Simulation of a Turbojet Engine on a Standard Electronic Analogue Computer for the Purpose of Control Study
Lilv
hool
SIMULATION OP A TURBOJET ENGINE ON A
STANDARD ELECTRONIC ANALOGUE COMPUTER
FOR THE PURPOSE OP CONTROL STUDY
by
Lt. C. H. Rockcastle, USN
Lt. A. J. Yates, USN
May 1, 1956
ilUH
SIMULATION OP A TURBOJET ENGINE ON A
STANDARD ELECTRONIC ANALOGUE COMPUTER
FOR THE PURPOSE OP CONTROL STUDY
SUMMARY
The simulation of a turbojet engine on a standard
electronic analogue computer is presented.
The non-linear
equation for the static and transient behavior of the engine
was developed and solved on the computer without reducing
the equation to a linear form.
A typical control was sim-
ulated and used to operate the engine under transient and
steady state conditions.
The results indicate that a reasonably accurate
simulation of an actual engine on a standard computer without special equipment is possible.
Control studies may be
made using a variety of engine parameters which are contin-
uously generated during transient as well as equilibrium
operation.
The control studied indicates that the control
coefficients should be made functions of engine speed to
obtain maximum acceleration.
It is also shown that fuel
flow can be positively limited to maintain a given maximum
turbine inlet temperature during acceleration in order to
obtain maximum acceleration.
-1-
s^( r^
.
INTRODUCTION
In a turbojet engine the speed responds compara-
tively slowly to changes in fuel flow.
A control charged
with the responsibility of accelerating the engine must
change the fuel flow at a rate that will bring the engine
to the new speed as quickly as possible, but not so fast
that excessive turbine temperatures are encountered.
Thus
to design an optimum control the designer must know the
response rate of the engine variables that he is controlling.
This requires a knowledge of the dynamic behavior
of the engine.
One approach to the problem has been to predict the
engine dynamics from steady state performance charts of the
engine components using the hypothesis that the thermody-
namics and flow processes are quasi-static; that is, they
act like a continuous series of equilibrium states (Ref. l)
Then further, the non-linear differential equation for
engine acceleration was linearized around a steady state
operating point.
At points far from the steady state condi-
tions the linear equation may result in significant error,
hence, in order not to exceed engine limitations, the control
must program fuel on the conservative side.
This procedure
results in an acceleration less than the optimum possible.
The study contained herein consists of an analogue
computer solution of the non-linear equation for acceleration.
The computer solution furnishes continuous informa-
tion on the response of all engine variables.
A control can
then sense the change during acceleration of various engine
parameters and generate a fuel flow which will maintain the
maximum allowable turbine inlet temperature throughout the
acceleration.
This procedure should provide the maximum
acceleration.
Engine design calculations are shown in Appendix B.
Equilibrium running calculations are shown in Appendix C.
This work was performed at the University of Michigan by Lt. Charles H. Rockcastle, USN and Lt. Andrew J.
Yates, USN under the supervision and guidance of Dr. James
E. Broadwell.
ANALYSIS AND PROCEDURE
Analysis
A typical configuration of a direct coupled turboJet engine is shown in Fig. 1.
When turbine torque equals compressor torque a turbojet engine will run at some equilibrium speed.
A change
in fuel flow causes a change in turbine inlet temperature
which results in a change in turbine torque.
The differ-
ence between turbine torque and the torque absorbed by the
compressor then accelerates or decelerates the engine according to the following equation:
4
Q
»
I°c
(1)
Equation (l) may be written:
pt
p
w T2
w T2
c
=
1
f
(2)
dt
where
pc
and
are turbine and compr
w T2
w T2
parameters as developed in Appendix C and N/y^^I Is the
speed parameter. 2
The use of these parameters keeps the
analysis quite general and permits altitude corrections to
the computer circuit by simply changing the effective
moment of inertia.
If engine speed and fuel flow are used as indepen-
dent variables then compressor power and turbine power can
be computed as will be shown In detail In the next section.
Hence, equation (2) may be solved on an analogue computer
as is indicated schematically on following page:
1
2
All symbols are defined in Appendix A.
Henceforth such parameters as these will be refer-
red to as simply "turbine power", etc.
found In Appendix A.
Definitions will be
FU£L FiOW
eo/^oi.
»-*
pfVIDEK
a//^JuJ"£
lNTE$*A TOH
y
/v//^
A standard electronic analogue computer, Reeves
Computer C-101 Mod 5, was used for the engine simulation.
No special equipment was necessary.
Engine Simulation on the Computer
As previously described, the basic equation used
for the operation of the engine is:
_
Pc
w TV
«
_
r
I
d(N/7%)
(N//T2)
w T2
d^
_
(3)
In the simulation of the components of this equa-
tion the two independent variables, fuel flow and engine
speed, were used for the following reasons:
(1)
fuel flow,
as it represents a quantity supplied by the control circuit
and, therefore, isolates the simulated engine from the con-
trol,
(2)
engine speed, as it is almost universally used as
a thrust control parameter and may be easily and accurately
measured in an actual engine.
In this study fuel flow was
supplied by the control circuit (to be described later) and
.
engine speed was determined by the engine circuit and fed
back as an input
For the engine simulation, circuits were set up to
simulate the compressor, the burner, and the turbine (In-
The operation of these components
cluding exhaust nozzle).
then determined the inputs to the basic engine equation
(Equation 3) which was solved for the engine running speed.
Thus, an actual engine was simulated and the same engine
parameters were made continuously available in the computer
solution that would be available in an actual engine.
Compressor Simulation .
For derivation of an equa-
tion to be used for the generation of compressor pressure
ratio, Fig. 3 was plotted from Fig. 2.
From this plot the
equation for compressor pressure ratio (for design turbine
inlet temperature) was approximated by:
(W
design
=
-
01419
(N/^J - 0.65
(4)
An empirical relationship was derived for compressor pressure ratio for an arbitrary value of turbine inlet temperature:
P /P
3
2 = (P3/P2) design +l T */*2 ' *'
(- 01 °35
N/VTi
-
l.*5)
This value was then used in the circuits for generation of
turbine Inlet temperature and compressor power.
The equation for compressor power (given in exact
form in equation (C7))was plotted versus compressor pressure
ratio in Fig. 5*
„ f
=
This was approximated by:
-0152 + .0305
(P /P
3
2)
(6)
(5)
Burner Simulation
Per burner simulation it was
.
necessary to convert fuel flow to turbine inlet temperature.
The equation used for this purpose was:
k^(w f A 2 Vri)
T 4 /T 2
=
(P /P 2 )7f
3
(7)
H
7p 2
(w a yT 2
)
At this point it was decided to include burner efficiency
with fuel flow for further calculations.
Prom Pig. 2, a
cross plot of air flow versus engine speed for different
turbine inlet temperatures (Pig. 4) was made.
The approxi-
mate equation below was found:
wa
T2
_ . .710+ .002825 (n/Vt^ - 310)
(8)
The term including the compressor pressure ratio was
approximated as follows:
(P /P )~^~
2
3
-
1.442 4 .11 (P /P 2
3
3.6)
-
An approximate value of k was assumed to be 80,000.
(9)
Thus,
if the input of fuel flow from the control circuit is multi-
plied by k, divided by airflow, and added to the function of
the compressor pressure ratio, the resulting value is the
turbine inlet temperature.
This output was then divided by
the design value of the turbine inlet temperature in order
to obtain a value in percent which was used in all subsequent
circuits.
Turbine Simulation
.
An expression for turbine power
was known as a function of P5/P4 and Tu/F
2
.
A graphical de-
termination was made of turbine power as a function of com-
8
pressor pressure ratio for T^/T 2 of 100$ (Pig. 5).
However,
in determining an approximate relationship to be used in the
computer circuit in the area of interest where the turbine
nozzles are choked, the turbine power was found to be a function of turbine inlet temperature only.
This relationship
was given by:
=
w-f^
.1372 (T 4 /T 2 )
(10)
% design
It will be noted that several complicated functions
have been approximated by straight line equations in order
to simplify their use on the computer.
If greater accuracy
is desired, function generators may be used for these expres-
sions.
Observation of this circuit in operation, however,
indicated sufficient accuracy for this study.
This completed the determination of equations for
the components of the basic engine equation.
The difference
between turbine and compressor powers was divided by engine
speed.
The resultant quantity, when divided by the moment
of inertia of the engine, was equal to the derivitive of the
engine speed which was integrated to determine engine speed.
The speed was then fed back to the circuit components where
needed.
Suitable scale factors were Introduced where neces-
sary throughout the circuit.
Pig. 6 shows the schematic dia-
gram of the computer circuit.
It will be noted that within this circuit are ex-
plicit outputs of compressor pressure ratio, turbine and
compressor powers, turbine inlet temperature, and air flow.
All may be plotted, observed while the engine Is running,
or made available for control parameters if desired.
Description of an Illustrative Control
For an illustrative control it was decided to use
the difference between the operating engine speed and the
desired or set engine speed, as determined by the throttle
position, as the controlling parameter.
The control de-
livers fuel flow proportional to the difference between
these speeds and proportional to the integral of the dif-
ference between these speeds.
The proportional control Is
used for speed of response and the integral control for
elimination of steady state errors.
The equation for the
control is:
/N s -
N
(11)
A preliminary theoretical analysis indicated that
the control coefficients /9 and € should each be approxi-
mately 0.50.
The control was then set up on a second com-
puter and applied to the previously determined engine circuit.
Runs were made with p set at one of several values
(of the order of magnitude indicated by the preliminary
analysis) with a range of values for 6 for each value of /3
In this manner an optimum combination was determined to be
O.65 for /3 and O.50 for €
.
10
An upper limit of turbine Inlet temperature Is
critical within the engine.
Therefore, some method of
limiting fuel flow Is necessary so that this predetermined
maximum level of temperature is reached and maintained, but
not exceeded, during acceleration.
A continuously generated
limit was obtained by rearranging equation (7) to give:
W
=
(WaVT2/P2)
T4
A2
-
rc
(P /P
3
2)
(12)
k
In this equation, when the turbine inlet temperature is made
a constant at the desired limiting value and is combined in
a circuit with the function of compressor pressure ratio and
air flow generated in the engine circuit, a limiting value of
the fuel flow parameter Is continuously generated.
This value is then introduced into the control circuit as an electronic limiter (equivalent to controlling a
by-pass valve in an engine).
Thus, when the control circuit
generates a value of fuel flow that exceeds the limiting
value, this limiting value takes over and controls the
engine.
Pig. 6 shows the control circuit with the limiter
in schematic form.
This circuit does not take into consideration
physical limitations of a specific burner.
Therefore, in a
given engine the maximum rate of introduction of fuel as indicated by this control circuit might result in burner blowout.
Such a condition could be avoided by changing the
values of fi and €
.
An extremely low value of fuel flow
11
might also cause burner blow-out.
A control llmlter based
on compressor pressure ratio could be incorporated to insure
against this condition.
No such electronic limiting has been
incorporated in this study.
Engine Operation
To adequately show the results of this study three
types of response of the engine were obtained.
The first
was the response of the engine to an accelerating step input
of set engine speed.
The second was the response of the
engine to an accelerating ramp input of set engine speed.
Then, to show the response of the engine to a decelerating
command, a decelerating step input of set engine speed was
made.
In each of these three types of response the engine
was accelerated or decelerated to several equilibrium run-
ning speeds.
In order to obtain the behavior of the engine at
maximum limits of acceleration and deceleration one of the
equilibrium engine speeds used was that determined by allowing the turbine inlet temperature to remain at 110$ of the
design operation.
A limiting turbine inlet temperature of
115$ of the design operation was used to allow for reason-
able time to accelerate to the maximum speed.
This was
thought compatible with the concept of "military power",
where engine running time would be limited but would be
available in case of emergency.
With this concept it will
be noted that minimum time to accelerate to the high speed
12
condition would be vital, so the control coefficients were
set to obtain a minimum accelerating time to this condition
at some sacrifice in acceleration
ta lower speeds.
Such
limits on an actual engine would, of course, be set with
regard to the specific burner and turbine in use on that
engine
RESULTS AND DISCUSSION
Engine Acceleration
-
Step Input
.
The engine was
accelerated from an initial equilibrium speed to several
final speeds by using step inputs to the control set speed.
The initial speed of N /yTj of 280 represents equilibrium
for a turbine inlet temperature of approximately 80$ of design operation.
The control set speeds, N 3 / ifTj, were se-
lected to obtain final turbine inlet temperatures of 90$,
100$, and 110$ of design operation.
Plots of the following
engine variables versus time were made:
1
Speed
2.
Fuel Plow
3.
Turbine Inlet Temperature
4.
Air Plow
5.
Compressor Pressure Ratio
6.
Accelerating Torque
The results are shown in Pigs. 7 through 12.
Pig. 7 shows the speed response of the engine.
slight overshoot for N s /yT^ of 345 may be observed.
A
This
overshoot was tolerated to reduce the time to accelerate to
the equilibrium speed.
It may be controlled
by varying the
13
control coefficients a and €
.
It will be noted that there
is no overshoot when accelerating to the lower equilibrium
speeds.
If the control coefficients were made to vary with
engine speed, however, a slight overshoot could result for
all values of N s / Vt^ and acceleration time would be improved
throughout the speed range.
The time to accelerate from N Q / Yt^ to N s / VTg" of
345 is approximately eight seconds.
This represents an ac-
celeration of about 1500 RPM under standard conditions.
Comparison with acceleration data given in Ref
.
2 shows that
this time is typical of an actual engine.
Pig. 8 shows the variation of fuel flow.
As would
be expected, the fuel flow rapidly increases to some maximum
value and then decreases to the new equilibrium value.
For
the set speed of 3^5 the fuel flow is limited (by-passed in
an actual engine) so as not to exceed the upper limit of tur-
bine inlet temperature.
A run without the llmiter was made
for comparison purposes.
It is noted that the limited fuel
flow is not constant but varies with time.
Pig. 9 shows the variation of turbine inlet temperature.
It is seen that the variable limited fuel flow shown
in Pig. 8 produces a constant maximum turbine inlet tempera-
ture of 115#.
The effect of limiting on the compressor pressure
ratio and torque may be observed in Pigs. 11 and 12.
Engine Acceleration
-
Ramp Input
.
Acceleration runs
between the same initial and final conditions listed above
were made for a slow ramp input to the control.
The set
14
speed was made to vary at the rate of approximately 100 RPM
per second.
Variation of engine speed, fuel flow and tur-
bine inlet temperature versus time were plotted.
These re-
sults are shown in Pigs. 13 through 15.
Pig. 13 shows that the engine speed parallels the
control set speed with a time lag of about one -half second.
Figs. 14 and 15 show that very little limiting is encoun-
tered for a slow ramp input.
Engine Deceleration
-
Step Input
.
Deceleration runs
were made using a step input of N s /yT^ of 280.
Initial
speeds for 110$, 100$ and 90$ of design turbine Inlet tem-
peratures were used.
Variation of engine speed, fuel flow,
and turbine inlet temperature versus time were plotted.
results are shown in Pigs. 16 through 18.
is about seven seconds.
The
Deceleration time
Comparison with data given in Ref
.
2
indicates that this time is typical of an actual engine.
Again, if the control coefficients were made to vary with
engine speed, time to decelerate from the lower speeds
would be reduced.
Pig. 17 shows the variation in fuel flow for decel-
eration.
As is pointed out in Ref. 3, deceleration is accom-
panied by the possibility of unstable burner operation or
blow-out.
Ref. 3 further indicates that the compressor dis-
charge pressure is a suitable parameter for controlling minimum fuel flow.
Although compressor pressure ratio is avail-
able for such a lower limit in the control, no effort was
made In this study to provide for such a limit.
15
It will be noted that the equilibrium running speeds
as determined by the computer are 3^5, 328, and 307 for oper-
ation at 110$, 100$, and 90$ of design turbine inlet temperature.
The speeds calculated in Appendix C for these condi-
tions are 3^9, 328, and 311.
The percentage errors are 1.14$,
zero, and 1.28$ respectively.
The foregoing results show that a reasonably accurate
dynamic and static simulation of a turbojet engine in the
region of greatest interest is possible on a standard electronic analogue computer without resort to special equipment.
The non-linear differential equation governing engine opera-
tion has been solved without reducing the equation to a linear
form.
As a result, operation far from equilibrium is made
possible.
The accuracy is satisfactory and the range of op-
eration of the simulated engine covers one of the important
ranges for control studies on an actual engine.
The accuracy
and range of operation could be improved, however, by use of
function generators in place of the straight line relationships used to simplify the computer circuit.
With this simulation it is possible to make control
studies on any engine whose characteristics (design or actual)
are known, using any of several continuously generated engine
parameters for control.
As has been shown in this study by
the illustrative control, the control may be applied to the
simulated engine and the time history of all important engine parameters plotted.
Then theoretical or empirical
changes may be made to the control to optimize it or to stay
16
within limits of engine operation.
In this way it was shown
that the control coefficients used on the illustrative con-
trol should be made variable with engine speed for optimization.
The need for the upper and, possibly, the lower limi-
ters on fuel flow also became apparent as the result of such
an analysis.
CONCLUSIONS
A reasonably accurate dynamic and static simulation
of a turbojet engine is possible on a standard electronic
analogue computer.
Such a simulation has the following
characteristics:
1.
No special equipment is necessary.
2.
No linearizing assumptions are necessary in the
equation for engine operation.
3.
Operation far from equilibrium is possible.
k.
Compressor pressure ratio, air flow, turbine in-
let temperature, turbine and compressor powers, and engine
speed are available as computer outputs for use as control
parameters, for observation, and for plotting.
5.
Parameters above are continuously generated.
6.
The simulation is equally applicable to engines
in the "hardware" or design stages of development.
7-
Accuracy and range of operation may be improved,
if necessary, by use of function generators in the circuit.
17
Improvement of the illustrative control may be accom-
plished by:
1.
Making the control coefficients vary with
engine speed.
2.
Incorporating a lower limiter to fuel flour to
prevent unstable burner operation or blow-out.
REFERENCES
1. Otto,
Edward W., and Taylor, Burt L., Ill:
Dynamics of
a Turbojet Engine Considered as a Quasi-3tatic System.
NACA TR No. 1011, 1951.
2. Bell,
Arthur H., and Parmer, J. Elmo:
Transient opera-
ting Characteristics of a Turbojet Engine When Subjected to Step Changes in Fuel Flow.
NACA RM No. E9K25A,
1950.
3.
Novlk, David, and Otto, Edward W.:
An Analysis of Con-
trol Requirements and Control Parameters for Direct
Coupled Turbojet Engines. NACA RM No. E7125a, 19^8
18
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APPENDIX A
Symbols
Ag
Exhaust nozzle area, sq. ft.
At
Turbine nozzle area, sq
Cp C
Average specific heat of gas passing through
.
ft.
compressor (assumed .240 BTU/lb ° R)
c pt
Average specific heat of gas passing through
turbine (assumed .270 BTU/lb. °R)
Polar moment of inertia of rotating parts
I
(20 lb-ft sec 2 )
K
k
Constant
Heating value of fuel
specific heat
N
Engine speed, rpm
Ns
Set value of engine speed, rpm
P
Total pressure, Ib/sq. ft.
p
Static pressure, lb/sq. ft.
Pt
Turbine power, BTU/sec
P
Compressor power, BTU/sec
c
,
(assumed 80,000 *R)
Q
Accelerating torque, lb-ft.
T
Total temperature,
t
Time, sec
wa
Air flow, lb/sec
wf
Fuel flow, lb/hr
Wg
Total gas flow, lb/sec
w
Equals w a and w_ by assumption
°R
A-l
A-2
Acceleration of engine speed, rad/sec 2
jS
Coefficient of Integral control component
h
Ratio of specific heats In compressor (1.40)
dT
R<tio of specific heats in turbine (1.3*0
t
t
Coefficient of proportional control component
\b
Burner efficiency, percent
Vi
c
Compressor efficiency, percent
t
Turbine efficiency, percent (85$)
>>
Subscripts
Ambient
1
Diffuser inlet
2
Compressor Inlet
3
Compressor outlet
4
Turbine Inlet
5
Turbine outlet
6
Exhaust
Parameters
N/^Tg
P c/ w
Engine speed
T2
w T2
Pt/
Compressor power
Turbine power
P^/P2
Compressor pressure ratio
T^/T 2
Turbine Inlet temperature
Wa/T^
Air flow
P
2
(b
Fuel flow
APPENDIX B
Engine Design Calculations
Fig.
1
shows the configuration of a typical direct -
coupled turbojet for which typical compressor characteristics (Pig. 2) are assumed to apply.
It is necessary to
choose a point of design operation for the compressor and
then to determine a reasonably sized turbine and tailpipe
area to complete the design of the engine.
The method used
to determine the turbine nozzle throat area consists of de-
termining the choked nozzle area necessary to pass the mass
Equating compressor and turbine powers gives tempera-
flow.
ture and pressure at the turbine outlet from which the tail
pipe area is calculated.
The following assumptions were made:
1.
Negligible velocity at stations 2, 3, ^ , and 5.
2.
No ram (p Q
3.
Burner pressure drop equals k%
« P
2)
(Pw/P-^ •
.96).
4. Turbine efficiency constant at O.85.
5. Weight of fuel neglected (w a =
wg
-
w)
6. Standard atmospheric temperature and pressure at station 2
7.
Tjj.
=
i960 'R at design operation.
8. Neglect nozzle and tailpipe losses,
9.
Neglect friction.
10. Turbine nozzles are choked.
B-l
i.e., lsentropic.
B-2
Nozzle Throat Area (A^)
t
For choked nozzles the following equation may be used:
g '
A
P
t
4
constant
=
(Bl)
4
which may be written:
**
^
w a VT"2
- v,
C .
=
P
P,
3
W
U T4
wa
p4
where C
-
O.525 °R*
g
X
(B2)
At
for rt r 1.3^.
/ sec
At design point "A" on the compressor chart the
following values are obtained:
w
z
w /T^
71.2 lbs/sec
=
.766
=4.0
P /P 2
3
Solving equation (B2) gives a turbine nozzle throat area of:
At = 0.737 sq. ft.
Tail Pipe Area (Ag):
Compressor power is given by the equation:
c
-
a
pc
v
(B3)
3
<=
which may be written:
fc-l
P
c -
"a
V-
T
>W
2
He
P
- 1
(B4)
2
Turbine power is given by the equation:
P^ = w.
t
g
c
Pt
^4 " TA 5K
(Tli
)
(B5)
At equilibrium, compressor and turbine powers are equal
which permits solving for Tj- by equating equations (B4)
and (B5) giving a Tj- of 1696 °R.
P_ is determined by
turbine efficiency as expressed by the following equation:
B-3
1
Tit
T 5 /T 4
-
- °- 8 5 =
1
(B6)
-Lip*"
Kl
Solution of equation (B6) gives a
Pc
of 4110 lbs/sq. ft.
P0A5 r O.516 but Pg/Pc = O.536 for a choked nozzle at
/t = 1.34.
and tne tailpipe is choked.
Therefore, Pg > P
Then the equation
wg
fr^ = .C = O.525
A6
p
°R
/z
/sec
5
may be solved for Ag giving a tailpipe area of
Ag = I.36 sq. ft.
(B7)
APPENDIX C
Equilibrium Running Calculations
An equilibrium operating point for a direct coupled
turbojet is determined by satisfying two conditions:
(1)
The mass flow through the turbine and compres-
sor must be equal
(2)
The powers of the turbine and compressor must
be equal.
For choked turbine nozzles:
w /T^
-—
At p 4
(CI)
—
constant
-
/sec for
where the constant is 0.525 ° R
-
1.3^.
8J.
Equation (Cl) may be written:
"f^I^k
P
2
where K - O.525
f
A
fc
K
T2
P
j?^
P
p
=
3
(C2)
2
(0.525)
(.737)
(.96) = .3715.
3
Equation (C2) is used to draw lines of constant
Tn/Tp on the compressor performance chart (Fig. 2).
Lines
for T4/T2 of 90$, 100$, and 110$ of design operation are
shown on Fig. 2.
On a given line of T^/T 2
=
constant, the
mass flow through the turbine and the compressor is equal.
An equilibrium running point is determined by finding the
point on the line where compressor and turbine powers are
equal.
C-l
C-2
A relation between turbine pressure ratio (P^/Pc)
and turbine temperature ratio (Vtv/T-) is determined
by-
turbine efficiency as expressed by the following equation:
1
-
1
-
T5A4
(C3)
s
1t
/P,
hi
For this analysis ^ t was assumed to be 85$ and *t was assumed
to be 1.34.
The resulting relation between turbine tempera-
ture ratio and turbine pressure ratio is plotted in Pig. CI,
Equation (Cl) may be expanded as follows:
W/T5
T
K
5
P
4
(c4)
also:
w
(C5)
S
then:
Po
(C6)
f
T
P
Prom equation (C6) values may be obtained for a plot of
Pc/p
as a function of Ph/Pc
of I.36 sq. ft.
**°r ar*
This relation is shown in Pig. C2.
Cl and C2 may then be used to obtain
P4/P5.
exhaust nozzle area
P|j./p
This relation is shown in Pig. C3.
Pigs.
as a function of
C-3
Compressor power is given by the equation:
p«
c
w c „ (To
pc
3
-
v
-
T
2
)
(C7)
1
(C8)
which may be written:
P
w
c
= c pc
2
\
&J
-
F
L
2
Turbine power is given by the equation:
p
t =
w c t
P
(
T
4
"
V
(C9)
which may be written:
K
T4
=
w
T 4l
*
pt>
,*S
P
1
100$
-
T"2l
1
5 S] (CIO)
>*
Entering the compressor performance chart (Pig. 2)
along a line of constant T^/ T 2
various N/f/Tg.
P
c
/wT 2 vs.
,
p ^/p 2
"^ ^e ODta ^ ne(
**
^
or
Equation (C8) is then used to determine
U/fY2 fQr a glven T4/Ap 2<
p^/p ^ ls obtained
from P^/Pg from tne relation:
P
P
3
2
P
4
(Cll)
'0
where Pg/Pn is 1»° and p li/p o is 9^$ in tn *s analysis.
Know-
ing Ph/pQ permits the determination of P,-/Ph from Pig. C3.
Equation (CIO) is then used to determine the turbine power
parameter vs. the engine speed parameter for a given Tw/T
.
A plot of both the turbine and compressor power parameters
vs. the engine speed parameter for T^/^ of 90$ and 110$ of
design operation is shown in Pig. C4.
The intersection of
the powers determines the equilibrium operating point.
Equilibrium points obtained are N/'/TV,
and
K/fY2 = 349 for T4/T2 of 110$.
on Pig. 2.
= 311
for T4/T4 of 90$
These points are plotted
C4-
2.4
sT
2.2
5
^o
3
2.0
/8
/
1.6
1.05
IJJ6
/.07
LOB
W9
Turbine temperature ratio, y7+/7s
Fig.
CI.
Relation between turbine
ratio
tota I -press u re
and total-temperature ratio
turbine etticiency ot o.QS.
at constant
UO
C5
1.0
A
d?
/8
S
1.4
J.O
0.75
loo
LZ5
(.50
1.75
PjPs
fWTT
r/G.CZ.
Re/ah on of exhaust-nozz/e pressure, rat/o
and turbine, tota/ pressure temperature rat/o
for
exhaust -noz 7 /e
area
of /.J6
oq.Fr.
Z.OO
C6
H-.U
36
S2
_o
2.8
2.4
?
n
/.5
/.6
/.7
Turbine pressure rat/o
FiG.CZ.
Relation between
Z.D
/-9
/.8
,
P*/Ps
engine.- pressure
ratio
and turbine -pressure ratio for exhaust
nozz/e area of /J6 3Q. FT.
C7
CD
3s
^
^
^
250
325
300
350
(a)
Q/4
,.,.
.
..
.
_.
1
00
D
/
CD
^ SO %
-
12.
a/j
/
1
1
i
'
a/2
/
*"
Equil
.
-
Zll
/
/
/
i
!
I
i
a//
250
275
300
3Z5
550
375
661
FlG.
C4.
Turbine,
and compressor pocuer parameters
ve rsus
sp eed p aram eter.
oOJ-*30
Rockcastle
Simulation of a turbojet engine on a standard
electronic analogue
computer for the purpose
of control study.
board
^Thesis
R664
33155
Rockcastle
Simulation of a turbojet
engine on a standard electronic
analogue computer for the
purpose of control study.
thesR664
Simulation of a turbojet engine on a sta
3 2768 001 94924 1
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