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A Complete Parametric Cycle Analysis for Ideal TurboFan Engine with Interstage Turbine Burner
A Complete Parametric Cycle Analysis for Ideal TurboFan Engine with Interstage Turbine Burner
S.L. Yang, Y.K. Siow, K.H. Liew, and E. Urip
Mechanical Engineering – Engineering Mechanics Department
Michigan Technological University
1400 Townsend Drive
Houghton, MI 49931-1295
906/487-2624
906/487-2822 (Fax)
C. J. Marek
Combustion Technology Branch, MS 5-10
NASA Glenn Research Center
21000 Brookpark Road
Cleveland, OH 44135
216/433-3584
216/433-3000 (Fax)
October 23, 2002
ABSTRACT
Today modern aircraft is based on air-breathing jet propulsion systems, which uses moving fluids as substances to transform energy carried by the fluids into output power. Throughout aero-vehicle evolution, improvements have been made on the engine efficiency and pollutants reduction. This parametric study focuses on a complete parametric cycle analysis of a turbofan engine with an Interstage Turbine Burner (ITB).
The major advantages associated with the addition of an ITB are the improvement of thermal efficiency and reduction in NOx emission. Lower temperature peaks in the main combustor results in lower thermal NOx emission and a lower amount of cooling air required.
The objective of this study is to make use of the engine component parameters, namely compressor or turbine polytropic efficiency, burner efficiency, pressure drop across the engine components, design limitation (burner exit temperature), and flight environment as inputs to calculate engine performance, specific thrust and thrust specific fuel consumption. This relation can provide guidance in identifying the characteristics of the engine’s components. The knowledge can subsequently be used to develop and optimize the performance and integration of each component.
In this study, each component in the engine is treated individually. The first-law energy equation, second-law, and conservation of momentum are then applied.
Visual Basic program, Excel macrocode, and Excel neuron code are used to facilitate Excel software to plot engine performance versus engine design parameters. This program computes and plots the data sequentially without forcing users to open other types of plotting programs. A user’s manual on how to use the program is also included in this report.
ACKNOWLEDGEMENTS
This project on evaluating the improvement in performance of an aircraft gas turbine by adding an Interstage turbine burner (ITB) is really challenging and would have not been accomplished without the technical assistance and suggestion given from our graduate advisor, Dr. S.L. (Jason) Yang.
Our sincere gratitude next goes to Dr. Cecil John Marek, for giving us this opportunity to employ our limited knowledge in engineering mechanics in solving such an interesting and practical problem related to gas turbine propulsion like this.
The authors would like to thank Dr. Paul Penko for being the grant monitor on this project. Appreciation also goes to Dr. Scott John at NASA Glenn Research Center for providing us the listing data for a two-spool high bypass turbofan engine with and without ITB, and suggesting some valuable improvement that can be done to the code.
Special thank is due to Mr. Jeffrey R. Herbon from Williams International for carefully reviewing and test-driving the code. Based on his experience and expertise in cycle analysis, he has given us many valuable suggestions on how the program could be enhanced for increased usefulness in the future.
TABLE OF CONTENTS
ABSTRACT ………..……………………………………………………………..ii
ACKNOWLEDGEMENT ……………………………………………………...iii
LIST OF FIGURES ……………………………………………………………...v
LIST OF TABLES …….………………………………………………………… viNOMENCLATURE ……………………………………………….…………….. vii
1. INTRODUCTION ……………………………………....…...………..….…...1
2. ENGINE PERFORMANCE PARAMETERS…………………………….…3
3. TURBOFAN with ITB CYCLE ANALYSIS……………………………….. 11
4. SUMMARY of EQUATIONS………………………………………………... 20
5. USER MANUAL …………………………………………………….……….. 24
6. DATA VALIDATION………………………………………………………... 327. RESULTS AND DISCUSSIONS…………………………………….………. 35
APPENDIX
A1 – EXCEL MACRO VISUAL BASIC CODE ………………………….. 44
A2 – FORTRAN 77 CODE ………………..………………………...……… 52
REFERENCES………………………………………………………………….. 78
LIST OF FIGURES
Figure 1.1 – A Gas Generator propulsion system……………………………………….. 1
Figure 1.2 – A TurboJet Engine…………………………………………………….…… 1
Figure 1.3 – A TurboFan Engine…………………………………………………….….. 2
Figure 3.1 – Turbofan with Interstage Turbine Burner………………………………… 11
Figure 5.1 – Excel Input sheet screenshot…………………………………………….... 25
Figure 6.1 – Turbofan Engine with ITB: specific thrust vs flight Mach number
(both Fortran77 and Excel) …………………………………….…………. 32
Figure 6.2 – Turbofan Engine with ITB: specific thrust vs flight Mach number
(both Fortran77 and Excel) …………………………………………….…. 33
Figure 6.3 – Turbofan Engine with ITB: specific thrust vs flight Mach number
(both Fortran77 and Excel) ……….………...…………………………….. 33
Figure 6.4 – Turbofan Engine ITB: specific thrust vs flight Mach number
(both Fortran77 and Excel) ……….………...…………………………….. 34
Figure 7.1 – Turbofan Engine: specific thrust vs flight Mach number……………….… 36
Figure 7.2 – Turbofan Engine: thrust specific fuel consumption vs flight Mach
number ……………………………………………………………………. 37
Figure 7.3 – Turbofan Engine: specific thrust vs compressor pressure ratio ………….. 38
Figure 7.4 – Turbofan Engine: thrust specific fuel consumption vs compressor pressure
ratio ………………………………………………………………..……... 38
Figure 7.5 – Turbofan Engine: fuel/air ratio vs flight Mach number …………..……... 39
Figure 7.6 – Turbofan Engine: specific thrust vs fan pressure ratio …………….…….. 40
Figure 7.7 – Turbofan Engine: thrust specific fuel consumption vs fan pressure
ratio ………………………………………………………..……….……. 40
Figure 7.8 – Turbofan Engine: specific thrust vs bypass ratio ……………………..….. 41
Figure 7.9 – Turbofan Engine: thrust specific fuel consumption vs bypass pressure
ratio …………………………………………………….………………... 42
LIST OF TABLES
Table 7.1 – Input Data for Engine Parameters (SI unit) …………………………...35
Table 7.2 – Input Data for Option 1 (SI unit) ……………………………………...36
Table 7.3 – Input Data for Option 2 (SI unit) ……………………………………...37
Table 7.4 – Input Data for Option 3 (SI unit) ……………………………………...39
Table 7.5 – Input Data for Option 4 (SI unit) ……………………………………...41
Table A.1 – Description of variables in MS Excel Macro Visual Basic code ……..44 Table B.1 – Description of Input Variables in ‘input’ file (English and SI unit)…..55
NOMENCLATURE
A
cross-sectional area
a
sound speed
CP
specific heat at constant pressure
D
drag force
e
polytropic efficiency
F
force uninstalled thrust
f
fuel/air ratio
gc
Newton’s constant
h
enthalpy
hPR
low heating value of fuel
M
mach number
0
·
m
F
0
m
F
mass flow rate
P
pressure
Pt
total pressure
Q
·
rate of thermal energy released or absorbed
R
universal gas constant
S
uninstalled thrust specific fuel consumption
T
temperature or installed thrust
TSFC
installed specific fuel consumption
Tt
total temperature
V
absolute velocity
W
·
power
α
bypass ratio
γ
ratio of specific heats,
v
p
c
c
m
h
mechanical Efficiency
ηO
overall Efficiency
ηP
propulsive Efficiency
ηT
thermal Efficiency
π
ratio of total pressure
πr(exception) ratio between total pressure and static pressure due to the ram effect,
P
P
t
τ
ratio of total temperature
(r(exception) ratio between total temperature and static temperature due to the ram effect,
T
T
t
τλ
ratio between total enthalpy and enthalpy at ambient condition
SUBSCRIPTS
b
main burner or properties between main burner exit and ITB
c
properties between upstream and main burner or engine core
d
diffuser
e
exit
f
fan
fn
fan-nozzle
HPC
high pressure compressor
HPT
high pressure turbine
ITB
interstage turbine combustors
LPC
low pressure compressor
LPT
low pressure turbine
O
inlet
n
nozzle
r
ram
tproperties between ITB exit and downstream or total/stagnation values of properties (i.e. temperature, pressure or enthalpy)
1. Introduction
This program is designed to give a complete parametric cycle analysis of an ideal Turbo Fan air-breathing propulsion system. In most common air-breathing propulsion engines, the “heart” of a gas turbine is the gas generator. It consists of three major components namely, compressor, combustor, and turbine as shown schematically in Figure 1.1.
Figure 1.1 – A Gas Generator Propulsion System
The idea behind a gas generator is to convert intake air mixed with fuel into high temperature and high pressure gas. Depending on the applications of the gas turbine, the energy provided is extracted and used for different applications (turbojet, turbofan, turbo-shaft, turboprop, and ramjet) through different mechanisms. A turbojet engine can be constructed by adding an inlet and a nozzle as shown in Figure 1.2.
m
·
Figure 1.2 – A Turbo Jet Engine
The nozzle converts the internal energy of the hot gas into kinetic energy or thrust. The work extracted by the turbine is used to drive the compressor. In the case of a turbofan, turboprop, and turbo-shaft engine, the work from the turbine is required to drive a shaft for the turbo-shaft, a fan for the turbofan, and a propeller for the turboprop in addition to driving the compressor. The ramjet engine consists of an inlet, a combustor, with a nozzle at the exit. It does not require the compressor because the inlet already uses a ram air-compressing mechanism such that intake air has sufficient kinetic energy to increase its pressure.
The main objective of this analysis is to determine the relationships between engine performance (primarily specific thrust
·
m
F
, thrust specific fuel consumption) to design parameters (compressor pressure ratio, fan pressure ratio, bypass ratio, etc), to design constraints (burner exit temperature, compressor exit pressure, etc), and to flight environment (Mach number, ambient temperature, ambient pressure, etc).
0
·
m
F
Figure 1.3 – A Turbo Fan Engine
2. Aircraft Engine Performance Parameters
Thrust
Thrust is the force used to sustain the flight (thrust = drag), accelerated flight (thrust > drag), deceleration (thrust < drag). Using Figure 1.3 for the control volume, we can apply a momentum balance to the control volume. Uninstalled thrust F of a jet engine (single inlet and single exhaust) is given by
)
(
)
A
P
P
g
V
m
V
m
m
e
o
e
c
o
e
o
fuel
o
F
-
+
-
+
ç
ç
è
æ
=
·
·
·
(2.1)
where
m
m
fuel
o
.
.
,
are mass flow rates of air and fuel respectively
V
V
e
o
,
are velocities at inlet and exit respectively
P
P
e
o
,
are pressure at inlet and exit respectively
For the ideal case, the hot gas is expanded to the ambient pressure which gives Pe = Po. Equation (2.1) then becomes
)
g
V
m
V
m
m
c
o
e
o
fuel
o
F
·
·
·
-
+
ç
ç
è
æ
=
(2.2)
The installed thrust T is given by
D
D
F
T
nozzle
inlet
-
-
=
(2.3)
where Dinlet and Dnozzle are the drag force from the inlet and the nozzle.
Specific Fuel Consumptions
The specific fuel consumption is the rate of fuel use by the propulsion system per unit of thrust produced. The installed specific fuel consumptions, TSFC, and the uninstalled specific fuel consumptions, S, are given by
T
m
TSFC
fuel
·
=
(2.4)
F
m
S
fuel
·
=
(2.5)
Efficiency of an Engine
Some of the following parameters will be used widely in this program namely, thermal efficiency, propulsive efficiency, and overall efficiency
The thermal efficiency characterizes the net energy output extracted (shaft work) from the engine divided by the available thermal energy (fuel).
Q
W
in
out
T
·
·
=
h
(2.6)
where,
÷
ø
ö
ç
è
æ
=
=
=
·
·
·
h
m
released
energy
thermal
of
rate
Q
engine
of
out
power
net
W
efficiency
thermal
PR
out
T
f
in
h
The propulsive efficiency defines the ratio between the engine power output and the power being used to run the aircraft.
W
V
T
out
P
·
=
0
h
(2.7)
where,
engine
of
out
power
net
W
aircraft
of
velocity
V
system
propulsion
of
thrust
T
engine
of
efficiency
propulsive
out
P
=
=
=
=
·
0
h
An overall performance of a propulsion system is given by the combination between thermal and propulsive efficiencies.
h
h
h
T
P
O
=
(2.8)
where,
efficiency
overall
O
=
h
Notations
Some useful quantity notations for compressible flow will be used in this report namely, stagnation temperature, stagnation pressure, and Mach number.
Stagnation temperature or total temperature Tt is defined as the temperature obtained when steadily flowing fluid is brought to rest adiabatically without extraction of work. Applying the first law of thermodynamic to a calorically perfect gas gives:
2
2
V
h
h
t
+
=
(2.9)
where,
h = static enthalpy
ht = enthalpy at stagnation condition
V = velocity
With the assumptions of constant specific heat coefficient, the above equation can be written as:
C
V
T
T
P
t
2
2
+
=
(
÷
ø
ö
ç
è
æ
-
+
=
M
T
T
t
2
2
1
1
g
(2.10)
where,
heats
specific
of
ratio
number
Mach
M
e
temperatur
static
T
e
temperatur
stagnation
T
t
=
=
=
=
g
Stagnation pressure or total pressure Pt is defined as the pressure reached when a steady flowing fluid is brought to rest adiabatically and reversibly. Using the isentropic relation, the total pressure is given by
÷
ø
ö
ç
è
æ
-
+
=
-
M
P
P
t
2
2
1
1
1
g
g
g
(2.11)
The ratio of total temperatures τ and the ratio of total pressure π across a component is denoted by the subscript: d for diffuser, LPC for low pressure compressor, HPC for high pressure compressor, b for main burner, ITB for inter-stage turbine burner, LPT for low pressure turbine, HPT for high pressure turbine, n for nozzle, and f for fan.
For example:
diffuser
entering
pressure
total
diffuser
leaving
pressure
total
d
=
p
diffuser
entering
e
temperatur
total
diffuser
leaving
e
temperatur
total
d
=
t
Exceptions
For the free stream, ram, we define τr as a ratio of total temperature/static temperature and πr as a ratio of total pressure/static pressure.
M
T
T
t
r
0
2
0
0
2
1
1
-
+
=
=
g
t
(2.12)
(
)
÷
ø
ö
ç
è
æ
-
+
=
=
-
M
P
P
t
r
0
2
2
1
1
1
0
0
g
p
g
g
(2.13)
A ratio between total enthalpy of the burner exit and ambient enthalpy, denoted by τλ, is defined such that it will be one of the input parameters.
(
)
(
)
T
C
T
C
p
t
p
ambient
for
exit
burner
b
,
0
=
-
t
l
(2.14)
(
)
(
)
T
C
T
C
p
t
p
ambient
for
exit
burner
stage
Inter
ITB
,
0
-
-
=
t
l
(2.15)
Component Performance
In this analysis it is acceptable to assume that the working fluid in the engine can be idealized as a perfect gas. Properties of an ideal gas strongly depend on the temperature. This cycle allows fluid properties variation across the engine which assumes constant fluid properties from the main burner entrance upstream (Cpc, γc), from ITB entrance to the main burner exit (Cpb, γb), and from ITB exit downstream (Cpt, γt).
Inlet and Diffuser
Pressure losses occur due to the friction with the inlet wall. The total pressure ratio, πd, is always less than 1.
In supersonic flight, the pressure losses cause shock waves which produce greater pressure losses. The inlet total pressure is defined as the product of the ram pressure ratio and the diffuser pressure ratio. Therefore the portion of the pressure loss due to the shock waves and wall friction is defined by:
h
p
p
r
d
d
max
=
(2.16)
From the Military Specification 5008B (Ref. 2), the following relation is obtained:
(
)
ï
ï
ï
î
ï
ï
ï
í
ì
<
+
£
<
-
-
£
=
M
M
M
M
M
r
0
4
0
35
.
1
0
5
935
800
5
1
1
075
.
0
1
1
1
0
0
h
(2.17a,b,c)
Compressor and Turbine
The compressor is measured through two type of efficiencies namely, isentropic efficiency and poly-tropic efficiency. The Isentropic efficiency is defined by
(
)
1
1
1
-
-
=
=
-
t
p
h
p
p
h
g
g
c
c
c
c
c
c
given
for
n
compressio
of
work
actual
given
for
n
compressio
of
work
ideal
(2.18)
The poly-tropic efficiency is defined as
change
pressure
al
differenti
a
for
n
compressio
of
work
actual
change
pressure
al
differenti
a
for
n
compressio
of
work
ideal
e
c
=
(2.19)
With the assumption of constant ec,we can obtain the relation between τc and πc:
(
)
(
)
p
t
g
g
c
e
c
c
c
c
1
-
=
(2.20)
Going through a similar procedure as the compressor, we obtain turbine isentropic efficiency, turbine poly-tropic efficiency and the relationship between τt and πt as follows:
(
)
p
t
h
p
p
h
g
g
t
t
t
t
t
t
given
for
work
turbine
actual
given
for
work
tubine
ideal
1
1
1
-
-
-
=
=
(2.21)
(
)
[
]
t
p
g
g
t
t
e
t
1
-
=
(2.22)
3. TurboFan-Seperate Exhaust Streams with ITB Cycle
Analysis
Figure 3.1 – TurboFan with Interstage Turbine Burner (ITB)
Assumptions:
1. Perfect gas upstream of main burner with constant properties γc, Rc, Cpc.
2. Perfect gas between station four and five with constant properties γb, Rb, Cpb.
3. Perfect gas downstream of inter-stage burner with constant properties γt, Rt, Cpt.
4. All components are adiabatic, no heat loss.
5. The efficiencies of the compressor-HPC, compressor-LPC, fan, turbine-HPT, and turbine-LPT are described through the use of polytropic efficiencies eHPC, eLPC, ef, eHPT, and eLPT, respectively.
Fan Stream
Step 1.
Uninstalled thrust of the fan stream Ff is given by
(
)
(
)
P
P
A
V
V
g
m
F
c
f
f
0
19
19
0
19
-
+
-
·
=
(3.1)
rearranging gives:
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
-
+
-
=
g
c
c
f
f
P
P
a
V
T
T
M
a
V
g
a
m
F
19
0
0
19
0
19
0
0
19
0
1
.
(3.2)
Step 2.
M
T
T
a
M
a
a
V
2
2
2
2
19
0
19
0
19
19
2
0
19
=
=
÷
ø
ö
ç
è
æ
(3.3)
Step 3.
(
)
ú
ú
û
ù
ê
ê
ë
é
-
÷
ø
ö
ç
è
æ
-
=
-
1
1
2
1
19
19
19
2
g
g
g
c
c
P
P
M
t
c
(3.4a)
where
p
p
p
p
fn
f
d
r
t
P
P
P
P
19
0
19
19
=
(3.4b)
Step 4.
(
)
g
g
c
c
P
P
T
T
T
T
T
T
T
T
t
t
t
t
1
19
19
0
19
19
19
0
19
0
19
-
÷
ø
ö
ç
è
æ
=
=
(3.5a)
where
t
t
f
r
t
T
T
=
0
19
(3.5b)
Engine Core Stream
Step 1.
Uninstalled thrust
(
)
P
P
A
V
m
V
m
g
F
c
c
c
0
10
10
0
10
10
1
-
+
÷
ø
ö
ç
è
æ
·
-
·
=
(3.6)
rearranging gives:
÷
ø
ö
ç
è
æ
-
+
÷
÷
ø
ö
ç
ç
è
æ
-
=
·
·
·
·
P
P
m
P
A
M
a
V
m
m
g
a
m
F
c
c
c
c
c
10
0
10
10
0
0
10
10
0
1
(3.7)
Step 2.
f
m
m
c
+
=
·
·
1
10
(3.8a)
where
m
m
m
m
ITB
b
c
·
·
·
·
+
+
=
10
(3.8b)
f
f
f
ITB
b
+
=
(3.8c)
m
m
f
c
b
b
·
·
=
(3.8d)
m
m
f
c
ITB
ITB
·
·
=
(3.8e)
Step 3.
÷
ø
ö
ç
è
æ
-
P
P
m
P
A
c
10
0
10
10
1
.
multiplied by
T
R
a
a
V
A
m
c
c
0
0
0
10
10
10
10
g
r
·
(3.9)
rearranging gives:
(
)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
+
=
÷
ø
ö
ç
è
æ
-
·
g
c
c
c
t
c
P
P
g
a
a
V
T
T
R
R
f
P
P
m
P
A
10
0
0
0
10
0
10
10
0
10
10
1
1
1
(3.10)
Step 4.
Uninstalled thrust for the engine core becomes:
(
)
(
)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
÷
ø
ö
ç
è
æ
-
+
+
-
+
=
·
g
c
c
t
c
c
c
P
P
a
V
T
T
R
R
f
M
a
V
f
g
a
m
F
1
1
1
1
10
0
0
10
0
10
0
0
10
0
(3.11)
Step 5.
T
R
T
R
M
a
V
c
c
t
t
0
10
10
2
2
0
10
g
g
=
ú
û
ù
ê
ë
é
(3.12)
From the total pressure and mach number relation
(
)
ú
ú
û
ù
ê
ê
ë
é
-
÷
ø
ö
ç
è
æ
-
=
-
1
1
2
1
10
10
10
2
g
g
g
t
t
P
P
M
t
t
(3.13)
(
)
÷
ø
ö
ç
è
æ
=
=
-
P
P
T
T
T
T
T
T
T
T
t
t
t
t
t
t
10
10
1
0
10
10
10
0
10
0
10
g
g
(3.14a)
where
p
p
p
p
p
p
p
p
p
n
LPT
ITB
HPT
b
HPC
LPC
d
r
t
P
P
P
P
10
0
10
10
=
(3.14b)
t
t
t
t
t
t
t
t
t
n
LPT
ITB
HPT
b
HPC
LPC
d
r
t
T
T
=
0
10
(3.14c)
1
=
d
t
(isentropic)
(3.14d)
1
=
n
t
(isentropic)
(3.14e)
LPC
HPC
r
t
b
T
T
t
t
t
t
0
4
=
(3.14f)
Step 6.
Applying the First Law of Thermodynamics and ideal gas relation to the main burner, neglecting kinetic and potential energy changes:
T
C
m
h
m
T
C
m
t
pb
b
PR
b
b
t
pc
c
4
4
3
·
·
·
=
+
-
h
(3.15)
Multiplying the above equation with
T
T
C
m
T
t
pc
c
t
2
0
0
·
, and re-arranging gives:
(
)
t
h
t
t
l
b
b
pc
b
b
PR
b
r
c
f
T
C
f
h
-
-
+
=
+
1
0
(3.16)
Solving for fb
T
C
h
f
pc
b
PR
b
b
b
r
c
b
0
-
-
-
-
-
=
h
t
t
t
t
l
l
(3.17a)
where
T
T
C
C
t
pc
pb
b
0
4
=
-
t
l
;
T
T
t
t
2
0
=
for the case adiabatic (3.17b)
LPC
HPC
c
t
t
t
=
(3.17c)
Step 7.
Applying the First Law of Thermodynamics and the ideal gas relation to the ITB, neglecting kinetic and potential energy changes:
T
C
m
h
m
T
C
m
t
pt
ITB
PR
ITB
ITB
t
pb
6
6
5
4
·
·
·
=
+
-
h
(3.18)
Multiplying the above equation with
T
T
C
m
T
t
pb
t
7
0
4
10
·
, and re-arranging gives:
÷
÷
ø
ö
ç
ç
è
æ
+
+
=
÷
÷
ø
ö
ç
ç
è
æ
+
+
-
-
f
C
C
f
f
T
C
h
b
pb
pc
ITB
b
ITB
pb
ITB
PR
ITB
b
c
d
r
f
1
1
1
0
t
h
t
t
t
t
l
(3.19)
Solving for fITB
[
]
f
T
C
h
C
C
C
C
f
b
pb
ITB
PR
ITB
ITB
pb
pc
ITB
pb
pc
b
c
d
r
ITB
+
÷
÷
ø
ö
ç
ç
è
æ
-
÷
÷
ø
ö
ç
ç
è
æ
-
=
-
-
-
1
0
h
t
t
t
t
t
t
l
l
(3.20a)
where
T
T
C
C
t
pc
pt
ITB
0
6
=
-
t
l
;
T
T
t
t
7
10
=
for the case adiabatic (3.20b)
LPC
HPC
c
t
t
t
=
(3.20c)
LPC
HPC
r
t
b
T
T
t
t
t
t
0
4
=
(3.20d)
1
=
d
t
(3.20e)
Step 8.
Applying the first law to each individual compressor and turbine, neglecting kinetic and potential energy changes:
Power balance for LPC:
(
)
T
T
C
m
W
t
t
pc
c
LPC
2
5
.
2
.
.
-
=
(3.21)
Power balance for HPC:
(
)
T
T
C
m
W
t
t
pc
c
HPC
5
.
2
3
.
.
-
=
(3.22)
Power balance for HPT:
(
)
T
T
C
m
W
t
t
HPT
m
pb
HPT
5
4
4
.
.
-
=
-
h
(3.23)
Power balance for LPT:
(
)
T
T
C
m
W
t
t
LPT
m
pt
LPT
7
6
6
.
.
-
=
-
h
(3.24)
Power balance for Fan:
(
)
T
T
C
m
W
t
t
pc
F
f
2
13
.
.
-
=
(3.25)
It is chosen that the HPT and HPC are connected by a single shaft; therefore for an ideal turbofan the work relation is given by:
HPT
HPC
W
W
·
·
=
(3.26)
Multiplying both sides by
T
C
m
t
pc
c
5
.
2
.
1
gives:
(
)
(
)
(
)
HPT
HPT
m
b
HPC
b
pc
pb
HPC
f
C
C
t
h
t
t
t
-
+
=
-
-
1
1
1
(3.27)
Solving for τHPT:
(
)
h
t
t
t
t
HPT
m
b
HPC
pc
pb
b
HPC
HPT
C
C
f
-
+
-
+
=
1
1
1
(3.28a)
where
c
r
t
b
T
T
t
t
t
0
4
=
(3.28b)
LPT, LPC, and the fan are connected by a single shaft; therefore for an ideal turbofan, the work relation is given by:
W
W
W
LPT
f
LPC
.
.
.
=
+
(3.29)
multiplying both sides by
T
C
m
t
pc
c
2
.
1
gives:
(
)
(
)
(
)
(
)
t
h
t
a
t
LPT
LPT
m
t
t
pc
pt
f
LPC
T
T
f
C
C
-
+
=
-
+
-
-
1
1
1
1
2
6
(3.30)
Solving for τLPT:
(
)
(
)
h
t
a
t
t
LPT
m
t
t
pc
pt
f
LPC
LPT
T
T
C
C
f
-
+
-
+
-
+
=
2
6
1
1
1
1
(3.31a)
where
t
t
t
t
t
ITB
HPT
b
HPC
LPC
t
t
T
T
=
2
6
(3.31b)
HPT
b
c
r
t
ITB
T
T
t
t
t
t
t
0
6
=
(3.31c)
Step 9.
The total uninstalled thrust per unit mass flow rate intake is given by:
(
)
a
a
+
+
=
1
.
.
.
0
m
F
m
F
m
F
F
F
c
c
(3.32)
Step 10,
The thrust specific fuel consumption S is give by:
(
)
(
)
m
F
f
f
S
ITB
b
.
0
1
a
+
+
=
(3.33)
4. Summary of Equations:
INPUTS:M0, T0, γc, Cpc, γb, Cpb, γt, Cpt, hPR-b, hPR-ITB, πd max, πb, πITB, πn, πfn, eHPC,
eLPC, ef, eHPT, eLPT, ηb, ηITB, ηm-HPT, ηm-LPT, P0/P10, P0/P19, Tt4, Tt6, πHPC,
πLPC, πf, α
OUTPUTS:
f
S
m
F
,
,
.
0
EQUATIONS:
C
R
pc
c
c
c
g
g
1
-
=
(4.1)
C
R
pb
b
b
b
g
g
1
-
=
(4.2)
C
R
pt
t
t
t
g
g
1
-
=
(4.3)
T
R
a
c
c
0
0
g
=
(4.4)
M
a
V
0
0
0
=
(4.5)
M
c
r
0
2
2
1
1
-
+
=
g
t
(4.6)
(
)
t
p
g
g
r
c
c
r
1
-
=
(4.7)
(
)
ï
ï
ï
î
ï
ï
ï
í
ì
>
+
£
<
-
-
£
=
5
935
800
5
1
1
075
.
0
1
1
1
0
4
0
0
35
.
1
0
0
M
for
M
M
for
M
M
for
r
h
(4.8)
h
p
p
r
d
d
max
=
(4.9)
T
T
C
C
t
pc
pb
b
0
4
=
-
t
l
(4.10)
T
T
C
C
t
pc
pt
ITB
0
6
=
-
t
l
(4.11)
(
)
(
)
p
t
g
g
HPC
e
HPC
c
c
HPC
1
-
=
(4.12)
(
)
(
)
p
t
g
g
LPC
e
LPC
c
c
LPC
1
-
=
(4.13)
(
)
(
)
p
t
g
g
f
e
f
c
c
f
1
-
=
(4.14)
LPC
HPC
c
t
t
t
=
(4.15)
c
r
t
b
T
T
t
t
t
0
4
=
(4.16)
1
1
-
-
=
t
t
h
HPC
HPC
HPC
e
HPC
(4.17)
1
1
-
-
=
t
t
h
LPC
LPC
LPC
e
LPC
(4.18)
1
1
-
-
=
t
t
h
f
f
f
e
f
(4.19)
T
C
h
f
pc
b
PR
b
B
b
r
c
b
0
-
-
-
-
-
=
h
t
t
t
t
l
l
(4.20)
(
)
h
t
t
t
t
HPT
m
b
HPC
pc
pb
b
HPC
HPT
C
C
f
-
+
-
+
=
1
1
1
(4.21)
[
]
f
T
C
h
C
C
C
C
f
b
pb
ITB
PR
ITB
ITB
pb
pc
ITB
pb
pc
HPT
b
c
r
ITB
+
÷
÷
ø
ö
ç
ç
è
æ
-
÷
÷
ø
ö
ç
ç
è
æ
-
=
-
-
-
1
0
h
t
t
t
t
t
t
l
l
(4.22)
f
f
f
ITB
b
+
=
(4.23)
HPT
b
c
r
t
ITB
T
T
t
t
t
t
t
0
6
=
(4.24)
t
t
t
t
t
ITB
HPT
b
HPC
LPC
t
t
T
T
=
2
6
(4.25)
(
)
(
)
h
t
a
t
t
LPT
m
t
t
pc
pt
f
LPC
LPT
T
T
C
C
f
-
+
-
+
-
+
=
2
6
1
1
1
1
(4.26)
(
)
[
]
t
p
g
g
HPT
e
HPT
t
t
HPT
1
-
=
(4.27)
(
)
[
]
t
p
g
g
LPT
e
LPT
t
t
LPT
1
-
=
(4.28)
t
t
h
HPT
e
HPT
HPT
HPT
1
1
1
-
-
=
(4.29)
t
t
h
LPT
e
LPT
LPT
LPT
1
1
1
-
-
=
(4.30)
p
p
p
p
p
p
p
p
p
n
LPT
ITB
HPT
b
HPC
LPC
d
r
t
P
P
P
P
10
0
10
10
=
(4.31)
LPT
ITB
HPT
b
c
r
t
T
T
t
t
t
t
t
t
=
0
10
(4.32)
(
)
÷
ø
ö
ç
è
æ
=
=
-
P
P
T
T
T
T
T
T
T
T
t
t
t
t
t
t
10
10
1
0
10
10
10
0
10
0
10
g
g
(4.33)
(
)
ú
ú
û
ù
ê
ê
ë
é
-
÷
ø
ö
ç
è
æ
-
=
-
1
1
2
1
10
10
10
g
g
g
t
t
P
P
M
t
t
(4.34)
T
R
T
R
M
a
V
c
c
t
t
0
10
10
0
10
g
g
=
(4.35)
p
p
p
p
fn
f
d
r
t
P
P
P
P
19
0
19
19
=
(4.36)
(
)
ú
ú
û
ù
ê
ê
ë
é
-
÷
ø
ö
ç
è
æ
-
=
-
1
1
2
1
19
19
19
g
g
g
c
c
P
P
M
t
c
(4.37)
t
t
f
r
t
T
T
=
0
19
(4.38)
(
)
g
g
c
c
P
P
T
T
T
T
T
T
T
T
t
t
t
t
1
19
19
0
19
19
19
0
19
0
19
-
÷
ø
ö
ç
è
æ
=
=
(4.39)
M
T
T
a
M
a
a
V
19
0
19
0
19
19
0
19
=
=
(4.40)
(
)
(
)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
-
+
+
-
+
=
·
c
c
t
c
c
P
P
a
V
T
T
R
R
f
M
a
V
f
g
a
m
F
c
g
1
1
1
1
10
0
0
10
0
10
0
0
10
0
(4.41)
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
è
æ
-
+
-
=
·
g
c
c
f
f
P
P
a
V
T
T
M
a
V
g
a
m
F
19
0
0
19
0
19
0
0
19
0
1
(4.42)
)
1
(
a
a
+
+
=
·
·
·
f
f
c
c
m
F
m
F
m
F
(4.43)
(
)
(
)
·
+
+
=
0
1
m
F
S
f
f
ITB
b
a
(4.44)
5. User’s Manual
The excel program is written in combination between spreadsheet neuron cells, Visual Basic, and macro code. These three combinations provide user-friendly software such that compilation and preprocessing are no longer necessary. The user obtains result plots right a way just by clicking some simple buttons.
The program is mainly comprised of six sheets namely CoverPage, Instruction, Input, plot sheet, data sheet, and Otape&Test.
CoverPage sheet
The CoverPage sheet contains the authors of this program. Any questions regarding the program can be addressed to us through email or phone.
Instruction sheet
First time users are strongly recommended to read this sheet before running the program. Since there are always possibilities of getting error computations such as division by zero, square root of a negative quantity, or over floating – under floating number, the program is written such that it will not crash if those errors are encountered during the computation. It will instead tell the user where the computation encounters those errors. In this sheet, you will find details of how to run the program and how to fix a problem if something goes wrong.
This sheet also explains several assumptions made in the equations so that the users are aware of some cases in the equations that have been idealized to simplify the problems.
f
f
m
F
·
Figure 5.1 – Excel Input Sheet Screenshot
Input sheet
This sheet is where most of the inputs are specified. The program will check input value in this sheet to make sure that all of the inputs are specified. It will tell the user if there are inputs that are not specified. There are three Combo Box in the Input sheet (Combo Box is a list box that displays a list of values and lets the users select one of the values in the list) namely ITB, Units, and Choose a Plot as shown in Figure 5.1. You need first to specify the value in combo box ITB and combo box Units before moving on to combo box Choose a Plot. Combo Box Units lets you specify the input and output unit system. Currently, the program can handle only two units systems, which are English or SI. Combo Box ITB lets you turn ON or OFF the Interstage Burner (ITB) feature. This feature provides a flexibility to choose two types of engine and they are engine with ITB - ON and engine with ITB - OFF. With this feature, you will be able to see how much additional engine performance you can get with ITB-ON or with ITB-OFF. Note that the equations used for ITB-OFF are not the same as the equations from the reference book, Elements of Gas Turbine Propulsion. This is because the cycle analysis in this program is based on two shafts engine, and the one in the reference book is based on one shaft engine.
When ITB-OFF is chosen, the program will execute as if there is no ITB added to the Turbofan engine. Accordingly, the following variables will be changed internally and automatically:
1. Cpt = Cpc
2. (t = (c
3. fitb = 0
4. (itb = 1.0
5. (itb = 1.0
Once all the inputs values are specified and combo box values on both ITB and Units are specified, you can specify a value in combo box Choose a Plot. This combo box provides several option lists to choose from, and they are:
1. Specific Thrust and TSFC Vs Compressor Pressure Ratio
2. Specific Thrust Vs Flight Condition Mach Number
3. TSFC Vs Flight Condition Mach Number
4. Specific Thrust Vs Fan Pressure Ratio
5. TSFC Vs Fan Pressure Ratio
6. Specific Thrust Vs ByPass Ratio
7. TSFC Vs ByPass Ratio
Once you select one of them, you will be directed to a new sheet. Basically once the value of the combo box changed, it will execute a certain macro code. The macro code associated with this combo box will open hidden plot sheets and hidden data sheets depending on the selection you made.
The following discussion describes the instruction of how to run the program for each different option.
Specific Thrust and TSFC Vs Compressor Pressure Ratio
When this option is chosen, the program opens four sheets namely ST_VS_PIc, SF_VS_PIc, DataSTPIc, and DataSFPIc. You will be directed from Input sheet to ST_VS_PIc sheet. ST_VS_PIc plots specific thrust and fuel-air mass fraction versus compressor pressure ratio, and SF_VS_PIc plots thrust fuel specific consumption versus compressor pressure ratio.
ST_VS_PIc
You expect to see result profiles similar to Figure 7-13a1 and Figure 7-13b1. You need to complete all the inputs parameters indicated in green cells of the Input sheet and ST_VS_PIc sheet. Click the ( combo box to select number of plots for different (’s, bypass ratio. New input parameters for different bypass ratio will be created indicated in green cells. Once all the green cells are filled, you will need to specify the relation between high total pressure compressor and low total pressure compressor ratios. To do this you need to click on the Enter Relation. Once the relation between the two compressors has been established, you need to click Generate to generate a table for compressor pressure ratio. At this point you are ready to run the program and generate the plots, simply by clicking Calculate button. Data computed will be stored in the DataSTPIc sheet.
SF_VS_PIc
You expect to see results profiles similar to Figure 7-13c1. Most of the inputs are taken from the Input sheet and ST_VS_PIc sheet except for the bypass ratio. You need to specify number of bypass ratio you want to plot, then you can click Calculate to perform computation and to plot the results. Data computed will be stored in the DataSFPIc sheet.
Specific Thrust Vs Flight Condition Mach Number or TSFC Vs Flight Condition Mach Number
When you choose this option, the program closes the plot sheets and data sheets from the previous use and opens one plot sheet and one data sheet, ST_VS_Mo and DataSTMo or SF_VS_Mo and DataSFMo. You will then be directed from Input sheet to ST_VS_Mo or SF_VS_Mo. This sheet is used to plot specific thrust or thrust specific fuel consumption versus flight condition Mach number with different bypass ratio.
ST_VS_Mo or SF_VS_Mo
You expect to see result profiles similar to Figure 7-14a1 or Figure 7-14b1. You need to complete all the input parameters indicate in the green cells of this sheet. Then you can click Calculate to run the program and plot the results. Data computed associated with this sheet will be stored in the DataSTMo sheet.
Specific Thrust Vs Fan Pressure Ratio or TSFC Vs Fan Pressure Ratio
Similar to Flight Condition Mach Number, this option opens ST_VS_PIf and DataSTPIf or SF_VS_PIf and DataSTPIf. It plots specific thrust or thrust specific fuel consumption vs fan pressure ratio with different bypass ratio. The steps to run the program in this sheet are the same from option two or three of the Choose a Plot combo box.
Specific Thrust Vs ByPass Ratio or TSFC Vs ByPass Ratio
This option opens ST_VS_Alp and DataSTAlp or SF_VS_Alp and DataSFAlp. It plots specific thrust or thrust specific fuel consumption vs bypass ratio with different fan total pressure ratio, f. The steps to run the program in this sheet are the same from option two or three of the Choose a Plot combo box.
In summary you need to do the followings to run the program:
1. Go to the Input sheet
2. Specify the Unit system (SI or English)
3. Specify the ITB switch. Is the computation for engine with ITB-ON or for engine with ITB-OFF?
4. Enter all the input parameters indicated in green cells (do not modify or change the value indicated in cyan).
5. Specify Choose a Plot. You will be directed to a new sheet depending on the selection.
If you select to engine performance (Specific Thrust and TSFC) Vs Compressor Pressure Ratio, do the followings:
ST_VS_PIc:
a. Specify number of bypass ratio (new green cells will be created).
b. Click Enter Relation (Enter relation).
c. Specify all input parameters indicated in green.
d. Click Generate to generate compressor total pressure ratio.
e. Click Calculate to compute and plot the results.
f. Repeat the above steps for different input parameters. If you want to change the input parameters in the Input sheet, you need to go to the Input sheet.
SF_VS_PIc:
a. Specify number of bypass ratio (new green cells will be created).
b. Specify all input parameters indicated in green.
c. Click Calculate.
d. Repeat the above steps for different bypass ratio. If you want to change the input parameters in the Input sheet or in the ST_VS_PIc sheet, you need to go back to one of those sheets and change the input parameters.
ST_VS_Mo, SF_VS_Mo, ST_VS_PIf, and SF_VS_PIf:
a. Specify number of bypass ratio (new green cells will be created).
b. Specify all input parameters indicated in green.
c. Click Calculate.
d. Repeat the above steps for different input parameters. If you want to change the input parameters in the Input sheet, you need to go to the Input sheet.
ST_VS_Alp and SF_VS_Alp:
a. Specify number of fan total pressure ratio (new green cells will be created).
b. Specify all input parameters indicated in green.
c. Click Calculate.
d. Repeat the above steps for different input parameters. If you want to change the input parameters in the Input sheet, you need to go to the Input sheet.
Discussions
The last sheet in the program is the Otape&Test; this is where the user can check a single quantity namely Specific Thrust or Thrust Specific Fuel Consumption. Most of the input parameters are from the Input sheet; additional inputs are inputs parameters that you see in the plot sheets. By clicking the Test, you will obtain Specific Thrust and Thrust Specific Fuel Consumption stored in cyan cells.
This program has been debugged several times. Therefore whenever you encounter computation errors due to either zero division or square root of a negative quantity, you will be notified by a pop up window indicating where the computation problem is. It will indicate which equation that the computational error is encountered. If you have any comments or bug problems you encounter in the program, you can report them to us for further improvement. Detail about the contact number can be found in the CoverPage sheet of the program.
It is also predicted that the plotting macro can encounter some problems in the future if the users are not fully understand the program. Therefore I have set up a way how to fix the problem. In case the plot starts giving problem, you need to do the followings (apply to all plot sheets: ST_VS_PIc, SF_VS_PIc, ST_VS_PIf, SF_VS_PIf, ST_VS_Mo, SF_VS_Mo, ST_VS_Alp, and SF_VS_Alpp):
a. Go to the plot sheet (e. ST_VS_Mo)
b. Go to cell D5
c. Note that you will not be able to click on that cell because it is lying on the back of the plot chart. You may want to use cursor button from your keyboard.
d. In that cell it reads “SUCCESS”. Change the value to any other type of character, “FAIL” for example.
e. Clear the series in the chart plot by clicking right mouse button on the chart plot.
f. Choose Source Data
g. Select Series and remove all the series. For the case ST_VS_PIc, you need to clear the series on both charts.
Users are not expected to understand macro code in the program. However the part that takes care the computation can be found in Module1 Sub Itb( ). In order to open it, you need to open Excel Visual Basic by pressing Alt + F11. Any modification can be made in subroutine Itb ( ). Do not add new variables into the program!
6. Data Validation
Since there is no experimental data for this application, a Fortran 77 code is written for the comparison between results computed by Excel and Fortran 77. All the equations used are the same in both codes and can be found in section 4 of ‘Summary of equations’.
Under the same turbofan engine operating conditions with ITB-ON, specific thrust (i.e.
·
m
F
, SI unit:
s
kg
N
/
) is computed using both codes and plotted versus various design parameters, namely flight Mach number (M0), compressor pressure ratio ((c), fan pressure ratio ((f), and bypass ratio ((). All the plots are in Tecplot 9.0 format as shown below.
c
c
m
F
·
Figure 6.1 – Turbofan engine with ITB: Specific thrust vs flight Mach number
Figure 6.2 – Turbofan engine with ITB: Specific thrust vs compressor pressure ratio
Figure 6.3 – Turbofan engine with ITB: Specific thrust vs fan pressure ratio
Figure 6.4 – Turbofan engine with ITB: Specific thrust vs bypass ratio
7. RESULTS AND DISCUSSIONS
Using data provided in eng_itb.dat (a sample listing data file for a two-spool high bypass turbofan provided by Scott Jones, currently at Propulsion Systems Analysis Office, NASA Glenn Research Center), results at different operating conditions are plotted using Tecplot 9.0 and shown as below.
Table 7.1 –Input data for Engine parameters (SI unit)
Pressure drop ratio
engine-core
P0/P10 =
0.9
T0 =
216.6667
K
engine - fan
P0/P19 =
0.9
Total pressure ratio
Maximum diffuser total pressure ratio
d max =
0.99
Properties of air and fuel
main burner total pressure ratio
b =
0.96
Specific heat at constant pressure
ITB - burner total pressure ratio
ITB =
0.96
region 0 --> 3
Cpc =
1.000645
KJ/Kg-K
engine nozzle total pressure ratio
n =
1.333
region 4 --> 5
Cpb =
1.159743
KJ/Kg-K
fan nozzle total pressure ratio
fn =
1.274
region 6 -->10
Cpt =
1.147183
KJ/Kg-K
Polytropic efficiency
Ratio of specific heat
eHPC=
0.9066
region 0 --> 3
c =
1.39999
eLPC=
0.9036
region 4 --> 5
b =
1.27284
eHPT=
0.9029
region 6 -->10
t =
1.27882
eLPT=
0.9174
Low heating value of fuel
ef=
0.8961
Burner
hPR-b =
7917.12
KJ/Kg
Burner efficiency
ITB
hPR-ITB =
7917.12
KJ/Kg
b =
0.999
ITB =
0.999
Turbine mechanical efficiency
m-HPT =
0.92
m-LPT =
0.93
Burner exit temperature
main-burner
Tt4 =
1785.594
K
ITB - burner
Tt6 =
1563.844
K
Table 7.2 - The input data for Option 1:
Figure 7.1 - Specific thrust vs flight M0
Figure 7.2 - TSFC vs flight M0
Mo =
0
To
3
1 =
0.5
Mo =
0.1
2 =
2
HPC =
25
LPC =
1.3
f =
1.3
Figure 7.1 – Turbofan engine: Specific thrust vs flight Mach number
Figure 7.2 – Turbofan engine: thrust specific fuel consumption vs flight Mach
number
Table 7.3 - The input data for Option 2:
Figure 7.3 - Specific thrust vs compressor pressure ratio
Figure 7.4 - TSFC vs compressor pressure ratio
Figure 7.5 - Fuel/air ratio vs compressor pressure ratio
Scalec =
19.23
LPC =
1.01
c =
19.62
to
50
HPC =
19.42
c =
1
f =
1.3
1 =
0.5
0.9
2 =
2
Figure 7.3 – Turbofan engine: Specific thrust vs compressor pressure ratio
Figure 7.4 – Turbofan engine: thrust specific fuel consumption
vs compressor pressure ratio
Figure 7.5 – Turbofan engine: fuel/air ratio vs compressor pressure ratio:
Table 7.4 - The input data for Option 3:
Figure 7.6 - Specific thrust vs fan pressure ratio
Figure 7.7 - TSFC vs fan pressure ratio
f =
1.01
To
10
f =
0.25
1 =
3
HPC =
25
2 =
5
LPC =
1.3
3 =
10
0.9
Figure 7.6 – Turbofan engine: Specific thrust vs fan pressure ratio
Figure 7.7 – Turbofan engine: thrust specific fuel consumption
vs fan pressure ratio
Table 7.5 - The input data for Option 4:
Figure 7.8 - Specific thrust vs bypass ratio
Figure 7.9 - TSFC vs bypass ratio
=
0
To
10
=
0.1
HPC =
25
f1 =
1.5
LPC =
1.3
f2 =
3.5
0.9
Figure 7.8 – Turbofan engine: Specific thrust vs bypass ratio
Figure 7.9 – Turbofan engine: thrust specific fuel consumption
vs bypass ratio
Discussions
The computations are successful in providing an estimation of how the turbofan performance is improved by adding an Inter-Stage Turbine Burner. As expected, the specific thrust is increased at four different engine design parameters, namely Mach number, compressor pressure ratio, fan pressure ratio and bypass ratio with the addition of ITB. Nevertheless, the specific fuel consumption is relatively increased.
Typical ranges for each engine design parameter is based on Chapter 7 of ‘Elements of Gas Turbine Propulsion’ by Dr. Jack Mattingly1.
In addition, more plots will be added in the future if necessary. For convenience purpose, a Visual-Basic code may be developed to act as an interface between fortran77 data file and MS Excel plot function. Then, user can plot the result data computed using Fortran77 on MS Excel without knowing how to use Tecplot 9.0 or any other equivalent post-processing software.
Appendix
A - MACRO VISUAL BASIC CODE in MS EXCEL
VB code
variable
VB code
variable
VB code
variable
a0
a0
gamab
b
T10_T0
T10/T0
alpha
gamac
c
T19_T0
T19/T0
Cpb
Cpb
gamat
t
tau_alp_itb
-b
Cpc
Cpc
hPRb
hPR-b
tau_alp_itb
-ITB
Cpt
Cpt
hPRitb
hPR-ITB
tau_b
b
ef
ef
M10
M10
tau_c
c
eff_hpc
HPC
M19
M19
tau_f
f
eff_hpt
HPT
pi_d
d
tau_hpc
HPC
eff_lpc
LPC
pi_f
f
tau_hpt
HPT
eff_lpt
LPT
pi_hpc
HPC
tau_itb
ITB
eff_r
r
pi_hpt
HPT
tau_lpc
LPC
effm_hpt
m-HPT
pi_lpc
LPC
tau_lpt
LPT
effm_lpt
m-LPT
pi_lpt
LPT
tau_r
r
ehpc
eHPC
pi_r
r
Tt10_T0
Tt10/T0
elpc
eLPC
Pt10_P10
Pt10/P10
Tt19_T0
Tt19/T0
fb
fb
Pt19_P19
Pt19/P19
Tt6_Tt2
Tt6/Tt2
fc_mc
rb
Rb
v0
v0
ff_mf
rc
Rc
v10_a0
v10/a0
fitb
fITB
rt
Rt
v19_a0
v19/a0
f_mo
s
S
Table A.1 – Description of variables in MS Excel Macro Visual Basic code
Sub itb()
Dim rc, rb, rt, a0, v0, tau_r, pi_r, eff_r, pi_d, tau_alp_b, tau_alp_itb, tau_hpc, tau_lpc, tau_f
Dim tau_c, tau_b, eff_hpc, eff_lpc, eff_f, fb, tau_hpt, fitb, tau_itb, Tt6_Tt2, tau_lpt
Dim pi_hpt, pi_lpt, eff_hpt, eff_lpt, Pt10_P10, Tt10_T0, M10, v10_a0, Pt19_P19, M19, Tt19_T0
Dim T19_T0, v19_a0, fc_mc, ff_mf, T10_T0
'zero-ing internal variables
rc = 0#: rb = 0#: rt = 0#: a0 = 0#: v0 = 0#: tau_r = 0#: pi_r = 0#: eff_r = 0#: pi_d = 0#
tau_alp_b = 0#: tau_alp_itb = 0#: tau_hpc = 0#: tau_lpc = 0#: tau_f = 0#
tau_c = 0#: tau_b = 0#: eff_hpc = 0#: eff_lpc = 0#: eff_f = 0#: fb = 0#
tau_hpt = 0#: fitb = 0#: tau_itb = 0#: Tt6_Tt2 = 0#: tau_lpt = 0#
pi_hpt = 0#: pi_lpt = 0#: eff_hpt = 0#: eff_lpt = 0#: Pt10_P10 = 0#: Tt10_T0 = 0#
M10 = 0#: v10_a0 = 0#: Pt19_P19 = 0#: M19 = 0#: Tt19_T0 = 0#
T19_T0 = 0#: v19_a0 = 0#: fc_mc = 0#: ff_mf = 0#
'empty-ing output variables
F_mo = Empty: s = Empty: f = Empty
Style = vbYes
Title = "ITB Cycle Analysis"
Style1 = vbYesNo
Msg = "Do you want to jump to the next alpha?"
Msg1 = "Division by zero ==> Refer to equations 1, 2, 3"
Msg2 = "Square root of a negative quantity ==> Refer to equation 4"
Msg3 = "Division by zero ==> Refer to equation 7"
Msg4 = "Division by zero ==> Refer to equations 10, 11, 12, 13, 14"
Msg5 = "Division by zero ==> Refer to equation 16"
Msg6 = "Division by zero ==> Refer to equations 17, 18, 19"
Msg7 = "Division by zero ==> Refer to equation 20"
Msg8 = "Square root of a negative quantity ==> Refer to equation 34"
Msg9 = "Square root of a negative quantity ==> Refer to equation 37"
If gamac = 0# Or gamab = 0# Or gamat = 0# Then
response = MsgBox(Msg1, Sytle, Title)
Msg1 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg1, Style, Title)
jump = MsgBox(Msg1, Style1, Title)
GoTo 10
End If
rc = (gamac - 1#) * Cpc / gamac '(1)
rb = (gamab - 1#) * Cpb / gamab '(2)
rt = (gamat - 1#) * Cpt / gamat '(3)
If gamac < 0# Or rc < 0# Or T_o < 0# Or gc < 0# Then
response = MsgBox(Msg2, Style, Title)
Msg2 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg2, Sytle, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
a0 = (gamac * rc * T_o * gc) ^ 0.5 '(4)
v0 = a0 * M0 '(5)
If gamac = 1# Then
response = MsgBox(Msg3, Sytle, Title)
Msg3 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg3, Style, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
tau_r = (gamac - 1#) * (0.5 * M0 ^ 2) + 1# '(6)
pi_r = tau_r ^ (gamac / (gamac - 1)) '(7)
If M0 <= 1# Then '(8)
eff_r = 1#
Else
eff_r = 1# - 0.075 * (M0 - 1#) ^ 1.35
End If
pi_d = pi_dmax * eff_r '(9)
If Cpc = 0# Or T_o = 0# Or gamac = 0# Or ehpc = 0# Or elpc = 0# Or ef = 0# Then
response = MsgBox(Msg4, Sytle, Title)
Msg4 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg4, Style, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
tau_alp_b = Cpb * Tt4 / (Cpc * T_o) '(10)
tau_alp_itb = Cpt * Tt6 / (Cpc * T_o) '(11)
tau_hpc = pi_hpc ^ ((gamac - 1#) / (gamac * ehpc)) '(12)
tau_lpc = pi_lpc ^ ((gamac - 1#) / (gamac * elpc)) '(13)
tau_f = pi_f ^ ((gamac - 1#) / (gamac * ef)) '(14)
tau_c = tau_hpc * tau_lpc '(15)
If T_o = 0# Or tau_r = 0# Or tau_c = 0# Then
response = MsgBox(Msg5, Style, Title)
Msg5 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg5, Style, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
tau_b = Tt4 / (T_o * tau_r * tau_c) '(16)
If tau_hpc = 1# Or tau_lpc = 1# Or tau_f = 1# Then
response = MsgBox(Msg6, Style, Title)
Msg6 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg6, Style, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
eff_hpc = ((tau_hpc ^ ehpc) - 1#) / (tau_hpc - 1#) '(17)
eff_lpc = ((tau_lpc ^ elpc) - 1#) / (tau_lpc - 1#) '(18)
eff_f = ((tau_f ^ ef) - 1#) / (tau_f - 1#) '(19)
If tau_alp_b = eff_b * hPRb / (Cpc * T_o) Then
response = MsgBox(Msg7, Style, Title)
Msg7 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg7, Style, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
fb = (tau_c * tau_r - tau_alp_b) / (tau_alp_b - (eff_b * hPRb / (Cpc * T_o))) '(20)
tau_hpt = 1# + (1# - tau_hpc) / ((1# + fb) * (Cpb / Cpc) * tau_hpc * tau_b * effm_hpt) '(21)
If Sheets("Input").ComboBox2.Value = "ON" Then '(22)
fitb = (1# + fb) * (tau_r * tau_c * tau_b * tau_hpt - (Cpc / Cpb) * tau_alp_itb) / ((Cpc / Cpb) * tau_alp_itb - eff_itb * hPRitb / (Cpb * T_o))
Else
fitb = 0#
End If
f = fb + fitb '(23)
If Sheets("Input").ComboBox2.Value = "ON" Then '(24)
tau_itb = Tt6 / (T_o * tau_r * tau_c * tau_b * tau_hpt)
Else
tau_itb = 1#
End If
Tt6_Tt2 = tau_c * tau_b * tau_hpt * tau_itb '(25)
tau_lpt = 1# + (1# - tau_lpc + alpha * (1# - tau_f)) / ((1# + f) * (Cpt / Cpc) * Tt6_Tt2 * effm_lpt) '(26)
pi_hpt = tau_hpt ^ (gamat / ((gamat - 1#) * ehpt)) '(27)
pi_lpt = tau_lpt ^ (gamat / ((gamat - 1#) * elpt)) '(28)
eff_hpt = (1# - tau_hpt) / (1# - tau_hpt ^ (1# / ehpt)) '(29)
eff_lpt = (1# - tau_lpt) / (1# - tau_lpt ^ (1# / elpt)) '(30)
Pt10_P10 = P0_P10 * pi_r * pi_d * pi_lpc * pi_hpc * pi_b * pi_hpt * pi_itb * pi_lpt * pi_n '(31)
Tt10_T0 = tau_r * tau_c * tau_b * tau_hpt * tau_itb * tau_lpt '(32)
T10_T0 = Tt10_T0 / Pt10_P10 ^ ((gamat - 1#) / gamat) '(33)
If gamat < 1# Or Pt10_P10 ^ ((gamat - 1#) / gamat) < 1# Then
response = MsgBox(Msg8, Style, Title)
Msg8 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg8, Style, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
M10 = (((Pt10_P10 ^ ((gamat - 1#) / gamat)) - 1#) * 2# / (gamat - 1#)) ^ 0.5 '(34)
v10_a0 = M10 * (gamat * rt * T10_T0 / (gamac * rc)) ^ 0.5 '(35)
Pt19_P19 = P0_P19 * pi_r * pi_d * pi_f * pi_fn '(36)
If gamac < 1# Or Pt19_P19 ^ ((gamac - 1#) / gamac) < 1# Then
response = MsgBox(Msg9, Style, Title)
Msg9 = "Occurs at " & "PIc,PIf,Mo,ByPass = " & pi_hpc * pi_lpc & "," & pi_f & "," & M0 & "," & alpha
response = MsgBox(Msg9, Style, Title)
jump = MsgBox(Msg, Style1, Title)
GoTo 10
End If
M19 = (((Pt19_P19 ^ ((gamac - 1#) / gamac)) - 1#) * 2# / (gamac - 1#)) ^ 0.5 '(37)
Tt19_T0 = tau_r * tau_f '(38)
T19_T0 = Tt19_T0 / Pt19_P19 ^ ((gamac - 1#) / gamac) '(39)
v19_a0 = M19 * T19_T0 ^ 0.5 '(40)
fc_mc = (a0 / gc) * ((1# + f) * v10_a0 - M0 + (1# + f) * (rt / rc) * (T10_T0 / v10_a0) * (1# - P0_P10) * (1# / gamac)) '(41)
ff_mf = (a0 / gc) * (v19_a0 - M0 + (T19_T0 / v19_a0) * (1# - P0_P19) / gamac) '(42)
F_mo = (fc_mc + alpha * ff_mf) / (1# + alpha) '(43)
s = f / ((1# + alpha) * F_mo) '(44)
10
End Sub
B - FORTRAN 77 code
Three separate files are needed to execute this code, namely itb.f, comitb.i, and input. All the equations used in this code are summarized in the section 4 - ‘Summary of equations’.
(A) Main program: itb.f
itb.f is the main program for this application. It comprises of a main driver and nine subroutines. Each subroutine has its own specific function and is described in detail below.
1) Subroutine START
START is the first subroutine called by main driver itb. It ensures that the ‘input’ file exists, and then the program will carry on. Or else, the program will terminate.
2) Subroutine RINPUT
Once the file input is provided at the same directory as itb.f is located, RINPUT will then be called and read in the data file.
3) Subroutine CONSTANT
All the constants will be initialized in this subroutine. This subroutine is called only once before performing necessary computation.
4) Subroutine SELECT
SELECT allows user to specify which plot they desire to get. Up to this point, there are only four options to choose, which include:
Option 1:
(a). Specific Thrust vs Mach Number (ST_M0.dat)
(b). TSFC vs Mach Number (TSFC_M0.dat)
Option 2:
(a). Specific Thrust vs Compressor Pressure Ratio (ST_PIc.dat)
(b). TSFC vs Compressor Pressure Ratio (TSFC_PIc.dat)
(c). Fuel-air Ratio vs Compressor Pressure Ratio (f_PIc.dat)
Option 3:
(a). Specific Thrust vs Fan Pressure Ratio (ST_PIf.dat)
(b). TSFC vs Fan Pressure Ratio (TSFC_PIf.dat)
Option 4:
(a). Specific Thrust vs Bypass Ratio (ST_Alpha.dat)
(b). TSFC vs Bypass Ratio (TSFC_Alpha.dat)
The selected plots will be plotted and stored in the same directory where the program is executed.
5) Subroutine MACH
MACH is called to plot engine performance parameters (for instance, specific thrust or thrust specific fuel consumption) vs flight mach number.
6) Subroutine COMPR
COMPR is called to plot engine performance parameters vs compressor pressure ratio.
7) Subroutine FANPR
FANPR is called to plot engine performance parameters vs fan pressure ratio.
8) Subroutine BYPASS
BYPASS is called to plot engine performance parameters vs bypass ratio.
9) Subroutine EQUATION
EQUATION is called to perform all necessary computations. It includes all equations here.
(B) Variable initialization: comitb.i
All variables will be declared and initialized before running main program and all subroutines. Variables are grouped into five common blocks, i.e. inputs, outputs, equations, constants, and chardat.
(C) Data file: input
Since two types of unit system will be used for computation, namely SI and English unit, user needs to specify the input parameter ‘unitsys’ to be either ‘0.0’ (English unit) or ‘1.0’ (SI unit).
Another important input parameter is ‘itbsw’. When ‘itbsw’ is set to be ‘1.0’, ITB will be turned on and vice versa (i.e. ‘itbsw’ is set to be ‘0.0’).
Table shown below is the brief descriptions of input variables in ‘input’ file:
Input Variables
Unit
program
literature
English
SI
unitsys
-
-
-
itbsw
-
-
-
T0
T0
(R
K
gamac
γc
-
-
cpc
cpc
R
lbm
Btu
×
K
kg
kJ
×
gamab
γb
-
-
cpb
cpb
R
lbm
Btu
×
K
kg
kJ
×
gamat
γt
-
-
cpt
cpt
R
lbm
Btu
×
K
kg
kJ
×
hprb
hPR-b
lbm
Btu
kg
kJ
hpritb
hPR-itb
lbm
Btu
kg
kJ
pidmax
πd max
-
-
pib
πb
-
-
piitb
πitb
-
-
pin
πn
-
-
pifn
πfn
-
-
ehpc
eHPC
-
-
elpc
eLPC
-
-
ef
ef
-
-
ehpt
eHPT
-
-
elpt
eLPT
-
-
effb
ηb
-
-
effitb
ηitb
-
effmhpt
ηhpt
-
effmlpt
ηlpt
-
p0op10
10
0
P
P
-
p0op19
19
0
P
P
-
Tt4
Tt4
(R
K
Tt6
Tt6
(R
K
Table B.1 – Description of input variables in ‘input’ file in Fortran77 code (both SI
and English units)
Main program: itb.f
*deck itb
program itb
c
c +++
c
c===========================================================
c
c ITB is the main driver.
c
c===========================================================
c
include 'comitb.i'
c
c//////////////////////////////\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
c
c +++
call start
c +++
call rinput
c +++
call constant
c
call select
c +++
if (plot.eq.1) call mach
c
if (plot.eq.2) call compr
c
if (plot.eq.3) call fanpr
c
if (plot.eq.4) call bypass
c +++
end
c
*deck bypass
subroutine bypass
c
c===========================================================
c
c BYPASS determines the range for bypass ratio for
c plotting.
c
c BYPASS is called by: itb
c
c===========================================================
c
include 'comitb.i'
c
c//////////////////////////////\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
c
write(3,*) 'Subroutine BYPASS'
c
open(unit=18,file='ST_Alpha.dat')
open(unit=19,file='TSFC_Alpha.dat')
c
WRITE(18,*) 'Title="','Specific Thrust VS Alpha"'
if (unitsys.eq.0.0) then
WRITE(18,*) 'Variables = "`a","Sp. thrust [lbf/(lbm/sec)]"'
else
WRITE(18,*) 'Variables = "`a","Specific thrust [N/(kg/s)]"'
endif
c
WRITE(19,*) 'Title="','TSFC VS Alpha"'
if (unitsys.eq.0.0) then
WRITE(19,*) 'Variables = "`a","TSFC [(lbm/hr)/lbf]"'
else
WRITE(19,*) 'Variables = "`a","TSFC [mg/(N-sec)]"'
endif
c
if (npif.gt.0) then
do 90 i=1,npif
pif=pi_f(i)
write(*,*) 'pif(',i,')=',pif
c
if (itbsw.eq.0.) then
write(18,314)'Zone T="`p_f = ',pif,'(no itb)"'
write(19,314)'Zone T="`p_f = ',pif,'(no itb)"'
else
write(18,314)'Zone T="`p_f = ',pif,'(itb)"'
write(19,314)'Zone T="`p_f = ',pif,'(itb)"'
endif
c
do 100 alpha = ialp,falp,dalp
c
alpp1 = alpha+1.0
ralpp1 = 1.0/alpp1
c
call equation
if (error.eq.1.0) then
write(*,*) 'error'
go to 90
endif
write(18,313) alpha,st
write(19,313) alpha,tsfc
100 continue
90 continue
endif
c
close(18)
close(19)
return
313 format(f10.5,2x,f13.8)
314 format(a15,f5.2,a9)
end
c
*deck compr
subroutine compr
c
c===========================================================
c
c COMPR determines the range for compressor pressure
c ratio for plotting.
c
c COMPR is called by: itb
c
c===========================================================
c
include 'comitb.i'
c
c//////////////////////////////\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
c
write(3,*) 'Subroutine COMPR'
c
open(unit=15,file='ST_PIc.dat')
open(unit=16,file='TSFC_PIc.dat')
open(unit=17,file='f_PIc.dat')
c
WRITE(15,*) 'Title="','Specific Thrust VS Pi Comp"'
if (unitsys.eq.0.0) then
WRITE(15,*)'Variables ="`p_c","Sp.thrust [lbf/(lbm/sec)]"'
else
WRITE(15,*)'Variables ="`p_c","Sp.thrust [N/(kg/s)]"'
endif
c
WRITE(16,*) 'Title="','TSFC VS Pi Comp"'
if (unitsys.eq.0.0) then
WRITE(16,*)'Variables = "`p_c","TSFC [(lbm/hr)/lbf]"'
else
WRITE(16,*)'Variables = "`p_c","TSFC [mg/(N-sec)]"'
endif
c
WRITE(17,*) 'Title="','f VS Pi Comp"'
WRITE(17,*) 'Variables = "`p_c","f"'
c
ipic = scalec*pilpc*pilpc
k=1
do 60 pic = ipic,fpic,dpic
term = pic/scalec
tpilpc(k) = sqrt(term)
tpihpc(k) = pic/tpilpc(k)
k=k+1
60 continue
write(3,*) pilpc,pihpc,pic
write(3,*) ipic,fpic,dpic
c
if (nalpha.gt.0) then
do 70 i=1,nalpha
alpha=alp(i)
write(*,*) 'alpha(',i,')=',alpha
if (itbsw.eq.0.) then
write(15,12)'Zone T="`a =',alpha,'(no itb)"'
write(16,12)'Zone T="`a =',alpha,'(no itb)"'
write(17,12)'Zone T="`a =',alpha,'(no itb)"'
else
write(15,12)'Zone T="`a =',alpha,'(itb)"'
write(16,12)'Zone T="`a =',alpha,'(itb)"'
write(17,12)'Zone T="`a =',alpha,'(itb)"'
endif
c
alpp1 = alpha+1.0
ralpp1 = 1.0/alpp1
c write(3,*) 'alpp1=',alpp1
c write(3,*) 'ralpp1=',ralpp1
c
j=1
write(3,*) pilpc,pihpc,pic
write(3,*) ipic,fpic,dpic
do 80 pic = ipic,fpic,dpic
pilpc = tpilpc(j)
pihpc = tpihpc(j)
write(3,*) pilpc,pihpc,pic
c
call equation
if (error.eq.1.0) then
write(*,*) 'error'
go to 70
endif
write(15,11) pic,st
write(16,11) pic,tsfc
write(17,11) pic,f
c count=count+1
j=j+1
80 continue
70 continue
endif
c
close (15)
close (16)
close (17)
return
11 format(f10.5,2x,f13.9)
12 format(a13,f5.2,a9)
end
c
*deck constant
subroutine constant
c
c===========================================================
c
c EQUATION performs all necessary computations.
c
c EQUATION is called by: mach,compr,fanpr,bypass
c
c===========================================================
c
include 'comitb.i'
c
c//////////////////////////////\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
c
write(3,*)
write(3,*) '*******Check constant begins!!******'
c
gc=32.174
if (unitsys.eq.1.0) gc = 1.0
c
ecvrn = 778.14
if (unitsys.eq.1.0) ecvrn = 1000.0
c
rgc = 1.0/gc
write(3,*) 'gc=',gc
write(3,*) 'rgc=',rgc
c
rgamac = 1.0/gamac
rgamat = 1.0/gamat
rgamab = 1.0/gamab
write(3,*) 'rgamac=',rgamac
write(3,*) 'rgamat=',rgamat
write(3,*) 'rgamab=',rgamab
c
gcm1ogc = (gamac-1.0)*rgamac
gtm1ogt = (gamat-1.0)*rgamat
gbm1ogb = (gamab-1.0)*rgamab
write(3,*) 'gcm1ogc=',gcm1ogc
write(3,*) 'gtm1ogt=',gtm1ogt
write(3,*) 'gbm1ogb=',gbm1ogb
c
rgcm1ogc = 1.0/gcm1ogc
rgtm1ogt = 1.0/gtm1ogt
rgtm1 = 1.0/(gamat-1.0)
rgcm1 = 1.0/(gamac-1.0)
write(3,*) 'rgcm1ogc=',rgcm1ogc
write(3,*) 'rgtm1ogt=',rgtm1ogt
write(3,*) 'rgcm1=',rgcm1
write(3,*) 'rgtm1=',rgtm1
c
Tt4oT0 = Tt4/T0
Tt6oT0 = Tt6/T0
write(3,*) 'Tt4oT0=',Tt4oT0
write(3,*) 'Tt6oT0=',Tt6oT0
c
cpbocpc = cpb/cpc
cptocpc = cpt/cpc
cpcocpb = 1.0/cpbocpc
write(3,*) 'cpbocpc=',cpbocpc
write(3,*) 'cptocpc=',cptocpc
write(3,*) 'cpcocpb=',cpcocpb
c
rehpc = 1.0/ehpc
relpc = 1.0/elpc
write(3,*) 'rehpc=',rehpc
write(3,*) 'relpc=',relpc
c
rehpt = 1.0/ehpt
relpt = 1.0/elpt
write(3,*) 'rehpt=',rehpt
write(3,*) 'relpt=',relpt
c
taud = 1.0
c
write(3,*) '******constant check ENDs!!******'
write(3,*)
c
return
end
c
*deck equation
subroutine equation
c
c===========================================================
c
c EQUATION performs all necessary computations.
c
c EQUATION is called by: itb
c
c===========================================================
c
include 'comitb.i'
INTEGER eqn
c
c//////////////////////////////\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
c
error=0.
eqn = 0
write(3,*) '-----------------------------'
write(3,*) 'Subroutine EQUATION'
write(3,*) '-----------------------------'
c
c --- equation 1,2,3