and robotics [21]. Outside of these areas the method has not been extensively used, but

the method is readily applicable to a broad spectrum of ODE problems. The method is

p o w e f il because it applies to higher order ODEs as well. Furthermore, the method can

potentially reduce the number of numerical calculations, significantly decreasing the

compu ter CPU time. Presently, Dr. Pasic is modifjrlng and expanding the m ethod. This

advanced method has great potential because it can be implemented using parallel

processing and high-speed computing. It is also extremely versa tile and can be used with

other numerical methods, providing many opportunities for hr th er investigating [22].

The simple Pasic method (as presented in this chapter) is a synthesis of several

methods in numerical analysis used to solve ODE s. This chapter first describes three

methods used in solving differential equations and then synthesizes the main

characteristics of each together with appropriate modifications to describe the Pasic

method. The last section shows a scheme for solving higher order differential equations.



3.2 Picard's Iterations by Successive Approximations

The eminent French mathematician Emile Picard developed an iteration technique

for differential equations. For example, the following initial-value problem consists of

the first order differential equation with the initial condition :

~ ' ( t )f ( t j~ ( t ) ) (3. a)

The Picard fixed-point iteration scheme is illustrated below [23], and shows the

successive formulas

Notice that the successive formulas are integrated over the same domain and that the

previous yn value is used for the next integration y,+l. The convergence requirement for

this fixed-point iteration, when starting with one solution y, and obtaining two successive

solutions y,+l and y,+z [24], is the following:



Pasic presents the convergence criteria for functions that satisfy the Lipschitz condition

[24]. The discussion is instructive because the convergence criteria are related to the

allowable size of the domain. The Lipschitz condition also verifies the existence of the

solution [ 25 ] . The important conclusion drawn from the convergence discussion is that

the domain can be selected small enough to force convergence, except for "stiff'

differential equations. This method applies to higher order differential equations, where

the number of times one integrates is the order of the differential equa tion. The

integration occu rs prior to iteration.

3.3 Collocation Method

The collocation method can solve either boundary-value problems or initial-value

problems for m-th order ODES with m either initial o r boundary conditions:

The approxim ate solution is the following:

T(*) =Coimi x )

where Qi are functions satisfying the initial or boundary conditions. The co efficients are

found by enforcing the error E ( x ) = L(") - to zero for each of the k + 1 collocation

points. Notice that the number of coefficients equal the number of collocation points. In

other words, the number of collocation points equals the number of equations, thus

allowing an exact solution at the collocation points. This requirement leads to a linear



54

algebraic system for the unknown coeficients ai. Hence, the net result is that (3.7) is

solved without evaluating the integral, such as in (3.5). As pointed out in [24], Newton's

method is normally used; however, this equation can be solved by fixed-point iteration

(3.7), similar to Picard's successive approxim ations.

3.4 Shooting M ethod

Shooting methods are commonly used for solving boundary-value ODEs by

transforming the problem into an initial-value one. The process of solving for a second

order ODE usually involves guessing the value of the derivative at the location of one

boundary condition and so lving over $he domain to the other boundary cond ition: but this

depends on the type of boundary conditions associated with the problem. One can use

Newton's method, the Runge-Kutta method, or any other method for solving ODEs. If

the differential equation is of order m, then m initial conditions need form ulation from the

boundary conditions. Typically, stability and convergence concerns cause one to use a

stepping process, which sequentially solves for each small sub-domain over the larger

domain space. At the end of the domain, the final value is com pared with the boundary

conditions to see if the guessed derivative at the initial-value is correct. Figure 3.1 (a)

shows the initial condition guess and the resultant solution for a first order ODE. Notice

that the end solution does not equal the prescribed end boundary condition. The

difference between the boundary condition and the end solution obtained will determine

the next guessed value for the initial condition. The next guess is made in Figure 3 .1 (b)

and a new end solution is found. Again the difference in the end solution and the

boundary condition necessitates another guess. Based on the tw o previous guesses a



5 5

better guess can be made, etc. Figure 3.1 (c) shows the better guess and the resulting

solution. Notice that the end solution and the boundary condition match for this case. If

the solution is not within the specified error range, then this selection of the initial

conditions based on the previous guesses continues and the iteration continues until a

solution is obtained within the prescribed error range.

Figure 3.1 The Shooting Method Changes a Boundary-value Problem into anInitial-value Problem [20]

4

Boundary Condition

b xA) First Solution Based on Guessed Initial Condition

A

First End Solution

I

&First Guess

I

IB) Second Solution Based on Next Guessed Initial Condition

A

First End S olution

III

-Second Guess

C) D esired Solution that Matches Boundary Condition



3.5 Synthesis of Methods to Form ulate the Pasic Method

The Pasic method when applied to boundary-value problems can be viewed as a

synthesis of the following methods: Picard's method of successive approximations, the

collocation method and the shooting method. As in Picard's method, one uses a fixed-

point iteration process until the solution converges. Oftentimes, smaller domains are

repetitively solved sequentially, until the whole domain is covered- imilar to the way

that is sometimes applied in the shooting method. The convergence, stability and error

requirements associated with the Picard method determine the size of the domain. The

collocation method is used, so that evaluation of the integrals is not required as in

Picard's method. The cD, functions are selected to approximate the actual solution.

Pasic uses the power series, but other functions, such as Fourier series, can be used. The

objective is to select the number of collocation points to match the number of unknown

coefficients. The shooting method applies when solving a boundary-value problem in

ODES. A guess is made for the initial conditions. The domain is solved (repetitively, if

sub-domains are used) and the resulting y value obtained by the guessed initial cond ition

is compared to the value of the boundary condition. The initial condition is re-adjusted

such that a closer value is obtained after the next time the ODE is solved over the

domain. This guessing and solving continues until the boundary condition and value at

the end of the domain, are within an allowable error range. If one wants to avoid the

shooting process, one needs to reformulate the boundary-value problem as an initial-

value problem.



57

Initially, we will formulate the Pasic method for only the initial-value problem.

Later we will discuss the procedure used for a boundary-value problem. Let the initial-

value problem of mth-order be

y ( m ) = f ( x , y , Y ' ,Y " ~ . . . ~ . Ym - 1 ) )( 3 . 9 )

~ ( 0 )C Q ( 3 . 1 0 )

Y ' ( 0 ) =Cl

y'"-')(0) = C m - I

Suppose that the solution has the form of a polynomial :

1y ( x ) = C o + c 1 x + - C 2 x 2 +...+

1x

2 ( 2 ) ( 3 ) ...(m - )

+A1xm+ 4 x m + l+A3xr2 + ...+A(k-m+l,x (3.11)

with unknow n coefficients Aj; while C i are the initial conditions. Equa tion ( 3 .1 1 ) written

in sum mation form is

where k > m. Notice that the number o f initial conditions, C, , s the same as the degree

of the ODE. Additionally, the number of coefficients, Aj, is (k - m + I), which needs to

be the num ber of collocation points. The points are at XO , X I , x2, ... ~ ( l c - m - 2 ) ~+-m-l ) , and are

equidistant for convenience - though this is not required. An example is in order here.

Consider the first order initial-value ODE problem,

Y ' = f ( x , y ) (3 . 13a )



A pow er series of fourth order (k = 4) is created to describe the function:

such that the initial condition is satisfied. Additionally, the fourth order power series

equation that models the first order ODE is differentiated. Notice that the coefficients are

unknown, and the equation is

The differentials are calculated from the actual equationf (x, y)

The length between each x value is h, while the domain 3h is the length for a given

iteration. The sm all domain forces the fixed-point iterations to converge and allows for

small error. The fou r points of the collocation method are used to find the four unknowns,

which in m atrix form is

where

The linear system to be solved is



which is symbolically solved as

The subscript n is the iteration counter. This means that the polynomial coefficients in

the n + 1 iteration are found based on the hnction from the previous, or n-th iteration.

Initially, the values for yl through y4 of equations (3.16) are guessed to obtain the f,

vector. Next, the A,+l matrix is solved and used to calculate the yl through y4 by

integrating equation (3.15) to obtain equation (3.14). Actually no integration is required,

since the integrated value is obtained by explicit solution. The new yl through y4 values

are used to calculate the new f h l vector by use of equations (3.16). Notice that subscript

n refers to the num ber of iterations that occur over the same domain. The old y,-1 values

are compared with the new y, values, during each iteration. The iterations are stopped if

a d ifference is within a tolerable error (i.e. E < h41 or sm aller, since the polynomial is of

fourth order). Typically, the domain is descretized into many sub-domains to minimize

error and insure convergence - similar to the shooting method . Once convergence is

reached, the domain is moved forward so that the value of the previous x4 collocation

point is assigned to the new X I collocation point. New values are ca lculated for x2, x3, and

x4 by respectively add ing h, 2h, and 3h to XI. The iterative process is again repeated until



convergence. This consecutive marching continues over the length of the domain to

solving for y.

If the ODE is a boundary-value problem, a guess is made at the initial condition

Coand the resulting y4 value of the last sub-domain is compared with the boundary-value

(if the boundary condition is at the end of the domain). The initial con dition CO s again

guessed based on the previous guess and the end solution. Again the ODE is solved and

th e y4 value of the last sub-domain calculated. If the boundary condition and they4 value

of the last sub-domain are not within the allowable error, then the solving of the ODE

continues. Normally, a Newton-Raphson, bisection, or other root finding method is used

to determine the next best guess after the first two guesses. The OD E is again solved and

the resulting error evaluated to determine if it is within the allowable tolerance. This

process continues until convergence.

3.6

Pasic M ethod for Higher O rder Differential Equations

The Pasic method applies to higher order ODES. The form for an initial-value

second order ODE, for example is.Y" = f ( x , ~ J ~ ' )

~ ( 0 )Co

Y ' ( 0 )=c,

The solution of the differential equation in terms of a fifth o rder polynomial is



where the coefficients can be found with four points by the collocation method. Notice

that the distance between successive points in this case is assumed to be a constant value

ofh .

The resulting solution for the coefficients is:

The application of the Pasic method is the same for the first order ODE case. The main

difference is that w e have more initial conditions and integration is carried out two times

rather than just once to form the above polynomial, which is the solution. If the problem

is a boundary-value problem, two guesses may be required rather than one, depending on

the boundary conditions.

The Pasic method also applies to third order ODES . The basic form for the

differential equation is

ym= f ( x , ,y', y" )

~ ( 0 )c,

~ ' ( 0 )c,

y" (0 ) =C2

The assumed polynomial solution is the following:

where i = 1, 2, 3, 4 . Solving symbolically for the unknown coefficients lead to



Once the coefficients are known the solution of the differential equation over the small

sub-domain is known.

All the higher order differential equations also perform the fixed-point iterations.

In other words, the new calculation for the y for each of the collocation points is used

again to calculate the values of the differential equations at the collocation points. Again,

the n + 1 subscript under the new coefficients refer to the fixed-point iterations. As in the

first order case, the iterative process is continued until the convergence criteria is met.

After convergence, the domain is shifted so that the new x, is assigned the value of the

old x4 . The shooting steps continue over the domain, as in the case w ith the first order

differential equations. Boundary-value problems also apply to higher order O DES where

more guesses are made for the added initial conditions. The differentials may also be

broken dow n into a system composed of several first order equations [26].

In conclusion, the Pasic method is simple to implement yet a potentially po w e f i l

tool in numerical analysis. This method is usefbl possibly in Stirling analysis where

differential equations are solved. Chapter 4 presents the application of the method to the

Ideal Adiabatic model.



Chapter 4

Application of the Pasic M ethod to the Ideal Adiabatic

Stirling Model

4.1 Introduction

This chapter presents the work in applying the Pasic method t o the Ideal Adiabatic

model. The simplest set of equations is specified in section 4.2, which define the

temperature differentials for the expansion and compression spaces. The temperature

differentials, DT, and DT,, can be solved by only considering those variables that are

found within the tw o equations. Section 4.3 presents some of the d ifficulties in solving the

Ideal Adiabatic model and the techniques applied to overcom e the difficulties. The two

main difficulties are the coupling of the differential equations and the problems associated

with the boundary conditions. Section 4.4 describes the algorithm of the "C" program

that implements the Pasic method in solving the Ideal Adiabatic model. Section 4.5

presents the program results of the Ford-Philips 4-215 and the Ross-90 Stirling engines.

Som e of the key parameters examined are the CPU solve time of the program, the error

Qr (it should equal zero), the total work of the cycle and the number of sub-domains.

Also, the resu lts are compared with the original program developed by Urieli that uses the

Runge-Kutta metho d. The last section states the conclusions.



4.2 Problem Description

Chapter 2 presents the formulations of the equations for solving the Ideal

Adiabatic model. Here, the focus is upon solving the two temperature differential

equations by implementing the Pasic method. No t all of the equations of Table 2.1 need

t o be solved to obtain the temperatures of the expansion and compression spaces from the

differential equa tions . Table 4.1 presents the required set of algebraic and differential

equations. Th e tw o basic equations that we want to solve are the following coupled first

order ODES:

DP c DVC Dmc )DT, = T(-P VC mc

which are in the form

DT, = f ( 8 , T c , T e )

D r , = f ( 9 , T , , T e ) .

Th e boundary conditions are the following:

T , ( 8 = 0 ") = T,(0= 360") (4.5)

T , ( 8 = O 0 ) = T e ( 8 = 3 6 0 0 ) . ( 4 . 6 )

The assumption is made that the system is at quasi-steady state. Th e initial temperature at

the time the crank angle is at 0" should equal the temperature at the time the crank angle is

at 360" for both th e expansion space and the compression space.



Table 4.1 Essential Equations Required to Solve Temperature Differential

Equations

DT, = T, (Dp/p+D y / y Dmc / mc ) Temperature Differential

DT, = T,(DpIp+DVe V, -Dm, l m, ) Equations

p =MRI(V c T, +Vk /Tk +Vr T , +V, IT, +Ve T,) Pressure

Dp = -Y P(DK /T,k + D c / & e )

[ y / L y(Vk/Rk y / T +Vh/T,)+VJThe]

m, = P K / (RT , ) Masses

me=PV, /(RT ,)

Dm, = (PDV, +VcD p/ y ) l (RT Cd Mass Accumulations

Dm, =(pDV,+ y e Dp/y)/(R&,)

gAck= - ~ m , Mass Flows

gAhe= Dm, = gA, -Dm,

i fgAck> 0 then Tckt c else Tckt k Conditional

ifgAhe>O then Th et Th else T h e t T e Temperatures

Table 4.1 shows the basic equations needed to solve the tw o differential equa tions

(4.1)and (4.2). The only equations omitted are the equations that describe the volumes of

the expansion and compression spaces over the cycle. These depend on the type of drive

and were given for several drive types in Chapter2

and in Appendix A.

Two conditional relations that cause the differential equations to be nonlinear are

found in Table 4.1. These conditional relations cause the upstream fluid tem perature to be

used to describe the fluid between the cooler and compression space. Also, it describes

the temperature of the fluid flowing between the expansion space and heater. Finkelstein



66

wa s the first to u se these conditional relations to more accurately describe the temperature

characteristics of the flowing fluid.

Equation (4.1) and (4.2) form the two-coupled differential equations that we desire

to solve. It is paramount that the variables specified by the equations in Table 4.1 and the

volume relations are quantified over the dom ain in orde r to solve the differential equations

over the same domain. This set of equations is the most basic one that describes the

differential temperatures. The othe r variables- uch as the mass and mass accumulation

of the cooler, heater or regenerator - nd six energy differentials are easily calculated

after the temperatures o f the compression and expansion spaces are found. Although the

heat differentials are still differentials, the analysist or programmer can integrate these

without having to perform the fixed-point iterations, thus saving CPU time on the

computer.

4.3 Com plications in the Application of the Pasic M ethod

Tw o main complications are found in attempting to solve the differential equations.

The first complication is the coupling of the equations. This means that both differential

equations are inter-related by having more than one dependent variable. In ou r case, the

two differential equations are a fbnction of Tc and T,: thus, they are coupled.

The way around this difficulty is to consecutively solve each differential equation

during the fixed-point iteration part of the Pasic method. Figures 4.1 and 4.2 show the

fixed-point iterations and the sequence of solving. Notice that th e sub-domain is the same



Tc Collocation Points are Calculated First

for Each Fixed-Point Iteration

+ st Fixed-Point

Iteration

+ nd Fixed-Point

Iteration

+ rd Fixed-Point

Iteration

+ th Fixed-Point

Iteration

+ th Fixed-Point

Iteration

0 3.3 6.6 10

Crank Angle (Degrees)

Figure 4.1 Graph Sh owing Fixed-point Type Iterations and Collocation

Points for Tem perature Tc over Same Sub-dom ain as Figure 4.2

for both Tc and T, in the figures. First, the four collocation points are found for T, in the

first fixed-point iteration. The first collocation point of T , is known and the other three

collocation points are initialized to this known temperature. The same initialization occurs

for T , . The differential temperatures, DT,, are calculated from the four points. The

coefficients of the power series are calculated by equation (3.25). Once the coefficients

are known the collocation points of Tc are calculated. The graph labeled "1st fixed-point

iteration" in Figure 4.1 show s the new collocation points. Nex t, the same process is



T e Collocation Points are Calculated Seco nd for IEach Fixed-Point Iteration 1

0 3.3 6.6 10


+ st Fixed-PointIteration

+ nd Fixed-PointIteration

+ rd Fixed-PointIteration

+ th Fixed-PointIteration

++ 5th Fixed-PointIteration

Figure 4.2 Graph S howing Fixed-point Type Iterations and Collocation Points

for Temperature T, ver Same Sub-domain a s Figure 4.1

performed for T,, but this time the newly calculated T , collocation points are used to

calculate the T, differentials at the collocation points. The coefficients of T , are found, as

before, using equation (3.25) fiom which the new Te collocation points are found. The

graph labeled "1" fixed-point iteration" in Figure 4.2 shows the new Te collocation points .

This who le process constitutes one fixed-point iteration. The next fixed-point iteration is

performed by solving in the same manner as already stated, but the new collocation points

as found in the first iteration are used to find the differentials DT,. First, DT, is calculated



69

and equation (3.25) is used to find the coefficients of the polynomial that are used to find

the collocation points Tc. The differentials DT, at the collocation poin ts are then found by

using the newly calculated collocation points Tc and the collocation points T, f iom the last

iteration. The new collocation points are calculated for T, by using the coefficients found

fiom the differentials DT, at the collocation points. This constitutes the second fixed-

point iteration. The iterations continue as shown in Figures 4.1 and 4.2 until convergence

is reached. The test for convergence is executed after each fixed-point iteration to see if

the previous value forT, and T,, in the last fixed-point iteration, are within a tolerable

error range, as compared with the new T , and T , values. Upon convergen ce of the fixed-

point iterations, the sub-domain is stepped ahead to the next sub-domain, similar to the

shooting method. The next set of collocation points is initialized for the new sub-domain,

and the fixed-point iteration process continues. This stepping sequence is continued over

the full domain with the fixed-point iterations occurring over each sub-domain.

The difference between the Pasic method as described in Chapter 3, and the one

described here, is important to note. Here the two-coupled differentials are solved over

the sub-domain within each fixed-point iteration. Chapter 3 solves only one differential

equation over the sub-domain, during the fixed-point iteration, since there is only one

differential equation w ithout any coupling.

The second complication in the application of the Pasic method is the boundary

conditions. A guess is made at the initial conditions in the application of the Pasic method

a

to boundary-value problems- s the case is here. The difference between the regular

Pasic method and the case here is that the regular Pasic method typically uses a root



70

finding technique as is often implemented with the shooting method. Here, we use the

thermodynamics of the system to find the next guess for the initial conditions for each

differential equation. The temperatures, T, and T,, obtained from the solution of the first

cycle (at the crank angle of 360")are used for the initial conditions of the next cycle. This

process continues until the temperatures at the boundary condition are satisfied within a

designated error. The thermodynamics of the system causes this convergence, such that

the beginning temperatures of the compression and expansion spaces are the same as the

temperature at the end o f the cycle. Urieli was the first to notice and utilize this method

P I .

4.4 Program Structure

A "C" program was written to solve the Ideal Adiabatic model by the Pasic

method . The objective of this section is to present the logic of the p rogram by explaining

the algorithms in Figures 4.3 and 4.4. Figure 4.3 shows the algorithm that describes the

overall logic without going into the details surrounding the solve process of the cycle.

Principally, this algorithm focuses on the logic for solving the boundary conditions, which

is similar to the "shooting method part" of the Pasic method. Figure 4.4 presents the logic

of the process for solving one cycle. This module contains the fixed-point iterations, the

collocation process and the consecutive stepping from the first sub-domain all the way

through to the last sub-domain.

Figure 4.3 shows the solution algorithm starting with the module called

"adiabatic". The first choice in the module is the method of solving. Either the Pasic



Figure 4.3 Adiabatic Module Algorithm

0

adiabatic No , diab

Initialize var[][]

dvar[l[l

Initialize

[Te(O= 0) =

stirling1Solve Stirling cycle

Te(O= 360°)]by Pasic method

[Tc(O= 0) =ATc= Tc (0° )- TC(36O0)

Tc(e= 360°)]AT, = T, (0")- T,(36O0)

Are ATc and AT,

within error?

YPC

7

solveht

Integrate the energy derivatives by Pasic Integration

printad

Print out var[][] and dvar[] I results

C

Return to calling hnction



method is selected or the Runge-Kutta method is selected which calls module "adiab".

This module was w ritten by Urieli; however, it was modified as com pared to appendix C.5

of [ I ] and rewritten in "C" . With the Pasic method, the "var[][ln and "dvar[][]" matrices

are initialized to zero while Tc and Te are initialized to Tk and Th, respectively, for the first

time of solving the cycle. The function "stirlingl" solves the Ideal Adiabatic model

over the full cycle of 360". At the end of the cycle, the difference in the initial and end

temperature of both Tc and T, a re computed . These differences are compared to see if

they are w ithin tolerance as shown below:

[abs(Tc(OO) TC(36O0)) abs(Te(OO) T,(36O0))]<= 1 O. (4.7)

The symbol "abs" refers to the absolu te value. The final values TC(36O0) nd T,(36O0) are

used to re-initialize Tc(OO) nd Te(OO)n the "var[][]" and "dvar[][]" matrices for the first

set of collocation poin ts. The function "stirlingl" is again called and the differences in

temperature are calculated after the cycle is solved. As long as the test criteria of equation

(4.7) is not met, the loop continues. This test constrains the model, so that the boundary

conditions are found- imilar to the shooting method described in section 3.4.

Function "solveht" is called after the differential equations are solved when the

looping process is stopped . This finctio n integrates the energy differentials similar to the

"npsolve" kn ct io n in finction "stirlingl". The main difference is that the energy

U

differential points are exactly known, so no fixed-point iterations are needed.

Additionally, an array of differential points is passed to the function and no calculation of

derivatives is needed. A curve-fit is applied to the differential points and the coefficients



73

are found for the power series. Once the coefficients are known the actual value of the

points are found from the power equation. A sequential stepping process is used over a

small sub-domain having four collocation points. Given that this process is similar to the

Pasic method it is called Pasic integration. The next hnct ion call is to "prntad", which

prints out the resu lts of the var and dvar matrices. This module was originally written by

Urieli in FORTRAN, but was later translated into "C" [I ]. Last of all, a return is made to

the calling hn ction.

Figure 4.4 shows the "stirlingl" hnction, which solves the temperature

differentials over the cycle. First, "s tirlingl" solves the derivative for the first collocation

point by calling hnction "dadiab2". Next, it enters the "for" loop and initializes the next

three collocation points for the sub-domain. Additionally, three calls to "dadiab2" are

made to calculate the derivatives for the next three collocation points. The hn ction

"npsolve" is called next. This hn ct ion first fits a polynomial curve to the derivatives of

the four collocation points of temperature T, by using equation (3.25) to find the

coefficients for the power series. Next, the T ,values for the three unknown collocation

points are found from the power series coefficients and the first known collocation point.

The new values that are found for the collocation points are like the points labeled "1st

fixed-point iteration" in Figure 4.1. Following, the hnction "npsolve" calculates the T,

derivatives for the three unknown collocation points and finds the coefficients for the

pow er series by equation (3.25). As was done before for T,, the T, values are found for

the three unknown collocation points. These new values that are found for the collocation



Q

Stirling1dadiab2

Calculate derivative for first collocation point

PerForm next sub-domain

Initialize next three collocation

points for current sub-domain

*dadiab2

Calculate derivatives for next three collocation points

npsolve - performs one fixed-point iterationCalculate coefficients for two power series that are

curve fitted, first for the derivative of Tc & then T,

Calculate the collocation points for T c & Te from the

power series

Fixed-point iterations

abs(T,(last fixed-point iteration) - Yes

No T,@resent fixed-point iteration))<

4 MAXERROR

(Same condition for T, nd in both

cases solved for all collocation points)

No

Return the var[][]& dvar[][] matrices

Figure 4.4 Stirlingl M odule Algorithm



75

points are like the collocation points labeled "1st fixed-point iteration" in figure 4.2. Tlis

action within "npsolve" constitutes one fixed-point iteration. The d iscussion of section 4.3

about the fixed-point iterations apply here. After the hn ct ion "npsolve", several test

criteria are evaluated. This ensures that the fixed-point iterations occur at least twice so

that the com parison is made between the same sub-domain and not the previous one. The

second test criteria ensures that the difference between the temperatures Tc and T, for all

collocation points is within the maximum allowed error (MAXERR). The last test criteria

is to ensure that if convergence is not obtained within fifty fixed-point iterations a warning

is printed to inform the user that the output is flawed since convergence was not attained.

After convergence of the fixed-point iteration, the test is made to see if the end of the

cycle is reached. If the end of the cycle is not reached, then the domains of the collocation

points are stepped forward by adding 3h to each point of the domain and adding three to

the index speciflm g the location in "var[][]" and "dvar[][]" matrices. Next, the last three

collocation points a re initialized to the fourth collocation point value of th e p revious sub-

domain and the derivatives are calculated. The "npsolve" module is re-executed and the

fixed-point iterations continue until it converges. The test criterion, as shown in Figure

4.4, determines the convergence. The next sub-domain is solved and this stepp ing process

continues over the full domain until the end of the cycle. The function "stirlingl" returns

the "var[][]" and "dvar[][]" matrices at this point in the cycle, which terminates the

"stirlingl" module.

The program modules that comprise the "adiabatic" and "stirlingl" modules are

found in Appendix B. These modules are used to obtain the results for the Ford-Philips 4-



76

215 and Ross-90 engines. The "define" modules, which were originally written by Urieli,

were re-written in "C" language. These modules specifL the pertinent dimensions and

inform ation for the S tirling engine that is to be analyzed. By using the "define" modules,

which Urieli originally wro te [ I ] , the relevant information for the Ford-Philips 4-215 and

Ross-90 Stirling engines were analyzed. The next section presents these results.

4.5 Results

The essential questions addressed by this thesis are:

Can the Pasic method be applied to the Ideal Adiabatic model and obtain

results?

Will the C PU solve time significantly decrease by use of the Pasic method?

Will an acceptable error result by use of this new m ethod?

The program modules as described in the previous section were developed to address the

central questions of the thesis. Adaptations were made on the program modules

developed by Urieli to analyze a specified engine. The Ross-90 and the Ford-Philips 4-

215 engines were analyzed by running the program s. Verification of the program was

accomplished by comparing the results obtained by using the Pasic method against the

results obtained by using the Runge-Kutta method. Figure B. 1 shows the results of the

temperatures T,and T, by using the Pasic and the Runge-Kutta method. The program

segmented the domain of one cycle into twelve sub-domains where the collocation points

hrt he r segmented the domain into three smaller segments of "length" h. Each sub-domain

therefore has a "length" of l o0 , since the total cycle is 360". The Runge-Kutta program



77

module calculates every degree but only records values every 10". Notice that the results

are nearly identical, but that the Pasic method has a much larger step size: 10" versus lo.

Table 4.1 summarizes the most important data. The value Q, is the regenerator heat that

is stored in the foil and wire mesh, respectively, for the Ross-90 and Ford-Philips 4-21 5

engines. This value should be equal to zero over the cycle since the stored heat is the

same amount that is given back to the working fluid. The percentages refer to the

maximum heat stored. Table 4.2 shows that the Runge-Kutta results is more accurate as

compared to the Pasic results, since Q, is closer to zero, but we must remember that the

step size of the Runge-Kutta method is significantly smaller. The error of the Pasic results

is nearly irrelevant, because the percentage error of the total work of the cycle between

the two methods is 0.086% and 0.075%, respectively, for the Ross-90 and Ford-Philips 4-

2 15 engines.

Table 4.2 Summary ofResults

Engine

Type

ROSS-

90

Ford-

Philips

Differential

EquationSolving Method

Runge-Kutta

Pasic

Runge-Kutta

Pasic

c p uTime

(see)7.22

0.60

3.12

0.41

QrEndofCycle

0.0015 J

.0384 W

0.0141%

.00892 J

.2230 W

.0822%

-.871 J

-47.88W

.0064%

-3.8

-207.53W

.0126%

WTotal Work of C ycle

1.16 J

29.064W

1.16J

29.089W

401 1.9J

220655 W

4008.9 J

220491 W

-



The time for the CPU to execute the program is the next concern. The Pasic

method is 12 times faster for the Ross-90 engine and 7.6 times faster for the Ford-Philips

4-2 15 engine . This is a significant improvement. For example, a design problem requiring

many executions of the program could easily take an hour of CPU time using the Runge-

Kutta method. Using the Pasic method the time would take approximately 5 t o 8 minutes.

Figure 4.5 shows the effects on temperature of decreasing the size of the sub-

domain for the Pasic method on the Ross-90 engine. Notice that even with two sub-

Shape ofT. Graph for Different Numbersof Sub-domains (Ross-90)

3 3 0 0 0 0 0 0 0 00 0 P 1 9

7 m 9 p a p l m m

Cran k Angle (Degrees)

12 Sub-domains

6 Sub-domains

4 Sub-domains

2 Sub-domains

Figure 4.5 Effects of N umber of Sub-domains on T emperature (Ross-90)



Q, vs. Number of Sub-domains(ROSS-90)

A

- - - - - - - - - - - - - - - - - - - .Qr Max = 10.85 J

- - - - - - - - - - - - - - - - - - - *Qr(360)

A A A6

Number of Sub-domains

Q, vs. Number of Sub-domains(Ford-Philips4-215 Engine)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Qr Maximum= 29,934- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

. . . . . . . . . . . . . . . . . . . . . . . . .

* A * A A- v - - v 4


Figure 4.6 Effects of Number of Sub-domains on Error



80

domains the temperatures over the cycle are very similar. The same is true for the Ford-

Philips 4-215 engine, but the deviation from the results is more pronounced at only two

steps (see Figure B.6). The heat of the regenerator over the cycle Q, is graphed versus

the number of sub-domains for both engines in Figure 4.6. Notice that the percentage

error- ompared to the maximum value of Q,- is under 1% even at three sub-domains.

Figure 4.7 shows the time versus the number of sub-domains for both engines.

Notice that the graphs are nearly linear with positive slopes above 4 subdomains. The

CPU Time vs. Number of Sub-domains

" , , . . .

- r n r n w O ' = ~ ~ ~ ~


Figure 4.7 Effects of Number of Sub-domains on Time



minimum times are 0.2 seconds for the Ford-Philips 4-215 engine and 0.35 seconds for the

Ross-90 engine and both minimum values are found at four sub-domains. Under four

sub-domains the number of times to solve the cycle increases, so the time increases.

Obviously, the use r would w ant to choose four or more sub-domains in applying the Pasic

method t o the Ideal Adiabatic model. The reason for the increased time of the Ross-90, as

compared to the Ford-Philips 4-215 engine, is most likely due to an increased number of

fixed-point iterations of the Ross-90 engine. The increase in the number of fixed-point

iterations may be caused by the non-sinusoidal volume variations of the Ross-90 engine,

but h rt he r analysis is needed to verifjl this.

The question of the optimal number of sub-domains is an important point to

discuss. There is no definite answer to this question, given that there are several

considerations that one has to make. For example, the time element plays a major role in

optimizing the Stirling model when multiple calculations of the cycle are required.

Conversely, the time element is not important where few calculations are required:

however, the accuracy may play a more important role. More sub-domains may be in

order for increased accuracy. The user needs to balance both time and accuracy

considerations. There a re other issues in discussing the optimal number o f sub-domains.

Particularly, there is the need to determine the number of points the user would like to

obtain over the cycle to describe the cycle. Fewer sub-domains provide fewer points.

There are ways around this problem, but they may require more CPU time. One possible

solution would be to use the coefficients that describe the power series to generate



additional points. (i.e ., every 10"). This solution would require more calculations, and

thus more time, but it may be beneficial in some applications. The current program would

require extensive revision to accomm odate this change.

4.6 Conclusion

The Pasic method solves the Ideal Adiabatic model that has two-coupled first-

order ODES with difficult boundary conditions. Several "C" program m odules were

written to verify the Pasic method and to compare it to the classical fourth order Runge-

Kutta method that was the numerical solver used in the past. The time required by the

CPU is significantly reduced in both test cases ran - he Ford-Philips 4-215 and the Ross-

90 Stirling engines. The Pasic method is 7.6 times faster in solving the Ford-Philips 4-215

engine and 12 times faster in solving the Ross-90 engine as compared to the classical

fourth order Runge-K utta method. The error was more using the Pasic method given that

the sub-domains were larger, but this error was not significant. Another area of

investigation was how the number of sub-domains effected both the error and the CPU

time. Even with just four sub-domains the error was not significant when using the Pasic

method. The CPU time significantly dropped to a low of 0.2 and 0.35 seconds,

respectively, for the Ford-Philips 4-21 5 and the Ross-90 engines.



Chapter 5

Conclusions and Future Research

5.1 Conclusion

A new method for solving differential equations was developed recently by Dr.

Pasic of Ohio University. The objective of the thesis was to apply the Pasic method to

the Ideal Adiabatic model. The Pasic method was programm ed in "C" and the numerical

method solved the Ideal Adiabatic Stirling Model. The primary application of the

method would be for use in the design and optimization of the Stirling engine: therefore,

part of the objective was to investigate if the new method would improve the speed in the

analysis without significantly decreasing the accuracy.

The Pasic method applied to the Ideal Adiabatic model is faster at solving the

model as com pared to the Runge-Kutta m ethod. The fixed-point iterations converge, thus

confirming that the model solves. Additionally, the effect of the step hn ct io n change

based on the direction of flow is relatively insignificant as compared to what was

originally hypothesized. The output shows a larger error for the Pasic method, but this is

the result of the s ignificantly larger step-size used with the me thod. The cost in time

compensates for the cost in accuracy. The user has the option to increase the number of

sub-domains to increase the accuracy if so desired.

Another area of investigation was the determination of the number of sub-

domains that minimize the CPU time while having little impact on the error. Both the

time and the error were in conflict given that the optimum of the one produced an adverse

effect on the ofher. Four sub-domains resulted in a m inimum time w hile the error related



84

to Qr diminished with more sub-domains. The small number of sub-domains has another

consequence that the user must realize. Few sub-dom ains result in few points that

describe the cycle.

5.2 Future Research

The Pasic method solves the formulated ODES for Urieli's Ideal Adiabatic m odel,

thus accom plishing the primary objective of the thesis. Several areas, however, deserve

fhrther research: the Pasic method as presented deserves closer research; the New Pasic

method needs to be applied to Stirling analysis; and the Pasic method applied to partial

differential equa tions deserves investigation.

First, the Pasic method deserves more research. Part of the increased speed was

attributed to the logic of the programming. Particularly, the temperature differentials

were solved using only the essential information while simultaneously solving the values

of the energy differentials at the collocation points. The energy differen tials were solved

after the temperature differentials by integration similar to the Pasic method without the

fixed-point iterations. The logic in the program, using the Rung e-Ku tta method, was

different in that all the differen tials were solved using the Runge-K utta method. This

increases the time t o solve the model. A better comparison of the Pasic method would be

to solve the overall model in the same manner without the variation in the method of

solving the model. Additionally, the investigation could include solving the Ideal

Adiabatic model using both methods such that the step-size was the same (i.e. hp = h ~

where P stands for Pasic and R for Runge-Kutta). The comparison of the erro r and CPU

solve time could better determine the strengths of the Pasic method. The case where 4*hp



equals hR also deserves closer analysis. Another possible area of investigation is to

include logic in the program to "remember" the past solutions. A guess could be made at

the next differentials in the stepping process based upon the previous solutions. This

could possibly decrease the number of fixed-point iterations.

Dr. Pasic, of Ohio University, has developed an advanced method for solving

differen tial equations. This method is described in the literature [19]. This advanced

method has some similarities to the simple Pasic method but in other respects it is

hnd am enta lly different. Like the simple Pasic method, guesses are made at points within

the sub-domain. The primary difference is that the "sub-domain" actually covers the full

domain. Treating the guesses as boundary conditions solves the ODE. Any method for

solving ODEs can be used in solving the sub-domains with the guessed boundary

conditions. After the O DEs are solved a "matching" occurs where the two slopes at the

guessed points are used to determine the next best guess for the point. Obviously, one

wants the slopes to be equal at the points. Each of the points is examined and new

guesses are made for the dependent variable values of the points. Again the sub-dom ains

are solved based on the boundary conditions given fiom the new guesses. This process

continues until the ODE is solved within an error as determined by the difference in the

slopes.

One key advantage of the method is that parallel processing can be utilized in

solving the problem. The advantage of parallel processing is that difficult problems are

solved with a computer that has many processors, which significantly decreases the

computational time. Another advantage is that many variations of this so lving process



86

exist and any method in solving ODES can be used to solve the sub-domain. This new

method of solving OD ES should be applied to the Ideal Adiabatic model.

The last area of possible future research is applying both the Pasic method and his

advanced method to partial differential equations (PDE s). The Pasic method utilizes a

polynomial as the shape fbnction. For PDEs, the shape fbnction has to incorporate two or

more independent variables so a two o r more dimensional polynomial would be requisite.

Pascal's triangle is a usefbl tool for formulating the polynomials. An example of a two-

dimensional third-order polynomial shape function is the following:

One can apply the Pasic method when the collocation points are defined at x and y values.

A ten-point element is required to solve the coefficients for a shape function of a third

order polynomial. Initially, a guess is made at the magnitude of the z-values for the

unknow n points. The fixed-point iterations comm ence and stop when the solution is

found within a specified error range. After converging on a solution, the next sub-

domain is solved by again using the fixed-point iterations. Further research will

determine if this method can com pete against FEM methods.

The Advanced Pasic method may prove useful for solving PDEs by segmenting

the dom ain, for example, on both the x and y-axis and solving these large sub-domains by

any of the various methods: FEM; finite difference; or Pasic method applied to PDEs.

The partial derivatives at the boundaries can then be used in the "matching" process to



87

determine the next best guess for the z dependent variable boundary points. This

continues until the slopes of the boundary conditions converge, which solves the PDE.

The Advanced Pasic method may prove beneficial in cases where large segments of the

model are connected by narrow boundaries. The Stirling engine is an ideal candidate for

investigating the method.

Clearly, this thesis is only the beginning in the analysis of the Pasic method and

the Ideal Adiabatic S tirling model. Possibly in the future, multiprocessor com puters will

analyze and optimize the S tirling engine performance in real time.



REFERENCES

1. Urieli, I., and Berchow itz, David M., Stirling Cvcle Ena ine Analysis Adam Hilger

Ltd., Bristol, 19 84: 2, 13, 12, 1, 104, 121, 122, 126, 127, 141, 126, 93, 169-252, 87.

2 Organ, Allan J . , Thermodynamics and Gas Dynamics of the Stirling Cvcle Machine

Cambridge University Press, Cam bridge, 19 92: 2,7- 9,9- 53, 209-21 1.

3 Senft, Jam es R., Ringbom Stirling Engines Oxford University Press, New York,

1993: 3.

4 Walker, G., Stirling Engines Oxford University Press, New York, 1980: 2, 56, 72.

5 Seminar Proceedings Stirling-Cycle Prime Mo vers June 14- 15, 1978: 10, iii, 166, 60,

125.

6 Simon , T.W ., Ibrahim, .M.B ., Kannapareddy, M., Johnson T., and Friedman, G.,

"Transition of Oscillatory Flow in Tu bes: An Emperical Model for Application to

Stirling Engines," 27' IECEC Proceedings, V ol 5, 1992 : 495-502.

7 Tew. Roy C., and Geng, Steven M., NASA Lewis Research Center Overview of

NASA Supported Stirling Therm odynam ic Loss Research, 27* IECEC Proceedings,

V ol5 1992: 489.

8 Cengel, Yunus A., and Boles, Michael A. Thermodynamics an Eng ineering Approach

2ndEd, M cGraw Hill, NY, 1994: 269, 15.

9 Berchow itz D M, Stirling Cycle Enaine D e s i ~ nnd Optimisation, PHD Dissertation

University o f the Witwatersrand, South Africa, 1986: 24, 26, 27, 3 1.

10 West, C. D., Oak Ridge National Laboratory, Principles and App lications of Stirling

Eng ines Van Nostrand Reinhold, New York, 1986 : 138, 140.



89

11 Urieli, Israel, "Current Review of Stirling Cycle Machine Analysis Methods"@

IECEC Proceedings, Paper 8391 13, 1983: 702-707.

12 Organ,A

J, "Perturbation Analysis of the Stirling Cycle" 1 7 ' ~ECEC Proceedings

1982: 1699-1704.

13 Rix,D H, An Enquiry into Gas Process Assvmmetry in Stirling Cvc le Ma chines .

PHD dissertation, University of Cam bridge, 1984 .

14 Isshiki, Naotsugu, "Sim ple Vector Analysis of Stirling Machine's Performance"xfiIECEC Proceedings Vol. 5, 1992: 59-68.

15 Huang, S.C. "HFAST -A Harmonic Analysis Program for Stirling Cycles" 27'hIECEC Proceedings Vol. 5, 1992: 47-52.

16 Calandrelli, Luigi and Rispoli, Franco, "Wave Propaga tion Method fo r Stirling

Eng ine Cycle Simulation: An Experimental Validation", 3 ofi IEC EC Proceedings

Vol. 5, 1995: 397-406.

17 Urieli I, and Kushnir M, "The Ideal Adiabatic cycle - a rational basis for Stirling

engine analysis", 17fi IECEC Proceedings Paper 829275, 1982: 1662- 1668.

18 Urieli I, Class notes on Stirling analysis, Ohio University, Athens Ohio, 1997 .

19 Ross, Andy, Making Stirling Engines, Ross Experimental, Colum bus, OH, 1993 42.

20 Steven, Chapra, and Raym ond, Canale, Numerical Methods for Engineers, 2ndEd,

McGraw-Hill, New York, NY, 1988: 603.

21 Annapragada, Madhu, Optimal N-body Operations for Chase and Ope rations in

S ~ a c e , HD Dissertation Ohio University, OH, 1998 (expected date).



90

22 Pasic, H., Zhang, Y ., "Parallel Solutions of Boundary Value Problem s in Ordinary

Differential Equations based on Local Matching", 8~ SIAM Conference on Parallel

Processing; for Scientific Computing, Minneapolis, 1997.

23 Burden, R ichard L ., and Faires, J. Douglas, Numerical Analysis 3rdEd., Prindle,

Weber & Schmidt, Boston, 1985: 200-205.

24 Pasic, H., "A Simple Numerical Solution of Boundary-Value Problem s in Ordinary

Differential Equations by a Fixed-Point Iteration", (submitted for publication in the

International Journal on Numerical Methods in Engineering) 1997.

25 Quinney, Douglas, An Introduction to the Numerical So lution of Differential

Equations Research Studies Press Ltd., Letchworth England, 1985: 126.

26 De mc k, William R., and Grossman, Stanley I., Elementary Differential Equations

with Applications: A S hon Course 2ndEd, Addison-Wesley Publishing Company,

Philippines, 1981 143.



Appendix A

Drive Mechanisms and Pictures of Stirling E ngines



I

III

I

I Basic constant parameters

b , = J L Z - ( e - r ) 2

b 2 = Jb , = J L 2 - ( e + r ) 2

6 , = J ( ~ + r ) ~ - e 2I!I

; Swept vo lumes

I( " , =2 A p (b 1 -b , )

V,, =A ,@ ,- b 2 )

I

Volum e vsriations

6, = j L 2 - (e +rc os O) 2 1: Vc = VC,,+ 2 A , (b 1 - 6,) I I cedI v. = v,, +A, (b , - b , - sl n U )

! dV,IdU = - 2Aprs in O(e +rcos 0 ) /b ,

! dV. Id0 = - d V c dO ) A, / ( 2 A p )-Adrcos U

Ij

Phase angle advance

a ~ [ ( ' - x - O * r u x ) + ( ~ m n - ~ - n ) ] / 2

= n - t a n - ' b.18) +tan - ( b , / e ) ] i 2

Figure A.1 Rhombic Drive and Equation Summ ary [I]



b , s i n ~ = r c o s U

b,, = ,;b b: - rcos 0 ) 2

Displacements

y, = r [ s i n O - c o s O ( b , l b , ) ] + b ,

y, = r [ s i n O + c o s 0 ( b 2 / b ,) ] + b ,

Volume variations

v, = VCIC A,,(Y",,. - y c )

a " , I . + A ~ ( Y ~ ~ - Y . )

dV,/dO = - A,r[cosO + sinO ( b , / b , ) + ( rs in 0 cosO ) /bU ]

dV,/dO= - A,I[COS

0-

in 0 ( 6 , b , )+

(rsrn Ocos 0 ) b, ]

Figure A.2 Ross Yoke and Equation Summ ary [I]



Figure A.3 Ross-90 Stirling Engine 1171



Figure A.4 Ross-90 Foil Regenerator a n d Cooler [ I 71



r 1 . 5 Ross-90 Engine Partiall? Dis :~ssembled 171



Heater

CompressionSpace

V-configuration DriveIigure A.6 V-Configuration Drive Mechanism



Schmidt Analysis Data:

Equivalent Sinusoidal approximation:

v, = r/,,, + 1 + cosO)/2

Ve = w e 1 + cos(8 + a ) ) / 2

Note: V,,,, V,,,, d,, d, are input data.

ac = x d c 2 / 4

K w c = a c (Y-- Ym,,,)

-K w e - a e ( Y m a x - ~ - )

a = X ( 8 m m c - Ommc,) + %(Omin, - Omin,)

The swept volume of the expansion and compression spaces are calculated by

knowing the maximum and minimum linear displacements of the corresponding piston.

Here we assume that there is symmetry about the drive shaft, thus resulting in the same

maximum and minimum linear displacements for each piston. Also, an approximate

phase advance angle is determined based upon the corresponding angle at the maximum

and minimum y displacement points for both the compression and expansion spaces. The

analysis of the Ford-Philips 4-215 engine assumes sinusoidal volum e variations.

Figure A.7 Schm idt Analysis Equations for Sinusoidal VolumeApproximations



Maximum Displacement

Assum ption: no lateral conrod movement

Thus: b2 co s $ = b,

Figure A.8 Max imum L inear Displacement of Inverted Ross Yoke D rive



Appendix B

Results ofAnalysis



Com parison of Pasic M ethod with

Runge-Kutta M ethod (Ross 90fTc)

290

" % 5 Z 0 8 % 8 Z S2 m m m n nCrank Angle (Degrees)

--tTc by PasicMethod

-Tc by Runge-Kutta

Com parison of Pasic Method with

Runga-Kutta M ethod (Ross 901Te)690

2 70w

650 , + e byQ, 1L Pasic3 630- Method ,

6 610+Teby iIE Runge- ,

Kutta

F '"570

= 2 % 5 ~ s = " ~ z g R %- - - m m m r ? C ? c ?


Figure B .l Ross-90 Engine: Comparison of Tem perature Results betweenRunge-Kutta and Pasic Method



Heat vs. Cr ank A ng le (Ross 90IPasic

M e t h o d )

-12 1 ICrank A ngle (Degrees)

Heat vs. Cr an k An gle (R oss 9OIRungeKutta)

I

.-Crank Angle (Degrees)

Figure B.2 Ross-90 Engine: Comparison of Heat Results between Runge-Kutta

and Pasic Method



W ork vs. Crank A ng le (R oss 9OIPasic

Method)


Work vs. Cr ank Angle (Ross

9OIRunge-Kutta)

C'ran k Angle (Degrees)

Figure 8.3 Ross-90 Engine: Comparison of Work Results between Runge-Kuna

and Pasic Method



Shap e of Tc Graph for Different

Num bers of Shooting Steps (Ross 90)

--t 2 shootingsteps

+ 6 shootingsteps

+4 shooting

i steps8 +2 shooting

steps


Shap e of Te G raph for Different

Num bers of Shooting Steps (Ross 90)

6 2 shootingsteps

+ shootingsteps

- - - - - - - - - - - - 4 4 hooting

steps- - - - - - - - - - - - -+2 shooting

steps

" S Z 2 % " 3 S 2 %- - N N r n m mCra nk Ang le (Degrees)

Figure 8.4 Effects of Number o f Sub-doma ins on Tc and T. Temperatures (Ross-90)



Work vs. Cran k An gle (Ford-PhilipsIPasic

I5000

Method)


W ork vs. Crank Angle (Ford-Ph il ipsRu nge-

I5000 Kutta) 1

Figure B.5 Ford-Philips 4-2 15 Engine: Com pariso n of Wor k Results between

Runge-Kutta and Pasic Method



Shape of Tc G rap h for Different

Num bers of Shooting Steps (Ford-

Philips 4-215)

---e 12. shooting

steps+6 shooting

steps+4 shooting

steps++ 2 shooting

steps- ~ ~


Shape of T e G raph for Different

Num bers of Shooting Steps (Ford-

Philips 4-2 15)

- - - - - - - - - - + ? s h o o t : .

steps

+6 shooting

steps

-A-4 stlooting- - - - - - - - - steps

+ \liooting

. . . - S I C [ h

C'runk ..\nglc ( D c g r c e s )

Figure B.6 Effects of Num ber of Sub-domains on Tc and Te Tem peraures

( Ford-Philips 4-2 15 )



Appendix C

Program Modules



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* FUNCTION: adiabatic - Stirling cycle machine simulation *

* Ideal A diabatic solution Analysis using Pasic m ethod or ** Runge-Kutta m ethod *

* FILENAME: adiabatic.^ ** PROGR AMM ER 1.Urieli (FORTRAN) ** E. M alroy (C Translation) *

* DATE: 11130194 (FORTRAN), 03/12/96 (C Translation) ** LAST MODIFIED: 8/01/97 ** INCLUDE: <stdio.h>,"adiabatic.hW ** GLOBAL VA RIABLES: none ** SYM BOLIC C ONSTA NTS: R O W , R O W , C OL *

* PROTOTYPE: void adiabatic(void); ** PRE: none *

* POST: none *

...................................................................#include <stdio.h>

#include <math.h>

#include <time.h>

#include "adiabatic. h"#include "newton.h"

#include "define.h"

void adiabatic()

{

const double epsilon = 1 O; /* allowable error in temp erature (K) *Iconst int maxit = 20; I* maximum number of iterations to converge */

double var[ROW ][CO L]; I* matrix of variables */

double dvar[ROWD][COL]; /* matrix of derivatives *Idouble terror;

long clock1, clock2;

float second , period

int i, j, method, num;

printf("\n\n\t\t Pasic method is 'p' or 'Pt\n");

printf("\t\t Runga Kutta method is 'r' or 'R' (or anything else) \nu);printf("\t\t Enter your choice.\nH);

Mush(stdin);

method = getchar();

Adiabatic M odule



if((method = pl)ll(method== 'PI)) {

angle0 = 0.0; /* these need to be commented out if used with pdefine.h */

printf("\nenter the number of shooting step sh" );

scanf("%dU,&count); I* set count to the number o f shooting stepslsub-domains *I

npoint = 3 *count; I* npoint = 3 *count *Iperiod = 2.0*PI;

h = periodinpoint;

clock1 = clock();/* start cpu clock coun ter (time in microseconds) *Iprintf("...now do ing an ideal Adiabatic Simulation by Pasic me tho dh ");

@rintf(printfile,"IDEAL AD IABATIC Simulation done by Pasic m eth od h" );

for(j=O; j<=npoint; j*)

for(i=O; i<ROWV; i++)var[i]fi]=O.O;

forQ=O; j<=npoint; j++)for(i=O; i<RO WD ; i++ )

dvar[i]u]=O. 0 ;var[TCK][O] = tk;

var[THE] [0] = th;var[TC][O] = tk ;

var[TE] [0] = th ;num = 1;

do {stirling 1 var, dvar);

printf("hiteration %2d: initial Tc = %.lf(K), Te = %.lf(K)\nU,

num ,var[TC] [O],var[TE] [O]);

fprintf(printfile,"\niteration %2d: initial Tc = %.lf(K),",num,var[TC] [O]);

fprintf(printfile," Te = %.lf(K )h",va r[TE ] [O]);

terror = fabs(var[TE][O] - var[TE][npoint]) +fabs(var[TC ] [0] - var[TC] [npoint]);

printflU\t\ttemperatureerror (del(Te) + del(Tc)) = %.3lf(K)\n",

terror);

fprintf(printfile,"\t\ttemperature rror (del(Te)+ del(Tc)) = ");

fprintf(printfile, "%.3lf(K)\nU,terror);if(terror > epsilon) {

var[TC] [0] = var[TC] [npoint];var[TE] [0] = var[TE] [npoint];

) /* end of "if ' statement */

num++;

) /* end of "do" loop */

while((terror > epsilon) && (num <= maxit));

Adiabatic Module (cont.)



for(i=O;i<npoint;i+=3) { I* pasic integration of heat derivatives */

solveht(var,dvar,h,i,QK, 5);for(i=i;j<(i+3);j++)

var[W]u] = var[WC] u] + var[WE]u];

) /* end o f for loop that does pasic integration */var[W] [npoint] = var[WC ] [npoint] + var[WE] [npoint];

clock2 = clock(); /* put cpu time in variable clock- */

prnt ad(var, dvar);) /*end of pasic method (if statement) */

else { /* Use Runge-Kutta method */

npoint = 36 ; I* set global variable so that prntad w orks */

clock1 = clock(); I* start clock (cpu time - microseconds) */

for(j=O; j<C OL ; j++) I* initialize var and dvar matrix to zero */

for(i=O; i<ROWV; i++) {var[i] U]=O.0;

dvar[i]fi]=O.O;

1adiab(var, dvar); I* adiab: Ideal Adiabatic analysis */

clock2 = clock(); I* put number of signls in clock2 */

prntad(var, dvar); I* prntad: Ideal Adiabatic printout routine */

1second = (float)(clock2-clockl)/CLOCKSPER-SEC;printf("The CP U time is % f sec.\nH,second);fprintf(printfile,"The C PU time is %f sec .\nH, econd);

Adiabatic Module (cont.)



. . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. .* FUNCTION: stirlingl ** used to solve tw o derivatives, solving for Te and Tc *

* FILENA ME: stirling 1 c ** PROGRA MM ER: Eric Malroy **DATE: 07/01/97 ** LA ST MOD IFICATION DA TE: 0710 1/97 ** PROT OTY PE: void stirling1 double var[][h4AXRAY], ** double dvar[] [MAX RAY ]) ** SYMBOLIC CONSTANTS: N U M D - Number of derivatives ** GLOBAL VARIABLES: ** myeng ....................enginge type ** vk ...................... olume cooler (cu.m) ** vr ...................... olume regenerator (cu.m) *

* vh ...................... volume heater (cu.m) ** vclc ....................compression clearance ** volume (cu.m) ** vcle .................... expansion clearance ** volume (cu.m) ** vswc .................... ompression swept vo!ume ** (cu.np) *

* vswe .................... xpansion swept volume ** (cu .m) ** alpha ................... phase angle (radians) **

tk ...................... cooler temperature (K) ** th ......................heater temperature (K ) ** mgas .................... otal mass of gas (Kg) ** mgas .................... otal mass of gas (Kg) ** myeng ...................engine type (s sinusoidal) ** npoint .................. ndex # for last value ** t r ......................regenerator temperature (K) ** PRE: NONE ** PO ST: var[][i] and dy[]lj] are fblly determined *.... . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . ..

#include <stdio.h>

#include <math.h>#include "adiabatic.h"#include "define.hU

void stirlingl(doub1e var[][COL],double dvar[][COL])

{Stirlingl Module



double d v l a s t m - D l ;double errmax;double err;

double x[MAXRAY];

int i, j, k, n;

int index;

int n-seg;

int counter;

index = 0;j=O;

vlast[O]=var[TC] [O];

vlast[l]=var[TE][O];

dadiab2(x,var,dvar,O);dvlas t [O]=dvar[TC] [O];

dvlast [1 =dvar[TE][O]; O

for(index = O;index<=npoint;index+=3) {for(j=l;j<=3;j++) {

for(i=O;i<NUMD; i++) {

var[i] [index+j]=vlast [i];

dvar[i] [index+j]=dvlast [i];1x[index+j]=x[index]+h*j;

dadiab2(x,var,dvar,index+j);

1counter = 0;

do {npsolve(dadiab2,x, h,index,var,dvar);errmax = 0.0;

for(i=O; i< W -D ; i + + ){

err = fab s((var[i] [index+3] - vlast [i])/var[i][index+3]);

if(err > errmax)errmax = err;

1

Stirlingl Module (cont.)



for(i=O; i< NUM D; i++) {

vlast[i] = var[il[index+3];dvlast[i] = dvar[i][index+3];

1

counter++;if(counter>50) {

printf("counter = %d \n ',counter);

printf("'do loop in stir lingl .c is not converg ing\n1');

printf("increase MA XER R in adiabatic.h\nU);

printf("or decrease h size by increasing se gme nts\nU );

11while((errmax>MA XERR )I [(counter== )) ;

11

Stirlingl M odule (cont.)



* FUNC TION : npsolve (var and dvar are not global) ** t c return the array of y values used in the Pasic method *

* FILENAME: npso1ve.c ** PROGRA MM ER: Eric Malroy *

* DATE: 03/18/97 *

* LAST M ODIFICATION DATE: 06/23/97 *

* PROT OT YP E: void psolve(doub1e deriv(doub1e x[], *

* double var[][], double dvar[][MAXRAY], int j), double x[], *

* double h, int i,double var[][MAXRA Y] double dvar[][MAXRAY]); *

* SYMBOLIC CONSTANTS: NLTM-D - Number of Derivatives *

* MAXRAY - maximum number of array elements in array *

* GLOBAL VARIABLES:NONE *

* PRE: fbnction fbn is defined, array argument x[] is defined, *

* double array var[][] argument is defined, double array *

* dvar[], double h and int i are defined*

* is defined, d ouble argument h is defined, and int *

* POST: another element in the array var[][] and dvar[][] *

* been defined *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

#include "adiabatic. h"

#include "npsolve.h"

void

npsolve(void deriv(doub1e x[],double var[][MAXRAY],double d v a r [ ] [ m 4 Y ] , i n t j I),

double x[], double h,int index,

double var[] [MAXRA Y], double dvar [I [MAXRA Y])

tdouble a[5];int i,j;

for(i=O;i<NUM_D;i++) (

for(j= 1 j<=3 j++)

deriv(x,var,dvar,index+j);

a[O] = var[i][index];

a[]] = dvar[i][index];a[2] = (-1.0*(1 l.O*dvar[i][index]-2.O*dvar[i][index + 31

-18.O*dvar[i][index+l]+9.0*dvar[i][index + 2])/(12.0*h));a[3] = (2.0*dvar[i][index]-dvar[i][index+3]-5.O*dvar[i][index+ I]+

4.0*dvar[i] [index+2])/(6.0* h* h);

Npsolve Module



a[4] - (dvar[i][index]-dvar[i][index+3]-3 .O*dvar[i][index+l ]+

3.O* var[i] [index+2])/(-24.O* h* * h);

Npsolve Module (cont.)



...................................................................

* FUNCTION: poly ** to return the va lue of the polynomial *

* FLLENAME: pso1ve.c ** PROGRAM MER: Eric Malroy ** DATE: 03/18/97 ** LAS T MOD IFICATION DATE: 03/27/97 ** PROTOT YP E: double poly(doub1e x,double a[]) ** SYMBOLIC CONSTANTS: NONE ** GLOBAL V ARIABLES: NONE ** PRE: argument x is defined as well as array a[] ** PO ST: sum is evaluated and returned *..................................................................

double poly(doub1e x, doub le a[])

{double x 1, sum;int j,i;

sum = 0.0;for(i=O;i<=4;i++){

x1 = 1.0;for(j=O;j<i;j++)

x l =x*xl ;sum += a[i]*xl;

1return sum;

Npsolve M odule (Function poly)



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* FUNCTION: solveht (var and dvar are not global) ** to return another element in the array var[][] by solving *

* for the heat variables ** (this is for solving the heat derivatives and is a *

* modification of npso1ve.c) ** FILENAME: so1veht.c *

* PROGRAMMER: Eric Malroy *

* DATE: 07/23/97 *

* LAST MODIFICATION DATE: 07/23/97 *

* PROTOTYPE: void solveht(doub1e var[][MAXRA Y], *

* double dvar[][MAXRAY], double h, int index, int nstart, *

* int num); ** SYMBOLIC CONSTANTS: ** MAXRAY - maximum number o f array elements in array *

* GLOBAL VARIABLES: NONE ** PRE: all values fo r var and dvar have been solved for except ** for the values for the heat variables; this function ** solves the derivitives of the heat variables. *

* POST: anoth er element in the array var[][] and dvar[][] ** been defined (the heat variables) *.... . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . .

#include "adiabatic.hW* for solveht function prototype */

#include "npso1ve.h" /* for poly function prototype */

voidsolveht(doub1e var[][MAX RAY ], /* variable matrix */

double dvar[][MAXRAY], /* derivative matrix */

double h, /* increment value for dom ain *Iint index, /* index for the dom ain */

int nstart, /* starting index for the derivatives to solve * /

int num) /* number of derivatives to solve *

{double a[5];int i,j;

for(i=nst art ;i<(num+nstart)$++) {a[O] = var[i] [index];a[ 1] = dvar[i] [index];

Solveht Module



Solveht Module (cont.)



The complete "C" program modules that were used to obtain the results for this

thesis can be obtained by contacting Dr. Urieli of Ohio University.

Address:

Dr. Israel UrieliDepartment of Mechanical Engineering

Ohio UniversityAthens, OH 45701

E-mail:urieliO,bobcat.ent.ohiou.edu

Phone:740-593-1560



Appendix D

Output from Programs



input data filename: ford.dataSinusoidal drive con figuration:com p. swept, clearance vols (m**3):

8.706e-04 2.142e-04exp. swept, clearance vols (m**3):8.706e-04 2.142e-04

phase advanc e angle "alpha" (rads):1.571

cooler:

homogeneous bundle of smooth pipesinside diam(m), length(m), num. pipes:

0.000900 0.087000 2968

void volume(cc) 164.27

tubular regenerator housingtube ext,int diam(m):

8.300000e-02 7.300000e-02

length(m), no.of tubes:3.400000e-02 8

mesh regen erator matrix:

stacked wire mesh m atrixporosity, wire diam(m):

0.620 3.600000e-05

hydraulic diam 0.059(mm )

total wetted area 4.806684e+O1 sq.m)void volume 705.82(cc)

mesh thermal capacity 950.8 15(joules/K)

heater:

homogeneous bundle o f smooth pipesinside diam(m), length(m), num . pipes:

0.004000 0.462000 88

void volume(cc) 5 10.90

gastype is hydrogen

mean pres(Pa), tk, th(K), freq(Hz):1.500000e+07 337.0 1023.0 55.0

effective regen erator temperature (K): 6 17.8

Total mass of gas (gm) 16.201

Ford-Philips 4-215 Output



theta Vc(cc) Ve(cc) V(cc) Tc(K) Te(K)

0 1084.8 649 .4 3115.2 316.6 859.9

10 1078.2 573.8 3033.0 318.8 865.2

20 1058.5 500.5 2940.1 321.5 872.5

30 1026.5 431.8 2839.2 324.9 881.7

40 983 .0 369.6 2733.6 329.0 892.8

50 929.3 316.0 2626.3 333.7 905.6

60 867.1 272.5 2520.6 339.0 919.8

70 798.4 240.4 2419.8 344.6 935.2

80 725.1 2 20.8 2326.9 350.6 951.3

90 649.5 214.2 2244.7 356.6 968.6100 573.9 220.8 2175.7 362.6 987.9

110 500.6 240.5 2122.1 368.3 1006.4

Ford-Philips 4-215 Output: Pasic method112 sub-domains (cont.)



Ford-Philips 4-215 Output: Pasic method112 sub-dom ains (cont.)



140 4.056 2.382 5.583 2.441 1 .739 16.201 1047.095 858.349 668.218

150 3.542 2.425 5.685 2.485 2.064 16.201 889.638 794.277 698.217

160 3.138 2.439 5.717 2.499 2.409 16.201 676.172 681.743 687.356

170 2.867 2.422 5.677 2.482 2.753 16.201 424.010 529.372 635.507

180 2.744 2.376 5.570 2.435 3.077 16.201 155.083 350.806 547.966

190 2.773 2.305 5.402 2.361 3.361 16.201 -125.434 150.572 428.603

200 2.949 2.212 5.184 2.266 3.591 16.201 -381.675 -45.245 293.654

210 3.250 2.104 4.931 2.156 3.760 16.201 -588.182 -215.969 158.977

220 3.645 1.989 4.662 2.038 3.868 16.201 -738.802 -353 .477 34.675

230 4.103 1.873 4.390 1.919 3.917 16.201 -835.246 -455.536 -73.038

240 4.595 1.761 4.128 1.804 3.913 16.201 -885.669 -526.409 -164.513

250 5.099 1.657 3.884 1.698 3.864 16.201 -899.817 -571.261 -240.294

260 5 .598 1.563 3.664 1.601 3.775 16.201 -886.157 -593.969 -299.635

270 6.079 1.481 3.471 1.517 3.654 16.201 -852.651 -599.516 -344.523

280 6 .534 1.410 3.305 1.445 3.507 16.201 -805.555 -592.154 -377.187290 6 .956 1.352 3.168 1.385 3.341 16.201 -749.402 -575.212 -399.743

300 7.342 1.305 3.058 1.337 3.159 16.201 -687.205 -551.103 -414.001

310 7.691 1.269 2.975 1.301 2.966 16 201 -620.735 -521.423 -421.383

320 7 .999 1.245 2.918 1.275 2.765 16.201 -550.790 -487.083 -422.908

330 8.266 1.23 1 2.885 1.261 2.559 16.201 -477.432 -448.427 -419.208

340 8.489 1.227 2.876 1.257 2.351 16.201 -400.161 -405.338 -410.553

350 8.667 1.234 2.892 1.264 2.144 16.201 -3 18.059 -357.325 -396.879

360 8.798 1.251 2.932 1.282 1.939 16.201 -229.906 -303.586 -377.807

The CPU time is 0.400000 sec.

Ford-Philips 4-215 Output: Pasic method112 sub-domains (cont.)



IDEAL ADIABATIC Simulation done by Pasic method

iteration 1 initial Tc = 337(K), Te = 1023(K)

temperature error (del(Te)+ del(Tc))=

152.904(K)

iteration 2: initial Tc = 32 l(K), Te = 886(K)

temperature error (del(Te)+ del(Tc)) = 29.277(K)

iteration 3 initial Tc = 3 18(K), Te = 860(K)temperature error (del(Te)+ del(Tc)) = 3.32400

iteration 4: initial Tc = 3 17(K), Te = 858(K)temperature error (del(Te)+ del(Tc)) = 0.3 19(K)

Ideal Adiabatic simulation results:Qk (W) = -143789.526, Qr (W) = -3828.5329, Qh (W) = 362460.177

Wc(W) =-138175.289, We (W)=357078.226, W (W)=218902.937eff (WIQh) = 0.604, COP(Qh/W)= 1.656

theta V(cc) P(bar) Qk(J) Qr(J) Qh(J) Wc(J) We(J) W(J)031 15. 2 106.75 0.00 0.00 0.00 0.00 0.00 0.00

30 2839.2 116.60 -144.47 -4081.63 799.75 -649.48 -2413.41 -3062.88

60 2520.6 135.18 -427.57 -5170.29 603.47 -2661.25 -4392.82 -7054.0790 2244.7 161.51 -1061.74 -2003.20 -547.50 -5877.32 -5233.66-1 1110.98

120 2085.4 189.62 -2322.94 6195.93 -2004.93 -9729.75 -4180.42-13910.17150 2085.4 206.89 -3640.78 17369.82 -2874.23-12880.88 -989.69-13870.57

180 2244.8 202.64 -4157.92 26946.02 -2675.68-14121.02 3525.26-10595.75210 2520.7 179.37 -3807.45 29870.59 -1465.32-13000.12 7703.53 -5296.59

240 2839.4 150.11 -3324.48 26732.73 44.12-10407.89 10354.93 -52.96270 3115.3 126.28-2935.11 20218.15 1612.11 -7408.81 11163.43 3754.62

300 3274.6 111.29 -2689.52 12862.63 3397.93 -4839.69 10489.21 5649.52330 3274.5 104.98 -2585.64 5838.88 5151.34 -3121.68 8780.78 5659.10360 3115.2 106.71 -2614.36 -69.61 6590.19 -2512.28 6492.33 3980.05

theta Vc(cc) Ve(cc) V(cc) Tc(K) Te(K)0 1084.8 649.4 3115.2 316.9 857.630 1026.5 431.8 2839.2 325.3 879.4

60 867.1 272.5 2520.6 339.3 917.1

Ford-Philips 4-215 Output: Pasic methodB subdomains (cont.)



theta mc(gm) mk(gm) mr(gm) mh(gm) me(gm) M(gm) gAk(gm/s) gA r( g d s) g A h( g d s)0 8.789 1.252 2.934 1.282 1.944 16.201 -229.880 -303.552 -377.764

30 8.852 1.367 3.205 1.401 1.377 16.201 87.175 -95.998 -280.517

60 8.311 1.585 3.715 1.624 0.966 16.201 496.191 199.866 -98.635

90 7.068 1.894 4.439 1.940 0.860 16.201 931.118 558.308 182.761120 5.269 2.223 5.21 1 2.278 1.220 16.201 1163.239 838.280 5 10.935

150 3.539 2.426 5.686 2.485 2.065 16.201 889.042 793.947 698.154

180 2.740 2.376 5.569 2.434 3.082 16.201 154.965 350.698 547.867

210 3.249 2.103 4.929 2.155 3.765 16.201 -587.985 -215.896 158.924

240 4.597 1.760 4.125 1.803 3.916 16.201 -885.203 -526.155 -164.471

270 6.072 1.481 3.470 1.517 3.662 16.201 -852.823 -599.812 -344.944

300 7.335 1.305 3.059 1.337 3.166 16.201 -687.566 -551.594 -414.624

330 8.259 1.231 2.885 1.261 2.565 16.201 -477.914 -449.046 -419.967

360 8.792 1.251 2.933 1.282 1.944 16.201 -230.381 -304.213 -378.586

The CPU time is0.190000

sec.

Ford-P hilips 4-215 Outpu t: Pasic method14 sub-dom ains (cont.)



Ideal Adiabatic simulation:

iteration 1 initial Tc = 337.000 , Te = 1023.O(K)

final Tc = 321.1@), Te = 886.7(K)temperature error (del(Tc) + del(Te))= 152.169(K)

iteration 2: initial Tc = 321.1(K), Te = 886.7s )

final Tc = 3 17.6(K), Te = 864.3s )

temperature error (del(Tc) + del(Te)) = 25.933(K)

iteration 3: initial Tc = 3 17.600, Te = 864.3s )

final Tc = 3 16.8(K), Te = 860.2(K)

temperature error (del(Tc) + del(Te)) = 4.9 16(K)

iteration 4: initial Tc = 3 16.8(K), Te = 860.2(K)final Tc = 316.60(), Te = 859.5s )

temperature error (del(Tc) + del(Te))= 0.946(K)

Ideal Adiabatic simulation results:Qk (W) = -137898.030, Qr (W) = -47.878 1, Qh (W) = 357613.254

Wc (W) = -137095.677, We (W) = 357751.142, W (W) = 220655.428

eff(W/Qh)= 0.617, COP(QhN)= 1.621

theta V(cc) P(bar) Qk(J) Qr(J) Qh(J) Wc(J) We(J) W(J)03115.2 106.71 0.00 0.00 0.00 0.00 0.00 0.00

10 3033.0 109.06 -38.59 -1605.73 345.99 -71.55 -814.88 -886.4320 2940.1 112.32 -92.16 -2974.35 606.72 -289.13 -1625.45 -1914.59

30 2839.2 116.55 -153.92 -4061.67 768.62 -656.26 -241 1.58 -3067.8540 2733.6 121.76 -224.14 -4820.75 820.38 -1 174.95 -3151.24 -4326.19

50 2626.3 127.96 -31 1.46 -5202.39 754.39 -1844.91 -3820.09 -5665.0060 2520.6 135.13 -426.85 -51 53.88 567.66 -2662.48 -4391.43 -7053.91

70 2419.8 143.19 -582.90 -4622.46 263.98 -3619.35 -4836.45 -8455.80

80 2326.9 152.03 -792.38 -3560.81 -144.00 -4701.08 -5125.07 -9826.1590 2244.7 161.44 -1065.94 -1940.71 -622.56 -5885.65 -5227.42-1 1 113.07

100 2175.7 171.13 -1409.03 237.95 -1 117.60 -7142.44 -51 16.13-12258.58110 2122.1 180.68 -1817.89 2962.70 -1605.51 -8431.66 -4769.33-13200.99

120 2085.4 189.58 -2276.64 6177.78 -2060.04 -9704.89 -4174.27-13879.16

130 2066.8 197.26 -2756.20 9774.83 -2452.38-10907.38 -3331.00-14238.38

140 2066.8 203.17 -3216.50 13592.70 -2754.40-1 1982.03 -2255.20-14237.23

Ford-Philips 4-215 Output: Runge-Kutta method




0 1084.8 649.4 3115.2 316.6 859.5

10 1078.2 573.8 3033.0 318.7 864.9

20 1058.5 500.5 2940.1 321.5 872.2

30 1026.5 431.8 2839.2 324.9 881.5

40 983.0 369.6 2733.6 329.0 892.6

50 929.3 316.0 2626.3 333.7 905.4

60 867.1 272.5 2520.6 338.9 919.6

70 798.4 240.4 2419.8 344.6 935.0

80 725.1 220.8 2326.9 350.5 951.2

90 649.5 214.2 2244.7 356.6 968.5

100 573.9 220.8 2175.7 362.6 987.8

110 500.6 240.5 2122.1 368.3 1006.3

120 431.8 272.6 2085.4 373.4 1021.6

130 369.7 316.1 2066.8 377.7 1032.4

Ford-Philips 4-215 Output: Runge-Kutta method (cont)



theta mc(gm) mk(gm) rnr(gm) mh(gm) me(gm) M(gm) gAk(gm /s) gAr(gm/s)g ~ w P / s )0 8.796 1.251 2.933 1.282 1.939 16.201 -230.001 -303.712 -377.963

10 8.875 1.279 2.997 1.310 1.741 16.201 -134.385 -243.220 -352.853

20 8.897 1.317 3.087 1.349 1.551 16.201 -29.514 -174.551 -320.652

30 8.858 1.367 3.203 1.400 1.373 16.201 87.568 -95.459 -279.830

40 8.752 1.428 3.346 1.463 1.213 16.201 214.695 -6.829 -229.980

50 8.573 1.500 3.517 1.537 1.074 16.201 351.614 91.865 -169.791

60 8.317 1.584 3.714 1.623 0.963 16.201 496.733 200.383 -98.143

70 7.981 1.679 3.935 1.720 0.886 16.201 646.547 317.269 -14.426

80 7.565 1.783 4.178 1.826 0.849 16.201 795.085 439.361 81.025

90 7.073 1.893 4.437 1.939 0.859 16.201 931.883 558.766 182.911100 6.516 2.007 4.703 2.056 0.920 16.201 1047.776 671.200 291.86

110 5.908 2.1 19 4.966 2.171 1.039 16.201 1130.057 768.377 404.042

120 5.274 2.223 5.210 2.277 1.217 16.201 1164 .561 839.050 511.150

130 4.645 2.313 5.421 2.370 1.453 16.201 1139.253 872.030 602.846

140 4.055 2.382 5.584 2.441 1.740 16.201 1047.317 858.491 668.279

Ford-Philips 4-215 Output: Runge-Kutta method



150 3.542 2.425 5.685 2.485 2.064 16.201 889.702 794.322 698.243

160 3.138 2.439 5.717 2.499 2.409 16.201 676.124 681.718 687.354

170 2.867 2.422 5 .677 2.482 2.753 16.201 423.928 529.324 635.493

180 2.744 2.376 5.570 2.435 3.077 16.201 155.051 350.789 547.964

190 2.771 2.305 5.402 2.361 3.361 16.201 -125.448 150.589 428.653

200 2.948 2.212 5.184 2.266 3.592 16.201 -381.710 -45.249 293.681

210 3.249 2.104 4.932 2.156 3.761 16.201 -588.256 -215.996 158.997

220 3.644 1.989 4.662 2.038 3.868 16.201 -738.887 -353.518 34.680

230 4.101 1.873 4.390 1.919 3.917 16.201 -835.350 -455.592 -73.047

240 4.594 1.761 4.128 1.804 3.914 16.201 -885.748 -526.441 -164.497

250 5.098 1.657 3.884 1.698 3.864 16.201 -899.872 -571.261 -240.237

260 5.597 1.563 3.66 4 1.602 3.775 16.201 -886.192 -593.943 -299.550

270 6.078 1.481 3.471 1.517 3.654 16.201 -852.668 -599.476 -344.424

280 6.532 1.410 3.306 1.445 3.508 16.201 -805.562 -592.109 -377.090

290 6.955 1.352 3.169 1.385 3.341 16.201 -749.406 -575.173 -399.661300 7.341 1.305 3.059 1.337 3.159 16.201 -687.214 -551.078 -413.943

310 7 .689 1.269 2.975 1.301 2.966 16.201 -620.754 -521.420 -421.357

320 7.998 1.245 2.918 1.275 2.766 16.201 -550.825 -487.106 -422.920

330 8.265 1.231 2.885 1.261 2.560 16.201 -477.485 -448.479 -419.26

340 8.488 1.227 2.877 1.257 2.352 16.201 -400.232 -405.421 -410.648

350 8.666 1.234 2.892 1.264 2.144 16.201 -318.147 -357.436 -397.012

360 8.797 1.251 2.932 1.282 1.940 16.201 -230.009 -303.721 -377.975


Ford-Philips4-215 Output: Runge-Kutta method



input data filename: ross9y.dat

Yoke drive configuration:comp clearance volume, exp clearance vol (mA3)8.0000e-06 1.0000e-05

Ross yoke lengths b l , b2&

r (m)0.0354 0.0354 0.0085

diam o f piston, diam of displacer -diamp & diamd (m)

0.056 0.056ymin = O.O23(m), ymax = 0.047(m)

alpha = 1.628(rad)vswc= 5.9171e-05(mA3), vswe 5.9171e-05(mA3)

cooler:

slot heat exchangerwidth and height of slot (m):

0.000532 0.0033 15heat exchanger length (m) & number of slots

0.048260 388

Void volume 33.02(cc)

Hydraulic diameter 0.92(mm )annular regenerator housing

housing ext,int diam(m):hou7.620000e-02 7.302500e-02

mat5.943600e-02 3.5 17900e-02:

foil regenerator matrix:

unrolled length of foil, foil thickness6.075000(m) 6.952971e-05(m )

hydraulic diam 0.326(mm)total wetted area 0.427425(sq.m)

void volume 34.87(cc)

porosity 0.701foil therm al capacity 52.676(joules/K)

heater:slot heat exchanger

widt0.000959 0.003556t (m):

heat0.037846 220gth (m)&

number of slotsVoid volume 28.3 9(cc)Hydraulic diameter 1.51 mm)

Ross-90 Inverted Yoke Drive Output



gastype is air

meal.O13250e+05 310.0 673.2 25.02):effective regenerator temperature (K): 468.4

Total mass of gas (gm) 0.138

Schmidt analysisWork(jou1es) 1.2, Power(watts) 30.5Qexp(jou1es) 2.3, Qcom(jou1es) -1 O

indicated efficiency 0.540regen. wall heat leakage(Watts) 96.0


iteration 1: itemperature error (del(Te)+ del(Tc)) = 30.50500

iteration 2: itemperature error (del(Te)+ del(Tc)) = 5.027(K)

iteration 3: itemperature error (del(Te)+ del(Tc)) = 0.848(K)

Ideal Adiabatic simulation results:

Qk (W)= -31.815, Qr (W)= 0.2230, Qh(W)= 60.521Wc (W) = -3 1.935, We (W) = 61.024, W (W) = 29.089

eff(W/Qh) = 0.481, COP(Qh/W)= 2.081

theta V(cc) P(bar)0 170.7 1.09

10 178.2 1.0320 185.7 0.97

30 193.0 0.92

40 199.8 0.87

50 205.8 0.8360 210.8 0.80

70 214.6 0.77

80 216.9 0.75

90 217.7 0.74100 216.9 0.73

110 214.6 0.73

120 210.8 0.74130 205.8 0.75140 199.8 0.77

150 193.0 0.80160 185.7 0.83

170 178.2 0.87

Ross-90 Inverted Yoke Drive Output: Pasic method112 sub-domains (cont.)



theta Vc(cc) Ve(cc) V(cc) T c s ) Te(K)

0 15.3 59.2 170.7 317.4 645.1)

10 19.3 62.6 178.2 311.6 635.520 24.0 65.4 185.7 306.7 625.5

30 29.2 67.5 193.0 302.7 615.5

40 34.7 68.8 199.8 299.3 606.3

50 40.3 69.2 205.8 296.6 598.0

60 45.8 68.7 210.8 294.4 591.0

70 51.0 67.3 214.6 292.8 585.2

80 55.7 65.0 216.9 291.8 580.9

90 59.7 61.7 217.7 291.4 578.1

100 63.0 57.7 216.9 291.6 576.7

110 65.3 53.0 214.6 292.3 576.9

120 66.7 47.8 210.8 293.6 578.6

130 67.2 42.3 205.8 295.4 581.6

140 66.8 36.7 199.8 297.8 585.9

150 65.5 31.2 193.0 300.6 591.4

160 63.4 26.0 185.7 303.9 597.9

Ross-90 Inverted Yoke Drive Output: Pasic method112 sub-dom ains (cont.)



Ross-90 Inverted Yoke Drive Output: Pasic method112 sub-domains (cont.)



180 0.058 0.034 0.024 0.013 0.009 0.138

190 0.056 0.035 0.025 0.014 0.008 0.138

200 0.053 0.037 0.026 0.015 0.006 0.138210 0.050 0.039 0.027 0.015 0.006 0.138

220 0.046 0.041 0.029 0.016 0.006 0.138

230 0.042 0.043 0.030 0.017 0.006 0.138

240 0.038 0.044 0.031 0.018 0.007 0.138

250 0.033 0.046 0.032 0.018 0.009 0.138

260 0.028 0.047 0.033 0.019 0.01 1 0.138

270 0.023 0.048 0.034 0.019 0.013 0.138

280 0.019 0.049 0.034 0.019 0.016 0.138

290 0.016 0.049 0.034 0.020 0.019 0.138

300 0.013 0.049 0.034 0.019 0.022 0.138

310 0.011 0.049 0.034 0.019 0.025 0.138

320 0.010 0.048 0.033 0.019 0.028 0.138

330 0.011 0.046 0.032 0.018 0.030 0.138

340 0.012 0.045 0.031 0.018 0.032 0.138

350 0.015 0.043 0.030 0.017 0.034 0.138

360 0.018 0.040 0.028 0.016 0.035 0.138


Ross-90 Inverted Yoke Drive Output: Pasic method112 sub-domains (con t.)




iteration 1 initial Tc=

3 10(K), Te=

673(K)temperature error (del(Te)+ del(Tc)) = 29.767(K)

iteration 2: initial Tc = 3 16(K), Te = 649(K)

temperature error (del(Te)+ del(Tc))= 4.672(K)

iteration 3: initial Tc = 3 17(K), Te = 646(K)

temperature error (del(Te)+ del(Tc)) = 0.759(K)

Ideal Adiabatic simulation results:Qk (W) = -31.468, Qr (W) = 0.9066, Qh (W) = 60.016

Wc(W)= -31.972, We(W)= 61.081, W (W)= 29.109eff (WIQh) = 0.485, COP(Qh1W)= 2.062

theta V(cc) P(bar) Qk(J) Qr(J) Qh(J) Wc(J) We(J) W(J)0 170.7 1.09 0.00 0.00 0.00 0.00 0.00 0.00

30 193.0 0.92 0.55 -0.87 0.49 1.39 0.84 2.23

60 210.8 0.80 0.96 -2.63 1.04 2.81 0.96 3.7690 217.7 0.74 1.15 -4.76 1.65 3.87 0.42 4.29

120 210.8 0.74 1.14 -6.98 2.23 4.39 -0.60 3.79150 193.0 0.80 0.95 -9.04 2.65 4.28 -1.87 2.41

180 170.7 0.91 0.59 -10.50 2.76 3.57 -3.05 0.52210 151.1 1.05 0.06 -10.77 2.50 2.34 -3.73 -1.39

240 138.3 1.19 -0.68 -9.42 2.10 0.76 -3.58 -2.82

270 134.0 1.30 -1.48 -6.61 1.80 -0.82 -2.53 -3.36

300 138.3 1.32 -1.95 -3.21 1.73 -1.92 -0.86 -2.78

330 151.1 1.25 -1.81 -0.71 1.95 -2.09 0.96 -1.13

360 170.7 1.09 -1.26 0.04 2.40 -1.28 2.44 1.16

theta Vc(cc) Ve(cc) V(cc) Tc(K) Te(K)0 15.3 59.2 170.7 316.9 645.730 29.2 67.5 193.0 302.4 616.2

60 45.8 68.7 210.8 294.2 591.590 59.7 61.7 217.7 291.2 578.7

120 66.7 47.8 210.8 293.4 579.2

Ross-90 Inverted Yoke Drive Output: Pasic method B sub-domains



theta mc(gm) mk(gm) mr(gm) mh(gm) me(gm) W g m ) gA.k(g&s) g A r( g ds ) gA W g d s)0 0.018 0.040 0.028 0.016 0.035 0.138 -2.404 -0.748 0.318

30 0.031 0.034 0.024 0.014 0.035 0.138 -3.106 -1.679 -0.758

60 0.043 0.030 0.021 0.012 0.032 0.138 -2.821 -1.956 -1.399

90 0.053 0.027 0.019 0.011 0.027 0.138 -2.176 -1.914 -1.745

120 0.059 0.028 0.019 0.011 0.021 0.138 -1.400 -1.699 -1.891

150 0.061 0.030 0.021 0.012 0.015 0.138 -0.425 -1.234 -1.755

180 0.058 0.034 0.024 0.013 0.009 0.138 0.862 -0.356 -1.141

210 0.050 0.039 0.027 0.015 0.006 0.138 2.323 0.920 0.016

240 0.038 0.044 0.031 0.018 0.007 0.138 3.485 2.229 1.419

270 0.023 0.048 0.034 0.019 0.013 0.138 3.642 2.965 2.528

300 0.013 0.049 0.034 0.019 0.022 0.138 2.260 2.518 2.685

330 0.011 0.046 0.032 0.018 0.030 0.138 -0.237 0.963 1.736

360 0.018 0.040 0.028 0.016 0.035 0.138 -2.404 -0.748 0.318


Ross-90 Inverted Yoke Drive Output: Pasic m eth od h sub-domains (cont.)



Ideal Adiabatic simulation:

iteration 1: initial Tc = 3 lO.O(K), Te = 673.2(K)final Tc = 3 16.O(K), Te = 648.8(K)

temperature error (del(Tc)+ del(Te)) = 30.399(K)

iteration 2: initial Tc = 3 16.00(), Te = 648.8(K)final Tc = 3 17.3(K), Te = 645.l(K)

temperature error (del(Tc)+ del(Te)) = 5.0066)

iteration 3: initial Tc = 3 17.3(K), Te = 645.1(K)final Tc = 3 17.6(K), Te = 644.5(K)

temperature error (del(Tc) + del(Te))= 0.844(K)

Ideal Adiabatic simulation results:

Qk (W) = -32.025, Qr (W) = 0.0384, Qh (W) = 60.910

Wc (W) = -31.961, We (W) = 61.025, W (W) = 29.064

eff (W/Qh) = 0.477, COP(Qh/W)= 2.096

theta V(cc) P(bar) Qk(J) Qr(J) Qh(J) Wc(J) We(J) W(J)0 170.7 1.09 0.00 0.00 0.00 0.00 0.00 0.0010 178.2 1.03 0.19 -0.15 0.17 0.43 0.36 0.79

20 185.7 0.97 0.38 -0.45 0.33 0.90 0.64 1.5430 193.0 0.92 0.55 -0.87 0.49 1.39 0.84 2.2340 199.8 0.87 0.71 -1.39 0.67 1.88 0.96 2.84

50 205.8 0.83 0.85 -1.98 0.85 2.36 1.00 3.35

60 210.8 0.80 0.96 -2.63 1.05 2.81 0.96 3.7670 214.6 0.77 1.05 -3.31 1.25 3.21 0.85 4.0680 216.9 0.75 1.11 -4.03 1.45 3.57 0.67 4.2390 217.7 0.74 1.15 -4.76 1.66 3.87 0.43 4.29

100 216.9 0.73 1.17 -5.50 1.86 4.11 0.13 4.24110 214.6 0.73 1.17 -6.25 2.06 4.28 -0.21 4.07

120 210.8 0.74 1.15 -6.99 2.25 4.38 -0.60 3.79

130 205.8 0.75 1.10 -7.71 2.42 4.42 -1.01 3.41140 199.8 0.77 1.04 -8.40 2.56 4.39 -1.44 2.95150 193.0 0.80 0.95 -9.05 2.68 4.28 -1.87 2.41160 185.7 0.83 0.85 -9.63 2.75 4.11 -2.29 1.82170 178.2 0.87 0.73 -10.13 2.79 3.87 -2.69 1.18180 170.7 0.91 0.60 -10.51 2.79 3.57 -3.05 0.52

Ross-90 Inverted Yoke Drive Output: Runge-Ku tta method




0 15.3 59.2 170.7 317.6 644.5

10 19.3 62.6 178.2 311.7 635.0

20 24.0 65.4 185.7 306.8 624.9

30 29.2 67.5 193.0 302.8 615.0

40 34.7 68.8 199.8 299.4 605.8

50 40.3 69.2 205.8 296.6 597.5

60 45.8 68.7 210.8 294.5 590.4

70 51.0 67.3 214.6 292.9 584.7

80 55.7 65.0 216.9 291.9 580.4

90 59.7 61.7 217.7 291.4 577.5

100 63.0 57.7 216.9 291.6 576.2

110 65.3 53.0 214.6 292.3 576.4

120 66.7 47.8 210.8 293.6 578.1

130 67.2 42.3 205.8 295.5 581.1

140 66.8 36.7 199.8 297.8 585.5

150 65.5 31.2 193.0 300.7 591.0

160 63.4 26.0 185.7 304.0 597.5

170 60.6 21.3 178.2 307.7 604.9

Ross-90 Inverted Yoke Drive Output: Runge-K utta method (cont.)



Ross-90 Inverted Yoke Drive Output: Runge-K utta method (cont.)



200 0.053 0.037 0.026 0.015 0.006 0.138

210 0.050 0.039 0.027 0.015 0.006 0.138

220 0.046 0.041 0.029 0.016 0.006 0.138

230 0.042 0.043 0.030 0.017 0.006 0.138240 0.038 0.044 0.031 0.018 0.007 0.138

250 0.033 0.046 0.032 0.018 0.009 0.138

260 0.028 0.047 0.033 0.019 0.011 0.138

270 0.023 0.048 0.034 0.019 0.013 0.138

280 0.019 0.049 0.034 0.019 0.016 0.138

290 0.015 0.049 0.034 0.020 0.019 0.138

300 0.013 0.049 0.034 0.019 0.022 0.138

310 0.011 0.049 0.034 0.019 0.025 0.138

320 0.010 0.048 0.033 0.019 0.028 0.138

330 0.011 0.046 0.032 0.018 0.030 0.138

340 0.012 0.044 0.031 0.018 0.032 0.138

350 0.015 0.043 0.030 0.017 0.034 0.138

360 0.018 0,040 0.028 0.016 0.035 0.138


Ross-90 Inverted Yoke Drive Output: Runge-Kutta method (cont.)



Abstract

This thesis presents the work in applying the Pasic method for solving ordinary

differential equations to the Ideal Adiabatic Stirling model. The model is presented along

with the formulation of the coupled first order differential equations. Also, the Pasic

method is presented along with some of the foundational numerical methods, which the

method is based upon. A "C" program was written to solve the Ideal Adiabatic model

utilizing the Pasic method. An explanation of the logic is given. The Pasic method is

shown to solve the Ideal Adiabatic model and the results are presented. Two areas of

concern are the solve time of the CPU and the error associated with the heat in the

regenerator - it should be zero over a cycle. A minimum time of 0.2 and 0.35 seconds

solves, respectively, the Ford-Philips 4-215 engine and Ross-90 engine with four sub-

domains. Even with the large sub-domains the error of Q, is under 1%. The Pasic

method is 12 and 7.6 times faster respectively for the Ross-90 and Ford-Philips 4-215

engine as compared to the Runge-Kutta method. One reason for the improved speed is

the logic of the program where the energy differentials are solved after the temperature

differentials, thereby eliminating the fixed-point iterations. Further comparison of the

Pasic method with the Runge-Kutta method may be warranted. The hture research

possibilities are significant: the Pasic method as presented deserves hrther research; the

Advanced Pasic method needs to be applied to Stirling analysis; and the Pasic method

applied to partial differential equations deserves investigation.

malroy eric thomas

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