4 Engine Selection: Parametric Cycle Analysis
4.2 Design Tools
The uninstalled engine thrust (F) and the uninstalled thrust specific fuel con- sumption (S) are the primary measures of the engine's overall performance. The uninstalled thrust (see Appendix E) for a single exhaust stream engine can be written as
F = --(th9V 9 1 -- th0V0) + A9(P 9 -- P0) (4.1) gc
where station 0 is far upstream of the engine and station 9 is the engine exit. The term gc appears in Eq. (4.1) as a reminder that the units must be watched carefully.
In this particular case, for example, if the units of rh are pounds mass/second and V are feet/second, then g¢ must be Newton's constant of 32.174 lbm-ft/(lbf-s 2) if F is to be measured in pounds force. If the units of rn were slug/s instead, then g~ would be unity. Because incorrect dimensions are responsible for more errors than any other thing, careful bookkeeping and double-checking are a worthwhile
ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS Table 4.1 Typical F/mo and S values
97
Compressor Fan
Engine pressure p r e s s u r e Bypass Tt7, I',4, F/tho, S, type ratio (zrc) ratio (zrf) ratio (or) °R °R lbf/lbm/s 1/h Turbojet 10-20
no A/B
Turbojet 10-20 with A/B
Turbofan 20-30 low u
no A/B
Turbofan 8-30
low u 10-30
with A/B
Turbofan 30--40 high ot
no A/B
2-4 0.2-1
3600
2000 54-58 1.0-1.1 3000 93-96 1.3-1.4 2000 94-101 2.0-2.2 3000 115-119 1.7-1.8 2000 23-47 0.85-1.0 3000 53-84 0.96-1.5 2-4 0.2-1 3 6 0 0 2000 75-98 2.1-2.7 3000 102-116 1.7-2.0 1.4-1.6 5-7.5 2000 5.5-12 0.76--0.97
1.4--4 5-10 3000 13-27 0.67-1.03
investment of time. The uninstalled thrust specific fuel consumption (S) is given by
S -- /~f -[- FhfAB (4.2)
F
where (rhf + rhfAB) is the total fuel flow rate to the main bumer and afterburner of an engine. Since one pound mass of fuel at (or near) the surface of the Earth weighs one pound force in the British Engineering system, the thrust specific fuel consumption may be regarded as pounds weight of fuel per hour per pound force of thrust, and is traditionally reported in units of 1/h.
Parametric cycle analysis employs the uninstalled thrust per unit mass flow of captured airflow (F/rho) or specific thrust and the uninstalled thrust specific fuel consumption (S) as the performance measures of primary interest. Representative values of these parameters for several common but different engine types are given for guidance in Table 4.1.
4.2.1 Station Numbering
Consider the generalized engine shown in Figs. 4.1a and 4.lb. The station numbers of the locations indicated there are in accordance with Aerospace
b l e e d air
Fig. 4.1a Reference stations--mixed-flow turbofan engine.
98 AIRCRAFT ENGINE DESIGN B)
2 13
1 are h I ~om~ ~om,
2.5 power extraction
PTOL PTOH
ID bleed air cooling air #2 Ib
| cooling air #1
3 /3.1 ,4 "~ 4 1 high-p ... turbine 4;4 ,~ 4.5
jrL[
l u~or[ I :Oxen: I I rotor [ ~ [rotor] [ mixer2 I high-pressure spool ]5 I
low-pressure spool
Fig. 4.1b Reference stations--bleed and turbine cooling airflows.
Recommended Practice (ARP) 755A (Ref. 3) and will be used throughout this textbook, and include:
Station Location
0 1 2 13 2.5 3 3.1 4
4.1
4.4 4.5 5 6 16 6A 7 8 9
Far upstream or freestream Inlet or diffuser entry Inlet or diffuser exit, fan entry Fan exit
Low-pressure compressor exit High-pressure compressor entry High-pressure compressor exit Burner entry
Burner exit Nozzle vanes entry
Modeled coolant mixer 1 entry
High-pressure turbine entry for zrtn definition Nozzle vanes exit
Coolant mixer 1 exit
High-pressure turbine entry for rtH definition High-pressure turbine exit
Modeled coolant mixer entry Coolant mixer 2 exit Low-pressure turbine entry Low-pressure turbine exit Core stream mixer entry Fan bypass stream mixer entry Mixer exit
Afterburner entry Afterburner exit Exhaust nozzle entry Exhaust nozzle throat Exhaust nozzle exit
ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS 99 4.2.2 Gas Model
The air and combustion gases are modeled as perfect gases in thermodynamic equilibrium, and their properties are based on the NASA Glenn thermochemical data and the Gordon-McBride equilibrium algorithm (see Ref. 7). Thus, this gas model includes the variation of specific heat at constant pressure Cp with tem- perature. In addition, the classical perfect gas model with constant specific heats (also know as a calorically perfect gas) is also included as a limiting case in this analysis.
For the perfect gas with variable specific heats, the required additional thermo- dynamic definitions (Refs. 1, 2, 4) are
h - - Cp d T (4.3a)
fr~ aT
(p - - C p --T- (4.3b)
ef
Pr--exp
( ~ - - ~ ) (4.3c)where ~b is the temperature dependent portion of entropy (s) and Pr (called the reduced pressure) is the pressure variation corresponding to the temperature change for an isentropic process (s = constant). The differential equation for entropy is the Gibbs equation T ds = dh - v dP, which becomes T d s = d h - R d P / P for a perfect gas (Ref. 4). Integration of this latter equation using the definition of q~ gives the following relationship for the finite change in entropy between states 1 and 2:
$ 2 - - S1 -~- ~b2 - - (~1 - - R In/°2 (4.3d)
P1 For an isentropic process, Eq. (4.3d) reduces to
~ 2 - - ~ 1 = R In P2 (4.3e)
P1
Using Eq. (4.3c) we can express the pressure ratio for an isentropic process in terms of the reduced pressure Pr as
P 2 nst = ~-,-1 (4.3f)
We note that for the simplifying case of a calorically perfect gas (CPG) or constant specific heats for the finite change between states 1 and 2
a) Eqs. (4.3a), (4.3b), and (4.3d) become
h 2 - h i = c p ( T 2 - T1) (4.3a-CPG)
(92 - - ~1 = Cp In T2 (4.3b-CPG)
T1
T2 _ R In P2 (4.3d-CPG)
s 2 - S l = C p In T1 P1
100 AIRCRAFT ENGINE DESIGN Table 4.2 Calling nomenclature for subroutine FAIR
Symbol Knowns Unknowns
FAIR(l, f, T, h, Pr, 4), Cp, R, y,a) f, T h, P~, (]), Cp, R, y,a FAIR(2, f, T,h, Pr, c]),Cp, R, y,a) f , h T, Pr, 4),Cp, R, y,a FAIR(3, f, T,h, P~,c]),Cp, R, y,a) f, Pr T,h, qb, Cp, R, y,a FAIR(4, f, T,h, P,.,qS, Cp, R, y,a) f, q5 T,h, Pr, Cp, R, y,a
b) Eq. (4.3f) becomes the familiar isentropic relationship
( - ~ l ) s = c o n s t = ( ~ l ) Y-zzT-I (4.3f-CPG) The effect of variable gas properties can be easily included in a computer analysis of gas turbine engine cycles, as follows. In addition to the ideal gas equation, two subroutines are needed: 1) a subroutine that can calculate the thermodynamic state of the gas given the fuel/air ratio f and one temperature dependent property; and 2) a subroutine that can calculate the compressible gas relationships Tt/T, Pt/P, and mass flow parameter (MFP) given the Mach number (M), total temperature (Tt), and fuel/air ratio ( f ) .
The subroutine FAIR was developed to calculate the temperature dependent properties given the fuel/air ratio f and one of the following properties: T, h, Pr, or ~p. Table 4.2 gives the calling nomenclature for the subroutine FAIR. The subroutine also determines the specific heat Cp, the gas constant R, the ratio of specific heats V, and the speed of sound a.
The compressible flow subroutine RGCOMP was developed to calculate the compressible flow relationships for this gas model with variable specific heats.
Table 4.3 gives the calling nomenclature for this subroutine. The equations used in RGCOMP are given in Appendix F.
4.2.3 Total Property Ratios
You will find it very important to be able to swiftly identify and thoroughly grasp the physical meaning of the quantities that are about to be defined in order to understand and manipulate the cycle analysis results. We therefore recommend that you spend enough time to completely familiarize yourself with the material
Table 4.3 Calling nomenclature for subroutine RGCOMP
Symbol Knowns Unknowns
RGCOMP(1, T,, f, M, T,/T, Pt/P, MFP) RGCOMP(2, Tt, f, M, TilT, P,/P, MFP) RGCOMP(3, Tt, f, M, Tt/ T, Pt/ P, MFP) RGCOMP(4, T,, f, M, T,/T, P,/P, MFP) RGCOMP(5, Tt, f, M, Tt/ T, P,/ P, MFP)
Tr, f , M Tr/T, Pt/P, MFP Tr, f, Tt/T M, Pt/P, MFP T,,f, Pt/P M, Tt/T, MFP T,,f, MFP T,/T, Pt/P,M < I T,,f, MFP Tt/T, Pt/P,M > 1
ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS 101 of Secs. 4.2.3 and 4.2.4, and then mark their location carefully so that you can find them quickly whenever necessary.
The ratio of total (isentropic stagnation) pressures rr and total (adiabatic stag- nation) enthalpies r are introduced, where
total pressure leaving component i
zri -- (4.4a)
total pressure entering component i total enthalpy leaving component i
ri -- (4.4b)
total enthalpy entering component i
For the case of the calorically perfect gas, we assume a zero reference value of the enthalpy at zero absolute temperature. Thus enthalpies are replaced by the specific heat times the absolute temperature, and ri becomes the ratio of total temperatures or
total temperature leaving component i
vi - (4.4b-CPG)
total temperature entering component i
Note should also be taken of the fact that for case of the calorically perfect gas the relationships of Sec. 1.9 apply to Eqs. (4.4a) and (4.4b).
Moreover, the Jr and v of each component will be identified by a subscript, as follows:
Subscript Component Station
A B Afterburner 6A --~ 7
b Burner 3.1 --~ 4
c Compressor 2 ~ 3
cH High-pressure compressor 2.5 ~ 3
cL Low-pressure compressor 2 ~ 2.5
d Diffuser or inlet 0 --~ 2
f Fan 2 ~ 13
- - Fan duct 13 --~ 16
m 1 Coolant mixer 1 4 --~ 4.1
m2 Coolant mixer 2 4.4 ~ 4.5
M Mixer 6 -~ 6A
n Exhaust nozzle 7 ~ 9
t Turbine 4 ~ 5
tH High-pressure turbine 4 --+ 4.5
tL Low-pressure turbine 4.5 ~ 5
Examples
zrc, rc = compressor total pressure, temperature ratios zrb, rb = burner total pressure, temperature ratios
102 AIRCRAFT ENGINE DESIGN
Exception. rr and Err are related to adiabatic and isentropic freestream re- covery, respectively, and are defined b y
h,o ho + V ~ / ( 2 & )
rr "-- - - (4.5a)
ho ho
['to Prto
Er r ~ ' -- - - (4.55)
Po g o
Thus, freestream total enthalpy hto = horr and freestream total pressure Pro = Po Err. For the simplifying case o f a calorically perfect gas, we have, in accordance with Sec. 1.9,
rto v 1
- 1 q- c - = - - - M ~ (4.5a-CPG)
Tr - - TO Z v
( Y-1 2) ~'--~
Err = Tr r - I = 1 + - - - - ~ M ; (4.5b-CPG)
Further exceptions. It is often desirable to w o r k in terms of design limitations such as the m a x i m u m allowable turbine inlet total temperature, Tt4. The term rx is thus used and is defined in terms o f the enthalpy ratio
rx -- - - ht4 (4.6c)
h0 Similarly, for the afterburner
r),aB --" - - ht7 (4.6d) h0
For a calorically perfect gas, Eqs. (4.6c) and (4.6d) b e c o m e
r ) ~ - Cp4Tt4 (4.6c-CPG)
C po To C p7 Tt7
r)~AB -- (4.6d-CPG)
C po To
Component Er and r . A complete compilation o f total pressure and total enthalpy ratios follow. Please note in the following that it has been assumed that Ptl3 = Ptl6, ht13 = htl6 a n d P t 3 = Pt3.1, hi3 = ht3.1.
Diffuser (includes ram recovery):
Fan:
Pt2 ht2
Erd = - - rd -- -- 1 (4.7a)
Pro hto
P t l 3 ht13
Erf = Pt2 "~f = ht2 (4.7b)
ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS 103 Low-pressure compressor:
7~cL ~. et2.5 ht2.5
75cL
P,2 ht2
High-pressure compressor:
2"fcH : Pt3 ht3
V:cH
Pt2.5 ht2.5
Compressor:
e t 3
7g c ~- - - ~__ 7~cLYgcH
P,2 ~c - -
ht3 - - "CcL TcH ht2
Burner:
e t 4 hi4
7~b = - - "Cb - -
e t 3 ht3
Coolant mixer 1:
P t 4 . 1
7"t'ml = ~'ml - -
/',4
ht4.1 ht4
High-pressure turbine:
~tH - - - -
et4.4 et4.4
- - ~ m l - -
P,4 Pt4.1 "ftH
ht4.4 ht4.1
Coolant mixer 2:
~m2 - - - - Pt4.5 et4.4
- - 1 ht4.5
"Cm2 - ht4.4
Low-pressure turbine:
YrtL = 1'15
Pt4.5 "~tL
ht5 ht4.5
Mixer:
~ M
Pt6A
e,6
ht6A
• M - - ht6
Afterburner:
Pt7 2"gAB _
Pt6A
ht7
~ A B - - ht6A
Exhaust nozzle:
Pt9
P~7
ht9 - 1 ht7
(4.7c)
(4.7d)
(4.7e)
(4.7f)
(4.7g)
(4.7h)
(4.7i)
(4.7j)
(4.7k)
(4.71)
Jr. -- rn -- (4.7m)
104 AIRCRAFT ENGINE DESIGN
For the calorically perfect gas, all of the component r except that of the burner and afterbumer become total temperature ratios. For example, rc = Tt3/Tt2 and rtH = T t 4 . n / T t 4 . 1 . The r for the bumer and afterburner become
C p4 Tt 4
-gb -- (4.7f-CPG)
C p3 Tt3
"gAB-- cp7Tt7 (4.71-CPG)
Cp6A Tt6A 4.2.4 Mass Flow Rates
The mixed-flow turbofan engine with afterbuming, bleed air, and cooling air is a very complex machine with numerous air and fuel flow rates. The cycle analysis of this engine includes those mass flow rates that have major importance in engine performance and, hence, cycle selection. Please note that the mass flow rate frequently changes between stations as flow is added or removed or fuel is added (see Fig. 4.2). The symbol rh is used for the mass flow rate with a subscript to denote the type as follows:
Subscript Description Station
b C cl c2 F f fAB
0--+9
Bleed air 3-3.1
Core airflow through engine 2.5, 3
Cooling air for high-pressure turbine nozzle vane 3-3.1, 4-4.1 Cooling air for remainder of high-pressure turbine 3-3.1, 4.1-4.4
Fan air flow through bypass duct 13, 16
Fuel flow to main burner 3.1-4
Fuel flow to afterbumer 6A-7
Flow rate at numbered station
M a s s flow ratios. In engine cycle analysis, it is often most effective to cast the calculations into dimensionless mass flow ratios. The most useful of these for the engine of Figs. 4.1 a, 4. lb, and 4.2 include the following:
Bypass ratio (or):
Bleed air fraction (fl):
bypass flow rh F
Ct --' -- (4.8a)
core flow rhc
Cooling air fractions (el and e2):
E l -
bleed flow rhb core flow rhc
82 ~"
(4.8b)
mcooll (4.8c)
rhc
rhc°°12 (4.8d)
rhc
ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS lO5
t • thcfl rhc¢ 2 •
mcEl b.
rhc(fl +el +~ 2 ) rhI[ v ~
3.1 • 4 41 high-p ... turbine 4i4 ,