”MWW‘A
L I B B: A P. Y

Michigan Staie
Univczsi'cy

i“

i”

ABSTRACT
A COMPREHENSIVE DIGITAL COMPUTER ANALYSIS
OF
THE POSITIVE DISPLACEMENT BRAYTON CYCLE
By

Raymond E. Trent

One of the most significant challenges now facing the engineer is
the solution of the many faceted pollution problem. This will require,
among other things, the deveIOpment of vehicular power plants that emit
significantly lower quantities of chemical pollutants than those we are
now using.

Currently, efforts to reduce automotive pollution involve either
the modification of our present spark ignition engine or the development
of an entirely different power plant. The alternative engines now under
consideration suffer from high cost, technical problems, performance,
or a combination of these factors.

The objective of this investigation was to study the performance
characteristics of a positive displacement engine Operating on a
modified Brayton cycle in an effort to ascertain whether this engine
would be a suitable alternative to our conventional vehicular engines.
The term modified Brayton cycle is used because the expansion process
occurs over a fixed volume ratio and not a fixed pressure ratio as in
the classical Brayton cycle. The use of positive diSplacement hardware
keeps the desirable low pollution combustion system of the Brayton

cycle gas turbine, but eliminates its expensive turbine, compressor,

Raymond E. Trent

and regenerator. If engine performance is comparable to that of
conventional vehicular power plants, an engine with the desirable
pollution characteristics of the gas turbine could be manufactured
at a fraction of the cost of a gas turbine.

The primary conclusions of this thesis are:

1. Mean effective pressures that are comparable to those of
conventional spark ignition engines can be obtained from
structurally feasible positive displacement Brayton cycle
engines.

2. Minimum brake specific fuel consumptions that are lower, and
part load brake Specific fuel consumptions that are signifi-
cantly lower than those of conventional spark ignition engines
can be obtained from structurally feasible positive displace—
ment Brayton cycle engines.

In one comparison the modified Brayton cycle engine exhibited brake
specific fuel consumptions from .35 to .6 lb./hp.—hr. over the entire
power range while a conventional spark ignition engine had brake
specific fuel consumptions of .5 to 4.0 lb./hp.—hr.

As analysed the engine consists of a piston cylinder compressor
discharging to a constant pressure burner which is followed by a
piston cylinder expander. All compression and expansion processes
were considered to be isentropic. Air during compression was considered
to be a Clausius gas with a "b" constant determined by the author.
After combustion the working fluid was considered to be a perfect gas.
A ten specie equilibrium composition working fluid was used at temp—

eratures above 1500°K, and temperature variable specific heats were

Raymond E. Trent

used in all calculations. Cubic polynomials were developed to
approximate empirical Specific heat data in temperature ranges from
298.15 to 1500°K and from 1500 to 3500°K. All thermochemical data
were taken from the "JANAF Tables."

A general digital computer program (FORTRAN IV) was deve10ped
which permits investigation of the effects of valve pressure drops,
differences between compressor and expander displacements and clearance
volumes, expander intake valve closure, and combustor heat loss and

pressure drOp.

A COMPREHENSIVE DIGITAL COMPUTER ANALYSIS
OF

THE POSITIVE DISPLACEMENT BRAYTON CYCLE

By , .3

Raymond E? Trent

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

1970

PLEASE NOTE:

Some pages have indistinct
print. Filmed as received.

UNIVERSITY MICROFILMS.

 

To my wife, Doris, and my sons, Reed and Blake,

who made it worthwhile

ii

ACKNOWLEDGMENTS

I wish to express my thanks to Professor J. E. Lay as chairman
of my guidance committee for the friendly encouragement he provided
during the course of this study.

I also wish to eXpress my appreciation to Professors C. R.
St. Clair,Jr. and E. A. Nordhaus for serving on my guidance committee,
and Professor J. J. Loeffler for sharing with me his knowledge of
computer programing.

Finally I am greatly indebted to Mrs. Dorothy HOOper, not only
for her invaluable editorial assistance and excellent typing, but also

for her persistent, good humored encouragement.

iii

TABLE OF CONTENTS

LIST OF TABLES O O O O O O O O O O O O O O O O O O O O O O O 0

LIST OF FIGURES O O O O O O O O O O O O O O O O O O O O O O O O

NOMENCLATURE

I 0 INTRODUCTION 0 O O O I O O O O O O I I O O O O O O O O Q

1.L
1.2
1.3

Obj ective O O O O O O 0 O O O O O O O O O 0 0
Review of Previous Investigations . . . . . . . .
Scope of This Paper . . . . . . . . . . . . .

II. THERMODYNAMIC PROPERTIES AND COMPOSITION OF WORKING FLUID

2.1
2.2

2.3

2.4

Air Composition and Equation of State . . . . .
Calculation of Ideal Gas Thermodynamic PrOperties.
2.2.1 Empirical Specific Heat Equations . . . .

2.2.2 Computation of Thermodynamic Functions
and Equilibrium Constants . . . . . . . .
The Composition of the Products of Combustion . .
2.3.1 The Assumption of Chemical Equilibrium. .
2.3.2 The Chemical Composition at Equilibrium .
Determination of Reactant Composition . . . . . .

III. SIMULATION OF THE ENGINE CYCLE ON A DIGITAL COMPUTER .

General . . . . . . . . . . . . . . . . . . . . .
The Cycle . . . . . . . . . . . . . . . . . . . .
Compressor Analysis . . . . . . . . . . . . . .

Burner Analysis . . . . . . . . . . . . . . . . .
Power Cylinder Analysis . . . . . . . . . . . . .
Engine Performance Analysis . . . . . . . . . . .

IV. CALCULATED RESULTS FROM ENGINE CYCLE SIMULATION . . . . .

General . . . . . . . . . . . . . . . . . . . . .
Effect of Compression Ratio on Engine Performance.
Effect of Heat Loss on Engine Performance . . . .
Effect of Valve Pressure Loss on Engine
Performance . . . . . . . . . . . . . . . . . .
Effect of Expansion Ratio and Clearance Volume
on Performance . . . . . . . . . . . . . . . . .

iv

Page
vi

viii

12
12

14
22
22
28
36

38

38
39
48
53
55
64

66
66
70
75
79

91

TABLE OF CONTENTS

4.6 Engine Performance and Concluding Remarks . . . .

APPENDICES .

APPENDIX A:

APPENDIX B:

APPENDIX C:

APPENDIX D:

APPENDIX E:

APPENDIX F:

APPENDIX G:

APPENDIX H:

APPENDIX 1:

APPENDIX J:

APPENDIX K:

APPENDIX L:

BIBLIOGRAPHY

POLYNOMIAL APPROXIMATION BY THE METHOD OF
LEAST SQUARES O O O O O O O C O O O O O O O I

COMPUTER PROGRAM FOR LEAST SQUARES POLYNOMIALS .

POLYNOMIALS FOR CONSTANT PRESSURE SPECIFIC
HEATS . O O O O O O I C O O O O O O I O O .

POLYNOMIALS FOR STANDARD FREE ENERGY OF
REACTION O O O O I O O O O O O O C O I O O O

CALCULATION OF PROPERTIES FROM EMPIRICAL
SPECIFIC HEAT EQUATIONS . . . . . . . . . . .

SOLUTION OF THE EQUATIONS OF CHEMICAL
EQUILIBRIUM O O O C O O O O O O O C O O O O O

DETERMINATION OF THE EQUILIBRIUM CONSTANT (KP)
AND THE STANDARD FREE ENERGY OF REACTION(AFO).

ITERATIVE METHODS FOR SOLVING NON-LINEAR
EQUATIONS O O 0 O O O O O O O O O O O O O O O

SUBROUTINE LISTINGS . . . . . . . . . . . . . .

FLOW CHART AND LISTINGS OF THE MAIN LINE
PRmRAM . C C O O O O O O O O O O O O O O O O

PLOTTER PROGRAM 0 O O O O O O O I O I O O O O O

SAMPLE COMPUTER PRINTOUT . . . . . . . . . . . .

Page
97

105

105

108

112

132

139

146

151

154

156

162
177
180

188

Table

C.l

C.2

C.3

C.4

C.5

C.6

C.7

C.8

C.9

C.10

LIST OF TABLES

Ideal Gas Empirical Specific Heat Equations . .
Fundamental Thermodynamic PrOperties .

Calculated and Tabular Values of Ideal Gas Enthalpy

O O
H -H298 o o o o o o o o o o o o o o o o o 0

Calculated and Tabular Values of Ideal Gas EntrOpy,
at 1 Atmosphere Pressure . . . . . . . . . . . . .

Calculated and Tabular Values of Log of the
.. 10
Equilibrium Constant, Kp . . . . . . . . . . .

Comparison of Available Engine Performance Data . .
Equilibrium Composition for a 15/1 Air-Fuel Ratio .

Thermal Efficiency Conversion to Indicated Specific
Fuel Consumption . . . . . . . . . . . . . .

Data and Equation for C of 02, 298 to 1500°K . . .

"U

Data and Equation for C of 02, 1500 to 3500°K .

O
"U

Data and Equation for of N2, 298 to 1500°K . . .

O
"U

Data and Equation for of N2, 1500 to 3500°K

('3
"C3

Data and Equation for of C02, 298 to 1500°K . .

0

Data and Equation for of C02, 1500 to 3500°K . .

O
'U "U "U "U "B

Data and Equation for of H20, 298 to 1500°K

0

Data and Equation for of H20, 1500 to 3500°K . .

0

Data and Equation for of CO, 298 to 1500°K .

0

Data and Equation for of CO, 1500 to 3500°K . .

"U

vi

Page
15

17

18

20

21

68

88

101
113
114
115
116
117
118
119
120
121

122

Table

C.11

C.14

C.15

C.16

C.17

Data and

Data and
Data and
Data and
Data and
Data and
Data and
Data and
Data and
Data and
Data and
Data and
Data and
Data and

Data and

Equation
Equation
Equation
Equation
Equation
Equation
Equation
Equation
Equation
Equation
Equation
Equation
Equation
Equation

Equation

LIST

for

for

for

for

for

for

for

for

for

for

for

for

for

for

for

OF

'U 'U 'U ’U 'U “U

'U 'U

AF°

APO

AF°

AF

AF°

AFO

Sample Computer Printout

TAB

of

of

of

of

of

of

of

of

of

of

of

of

of

of

of

vii

LES

NO,

0, 298 to

298 to 1500°K .
1500 to 3500°K
298 to 1500°K .
1500 to 3500°K
298 to 1500°K .

1500 to 3500°K

1500°K

O, 1500 to 3500°K .

AIR, 298 to

C02 '* C0 +

:I:
N
O
+
:3
N

<3
N
+

hflhlkflh*hflh4hflh‘
:n
N
+
:2

<3
N
+
C> hflk‘hflh‘

1500°K

Page
123
124
125
126
127
128
129
130
131
133
134
135
136
137
138

181

LIST OF FIGURES

Figure Page
1 Typical Gas Turbine Combustor . . . . . . . . . . . . 24

2 Schematic of a Positive Displacement Brayton Cycle

Engine . . . . . . . . . . . . . . . . . . . . . . 42
3 Typical Compressor Cycle Diagram . . . . . . . . . . 43
4 Typical Expander Cycle Diagram . . . . . . . . . . . 45
5 Typical Engine Cycle Diagram . . . . . . . . . . . . 47
6 System Boundary for Power Cylinder Intake Process . . 56

7 Effect of Compression Ratio on Thermal Efficiency,
500 pSia Burner 0 O O O O O O O O O O O O O O O O O 71

8 Effect of Compression Ratio on Mean Effective
Pressure 0 O O O O O O O O O O O 0 O O O O O O O O 72

9 Effect of Compression Ratio of Thermal Efficiency,
800 and 200 psia Burner . . . . . . . . . . . . . . 73

10 Effect of Burner Heat Loss on Thermal Efficiency . . 77

11 Effect of Intake Process Heat Loss on Thermal
EffiCienC—y o o o o o o o o c o o o o o o o o 0' o c 78

12 Effect of Burner Heat Loss on Mean Effective Pressure 80

13 Effect of Compressor Intake Valve Loss on Thermal
EffiCiency O O I O O O O I O O O 0 O O O O O O O O 82

14 Effect of Compressor Exhaust Valve Loss on Thermal
EffiCiency O O O O O O O O O O O O O I O O O O O O 83

15 Effect of Power Cylinder Intake Valve Loss on
Thermal Efficiency . . . . . . . . . . . . . . . . 85

16 Effect of Power Cylinder Exhaust Valve Loss on
Themal EffiCiency O O O O O O O O O O O O O O O O 86

viii

LIST OF FIGURES

Figure Page

17 Effect of Power Cylinder Intake Valve on Mean
Effective Pressure . . . . . . . . . . . . . . . . . 89

18 Effect of Extending Power Cylinder EXpansion on
Thermal Efficiency . . . . . . . . . . . . . . . . . 92

19 Effect of Extending Power Cylinder Expansion on
Mean Effective Pressure . . . . . . . . . . . . . . 93

20 Effect of Power Cylinder Clearance Volume on

Thermal Efficiency . . . . . . . . . . . . . . . . . 95

21 Effect of Power Cylinder Clearance Volume on Mean
Effective Pressure . . . . . . . . . . . . . . . . . 96
22 Engine Performance Without Pressure or Heat Losses . . 99
23 Engine Performance with Pressure and Heat Losses . . . 100
F.l Flow Diagram for Subroutine PD08CC . . . . . . . . . . 150
J.1 Flow Diagram for Main Line Program . . . . . . . . . . 163

ix

AC

E

A0

AFR

CR

PDl

NOMENCLATURE
Number of mole atoms of carbon
Number of mole atoms of hydrogen
Number of mole atoms of nitrogen
Number of mole atoms of oxygen
Mass air-fuel ratio
Clausius constant
Compression ratio ( ratio of maximum to minimum cylinder volume)
Constant pressure Specific heat
Constant volume specific heat
A constant, F = RT/V = P/N
Gibbs free energy
Enthalpy
Degrees Kelvin
Equilibrium constant based on partial pressures
Total number of moles
Number of moles of air entering compressor
Polytr0pic process exponent
Number of moles of component "i"
Pressure
Partial pressure of component "i"

Amount of heat transfered

Pressure drop in compressor inlet valve

PDZ
PD3

PD4

AF

AHf

ATM

NOMENCLATURE
Pressure drOp in compressor discharge valve
Pressure drop in exPander inlet valve
Pressure drop in expander exhaust valve
Universal gas constant
Entropy
Temperature
Internal energy
Volume
Specific volume
Work
Square root of the partial pressure of oxygen
Square root of the partial pressure of hydrogen

Chemically correct air-fuel ratio devided by the actual
air-fuel ratio

Square root of the partial pressure of nitrogen
Standard free energy of reaction (Gibbs)
Enthalpy of formation

Joule-Thomson coefficient

SUPERSCRIPTS
Ideal gas state
SUBSCRIPTS
Atmospheric
Enthalpy
xi

PROD

IEHCT

289

NOMENCLATURE
SUBSCRIPTS
Pressure
Products
Reactants
Temperature
Internal energy
Volume
Reference state

298.15

xii

I. INTRODUCTION

One of the most significant challenges currently facing mankind
in general, and the engineer in particular, is the solution of our many
faceted pollution problem. If we are to avoid the "ecological dooms—
day" now being prophesied we must, among other things, develop vehicle
power plants that emit significantly lower quantities of chemical pol—
lutants than those we are now using. One power plant receiving con—
siderable attention is the Brayton cycle gas turbine. This type of
engine has excellent emissiOn characteristics with reSpect to carbon
monoxide and hydrocarbons, and has attained thermal efficiencies com-
parable to the major vehicular power plants. The Brayton cycle turbine
appears to have two major obstacles it must overcome before it can be
considered a solution to our automotive exhaust pollution problem.
These obstacles are relatively high cost, when compared to the spark
ignition Otto cycle engine, and higher than desired emission of nitrogen
Oxides. The solution to the oxides of nitrogen problem is certainly
“0 mOre difficult in the case of the turbine than it is for our present
Otto Cycle engine, but reducing the cost disparity will be a formidable
task, The current high price of vehicular turbines is due to the
material, manufacturing, and development costs. This high initial cost
Can Only be justified in applications, such as the trucking industry,
where the turbine's characteristics of low maintenance, long life, and

h 0
18h Power-to—weight ratio can be fully exploited.

 

 

The turbine engine is now making its way into the commercial truck
market but it is envisioned as a slow movement that will hold Diesel
engine production constant for some time before reducing it. To produce
Brayton cycle turbines as a replacement for our automotive spark ignition
engines would require massive retooling expenses, as well as complete re—
education of manufacturing and service personnel. It appears that the
Brayton cycle turbine is not a rapid or cheap solution to our problem;
however, the Brayton cycle applied to different hardware may hold some
promise. The most obvious change in current Brayton cycle hardware
would be to use positive displacement compressors and/or expanders. A
recent study of a reciprocating Brayton cycle engine by Warren and
Bjerklie [4] indicates that the positive diSplacement Brayton cycle does

indeed merit further study.

1.1 Objective

The objective of this investigation was to study the performance
characteristics of a positive diSplacement engine operating on a modified
Brayton cycle. The term modified Brayton cycle is used to indicate that
the application to positive displacement hardware requires the expansion
process to occur over a given volume ratio and not over a given pressure
ratio as in the classical Brayton cycle. No attempt was made to model
any given hardware configuration but instead every effort was made to
develop a general thermodynamic analysis of the cycle by avoiding any
<kfiinition of hardware. This decision resulted in the deveIOpment of a
general digital computer program which allows investigation of the effects
of valve pressure drops, differences between compressor and expander

displacements and clearance volumes, expander intake valve cut off ratios,

and combustor heat loss and pressure drOp. The program also includes
variable specific heat effects throughout the cycle and a ten component

equilibrium composition working fluid at temperatures above 1500° Kelvin.

1.2 Review of Previous Investigations

A complete review of previous investigations must start with the
work of Nicholas Otto in 1876 when be constructed the first successful
gas engine which employed compression prior to the combustion of the
charge. An excellent review of work done between 1876 and 1960 has
been presented by Patterson [3] so only a brief outline of that period
will be presented.

The first efforts to analyze the Otto cycle were based on the work
of Sadi Carnot and evolved into what is commonly known as the Air Stan-
dard cycle. This type of analysis consists of idealizing the engine
cycle processes and Operating with a working fluid that has the properties
Of air at standard conditions. The air Otto cycle was adOpted in 1905
by the Institution of Civil Engineers [5] as the standard with which
measured engine performance should be compared. Measured efficiencies
were from 50% to 70% of the calculated air cycle values. Clerk [6], and
Hopkinson [7] improved the cycle analysis by using improved heat capacity
data and the Hopkinson analysis in 1908 included exhaust gas residual and
quantitatively predicted the influence of fuel-air ratios on performance.

In 1921 Tizard and Pye [8] assumed a working fluid composed primarily
of air during compression, but after combustion and during eXpansion the

working fluid consisted of an equilibrium mixture 02, N C0

H20, CO,

2’ 2?

and H2. The best measured thermal efficiencies were then 80 to 85 per—

cent of those calculated by the methods of Tizard and Pye.

In 1927, Goodenough and Baker [9] presented extensive calculations
on the Otto cycle using an analysis similar to Tizard and Pye, but they
included the effects of mixing exhaust residual with the fresh charge.

In 1935 the first edition of the "Hottel" charts IHHB presented by
Hershey, Eberhardt, and Hottel [10]. These charts eliminated the need
for much of the calculations of Goodenough and Baker and considered the
combustion products to include 02, N2, C02, H20, CO, H2, OH, NO, 0, and
H. These Charts were a significant step in cycle analysis and are still
used in revised and expanded form.

Charts, however, are limited to a given fuel and a specified number
of air-fuel ratios and while they are desirable and useful, they limit
any cycle analysis. In this study the same exhaust products are con—
sidered as the original "Hottel" charts but the use of the digital com-
puter permits the consideration of any hydrocarbon fuel and any air—fuel
ratio. By the mid 1930's the thermochemical aspects of ideal combustion
in the reciprocating engine was well understood and with the Hottel
charts to ease calculations many studies were produced. With the improve-
ments in both engines and calculations the measured thermal efficiencies
were then between 80 and 90 percent of those calculated. The difference
existed because important phenomena, such as flame propagation and heat
transfer, had been neglected and because of the quality of the thermo—
chemical data available.

Since the mid 30'S work has continued with the improvement of thermo-
chemical data and numerous attempts to improve the analysis of the cycle.
Many authors contributed important work during the 40's and 50's (see

Reference 3 for details) but, due to the sheer weight of calculations,

it was not until the advent of the digital computer that investigations
considering the combined effects of variable specific heats, mixture
composition, heat transfer, flame prOpagation, and dissociation were
seriously contemplated. Rapid development of the digital computer and
its application to thermochemical calculations and cycle analysis led
the Society of Automotive Engineers to publish references 11 through 17
in a single volume entitled "Digital Calculations of Engine Cycles."
This excellent collection of papers, published in 1964, contains two
investigations [16, 17] that include the effects of heat transfer and
finite combustion rates on the Otto cycle.

The investigations mentioned have all concerned analyses of an
Otto cycle engine while the subject of this thesis is purported to be
the analysis of a modified Brayton cycle. Brayton cycle analyses, with
the exception of reference 4, have been concerned with applications to
aerodynamic compression and expansion devices and not to positive dis—
placement hardware. Due to this fact, the study of Otto cycle litera—
ture proved to be more germane to the development of the analysis in

this thesis than did the Brayton cycle literature.

1.3 Scope of This Paper

The remainder of this paper is devoted to a description of the
analysis and digital computer calculation of the performance of a positive
displacement modified Brayton cycle engine. Chapter 11 describes in
general the methods used for the determination of the thermodynamic
properties and composition of the working fluid. Chapter III describes
an analytical model for the engine cycle calculation that employs the

methods used in Chapter II to describe the working fluid.

Chapter IV presents results of computer calculations made by using
the analytical model described in Chapter III. The results are presented
as engine performance maps and are compared with the calculated data of
Warren and Bjerklie [4]. Details of all calculations, computer programs

and flow charts, and sample computer output are presented in appendices.

II. THERMODYNAMIC PROPERTIES AND COMPOSITION OF WORKING FLUID

2.1 Air Composition and Equation of State

Atmospheric air contains many substances other than oxygen and
nitrogen but only water vapor exists to any appreciable extent. Since
the concentration of water vapor varies widely in atmospheric air a
reference standard of dry air is assumed for all calculations. The
composition of the dry air used in these calculations was as follows
[18]:

DRY AIR COMPOSITION-MOLES/IOO MOLES

02 20.99 20.99

N2 78.03

A 0.94

C02 0.03

H2 0.01 79.01
100.00 100.00

In this work, as has been done by many investigators, the argon,
carbon dioxide, and hydrogen are assumed to have the pr0perties of
rfitrogen. Having made this assumption, air is considered to be com—
;xmed of 79.01 moles of nitrogen and 20.99 moles of oxygen per 100 moles
Of air. In the combustion process we will then consider that 3.764 moles
Of nitrogen must be introduced into the reaction for every mole of oxygen

derived from air.

In the Otto and Brayton cycle literature reviewed in preparation
for this study, air was considered to be a perfect gas (pv = RT) whose
Specific heat was a function of temperature. In the bulk of these studies
the air pressure, prior to combustion, rarely exceeded 400 psia and the
assumption of ideal gas behavior appeared to be justifiable. In this
study anticipated air pressures can be as high as 1200 psia, and since
precise volume calculations are important in positive diSplacement de—
vices, it was felt that the ideal gas law would not yield the degree of
precision desired. The Clausius gas (p(v—b)=RT) is almost as simple to
handle as the ideal gas and, with an appropriate value for the "b" con—
stant, its use should significantly increase the accuracy of volume
calculations. The "b" constant was calculated to give zero volume error
at a pressure of 1200 psia and 1800° Rankine, which is the approximate
end point of an isentropic compression starting at 537°R and l atmOSphere
pressure. A series of volume checks were then made for pressures from
50 to 1200 psia with temperatures above and below the expected isentrOpic
temperature for each pressure. The error in using the ideal gas law for
volume calculations ranged from essentially zero at low pressures to a
nmximum of 2.7 per cent at 1200 psia, while the Clausius gas gave errors
ranging from zero to .18 percent. This was considered a significant
improvement in the accuracy of volume calculations without undue mathe—
matical complication.

A further investigation of the Clausius gas was conducted to ascer-
tain what effect its use might have on the calculation of energy related

Properties of air.

In the following derivations all basic relationships were taken
from Lay [22] and the number after the equation indicates the page
number where the equation is located in this text.

The effect of pressure and volume on internal energy was evaluated

as follows:

(3%) = TEA) — p <272>

T v

for a Clausius gas

p(v—b) = RT
9.2) =_.Ii_
3T v-b
1)
Therefore
(Bu) ___TR _ = _ =0
3v T v-b p p p
and
(E) “4:9 -p(3-V-) <27»
pT p pT
for a Clausius gas
(131) =3 (3v) __ v—b)
an P 313T p
Therefore
(£3.11) =_IE+(V_b)=IE_IE=O
813T p P p

These results show that the internal energy of a Clausius gas is
“Ct a function of pressure and volume and in this respect it is the same

as an ideal gas.

 

10
The enthalpy behavior of the Clausius gas was evaluated as follows:
3v
dh = deT + [v - T(§T) ]dp (274)
P

for a Clausius gas

(a, =—:

Therefore

dh

TR
C dT + v - ——'d
p [ plp

dh

deT + [v - (v-b)]dp

C dT + bdp
P

It is apparent that unlike the ideal gas, the enthalpy of a Clausius
gas is influenced by pressure. An order of magnitude evaluation was made
and it was found that for an isentropic compression from 537§R and 1
atmosphere pressure to 1200 psia the "bdp" term was slightly less than
one percent of the enthalpy change. Based on this calculation it was
decided to use the ideal gas to calculate the enthalpy air in this in—
vestigation.

An isentrOpic compression process will be used during the analysis
so the following investigation of entrOpy change for the Clausius gas
was made:

Tds = dh - vdp

For a Clausius gas

db 8 deT + bdp
Therefore

Tds = deT + bdp - vdp = deT - (v-b)dp

u
r:
pm
hi
I
1‘
a:
Q.
'o

ds

and

so that

ds = C QI,_ R'QE
PT P

This is the same result as the ideal gas and the Clausius gas does not
interject any complications in entropy calculation.

The analysis of the positive displacement Brayton cycle involves
flow through valves (constant enthalpy processes) so the Joule-Thomson

coefficient was evaluated as follows:

u =‘(13- [Te-X?) - V] (269)

For a Clausius gas

(Hg?-

Therefore

1. TR 1
u-C [I) -V] -C [v-b-v}
P P
b
H = ‘ ET'
P
By definition
’aT)
U = '_—
Aap h

dT = udp = -

12

Calculations indicated a temperature change of approximately .07°
Kelvin for a 10 psia pressure drop and it was decided to consider the
Joule-Thomson coefficient to be zero for air under the conditions of
this investigation.

The result of this analysis was the decision to treat air as an
ideal gas in all respects except for its equation of state which will
be that of a Clausius gas with a ”b" value of .02711 liter per gram
mole. All products of combustion will be treated as ideal gases in

every reSpect.
2.2 Calculation of Ideal Gas Thermodynamic Properties

2.2.1 Empirical Specific Heat Equations

In an engine cycle analysis of this type the thermodynamic
properties of a number of different compounds under a variety of con-
ditions must be known in order to compute the apparent properties of
the mixture of gases that constitutes the cycle working fluid. The
thermodynamic prOperties of pure compounds are normally presented in
tabular form or calculated from equations based on the tabular data.
It is now common practice in cycle analysis, when possible, to utilize
equations to calculate property values. The use of equations saves the
large number of computer storage locations needed for tables and obviates
the need for interpolation routines. Almost every thermodynamics text
published in the past 10 years has presented a number of specific heat
equations which have been gleaned from various sources. It was decided
to develop a set of equations for this study from the most consistent data
available because the valid temperature ranges and accuracies of the text

book equations of heat capacity vary widely. The data source selected was

13

the "JANAF Tables” [19] which were compiled and published by the Joint
Army—Navy-Air Force Panel on Thermochemical PrOperties. These tables

are an extension of previous National Bureau of Standards work and are
considered by most investigators to be the most comprehensive and re—

liable data currently available [11].

It was demonstrated by Patterson [3] that cubic polynomials could
adequately approximate empirical Specific heat data in temperature ranges
from 298.15 to 1500°K and from 1500 to 3500°K. Patterson [3] used the
method of least squares to develop equations to approximate data from
National Bureau of Standards Circular 564 [20] and American Petroleum
Institute Report 44 [21].

In this study a general library program for least squares curve
fitting was used to approximate specific heat data from the "JANAF
Tables" [19]. The method of least squares curve fitting is outlined in
Appendix A and the program used is presented in Appendix B. The constants
(a's) were calculated to five significant figures and the equations for

each substance are of the form:

C =a +aT+aT2+aT3
p o 1 2 3

Table 1 gives the a's for each temperature range, the maximum per
cent deviation from the tabular value, and the standard error of the
estimate (defined in Appendix A) for each of the components involved in
this analysis. The Specific heat data used to develop the equation for
air shown in this table was obtained by using a mole weighted average
0f Specific heats of oxygen and nitrogen. The apparent Specific heat

of air was evaluated at all temperatures for which oxygen and nitrogen

14

data were available and was based on the composition of air presented
in section 1 of this chapter.

The greatest single point deviation in the 298 to 1500°K range was
.70 per cent with most points significantly better than this value. The
greatest Single point deviation in the 1500 to 3500°K range was .064
per Cent. This value is within the range of accuracy quoted for the
tables used. The computer print out from the curve fit program is pre—
sented in Appendix C and gives data input values, calculated values,

differences and per cent differences.

2.2.2 Computation of Thermodynamic Functions and Equilibrium Constants
Values of the ideal gas thermodynamic functions and equilibrium
constants can be computed in a number of different ways. The most
obvious method is to calculate all functions and the equilibrium con—
stant from empirical heat capacity equations. One system utilizing this
method arranges the required data in matrix form and reduces the cal—
culations to a series of matrix multiplications. This procedure was
considered for use in this analysis until it was found that a more precise
calculation of the equilibrium constant could be made by developing
equations to approximate the tabular values of the Gibbs function than by
obtaining equilibrium constants from the Specific heat equations. This
discovery, coupled with the fact that the (g? v and the(%%)p can be easily
calculated by an alternate method, led to the decision to make the cal-
culations in the Simplest (mathematically) straightforward manner possible.
This decision was also an example of serendipity.' The author teaches

undergraduate thermodynamics and the straightforward subroutines developed

in this study can be utilized in his courses Since the thermodynamic

15

 

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o
MolmHOE\Hmo {and + «me + Haw + m I o

mone<=cu Ham: QHEHummm q<oHeHmzm mew q<moH

H wands“.

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sooo.o oao.o- aaumommka.o moumoqokn.o- Scum~m00~.o- Ho+ms-om.o o
HNoo.o oao.ou oH-mSH~ma.o oo-m~mems.o- Noumsmoma.o Ho+mnoafin.o oz
oaoo.o -o.o- oaumawaam.o Soummwmom.on Noumwowm~.o Ho+mehoqn.o so
Smoo.o mmo.ou oaum~flamm.o soummaoqs.o- Noummmaaw.o Ho+maommm.o N:
«Noo.o «mo.on oaumnmmem.o ooumwaamm.ou Noumomoma.o Ho+moooao.o oo
o~oo.o «mo.ou ooumwooma.o moumhasqa.o- No-mmaomm.o Ho+mM~emm.o cam
maoo.o «so.o- caumkmmoa.o soumnmomm.o- Noumamflqm.o ~¢+mwoaoa.o Noe
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moo.o no.o ooumwaowa.ou moummwom~.o Noumsmom~.o- Ho+mowooa.o mo
kHo.o om.o moummqmqm.o ooamammam.o- moumaqqmm.o Hc+mmmomo.o N:
sao.o oc.o- moumkmaoa.o- nonmamqow.o mo-mHHo~m.o- Ho+mkoamo.o ou
moo.o -.o- moumommoa.ou moummnaom.o Soumoomom.o Ho+mqasna.o ON:
mmo.o mm.o moammmom~.o moumomamo.ou HOImmmmma.o Ho+mkmmam.o moo
mao.o om.o- mormqoqaa.o- mo-m-qom.o ~o-momqaa.o- ~o+mwoso~.o ~z
omo.o So.o- oaumhoomn.o- ooummooaa.o- ~oumm-a~.o ao+meamao.o No
nouum .vum .>0a N.xaz me an an egg moz<smm=m
a AxooonHISSNV

 

l6

principles are unobscured by programing or the mathematical techniques
used.

The calculation of thermodynamic functions requires the use of basic
property information and unless otherwise noted this data was taken from
the "JANAF Tables" [19]. Table 2 is a listing of enthalpies of formation
and other fundamental data used in this study. Table 3 lists the cal-
culated and tabular values of enthalpies for temperatures ranging from
298.0 to 1500.0°K. The maximum deviation of the calculated enthalpy
value from the tabulated value in this range of temperatures is 0.4 per
cent. The comparison for the 1500.0 to 3500.0°K range is not presented
since the maximum per cent deviation in this high temperature range is
an order of magnitude less than the low temperature range. The value
of the calculated enthalpy at 1500°K using the high temperature range
e<luations is identical to the tabular value. This leads to a minor
enthalpy discontinuity at 1500°K which could be removed by adjusting
constants. This discontinuity was not removed because it was considered
negligibly small and its removal would have affected the accuracy at
Other Points.

The enthalpy values were calculated in the following manner:

Where
0 O T
H - H = J C dT
o P
a
nd To * 298.15°K for the low temperature range and To = 1500.0°K for
O

t
be high temperature range. The quantity (H; — H298) is the tabular
0

Val
ue at the apprOpriate "To."

l7

 

TABLE 2

FUNDAMENTAL THERMODYNAMIC PROPER'I‘ IES

SO

SO

H

O

H

0

ABC

 

Specie Specie 298 o 1500 o 1500' 298 f298
umber cal/mole— K cal/mole- K kcal/mole kcal/mole
1 C8H18 - - - -59.740*
2 02 49.004 61.656 9.706 0
3 N2 45.770 57.784 9.179 0
4 co2 51.072 69.817 14.750 —94.0540
5 H20 45.106 59.859 11.495 -57.7979
6 co 47.214 59.348 9.285 -26.4165
7 H2 31.208 42.716 8.668 0
8 on 43.918 55.594 8.800 9.330
9 NO 50.347 62.763 9.496 21.580
10 0 38.468 46.642 6.046 59.559
11 8 27.392 35.419 5.971 52.102

_..‘_

*Reference 31

All other values Reference _1_9

 

00:00; 0603000

mosam> woumasoamu

 

 

000.0 000.0 000.0 000.0 000.0 000.0 000.00 000.00 000.0 000.0 0.0000
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000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.00 000.0 000.0 0.0000
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.00 000.0 000.0
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 0.0000
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 0.000
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 0.000
000.0 000.0 000.0 000 0 000.0 000.0 000.0 000.0 000.0 000.0
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 0.000
000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0
000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 0.000
000. 000. 000. 000. 000. 000. 000. 000. 000. 000.
000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 0.000
000.- 000.- 000.- 000.- 000.1 000.- 000.: 000.- 000.: 000.-

0 0 oz 00 00 00 000 000 0z 00 00.0 0500

mHoE\Hmox "mafia:
000

 

 

00 u 00 .00000020 000 00000 00 000000 0000000 020 0000000000
m H.329

 

19

Table 4 is a listing of the calculated and tabular values of the
entrOpy, SO, at one atmosphere pressure for the temperature range from
298.0 to 1500.0°K. The maximum deviation of the calculated entropy
value from the tabular value in this temperature range is 0.032 per cent.
The comparison for the 1500.0 to 3500.0°K temperature range is not
presented because the maximum absolute deviation from the tabulated value
is .001 cal/mole-°K. In the high temperature range over 80 per cent of
the calculated values were identical to the tabulated values and the
maximum per cent deviation was .0027 and occurred for monatomic hydrogen.

The entrOpy values in Table 4 were calculated in the following

manner :
o = o _ 0
ST (ST ST ) + S
o o
where
. T 92
ST — ST = J 'T dT
o T

The cluantity S0 is the tabular value at the apprOpriate "To."

T
o

A-listing of calculated and tabular values of the log10 Kp for

selectexl temperatures is shown in Table 5. Three ways for calculating

the equilibrium constants were checked before a method was selected for
use in tfliis study. A linear approximation of the tabular values of
K- a calculation using the specific heat equations, and a least—

10 p’

Squares, third-degree, approximation of tabular Gibbs function (AFC)

10g

d .
ata were all investigated before a decision was made. The most prec1se
m
Ethod IJroved to be the third degree approximation of the Gibbs function.
Th
e equilibrium constant was then calculated from the following equation:

Kp = exp (—AF°/RT)

20

000060 0603060

mooam> wmumHSono

 

 

000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.0000
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 6000.00
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.0000
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.0000
000 00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.000
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.000
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.000
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.000
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 0.000
000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00 000.00
0 0 0 0 0
0 0 oz 00 0 00 0 0 00 z 0 0000 0000
00000000 0000000200 0 00 .00 .0000020 000 00000 00 000000 0000000 020 0000000000

00.-m0650006 "000000

.0 0001000de

 

21

 

CALCULATED AND TABULAR VALUES OF LOG10 OF

TABLE

5

THE EQUILIBRIUM CONSTANT, Kp

 

Reaction Temp (°K) 1500.0 2000.0 2500.0 3000.0 3500.0
CG + 1/2 02 = 002 5:358 2:9? 1:235 0;??? "of?"
5.7255 3.5404 2.2237 1.3430 0.7129
H =
2-+.1/2 02 H 0 5.725 5.540 2.224 1.345 0.712
—O.5586 -o.2358 -o.0457 0.0783 0.1648
1 =
/2 H2_'+ 1’2 02 0“ -0.559 —0.236 -0.046 0.078 0.165
-2.4889 -1.6990 -1.2268 —o.9131 -o.6904
1 2 =
/ N2'+ 1/2 02 No —2.487 -1.699 -1.227 -0.913 -0.690
112 0 := 0 —5.3954 —3.1779 -1.8423 -o.9493 -0.3106
2 “5.395 ”30178 "10842 ”00949 ”00310
112-H .0 H -4.7566 -2.7907 -1.6015 -o.8035 —o.2313
2 —4.757 -2.791 —1.601 -0.803 -0.231

5

Calculated Values
Tabular Values

22

The computer output from the least squares curve fit program is
given in Appendix D and lists the constants for third degree equations,
the tabular data input, the calculated values, the difference, the per

cent deviation, and the standard error of the fit.

The equations for the Gibbs function as a function of temperature
were derived only for the temperatures between 1500.0 and 3300.0°K. The
lower limit was set because at temperatures below 1500.0°K the work fluid

will be considered to have a constant composition and equilibrium cal-

culations are not required. The upper temperature limit was reduced

fnm113500.0°K to 3300.0°K to improve the precision of the approximating

equatjxyns. This temperature reduction was justified because it had been

deteruuined that 3300.0°K was sufficient to cover the temperatures that

would be encountered in this study.

Ekatails for the calculation of enthalpy, entropy, the (gng and the

gas

‘Silp Eire presented in Appendix E. The details for the calculation of

the eqinilibrium constant are presented in Appendix G.

24§__11§;_Composition of the Products of Combustion

2°3¥1 'The Assumption of Chemical Equilibrium

fitll of the techniques have been deve10ped which are required to
calculatne the thermodynamic properties of a system whose chemical compo-
sitHMJ is known. We must now investigate high temperature combustion
in ordeflf to establish the chemical composition of the combustion products.
In Cy¢143 analysis the conditions under which combustion occurs are always
idealizEuj for the purpose of calculation. The assumptions of steady

st
ate, lHomogeneous composition, and absence of all wall effects normally

23

made in Otto cycle analyses will be made in this study. These assump-
tions are made to permit use of the methods of steady state thermo-

dynamics and a discussion of their validity and effect on calculated

performance follows.

Homoggneity and Wall Effects

Numerous works ([3], [23], [24], [25], and others) have established
that the working fluid in an Otto cycle engine is not entirely homo—
geneous. The complex processes involved in adding fuel to the air stream,
inducting the charge (fuel air mixture) into the cylinder, and mixing the
charge with the residual (products of combustions still in the cylinder
after the exhaust process) can lead to a combustible mixture that is not
homogeneous. If this nonhomogeneous mixture is then ignited, a flame
front will progress through rich and lean volumes of mixture and the
resulting product mixture will be hetrogeneous. In addition to the non-
homogeneity produced by fuel distribution, we also have nonhomogeneity
produced by the extinction of the flame (the quench or wall effect) as
it nears the chamber wall. The inhomogeneity in the unreacted mixture
can be reduced or eliminated by more efficient mixing but the quench
effect is more difficult to eliminate and is the source of a major portion

of unburned hydrocarbon in the exhaust.* There are two sources of

\

 

:itui: now generally accepted that the quench effect is the major producer
motif: urrled hydrocarbons in the automotive exhaust. At the 1970 SAE Auto—
that e Ecgineering Congress and Exposition, Daniel [26] presented a paper
chambrevlews some of the work in this area and attempts to model the engine

er environment, including quench, in an effort to calculate emissions.
from in 1957 the author's [27] measurements of hydrocarbon emissions
probab‘irbulent and partially quenched flames indicated that quench was

y the most important factor in producing the unburned hydrocarbon

in a
utomOtive exhaust gas.

24

unburned hydrocarbon resulting from wall quenching of the flame, (l) the
walls in the Open volume of the combustion space, and (2) the crevices
separated from the open part of the chamber by restrictive passages. An
eXMple of the second source would be the volume behind the tOp ring of
a piston. It appears that to solve the problem of incomplete reaction
due to wall quench we must eliminate the walls, prevent reactive mixture
from contacting the walls, or devise a method to react the quench volume
gas later in the cycle. Of course we cannot completely eliminate the
solid boundaries from any engine and difficulties in implementing the last
two methods of reducing quench effects in the Otto cycle engine are part
of the reason this study was undertaken.

In the modified Brayton cycle the combustion process is quite
different from that of an Otto cycle engine. It is envisioned as being
esSentially the same as the Brayton cycle gas turbine, except for higher
operating pressures and the possibility of cyclic pressure variations.

A Sketch of a typical gas turbine combustor is shown in Figure 1.

fuel

\ / Cooling \Air\_. \ \ ‘

A// Primary I

Secondary -—--h- I

\ Reaction Zone I

V Zone

WWW/V I

._._.. 5

air: .___._

FIGURE 1. TYPICAL GAS TURBINE COMBUSTOR

 

25

It is obvious from this sketch that we are dealing with a diffusion

flame to a greater extent than a premixed flame. (The turbulence of the

flow prevents defining the reaction as a pure diffusion flame.) It is

also apparent that the wall quench is significantly reduced or eliminated

by the flow of air along the combustor liner and by the stabilizing of

the reaction zone away from solid boundaries. As the gas leaves the

primary reaction zone, preheated excess air is added and the reaction

can continue to completion. If the burner is properly designed, turbulent

mixing of secondary air and the exhaust from the primary reaction zone

results in a near homogeneous mixture. The overall air—fuel ratio for

tries gas turbine is considerably greater than the Otto cycle and this will

also be true for the positive displacement Brayton cycle. These factors

should contribute to lower CO and hydrocarbon emissions for the Brayton

CYCle than for the Otto cycle, and this is confirmed by research. Brayton

cycle turbines produce 1 to 10 per cent of the carbon monoxide, and l to
20 Per cent of unburned hydrocarbons of an uncontrolled spark ignition
engine [2]. This comparison is made on the basis of mass of exhaust
Specie per mass of fuel, and an "uncontrolled engine" indicates an engine
not equipped with our recently developed emission control devices.

It

seems apparent that assumption of homogeneity of mixture (at plane 5,

F—
lg‘l‘li‘e 1) is certainly better for the Brayton cycle device than the Otto

c
Yele and may, in fact, be a reasonably precise one.

W

Assumption of steady state implies that a system (the combustion

Dr .
oducts) has remained in a given state of temperature and volume long

26

enough to reach a homogeneous condition and that all chemical reactions
have proceeded to the point where the forward and reverse rates are
equal. This then is a condition where the Gibbs function is a minimum
and are therefore in a state of chemical equilibrium. Since we are in
a state of equilibrium the composition can be determined by the methods
of classical thermodynamics. The critical item in this assumption is,
of course, time. If the time available for the attainment or adjustment
of equilibrium is short, or the reaction rate low, then true equilibrium
may not be reached.
Residence time is the time during which a system is maintained at
constant condition and, strictly speaking, would be zero in any engine.
The state of any given quantity of working fluid is continuously
changing in an engine and the question becomes one of comparing rate
Of Change of state to the specific chemical reaction rates. The Specific
reacition rates are functions of temperature and it is generally accepted
that at the end of the combustion process the temperature is high enough
to insure equilibrium. This assumption of initial equilibrium is supported
by InIflnerous authors including Starkman and Newhall [28] with respect to
Spark ignition engine cycles, and by Cheng and Lee [29, 30], with respect
to I‘Ocket engine calculations. The main questions associated with the
assumption of equilibrium then become, at what point during an eXpansion
process do we start getting significant deviations from equilibrium?
what criterion do we use to establish this point?, and how do we handle
QOmPOsition once we have reached this point? Cheng and Lee [29, 30]
descr:lbe work on two models for the expansion process in supersonic

r0
(”‘9': nozzles; the first, an eXpansion that consists of an equilibrium

 

Efffi
v. 2...

27

composition to some point at which the composition is "frozen" and then
the expansion proceeds with this constant "frozen" composition, and the
second model takes equilibrium composition expansion to a point where the
reaction rate equations take over from the equilibrium constant and de-
termine the composition for the remaining portion of the expansion.
Prior to the mid 1960's the expansion processes had been considered to
have a "frozen" composition, an equilibrium composition, or an equilibrium
composition followed by frozen composition with changeover point being
determined by the temperature of the system. Consideration of the re-
action rate constants began very recently and then only for simple
Systems. Treating the expansion as equilibrium followed by frozen
composition has been the most pOpular system used in recent cycle anal—
YSis - In 1964 Patterson and Van Wylen [16] published the results of an
0t tO cycle analysis which included heat transfer, flame propagation,
equilibrium composition above 1500°K and "frozen" composition below
1500°K. This was an effort to simulate the operation of a given engine
and the measured thermal efficiency of the engine was 98 per cent of the
c
alculated value at one Operating condition. This is not to imply that
al
1 of the problems in Otto cycle analysis have been solved because other
Cal . .
Culated parameters did not demonstrate this degree of prec151on. It
d°es . . . . .
’ however, indicate that the assumptions on chemical compOSition are
not ‘
gIrossly incorrect and do, in fact, provide a good thermodynamic
repr
esentation of the working fluid.
After evaluating the available literature, and the combustion and
expans - . . .
lon processes involved in this analysis, it 18 felt that the
3.88%
13tion of chemical equilibrium is more justifiable in this analysis

tha
1'1
in either rocket or Otto cycle work where it has been extensively

 

 

’Q Its
'. c
.J“.

28

applied with generally good results. Based on the previous discussion,
this study will consider chemical equilibrium composition of the working
fluid above 1500°K and a "frozen" composition below 1500°K. The methods
of calculating both of these compositions will be discussed in the next

section of this investigation.
2 - 3 - 2 The Chemical Composition at Equilibrium

General

 

The first step in devising a method for calculating the composition
of the products of a chemical reaction is to determine what species will
be present in quantities considered significant to the achievement of
the objective of the calculation. The objective of the equilibrium
calculation in this study is to determine the thermodynamic properties
of the working fluid in an effort to analyze the performance of the
POSitive diSplacement Brayton cycle. It is apparent then that we need
COtlSider only those of the possible species that will be present in
SuffiCient quantities to affect the thermodynamic behavior of the mixture
and We need not consider a specie simply because of toxicity, corrosive-
Hess Or other undesirable characteristic. With the criterion for in-
clusion of a specie in our calculations established we must then determine
which of the many possible reaction products need to be included. In
general this determination may require trial calculations, a literature

SQa
rch or both. In the case of the hydrocarbon air system with which

We
a
1:63 concerned there is an abundant supply of literature available and

tria
3‘ Calculations are not required. While any one of a number of papers

Col-ll
d Provide the information needed, the one used in this study was

 

29

publ ished in 1964 by KOpa, Hollander, Hollander, and Kimura [12]. In
this excellent study 18 possible products from a hydrocarbon—air reaction
were considered and equilibrium gas compositions were calculated for a
wide range of pressures, temperatures and air—fuel ratios. The possible
products considered in this study were 02, N CO

2, 2, H20, co, H2, OH,

NO, 0, H, N, C, CH“, 03, N02, NH3, HN03, and HCN. This study provided
.ac1i_£1t)eatic flame temperature data that indicate temperatures will not ex—
ceaeeci 23100°K in the study and that at 3000°K only the first eleven of the
‘rteaaczt:zants considered have mole fractions that exceed 10'5. Of the eleven
:reaza<:t:£ants that might reasonably be eXpected to influence the thermo—
<13711£inntic properties of the working fluid, it was found that the mole
f1r21crtxion of atomic nitrogen (N) does not exceed 10"+ and it was removed
3EITCIDJ (:onsideration. As a result of the above evaluation we have reduced
Ollir Iltlst of significant products to the 10 that were used by Hershey,
jEbelfliardt, and Hottel [10] in 1936 in the first edition of the so—called

'I
I31‘3’ttel Charts." The products and reactants considered in this study

are liSted below.

REAC .
TANTS. C8H18 , Air (02, N2)

1?I{()I)UCT3; 02, N2, CO2, H20, CO, H2, OH, NO, 0, H

3:11 order to determine the amount of each specie present a system

Of l
C) EEquations must be solved. One set of conditions that must be

satiSfi . . .
ed in a reaction is that the mass of each atomic speCie remain

For the C, H, N, 0 system this requirement of constant mass

iczt
a O O I C
tlets the use of four mass balance equations. The remaining Six

3O

equations in our system then become equilibrium equations. Some of the
equilibrium equations are nonlinear and we are forced to use some type
of an iterative solution, because an explicit solution cannot be obtained.

(3ver the years many solution techniques have been developed and a
number of them were reviewed for use in this study. Some of the tech—
niques are very general and are capable of treating a large number of
reactant and product species in the liquid, solid and gaseous phase. In
1951 Huff, Gordon and Morrell [31] presented a solution of this type which
is the basis for most of the general solutions used since its publication.
The literature in this area also revealed a number of solution techniques
that are restricted to the C, H, O and N system used in this study. One
SUCh technique was presented by Patterson [3] and involves only the ten
prOducts felt to be important in this work.

The method used by Patterson [3] was based on the work of Ritter
V011 Stein [32] and Schmidt [33]. This technique was selected as being
the most compact and fastest available for computer calculation of the
equilibrium composition of the ten specie system used in this work. This
method is presented in the next section of this paper and the author's
only Contribution is the correction of minor typographical errors in the

O):
1g 1Tit-11 work. The presentation and application techniques are, of

C011
rse’ the author's.

CQm
Wtion of Equilibrium Composition

The following assumptions have been stated or implied and are
esse
ntial to the method of computation which follows:
(1) The products of the reaction are in a state of chemical

equilibrium.

 

31

(2) All product species are considered to be ideal gases.

(3) The products are homogeneous with respect to both
composition and properties (i.e., no property gradients).

(4) The fuel is a known pure hydrocarbon.

(5) Air is a defined mixture of O2 and N2.

(6) Only the 10 previously defined species exist in the product
mixture.

TFhe general reaction equation can be written as:
can +59(0)+¥A—1‘1(N)+n(0)+n(N)+n (CO)+n (H0)
11 1121 2 2 2 2 02 2 N2 2 002 2 H20 2

-+- nCO(CO) + nH2(H2) + n0H(OH) + nN0(NO) + n0(0) + nH(H) (2—1)

Vatleairee AC, AH, A0 and AN are the mole atoms of carbon, hydrogen, oxygen
aIIC1 Incitrogen respectively in the reactants and the "n's" are number of
tnc>3L€3£3 of the various product species.

TFhe number of mole atoms of an element in the reactants must be
ecllléiJL to the mole atoms of that element in the products, considering

C51 .
rbon we can then write:

AC = 116,02 + n00 (2-2)

a171d . . . .
reStating the relationship in terms of the partial pressures we

AC V P ) (2-3)

= if “’00, + co
hfi1e=
ITEE ‘7 and T are the volume and temperature of the system and R is the

gas
(:(3tlstant per mole. We can now derive the following four mass balance

 

E? :1"

32

eq uations:

F . AC = (:02 + co (2-4)
F o AB = 2H20 + 2112 + OH + H (2-5)
F - A0 = 2002 + H20 + 202 + OH + NO + co + o (2—6)
F - AN = 2N2 + NO (2-7)

where F = RT/V and specie formulae are interpreted to be the partial

pressure of the particular specie. It should be noted that F is also

 

equal to Pm/Nm where Pm is the mixture pressure and Nm is the total
nimber of moles of products. Each of these definitions of F will be
utilized in calculations depending on whether the system volume or
pressure is known for a given state.

The next consideration is the development of six equilibrium

e<1ll€=1tions and to accomplish this the following reactions will be

ut 11 ized:

(1) (:02 + C0 + 1/2 02
(2) H20 + H2 + 1/2 02
(3) 0H + 1/2 H2 + 1/2 02
(4) NO + 1/2 N2 + 1/2 02
(5) 0 + 1/2 02

(6) H + 1/2 H2

We
can write the equilibrium constants in terms of the partial pressures

and b
y letting 02 = X2, H2 = Y2 and N2 = 22 they have the following form:

= co - x

1
CO2

K (2-8)

33

 

 

K2 = Y_2§-_O_)g (2-9)
2

1‘3 = Yo}; X (2-10)

K1+ = 2M; X (2-11)

K5 = 23. (2-12)

K6 = {1' (2-13)

Equations (2-4) through (2-13) must now be solved for our 10 partial
Pressures.
By substituting the equilibrium equations (2-8) through (2—13) into

the mass balance equations (2-4) through (2—7) we have:

F - AC = c0(1 +E)5—) (2-14)
1
2
F-AH=2(Y2+l—X)+fl+—Y— (2-15)
K K K
2 3 6
F~Ao=2(99——)5)+3—’$+2x2+33—+-Z—)5+co+l (2-16)
K K K K K
1 2 3 1+ 5
F . AN= 222 +fz} (2-17

1,

Solving Equation (2-17) for the positive value of Z, and Equation (2—15)

for the positive value of Y yields

z= «WW—K— “ME AN (2-18)

(L+-1—) +\/(—)-(—+—1-)2+8(1+-Z(—) - F - AH
K3 K6 K3 K6 K2
Y I X (2-19)
4(1 +-)
K2

 

 

 

A .

34

The substitution of Equation (2-14) into Equation (2—15) yields:

_ _x_ .._- __F_._:_A_c .Y__2x 2 x_Y a i
F AO- (K1+X) +K2+2X +K3+KH+F AC+K5 (220)

If we observe from Equations (2—18) and (2—19) that Y and Z are functions
of X only, then it is apparent that Equation (2-20) is a function of X
only. If F, the system temperature, and the mole atoms of reactants are
known, then we need only obtain a positive value of X which satisfies
Equation (2-20). A method of finding this value of X is presented in
Appendix F.

Once the quantity X has been determined, Y and Z can be evaluated
from Equations (2-18) and (2-19). The partial pressures can be calculated
from the carbon balance, the definition of X, Y and Z, and the equilibrium

e quations as follows:

02 = x2 (2-21)
N2 = 22 (2-22)
(302 = F - AC ° X/(Kl + X) (2-23)
1120 = XYZ/K2 (2—24)
co = co2 . Kl/X (2-25)
Hz a Y2 (2—26)
OH = XY/K3 (2-27)
N0 = XZ/K” (2-28)
0 = X/K5 (2—29)
in = Y/K6 (2—30)

The number of moles of each Specie can now be simply calculated

b
y dividing the partial pressure by the factor "F."

 

35

This method of determining equilibrium compositions was designed
for use where the system volume and temperature were known; therefore,
the "F" factor was known. In this particular study calculations of
equilibrium compositions at a defined pressure were also required for
the analysis of the positive diSplacement Brayton cycle. Once the
multiplicity of definitions for the "F" factor is recognized, however,
the development of a rapidly converging method for calculating equi-
librium composition presents no great problem. Briefly the system used
consisted of utilizing the fact that "F" equals the total mixture pressure
divided by the total number of moles in the mixture. The factor "F"
Was first estimated by using the desired mixture pressure and the total
I'1"--1‘Inber of moles of mixture calculated by assuming no dissociation of the
Products. This estimate of "F" was then used to calculate the partial
Pressures and number of moles of the Species present in the product
mixture. The sum of the calculated partial pressures was then compared
to the desired pressure and if the difference between these pressures
did not meet a defined limit of acceptability, then a new value of "F"
was determined using the desired pressure and number of moles of products

calGillated from the previous value of "F". The process was then repeated

until the calculated pressure and the desired pressure were within a

Specified amount of each other. The desired result was usually achieved

w
ith two or three iterations; only very rarely was a fourth iteration

re(“lirech

 

36

Frozen Equilibrium Comgosition

The products of combustion are considered frozen for all processes
l)€3:1uC)VJ 1500°K. This means that a composition is determined and that it
remains constant and unaffected by temperature changes below 1500°K.
thijlss study considers only fuel lean and chemically correct mixtures,

Eitlci eat temperatures equal to or below 1500°K only 02, N C02, H20 are

2’

215353113ned to be present. The mass balance equations are now sufficient

t1C> (isstermine the composition and can be stated as follows:

n00 = AC
2
n = AH/2
H 0
2
n0 = AO/2 - AC - AH/4
2
n = AN/2
N2

UDI1€3 {Dartial pressure of each constituent can now be calculated by multi—

F33-3721I1g the number of moles of the constituent by "F."

2£:_5£____Determination of Reactant Composition

The preceding two sections have considered the determination of
comI><)sition when the mole atoms of carbon, hydrogen, oxygen and nitrogen
aITEE Icnown. This section describes the method used to determine these
‘Véileless in this study. One of the primary variables in this study is air—
f11‘3:L- ratio and we must then relate the air-fuel ratio to the number of
m()]_€3 atoms of reactants. For reasons that will become apparent in sub-
sequ€nt sections of the paper it was decided to hold the number of mole

a"20:3 .
““53 of oxygen and nitrogen constant for all calculations and to vary

til
E3 flle1 quantity. Since only normal octane is used as a fuel in the

37

study, the following stoichiometric reaction equation was used as a basis

fo r all calculations:
C8H18 + 12.5(02 + 3.764N2) + 8C02 + 9H20 + 47.05N2 (2—31)

For this reaction the number of mole atoms of oxygen is 25 and the number

of mole atoms of nitrogen is 94.1. These values we then used and AC - 25

and AN = 94.1 for the combustion reaction. These values are, of course,

apprOpriately modified for the inclusion of exhaust gas residual during
the expansion in the power cylinder but are used for all calculations in
tlTle compressor and combustor of the engine. The mass air—fuel ratio for

Equation (2—31) is 15.042/1. Since the ratio of the stoichiometric air—

fLlel ratio to any other air-fuel ratio is the same for both mass and

mole ratios we can write our reaction equation as

(2—32)
Y1 ° C8H18 + 12.5 02 + 47.05 N2 + Products

Where

Y1 = 15.042/Actual mass air—fuel ratio (2-33)

We See that we can establish values for both the mole atoms of carbon

(AC) and the mole atoms of hydrogen (AH) based on the actual air—fuel

ratios as follows:

AC Yl°8

AH

Yl°18

I - .
t 18 eVident that we can now determine the mole atoms of reactants

e
Iitering our combustion process by specifying the air—fuel ratio per

unit mass.

 

III. SIMULATION OF THE ENGINE CYCLE ON A DIGITAL COMPUTER

3 - 1 General

During the course of this study a total of four computer programs
were written. The first program was a simple analysis of the classical
Brayton cycle extended to higher pressure ratios than are normally avail—
able in the literature. This analysis considered working fluid to be
an ideal gas with constant specific heats and the prOperties of air
Standard conditions. This program served only to familiarize the author
With FORTRAN IV and the IBM 360 computer used in this study.

The second program written was again an ideal gas constant specific
heat analysis to ascertain the steady state burner pressure that could
be achieved on motoring the engine without heat addition. This analysis
indicated that pressure levels adequate to start the engine could be
Obtained. The basic cycle used was the same as the one discussed in
detail in section 2 of this chapter.

The third program in the series utilized an ideal gas-constant
SPQCific heat working fluid with the prOperties of air at standard con—
dit ion. This program was written to obtain a better knowledge of the
probable operating characteristic of the engine and to attempt to define
what form the final program should take. In this program the engine
displacement, (volumes of compression and eXpansion Spaces) clearance
vold-uues, valve openings, and burner pressures were defined. To obtain a

St
able operating condition then required iteration to determine the heat

38

39

addition required to match mass flow rates through the compressor and
expander sections of the engine as well as iterations for each quantity
of heat added to determine the effect of exhaust residual.

While the author was successful in writing the program, it appeared
that the inclusion of a realistic working fluid with specific heats vary-
ing as a function of both temperature and composition would lead to ex-
cessive computer time if the task could be accomplished at all. At this
point a revaluation of the procedure used led to the development of the
relatively simple system used in the fourth and final program in this
Series.

The analysis used in the develOpment of the final program of this
Series will be presented in detail in the subsequent sections of this
Chapter and a listing of the program is presented in Appendix J. The
fj-I‘St three programs written will not be presented or examined in detail
in this thesis. They are mentioned only to indicate that a significant

amount of preliminary work went into the development of techniques used

in the analysis of this engine.

3"\2 The Cycle

In defining the cycle used in these calculations we should first
COl'lszider what is meant when we use the term positive displacement engine.
In the context of this thesis we mean that the major pressure changes
Occlll‘ring in the cycle will be the result of the movement of physical
boundaries that surround a trapped mass of gas. Both the modern Otto
Cycle and Diesel cycle engines are positive diSplacement engines by this

de .
fit‘lltion. In each of these engines the compression process is the

40

result of a piston reducing volume of a cylindrical cavity in which we
have a trapped mass of gas and the expansion process occurs when an
increase in energy of the trapped mass (due to combustion) produces a
reverse motion of the piston and a volume increase. In positive dis-
placement engines the eXpansions and compressions normally occur over
fixed volume ratios. The modern Brayton cycle gas turbine is not a
positive displacement device and Operates as near as possible over the
fixed pressure ratios of the classical Brayton cycle. Application of the
Brayton cycle to a positive displacement device requires some modification
Of the classical cycle since expansion over a fixed volume ratio may not
Permit expansion of the working fluid to its original pressure.

In January of 1969 Warren and Bjerklie [4] proposed using one bank
of a 440 cubic inch displacement V—8 engine as compressors and the other
four cylinders as expanders. Between the compressors and expanders they
Placed a burner similar to a gas turbine burner. This is, of course,
one possible hardware configuration for the cycle analyzed in this study
but the cycle can also be applied to other positive diSplacement hardware
Sneh as the rotary Wankel engine.

In order to make a more complete study of the potential of the pos-
itive displacement Brayton cycle we have removed the hardware restriction
imposed by Warren and 3;] erklie [4] and evaluated operating conditions
that could not be met by their prOposed engine configuration.

This study then is not restricted to piston cylinder devices but
in the eXplanation of the cycle and develOpment of the analysis it is
advantageous to utilize the piston cylinder arrangement.

For the purposes of analysis we will consider that our engine con—

318
ts of a single cylinder compressor followed by a constant pressure

41

c:<3a11t>ustor (pressure drops can occur but pressure oscillations are not
considered) and then a single cylinder expander.

Figure 2 is a schematic of this type of engine. This configuration
i_s; riot seriously proposed as a workable design, but is used only to
ES:iJIlI)11fY‘the analysis. The important aspects to consider are that the
t>111rt1er is considered to be a steady flow, steady state device and that
titlee compressor and expander each go through a complete cycle every revo—
lution Of the engine.

Figure 3 is a pressure volume diagram of the compressor used in this
engine analysis. The compressor cycle consists of the following idealized
processes:

Process 1-2. An isentropic compression of the working fluid
from pressure P1 tO P2.

Process 2-3. At point 2 a pressure differential actuated valve
Opens and a constant pressure and temperature
adiabatic exhaust process occurs from point 2 to
point 3.

Process 3-4. At point 3 the piston motion reverses, the exhaust
valve closes and an isentropic expansion Of the
clearance volume gas occurs from pressure P3 to P4.

Process 4-1. At point 4 a pressure differential actuated intake
valve Opens and a constant pressure and temperature
adiabatic intake process occurs from point 4 to
point 1. At point 1 the intake valve closes.

Figure 4 is a pressure volume diagram Of the eXpansion cylinder

Prt:
cesses assumed to occur in an analysis Of this engine. The expander

 

42

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Hmcaaaso uwcaaasu
Cowmamaxm . Hommmuaaoo

 

 

aoumfim aOumHm

 

Hoauam

 

T

    

umsmnxm

 

m “aflom

Hmsm

43

E<MO<HQ mauwo mommmmmzco A<Uhmwh m MMDUHm
AmosoaH 0H930v osbao>

oom

00¢

com

com

ooa

 

om u m>\i> OHH<m onmmmmmzoo
mmnoaa cease owe Hzmzmo<qmmHn
mammoq mesmmmmm mo summzm oz

u

 

 

 

 

N _

_mL

00H

com

com

ocq

8m

 

(VISd) aunsssad

 

44

consists of the following idealized processes:

Process 7-8.

Process 8-10.

Process lO-ll.

Process 11 to

The intake valve closes at point 7 and an isen—

trOpic expansion of the working fluid occurs from
volume V7 to V8.
At point 8 the exhaust valve Opens and the cylinder

pressure is reduced at constant volume by a reduction

in the mass of fluid in the cylinder. The mass

It» JLqT

remaining in the cylinder during this process is

assumed to undergo an isentropic process. (Point 9

 

is the end point of an isentropic expansion of all
the working fluid from point 8 to pressure 10 and
describes the state change of the fluid remaining
in the cylinder.

With exhaust valve open the piston moves to tOp
dead center and a constant pressure and temperature
exhaust process occurs from point 10 to 11. (The
prOperties at point 11 can be most easily Obtained
by considering a constant pressure and temperature
process from point 9 since the thermodynamic state
at points 9, 10, and 11 are identical. This process
is always adiabatic.)

6. At point 11 the exhaust valve closes and the
intake valve Opens. The process from 11 to 6 is
a constant volume intake process from pressure P9 to

P7.

45

2<MQ<HQ mquwo mmnz<mxm A<onwH q mmaon
Amonoaw OHn=OV OESHO>
OOOH oom cow oom cow OCH 0
. \P
.........n.: hi-

:- OH HH

w

 

.OOH

.oom

.oom

om n m>\~> oHa<m onmmmmmzoo
mmnuaa canoe one ezmzmo<gmmHo .ooe

H\mm oHH<m amomumH<
mommoq mmommmmm mo wommzm oz

 

 

l. as

(VISd) aunssaua

46

A constant pressure intake process occurs from

Process 6 to 7.
volume V6 to V7. (The process from 11 to 6 is

adiabatic unless indicated otherwise for a

specific data set.)

Figures 3, 4, and 5 are plotted from data for 35 to 1 air-fuel

ratio with no pressure or heat losses. The numbered state points are

taken from calculated data and the processes were approximated by
constant) and determining a value

assuming a polytropic process (Pvn

of "n" to fit the known process end points.

Figure 5 is a combination of Figures 3 and 4 and it should be
remembered that Area 1-2-3-4 is negative and Area 6-7-8-10-11 is pos-

Combining the two cycle diagrams gives us the net positive out-

itive.
We can also see

put of the engine as Area 1-2—7-8 plus Area 3-4—11.

from Figure 5 that Area l-2-7-9 is the classic Brayton cycle and that

we are reducing its work per cycle and thermal efficiency by cutting

Off the toe of the cycle (Area 8-9-10). It is also apparent that we are

gerining area 3-4—11 which is not available in the conventional Brayton

cyn21e and thus our reduction in thermal efficiency and net work per

CyW21e is not as great as would be expected. In fact, cutting Off the

t<>e= of the Brayton cycle in this instance approximately doubles the mean

effective pressure (net work per cycle divided by volume displaced) and

red“files the work by a relatively small amount. It would appear that for

Scans: situations the friction power would be reduced enough to produce

emu Iitlcrease in brake work and brake thermal efficiency.

47

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Amosocfi OHnDOV OBSHO>
coca oom ooa com com ooa

 

m l-’"' '
'

oo u m>\z> oHH<z onmmmzmzoo
moses“ oaooo omo azmzmo<aomHo

H\mm oHsoz amozqu<
mammoa mzommmzm mo wozmzm oz

 

 

 

 

 

OOH

com

com

ooa

8m.

(VISJ) aunssaua

48

3.3 Compressor Analysis

In the following analysis of the compressor section of the engine
the subscripts will refer to the state points defined in Figure 3. It
was decided to define the compression ratio (CR) as ratio of the maximum
volume to the minimum volume (VI/V3) as is the convention in engine
analysis and not Speak in terms of the per cent clearance as is common
in conventional piston compressors. This leads to the definition of
effective compression ratio, the volume ratio over which the compression

process actually takes place, as Vl/V2°
From Figure 3 it can be seen that pressure during the intake process

remains constant (i.e., P = P4) and this pressure is calculated in the

1

following manner:
P1 = PATM - PDl (3-1)

where PAmM is the atmospheric pressure and FBI is the pressure drop
produced by flow through the compressor intake valve. The pressure
chiring the exhaust stroke of the compressor also remains constant
(P2 = P3) and is defined in the following manner:
P2 = PI + PD2 (3—2)
where PI is the burner outlet pressure and PD2 is pressure drOp through
the compressor exhaust valve and burner. The values of the following

variables are known inputs to the compressor program.

P = 14.696 psia

ATM
PD1_: 0 defined for a given situation
NI = 59.55 moles of air through the compressor for all runs

P1 = Define burner outlet pressure

49

Pressure loss through compressor exhaust valve and the

PD2 =
burner.
CR = the compression ratio (VI/V3)
T1 = 298.15°K the temperature at point 1
The molar Specific volumes can now be calculated as
RT1
v = -—— + b (3-3)
1 p
1
RT2
v2 = T + b (3-4)
2
and
Using the definition of compression ratio
N1 _ Y1 _ (13 i)
V2 ‘ V2 ‘ CR ‘ 1 v1 ’ CR
NI V V2 1 5
V2 ‘1VI‘CR (3-)
and therefore
v2NI
V1 = —'—""——v (3_6)
(.2. _ _1-)
v1 CR
'The: volume V1 can now be calculated if specific volume v2 can be
requires

determined from Equation (3-4). The determination of v2

k“owing T2 and this requires knowledge of the type of compression
This process has been defined

prOCess used from point 1 to point 2.
as an isentropic process so our problem reduces to finding T2 such that

s
(3-7)

 

50

where the E2 used in this study was 0.0001 and a temperature check
showed T2 to be within .l°K of the temperature required for zero

entropy change condition.
The molar entrOpies were then calculated as follows:

Tds

dh - vdp

C dT C dT
ds=_p___£12=_p__&1_2
T T T p

2 C dT P2
1A28 = J T - Rln P:-
1
where
- 2 3
Cp — a0 + alT + a2T + a3T

The values for a's for air are Shown in Appendix C and in Table 1.
Substituting the Cp expression in the entrOpy change equation and

integrating we have
T2 a2 2 a3 3 P2
.. __ _ _ _ 2 __ 3 _ _ __
1A28 - aozn T1 + a1(T2 T1) + 2 (T2 T1) + 3 (T2 T1) R£n P1

arui for calculation S1 and 82 are taken to be

a a
_ _2 2 _3 3_ _
Sl-aofinT1+alTl+ 2 T1+ 3 T1 RILnPl (3 8)
a a
s =a£nT +aT +-3T2+-3T3-R2np (3-9)
2 o 2 12 2 2 3 2 2

we ulust now determine T2 to meet the criterion established by Equation

(3m-7r)

51

The Newton-Raphson iteration technique is explained in Appendix H

and is used to determine successive estimates of T2 as

 

 

T 15mS
T2(n+1) - 2n - as (3’10)
(3?
P
where in general
88 EB_ aO [—
__. = = ___ 2
(3T)p 'T 'T +a1 + aZT + a3T i
and in particular [
35 ao
__ =_+ 2 _
(8T)p T2n a1 + a2T2n + 33(T2n) (3 11)

To start the iteration process we must have an initial estimate of
T2 which was Obtained from the following relationship for an ideal gas
with constant Specific heats:

k-l
4P2)”—

2 1 (5" k , where k = 1.4 (3-12)
1

After an initial estimate of T2 is made using Equation (3-12)
the entrOpy at point 2 is calculated from Equation (3-9) and tested in
Equation (3-7). If Equation (3-7) is not satisfied a new temperature
is determined for Equations (3-10) and (3—11) and the process is re-
peated until Equation (3-7) is satisfied.

Since T2 is now defined, the Specific volume at point 2 (v2) can
now be calculated from Equation (3—4) and V1 can then be calculated from

Equation (3—6).

52

We can now calculate V from

3
V1
V3 =IEE (3-13)
and piston displacement (VPD) is calculated from
VPD = V3 - V1 (3—14)

The conventional volumetric efficiency (VEF) can be calculated from

leI
VDP

 

VEF = (3-15)

and volume at the Opening of intake valve (V4) is calculated from the

following equation:
V1+ = V1 - leI (3-16)

From Steady flow adiabatic energy analysis of the compressor,
assuming no change in kinetic and potential energy, we can Obtain the

magnitude of the compressor work (We).

w = NI(H° - H°) = NI 2 c dT
c 2 1 1 p

a a a
1 2 3
c NI[aO(T2 Tl) + 2 (T2 T1) + 3 (T2 T1) + 4 (T2 Tl)] (3 17)

The pertinent equations in this section were programed in FORTRAN IV
and formed part of the main line program Shown in Appendix H. The burner
outlet pressure is input to the computer in psia but the pressure drOps
must be input in atmospheres. All calculations are performed in atmos—
phere, liter, Kelvin, calorie units and outputs are presented in these

units and in psia and degrees Fahrenheit.

 

53
3.4 Burner Analysis

In the previous section the compressor discharge temperature has
been defined and, if the flow through the compressor exhaust valve is
assumed to be isenthalpic, the burner inlet temperature is equal to
the compressor discharge temperature. In order to calculate the burner
exhaust temperature and composition we must know the air-fuel ratio
(AFR), the burner heat loss, the burner exhaust pressure, and burner in-
let temperature. All Of these variables are either input data or are
defined by previous calculations.

The solution to the burner exhaust temperature and composition
then requires that a value of the exhaust temperature T5 be found that

satisfies the following energy equation.

0 0 O
z I -
n1 (HT H + AH )

O 0 )
298 f298

O
. - Zn.(H - H + AH .
t(PROD) ‘ T 298 37298 7’ (REACT)

_ Q = 0 (37718)
TWO subroutines, PDOBHH and PDOSHL (see Appendix E for details), were
written to evaluate the first summation in Equation (3-18) and its value
will be named HHH for convenience. Thus we have-

0 + AHO ) — Q = o

O
HRH 7 Zni(HT 7 H298 f298 i
(REACT)

and since reactants consist only of air and fuel we can write

0 O O O
H ) - nfueZ(HT H +

0 0
Han 7 n0 (HT 7 H ) n (HT 7 298 N2 298

2 298 02 N2

0 — .-
77f298)fuel7 Q 7 0 (3 19)

54

The value of the sum of the enthalpies for oxygen and nitrogen can be
calculated from subroutine PD08HL since the compressor exhaust temper-
ature never exceeds 1500.0°K. This is accomplished by setting the number
of moles of all constituents equal to zero except for oxygen and nitrogen.
For convenience this value is named HHL and we can restate Equation

(3-19) as

HHH - HHL - n ( ° °

0
H + A ) -
fuel T 298 f298 fuel

The only fuel considered in these calculations is normal octane and it

Q 0 (3-20)

is always injected as a liquid at 298°K, hence

HHH - HHL — Y1 ~Aa;298 — Q = 0 (3—21)

where "Y" is the number of moles of fuel and has been previously defined

as

Y1 = 15.042/AFR (2-33)

The heat loss term "Q" is defined during calculation as some fraction
of the enthalpy of reaction of the appropriate quantity of fuel.

To find the value of the exhaust temperature T5 that satisfies
Equation (3-21) a variation Of Newton's method of iteration known as
the regula falsi method is used (see Appendix H). In this method the
derivative of the function is approximated by the difference quotient
formed from two preceding approximations. To utilize this method

Equation (3-21) is restated in the following form:
HHN = HHH - HHL - Y1 oAH;298 - Q (3-22)

where the 100 calorie test is approximately 0.1 per cent of HHH and

55

insures the temperature is within .2°Kelvin Of that required for a zero
value of HHN.

The initial value of T5 is estimated to be 1.2 times the value of
the compressor discharge temperature. We then calculate HHN and if its
value does not satisfy Equation (3-23) a new exhaust temperature is
estimated as T5 j;lO°K and a new value of HHN is calculated and tested
in Equation (3-23). If this second estimate does not satisfy Equation
(3-23) then all subsequent temperature estimates are made using the

following:

T _ T _ (Tsn 7 T5(n-1))HHNn
5(n+1) 5n (HHNn ' HHN(n_1))

 

(3—24)

The analysis of the burner process is complete at this point but
programing the solution for a digital computer is very tedious. Since
the thermodynamic prOperty calculations were broken into two ranges,
one below 1500°K and one above 1500°K, frequent checks of the exhaust
temperature are required to select the prOper subroutine for calculation.
It should be further noted that the calculation of composition at a given
pressure and temperature using subroutine PD08CC also requires use of
an iteration technique to match the desired pressure. The details of
the calculation are given in the flow chart for the main line program

in Appendix J.

3:5 Power Cylinder Analysis
In the analysis of Power Cylinder we must first consider the intake
Process. To facilitate the analysis of the intake process we define the

general Open thermodynamic system (a region in space) shown in Figure 6.

56

Intake Valve Exhaust Valve

      
  
       
 
  

  

Point 5

—*

Point 6

System —— TOp Dead
Boundary Center
Point 7
Intake Valve
Closure

 

//////////////i

 

 

 

FIGURE 6. SYSTEM BOUNDARY FOR POWER CYLINDER INTAKE PROCESS

This system is defined as the volume in the cylinder above the piston.
The following assumptions are made with respect to this system:

1. All material entering the system through the intake valve
has the prOperties of the burner exhaust. (Point 5)

2. The mass of fluid entering the system is equal to the mass of
air delivered by the compressor (this is constant for all cycles)
plus the mass Of fuel added in the burner.

3. The intake valve opens at the piston top dead center position
(Point 6) and remains Open until the prescribed quantity of

material has entered the system and then closes. (Point 7)

57

4. All previous assumptions regarding ideal gas nature of the
working fluid are in effect in this analysis.
The state points used in this analysis are taken from Figures 2 and 4
and are repeated here for convenience.

Point 11. Piston is at tOp dead center, the exhaust valve has
closed, the gas remaining in the cylinder has the specific
prOperties which are the same as point 9, and the intake
valve opens.

Point 6. The piston is at tOp dead center, the intake valve is
open and the pressure has been raised to P6 by the inflow
of gas from the burner.

Point 7. The piston has moved during a constant pressure process
at a volume V7 and the intake valve closes. The mass
in the cylinder at this point is equal to the mass air
provided from the compressor plus the mass Of fuel added
in the burner and the residual mass present at point 11.

Point 5. This is the exhaust condition of the burner and defines

the state of the working fluid entering the system.

The energy relation for the general Open system states that the
energy entering the system must be equal to the energy leaving the system
plus any change in energy that occurs within the system boundaries. By
neglecting changes in kinetic and potential energy we can write the en—

ergy equation for our system in the following form:
0 o O
Q + (wings — (EniUi)7 - (miui)11 + w (3 25)

Where temperature and composition are in general different at points 5,

58

7, and 11. The normal thermodynamic Sign convention for heat (Q) and
work (W) are used in Equation (3-25). The work done by this system is
due to the movement Of the system boundary and can be eXpressed as
7
W = [ Pdv
6

and since pressure is constant during this process

W = P7 (V7 - V6). (3-26)

By substituting (3—26) into (3-25) and using the definition of enthalpy
(Ho) to express the internal energy (U0) we can rewrite Equation (3-25)

in the following form:

Q + (2n.H?)5 = (2n.(H9 - RT>> - (zn.(H? - RT>) + P (v — v )
L z z t 7 z z 7

11 6

By rearrangement
.O _ O _ _ o _ _ _
Q + (Enidi)5 - (ZniHi)7 (RTzni)7 (zniHi RTZni)11 + P7V7 P7V6 (3 27)

Since

P7v7 = RT7(Zni)7

we find that Equation (3-27) becomes

0 = O _ O _ _ _
Q + (zniHi)5 (EniHi)7 (ZniHi RTzni)11 P7V6 (3 28)

Since we have assumed the working fluid inside the cylinder continues

an isentrOpic expansion once the exhaust valve Opens, it is evident that
the state of the working fluid at point 9 is determined by an isentropic
expansion from point 7. The process from point 10 to point 11 is a pure

mechanical exhaust process with no change in thermodynamic state. The

59

thermodynamic state Of the working fluid is, in fact, identical at point
9, 10 and 11, and differs only in the quantity of mass involved. We can

make use Of this condition constant specific properties in the following

 

 

manner:
v = =
9 10 11
and
V9 V11
v9 = fi-= N__. where Nx = (Zni)x
9 11
therefore
N V
11 11
Er—-=‘V—— (3-29)
9 9
O O
and Since U11 = U9 then
0 o
(EniHi - RTZni)11 (ZniHi - RTZni)9
N = N (3-30)
11 79
and combining Equations (3—29) and (3-30) we have
V
.9- . = .9_T;;.)-—1i (3—31)
(anHt RTZnZ)11 (anHz R n1 9 V9

Utilizing the fact that V11 = V6 and that (ZniH2)5 has been previously

defined as HHH, Equation (3-28) can now be written as

V
o O 6
Q + HHH = (ZniHi)7 - (EniHi - RTzni)9'V;'- P7V6 (3-32)

For convenience then, let

0
(zniHi)7 = HH7
O
(ZniHi)9 = HH9
(Eni)9 = N9

60

Equation (3-32) can be expressed in the following manner:

V

6
+HHH= H — HH — T —- —
Q H 7 ( 9 R 9N9) v9 P7V6 (3 33)

Equation (3-33) is deceptively Simple in appearance but its solution
presents some problems. The heat transfer (Q) may be expressed as some
fraction of enthalpy of combustion, the volume at point 6 (V6) can be
expressed as the compressor clearance volume (V3) multiplied by a constant,
the entering enthalpy (HHH) is known, and the pressure P7 can be calculated
from the burner pressure and a defined inlet valve pressure drop. The
prOperties at point 9 are unknown but the thermodynamic state at point 9
is the result of isentropic expansion from point 7 over a defined pressure
ratio. There are two unknown quantities involved in calculating HH7,
the temperature and the number of mole atoms of the elements present in
the mixture. The number of mole atoms of the elements present is equal
to the number of mole atoms present at point 5 plus the number present
in the residual gas at point 11. Clearly the solution to Equation (3-33)
will require multiple iterations starting with an estimate of the temp—
erature at point 7 and the mole atoms of elements present in the residual
gas. TO obtain the solution, Equation (3-33) is restated in the follow-
ing form:

V

6
ERR7 8 BER - HH7 + P7V6 + (HH9 - RT9N9) V9 + Q (3-34)

where ERR7 will be made to approach zero within a small limit.
If the volume ratio V6/V9 is very small the residual could be assumed
to be zero and Equation (3-34) could be expressed as

ERR7 = HHH - HH7 + 127v6 + Q (3-35)

61

The assumption of no residual also defines the number of mole atoms at
point 7 as being the same aS the number of mole atoms entering the
system from the burner.

If we consider the Operating condition used in Figure 4, we find
that the volume at point 6 is about 0.5% Of the volume at point 9.
While itis realized that this value will change with power cylinder
clearance volume, burner pressure and air—fuel ratio, the assumption
of no residual is justifiable.

The solution to Equation (3-33) can now be obtained through the
following sequence of steps: 4

1. Estimate the temperature at point 7 and determine composition

at the desired pressure.
2. Calculate HH7 and test ERR7 from Equation (3~35). The test

used is

|ERR7i - 100 5 0 (3—36)
If Equation (3-36) is not satisfied, estimate a new temperature as
T7 i 10°K and return to step 1. If Equation (3-36) is not satisfied
with the second temperature estimate the regula falsi iteration on temp-
erature is used until Equation (3-36) is satisfied.

3. Calculate volume V7 from known temperature, pressure and
composition. Expand the working fluid isentrOpically under
equilibrium conditions from V7 to V8 and from V7 to the ex—
haust pressure P9.

4. Calculate V9 from known temperature, pressure and composition.

62

5. Calculate the number of mole atoms of the elements in the
exhaust gas residual and use them to correct the number of
mole atoms at point 7.
6. Return to step 1 and repeat process using the definition of
ERR7 given by Equation (3-34) in all subsequent iterations.
Through the use of the preceding six steps all points on the ex-
pansion cylinder diagram can be determined. The isentrOpic expansion
process in step 3 is accomplished by first calculating the entrOpy at
point 7, (S7) and calculating the entrOpy at point 8 (88) from an
estimate of the temperature at point 8. These calculations are per-
formed using the appropriate subroutines.
A Newton—Raphson iteration on temperature is then used to satisfy

the following relationship:

'"ET——*' - E1 £_0 (3-37)

and the recurrence formula used for T8 is

 

 

(Sn — S ) Tn
TYH'I = Tn " C . N (3‘38)
v n
(Sen 7 S7) Tan
T = T - c . N
8(n+1) 8n v en

where CD is specific heat per mole at Ten and Nan is total moles of

mixture at Tan .

63

The expansion process from point 7 to point 9 is conducted in a

similar manner where the test is then

and the recurrence formula for T9 is

(S9n 7 S7) T9n

n 7 c - N
p 9"

 

T9(n+1) = T9

The initial temperature estimate, in each case, is made by assuming
a polytropic process (Pvn = C) and using an apprOpriate value of "n."

The mole atom correction indicated in step 5 is made by multi-
plying the mole atoms present at point nine by the volume ratio VG/V9'
Care must be taken to add this correction to the mole atoms entering
the cylinder (Point 5) and not to the mole atoms present in the cylinder
at point 7 Of the just completed cycle. The latter situation would lead
to an ever-increasing mass in the power cylinder, since each correction
would be added to the sum of the mass at point 5 and all previous cor-
rections.

Two methods are used to insure valid results from the calculations
outlined in steps 1 through 6 of the power cylinder calculations. The
first method is to force the calculation to be made four times using
the previous value of a variable as the new estimate and the second is
to provide suitable convergence tests at apprOpriate locations in the

program.

64
3.6 Engine Performance Analysis

The preceding sections of this chapter have outlined methods used
in calculating the thermodynamic state of the working fluid at all
numbered points on Figure 5. When this has been accomplished we can
then calculate engine performance and any other engine parameters de-
sired.

The engine work per cycle is calculated by taking the algebraic
sum of the work of the engine power section and compressor section.
The compressor work calculation was described in section 3.3 and power
cylinder work is calculated by taking the algebraic sum of the areas
below the curves in Figure 4. The following equation is used to cal-
culate the power cylinder work (WP).

WP = P7(V7 - V6) + P9(V6 - V8) + (HH7 - RN7T7) -

(HH8 — RN8T8) (3-40)

where

(HH7 — RN7T7) - HH8 - RNBTB) = U7 - U8
and for a closed system isentrOpic process
8
U7 - U8 = [ PdV
7
The new work (WN) is then calculated as
W” = wp - lwcl (3—41)

The thermal efficiency is

wN

“a. = m (3'42)

65

where LHV is the lower heating value of the fuel per mole and Y1 is
the number of moles of fuel used.
The indicated mean effective pressure (MEP) is

W

IL
MEP =
(VI—V3) + (V87V6)

 

(3-43)

This definition of MEP is based on a two—stroke cycle Operation of the
engine and the power is then calculated as the total displacement
(compressor plus power cylinder) multiplied by the engine rotational Speed
and the MEP. This definition is not the same as that used by Warren and
Bjerklie [4] who based their MEP on a four-stroke cycle basis. For
direct comparison of the indicated mean effective pressures with those
of Reference 4, the indicated mean effective pressures in this thesis
Should be multiplied by a factor of two.

One other important engine parameter which must be calculated is
the time which the intake valve and the power cylinder remain open.
This is expressed in terms Of the cutoff ratio (COR) which is defined
in the following manner:

(V7 - V6)

(V8 - V6)

COR = (3-42)

The program written to perform the calculations described in this
chapter is presented with a flow chart in Appendix J and sample output
from the computer is presented in Appendix L. The program also pro-
vided punched card output which was fed to the plotter program presented
in Appendix K. The plotter program provided point plots of MEP, nth
and combustion temperature which were used to generate the performance

maps shown in chapter 4 of this thesis.

IV. CALCULATED RESULTS FROM ENGINE CYCLE SIMULATION

4.1 General

This chapter will present a discussion of the effects of selected
engine variables on the performance of the positive diSplacement Brayton
cycle. A computer program, based on the analysis presented in Chapter 3,
was used to evaluate engine performance. The program was written in
FORTRAN IV and is presented in Appendix J. An IBM 360 system was used
to perform the calculations and sample output from this program is
presented in Appendix L. A brief discussion of the validity of the
calculations is presented in this section, and subsequent sections of
the chapter will discuss the effects on performance of compression ratio,
heat loss, valve pressure drops, expansion ratio and clearance volume.
Performance maps and a brief examination of the engine's potential will
conclude the chapter.

The performance calculation program consists Of a mainline program
and series of subroutine programs. The precision Of the subroutines
used to calculate enthalpy, entrOpy, and equilibrium constants is in-
dicated by Tables 2, 3, and 5 in Chapter 2. Subroutine PD08CC (see
Appendix I) was used to calculate equilibrium compositions of the work—
ing fluid. The compositions calculated were compared with those presented
in References 3, 11, and 12, and the extremely minor differences in specie

concentrations were all attributable to Slight differences in basic

66

67

thermodynamic property data. The reaction temperatures calculated in
this study were found to be in agreement with those presented by KOpa
[12]. All differences were eXplainable by differences in the state of
the fuel used and slight differences in equilibrium composition.

To verify the validity of the entire program, complete hand calcu—
lations were made of two computer runs. These calculations were made
using Huff, Gordon and Morrell [31] as a data source. In these cal—
culations, the values of variables from the computer runs were used
as first estimates for all iterations. The small differences between
the hand calculated and computer values were attributable to slight
differences in basic thermodynamic data. (The JANAF Tables [19] were
used for computer calculations.)

After these checks on the analysis and the program had been com—
pleted, a comparison was made with the performance data of Warren and
Bjerklie [4]. This comparison is Shown in Table 6. Details of Warren
and Bjerklie calculations were not presented in reference [4] and an
exact definition is not given for some of the points in Table 6. One
point in question is the temperature of gas to the engine. This could
be considered the burner exit gas temperature or the cylinder gas
temperature, and the end power cylinder intake process. Both temper-
atures from this investigation have been included, and the lower Of the
two temperatures is burner exit gas temperature. The working fluid
always undergoes a temperature rise during intake process. This
temperature rise varies from 20°F to 200°F depending primarily on
compression ratio, burner pressure, and air-fuel ratio. The com-

parison appears to Show remarkably good agreement for two independent

68

TABLE 6
COMPARISON OF AVAILABLE ENGINE PERFORMANCE DATA

NOTES:

The underlined variables were defined in this study to match those
Of Reference 4 for this comparison.

Compression Ratio for Reference 4 estimated to be 100/1 as defined
in this study. Compression Ratio 100/1 for data presented for
this study.

*Depreciation Factor, Compression Efficiency, and Expansion Effi-
ciency not used in this study.

**Indicated Mean Effective Pressure (IMEP) calculated in this study
is one-half the value shown. The calculated value was doubled for
purposes of comparison because a two-stroke cycle definition was
used in these calculations and a four-stroke cycle definition was
used in Reference 4.

69

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70

studies. This study produced a small increase in indicated mean effect—
ive pressure and a slight decrease in indicated specific fuel consumption.
For a given air-fuel ratio, of course, either Of these changes leads to
the other change. Possibly the most significant change is the differ-
ence in cut off ratio. The Warren study was conducted using fixed cut
off ratios, while in this study the cut off ratio was calculated from
other engine variables. A reduction in cut Off ratio would be produced
by a reduction in engine gas temperature, which is the case in this
study. A complete evaluation Of the reasons for differences between the
two studies was not possible due to a lack of detailed knowledge of the
calculations in Reference 4. It is apparent, however, that the two
Studies are in substantial agreement with reSpect to engine performance

for the stated Operating conditions.

4.2 Effect of Compression Ratio on Engine Performance

The effects of compression ratio on engine efficiency are shown
in Figures 7 and 9. In both figures thermal efficiency is plotted versus
cut Off ratio for compression ratios of 30, 60, 90 and 120/1. This range
of compression ratios was chosen because construction of an engine with
a compression ratio of 120/1, or greater, is improbable and it would be
undesirable, from efficiency considerations, to drOp the compression
ratio below 30/1. In Figures 7 through 9 no heat losses or valve pres-
sure drops are considered and the expansion and compression ratios are
EQUSI. Calculations were made for burner pressures from 200 psia to
1200 psia in increments of 100 psia. The effects of compression ratio
were similar throughout the entire range Of pressures and, therefore,

only the curves for three pressures are presented. Figure 7 shows the

71

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72

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74

curves for a burner pressure Of 500 psia and Figure 9 shows curves for
800 and 200 psia. For each burner pressure it can be seen that, for a
fixed cut Off ratio, increasing compression ratio increases the thermal
efficiency, but at a decreasing rate. With a 500 psia burner pressure,
and a cut off ratio Of .10, doubling the compression ratio from 30/1 to
60/1 increases the thermal efficiency 5.3 per cent, but doubling from
60/1 to 120/1 only increases efficiency 2.6 per cent. By considering
both Figure 7 and Figure 9 it is evident that the effect of compression
ratio on the thermal efficiency is somewhat more pronounced at higher
burner pressures. It is also evident that the efficiency difference
between a compression ratio of 90/1 and 120/1 never exceeds one per cent
at any cut off ratio or burner pressure. With regard to thermal
efficiency, it seems Obvious that increasing compression ratio is de-
sirable, but little is to be gained by exceeding compression ratios of
90 or 100/1.

Thermal efficiency is not the only important parameter in engine
design and we must also consider the effect of compression ratio on work
per unit of displaced volume. This can be accomplished by considering
the effect Of compression ratio on mean effective pressure. Figure 8
shows this effect for burner pressures of 500 and 200 psia. The mean
effective pressure in this figure is given in the normal units of pounds
per square inch, which is equivalent to inch-pounds of work per cubic
inch of diSplaced volume. In this investigation the mean effective
pressure is based on two stroke cycle operation of the engine and to
calculate the power of an engine we need only multiply the mean effective

pressure by the total engine diSplacement (compressor cylinder plus power

75

cylinder displacement) and the engine revolutions per unit of time. It
is evident from Figure 8 that increasing compression ratio, for a fixed
cut off ratio, decreases mean effective pressure at a decreasing rate.
It is also apparent that this effect is more pronounced at higher
pressures. One other effect of increasing the compression ratio is
that the maximum possible mean effective pressure, for any given burner
pressure, is increased and that this maximum occurs at larger cut Off
ratios. This maximum is defined by the 15/1 air-fuel ratio line.

From Figures 7, 8 and 9 it is apparent the design of the positive
displacement Brayton cycle engine will involve the usual compromise
between efficiency and power production. It should be noted that while
the magnitude of compression ratio effects may be modified by the intro—
duction of pressure and energy losses, the general trends will remain

unchanged.

4.3 Effect of Heat Loss on Engine Performance
In this investigation the possibility of heat transfer was per-

mitted during the combustion process in the burner and during the power
cylinder intake process. The program was constructed to permit the
quantity of heat loss to be varied independently between these parts

of the engine cycle and to Specify the heat loss as a fixed quantity Of
energy or as a fixed percentage of the lower constant pressure heating
value of the fuel. In all the heat loss calculations presented, the
quantity of heat loss was Specified as some per cent of the heating
value of the amount of fuel used. This means that the quantity of
energy lost will be directly prOportional to the working fluid temper-

aturwz. This assumption is commonly made in a cycle analysis of this

I
‘

76

type and is more reasonable than the fixed heat loss assumption.

To evaluate the effects of heat loss on thermal efficiency and
mean effective pressure a series of computer runs were made for com-
pression ratios Of 30/1 and 90/1 over a range of pressures from 200 psia
to 1200 psia. In this set of calculations no valve pressure losses
were considered and the compression and eXpansion ratios were the same.
Heat loss values were set at O, 5, 10 and 15 per cent of the lower
heating value and performance was calculated for loss from the burner
only, loss during the intake process only, and loss during both proc-
esses. These calculations Show that for a given total heat loss the
engine mean effective pressure and thermal efficiency are identical
whether the loss occurs during the burner process, intake process, or
during both processes. This result was not unexpected since the quantity
of energy in the working fluid at the end of the intake process should
be the same regardless of where the loss occurred. The only differences
between the aforementioned calculations were in the burner exhaust temper-
atures which were of course decreased by burner energy losses.

Figures 10 and 11 Show the effect of heat loss on efficiency for
two burner pressures and two compression ratios. The titles on these
figures may be somewhat misleading in light of the preceding discussion,
since they tend to indicate that a difference exists between the effect
0f intake process heat loss and burner heat loss. The titles only in-
dicate the exact source of the data used to plot the curves and an inter-
Change of titles would not affect the validity Of the curves. Exam-

ination of the curves shows a five per cent heat loss produces an

77

 

 

 

 

 

 

 

 

 

 

 

 

  

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79

efficiency loss Of from 2 to 3 per cent over the range of both Figures 10
and 11. In fact, a more complete evaluation of available data indicates
that, in general, the efficiency decrease is from 40 to 60 per cent of
the heat loss percentage, and that the efficiency loss increases with
increasing pressure and air-fuel ratio.

The effect of heat loss on mean effective pressure is shown in
'Figure 12. The lines of constant burner pressure remain in essentially
the same position but are shortened by heat transfer. The dashed lines
.identified as per cent heat loss lines, are also lines of constant 15/1
aiir-fuel ratio. The per cent heat loss lines then indicate the terminal
IDCDintS of burner pressure lines for the given loss percentages. If we

aatztempt to maintain a given burner pressure and mean effective pressure,
IfC)I a fixed cut Off ratio, heat loss requires that our air—fuel ratio

be reduced.

ELL4I Effect of Valve Pressure Loss on Engine Performance

In the positive displacement Brayton cycle engine there are four
ENDiIIts where valve pressure losses can occur. The effect of these
Priassure losses on thermal efficiency and mean effective pressure will
be IDresented for a compression ratio of 90/1. The calculations of valve
1°85; effects were made for an engine which had equal eXpansion and
Compression ratios, no heat losses, and zero pressure loss for all
vaJJ'es except the valve being investigated. The valve pressure losses
werfi expressed as a percentage of the pressure on the high pressure
Sidii of the valve except for the power cylinder exhaust valve which is
expressed as a percentage of atmOSpheric pressure. The effect on

th 0 C I C O
erjnal eff1c1ency of pressure loss in the various valves is shown in

 

80

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(ISd) ainssala aArqoaggg ueaw

81

Figures 13 through 16. For a given valve, the pressure loss effect was
similar in character for all burner pressures and tended to increase
slightly in magnitude with increasing burner pressure. Only the results
of the calculations for a 500 psia burner are presented because they
clearly describe what is occurring.

The effect of compressor intake valve pressure loss on thermal
efficiency is shown in Figure 13. It can be seen from this figure that,
:for a fixed cut Off ratio of .15, the thermal efficiency loss for a
L12 per cent pressure loss (1.764 psi) is approximately 1.5 per cent.

It: is also evident that the efficiency loss is greater at both larger
earud smaller cut off ratios. This 1.5 per cent is the greatest per-
crerltage loss in efficiency observed for any Of the valves at this cut
c>f E ratio and burner pressure. The maximum efficiency loss for this
\LaLLve, which occurred at low cut off ratios, was less than that of the
<>t11er'valves, but for cut Off ratios exceeding .135 it produced the
highest efficiency loss for a given per cent pressure loss. For a fixed
(“It Off ratio, increased compressor intake valve pressure loss leads to
a Etignificant reduction in air—fuel ratio. This trend is almost non-
exiéStent for losses in the other valves and is, in fact, the reverse
forthe power cylinder intake valve. This decreasing air-fuel ratio
wit}! increasing pressure loss leads to a Slight increase in mean
effeetive pressure at a fixed cut Off ratio. This increase is so slight
(apfrroximately 2 psi) that no curve of compressor intake valve loss
effeutt on mean effective pressure was prepared.

The effect of compressor exhaust valve pressure loss on thermal

efficiency is shown in Figure 14. The effect of valve pressure loss

82

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

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(Z) KOUBIDIJJH IBNJGQL

84

increases as cut Off ratio decreases and air—fuel ratio increases. A
compressor exhaust valve loss of 12 per cent (68.2 psi) is less sig—
nificant, except at the lowest cut Off ratios, than a 12 per cent intake
valve loss even though its absolute magnitude is 39 times as great.

The air—fuel ratio lines are nearly parallel to the lines of constant
cut off ratio and, as would be expected, the valve pressure loss pro-
duces a reduction in mean effective pressure. This decrease is Slight
(approximately 2 psi) and no curve Of the effect of compressor exhaust
xlalve pressure loss on mean effective pressure was prepared.

The effect of power cylinder intake valve pressure loss on thermal
sufficiency is shown in Figure 15. For this valve the effect of pressure
.1c>ss on thermal efficiency is significantly different from that for
trieeother engine valves. In Figure 15 the lepe Of the air-fuel ratio
].iJnes is the negative of that observed in Figures l3, l4, and 16. The
liJnes of constant valve pressure loss show the expected decrease in
Gaffiiciency with increasing pressure loss at high air-fuel ratios, but the
‘r3\flerse is true at air-fuel ratios near stoichiometric. Although the
L“lique characteristics of the curves in Figure 15 may seem somewhat
suI7Prising at first, their explanation is relatively simple. At high
aiI“-fuel ratios the equilibrium reaction temperatures are low and no
diSS30ciation occurs. In this region an isothermal pressure reduction
would produce no change in the number of moles of gas in the system.

AS tflne air—fuel ratio is decreased, system temperatures rise and dis-
SOCjJition becomes increasingly important. When we near stoichiometric
Conttitions an isothermal pressure reduction produces increased dis-

S o
0C1ation and hence an increase in the total number Of moles in the

 

85

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86

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87

system. This effect, at the stoichiometric air-fuel ratio, is illus-
trated by the composition data shown in Table 7. A comparison of the
three columns of Table 7 shows that even with a temperature drOp, a
pressure drOp increases the mole fractions of all Of the minor species.
This change necessitates an increase in the total number of moles
present in the system. For an engine Operating with a fixed cut off
ratio the system volume (V7) at the end of the intake stroke must remain

constant. Since

 

v7 = (4—1)

we can see that any reduction in P7 requires a reduction in the product
of system temperature (T7) and the total number Of moles present in the
system (N7). At high air-fuel ratios a reduction in fuel flow reduces
both N7 and T7 in a straightforward manner (no dissociation effects)

and this explains the decrease in efficiency and air-fuel ratio for
pressure losses in this region. In the region of Operation near stoich-
iometric conditions a reduction in fuel flow is still required to main-
tain a constant V7 as P7 is reduced, but the effect of pressure reduction
on dissociation in this region, and the fact that energy of dissociation
can be recovered during expansion, combine to produce an increase in
thermal efficiency.

The effect of power cylinder intake valve pressure loss on the
magnitude of the thermal efficiency is not pronounced, except at high
air-fuel ratios. The effect Of this pressure loss on mean effective
pressure, however, is pronounced at all air-fuel ratios as is shown in

Figure 17. For a 500 psia burner pressure a 12 per cent valve pressure

 

TABLE 7

EQUILIBRIUM COMPOSITION FOR A 15/1 AIR-FUEL RATIO

 

 

MOLE FRACTION

Chemical 4192.7°F 4214.7°F 4212.1°F
Specie 500 psia 485 psia 470 psia
O2 .5150E—02 .5406E—02 .5442E—02
N2 .7250E 00 .7245E 00 .7245E 00
C02 .1093E 00 .1086E 00 .1085E 00
H20 .1346E 00 .1343E 00 .1343E 00
CO .1465E-01 .153lE—01 .1538E-01
H2 .2861E—02 .2989E—02 .3006E-02
0H .3656E-02 .3859E-02 .3879E—02
NO .4179E—02 .4365E—02 .4369E-02
0 .2649E-O3 .29l4E-03 .2951E-03
H .3292E-03 .359lE-03 .3637E-03

 

 

 

 

89

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(ISd) ainssaid anrnoaggg ueaw

90

loss produces a 40 per cent loss in mean effective pressure at high
air-fuel ratios and 20 per cent loss at low air-fuel ratios (i.e., near
stoichiometric). The same trend is indicated at 800 psia and 200 psia,
with the losses increasing Slightly with increasing burner pressure.
Engine power is very sensitive to pressure loss in this valve and care
must be exercised to minimize this loss if output is to be maintained
at a high level.

The effect of power cylinder exhaust valve pressure loss on thermal
efficiency is shown in Figure 16. For a given per cent pressure loss
the curve in Figure 16 is essentially the same as that in Figure 14,
although the absolute magnitude of the pressure losses are Significantly
different. Pressure loss in the power cylinder exhaust valve leads to
a slight decrease in mean effective pressure which is of the same magni—
tude as that of the compressor exhaust valve.

Figures 13 through 17 illustrate the effects of valve pressure
losses on performance of a 90/1 compression ratio engine. A complete
analysis of a 30/1 compression ratio engine was also made and pressure
loss effects on performance were the same in character as with the 90/1
engine and differed only slightly in magnitude. In general, the ab-
solute magnitudes of the losses were the same or slightly less and
the percentage loss was the same or slightly greater.

The results of the analysis of the effect Of valve pressure loss
indicate that a compressor intake valve loss probably has the most detri—
mental effect On thermal efficiency in a normal engine Operating range

and that a power cylinder intake valve has the most adverse effect on

91

mean effective pressure. These conclusions are based on the maintenance
of a constant burner pressure and could be altered if another standard

were applied.

4.5 Effect of Expansion Ratio and Clearance Volume on Performance

The effects on engine performance Of the expansion cylinder piston
displacement being greater than that of the compressor, and of utilizing
different clearance volumes for the expansion and compression cylinders,
are examined in this section. NO pressure or heat losses are considered
and the only changes from the basic no loss, equal clearance, equal
compressor and expander displacement engine are those noted on Figures
18 through 21. Only representative curves from the 90/1 compression
ratio analysis are presented.

The effect of the expansion cylinder piston displacement (VB—V6)
being greater than the compression cylinder piston displacement (V17V3)
is shown by Figures 18 and 19. In Figure 18, for cut off ratios greater
than .12, the thermal efficiency decreases as displacement ratio
((V8-V6)/(Vl-V3)) increases and in Figure 19 the mean effective pressure
increases with increasing diSplacement ratio. This behavior of mean
effective pressure and thermal efficiency at first appears to conflict
with the discussion presented on page 46 of Chapter 2. The discussion
on page 46 related to Figure 5 and was predicated on a fixed cut off
volume (V7) and not on fixed cut Off ratio ((V7-V6)/(VB-V6)). In both
Figures 18 and 19 the lines of constant air-fuel ratio are equivalent

to lines of constant cut Off volume since the volume required to admit

a fixed quantity of gas is determined air—fuel ratio and burner pressure.

r-_:

92

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(18d) alnssaid aArnoa;;3 usan

94

(Other factors which affect this volume, such as clearance volume, heat
loss and pressure loss are either zero or constant in these figures.)
For a constant air-fuel ratio the thermal efficiency increases and the
mean effective pressure decreases with increasing displacement ratio.
This trend then supports rather than contradicts the discussion pre-
sented in Chapter 2.

In Figures 18 and 19 the lines of constant displacement ratio
terminate at successively lower air-fuel ratios as the displacement ratio
increases. This termination was made whenever the bottom dead center-
cylinder pressure dropped below the exhaust pressure because the
assumptions made in the calculation of cycle work are of questionable
validity beyond this point.

It is apparent that significant increases in mean effective
pressure may be obtained by increasing the displacement ratio while
maintaining a constant cut off ratio. It is also apparent that this
could result in serious efficiency penalties at cut off ratios greater
than .12.

The effect on performance of increasing the clearance volume ratio
(V6/V3) above a value of one is shown in Figures 20 and 21. Compression
ratios of 30/1 and 90/1 were investigated over a range of burner
pressures for an engine with a displacement ratio of unity and no heat
or pressure losses. Only representative curves from the 90/1 compression
ratio analysis are presented. Figure 20 indicates that increasing the
clearance volume ratio produces a decrease in thermal efficiency. For
a given burner pressure the efficiency loss increases with increasing

air-fuel ratio, and for a given change in clearance volume ratio the loss

95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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96

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(ISd) alnssala aAtnoaggg ueaw

97

imcreases with increasing burner pressure. The effect on mean effective
pressure of increasing the clearance volume ratio above a value of one
is shown in Figure 21. The mean effective pressure increases with
increasing clearance volume ratio for a given cut Off ratio. For a
given change in clearance volume ratio the mean effective pressure
improvement increases with decreasing cut Off ratio and with increasing
burner pressure.

In general, a thermal efficiency decrease occurs when either
displacement ratio or clearance volume ratio are increased above a value
of one. Figures 19 and 21 Show that the mean effective pressure
increases as previously described but they also Show a decrease in
maximum mean effective pressure possible for any given burner pressure.
This decrease in maximum possible mean effective pressure would lower
the tOp bounding curve on a performance map, such as Figure 22 or 23,
and the result is a decrease in the maximum mean effective pressure
available at any given cut Off ratio. For these reasons it is probably
undesirable to increase either the displacement or the clearance volume
ratio above a value Of one. Further study of the effects of displacement
ratios greater than one might be warranted for an application where

maximum burner pressures are to be restricted.

4.6 Engine Performance and Concluding Remarks

The Ojective of this investigation was to study the performance
characteristics of a positive displacement engine Operating on a
modified Brayton cycle. The effects of isolated changes in engine
variables on mean effective pressure and thermal efficiency have been

discussed inthe preceding sections of this chapter. This concluding

98

section presents two engine performance maps and a brief evaluation of
the potential of positive displacement Brayton cycles.

Figures 22 and 23 are plots of mean effective pressure versus cut
Off ratio for a 90/1 compression ratio engine and they include lines of
constantburner pressure, constant burner exhaust temperature, constant
air—fuel ratio, and constant thermal efficiency. Both figures are for
engines with displacement ratios and clearance volume ratios equal to
one and both include the effects of dissociation and variable Specific
heats.

Figure 22 is a plot of maximum possible performance and does not
include any heat loss or valve pressure losses. Figure 23 includes the
effects of both heat losses and valve pressure losses, and the losses
are those listed in column 2 of Table 6 (page 69). These loss values

are repeated below for convenience.

Compressor intake valve loss .696 psia
Compressor exhaust valve loss 6.04 percent

Power cylinder intake valve loss 3.00 percent
Combustor heat loss 9.40 percent Of the

lower heating
value of the
fuel.

Pressure losses are generally a function of engine Speed and the
losses used are probably high for low load, low speed Operation and may
be slightly low for a maximum output condition. The compressor exhaust
valve loss includes a combustor pressure loss estimate of 3 percent.
The combustor heat loss estimate may be slightly low at low speed and

load and somewhat high at maximum power. Improved estimates of the

engine losses could only be Obtained by a detailed analysis of engine

Oq.

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(ISd) alnssela anrnaaggg use“

101

hardware and such an analysis is beyond the scope of this study. These
losses are not considered by the auther to be overly Optimistic and
Should provide a reasonable estimate of the performance of the modified
Brayton cycle engine for comparison with conventional positive displace-
ment engines.

Table 8 is presented for readers who prefer specific fuel
consumption instead of thermal efficiency as an engine efficiency
parameter. The indicated specific fuel consumption (ISFC) is based on

the lower heating value of normal octane.

 

TABLE 8

THERMAL EFFICIENCY CONVERSION TO INDICATED
SPECIFIC FUEL CONSUMPTION

 

 

Thermal Efficiency ISFC
Z lb./hp.-hr.
30 .444
35 .381
40 .333
45 .296
50 .266
55 .242
60 .222

 

The analysis and the calculated performance data presented in this
study are not restricted to a piston cylinder engine, however, for
comparison with conventional power plants, a piston cylinder engine
configuration will be considered. The simplest arrangement of a piston

cylinder engine would be designed with a fixed cut Off ratio ( i.e.,

102

fixed power cylinder valve Opening volumes).

If we consider the operation of a fixed cut Off ratio engine, it
is apparent from Figures 22 and 23 that the engine thermal efficiency
changes by less than 5 percent between minimum and maximum power at any
fixed cut off ratio. This is, Of course, a highly desirable engine
characteristic.

For piston cylinder hardware, it could prove difficult to achieve
cut Off ratios less than .15 (Reference 4 uses .14 as a cut off ratio)
and this value will be used for comparison with conventional engines.
The conventional engine data for comparison with the data presented in
Figure 23 was taken from Taylor and Taylor [36]. Although the data in
Taylor and Taylor [36] is at least ten years Old, it is considered valid
since recent spark ignition engines have undergone a decrease in
performance due to the addition of exhaust emission controls.

Figure l7-3 of Reference 36 indicates automotive Spark and
compression ignition engines have maximum two-stroke brake mean
effective pressures ranging from 45 to 65 psi. A maximum two-stroke
indicated mean effective pressure of 82 psi is shown in Figure 23 of
this study. Considering the frictional effects, which could produce
brake mean effective pressures from 55 to 85 percent of the indicated
mean effective pressure, the positive displacement Brayton cycle engine
would have a brake mean effective pressure from 45 to 70 psi. This
comparison indicates that, with respect to power per unit of displaced
volume, the modified Brayton cycle is probably competitive with
conventional engines.

Figure 18-10 of Reference 36 indicates that automotive spark

103

ignition engines have brake Specific fuel consumption from .70 to
.52 1b./hp.-hr. at full throttle, minimums of slightly less than
.5 lb./hp.-hr. and part load brake specific's as high as 4.0 lb./hp.-hr.
The .15 cut off ratio engine with the losses assumed in Figure 23 has
indicated Specific fuel consumptions from .300 to .333 lb./hp.-hr. over
the entire power range. The ratio of brake to indicated specific fuel
consumption is inversely prOportional to the ratio Of brake to indicated
mean effective pressure at any given operating condition for an engine.
Using this prOportionality, and an estimated friction mean effective
pressure of 15 percent of the indicated mean effective pressure, the
brake Specific fuel consumption range would be from .35 lb./hp.-hr. to
.39 1b./hp.-hr. over the entire power range. If we use the 45 percent
estimate of friction mean effective pressure the brake specific fuel
consumptions vary from .55 to .60 lb./hp.-hr. over the entire power
range. This means that we might logically eXpect brake specific fuel
consumptions to range from a minimum of .35 to a high of .6 1b./hp.—hr.
If these estimates are reasonable they show significant improvement in
minimum brake specific fuel consumption and a dramatic reduction in
part load brake specific fuel consumption over that of conventional
spark ignition engines.

It is Obvious from Figure 23 that high compression ratios, low
cut Off ratios, high temperatures, high burner pressure, and the
wide range of air-fuel ratios required will present the designer with
some challenging problems. None of the problems, however, appear to
be insurmountable.

The positive displacement Brayton cycle investigated in this study

104
can achieve power per unit volume that is competitive with conventional
passenger car engines and specific fuel consumptions that are superior
to conventional engines. These two factors, coupled with the
possibility of significantly reduced exhaust gas pollutants, indicate
that this power plant should receive serious consideration as a possible

alternative to our present vehicular engines.

A

APPENDIX A

APPENDIX A

POLYNOMIAL APPROXIMATION BY THE METHOD OF LEAST SQUARES

A study of the problem of curve fitting leads to the possibility
of two somewhat different interpretations. In one case we could require
finding the equation of a curve which will pass rigorously through each
point of a set. In the other case, we could weaken this requirement and
ask for a simpler curve which can not possibly pass exactly through each
point, but which comes "as close as possible" to each point. The measure
of "as close as possible" is usually taken to be the least-square cri-
terion, and the process of applying it is known as the method of least
squares. This method is not limited to use with polynomials, where the
minimizing or normal equations are linear, but we will restrict our
discussion to this application. For a complete treatment of least squares
consult a work on numerical analysis or a text on engineering mathematics
such as Wylie [1].

If f(x) is Specified by a set of m discrete points of equal reli-
ability, x1, x2, ...xm, an n'th degree polynomial may be used to approxi-
mate f(x).

f(x) = a0 + alx + a2x2 + .... = :E:akxk = y(x)
=0

If we evaluate the above polynomial for some xi we will in general
not get f(xi) but instead get y(xi) which differs from f(xi) by 6i' In

other words

f(xi) - Y(xi) = 6i # 0

105

106

If we Simply take the sum of the Qi's as a measure of how closely
y(x) approximates f(x) we could arrive at an unwarranted impression of
accuracy due to cancelling of positive and negative G's. To avoid this
we use the sum of the squares Of the 6's as our criterion for curve
fitting. The method of least squares defines the ak's such that the
sum of the 62's is a minimum. This can be stated formally as follows:

m
E7- = :(Oi)2 = Z [f(xi) - : akxfi]2 = Minimum
I=1

I=1

-O
For a specified set of XI'S’ the above expression is a function of the
ak's only.

To find the minimizing values for the ak's, we apply the usual
conditions for minimizing a function of several variables, and equate

the partial derivatives to zero. That is to say

3E2 _ 3E2 3E2 _
3a — 0, 3a] 0, "°aa — 0
O n

Letting f(xi) = yi we can express the sum of the squares as

m m .
72
E2 = ' 2 = + o + 2- + o o o o "' o 2
Z (67,) 2(80 alx’b 82x7, £7an 7’1)
i=1 i=1

If we differentiate and set the first partial derivatives equal to zero

we Obtain the following equations.

m
SE: 2 n _
Ba 22(70 + 31x1: + 32x1: + anxi yi) l - 0
0 i=1
.3;
3E2 2 n
.77—3a1 A 2(ao + 31x1: + azxi + anxi yi) xi 0
I=1
m
2
figs—=ZZ(3+ax.+ax2.+...axr.‘-y.)xr.z=0
Ban 0 1 I 2 I n I I I

I=1

107

We have now Obtained a system of n + 1 linear equations. These equations
are known as the minimizing or normal equations and their Solution is
straightforward.

There are currently available a number of least squares polynomial
computer programs. The author is including the program used in this work
in Appendix B which will fit 2 through 100 points with a polynomial of

degree 1 through 11.

In this program standard error Of estimate is defined as:

 

 

 

m _ 1
I" 2 —
2% 2
I'--1
STANDARD ERROR — (m _ (n+1) )
.. _J

APPENDIX B

APPENDIX B

COMPUTER PROGRAM FOR LEAST SQUARES POLYNOMIALS

A computer program utilizing the method of least squares for
determining a polynomial approximation of tabular data is presented in
this appendix. This program is part of a library of scientific sub-
routines. It is written in FORTRAN IV and the author's only contri—
bution was format changes which allow the computer output to be used
directly as tables in Appendices C and D. It is included only to pro-

vide complete documentation Of the programs used in this thesis.

108

[JOB

55
60
TO

310

316

320

322

326

326
330

352

362

378
380

109

GO

DIMENSION AIISI.8(IS).S(IS).G(153.U(15)
DIMENSION QIIOOI.P(IOO).X(100).Y(100).C(100)
READISIfloN

1:0

O‘I

K812

N=N01

IFIN-III5595S.576

IFIH-NI616.60.60

IFIN-IOOI?O.70,570

T731

T881

H781

DO 310 l=l.H

READISIXII).Y(IJ

HT=H7*XII)

T7=T7tYIII

T8=T80YII)“2

CONTINUE

T98IH’T8-T7**2)/(H’*2-H)

HRITEI6.316I

FORMATII' L E A S T - S Q U A R E S P O L Y N O N l A L S'./)
HRITEI6.3ZOIH

FORMATI' NUMBER OF POINTS =‘.IIZI
AHEAN=HTIM

HRITEI6.3ZZIAHEAN

FORMATII' MEAN VALUE OF X ='.E12.SI
YHEAN'TTIN

HRITEI6.32§IYNEAN

FORMATII' MEAN VALUE OF Y ='.E12.5)
STOE‘SQRTIT9) '
HRITEI6Q326ISTDE

FORNATII' STD ERROR OF Y 8'.E12.5I
HRITEI6933OI

FORMATII' NOTE. . .CODE FOR "HHAT NEXT?" IS'.I.9X.

I'O - STOP PROGRAN'./.9X.‘I = COEFFICIENTS ONLY‘.I.9X.
2‘2 3 ENTIRE SUNNARY'./.9X.'3 = FIT NEXT HIGHER DEGREE7DIII

00 352 I'le
FIII‘Z
QIII‘O
CONTINUE

00 362 I'loII
AIII'l

OIII‘Z

SIII'I
CONTINUE
El'l

F132

NIIN

NQIK

I'I

KI'Z
IFINI37803809378
KI'NQ

“'1

00 386 L'IpN

386

392

398

#16

£28
632

«.4
«a
«a

660
‘62
#64

#82

$8.
492

#96
49B

110

N=H§YILI‘QILI
CONTINUE

SIII‘N/NI
IEII’N4I39294280428
E131

DO 398 L=IOH
EI‘EI’XILI‘QILI‘QILI
CONTINUE

EI'EI/HI

AII‘II3EI

N31

00 ‘16 L=IoN
V8IXILI-EII.QILI-F1‘PILI
PILI‘QILI

QILI‘V

NtHOV‘V

CONTINUE

FI‘H/HI

BII§2I3FI

HI‘H

I3I9I

GO TO 380

00 “32 L3201I
GILI*Z

CONTINUE

GIII30

DO 465 J=ION

SI‘Z

OO “48 L‘IoN
IFIL’II45404‘69444
GIL)‘GILI’AILI’GCL’II-BILI‘GIL’ZI
SI‘SI‘SILI‘GILI
CONTINUE

UIJI‘SI

L3N

OO ‘60 I2=ZoN
GILI3GIL-II

L3L‘I

CONTINUE

GIII‘Z

CONTINUE

Til

O0 588 L'IoN
CILI‘I

J3N

OO ‘82 I231!"
CILI‘CILI‘XILI’UIJI
J‘J‘I

CONTINUE
T33YIL2’CIL)
T3T0T3.‘2

CONTINUE
IFIN-NI496.‘9Z.496
T530

50 TO ‘98
T53T/IN-NI
QT’I’ITS/T9I
NN'N‘I

502

512
516
516
517
520

526
526

531
530

568
560

566
56?
550
556
558
560
566
568
570
572

576
578

616
618
628

111

HRITEI69502INN907
FORMATI/I' POLYFLT OF DEGREE'.I3.' INDEX OF DETERM 8'9F10.4.

Illo'HHAT NEXT 'I

READI9IIR

IFIIRI516962805I6

IFIIR-33516.564'516

HRITEI69517I
FORNATI/1911Xc'TERM'.4X.'COEFFICIENT'.II
DO 526 JSIoN

I28J-I

HRITEI60526II29UIJ)

FORNATIIIS.E15§5)

CONTINUE

IFIIR-1I5319558.531

HRITEI6o5303
FORMATI/I.6X.'X~ACTUAL‘.6X.'Y-ACTUAL'.8X.'Y-CALC'.7X.'DIFF'.8X.

1'PCT¢DIFF'.II

DO 550 L319"

Q88YILI-CILI

IFICILI)546.568.544
HRITEI6¢560IXILIoYILInCILIuQB
FORMATI6E15.6.6Xp‘INFINITE')

GO TO 550

CL3100.008/CILI
URITEIboSbTIXILIwYILIaCILIoQBoCL
FORMATISEIfiobI

CONTINUE

STSISORTITSI

URITEI60556IST5

FORMATI/I' STD ERROR OF ESTIMATE FOR Y ='oF10.¢/I
IFIK-NI55896280558

HRITEI6¢56OI

FORMATI' HHAT NEXT7'I

READI9IIR

GO TO 512

N‘NO1

IFIM-NI61605689568

GO TO 628

NRITEI6.572I

FORNATI'PROGRAM SIZE LIMIT IS 100 DATA POINTso'I
GO TO 628

HRITEI69578I

FORNATI'ELEVENTH DEGREE IS THE LIMIT'I
GO TO 628

NN‘N-I

HRITEI6o618INN

FORMATI' TOO FEH POINTS FOR FITTING DEGREE'oISI
GO TO 628

STOP

END

[DATA

APPENDIX C

APPENDIX C

POLYNOMIALS FOR CONSTANT PRESSURE SPECIFIC HEATS

This appendix contains a complete summary of the polynomials used
to approximate the specific heat data from the "JANAF Tables" [19].
Tables C.l through C.19 are direct output from the program presented
in Appendix B and contain the actual Cp's, the Cp's calculated from
the polynomial approximations, the difference and per cent difference
between the actual and calculated values, and the constants for the
polynomials. The standard error of estimate as defined in Appendix A
is also included in each table. Tables are included for all Species
considered in this thesis except normal octane which is always injected
at 298.15°K and atomic hydrogen whose specific heat remains constant
at 4.986 cal/mole-°K.

In these tables

CP = Aw + Al*T + A2*T**2 + A3*T**3

is the FORTRAN equivalent of
Cp - a0 + alT + a2T + a3T

which is used in the text of this thesis.

112

113

TABLE C.l
DATA AND EQUATION FOR Cp OF 02, 298 T0 1500°K

L E A S T - S Q U A R E S P O L Y N O M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT 0F DEGREE 5 INDEX OF DETERM : 0.9982
CP = A0 + A1*T + A2*T**2 + A5*T**5

TERM COEFFICIENT

A0 0.618968 01

A1 0.29278E-02

A2 -0.71009E-06

A5 -0.78607E-10
T-KELVIN CP-ACTUAL CP-CALC DIFF PCT-DIFF
298.00 7.020000 6.996927 0.025072 0.529748
500.00 7.025000 7.001891 0.021109 0.501470
400.00 7.195999 7.242056 -0.046057 -0.655962
500.00 7.451000 7.466155 -0.055155 -0.470570
600.00 7.669999 7.675650 -0.005651 ~0.047574
700.00 7.882999 7.864155 0.018865 0.259882
800.00 8.065000 8.057117 0.025885 0.522040
900.00 8.212000 8.192125 0.019876 0.242629
1000.00 8.555999 8.528685 0.007515 0.087825
1100.00 8.458999 8.446527 -0.007528 -0.086760
1200.00 8.526999 8.544580 -0.017580 -L.205745
1500.00 8.605999 8.622971 -0.018971 -0.220011
1400.00 8.674000 8.681029 -0.007050 -0,080976
1500.00 8.758000 8.718285 0.019717 0.226159

STD ERROR OF ESTIMATE FOR CP 2

0.3264

114

TABLE C.2
DATA AND EQUATION FOR Cp OF 02, 1500 TO 3500°K

L E A S T - S O U A R E S P O L Y N 0 M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE INDEX OF DETERM 3 1.8888
CP 2 A8 + A1*T + A2*T**2 + A5*T**5

TERM COEFFICIENT

A8 8.78858E 81

A1 8.52414E-85

A2 8.58596E-87

A5 -8.15471E'18
T'KELVIN CP-ACTUAL. CP'CALC DIFF PCT-DIFF
1688.88 8.799999 8.798745 8.881254 8.814255
1788.88 8.858888 8.856858 8.881158 8.812986
1888.88 8.915999 8.914595 8.881487 8.815779
1988.88 8.975888 8.971892? 8.881187 8.812541
2888.88 9.828999 9.828668 8.888551 8.885665
2188.88 9.884888 9.884848 -8.888848 -8.889248
2288.88 9.159888 9.148526 '8.881527 '8.814515
2588.88 9.195999 9.195845 '8.881846 '8.811578
2488.88 9.247999 9.248919 '8.888919 ‘8.889948
2688.88 9.555999 9.555888 8.888199 8.882151
2788.88 9.485888 9.484647 8.888555 8.885752
2888.88 9.455888 9.454525 8.888677 8.887162
2988.88 9.582999 9.582748 8.888251 8.882659
5888.88 9.551888 9.549841 8.881159 8.812155
5188.88 9.596888 9.595521 8.808479 8.884989
5288.88 9.659999 9.659788 8.888292 8.885827
5588.88 9.681999 9.682528 '8.888528 -8.885589
5488.88 9.725888 9.725277 '8.888278 '8.882854
5588.88 9.761999 9.762497 '8.888498 '8.885899

STD ERROR 0F ESTIMATE FOR CP

: 8.9811

115

TABLE C.3

DATA AND EQUATION FOR Cp OF N2, 298 TO 1500°K

L E A S T - S Q U A R E S P O L Y N O M I A L S

POLFIT OF DEGREE 5 INDEX OF DETERM : 0.9985
CP 2 A8 + A1*T + A2*T**2 + A5*T**5

TERM COEFFICIENT

A0 0.70498E 01

A1 -0.11450E-02

A2 0.50422E-05

A3 -0.11464E-08

T-KELVIN CP-ACTUAL CP-CALC DIFF PCT-DIFF
298.00 6.960999 6.948456 0.012544 0.180525
300.00 6.960999 6.949189 0.011810 0.169952
400.00 6.990000 7.005229 -0.015229 -0.217598
500.00 7.068999 7.094601 -0.025601 -0.560857
600.00 7.195999 7.210425 -0.014426 -0.200075
700.00 7.549999 7.545825 0.004174 0.056825
800.00 7.511999 7.495921 0.018078 0.241255
900.00 7.669999 7.647857 8.022162 0.289787
1000.00 7.815000 7.800692 0.014508 0.185419
1100.00 7.945000 7.945608 -0.000608 '0.007658
1200.00 8.061000 8.075709 -0.014709 -0.182145
1500.00 8.162000 8.184114 -0.022115 -0.270216
1400.00 8.252000 8.265947 -0.011947 '8.144564
1500.00 8.550000 8.508528 0.021672 0.260850
STD ERROR 0F ESTIMATE FOR CP 2 0.0194

FOR CONSTANT PRESSURE SPECIFIC HEATS

116

TABLE C.4
DATA AND EQUATION FOR Cp OF N2, 1500 TO 3500°K

L E A S T - S Q U A R E S P O L

POLFIT

T-KELVIN

1500.00
1600.00
1700.00
1800.00
1900.00
2000.00
2100.00
2200.00
2500.00
2400.00
2500.00
2600.00
2700.00
2800.00
2900.00
5000.00
5100.00
5200.00
5500.00
5400.00
5500.00

OF DEGREE

TERM

A0
A1
A2
A5

CP-ACTUAL

8.550000
8.598000
8.457999
8.511999
8.558999
8.601000
8.658000
8.672000
8.702999
8.751000
8.756000
8.778999
8.799999
8.820000
8.857999
8.855000
8.870999
8.886000
8.900000
8.914000
8.926999

INDEX OF DET

COEFFICIENT

0.64477E 01
0.19796E-02
-0.57055E-06
0.59247E-10

CP-CALC

8.555880
8.597759
8.455921
8.508718
8.556508
8.599646
8.658486
8.675586
8.704699
8.752781
8.757989
8.780677
8.801202
8.819916
8.857178
8.855544
8.868767
8.885802
8.898808
8.914157
8.950145

STD ERROR OF ESTIMATE FOR CP

Y M O M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

CP : am + A1*T + A2*T**2 + A5*T**3

ERM : 0.9999
DIFF PCT-DIFF
-0.005881 -0.046565
0.000240 0.002862
0.002078 0.024575
0.005281 0.058556
0.002491 0.029112
0.001554 0.015747
~0.000486 -0.005650
-0.001586 -0.0l5976
-0.001699 -0.019525
-0.001781 -0.020400
~0.00l989 ~0.022715
-0.001678 ~0.019105
-0.001205 -0.015664
0.000084 0.000952
0.000821 0.009292
0.001656 0.018700
0.002255 0.025175
0.002197 0.024755
0.001192 0.015596
-0.000157 -0.001541
-0.005146 -0.055251
: 0.0022

117

TABLE C.5
DATA AND EQUATION FOR Cp OF C02, 298 TO 1500°K

L E A S T -

S Q U A R E S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE INDEX OF DETERM :

CP : A0 + 41*T + AZ*T**2 + A5*T**5

TERM COEFFICIENT

A0 0.515578 01

Al 0.15558E-01

A2 -0.98480E-05

A5 0.25685E-08

T-KELVIN CP-ACTUAL CP-CALC DIFF
298.00 8.874000 8.892426 -0.018426
500.00 8.896000 8.912595 -0.016595
400.00 9.877000 9.844629 0.052571
500.00 10.666599 10.656549 0.050050
600.00 11.509999 11.502565 0.007457
700.00 11.846000 11.856885 -0.010885
800.00 12.292999 12.515720 -0.020720
900.00 12.667000 12.687285 ~0.020285
1000.00 12.980000 12.991788 -0.011788
1100.00 15.242999 15.241456 0.001565
1200.00 15.466000 15.450449 0.015551
1500.00 15.655999 15.655051 0.022968
1400.00 15.815000 15.805595 0.011604
1500.00 15.952999 15.975754 -0.022755
STD ERROR OF ESTIMATE FOR CP : 0.0227

P O L Y N O M I A L S

0.9998

PCT-DIFF

'0.207209
”0.186175
0.528814
0.282519
0.065797
-00091789
“00168271
-00159867
-60090757
0.011804
0.115614
0.168475
0.084068
-00162815

118

TABLE C.6
DATA AND EQUATION FOR Cp 0F C02, 1500 TO 3500°K

L E: A 53 T - S (D U 11 R 15 S P (3 L 1' N (3 M 11 A 1. S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 5 INDEX OF DETERM : 8.9998
CP : A8 + A1*T + A2*T**2 + A5*T**5

TERM COEFFICIENT

A8 8.18788E 82

A1 0.54197E-02

A2 -8.98951E-86

A5 8.185978-89

T-KELVIN CP-ACTUAL CP-CALC DIFF PCT-DIFF
1588.88 15.952999 15.961965 -8.888965 -8.864214
1688.88 14.875999 14.872146 8.881855 8.815168
1788.88 14.176999 14.172519 8.884488 8.851615
1888.88 14.268999 14.265786 8.885295 0.857107
1988.88 14.551999 14.546551 8.885669 0.059515
2888.88 14.424888 14.421818 8.882982 8.828679
2188.88 14.488999 14.488591 8.888688 0.034200
2288.88 14.547888 14.549875 -8.882875 ~0.814250
2588.88 14.599999 14.605689 -8.885698 -8.825266
2488.88 14.648888 14.652865 -8.884865 -8.855187
2588.88 14.691999 14.697216 -8.885217 -8.855494
2688.88 14.755999 14.757575 -8.885576 -8.822988
2788.88 14.771888 14.775962 -8.882962 -8.020050
2888.88 14.886999 14.887682 -8.888685 -z.084078
2988.88 14.841888 14.858919 8.882881 8.014825
5888.88 14.872999 14.868554 8.884465 8.858851
5188.88 14.901999 14.897075 8.884925 0.055059
5288.88 14.929999 14.925162 8.884857 8.852489
5500.00 14.955999 14.955421 8.882579 0.017245
5488.88 14.981999 14.982475 -8.888476 -0.085176
5588.88 15.886888 15.812948 -8.886948 '0.046285
STD ERROR OF ESTIMATE FOR CP : 8.8848

119

TABLE C.7
DATA AND EQUATION FOR Cp OF H20, 298 TO 1500°K

L E A S T - S Q U A R E S P O L Y N O M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 3 INDEX OF DETERM = 8.9999
CP : A8 + 41*T + A2*T**2 + A3*T**3

TERM COEFFICIENT

A8 8.77474E 81

41 8.88868E-84

82 8.38752E-85

A3 -8.18556E-88

T-KELVIN CP-ACTUAL CP-CALC DIFF PCT-DIFF
298.88 8.825888 8.816683 8.888396 8.184734
388.88 8.826999 8.819876 8.887123 8.888817
488.88 8.186888 8.284168 -8.818168 -8.221454
588.88 8.415888 8.424638 -8.889638 -8.114318
688.88 8.676888 8.674927 8.881873 .8.812368
788.88 8.954888 ‘ 8.948728 8.885272 8.858912
888.88 9.245999 9.239697 8.886382 8.868284
988.88 9.547888 9.541583 8.885497 8.857611
1888.88 9.851888 9.847812 8.883188 8.832374
1188.88 18.151999 18.152289 -8.888298 -8.882856
1288.88 18.443999 18.448683 -8.884683 -8.844857
1388.88 18.723888 18.738419 -8.887428 -8.869145
1488.88 18.987888 18.991484 -8.884484 -8.848868
1588.88 11.233888 11.225222 8.887778 8.869292
STD ERROR OF ESTIMATE FOR CP : 8.8898

120

TABLE C.8
DATA AND EQUATION FOR C? OF H20, 1500 TO 3500°K

L E A S T - S O U A R E S P O L Y M O M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 3 INDEX OF DETERM 2 1.8888
CP 2 A“ + A1*T + A2*T**2 + A3*T**3

TERM COEFFICIENT

A8 8.53623E “1

Al 8.58813E-82

A2 -8.14617E-85

43 8.13668E-89
T-KELVIN CP-ACTUAL CP-CALC DIFF PCT-DIVE
1588.80 11.233888 11.236883 -9.003883 -8.833846
1688.89 11.462888 11.462369 -8.088369 -O.883228
1788.8” 11.674888 11.671818 8.882182 8.218695
1888.88 11.868999 11.865976 8.883823 8.825477
1988.88 12.847999 12.845663 8.882337 8.819397
2888.28 12.214888 12.211699 8.882381 8.818844
2188.88 12.365999 12.364897 8.881182 8.888916
2288.88 12.584999 12.586888 -O.881889 ~8.888789
2388.88 12.634888 12.636887 -fl.882888 -8.816521
2482.88 12.752999 12.755715 -9.882716 ~8.021293
2588.88 12.863888 12.865786 -8.882786 -G.8?1652
2688.88 12.964999 12.967138 -3.282131 -0.0l6432
2788.88 13.858999 13.868563 -2.881564 -8.811975
2288.88 13.146888 13.146983 -O.888983 -8.886872
2988.80 13.228288 13.226966 8.281234 8.837816
3888.88 13.384888 13.381584 8.882416 8.218161
3188.28 13.374888 13.371572 8.882428 8.818158
3288.WO 13.441888 13.437747 8.883253 8.824288
3328.82 13.582999 13.588933 8.892867 8.815387
3498.88 13.561999 13.561936 8.880863 8.888464
3588.88 13.617888 13.621598 -8.884599 -8.033768

STD ERROR OF ESTIMATE FOR CP = 8.9325

121

TABLE C.9
DATA AND EQUATION FOR Cp OF CO, 298 TO 1500°K

L E A S T - S Q U A R E S P O L Y N O M l A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 3

INDEX OF DETERM : 8.998
CP : A8 + A1*T + A2*T**2 + A3*T**3

TERM COEFFICIENT

8 8.69187E 81

A1 -8.57811E-83

A2 8.26491E-85

A3 '8.18737E-88
T-KELVIN CP-ACTUAL CP-CALC DIFF PCT-DIFF
298.88 6.964999 6.947684 8.817515 8.249218
388.88 6.964999 6.949136 8.815863 8.228279
488.88 7.815888 7.857836 '8.824837 -8.552988
588.88 7.128999 7.155749 “8.832758 -8.457884
688.88 7.275999 7.298434 -8.8l4455 “[0197997
788.88 7.458888 7.441448 8.888552 8.114917
888.88 7.624888 7.688348 8.823652 8.311197
988.88 7.785999 7.768694 8.825586 8.326876
1888.88 7.931888 7.916848 8.814959 8.180975
1188.88 8.856999 8.859948 -8.882949 -8.836585
1288.88 8.167999 8.185971 -8.817972 -8.219546
1588.88 8.263888 8.287678 '8.824671 -8.297672
1488.88 8.546888 8.358684 -8.812685 -8.158799
1588.88 8.417888 8.392327 8.824673 8.293989

STD ERROR OF

ESTIMATE FOR CP

: 8.8238

122

TABLE C.10

DATA AND EQUATION FOR cp OF co, 1500 to 3500°K

L E A S T “ S Q U A R E S

POLFIT OF DEGREE 5

T-KELVIN'

1500.00
1600.00
1700.00
1800.00
1900.00
2000.00
2100.00
2200.00
2500.00
2400.00
2500.00
2600.00
2700.00
2800.00
2900.00
3000.00
3100.00
3200.00
3300.00
3400.00
3500.00

CP : A0 + AI*T + A2*T**2 +

P 0 L Y N

INDEX OF DETERM :

TERM COEFFICIENT

A0 0.67060E 01

41 0.180508-02

A2 -0.52112E-06

A5 0.54585E-l0
CP-ACTUAL CP-CALC
8.417000 8.421507
8.480000 8.479468
8.555000 8.552229
8.582999 8.580115
8.625999 8.625452
8.664000 8.662567
8.698000 8.697784
8.728000 8.729455
8.756000 8.757857
8.780999 8.785525
8.804000 8.806222
8.825000 8.826855
8.844600 8.845547
8.865000 8.862629
8.879000 8.878425
8.895000 8.895261
8.910000 8.907464
8.924000 8.921561
8.956999 8.955277
8.948999 8.949559
8.960999 8.964475

STU ERROR OF ESTIMATE FOR OR

FOR CONSTANT PRESSURE SPECIFIC HEATS

0 M I A L S

0.9998
AS*T**5
DIFF PCT“DIFF
“0.004507 “0.053519
0.000531 0.006265
0.002770 0.032470
0.002884 0.033612
0.002547 0.029539
0.001432 0.016536
0.000216 0.002478
“0.001433 “0.016420
“0.001838 “0.020984
“0.002326 “0.026482
“0.002222 “0.025233
“0.001547 “0.017487
0.000371 0.004186
0.000575 0.006477
0.001739 0.019549
0.002536 0.028468
0.002639 0.029579
0.001722 0.019276
“0.000540 “0.006031

: 0.0024

123

TABLE C.ll
DATA AND EQUATION FOR Cp OF H2, 298 T0 1500°K

L E A S T - S O U A R E S P O L Y N O M I A L S
FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 5

INDEX OF DETERM : 0.9961
CP = A0 + AI*T + AZ*T**2 + A3*T**5

TERM COEFFICIENT

A0 0.68053E 01

A! 0.38441E-03

42 “0.21831E“06

A3 0.24845E-09

I“KELVIN CP“ACTUAL CP“CALC DIFF PCI“DIFF
300.00 6.893999 6.907684 “0.013685 “0.198116
400.00 6.974999 6.940037 0.034963 0.503782
500.00 6.992999 6.973985 0.019014 0.272647
600.00 7.009000 7.011021 “0.002021 “0.028824
700.00 7.035999 7.052635 “0.016636 “0.235882
800.00 7.087000 7.100317 “0.013317 “0.187557
1000.00 7.219000 7.219851 “0.000852 “0.011796
1100.00 7.299999 7.294683 0.005316 0.072872
1200.00 7.389999 7.381548 0.008451 0.114494
1300.00 7.490000 7.481935 0.008065 0.107796
1400.00 7.599999 7.597333 0.002666 0.035097
1500.00 7.719999 7.729236 “0.009236 “0.119499
STD ERROR 0F ESTIMATE FOR CP 2 0.0166

124

TABLE C.12

DATA AND EQUATION FOR Cp OF H2, 1500 TO 3500°K

L E A S T

POLFIT OF DEGREE

T-KELVIN

1500.00
1600.00
1700.00
1800.00
1900.00
2000.00
2100.00
2200.00
2300.00
2400.00
2500.00
2600.00
2700.00
2800.00
2900.00
3000.00
3100.00
3200.00
3300.00
3400.00
3500.00

STD ERROR

TERM

A0

A1

A2
A3

CP-ACTUAL

7.719999
7.823000
7.921000
8.016000
8.108000
8.195000
8.278999
8.358000
8.433999
8.506000
8.575000
8.639000
8.700000
8.757000
8.809999
8.858999
8.910999
8.962000
9.011999
9.061000
9.110000

- S Q U A R E S P O L

INDEX OF DETERM 2

COEFFICIENT

0.53309E 01
0.21733E-02
-0.44619E-06

CP-CALC

7.715560
7.822051
7.923276
8.019465
8.110847
8.197649
8.280101
8.358431
8.432868
8.503640
8.570977
8.635107
8.696257
8.754659
8.810538
8.864126
8.915650
8.965339
9.013421
9.060125
9.105681

OF ESTIMATE FOR CP 2

Y N 0 M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

CP 2 A0 + A1*T + A2*T**2 + A3*T**3

0.9999
DIFF PCT-DIFF
0.004439 0.057538
0.000949 0.012131
“0.002276 “0.028731
“0.003466 “0.043216
“0.002848 “0.035109
“0.002649 “0.032318
0.001131 0.013412
0.002359 0.027746
0.004023 0.046933
0.003893 0.045082
0.003743 0.043043
0.002341 0.026743
“0.000539 “0.006116
“0.005127 “0.057839
-0.004651 “0.052167
“0.003339 “0.037241
~0.001422 “0.015776
0.000875 0.009652
0.004318 0.047424
0.0034

125

TAEUEC.13
DATA AND EQUATION FOR Cp OF OH, 298 TO 1500°K

L E A S T - S Q U A R E S P O L Y N O M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 3 INDEX OF DETERM : 0.9999
CP : A0 + A1*T + A2*T**2 + A3*T**5

TERM COEFFICIENT

A0 0.76080E 01

A1 -0.23624E-02

A2 0.28685E-05

A3 ~0.78612E-09
T-KELVIN CP-ACTUAL. CP-CALC DIFF PCT-DIFF
298.00 7.136000 7.137953 “0.001953 -0.027363
300.00 7.134000 7.136237 -0.002237 -0.031352
400.00 7.077000 7.071713 0.005286 0.074752
500.00 7.049000 7.045693 0.003306 0.046928
600.00 7.052999 7.053461 “0.000462 “0.006544
700.00 7.087000 7.090300 “0.003300 -0.046538
800.00 7.148000 7.151491 “0.003491 -0.048821
900.00 7.231999 7.232320 -0.000320 -0.004431
1000.00 7.329000 7.328070 0.000930 0.012689
1100.00 7.436000 7.434024 0.001976 0.026581
1200.00 7.547999 7.545465 0.002535 0.033595
1300.00 7.657000 7.657677 “0.000677 -0.008842
1500.00 7.865999 7.865542 0.000457 0.005808

STD ERROR 0F ESTIMATE FOR CP : 0.0029

L E A S T “ S O

POLFIT OF DEGREE

T-KELVIN

1500.00
1600.00
1700.00
1800.00
1900.00
2000.00
2100.00
2200.00
2300.00
2400.00
2500.00
2600.00
2700.00
2800.00
2900.00
3000.00
3100.00
3200.00
3300.00
3400.00
3500.00

126

TABLE C.14
DATA AND EQUATION FOR Cp OF OH, 1500 TO 3500°K

U

R E S

P O L Y N 0

FOR CONSTANT PRESSURE SPECIFIC HEATS

INDEX OF DETERM :

CP : A0 + A1*T + A2*T**2 + A5*T**3

TERM

CP-ACTUAL

7.865999
7.962999
8.051999
8.136000
8.212999
8.285000
8.351999
8.414000
8.469999
8.523000
8.573000
8.620999
8.664000
8.704000
8.742000
8.778000
8.811000
8.842999
8.872999
8.900999
8.929000

A0
A1
A2
A3

COEFFICIENT

0.54676E 01
0.23292E“02
“0.56385E“06
0051721E-l@

CP-CALC

7.867273
7.962689
8.051793
8.134895
8.212308
8.284339
8.351298
8.413499
8.471250
8.524861
8.574642
8.620906
8.663960
8.704117
8.741686
8.776977
8.810301
8.841969
8.872289
8.901574
8.930133

STD ERROR OF ESTIMATE FOR CP

DIFF

“0.001274
0.000310
0.000206
0.001104
0.000691
0.000661
0.000701
0.000501

“0.001250

“0.001862

“0.001642
0.000093
0.000039

“0.000117
0.000314
0.001023
0.000699
0.001031
0.000710

-80000575

‘60091155

: 0.0010

M I A L S

1.0000

PCT-DIFF

-60016195
0.003892
0.002558
0.013576
0.008419
0.007978
0.008393
0.005951

-00014759

“0.021837

-0.019152
0.001084
0.000451

“3.001548
0.003589
0.011659
0.007934
0.011659
0.008008

“0.006460

“0.012687

127

TMBLE(L15
DATA AND EQUATION FOR Cp OF NO, 298 TO 1500°K

L E A S T - S Q U A R E S P O L Y N O M I A L S
FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT 0F DEGREE 3 INDEX OF DETERM 2 0.9957
G? 3 A0 + A1*T + AZ*T**2 '1' A3*T**3

TERM COEFFICIENT

A0 0.70069E 01

A1 “0.32731E“03

A2 0.25026E-05

A3 “0.10745E“08

T-KELVIN CP“ACTUAL CP-CALC DIFF PCT-DIFF
298.00 7.131000 7.103151 0.027848 0.392055
300.00 7.132000 7.104914 0.027086 0.381233
400.00 7.157000 7.207608 “0.050609 “0.702156
600.00 7.466000 7.479340 “0.013340 “0.178358
700.00 7.655000 7.635484 0.019516 0.255596
800.00 7.832000 7.796550 0.035450 0.454688
900.00 7.988000 7.956092 0.051908 0.401052
1000.00 8.122999 8.107661 0.015338 0.189178
1100.00 8.238000 8.244811 “0.096811 “0.082611
1200.00 8.335999 8.361094 “0.025095 “0.300140
1300.00 8.419000 8.450066 “0.031066 “0.367641
1400.00 8.490999 8.505276 “0.014277 “0.167855
1500.00 8.551999 8.520278 0.031721 0.372301
STD ERROR OF ESTIMATE FOR CP 3 0.0350

128

TABLE C.16

DATA AND EQUATION FOR Cp OF NO, 1500 TO 3500°K

L E A S T - S Q U A R E S

POLFIT OF DEGREE 3

T-KELVIN

1500.00
1600.00
1700.00
1800.00
1900.00
2000.00
2100.00
2200.00
2300.00
2400.00
2500.00
2600.00
2700.00
2800.00
.2900.00
3000.00
3100.00
3200.00
3300.00
3400.00
3500.00

P O L Y N O M I A L

FOR CONSTANT PRESSURE SPECIFIC HEATS

CP 2 A0 + A1*T + A2*T**2 +

TERM COEFFICIENT
A0 0.71197E 01
A1 0.15094E-02
A2 -0.43653E-06
A3 0.45714E-10

INDEX OF DETERM =

CP-ACTUAL CP-CALC
8.551999 8.555911
8.605000 8.604486
8.650999 8.648720
8.691999 8.688886
8.726999 8.725258
8.759000 8.758112
8.787999 8.787721
8.813000 8.814359
8.837000 8.838300
8.858000 8.859819
8.877000 8.879191
8.895000 8.896689
8.912000 8.912588
8.926999 8.927163
8.941000 8.940687
8.955000 8.953435
8.967999 8.965681
8.980000 8.977697
8.990999 8.989762
9.002000 9.002148
9.011999 9.015128

STD ERROR OF ESTIMATE FOR CP

S

0.9998

A3*T**5

DIFF PCT“DIFF
0.000513 0.005963
0.002279 0.026354
0.003114 0.035836
0.001741 0.019958
0.000888 0.010138
0.000278 0.003169
“0.001359 “0.015418
“0.001820 “0.020538
“0.002192 “0.024682
“0.001690 “0.018995
“0.000588 “0.006602
“0.000164 “0.001837
0.000313 0.003499
0.001565 0.017479
0.002318 0.025858
0.002302 0.025643
0.001237 0.013759
“0.000148 “0.001642
2 0.0021

129

TABLE C.17
DATA AND EQUATION FOR Cp OF 0, 298 TO 1500°K

L E A S T - S Q U A R E S P O L Y N O M I A L S

FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 3 INDEX OF DETERM 2 0.9891
cp : Am + A1*T + A2*T**2 + A5*T**3

TERM COEFFICIENT

A0 0.55988E 01

A1 “0.16430E“02

A2 0.14835E-05

A3 “0.44398E“09

T-KELVIN CP“ACTUAL CP-CALC DIFF PCT“DIFF

298.00 5.237000 5.229212 0.007788 0.148927
300.00 5.235000 5.227462 0.007538 0.144197
500.00 5.080999 5.092712 “0.011713 “0.229996
600.00 5.049000 5.051195 “0.002195 “0.043462
700.00 5.028999 5.023364 0.005635 0.112181
800.00 5.014999 5.006557 0.008443 0.168636
900.00 5.006000 4.998109 0.007891 0.157874
1000.00 4.999000 4.995356 0.003644 0.072948
1100.00 4(993999 4.995634 “0.001635 “0.032721
1200.00 4.990000 4.996280 “0.006280 “0.125692
1300.00 4.987000 .A.994629 “0.007629 “0.152752
1400.00 4.983999 4.988019 “0.004020 “0.080588
1500.00 4.981999 4.973783 0.008216 0.165184

STD ERROR OF ESTIMATE FOR CP

: 0.0093

130

TABLE C.18

DATA AND EQUATION FOR Cp OF 0, 1500 TO 3500°K

L E A S T - S Q U A R E S P O L Y N O M I A L S

POLFIT

T-KELVIN

1500.00
1600.00
1700.00
1800.00
1900.00
2000.00
2100.00
2200.00
2300.00
2400.00
2500.00
2600.00
2700.00
2800.00
2900.00
3000.00
3100.00
3200.00
3300.00
3400.00
3500.00

OF DEGREE

TERM

A0

A1

A2
A3

CP-ACTUAL

4.981999
4.981000
4.978999
4.978999
4.978000
4.978000
4.978000
4.978999
4.980000
4.981000
4.983999
4.985999
4.990000
4.993999
4.999000
5.004000
5.009999
5.016999
5.025000
5.040999

COEFFICIENT

0.50226E 01
“0.26032E“04
-9077646E'68

CP-CALC

4.982230
4.980676
4.979425
4.978509
4.977955
4.977792
4.978048
4.978754
4.979937
4.981626
4.983850
4.986636
4.990016
4.994017
4.998667
5.003996
5.010033
5.016806
5.024343
5.032674
5.041827

FOR CONSTANT PRESSURE SPECIFIC HEATS

CP : A0 + A1*T + A2*T**2 + A3*T**3

DIFF

-30000251
0.000324
“0.000426
0.000490
0.000045
0.000208
-0.000049
0.000245
0.000063
“0.000626
0.000150
“0.000637
“0.000016
“0.000017
0.000333
0.000004
“0.000033
0.000194
0.000657
0.000326
“0.000828

STD ERROR OF ESTIMATE FOR CP 2 0.0004

INDEX OF DETERM : 0.9996

PCT-DIFF

'30004632
0.006510
“0.008561
0.009846
0.000900
0.004177
-00000977
0.004923
0.001264
“0.012558
0.003004
“6.012775
“0.000325
“0.000344
0.006658
0.000076
“0.000666
0.003859
0.013078
0.006481
-0.016418

131

TABLE c.19
DATA AND EQUATION FOR cp OF AIR, 298 to 1500°K

L E A S T “ S Q U A R E S P O L Y N O M I A L 3
FOR CONSTANT PRESSURE SPECIFIC HEATS

POLFIT OF DEGREE 3

INDEX OF DETERM : 0.9985
G? = A0 + A1*T + A2*T**2 + A3*T**3

TERM COEFFICIENT

A0 0.68692E 01

A1 “0.29008E“03

A2 0.22547E“05

A3 “0.92234E“09
T“ KELVIH CP“ ACTUAL CP“ CALC DI FF PCT“D IFF
298.00 6.973399 6.958619 0.014780 0.212399
300.00 6.974000 6.960240 0.013760 0.197689
500.00 7.145000 7.172590 “0.027591 “0.384669
600.00 7.295500 7.307670 '“0.012170 “0.166535
700.00 7.461900 7.454638 0.007261 0.097406
800.00 7.627700 7.607963 0.019737 0.259429
900.00 7.783799 7.762111 0.021688 0.279414
1000.00 7.924399 7.911546 0.012854 0.162467
1100.00 8.048699 8.050735 “0.002036 “0.025291
1200.00 8.158799 8.174145 “0.015346 “0.187733
1300.00 8.254800 8m276240 “0.021441 “0.259061
1400.00 8.340599 8.351487’ “0.010888 “0.130373
1500.00 8.415600 8.394351 0.021249 0.253132

STD ERROR OF ESTIHATE FOR CP

3 0.0203

APPENDIX D

APPENDIX D

POLYNOMIALS FOR STANDARD FREE ENERGY OF REACTION

This appendix contains a complete summary of the polynomials used
to approximate the standard free energy (Gibbs Function) data from the
"JANAF Tables" [19]. Tables D.l through D.6 are direct output from the
program presented in Appendix B and contain the actual AFO's, the AFO's
calculated from the polynomial approximation, the difference and per cent
difference between the actual and calculated values, and the constants
for the polynomials. The standard error of estimate as defined in
Appendix A is also included in each table.

In these tables

F = A0 + A1*T + A2*T**2 + A3*T**3

is the FORTRAN equivalent of

0 = 2 3
AF a0 + alT + a2T + a3T

which is used in the text of this thesis.

The reverse of the reactions shown in Tables D.3 through D.6 were
used to determine the equilibrium composition and, therefore, the neg-
atives of the free energy polynomials shown in these tables were used

in the equilibrium constant subroutine (PDOBEK).

132

133

TABLE D.l

DATA AND EQUATION FOR AFO OF 002 + co + é— 02

L E A S T - S Q U A R E S P O L Y N O M I A L S

FOP STANDARD FREE ENERGY OF REACTION

STD ERROR OF ESTIMATE FOR

: 0.0012

POLFIT 0F DEGREE 3 INDEX OF DETERM : 1.@@g@
F 1 0.0 + 01*T + 42*Takak2 + A3*T**5

TERM COEFFICIENT

A0 0.67939E 02

A1 -0.21677E-01

“2 0.52763E-06

A3 -0.36904E-10

T-KELVIN F-ACTUAL F-CALC DIFF PCT-DIFF

1500.00 36.487000 36.485352 0.001648 0.004517
1600.00 34.454987 34.454575 0.000412 0.001196
1700.00 32.431000 32.430801 0.000198 0.000612
1800.00 30.412994 30.413803 -0.000809 “0.002659
1900.00 28.401993 28.403397 -0.001404 -0.004942
2000.00 26.398987 26.399323 -0.000336 “0.001272
2100.00 24.399994 24.401367 -0.001373 “0.005628
2200.00 22.408997 22.409317 -0.000320 50.001430
2300.00 20.425995 20.422943 0.003052 0.014943
2400.00 18.441986 18.442047 -0.000061 “0.000331
2500.00 16.466995 16.466370 0.000626 0.003799
2600.00 14.495999 14.495712 0.000287 0.001980
2700.00 12.530000 12.529861 0.000138 0.001104
2800.00 10.568999 10.568588 0.000411 0.003889
2900.00 8.610999 8.611649 -0.000649 ' “0.007542
3000.00 6.658000 6.658844 -0.000844 “0.012675
3100.00 4.709000 4.709961 -0.000961 “0.020410
3200.00 2.764999 2.764755 0.000244 0.008830
3300.00 0.824000 0.823029 0.000971 0.118025

134

7%BLEIL2
1

DATA AND EQUATION FOR AFC OF H20 + H2 + 5 02

L E A S T - S O U A R E S P O L Y N O M I A L S

FOR STANDARD FREE ENERGY OF REACTION

 

POLFIT OF DEGREE 3 INDEX OF DETERM = 1.0000
F : A0 + A1*T + A2*T**2 + A3*T**3

TERM COEFFICIENT

A0 0.59366E 02

A1 “0.12969E“01

A2 “0.32107E“06

A3 0.32091E-10
T-KELVIN F-ACTUAL F-CALC DIFF PCT-DIFF
1500.00 39.296997 39.298340 “0.001343 “0.003417
1600.00 37.926987 37.925064 0.001923 0.005069
1700.00 36.548996 36.548447 0.000549 0.001503
1800.00 35.169998 35.168686 0.001312 0.003731
1900.00 33.785995 33.785965 0.000031 0.000090
2000.00 32.400986 32.400482 0.000504 0.001554
2100.00 31.011993 31.012436 “0.000443 “0.001427
2200.00 29.620987 29.621994 “0.001007 “0.003400
2300.00 28.228989 28.229385 “0.000397 “0.001405
2400.00 26.831985 26.834778 “0.002792 “0.010406
2500.00 25.438995 25.438354 0.000641 0.002519
2600.00 24.039993 24.040344 “0.000351 “0.001460
2700.00 22.640991 22.640900 0.000092 0.000404
2800.00 21.241989 21.240250 0.001740 0.008190
2900.00 19.837997 19.838562 “0.000565 “0.002846
3000.00 18.437988 18.436035 0.001953 0.010594
3100.00 17.033997 17.032867 0.001129 0.006629
3200.00 15.629999 15.629242 0.000757 0.004845
3300.00 14.223000 14.225357 “0.002357 “0.016572

STD ERROR OF ESTIMATE FOR 2 0.0015

135

TABLE D.3
l l

0
DATA AND EQUATION FOR AF OF '2- 02 + “2- H2 + OH

L E A S T “ S O U A R E S P O L Y N O M I A L S

“FOR STANDARD FREE ENERGY OF REACTION
POLFIT 0F DEGREE 3 INDEX OF DETERM : 1.0000
F : A0 + A1*T + A2*T**2 + A3*T**3

TERM COEFFICIENT

A0 0.91446E 01
A1 “0.36978E“02
A2 0.11264E-06
A3 -0.51704E“11

T-KELVIN

1500.00
1600.00
1700.00
1800.00
1900.00
2000.00
2100.00
2200.00
2300.00
2400.00
2500.00
2600.00
2700.00
2800.00
2900.00
3000.00
3100.00
3200.00
3300.00

F~ACTUAL

3.834999
3.495000
3.158000
2.822000
2.490000
2.158999
1.827999
1.499000
1.172999
0.848000
0.523000
0.202000
“0.120000
-0044gggg
“0.758000
“1.075000
“1.391000
“1.705000
“2.016000

STD ERROR OF

F-CALC

3.833889
3.495300
3.158467
2.823360
2.489947
2.158197
1.828080
1.499564
1.172619
0.847212
0.523314
0.200892
-m0129085
“0.439642
“0.757818
“1.074641
'1059@141
“1.704351
“2.017301

ESTIMATE FOR

DIFF

0.001110
“0.000300
”00000467
-00061361

0.000052

0.000802
“0.000081
“0.000565

0.000381

0.000788
’GQQQMSIA

0.001108

0.000083
“0.000358
“0.000182
“0.000359
“0.000858
“0.000648

0.001301

F : 0.0008

PCT-DIFF

0.028954
-00058595
“0.014795
“0.048201

0.002107

0.037162
“0.004434
“0.037649

0.032450

0.093029
“0.059922

0.551297
“0.069044

0.081440

0.023981

0.033368

0.061742

0.038050
“0.064483

L E A S T - S Q U A R E S

T-KELVIN

1500.00
1600.00
1700.90
1800.00
1900.00
2000.00
2100.00
2200.00
2300.00
2400.00
2500.00
2600.00
2700.00
2800.00
2900.00
3000.00
3100.00
3200.00
3300.00

136

TABLEIL4

FOR STANDARD FREE

DATA AND EQUATION FOR AFC OF

02 +-l N2 + NO

2

ENERGY OF REACTION

P O L Y N O M I A L S

POLFIT OF DEGREE INDEX OF DETERM 3 1.0000
F 2 A0 + A1*T + A2*T**2 + A3*T**3
TERM COEFFICIENT

A0 0.21598E 02

Al “0.29704E“02

A2 “0.47781E“07

A3 0.10291E“10
F-ACTUAL F“CALC DIFF PCT-DIFF
17.068985 17.069244 “0.000259 “0.001520
16.764999 16.764801 0.000198 0.001183
16.460999 16.460403 0.000595 0.003615
15.853000 15.851953 0.001047 0.006606
15.547999 15.548018 “0.000018 “0.000117
15.243999 15.244363 “0.000363 “0.002384
14.941000 14.941049 “0.000049 “0.000326
14.636999 14.638138 “0.001139 “0.007779
14.335999 14.335691 0.000308 0.002149
14.033000 14.033771 “0.000771 “0.005491
13.731999 13.732439 “0.000440 “0.003201
13.431999 13.431756 0.000243 0.001811
13.132000 13.131784 0.000216 0.001641
12.834000 12.832586 0.001413 0.011014
12.535000 12.534224 0.000776 0.006193
12.237000 12.236757 0.000242 0.001980
11.940000 11.940249 “0.000250 “0.002093
11.643999 11.644762 “0.000763 “0.006552

STD ERROR OF ESTIMATE FOR F : 0.0007

137

TABLE D.5
DATA AND EQUATION FOR AFC OF % 02 —> O

L E A S T “ S Q U A R E S P O L Y N O M I A L S

FOR STANDARD FREE ENERGY 0F REACTION

POLFIT OF DEGREE 5 INDEX OF DETERM : 1.0000
F 2 A0 + A1*T + A2*T**2 + A3*T**3

TERM COEFFICIENT

A0 0.60314E 02

A1 “0.15148E“01

A2 “0.29262E“06

A3 0.29442E-10
T-KELVIN F-ACTUAL F“CALC DIFF PCT-DIFF
1500.00 37.031998 37.032227 “0.000229 “0.000618
1600.00 35.447998 35.447891 0.000107 0.000301
1700.00 33.862000 33.860550 0.001450 0.004281
1800.00 32.270996 32.270340 0.000656 0.002033
1900.00 30.677994 30.677460 0.000534 0.001741
2000.00 29.081985 29.082077 “0.000092 “0.000315
2100.00 27.483994 27.484390 “0.000397 “0.001443
2200.00 25.883987 25.884552 “0.000565 “0.002181
2300.00 24.281998. 24.282745 “0.000748 “0.003079
2400.00 22.678986 22.679153 “0.000168 “0.000740
9500.00 21.072998 21.073944 “0.000946 “0.004489
2600.00 19.466995 19.467316 “0.000320 “0.001646
2800.00 16.250992 16.250427 0.000565 0.003474
2900.00 14.641999 14.640533 0.001466 0.010012
3000.00 13.030999 13.029922 0.001077 0.008263
5100.00 11.419999 .11.418747 0.001252 . 0.010966
«3200.00 9.806999 9.807205 “0.000206 “0.002100
£3300.00 8.193999 8.195450 “0.001451 “0.017699

STD ERROR OF ESTIMATE FOR F : 0.0009

138

TmEuaD.6
DATA AND EQUATION FOR AEO OF'% H2 + H
L E A 8 T “ S 0 U A R E S P O L Y N O M I A L 8

FOR STANDARD FREE ENERGY OF REACTION

POLFIT OF DEGREE 5 INDEX OF DETERM : 1.0000
F : A0 + A1*T + A2*T**2 + A5*T**5

TERM COEFFICIENT

A0 0.52955E 02

Al -0.12852E-01

A2 -0.52518E-06

A5 0.50422E-10
T-KELVIN F-ACTUAL F-CALC DIFF PCT-DIFF
1500.00 52.646988 52.647585 -0.000595 -0.001825
1600.00 51.257000 51.256542 0.000458 0.001465
1700.00 29.820999 29.819870 0.001129 0.005787
1800.00 28.598987 28.597888 0.001099 0.005869
1900.00 26.971985 26.970871 0.001114 0.004150
2000.00 25.558986 25.559159 -0.000155 -0.000597
2100.00 24.101990 24.105012 -0.001022 -0.004242
2200.00 22.661987 22.662766 -0.000778 -0.005454
2300.00 21.217987 21.218719 -0.000752 -0.005452
2400.00 19.770996 19.771149 -0.000155 -0.000772
2500.00 18.518985 18.520589 -0.001404 -0.007665
2600.00 16.866989 16.866750 0.000259 0.001558
2700.00 15.410999 15.410461 0.000558 0.005490
2800.00 15.952000 15.951920 0.000080 0.000574
2900.00 12.492999 12.491564 0.001656 0.015095
5000.00 11.050000 11.029129 0.000871 0.007895
5100.00 9.566000 9.565506 0.000494 0.005164
5200.00 8.101000 8.100784 0.000216 0.002661
5500.00 6.654000 6.655500 -0.001500 -0.019590

STD ERROR OF ESTIMATE FOR F : 0.0010

APPENDIX E

APPENDIX E

CALCULATION OF THERMODYNAMIC PROPERTIES FROM EMPIRICAL SPECIFIC HEAT

EQUATIONS.

E.l General

One of the problems which had to be solved in the course of this
study was the calculation of the reaction temperature for a constant
pressure steady flow reaction. This requirement was considered in the
calculation of the enthalpy in the following manner. The steady flow

energy equation for reaction can be written as

 

o o o _
Q + zni(HT — H298 + AHf298)i (Reactants) -
2n.(H° - H0 + AHO ). + w + AKE + APE (E-l)
z T 298 f298 z (Products)
The enthalpy calculated then will be
_ o o 0
HT — Xni(HT — H298 + Aszge) (E-2)
and not the sensible enthalpy Hg.
E.2 Ideal Gas Enthalpy Computation
_ o _ o 0
HT - Zni(HT H298 + AHf298)
Considering only a single specie
o o o
H. = . H - H + A . E-3
t nz( T 298 f298)t ( )
and
HT = ZHi (E-A)

for the mixture of species.

139

140

Considering Equation (E—3) and

o o o o o o
— = - + H — H
HT H298 HT HT ( T 298)
o o
where the quantity (H; — H398) is the tabular value at the apprOpriate

O

"T " we have
0

o o o o 0
Since
T
o
H; — HT = c dT (E-6)
o P
T
0
we have
T o o o
H. = . C T + H - H + H . E-
1 nt( [ pd ( To 298) A f298>$ ( 7)

T
o

and substituting the equation for Specific heat yields

T
= 2 3 O _ O O _
Hi ni( J (a0 + alT + a2T + 33T )dT + (HTO H298) + AHf298)i (E 8)
T

0

Integrating we have

a 32. as 31 32 333
,= , 2 __ 3 _ _ _ 2 _
H1, n7,(T(ao+ 2 T+ 3 T + 4 T) To(ao+ 2 To+ 3 To+ 4 To)
0 o o
+ - .
(HT H298) + AHf298)7,
0
which can be expressed as
81 32 2 a3 3
Hi = ni(T(aO +'7f T + ?r'T +-7T'T ) - HK)i (E-9)
where
HK -T( +21: +:2—T2+33T3) HO H°) AH° (E10)
{"030 2 O 3 o 40i'(To'298£‘ f298i

141

and is a constant whose value depends on the Specie and the temperature

range. The values of (H; - H398) and AH;298 for the individual con-
. o

stituents can be found in Table 2. Equations (E-9) and (E—4) were used
in a straightforward manner to write computer subroutines for the cal—
culation of enthalpy. These subroutines, PD08HH for the temperature
range from 1500.0 to 3500.0°K and PD08HL for the temperature range from
298.15 to 1500.0°K, are presented in Appendix I and the values of the
HKi's can be obtained from these programs.
E.3 Ideal Gas EntrOpy Computation

The general expression for the entropy of an ideal gas mixture

with a reference pressure of one atmosphere is

o
298.
t

O

0

where for the temperature range from 298—1500°K

 

 

C
o o T pi
2ni(ST - 8298)i = ZniI -Ef'dT (E-12)
298
and from 1500-3500°K
C dT
7n (3° - s° ) = 2n [ T pi + (5° - s° ) (E-13)
” i T 298 i 1: T 1500 298 i]
1500
o o
where 81500 and 5298 are the apprOpriate tabular values.
For a single component
0 -T C 7:dT
Si = ni(3298. + J T - R£nPi)

142

 

 

and ,
JT LpidT [T (a0 + alT + a2T2 + a3T3)dT
298 T - 298 T
a2 2 a3 3 32 ? a3 3
= -——-— — - +—-——+—
aOQnT + alT +- 2 T +3 T (aORnTO + alTO 2 TO :3 To)
therefore
a2 2 a3 3
S. = . + ——' ——- . — . - . E-14
$ nl((aognT alT'+- 2 1T +-:3 T )1 KSt RQnPZ) ( )
where
82 333 O
= ——- 2 ——- — . E-
KSi (aOQnTO + alTO +- 2 T6 +-3 Tb 3298)t ( 15)
for 298 < T :_lSOO°K with To = 298.15 °K.
In the temperature range from 1500 to 3500°K we find
32 a3 0
= v v___v2 __ v3_ __
KSi (aoznTO + alnTo +- 2(To) +3 (To) 81500)i (E 16)
where T; = 1500°K.

The value of KSi is a known constant for a given Specie and temperature

range. The entropy of the mixture is then
s = 2.5. (E—I7)

The subroutine (PD08CC) used to calculate the mixture composition provides
the number of moles of each component (ni), the total mixture pressure Pm,
and the total number of moles of mixture (Na). The following method is

used to calculate the Pi's.

Pm Pi
F=V=E
m ’L
and
Pm
Pi = nz'fi— — niF

 

143

therefore
RRnP. = R£n(n.F) (E-l8)
L t
hence
a2 2 a3 3
Si = ni((a02nT + alT +'3r T +-§- T )i — KSi — R£n(niF)) (E—l9)

Equations (E—l7) and (E—l9) were used in a straightforward manner, with
the apprOpriate KSi's, to write computer subroutines for the calculation
of entropy. These subroutines, ENTROH for temperature range from 1500.0
to 3500.0°K and ENTROL for the temperature range from 298.15 to 1500.0°K
are presented in Appendix I and the values of the KSi's can be obtained
from these programs.

Due to the method used in calculating frozen equilibrium composition
and computer round-off, it is possible to have zero or negative moles of
oxygen in the exhaust products below 1500.0°K for a stoichiometric air-
fuel ratio. The method of calculating the entrOpy would then require
the calculation of the logarithm of a negative orzero number which, of
course, cannot be accomplished by the computer. To avoid this possibility
a special abbreviated subroutine ENTROI was written and is shown in
Appendix I. Subroutine ENTROl is used only for entropy calculations

below 1500°K for air-fuel ratios of 15 to 1.

as as
E.4 Computation of (3T)v and (3T)p

Newton's Method (Appendix H) for solution of implicit functions
of a single variable is used during the compression and expansion processes

and requires the computation of 63% v and (3% Through use of the ideal

)p'

144

gas relationships, this computation can be accomplished in the following

manner:
c dT d
ds = £— - 11’- (E—20)
T T
also
ds = (3333—) dT + (3%) dp (E-21)
3T p ap T

and comparison of the coefficients of (E—20) and (E—21) shows that
(3%) = E9 (E-22)
3T p T
Equation (E-22) is valid for a single ideal gas and since a mixture
of ideal gases can be treated as a single gas with average or apparent

prOperties, the computation of (%%- becomes simply a matter of calcul-

)
P
ating the apparent specific heat for the mixture.

The specific heat of the mixture (Cp ) is calculated in the follow-

m
ing manner.

 

Zn.C
1 pi
C = - (E-23)
pm Nm
where n.C = n.(a + a T + a T2 + a T3). (E-24)
a pi t o l 2 3 1
and N = 2.11. (E—ZS)
m ’L ’L

Two subroutines, CPHIGH for the temperature range from 1500.0 to
3500.0°K and CPOLOW for the temperature range from 298.15 to 1500.0°K
were written in a straightforward manner using equations (E-23), (E-24),

and (E—25). These subroutines are presented in Appendix I.

145

For an ideal gas
C dT

 

ds = g + pdv (E-26)
and
ds = (93) dT + (33% dv (E-27)
3T v 8v T

and comparison of the coefficients of (E-26) and (E-27) shows that

C
(E) = .2.
3T U T

hence for a mixture of ideal gases

(E—28)

From the definition of enthalpy and the ideal gas equation of state

h = u + pv = u + RT

differentiating
db = du + RdT
and
C dT = C dT + RdT
p U
hence
C = C + R
p v
eund
c = c - R (E-29)
v P
m m

The gas constant per mole (R) is the same for all ideal gases and

(:1, and hence (2% U can be computed using Equations (E-28), (E—29),

m

arid the appropriate Cp .
m

APPENDIX F

APPENDIX F

SOLUTION OF THE EQUATIONS OF CHEMICAL EQUILIBRIUM

The determination of the amounts of the various species present in
the combustion product mixture requires the solution to Equations (2—18)

through (2—20). These equations are repeated here for convenience.

 

" x 2 F - AN

2 — - 35— + / (4m) + 2 (2-18)

4K
l+ ,
—(—)-‘-+-1-)+./ (-X—+—l-)2+8(1+-)-(-)F-AH
K3 K6 K3 K6 K2

Y = x _ (2-19)
4(1 + E—'
2

_ ___X°F°AC __sz 2 _X1 _Zl . i. 7 N
F Ao-(K1+X)+K2+2x +K3+KH+F AC+K5 (220)

Both Z and Y are functions only of X and hence the problem is to find a
value of X which satisfies Equation (2—20). In general we can restate

equation (2-20) as follows:

f(X) = O . (F—l)

For any initial estimated value of X, the right hand side of (F-l) in

general, is not zero.

f(Xl) # O

FVEE must then make a correction Af(X1) in order to achieve our desired
ze‘r'o.

f(Xl) + Af(X1) = O (F-2)

VJGE can now use (F-2) to obtain a correction for X1 that will produce

146

147

value X2 which is closer to our correct answer. The correction selected
is

. ._Q£i§l. . _
Af(X1) _ [d(2nx)]1 AlnX (F 3)

Substituting (F-3) into (F-2) we have,

_ _ _Qiiél . - _ .
f(Xl) — [d(2nX)]1 ARnX _ X1(f (X1))A2nX

and rearranging

 

x2 f(Xl)
QHXZ - 2nX1= 211 2'1- = - X1. f'(Xl)
or
-f(X1)
X2 = x1 ' eXp le'(xl)

and in general, for the i-th trial,
-f(X.)
X. = X. . exp'--—4k-
1+1 1 X.f'(X.) (F-4)
z t
Finding X1.+1 in this manner assures that for an initially positive
‘Value of X1 all subsequent values of X will be positive. We are now
.able to eliminate the possibility of negative partial pressures simply
IDy'insuring the initial estimate of X is positive.
In the following equations for the determination of f'(X) the

argument X will be drOpped for simplicity.

_d_f-af .41 Age 2:

t _ . _
f " dX ‘ aY dX + 32 dX + ax (F 5)
F1Z‘O'mEquation (2-20)
3f _.2£ _
az ‘ K1+ (F 6)
3f 2XY x
aY ‘ K + K (F'7)

148

F'AC'K

 

2
2£=———-———1—+4X«I--‘}-{—+—Y—+—Z‘—+—L (F—8)
3X (X+K1)2 K2 K3 Kn K5
The evaluation of dY/dX can be eased by using the following:
X 1
B = (-—-+--0
K3 K6
C = 4(1 + X/KZ)
D = / 132 + 2-C-F-AH
Equation (2-19) can now be written as
— - 2 o o o --
Y=B+/B+2CFAH= B+D (F_9)
C C
From equation (F-9)
dY l 1 l B 4-F-AH 4
dX ‘ CZ{"C[K ' D(K + K )1 ' K (D’BH (F 10)
3 3 2 2
If we set
A = X/(4KH)
E = /‘A2 + F-AN
2
then from (2-18)
2 = -—A + E (F-ll)
Eitldl
dZ A - E
dx “ 4°E°Kl+ (F 12)

lfic’lf some initial estimate of X the value of f(X ) and f'(X ) are de-
l l l

tetTrained from Equations (2-20) and (F-S) and new value of X, X2, can be

149

calculated from Equation (F-4). The process is then repeated until

is less than E2. In this study a value of 0.0005 for E2

was found to give the desired degree of precision.

When the partial pressures of the constituents are expressed in
the form 0.xxxxxxx 10xx a change in E2 from .001 to .0005 resulted in
a change in the seventh digit to the right of the decimal if a change
occurred. (Evaluated at 2400°K and 300 psia).

A flow diagram of the computer subroutine written to perform these

calculations is shown on the next page and it should be noted that a

f(Xi)

limit is placed on the value of the quantity' -—1—-——' in order to
Xif (Xi)

prevent any wild fluctuation of the successive estimates of X.
The variables A, B, C, D and E, defined in this appendix, are used
in programing subroutine PD08CC and the subroutine is presented in

Appendix I. This subroutine normally converged within 4 iterations for
a reasonable estimate of X and, as was noted by Patterson [3] who used

essentially the same system, it never failed to converge on the correct

answer .

150

 

Enter
Subroutine PD08CC

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 =
U
Evaluate:
Y.7-.f(X).F'(X)
From Equations Evaluate:
%"18’2—19’2'20’ Partial Pressures
F”) From Equations
i 2-21 Thru 2-30
r
EO " ~‘%§%%?§T Evaluate Number of
Moles of Each Specie
P;
n, --—F-
[N = In
m l
H = 3 .
f I Return
K2 — x '
x J
I
E2 = .0005

 

 

 

 

 

FIGURE F.1 FLOW DIAGRAM FOR SUBROUTINE PDOSCC

 

APPENDIX G

'1‘“

APPENDIX C

DETERMINATION OF THE EQUILIBRIUM CONSTANT (KP) AND THE STANDARD FREE

ENERGY OF REACTION (AFC)

G.1 Equilibrium Constant Computation

The basic equation relating to the standard free energy of reaction

(Gibbs Function) and the equilibrium constant is

AFO

-RT£nK
P

and hence
(G-l)

0
AF
KP exp-(iffv

For reasons discussed in chapter 2, section 2, the tabular values

of the standard free energies of the reactions were approximated as

° (c-2>

= 2 3
AF a0 + alT + aZT + a3T

and the equilibrium constants were calculated using Equations (G-1) and
(G-Z). A subroutine, PDO8EK, was written to perform the calculations and

is presented in Appendix I. If a comparison is made between the free

energy equations presented in Appendix D and those used in subroutine

IPDOSEK, some differences in signs will be noted. This difference occurred

£18 a result of using the reverse of some of the reactions shown in Appendix
I) in making the equilibrium calculations. The standard free energy of a
3E1!verse reaction is, of course, equal in magnitude and opposite in sign to
tiliat for the forward reaction. Standard free energy data were available in
t111e2"JANAF tables" for all reactions desired except the dissociation of CO2

into CO and 02. The method used to determine the standard free energy for

t:l‘lis reaction is discussed in the next section.

151

152
(3.2 Standard Free Energy of Reaction Computation

The criterion for equilibrium for a chemical reaction is that the
free energy change must be zero. The free energy change implied is the
free energy of the products minus the free energy of the reactants. The
particular reaction of interest here is

1
CO2 + CO +'§ 02

The eXpression for the free energy change of this reaction is

1
F + —-F - F = 0
co 2 02 co2

where F is the partial molar free energy. The partial free energy of

each specie (considering the specie to be ideal gases) is given by

F. = F? + RT2np.
’L ’L 1

Introducing these into the previous expression gives

0
CO

0

F 002

l<Fo

+ RTzanO + 2 O
2

+ RTznpOZ) - (F + RTzanOZ) = 0

and collecting similar terms yields

0 1

o o l
(FCO + 2 F02 - FCOZ) + RT(2anO + 2Rnp02 - finpcoz) - 0

where the first group is standard free energy of reaction AFO.

Therefore
AF° (F80 + %-F82 - F802) = -RT9.nKp (G—3)
where
1
pco<Po ’7
K =
p pco2

and K.p is the equilibrium constant. To find AFO for the reaction in

153

question we must then find a method of evaluating

(F0 + 1

c0 2' )

o o
F — F
0 CO2
and to maintain consistency we should use only data from the "JANAF
tables." This can be accomplished by considering the following re-

actions for which data are available.

1
5'02 + C + C0 (G-4)
C + 02 + CO2 (G-5)

The standard free energies for (G-4) and (G—5) respectively are

0 1' o _ o _ o _
(FCO - 2 F02 FC) — AF(G_4) (G 6)
o o o = o _
(FCO2 - F02 FC) AF(G_5) (G 7)

o l_ o _ o _ o o o _ o _ 0
(Foo ' 2 F02 Fc Fco2 + F0 + F0) ‘ AF(G-4) AF(G—S)
simplifying, we have
0 l_ o o _ o _ o A _
(FCO + 2 F02 FCOz) - AF(G_4) AF(G_5) (G 8)

which is the solution to our problem. The data for the standard free
energies of reaction for reactions (G-4) and (G-5) were used in Equation

(G-8) to calculate the standard free energy of the desired reaction.

 

APPENDIX H

APPENDIX H

ITERATIVE METHODS FOR SOLVING NON—LINEAR EQUATIONS

H.1 Newton-Raphson

 

The general problem is to find the solution or zeros of the

equation

f(x) = O (H—l)
where f(x) is in general non-linear. The problem reduces to finding
the zeros of (H-l) by using some iterative sequence. One method of
accomplishing this is to choose x0, and determine the sequence xn from

the recurrence relation

f(xn)
x = x

n+1 n - _TT(;;)—.’ n = 0,1,2,... (H-2)

This method is generally known as the Newton—Raphson method and
it converges quadratically.

Equation (H—2) can be interpreted very simply graphically. The
graph of the function f is approximated by its tangent at the point Xn’

that is, f(x) is replaced by
__ '
f(xn) + (x Xn)f (xn)

By setting this expression equal to zero and solving for X, we
have Equation (H—Z). While this graphical interpretation is intuitively
appealing, it tells us nothing about the nature of convergence and is

not easily extended to the case of systems of equations.

154

 

155

H.2 Regula Falsi

 

There are many modifications of Newton's method and a number of

them deal with changes which eliminate need for the derivative of the

function. One such method used in this thesis approximates the de—

rivative f'(x) by the difference quotient

f(xn) - f(xn_1)

Xn - Xn_1

 

(H-3)

formed from the two preceding approximations. When (H-3) is sub-

stituted in (H—2) we have

(Xn ‘ xn-1) f(xn)

 

x = x -
n+1 n

F(xn) - f(xn_1)

which is known as the regula falsi. This method can be interpreted

graphically in a manner similar to the Newton-Raphson method. The

regula falsi method is somewhat less convergent than the Newton-Raphson
method and can, in some cases, lead to serious convergence problems.

such problems were encountered in the course of this thesis since the
method was judiciously applied to well behaved functions. For a more

complete examination of these and other iterative methods consult any

text on numerical analysis such as Henrici [34].

APPENDIX I

APPENDIX I
SUBROUTINE LISTINGS

Listings for all subroutine programs are provided in this appendix.
PD08CC calculates the equilibrium composition of the working fluid for
temperatures above 1500°K. ENTROH calculates the entrOpy of the working
fluid for temperatures above 1500°K and PD08HH calculates the total P
enthalpy of working fluid for temperatures above 1500°K. CONLOW
calculates the composition of the working fluid for temperatures below
1500°K. CPHIGH calculates the average GP for temperatures above 1500°K
and PD08EK calculates the required equilibrium constants for temper—
atures above 1500°K. ENTROL calculates the entrOpy for temperatures
below 1500°K for all air—fuel ratios other than 15/1 and ENTROI
calculates the same value for 15/1 air-fuel ratios. PD08HL calculates
the total enthalpy for temperatures below 1500°K and CPOLOW calculates
the average Cp for temperatures below 1500°K.

A flow chart for PD08CC is presented in appendix F. Flow charts

for the other subroutines are not presented because they contain no

test statements or complex programing techniques.

156

157

SUBROUTINE PDOBCC(T39F.AC.AH.AO.AN.EK19£KZ.EK3.EK4.EKS,EKprN2.CN3
lgCNfi.CNS.CN6.CNT.CN8,CN9.CNIO.CN11.PI.CNI.X.ll

20
22

25

3O

45

|F(X)9:9110
X=l
A=XIl4.*EK4)

E 8 SQRIlA8A+F8AN/2.)

l s -A+E

8 = X/EK30l./EK6

c s 4.8(l.+X/EK2)

D 8 SQRT(888+2.8C8F8AHI
Y = (-8+Dl/C

FY = 2.8x8Y/EK2 + X/EK3
Fl 3 XIEK4

rx a FtAttEKiltx+EK1Ittz.+4.tx+thIEK2+VIEK3+zIEK4+1.IEKS
vx a (oat:1./EK3-(315K3+4.tFtAH/EK2)10)-4.ttD-a:/EK2)/(ttt)

IX 8 IA/E -l.)/(4.*EK4I
DFX 8 FY8YX+FZ‘ZX+FX

FFX 8 X8CF8AC/(EK1+X)+Y8Y/EK2f2.8X+YIEK3+ZIEK4+loIEKS)+F8(AC-AU)

E0 = -FFX/IX80FX)
IF(ABS(EQ)-16.)22g22.20
E0 2.8E0/ABSIEQI

X2 X*EXP(EQ)

XI (XZ-XDIX

52 .0005
lF!ABS(XF)-EZ)30o30v25
X8 X2

1 8 [01

GO TO 10

P02 8 X8X

PNZ 8 1’2

PHZ 8 YtY

PCOZ 8 F8ACI(1.0EKl/X)
PHZO 8 X8Y8Y/EK2

PCO 8 PC028EKl/X

POH 8 X8Y/EK3

PRO 8 X8l/EK6

P0 8 X/EKS
PH 8 Y/EKO
CNZ 8 POZ/F
CN3 8 PNZ/F
CN‘ 8 PCOZIF
CNS 8 PHZOIF
CNb 8 PCOIF
CH? 8 PHZ/F
CNO 8 PDHIF
CN9 8 PNOIF
CNIO8 PO/F
CNll8 PH/F

PF 8 POZOPN28PHZOPC02+PCO+PH20*POH+PNOOPO+PH
CNT 8 CNZ‘CN30CN§+CN5+0N6+CN7§CNB+CN9+CNIOOCNll
RETURN '

END

158

SUBROUTINE ENTROHII39F,CN29CN3oCNQoCNSpCNprNTgCN89CN99CNIOQCNllo
ITS",

R8l.98726

'31 8 ALOGII3)

I32 8 T3‘t3

I33 3 I32‘T3
SHZ‘CNZ“.7885851*I3l*o52§l4t'3‘T3*.25298E-7‘t32‘.4§9035'll‘t33
l-R‘RLOGICNZ’F)‘3.I57,
SH3=CN3‘C.64477Eltt318019796E’Z‘T38.285165-6.T32*o19749E‘lotr33
l‘R‘RLDG‘CN3’F)+8.Z36’
SH4=CN48(.10708828T31+.34197E-28T3-.49676E-btt32+.346516-108133
l-RtALDGICthFi-IZ.626D
5H53CN5“o5362351‘131*.580l3E-2‘73'o73085E86‘I32.o455605-10’t33
l‘R‘RLUG‘LNS‘F)*l3043Z’ '
5H68CN6*(.6706OEI.T319.18030h-2‘T3*o26056E-6RI320.lElZBE'IO‘I33
l-R‘ALOG‘CN6*F)*8o126)
SH73CN7"05330951‘131*021733E‘2‘I3-022309E'6.132*0[270‘E-10‘t33
l‘R‘RLUG‘CNT‘F)+o929’
SH8=CN8*(.54676EI‘I3I*.23292E’2*I3-.28193E’O‘T3Z*o17240E-10.I33
l‘R‘ALUG‘CNB‘F’+lZ-69l)
5H9=CN9.Io7ll97El‘T3l80lSO9fiE'Z‘T3-.ZlBZTE-b‘I3ZOo152385-10‘I33
l-R*RLUG(CN98F’*8o87l)
5H10=CN10‘(.5022651‘T3l‘oZbOJZE’4’I3’o38823E'3‘I32’olS975E‘ll’I33
l’R‘ALOG‘CNlO’F)‘9-953l
SHII=CNII.(.496305l‘t3l‘R‘RLUG‘CNll‘F"o9l3’

IS" 8 SHZ‘SH3*SHR§SH5*SH6*SH7*SH8+SH985H10*$H1l

RETURN

END

SUBROUTINE PDOBHHIT3.CNZ.CN39£N4.CNS.CN6.CN79CN89CN9.CNIOoCNll.
lHHT’

732 8 T38Y3

T33 8 T32‘T3

HZH 8CN284T38l7.8858+.26207£-38T39.168655-78132-o33678E-118T33D
1'2752o2)

H3H 8CN3*(73.(6.4477+.989BOE-3*T3-.190115868F32+.148126-108T33)
l‘2153.03

HQH 8CN4‘tT3‘l10.7080.17099E‘2‘T3-.329845-687320.25993E-108T33)
l-9823l.6)

HSH 8CN5*(T3‘(5.3623*.29007E-2'T3-.487235-68Y32+.34l70E-108T33)
1-59401.5D

HbH 8CN6*(T38(b.7060t.9OISOE-3¥I3-.l737lE-681328.l3596E-108T33)
1-28702.0)

HTH 8CN7*(T3*I5.3309+.10867582813-.148736-68132§.952805-l1*T33l
l-l3l9.6l

HOH 8CN8*(T38|5.4676+.[16466-28I3-o18795E-68T32*.12930E-108T33)
107877.!)

H9H 8CN9‘IT38C7.1197+.75470E-38T3-.145515-68132*.11429E-108T33)
1919131obl

HIOH 8CN10‘!'3‘!5.0226-.13016E-48T3-o25882E-88t320.ll982E-l18'33)
[058103.0)

HllH 8CN118lT384.968050620.7l

NH? 8 H2HOH3H+H4H*HSH+H6H*H7H+H8H+H9H+H10HOHl1H

RETURN

END

 

159

SUBROUTINE CONLOHTACpAH.A0.AN.CN2.CN39CNQ'CN59CN6.CN71CN89CN9.
lCNlOoCNllcCNTT
CN28AD/Zo-(AC+AH/4.)
CN3=RNI2o

CN48AC

CN5=RHI2

CN680.

CN780.

CN880.

CN9=0o

CN1080.

CNll=00

CNT 8 CNZ‘CN3OCN48CN5
RETURN

END

SUBROUTINE CPHIGH‘T3.CN2.CN3.CN4.CN5.CN6.CN7.CN8.CN9.CNIO.CN1l.
ICNT.CPA)
5 T328 T38T3
T338 T38T32
CPHZ 8.78858E1*.524l4E-38T3+.50596E-7‘T32-.134712-108T33

CPH3 =.64477El*.19796E-28T3-.57033E-68T32+.59Z47E-108T33
CPH‘ 8.lOTOBEZ’.34l9TE-28T3-.98951E’68T32+.lO397E-98T33
CPHS =.53623E1+.58013E~28T3-.lfiblTE-58T32*.l3668E-98T33
CPH6 8.67060EIO.18030E-28T3-.52112E-68T32+.563BSE-108T33
CPHT 8.53309El8.21733E-Z8T3ro446195-68T328.381125‘10‘T33
CPHB 8o54676E1+.23292E-28T3-o563855-68T32+.SITZIE-108T33
CPH9 =.71197El+.l509QE-Z8T3-.43653E-68T32+.657146-108T33

CPH108.50226E18.260325848T3-.77646E’8’T32+.4T926E-lI8T33
CPH118.496861

CPA 8 iCN28CPH20CN38CPH3OCNQ8CPH4*CNS‘CPHS+CN6*CPHb‘CNT’CPH7+
lCNB‘CPHB+CN98CPH9+CN108CPHIO+CNll‘CPHlll/CNT

RETURN

END

SUBROUTINE PDOBEKTT3gEKl.EKZoEK3oER4oEKsuEK6)

5 T32 8 T38T3
T33 8 T328T3
R 8 1.98726
RT 8 -R8T3/1000.
Fl 8 0.6793952 -0.2167TE-18T3+0.52763E-68T32-0.36904E-108T33
F2 8 0.59366E2 -O.12969E-I8T3-0.32107E-68T32+0.3209lE-108T33
F3 8-0.91446El +0.36978E-28T3-0.11264E~68T32+0.51704E-118T33
F4 8-O.21598E2 00.297065-2‘T3’0.677815-78T32-0.1029lE-108T33
F5 8-0.60314E2 +0.lSlfiBE-l‘T3*0.29262E-6.T32-0.29442E-108T33
F6 8-0.52933E2 *0.IZBSZE-I’T3+0.523185-68T3280.504225-108T33
ELKI 8 Fl/RT

ELKZ 8 FZIRT
ELK3 8 F3/RT
ELKQ 8 FQIRT
ELKS 8 FSIRT
ELK6 8 F6IRT

ER! 8 EXPTELKID
EKZ 8 EXPTELKZ)
EK3 8 EXPlELK3)
EKQ 8 EXPTELR6)
EKS 8 EXPIELKS)
EK6 8 EXP‘ELK6)
RETURN

END

160

SUBROUTINE ENTROL(T39F'CN20CN3ICN9oCNSoCNbgCNTgCNBpCN9gCNIOpCNllp
ITS")

R8lo98726

T3l= ALOG‘T3)

T32: T3'T3

T33: T32‘T3
SZ:CN2.‘.61896E1‘T3l*029278£‘2‘T3-035505E’6’t32’0262026’10‘T33
l‘R‘ALOG‘CNZ‘F}*o12901E2)
$3:CN3.‘010“98t1.T31-0l1450E-2‘T3*01521lE'S‘T32’o38213E-9‘133
l‘R‘ALOG‘CN3‘F)O.58229EI’
S“=CN4‘T-5l337£l*T3l*.15338t'1’T3'.4924OE'5‘T3Z*o789505'9.T33
I‘R*ALOG(CN4*F)+.17671EZT
55=CN5*(o77474El‘T3l*o80860t’4‘T38.15376E’5‘T32’o351875‘9’T33
l“R*AL05(CN5‘F)+.81701)

T5" 8 52053954055856+S7+58*S9*S[0*5ll

RETURN

END

SUBROUTINE ENTROI(T3.F.CN3.CN40CN5.TSH’

R31098726

T313 ALUb‘T3,

T32: T3‘T3

T333 T32‘T3
S3‘CN3“.70498£1*T318¢[[450h82’T3+o1521lE‘S‘T328-38213E-9'T33
l-R’ALUG‘CN3‘F".58229EIT
S‘:CN4“051337E1.I31*015338E’1‘T3‘.“9240E’5‘t32+0789508-9‘133
I'R‘ALUG‘CNQFF’801767152,
$5=CN5’(.TTQTQE1‘T3l*080860E-4*T3*o15376E‘S‘T32‘o35187E-9‘T33
l‘R‘RLOG‘CN5‘F)*081701)

T5" 3 $2*S3+S4+55*Sb+57*58+59*5[0*51l

RETURN

END

SUBROUTINE PDOBHLCT39CN2gCN3.CN4.CN5,CN6.CN7.CN8.CN9.CNIO.CN1lg
lHLT)

T32 8 T38T3

T33 8 T328T3

HZL 8 CNZ‘IT3‘(6.18960.l4639E-2‘T38.2367OE-68T32-.196516-10‘T33’
l-l969.2)

H3L 8 CN3‘TT3*(7.0698-.57255-38T3+.lOlblE-58T32-o28665-98T33)
182075.69)

H6L 8 CN48(T3.(5.1337+.0076698T3-.32827E-58T32+.59ZIE-98T33T
1-9618401,

HSL 8 CNS.lT3.(7.7476+.40§3OE-48T3+.10276E-58T32-o26396-98T33)
H6L 8 CNb“13“6.9107-028506E‘3.T3+08830E‘6‘T32‘026883E‘9’T33)
1-28473.4)

HTL 8 CN7*(T3*(6.8053*.l9221E-3*T3-.727lE-78T32+.621l3E-108T33)
l-204#.7)

H8L 8 CN8‘(T3‘(7.6080-ollBlZE-28T3+.956lTE-6‘T32-ol9653E-9’T33)
H9L 8 CN9*(T3*(7.0069-.16366E-38T3+.834205-6‘T32-o26863E-98T33)
[019485.4)

HIOL 8 CN10‘(T38!5.5988-.82lSUE-3tT3+.69450E-6‘T32-ollO995E-9‘T33)
1057950.5)

HllL 8 CNIl‘I4.968*T3+50620.7)

HLT 8 H2L+H3L+H4L+HSL0H6L+H7LOH8L+H9L+H10L+H11L

RETURN

END

161

SUBROUTINE CPOLUH(T39CN2pCN3pLN49CN59CN69CN79CN8'CN99CNIO,CNIl9
lCNTvCPA)

5 T328 T3‘T3
T338 T3*T32
CPZ=o6189651*029278E‘2*T3-o71UO9E-6’T32-o78607E810*T33
Cp3=o70498E18ol1450E’2‘T3+-3U422E-5‘T32‘o11464E'8*T33
CP4=.5133TE1+.15338E’l*T3-.98480E-5’T328o23685E’8‘T33
C958o77474El+o808605-4‘T3+o307525'5‘T328.[05565-88T33
CP68.69107E18.5701[E-3‘T3+o2649lE-5’T328.1073TE'8*T33
C978.68053E1*.3844lE-3'T3-o2lB3lE-b‘T32+o24845h-9‘T33
C988.76080E18.23624E‘ZFT3*o28685E‘5‘T32‘o78612589‘T33
CP93070069E1-032731E'3’T3‘.25026E‘b‘T32’.IOTQSE-8‘T33
CPIO=o55988518o16430E82tT3+o14835E'5*T328.44398E89’T33
CPA:TCNZ‘CPZ*CN3‘C938CN49CP4+CN5‘CP5‘CN6‘CPb‘CN7*CP7+CN8‘CP8*CN9*
1CP9+CNIOFCPIU*CNIIFCPIIT/CNT
RETURN
END

APPENDIX J

Kr

APPENDIX J

FLOW CHART AND LISTING OF MAIN LINE PROGRAM

This appendix contains a flow chart of the main line program
(pages 163 thru 169) used to calculate engine performance and a

listing of the program(pages 170 thru 176).

162

163

 

Start:
Set values of all
initially defined variables

J

Calculate entrOpy(Sl)
Estimate temperature(T2)

*8 :1—8

[Calculate entropy(82) J

 

666

 

 

 

 

Estimate new TZA]

 

 

 

Calculate all compressor
state points and the
compressor work.
Estimate combustion

temperature(TI)
j a {9

Set mole atoms of carbon and hydrogen.
Estimate composition of reaction products.
Calculate: F,X,V6, and V8

 

 

 

 

 

#
fi

 

 

Calculate equilibrium
constants, composition, b
pressure(PT), and total

moles.
Calculate:
, F

  

 

TI — 1500.

  

 

 

 

   

 

 

 

Calculate composition
total enthalpy(HHH).

 

 

 

A J ‘
o 1gCalculate total enthalpy(HHH).]

FIGURE J.l FLOW DIAGRAM FOR MAIN LINE PROGRAM

164

Calculate total enthalpy of
reactants(HHL).

 

Calculate energy balance
parameter(HHI).
HHI = HHH-HHL- heat transfer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

Calculate composition 0 Calculate equilibrium H
and enthalpy(HHJ). constants
* L 4; ' i

 

 

F _ .— 8

Calculate composition,
pressure(PT), and total 1
moles

Calculate energy balance
parameter(HHJ).
HHJ = HHH-HHL- heat transfer 4

 

 

 

 
    
      
 

 

Calculate F

 

 

 

  

' T—PII - .002? r

5..

Calculate enthalpy(HHH)]

 
 

 

 

 

 

 

 

 

'Estimate TK based on TI,TJ,HHI, A
and HHJ.
Set: TI = TJ, TJ 8 TR, and

HHI = HHJ

 

 

 

 

FIGURE J.l (cont'd.)

. 165
Calculate mole fractions and intake
pressure(P7).

Set: M = 1., TEST = 0.

 

Estimate temperature at end of the
intake(T7).

 

 

 

 

 

 

 

 

 

 

 

150
Calculate equilibrium
constants
, T TO , 0 1 ,
Calculate composition,F, Calculate F, composition,
and enthalpy(HH7). pressure(PT), and total

 

moles.

 

     
    

.002?

 
    

<’T-P7|-

__‘_\——[Calculate enthalpy (HH7) . I

—— FCalculate energy balance
Wparameter with residual
Texhaust gas effect(ERR7).

Calculate energy balance
parameter without residual
exhaust gas effect(ERR7).

 

 

 

 

 

 

 

 

 

 

| TT = T7 - 10.] [TT = T7 + 10.]
390 270 <: 1

FIGURE J.l (cont'd.)

 

 

 

 

166

 

 

Calculate equilibrium
constants

{ i

 

 

 

 

 

 

 

_ 7 - . r _w

lCalculate composition, F,] Calculate composition,

7 and enthalpy(HHT). pressure(PT), and total
moles.

 

 

if

 

 

 

 

 

Calculate energy balance
parameter without residual
exhaust gas effect.(ERRT)

 

 

 

 

 

Calculate energy balance parameter with residual
exhaust gas effect.(ERRT)

 

 

 
    

 

 

Estimate new T7 and
call it TX.

 

l 7

T7 = TT

 

 

HH7 8 HHT
ERR7 8 ERRT T7 8 TT
' l7 TT 8 TX

 

 

 

 

 

ERR7=ERRT
o + - ~

 

 

 

<ERR7-TEST -100 + 48.: 400
r' T: .
Print air fuel ratio(AFR) and M.
Calculate IMEP, thermal efficiency, and cut off ratio. 0
Print and punch selected variables

 

FIGURE J.l (cont'd.)

167

   

 
 

TEST 8 ERR7
Calculate mole fractions and

       
   

total moles

  

+

        
 
   
 
   
  

 

T7 - 1500. Calculate entrOpy(S7)

and volume(V ).

  

Calculate entrOpy(S7)
and volume(V ).

Estimate temperature (T8)

Estimate temperature (T )

 
    
 
   

Calculate F and entropy(S8
Calculate composition

 
  

Calculate enthalpy(HH8)
and pressure(P ).

 
  

+

       
   
   
  
   

AFR - 15.042

Calculate F and entrOpy(S )

Estimate new temperature(T )

 
   

+
AFR - 15.042

FIGURE J.1 (cont'd.)

168

f

and entrOpy(88) at point 8.

  
   

 

 

435 =
{y

 

[Calculate equilibrium composition, enthalpy(HH8),J

 

Calculate mole fractions and
total moles at point 8 and
pressure(Pg).

 

A
‘

 

 

  

 

 

 

 

[Estimate temperature (T9)

 

 

[7
Calculate composition

,

1

[Calculate F}? =8

 

 

 

 

 

 

 

 

 

’Set moles of 0

 
 

  
  

 

 

Calculate F, equilibrium
constants, composition,
entropy(89), and enthalpy
(HH9).

 

 

 

 

 

IEstimatenew T9}____.

 

 

 

Calculate F,
entrOpy(Sg), and enthalpy(HH9)

 

 

VCalculate entropy
($9),and enthalpy

 

FIGURE J.1 (cont'd.)

 

Calculate volume(V ), mole
fractions, and total moles

Correct mole atoms for residual

 

Set: M = M + l.

 

exhuast gas

169

 

 

 

 

450

 

 

 

 

 

 

 

- +

FR - 55. =: {AFR = AFR - 20,, 1
:llllllih 0
- + ._ I

FR - 25 ., LAFR = AFR - 10. 1

 

 

 

' 7

 

 

+
AFR - 20. =. { AFR AFR - 5.

'6
U

 

7

 

AFR 8 AFR - 2.

 

0 .

P2 8 P2 + 100.
IJK a 0. 7

 

FIGURE J.1 (cont'd.)

170

[ID PDOBQOoTRENTpP008gTRENT

[JOB
IFTC

666

820

840

860

GU.T1HE810
LIST

REAL LOSS

REAL N1

IJK 8 0

E1 8 .0001

Y2 8 1.

OUT 8 .094

CR = 900
HRTTE‘799031CR
Pl 8 200.

Pl 8 PZ/14.696
P01 8 .696/10.696
PDZ 8 .0604’91
P03 8 .03‘P1
P04 8 .304/14.696
AFR 8 175.

DEL = 10

LOSS 8 0.

TA 8 298.15

R 8 1.98726

G 8 1.4

PA 8 1.

N1 8 59.55

Pl 8 PA-PDI

P2 8 P1+PDZ

B 8 .02711

518 (5-1.1/5
T1 8 TA

STAl 8 .68692E1*ALOG(T1)-.29008E-3‘lT1)*.11274E-5*(T18T1)
STBl 8 -.30765£-98(T1*T1*T1)-R8ALOG(P11

ST1 8 STA1+ST81

T2 8 T18(P2/P1)**Gl

STAZ 8 .68692518ALOG(TZ)-.29008E-3*(T21*.11274E-5*(T2*T2)
STBZ 8 -.30745E-9‘lTZ‘T28T21-R’ALDGTP2)

STZ 8 STA2*ST82

1F1A851(ST1-ST2)/ST11-E118609860.840

DSTZ 8 .68692E1/T2—.290085-3+.22547£-58TZ-.92234E-98T28T2
T2 8 T2-(ST2-ST11/OST2 ‘

GO TO 820

5V1 8.820576-1‘T1/P1+B

5V2 8.82057E-1‘T2/P2t3

V1 8 SVZ‘NIICSVZ/SV1-1.ICR1

V3 8 V1/CR

VPD 8 V18V3

V2 8 V30SV28NI

VEF 8 SV1*NIIVPD

NC 8 N1‘(.6869ZE1.(TZ-T11-.145045-38(TZ‘TZ-T18T1)O.TSISTE-6
1*(T28‘3.-T1‘*3.)-.23059E-9.(T2884.-T1‘84.11

NCN8 HC/NI

CRE 8 VIIVZ

V6 8 V185V1‘N1

VOLEF 8(V1-V61/VPD
T1181T1‘9./5.1-€59.67
T213‘T2’9o,5o”‘59067
TL 8 T2

 

 

171

T1 8 1.2‘TL
1 CONT1NUE

AC 8 8.

AH 18.

A0 25.

AN 94.1

Y1 15.042/AFR

Q 8 -Yl.191000*l1‘022“2520l§5306

AH 8 Y1‘AH

AC 8 Y1‘AC

CALL CONLOH(AC.AH.AO.AN.CN2.CN3oCN4.CN5.CN6.CN7.CN8.CN9.CN10.CNll.

1CNT1

F 8 PTICNT

X 8 SQRT‘ABS‘CNZ)’

V6 8 DEL‘V3

V8 81V1'V31‘Y2‘V6

1F1T1'1500.149993
3 CALL PDOBEK1T1'EK1oEKZpEK3oEK4gEK5.EKb)

1 8 1 .
7 CALL PDOBCC‘T1chAC9AHoAUgANoEK1cEKZ'EK3oEKRpEKSQEK69CNZFCN3oCN40

1CN5oCN69CN79CN89CN99CN109CN11.PToCNT'X.11
8 1F‘ABS‘PT'PI1‘.002‘P1130930910
10 F 8 P1/CNT

50 T0 1
30 CALL PDOBHH1T1oCNZoCN3oCN49CN5oCNboCNT'CN89CN9oCN1OoCN119HHH)

50 T0 35
R K81

CALL CONLOH‘ACpAHpAO,ANFCNZpCN39CN49CN59CN60CN79CN89CN99CN10,CN119

1CNT1

N

F8P1/CNT
CALL PDOOHL‘TI.CN29CN3oCNfioCNSoCNboCNTcCNBoCN99CN10oCNllgHHH1
35 AN2 8 AOIZ.
AN3 8 AN/Z.
AN‘ 3 0.
AN5 ' O.
AN6 3 0.
RN? ‘ 00
ANB ‘ 00
AN9 3 00
AN108 0.
RN11: 0.

CALL PDOBHL1TL.ANZ.AN3.AN6.AN5.AN6.ANTcAN8cAN9nAN100AN110HHLI
HHT 8 HHH-HHLOY1‘591§O.-0‘OUT
IFCABSIHHI1-100.161u61060
61 TJ 8T1
60 T0 130
60 1F!HH1160960.50
50 TJ 8T1-10.
GO TO 62
60 TJ 8T1+10.
62 1F(TJ-1500.I67p67.65
65 CALL PDOOEKITJ.EK19£K2¢EK3gEK4gEKSgEK61
70 CALL PDOBCCTTJchACcAHoA0.ANgEchEKZFEK39EK4.EK5¢EK6¢CN2.CN3.CN6.
1CN59CN6oCNTuCNBcCN9.CN109CN11.PTgCNTglgI!
1F!AB$1PT-P11-.0028P11100.100.80
80 F 8 PIICNT
GO TO 70
100 CALL PDOBHHTTJoCN2cCN3¢CN§cCN5.CN60CNTuCNOoCN9oCN10.CN11.HHH)

 

 

 

172

GO to 115
61 L = 1

CALL CONLowtAc.AH.A0.AN.CN2.CN3.CN4.CN5.CN6.CN7.CN8.CN9.C~10.CN11.

1CNI1

FaPl/CNI
110 CALL PooaHLtIJ.CN2.C~3.CN4.CN5.CN6,CN7.CN3,CN9.CN10.CN11.HHH)
115 HHJsHHH-HHL+v1t59740.-Qt001

lFIABSlHHJl-IOO.)130.130.120
120 SL8(HHl-HHJ)/(Tl-TJ)

TK =TJ-(HHJ/SL)

11 =14

TJ =rx

HHI8HHJ

GO to 62
130 CONTINUE

CJZ =CN2/CNT ‘

CJ3 =CN3/CNI

CJ4 =C~41CNI

CJS =CN5/CNT

CJ6 =CN6/CNT

CJ7 =CN7/CNT

CJB =CN8/CNI

CJ9 =CN9/CN1

CJlO=CNlOICNT

CJ11=CNIIICNT

CNTJ = cur

T? = 1.038TJ

P? = P1-PD3

P7V6 = 978V681.98726/.082057

:8

M 8 1

TEST 8 0.
AC1 8 AC
AH1 8 AH
A01 8 A0
AN1 8 AN

150 lFlTT-l500.)160.160.170

150 CALL CONLOHtACI.AHl.AOl.AN1.CN2.CN3.CN4.CNS.CN6.CN1.CN8.CN9.CNIO.
1c~11.c~r1 .
F=PTICNT
CALL PDOBHLtT7.CN2.CN3.CN49CN5.CN6¢CN7.CNB:CN9.CN10.CN11.HH7)
co to 200

170 CALL PDOBEKCT?.EKl.EK2.EK3.EK4.EKS.EK6)

180 F s P1/CNI
CALL Pooacc111.F.Ac1.AH1.A01.A~1.EKL.5K2.EK3.EK4.EK5.EK6.C~2.C~3.
1CN4.C~5.CNb.CN1.CN8.CN9.CN1o.CN11.Pt.c~r.x,11
IF!ABS(PI-P7)-.0028P7)190.190.180

19o CALL PooaHH111.c~2.CN3.CN4.CN5.C~6.CN7.CN8.C~9.CM10.CN11.HH71

zoo lF(H-l.)220.220.210

210 ERR7 . 0*LOSS+HHH -HH7+ p1voo(Hug—1.98726tr9tcnr91tVb/V9
so to 230

220 ERR7 = 0tLoss+HHH-HH1+P7VL

230 [FLABSTERRTJ-IOO.1390.390.240

,240 IFLERR11250.250.260

250 It = 17-10.
60 T0 270

260 It a t7+10.

27o 1F111-1soo.)zeo.2ao.29o

zoo CALL C0~Low1Ac1.AH1.A01.AN1.c~2.c~3.CN4.CN5.CN6.c~1.cua.CN9.CNLO.

 

 

173

1CN119CNT1
F 8 P7/CNT
CALL PDOBHLTTTgCNzoCN3oCN4gCN59CN60CN7oCN8oCN99CN1ovCN119HHT1
GO TO 330
290 CALL PDOBEKTTT'EK1pEKZoEK3gEK69EK5gEK61
300 CALL P008CC1TT.F.AC1'AH19AU1FAN19EK1oEKZoEK39EK49EK5oEKprNZgCN3p
1CN40CN50CN60CN79CN89CN99CN1UFCN119PT9CNTFXQ11
1F1A8$1PT8P71‘.002‘P7132003200310
310 F 8 P7/CNT
GO TO 300
320 CALL PDOBHH1TT.CNZ9CN39CN4'CN59CN69CN79CNByCN9oC‘1ouCN11.HHT1
330 TETH-1.)350935093‘0
340 ERRT =Q‘L0558HHH8HHT8P7V681HH9-1.98726'79‘CN791‘V6/V9
GO TO 360
350 ERRT 8 Q’LOSS’HHH-HHT9P7V6
360 1FTABSLERRT1’250.)38093809370
370 SL 8TERR7-ERRT1/‘T7-TT1
TX 8 TT-ERRT/SL
1F1ABSTT7'TT1‘oO3)38093809371
371 CONT1NUE
T7 8 TT
TT 8 TX
ERR7 8 ERRT
50 T0 270
380 T7 8 TT
HH7 8 HHT
ERR7 8 ERRT
390 1FTN‘30140094000395
395 1FTABS‘ERR7-TEST1’100.190099000400
‘00 TEST 8 ERR7
C72 8CN2/CNT
C73 8CN3/CNT
C79 8CN4/CNT
C75 8CN5/CNT
C76 8CN6/CNT
C77 8CN7/CNT
C78 8CN8/CNT
C79 8CN9/CNT
C7108CN10/CNT
C7118CN11/CNT
CNT7 8 CNT
1F1T7‘1500.141094109405
‘05 CALL ENTROHTT7.FpCN2,CN3oCN“FLNSoCNboCN7oCN80CN99CN1ooCN11oTSN7)
V78 o082057‘T7‘CNT/P7
1F1N’10141BOC189919
‘18 T88 T7‘1V7/V81"o3
‘19 1FTT8‘1500.)‘200420¢§25
‘10 CALL ENTROLTITOFOCNZOCN3'CN‘OCN5'CN6'CN7!CNBQCN9'CNAOOCN11.ISM?)
V78 oOCZOS7‘T7‘CNT/P7
1F‘H‘1o101309130915
513 T88 T7‘1V7/V81..o3
‘15 F 8 0082057‘T8/V8
CALL ENTROLTTO.F.CN2.CN3.CN4.CNS.CN6.CN7.CN8.CN9.CN10.CNll.TSH8)
‘17 CALL PDOBHLlT8.CNZ.CN3.CN&.CN59CN6.CN7.CN8.CN9.CN10.CN11.HHBD
988 F’CNT
1FTABSTLTSHB-TSH71/TSH71‘E1193594350430
‘20 CALL CONLOH‘ACAO‘HLV‘OA'ANA'CNZOCN30CN“OCNSOCN6OCN7DCNBOCN9'CN109
1CN119CNT1

 

 

421

425

430

432
435

442
443
500

510

440

445

450

452

454

174

lFTAFR-IS.O4ZI421.421v415

CNZ = 0.0 -

F = .oazosvtra/va

CALL ENTRO!(T8.F.CN3.CN4.CNS.ISHB)

so to 417 _

F = .0820578t8/V8

CALL PDOBEKTT8.EK1.EKZ.EK3.tK4.EK5.EK6)

CALL P008CC118.F.AC1.AH1.A01.AN1.EK1.Exz.EK3.EK4.EK5.£K6.CN2.CN3.

1CN‘0CN59CNvaN70CN80CN9OCN109CN11098'CNT9X,11

CALL ENTROHTT8oFuCNZoCN3.CN4pCN59CN6.CNTFCN8gCN9gCN10.CNllgTSHBJ
CALL PDOBHHTT8.CN29CN3oCN49CN5oCN69CN7gCN8yCN9gCN10oCNllgHHB1
IFTABSTTTSNB-TSN?)ITSN7)-Ell435.435.440

CALL CPOLOHTT8.CN2.CN3.CN4.CN5.CN6.CN7.CN8,CN9.CV10.CN11.CNT.CPAl
CUA8CPA-1.98726

T8 8 T881TSN8-TSH718T8I1C0A8CNT1

[FITS-1500.1432v432n425

IF!AFR-15.0421421.421a415

P9 8 PA+PD4

C82 8CN2/CNT

C83 8CN3/CNT

C84 8CN4/CNT

C85 8CN5/CNT

C86 8CN6/CNT

C87 8CN7/CNT

C88 8CN8/CNT

C89 8CN9/CNT

C8108CN10/CNT

C8118CN11/CNT

CNTB 8 CNT

1F!N-1.)4429442p443

T9 8 T88TP9/P8188.286

TFIT9-1500.I445o4459500

F 8 P9ICNT

CALL PDOBEKlT9.EK1.EK2.EK3.EK4.EKS.EK61

CALL PDOBCCTT9uFuAC1oAchAOIpANlcEKloEKZ.EK3FEK4oEK50EK6FCN2gCN39

1CN89CN50CNboCN70CN8oCN9cCN1OFCN110P90CNTFxv11

CALL ENTROH!19oF.CN2.CN3.CN4.CN5.CN6.CN?.CN8.CN9.CN10.CN11.TSH9)
CALL PDOBHH‘T9gCNZoCN3oCN‘9CN59CNboCN7.CNBCCN9'CN10'CN110HH91
1F1ABSTTTSH9'TSH71ITSH71’E11655'4559510

CALL CPHIGH‘T9'CN2'CN30CN49CN5pCNbgCN7gCNBoCN9pCN10gCN11oCNTgCPA)
T9 8 T9’1TSH9‘TSH71‘T9/TCPA‘CNT)

GO T0 483

CALL clecH‘TOOCNZQCN3QCN‘QCN5'CN6QCN7'CNBOC~9OCNAOOcull'CNT’CPA1
CUA 8 CPA-1.98726

T8 8 T8“TTSHB'TSH71‘Ta/TCUA‘CNT1

1FTT8’1500.142004200425

CALL CONLOHTAC1oAH19A019AN10CN20CN39CN‘ICNSCCNbcCN79CN80CN90CN109

1CN11oCNTl

F 8 P9/CNT

1F1AFR'15.042145204520454

CNZ ‘ 0.0

F 8 P9/1CN38CN48CN51

CALL ENTR01TT9gEgCN3oCN49CN59TSH91

CALL PDOBHL1T90CN20CN39CN‘QCN50CN60CN70CNBQCNguCNIOQCN110HH91

GO TO 456

CALL ENTROLTT9.FoCNZoCN39CN40CN50CN6.CN7QCNBQCN9QCN109CN11gTSH91
CALL PDOBHL(T90CNZoCN3oCN‘0CN51CN69CN7oCNUQCN99C‘103CN110HH91

456 IFIABSTTTSN9-TSNTI/TSN71-E11455.455o460

:

 

175

460 CALL CPOLOHTT9oCNZ.CN3.CN4.CN5.CN6.CN7.CN89CN9gCN10.CNl1.CNT.CPA)

T9 8 T9-TTSH9-TSN718T9/(CPA8CNT1
1F(T9-1500.)450o4509500

455 V9 8.0820578T9/F

900

902
901

903
904

CNT9 8 CNT
H8N+1
C928CN2/CNT
C938CN3/CNT
C948CN4/CNT
C958CN5/CNT
C968CN6/CNT
C978CN7/CNT
C988CN8/CNT
C998CN9/CNT
C9108CN10/CNT

C9118CN11/CNT
CNT9 8 CNT

RAC 8 AC1’V6/V9
RAH 8 AH18V6IV9
RAD 8 A018V6/V9
RAN 8 AN1‘V6/V9
AC1 8 AC+RAC
AHI 8 AH+RAH
A01 8 A0+RAO
AN1 8 AN+RAN

GO TO 150
HRITET6g20021AFR9H

TJ18TTJ89./5.)-459.67

T718TT789.I5.)-459.67

T818!T8*9./5.)-459.67

1913‘79‘90’501-‘59067

P118P1814.696

P21892814.696

P118P1814.696

P71897814.696

P818P8814.696

P918P9814.696
HP18IPTRTVT-V6)+P9*(V6-V81181.98726/.082057
HPZ 8 THHT-HHB1-1.987268TCNT78T7-CNT8‘T81

HPX 8 HP18HPZ

HP 8 HPX~HC

BNEP 8 (HPT/(VPD‘VB-V61
DHEDI8T(SHEP828.3171/252)8778.16/144.

CDR 8TV7-V61/lV8-V61
TEF8T(NP8453.6)/(19100.8114.224*Y1*252118100.
DEF8TTHP8453.6)I(20591.8114.224*Y18252)18100.
HPX18HH7-HH881.98726*(V88(P8-P9)-V6*TP7-P9))/.082057
V8THHH-THH7-CNT78R‘T7)*THH9-CNT98R8T918V6/V9)8.082057!(P781.98726)
CDR1 8 V/(V8-V6)

IFTIJK190109029901

HRITEI79903)P11

1JK 8 1

CONTINUE

HR1TET799041AFR9TEF0C0R03ME01'TJ1
FDRHATTF10.21

FDRNAT15F15.4)
HRITE(6.40001T19T11vT2uTZloTJcTJloT7uT71oT8gT81yT9pT91
HRTTET694001IPloPllcPZcPZIoPI.PllgPT.P71

 

176

HRITE16.4002)P89P819P9.P91.CR.CRE.VDLEF
HRITETboZOOO)
HRITET6.20011
HRITET6.1000)CJZuCJB.CJ4FCJ59CJ6.CJ7.CJB.CJ9.CJ10.CJ11
HRITET6.10011C72.C739C74.C75.C76.C71.C78.CT9.C710.C711
HRITEtbo1002)C82.683.C84.C85.CB6.C879C88¢CB9.C810.CBl1
HR1TE16010031C920C93oC949C959C969C979C98'C99gC9109C911
HRITEI6g4003)COR.TEF.BNEDl
HRITElboSOOOD
lFlAFR-55.1550v550c555

555 AFR 8 AFR-20.
GO TO 1

550 1FlAFR-25.1556.556v551

551 AFR 8 AFR-10.
GO T0 1

556 IFTAFR-20.1570.510.560

560 AFR = AFR-5.
GO T0 1

570 1FlAFR-15.)800,800.580

580 AFR = AFR-2.5 ‘8—
GO TO 1

800 P2 8 Pl 9100.
1JK 8 0
IFTPZ-IZOO.)6669666.888

2000 FORMATT/g45X.‘H O L E F R A C T I O N 5'9!)

2001 FORMATTSX.'TENP'.6X.'OZ'.9X.'NZ'.9X,'COZ'.8X,'H20'.9X.'CO'.
19X.'HZ'.9X.'OH'.9X.'NO'.10X.'O'.10X9'H'l

2002 FORMATTloxg'AlR FUEL RATIO 8'.F6.2.7X.'H =‘.15./)

1000 FORMATTZXo'COMB EX'91X910E11.4)

1001 FORMATIZX9°POTNT 7'a1Xo10E11.4)

1002 FORHAT!2X.'POINT 8'n1Xn10E11.4)

1003 FORMATTZXy'POINT 9'a1X910E11.4o/)

4000 FORNATT4X.'T18'.F6.1.'K'.'8'.F6.1.'F'.2X.'T28'.F6.1.'K'.'8'.F6.1.
l'F.QZX9.TC3.'F6019.K.u.3.996019.F.pZXo'T73.oF6019.K.9'8..F601'.F.'
12x..788.9F6019'K'9.8'QF6011'F.QZXQ'T98'9F6.19'K.9.8'9F6019.F'0/1

«001 FORMATIbX.'Pl8'.F5.2.'AIH','8'.F6.3.'PSIA'.5X.'P28'.F6.2.'ATH'.‘8'
1.F7.2.'PSIA'.SX.‘PC='.F6.2.'ATR'.'='.F7.2$‘PSIA'.SX.'P7='.F6.2.
1'ATN'.'8'.F7.2.'PSTA'9/)

#002 FORNATTbx..P88.oF502g'RTH'..3.9F702"PS'R'.5XQ'P98‘.F5029'ATH'g'8'
1.F5.2.'PSIA'.5X.'ACT CR8'.F5.1.5X .‘EFECT CR8'.F6.2.5X.'VOL EFF8'.
1F5.2.'!'pli

4003 FORRATTbxg'CUT OFF RATIO8'.F5.4.5X.'THERNAL EFF8'.F5.2.'3'.6X.
1'1"EP8'.F6.2.'PSI'.//lll

SOOOFORH‘71/”6XQ.00......00.000.00.000...
1.000.000.0000ooooooooooooooooo.’/l1

one Stop
END

APPENDIX K

MC‘

 

APPENDIX K
PLOTTER PROGRAM
A listing of the plotter program that was used in the preparation
of the performance plots presented in this thesis is shown in this
appendix. No flow chart is presented because the program was only
used as an aid in preparing graphical presentations of data and

was not part of the cycle analysis.

 

 

177

178

// JOB P008gTRENToO3000E111,TRENT DOS PLOT
ll ASSGN SYSOO99X'04O'

II OPTION LINK

INCLUDE ILFGHTAB

II EXEC FFORTRAN

50

1000

24

DIMENSION 91(1119AFR111a13)9COR11191319BM(11u13)

DIMENSION TEFI11913)9TH(11.131oIBUF11000)

READ (591009END8500)CR

DO 1 181.11

READ (5.1009END8ZIPIIII

READ IS'IDIIIAFRIIoJ19TEFII9J).CORII.J).BM(I.J).THII.J)pJ81.131
CALL PLOTS ‘IBUFIIOOOI

CALL PLOT IO..‘11-u’31

CALL PLOT 100.006'-3) ' Ii
CALL AXIS IO.90.,'CUT OFF RATIO"°13.12..U.,0.9.O51

CALL AXIS I0..O..'THERMAL EFFICIENCY'118,10.o90..20..5.1

N8I-1 '
Imeo=o 1...
DO 5 I81oN -
DO 3 J81913

IFITEFIIoJI‘ZD.)3920p20

IFITEFII.J)870.)21121¢3

IFICORIIcJ113022922

IFICORIIoJJ-.6123923u3

X 320..CDRIISJ)

Y 8 .2*TEFIIoJ)-4.0

CALL SYMBOL IXpYroo7'INTEQIOOQ‘1)

CONTINUE

INTEQ8INTE091

IFIINTEQ’815'4.5

INTEQ810

CONTINUE

HRITE 16.1000) IITEFII. J19J81 13) 181:9)

NRITE 16.1000) IICORII. JIvJ81913) I81 9)

FORMAT IBF10.4/5F10.41

CALL PLOT112.590.¢'31

CALL SYMBOL I0.p9&8g.29'CR 8 ..0.'5)

CALL NUMBER I999..999.9.29CR.O..21

Y 3 9.8

INTEQ 8 0

DO 7 I 81.N

Y'Y‘Ooz

CALL SYMBOL I0..Yg.0791NTEQ,O.o'II

CALL SYMBOL I999.p999.o.1g' PC 8 .900971

CALL NUMBER I999..999...1.PII11.0..2)

INTEO 8 INTEQ’I

IFIINTEQ‘BIToboT

INTEO 8 10

CONTINUE

CALL PLOT (3.00.983)

CALL AXIS ID..O..'CUT OFF RATIO'p-13g12..0.90...051
CALL AXIS ID..O.9'IMEP IPSIAI'911.6.99O..U..20.1
INTEO 8 0

DO 10 I810N

00 8 .181013

IFIBMIIQJ1’160&12902898

IFIBMIIoJ11 8925925

 

25
26
27

28
29
30
31

11

12
I3

500

100
101

It
It

179

IFICORIIoJII 8.26.26
IFICORII.JI-.6)27o27.8

X 820.8CORII.JI

Y 8 .058BMIIoJI

CALL SYMBOL Ixon.07.INTEQ.O..‘I)
CONTINUE

INTEQ 8 INTEQLI

IF IINTEO-8110.9.10

INTEQ 810

CONTINUE

HRITE (6.1000) IIBMII.J).J=I.13I.I=1,9I
"RITE 16.10001 IITHII'JI'J=1913IOI=199I
CALL PLOT I15.p0..-3I

CALL AXIS SOOOOOOTCUI OFF RATED..-13'120,00'00'005,
CALL AXIS I0..0..'TENPERATURE IFI'.15,9.0,90..0..500.)
INTEQ 8 0

DO 13 I819N

DO 11 J81o13

IFITHIluJI-SOOO.128928.11
IFITHII.JIIII.29929
IFICORII.JIIII.30.30
IFICORII.JI-.6)31.31.11

X8 20.8CORIIoJI

Y 8 .0028THI11JI

CALL SYMBOL (X’Yroo7'IN7801000-1)
CONTINUE

INTEO 8 INTEQ+1

IF IINTE0-8113912gl3

INTEQ 810

CONTINUE

"RITE (6.1.0001 IIIHII'J)OJ=1913’!I=l097
CALL PLOT (12.90..83I

GO TO 50

CALL PLOT I0..0-.999.I

STOP

FORMAT IF10.2I

FORMATI5F15.4I

END

“F

 

 

APPENDIX L

 

 

APPENDIX L
SAMPLE COMPUTER PRINTOUT

This appendix contains sample computer printout from the main
line program for a pressure of 500 psia and air-fuel ratios from 175/1
to 15/1. The main line program normally starts with an assigned
burner pressure of 200 psia and an air-fuel ratio of 175/l. The
program changes air-fuel ratio in a defined manner until it reaches
an air—fuel ratio of 15/1 and at that point it increases the pressure
by 100 psia. This process is repeated until a burner pressure of
1200 psia is reached. The total computer time involved in calculating
performance for 11 burner pressures and 14 different air-fuel ratios
at each pressure was approximately three minutes. The calculations

were made on an IBM 360 system.

180

 

 

 

 

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