possible to generalize the performance of a given .... The combustor pressure drop value is established ... found that this could best be correlated as a function.
U
81-GT-202
AN ASME PUBLICATION
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-
R. K. Agrawal
M. Yunis Computing Analyst Pratt & Whitney Aircraft of Canada Ltd., Longueuil, Quebec, Canada
A Generalized Mathematical Model to Estimate Gas Turbine Starting Characteristics The paper describes a generalized mathematical model to estimate gas turbine performance in the starting regime of the engine. These estimates are then used to calculate the minimum engine starting torque requirements, thereby defining the specifications for the aircraft starting system. Alternatively, the model can also be used to estimate the start up time at any ambient temperature or altitude for a given engine/aircraft starting system combination.
NOMENCLATURE
Ah p 0 p
mass flow rotational speed static pressure total pressure pressure ratio gas constant static temperature total temperature corrected pressure specific work efficiency corrected tereperature static density torque
detailed performance maps for the various components in the engine. These maps, however, generally start to get well defined only beyond the engine idle speed and are therefore of not much use in the starting regime. In addition, differences in characteristics of individual component types from one engine to another which are so pronounced at higher speeds, have been found to become relatively small at the low speeds encountered in starting regime. It is thus possible to generalize the performance of a given component type, e.g. compressor, by a single generalized map without introducing gross errors. Using these generalized maps, it is then possible to carry out transient performance analysis to estimate the engine starting characteristics. This then is the approach followed in this model. AN OVERVIEW OF THE MODEL (This is presented in Fig.l)
Subscripts amb ACCEL c comb HPC R t
The Society shall not be responsible for statements or opinions advanced in papers or in iiss^i:sslon at meetir - of the Society or of its Divisions or Sections or printed in its publications D , scussion is printed only if the paper is publ .hed in an ASME -Journal or Proceed•nry.c R le, „:1 for genera publication upon presentation. Full cr u., should be qis •r, is , :.,4"[ the Techin.: D arson. and the author(s)
Copyright © 1981 by ASME
Senior Performance Analyst Assoc. Mem. ASME
in N p P PR R t T
THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS
ambient acceleration compressor combustor high pressure compressor reference turbine
INTRODUCTION This paper describes a mathematical model developed to estimate gas turbine performance in the starting regime. Starting regime, here, is defined as that part of the operating range which lies between zero and engine idle speed. Estimation of engine performance requires
Generalized Component Characteristics Compressor. The compressor as well as other engine component characteristics used in the model are presented in the form of normalized parameters defining the performance of that component. Thus, for the compressor these parameters would be flow, specific work and efficiency as functions of speed. To obtain these functional relationships, aero/thermodynamic analyses of these components were carried out. The compressor flow relation, for example, is derived as follows: The compressor mass flow through an annulus of Area Aann is given by -
m=
Contributed by the Gas Turbine Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for presentation at the Gas Turbine Conference & Products Show, March 9-12, 1981, Houston, Texas. Manuscript received at ASME Headquarters December 22, 1980.
Ac
--
\a =
Rt
A.,VQ
(c
where V a is the axial flow velocity and is related to
the compressor flow factor (^), speed (N) and diameter (Dc) by
Copies will be available until December 1, 1981.
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0.8 0.7
Turbine Compressor
0.6
component
Engine I aerodynamic characteristic
Flight conditions
I
w 0.5
Starter motor characteri:
F 0.4
Fuel schedule
0.3
matching calcs. engin e starting torque
Gas turbin e startingg model
I
Oil viscosity effects
Engine mechanical characteristics
0.2 0.1
Engine start up time
U
Parasitic losses Windage
Figure 1:
( ) (
From (i) and (ii) and normalizing p and t
rrl = An 7r N Dc /1.696 5/8.7
72o R
(zz7)
Denoting the parameters in (iii) with a subscript R at a given engine reference condition,
SR
A cin n 7(• Dc
/4.696 NRr^s (iv)
720 R
BR
sig•7
a
From (iii) and (iv) then,
)
1
(N/ie) R
(N
/a)
= K^(N R
(i)
fg ) R
R
^
Figure 2 shows this relation plotted based on steady state data from a number of engines. Similarly, it can be shown that the steady state compressor work and efficiency will be represented 2 by:
0
0.6
0.8
1.0
Compressor Inlet Flow Variation
and
720
(
0.4
(NI /)/ (NI 1)R
An Overview of the Model
= N D, (ii)
=
0.2
Figure 2:
Frictional
Va
R
^,^^g _ KWC r()
(i..L/e)R /R
(2)
Cc)R
k2< ( /
(3)
/ N )
I;r/ Z
The functions K YG and K^•t c are once again determined empirically. It should, however, be noted that since the engine does not operate at the steady state condition in the starting regime, the values obtained from equations (1) to (3) are used only as the first guess values in the engine performance calculations. The final values have to be obtained based on the flow match between the turbine (s) and the nozzle areas as will be described later. Equations (1) through (3) now define the equations governing the performance of the compressor and are used to calculate the compression pressure ratio, total compressor aerodynamic work and torque required which is to be supplied either by the starter motor or the turbine or both. Combustor. The main performance parameters here are the combustion efficiency and the combustor pressure drop. The combustor pressure drop value is established based on data from a number of combustors and is presented to the model as a constant percentage of the combustor inlet pressure. In addition, an estimate of the combustion efficiency is required, after the engine is lit up, to calculate the turbine inlet temperature. It was found that this could best be correlated as a function of the combustor air loading parameter and the fuel/ air ratio. Thus,
K Downloaded From: https://proceedings.asmedigitalcollection.asme.org/ on 05/10/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Li:]
comb = f ^^ZR ^^r^)
T u r bine . Since the engine, in the starting sequence, is not in steady state, the turbine work is not equal to the compressor work as is the case for steady state operation. Thus, the turbine work and hence PR cannot be determined from the knowledge of compressor work alone. In this model, the turbine PR is obtained by the following method:
(4)
where, f/a is the fuel air ratio and is calculated from the prescribed fuel flow schedule, and -n is defined as
^_m
P
R
t
= 4 P = P
p5
P. /•8 T-/340 P /co^,b
P3 . P¢ P7 Po Pi
p3
e
Ps
()
P7
In this identity: where m, P and T are the combustor inlet mass flow, pressure and temperature and Vcomb is the combustion chamber volume. The functional relationships were once again obtained empirically and are shown in Figures 3 and 4. The combustion efficiency is then calculated from:
`^ Com6
^OrY, fi9.2^ ^CenbR LC o.., L (f
(5)
where Ceo.nb-R is the combustion efficiency of the engine being analysed at a known reference point, e.g. the engine Take Off condition.
1.
P1/PO = Inlet Pressure Ratio, P3/P1 = Compressor Pressure Ratio (PR c ), P4/P3 = Combustor Pressure Loss (Assumed Constant), P5/P7 = Power Turbine or Second Turbine Pressure Ratio and P7/PO = Nozzle Pressure Ratio. For a single spool gas generator, Pl/PO = P7/PO 1.0 (for a two spool gas generator P1/P0 is obtained as described in a later section). Also, as shown in Figure 5, since the turbines and nozzles are operating in series, the flow match between the turbines and the nozzle result in a unique value of P5/P7 for each value of P4/P5. Thus, for each compressor pressure ratio, PR c , there is a unique value of PRt which satisfies the engine area restrictions. PRt can then be expressed as:
PRt of PR C
0. or,
PRt
_ kP PRc PRC-R
PRt-R k,7comb = ncc
Or,
Occ-R
0.94
P RE -I
K PRc-I (7 l P / PRc-R
PR t _Q
0.93 0.92 0.91
0.89 0 1 2 3 4 5 6 7 8 9 10 11 12
S2/S2R
Figure 3:
Combustion Efficiency Variation with S2
U
This relation is plotted in Fig. 6 for various engines with differing design Compressor Pressure Ratios. From this figure it is clear that the value of Kp in equation (7) is a function of the design compressor pressure ratio. In this mathematical model therefore, a series of these lines have been built to cover a wide range of present and probable future gas turbine engines. The variation of the turbine flow corresponding to the pressure ratio used in equation (7) is shown in Figure 7. This flow map was generated at varying power turbine or the second turbine speed (in case of a 2-spool gas generator) to simulate the operation in the starting regime where the second turbine operates at only a fraction of its design speed. This map should therefore be representative of the operation of two turbines in series in the starting region and is assumed as the generalized turbine flow map in the model. The only other parameter required to assess the turbine performance is the turbine efficiency. This correlates best against the turbine Pressure Ratio and is given by:
M Figure 4:
Combustion Efficiency Correction
(P
t ^t-e
-I)
^PRt e
-)
(8 )
for Fuel Air Ratio
C Downloaded From: https://proceedings.asmedigitalcollection.asme.org/ on 05/10/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use
'ut
n
P7
5 P4 P
P7
Po
LP turbine
Nozzle
P5
HP turbine Figure 5:
Series Operation of Turbine(s) and Nozzle
Land
,
Bon
^ I
is b e
I/ N s K9 ` f r`^, N
R
)
H PC
(10)
HPC R
where b R and 9 R are the normalized pressure and temperature to the HP compressor inlet at the engine reference point. Kg and K e are once again determined empirically. Figure 6:
Turbine PR Variation with Compressor PR (m ✓T/P) / (m fT/P)R In flow
0.8 0.7
0.4
0.1j 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (PRt-1) / (PRtR-1)
Figure 7: Turbine Flow Variation with Pressure Ratio Equations (7) and (8) now enable the calculation of the Turbine total aerodynamic work and the Torque which is supplied by the engine.
Transient Matching Calculations The transient matching calculations essentially involve matching the turbine flow with the compressor flow. The turbine flow is a function of the turbine pressure drop (Figure 7) which in turn is a function of the compressor pressure ratio (Figure 6). At any instant of time, both the compressor pressure ratio and flow are dependent on the engine operating point at that time. As shown in Figure 8, in general during starting, the engine operates closer to choke when being cranked and closer to surge when lit up. The problem then is to find the position of this operating point on the compressor map. Now, the compressor map as defined by equations (1) to (3) is based on steady state engine data. Since, as shown in Figure 8, the engine is not operating on steady state, the map has to be modified as the engine moves away from the steady state operations. For the compressor efficiency, the steady state to transient modifier is correlated as:
Two Spool Gas Generator
(C.
It was found that, in the starting region, the LP spool which is only aerodynamically coupled to the HP spool has only a small effect on the starting of the engine. In this model, this effect is accounted for by modifying the pressure and temperature into the HP compressor as per the following relationships:
mf S Is
trans:snt ()I)
S
slc4.ry sE.fe
and cmod) 2C CEYQns rn) = ^ CStewly st.Ee ) [G
I __ r bcw _i
= Kb
L
SR
(mod) c {C^ 'ic.
HPC.
'I
)
-Al Mn-2
N
HPc
N f HPC R
(9) where the transient has to be obtained through iteration to match the compressor and turbine mass flows.
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the final engine operating point.
ZCt
Effcy
ENGINE MECHANICAL CHARACTERISTICS
Light up Surge line
PR
Steady state Ni
•,\ N2 0
. Cranking
m`/Q
a
Figure 8:
The engine mechanical configuration gives rise to mechanical losses which adds to the compressor aerodynamic torque requirements to result in the total engine torque requirements to be supplied either by the starter or the turbine or both. Mechanical losses in the engine are incurred due to mechanical frictional loss in bearings and gears and aerodynamic windage loss in the compressor and turbine discs as well as on the impeller face. In addition some work is also required to power the fuel control unit, fuel pump and the oil pump. However, except at very low ambient temperatures, the proportion of the mechanical loss is generally less than 5% of the compressor aerodynamic power requirement and is sometimes ignored in calculating engine starting torque requirements. In this model, however, this is not the case and the mechanical losses are accounted for by splitting them into two categories: (i) Basic parasitic loss, At normal operating temperatures (oil at temperatures above 130 F). (ii) Additional viscosity effects to account for low oil temperatures.
°
Engine Operating Lines on Compressor Map
Basic Parasitic Loss These are further divided into two components Frictional and Windage where the frictional losses would be a function of the engine mechanical speed and the windage loss that of the aerodynamic speed. Additionally, in this model the parasitic losses are correlated as a fraction of the required compressor torque calculated in the previous sections. Thus, When the engine is cranking (near choke), the compressor steady state specific work is modified similar to the efficiency, so that, (0'^cl
e
. (C$^l
(APuI
/{ran
(3)
` 6 / rnod
B /shoddy Skate
(t/'Ce.c,, )
(z/ r Clnd, Wkere,
yv iN DA4E
P^ FR/c710N = Bo i- `t orasikic = tw"JD+ ZFRIc
.
C Ea^yn 3^
-
B / wood
K
L iii..e1
s;e
1
(,+)
(m/91 r-
—
/ sb
