5 Engine Selection: Performance Cycle Analysis
5.2 Design Tools
5.2.1 The Performance Problem
The performance of a selected design point mixed flow turbofan engine of the type shown in Figs. 5.1a and 5.1b is desired at any flight conditions, throttle settings, and nozzle settings. It is assumed that a parametric (design point) cycle analysis has been performed for the reference point engine using the methods of Chapter 4 to give so-called reference conditions (subscript R) for the engine (SR, [F/FnO]R, etc.), for each engine component (TrfR, r/R, etc.), and for the flight conditions (MoR, POR, and TOR).
To better understand the performance problem, it is instructive to review how the parametric analysis proceeds. In the parametric cycle analysis, Eqs. (H. 1-H. 16) of Appendix H constitute 16 independent equations for directly finding values of the B) power extraction
PTOL PFOH
2 r
I~ bleed air 13
ire t i corn 2.5
I cooling air #2 I~
cooling air #1
3.1 4 ~.L 4. l high-pre ... turbine 4.4 J. 4.5 5
high-pressure spool I /
low-pressure spool
Fig. 5.1b Reference stations--bleed and turbine cooling air flows.
ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS Table 5.1 Engine performance variables
141
Independent Constant Dependent
Component variable or known quantity variable
Engine M0, To, P0 /3 rn0,
Diffuser zra = f(Mo)
Fan Of 2gf, "~f
Low-pressure compressor 0eL rrcL, r~c
High-pressure compressor Ocn ZrcH, rc~
Burner T,4 ~rb f
Coolant mixer 1 e~ "Cm 1
High-pressure turbine 0tin M4 zr,H, rt~
Coolant mixer 2 e 2 rm2
Low-pressure turbine r/tL, M4,5 rCtg, rtL
Mixer zrM max, A6, JI'M, "gM, fit,
AI6, A6A m6, M16, m r a
Afterburner T, 7 Yr aBdry, JT AB f AB
Exhaust nozzle Po/ P9 re,, As a,y Ms, M9
Total number 6 24
16 dependent component performance variables in the order r f , rcL, r.cH, f, rml, rtH, zrtH, rm2, rtC, ZrtL, Or', rM, MI~, M6A, zrM, and fAB for given values of the inde- pendent quantities consisting of flight conditions, aircraft system parameters, and design choices. With values of these 16 component variables in hand, the engine reference point performance in terms of F/rho, S, tip, and tire/is readily found.
Similarly, to find the engine performance, the operational performance values of the 24 dependent variables listed in Table 5.1 must be determined and then, in turn, related by 24 independent equations to obtain a solution for each component performance variable. Because performance analysis is an indirect problem as opposed to the direct problem of parametric analysis, the solution of the 24 perfor- mance equations is not as straightforward as in the parametric case. Regardless of the difficulty in solving the equations, once the values of the 24 dependent variables in Table 5.1 are known, the engine performance in terms of f , fAB, F / m o , S, tie, and tirH follows immediately from Eqs. (4.18), (4.19), (4.14), (4.31), (4.32a), and (4.32b), respectively.
Please notice in the "Constant or known variable" column of Table 5.1 that Tgb, gAB, and zr, are assumed to remain constant, that "Cd, 7rm2 , and rn are assumed equal to one, and that 7gml is contained in rrtH. Also please be aware that, because power extraction is considered in the analysis, a preliminary estimate of rn0 {Eq.
(4.33) and Table 4.1} is required to proceed with the performance analyses. The correct value of rn0 is finally determined when the engine is sized (see Chapter 6).
5.2.2 Assumptions
The engine of interest is shown in Figs. 5.1a and 5.1b, where the notation of Chapter 4 has been retained. In addition to the assumptions summarized in Sec. 4.2.6, the following assumptions are employed in performance analysis:
142 AIRCRAFT ENGINE DESIGN
1) The flow areas are constant at stations 4, 4.5, 6, 16, 6A, and 8 dry (afterburner off).
2) The flow is choked at the high-pressure turbine entrance nozzles (choking area 4), at the low-pressure turbine entrance nozzles (choking area 4.5), and at the exhaust nozzle (station 8). Because the exhaust nozzle may unchoke at low throttle settings and affect the fan operating line (see Fig. 7.E 10), the case of the unchoked exhaust nozzle (station 8) is also included in this analysis.
3) The component efficiencies (Of, OcL, OcH, Oh, OtH, rltL, 1TAB, OmL, rlml-l, ~]mPL, and rlmeH) and total pressure ratios (rOb, Zrgmax, rCaBary, and rrn) do not change from their design values.
4) Bleed air and cooling air fractions are constant. Power takeoffs are constant.
5) The air and combustion gases are modeled as perfect gases in thermody- namic equilibrium, and their properties are based on the NASA Glenn thermo- chemical data and the Gordon-McBride equilibrium algorithm reference, s
6) The simplifying gas model of a calorically perfect gas is included in the analysis. It assumes that the gases are calorically perfect upstream and downstream of the bumer and afterburner and values of gt, Cpt, FAB, and CpZ B do not vary with throttle setting, but included is the variation of g and Cp due to mixing with bypass ratio. For this model, fuel-air ratios f and faB are ignored when compared with unity.
7) The exit area A9 of the exhaust nozzle is adjustable so that the pressure ratio Po/P9 can be set to a predetermined value.
8) The area at each engine station is constant. However the area of station 8 changes with the afterburner setting to maintain constant pressure at the mixer exit or nozzle entrance.
9) The diffuser total pressure ratio, 7rd, is given by Eqs. (4.12a-4.12d).
5.2.3 Referencing and the Mass Flow Parameter (MFP)
Two techniques are worthy of discussion at this point to prepare the path for their efficient and frequent use in the analysis to follow. The first is called referencing and involves the use of reference point conditions to evaluate constants appearing in equations for the dependent performance variables. The second exploits the mass flow parameter, introduced in Sec. 1.9.3, to capitalize on the law of conservation of mass.
Referencing.
The functional relations for engine cycle analysis are based on the application of mass, energy, momentum, and entropy considerations to the one-dimensional steady flow of a perfect gas at an engine reference or off-design steady state operating point. Thus, if at any off-design pointf ( r , zr) = constant
represents a relationship between the two performance variables r and Jr at a steady state operating point, then the constant can be evaluated at the reference point, so that
f ( r , 7r) = f ( r e , ZrR) = constant
ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 143 because f ( v , zr) applies to both on-design and off-design points. This technique for replacing constants with reference point values is used frequently in the per- formance analysis to follow.
Mass flow parameter. The mass flow parameter (MFP) is defined as the grouping rn 4 ~ t / P t A that can be written as
MFP-- p----~ V R Pt V T
The static to total pressure and temperature ratios (P/Pt, T~ Tt) are functions of the total temperature of gas, chemical equilibrium properties, and the Mach number.
Thus the mass flow parameter can be written in the following functional form:
FnVr~t -- M. V/-~ P . / ~ = MFP(M, Tt, f ) (4.24) MFP-- ~tA V R P,V T
The subroutine RGCOMP that is used to calculate the compressible flow functions (P/Pt, T/Tt, and MFP) is given in Appendix F. For the simplifying case of the calorically perfect gas, the properties P/Pt, T/Tt, and MFP are given by
T 1 + Y - (1.1)
T, 2
P ( Y - 1 M 2 ) # '
= 1 + 2 (1.2)
~'~ ~/~-t ~ / ~ ( 1 ) 2(1-)/) y+l
MFP-- P t ~ - - M l + Y 2 - M2 (1.3)
5.2.4 Performance of Turbines with~without Coolant Mixers
The first step in determining the performance of the entire engine at conditions away from the reference point is to analyze the behavior of the high- and low- pressure turbines. This is greatly expedited by the fact that they are both deliberately designed to be choked in their entrance stator airfoil or vane passages, and that the static pressure downstream of the low-pressure turbine is tied to the mixer entrance conditions. As you will see in the development that follows, this both restricts turbines to a very narrow range of operation and furnishes us with a straightforward method of solution. The remainder of the engine performance analysis flows directly from this step because the turbines, in conjunction with the throttle setting (i.e., T,4), provide the power for the fan and compressors and control the fan and compressor mass flows (see Sec. 5.2.5).
The performance (rrt and rt) of a turbine at off-design is primarily determined by the efficiency and mass conservation relationships. It is shown in Ref. 1 that zrtH and VtH remain constant for an uncooled turbine with constant specific heats (calorically perfect gas). Similarly, as shown below, these ratios can be considered constant for a cooled turbine with constant specific heats (calorically perfect gas).
144 AIRCRAFT ENGINE DESIGN
When a cooled turbine with variable specific heats is modeled, it is found that Zrtn and Ttn vary only slightly with engine operating condition.
It is also shown in Ref. 1 that 7rtL and TtL are constant in an uncooled low- pressure turbine with calorically perfect gas for a turbofan engine having choked separate (unmixed) fan and core streams. For a mixed flow turbofan, on the other hand, these ratios cannot be considered constant because ~rtL (and hence rtL) must modulate at off-design conditions to maintain P6 = P16, the Kutta condition, at the mixer entrance.
Variable specific heats. Consider a high-pressure turbine with cooling air fractions el and e2 modeled as in Fig. 5.lb. Let the nozzle throat stations just downstream of stations 4 and 4.5 be denoted by 4' and 4.5'. With choked flow at stations 4 t and 4.5', Eq. (4.24) yields, assuming Pt4 = Pt4' and Pt4.5 = Pt4.5,
l~t4' Pt4.5/Pt4 A 4 . 5 , MFP(M4,, Tt4, f ) rh4.5' T~-~-t4.5/Tt4 a4, MFP(M4.5,, Tt4.5, f4.5) or, since Y/'m2 = l, 7rJtH = Pt4.4/Pt4 and
rn4.5, rh4 then
YgtH [ 1 +
I
/4/4 @ /'hC(E'I + e2) (el + e2)
= = 1 +
/h 4 (1 - / 5 - el - e2)(1 + f )
(_~l -~_ 82 A I A4, MFP(M4,, Tt4, f ) (1 - fl - el - e2)(1 + f ) J A4.5, MFP(M4.5,, Tt4.5, f4.5)
(5.1) where the right-hand side o f the equation is essentially constant for the assumptions of this analysis, namely A n , / A 4 . 5 , = constant, M4, = 1, M4.5, = 1, and constant bleed and cooling airflow fractions. Since
f4.5 = f (4.8j)
1 + f + (el + e 2 ) / ( 1 - fl -- el -- E2)
then for a specific value of Tt4 and f , Eq. (5.1) can be used to calculate Jrtn for an assumed value of Tt4.5.
Likewise, for given values of Tt4, f, rrtn, and Orn, the resultant turbine exit temperature (Tt4.5) can be determined as follows:
1) The total enthalpy at state t4.1(ht4.1) follows from that at state t4(ht4) and the total enthalpy ratio o f coolant mixer 1 (rml).
~rn l and
ht4.1
ht4.1 = ht4~-'- = ht4"~ml nt4
where r,,i is given by
(1 - fl - el - e2)(1 + f ) + elrrrcLrcH/rX (1 - / 3 - el - e2)(1 + f ) + el
f
(5.2a)
(4.20a)
f4.1 = (4.8i)
1 + f + e l / ( 1 -- 13 - s 1 - S2)
ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 145 2) With state t4.1 known and using Eq. (4.9d), the reduced pressure at the ideal exit state t4.4i (Pr t4.4i) follows from its value at state t4.1 (Pr t41) and the turbine total pressure ratio (zrtH).
Pr t4.4i ~-- ~tHPr t4.1 (4.9d)
3) With Prt4.4i known and noting that f4.4 = f4.1, then ht4.4 i is known using the subroutine FAIR. The total enthalpy ratio of the ideal turbine (rim) follows directly using its definition
rtHi = ht4.4i/ ht4.1 (4.9d)
4) By the assumption of constant efficiency of the high-pressure turbine, Eq. (4.9d) can be solved for the high-pressure turbine enthalpy ratio (rtn)
~tH = 1 - rhn(1 - "CtHi) (5.2b) 5) With the total enthalpy ratio of coolant mixer 2 (rm2) given by
(1 - ~6 -- el - e2)(1 + f ) + el + e2{rrrcHrcL/(rZ'rmlrtit)}
rm2 = (4.20b)
(1 - / 3 - e l - e2)(1 + f ) + E1 n t- e2
and f4.5 given by Eq. (4.8j), the total enthalpy at station 4.5 (ht4.5) can be calculated using
ht4.5 = ht4 hJ 4"1 ht4"4 ht4"----~5 - ht4rml'Ctnrm2 (5.2c) t'It4 ht4.1 ht4.4
6) With ht4.5 and f4.5 known, Tt4.5 follows directly using the subroutine FAIR.
We will refer to this system of equations used to determine Tt4.5 as Eq. (5.2).
Equations (5.1) and (5.2) give us a system of equations to satisfy that determine the high-pressure turbine performance rtH and rctH. This system of equations is programmed into the cooled turbine subroutine TURBC outlined in Appendix E Subroutine TURBC solves this system by first assuming an initial value of Tt4.5 and using Eq. (5.1) to find ~rtH. Using this value of zrtH, Eq. (5.2) gives a new value of total temperature at station 4.5 called Tt4.5,. This new value is input into Eq. (5.1) and calculations are repeated until successive values are within 0.01.
Figure 5.2 shows the variation of rrtn with throttle setting (Tt4) for a typical mixed flow turbofan engine. Note that 7rtn only varies 0.9%, which shows the common assumption used in the constant specific heat analysis that zrtrl = constant is a very good approximation.
A similar, but simpler, set of equations can be solved for the uncooled low pressure turbine. Writing Eq. (5.1) for the low-pressure turbine gives
rrtL A4.5, MFP(M4.5,, Tt4.5, f4.5)
, V / ~ / T t 4 . 5 A5 MFP(M5, Tt5, f4.5)
Because the flow is assumed to be isentropic from engine station 5 to station 6 and we know the area at station 6, we use the Mach number and area at engine station 6 for the low-pressure turbine exit and write
YrtL A4.5' MFP(M4.5,, Tt4.5, f4.5)
(5.3)
~¢/~/Tt4.5 A6 MFP(M6, Tt5, f4.5)
1 4 6 A I R C R A F T E N G I N E D E S I G N
et4.5
e,4 0.34
0.33
0.32
0.31
' I I I I
Fig. 5.2 Variation of high pressure turbine pressure ratio with throttle setting (Tt4) for a typical mixed flow turbofan engine.
where A4.5,/A6 ---- constant and M4.5, = 1. Then for a specific value of Tt4.5, f4.5, and M6, Eq. (5.3) can be used to calculate :rtL for an assumed value of Tts.
Likewise, for given values of Tt4.5, f4.5, 7rtL, ~tL, and Mo, the resultant turbine exit temperature (Tts) can be determined as follows:
1) Using Eq. (4.9e), the reduced pressure at the ideal exit state t5i (Pr tsi) follows from its value at state t4.5 (Pr t4.5) and the turbine total pressure ratio (TrtL).
Pr t5i : 3TtL Pr t4.5 (4.9e)
2) With Pr t 5 i k_rlown, then ht5i is known using the subroutine FAIR. The total enthalpy ratio of the ideal turbine (~Li) follows directly using its definition:
"~tLi : ht5i / ht4.5 (4.9e)
3) By the assumption of constant efficiency of the low-pressure turbine, Eq. (4.9e) can be solved for the low-pressure turbine enthalpy ratio (rtD
rtt = 1 - tilL(1 - - "CtLi) (5.4) 4) Tt5 follows directly using the subroutine FAIR.
We will refer to this system of equations used to determine Tt5 as Eq. (5.4).
Equations (5.3) and (5.4) give us a system of equations to satisfy that determine the low-pressure turbine performance rrtL and rtL. This system of equations is programmed into the uncooled turbine subroutine TURB outlined in Appendix E Subroutine TURB solves this system by first assuming an initial value of Tts and using Eq. (5.3) to find ZrtL. Using this value of :rtt, Eq. (5.4) gives a new value of
0.30 = i i I i
2000 2200 2400 2600 2800 3000 3200
T~ 4 (°R)
ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 147 total temperature at station 5 called Tthn. This new value is input into Eq. (5.3), and calculations are repeated until successive values are within 0.01.
C o n s t a n t s p e c i f i c h e a t s . In the case of a perfect gas with constant specific heats [calorically perfect gas (CPG)], the equations that model the performance of the high-pressure and low-pressure turbines are simplified. The mass flow para- meter for a CPG is given by Eq. (1.3). Since the M F P is the same at stations 4' and 4.5' when M4, : M4.5' = 1, Y4' : Y4.5' ~--- ~/t, and R4' = R4.5' --- gt, then Eq. (5.1) becomes
7Z't/4 - - { 1 + ( 8 1 q " 8 2 ) } a A ~ 4 ' ~ (5.1-CPG) (1 -- ~ - - 7 1 --~-2)(1 + f ) za4.5'
where
A4' = constant A4.5,
and where, by the assumptions of Sec. 5.2.2, (El + E2)
1 + = constant
(1 - 13 - 81 - e 2 ) ( 1 + f )
and where, combining Eqs. (4.20a) and (4.20b),
rmlrm2 = 1 + (1 - ~ - e l - e2)(1 + f ) . ] (1 - ~ --e-i ~- e-2)(1 + f ) (5.1 b-CPG) Consider the order of magnitude of the second term in the numerator of Eq. (5. lb-CPG) compared with unity. Noting that, for the performance operation of interest, 7JtH ~ 0.85 and
r~. - - c p t T t 4 ~ 2 ---> 4 12r12cL12cH Cpc Tt3
the second term in the numerator of Eq. (5. lb-CPG) will be of the order of e or less and its variation at engine off-design can be considered small in comparison with unity. The denominator is constant, whence the product 72mllhm2 can be considered constant and Eq. (5.1-CPG) becomes
712tH
-- constant (5.1c-CPG)
when e is an order of magnitude less than unity.
For a calorically perfect gas, Eq. (5.2b) can be written as
"CtH : 1 -- t h r i l l -- 7r~H Y'-I)/y' }
and, by the assumption of constant efficiency of the high-pressure turbine, can be rewritten as
1 - - "~tn
(e,-1)/y, -- r/tH = constant (5.2b-CPG) 1 - 7rtH
148 AIRCRAFT ENGINE DESIGN
To satisfy both Eqs. (5. lc-CPG) and (5.2b-CPG), the total pressure ratio and the total temperature ratio of the high-pressure turbine must be constant, or
2"gtH ~ constant
75tH ~ constant
When the same order of magnitude analysis used with Eq. (5. lb-CPG) is applied to Eqs. (4.20a) and (4.20b) separately for rml and rm2, the coolant mixer total temperature ratios are found to be approximately constant, or
rml = constant
Z ' m 2 ~ constant
For the low-pressure turbine with a CPG and M4.5 = 1, Eq. (5.3) can be written
a s
• y~+l
{ ( 62)/"'"
~rtL A4.y 1 2 ?'t -- 1 M (5.3-CPG)
A6M y-T 1+ 2
where
A4.5, / A 6 = constant Likewise, Eq. (5.4) can be written as
(×,-b/×, / rtL = 1 -- OtL{ 1 -- ZrtL I
and, by the assumption of constant efficiency of the low-pressure turbine, can be rewritten as
1 - "(tL
(×,-1)/×, - - OtL = constant (5.4-CPG)
1 - 7 r t L
For a given value of the Mach number at station 6 (M6) and the constant values of the area ratio ( A 4 . 5 , / A 6 ) and the turbine efficiency (rhD, there is only one set of low-pressure turbine total properties (rtL, rCtL) that satisfies both Eqs. (5.3-CPG) and (5.4-CPG).
5.2.5 Component Performance Analysis
The performance of the mixed flow turbofan engine in Figs. 5.1a and 5.1b can now be analyzed using the assumptions of Sec. 5.2.2 and the techniques of Secs. 5.2.3 and 5.2.4. The goal is to obtain the 24 independent equations required to determine the dependent performance variables of Table 5.1. To underscore the logic of the iteration solution sequence that follows in Sec. 5.2.6, the required equa- tions will be developed in the order that each arises in the solution flowcharts of Figs. 5.3a and 5.3b. The equations to be developed next are represented in turn by the following 24 functional relationships between the 24 dependent performance
ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 149
A)
Calculate new value of a ' using Newtonian
iteration
Initial values of f , a ' , M6, M 8 , th o
Fig. 5.3a
Calculate
HPT ~ , , rtn, ~:ml, r.,2' T,4.5 Eq. (5-1,2) LPT ~,L, "r,L, T,s Eq. (5-3,4)
~r s Eq. (5-5a)
~ j Eq. (5-5b)
~'~L Eq. (5-6b)
:rL Eq. (5-6d)
"t'cH Eq. (5-7)
~ H Eq. (5- 8b)
f Eq. (4-15)
M~6 Eq. (5-9)
a ',e,' Eq. (5-10)
, _ a'°ew-a'l a er~o~ --
a Eq. (5-11)
' > 0.001 9
Yes Is ~ e r r o r "
~
OF l o w c h a r t of iterative solution s c h e m e (Part I).
©
v a r i a b l e s o f interest:
Zrtn = f l ( Z m l , rtn, Tin2, f ) Zml = f3(rcL, rcn, f ) ZrtC = fs(rtL, f )
Tf = f7(rtH, rtL, TcL, rcn, Ol, f , l~lo)
"rcL = f 9 ( r f )
rcH = f l l ( r t H , rcL, r.cH, Or, f , fnO) f = f13(rd,, ZcH)
Ol t = f l5( Pt6/ Ptl6, Tt6/ Ttl6, m 6 , m16) rM = f l 7 ( ' g f , "gml, ~TtH, rm2, rtL, O lt) ZrM = f l 9 ( r M , M6, M6A, or') M6 = f21(rM, JrM, M8, a')
fAB =
f23('~ml,
rtn, rm2, rtL, rM, f )rtH = f2(zrtu, f )
rm2 =
f4(rcL,
rcm rml, r~n, f ) r,/~ = f6(~r,L, f )~rl = f 8 ( r D zr~L = f l o ( r ~ D
~r~/ = flz(r~H)
M16 = f14(:rrf, ~rcL, 7rcH, Zrb, 7rtn, ZrtL, M6) a = fl6(Ot')
M6A = f l s ( M 6 , M I 6 , o~ t)
M8 = f20(zrcL, ZrcH, 7rb, zr,n, zrtL, JrM, f ) rho = fz2(ZrcL, zr~/~, a )
M9 = f24(TrcL, zr~H, Zrb, :r,m zrtL, rrM, f )
150 B)
AIRCRAFT ENGINE DESIGN
If M 6 > M 6"ew then
M 6 = M 6 - 0 . 0 0 0 1
else
M 6 =M 6 +0.002
I Yes
z" M Eq. (5-12)
M6A Eq. (5-13)
~M Eq. (5-14)
M8 Eq. (5-15)
M6,,e w Eq. (5-16)
M ... =IM6,e~.-M6[
Is M 6 ... > 0.0005 ? No
rh 0 ... Eq. (5-17) t~/0new --~/o th0error =
Is rn o ... > 0.001 ? No Remainder o f Calculations
(Appendix I)
Yes
?nO = mo new
Is an engine control limit (~., ~ Reduce Tt4 ]
/
Tt3, Pt3, etc.) exceeded? I t J
Fig. 5.3b Flowchart of iterative solution scheme (Part II).
Please notice that each equation can be solved, in principle, in the order listed for given initial estimates of the three component performance variables or', M6, and rn0 (see Sec. 5.2.6). As the solution progresses, these estimates are compared with their newly computed values and iterated if necessary until satisfactory conver- gence is obtained.
It is very important to recognize that the first six quantities of the solution se- quence (i.e., rCt~l, rtH, ~'ml, "gm2, ~tL, and rtH ) c a n be determined by the methods presented in Sec. 5.2.4. The fuel/air r a t i o s f a n d fAB are found using Eqs. (4.18) and (4.19), respectively. Additionally, the fuel/air ratios faA, f4.5, and f 6 a at sta- tions 4.1, 4.5, and 6A, respectively, are required to solve the system of equations for variable specific heats. Equations (4.8i), (4.8j), and (4.8k) give the needed relationships for f4.1, f4.5, and f6a in terms of f, faB, Or, and ft.
ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 151 Fan temperature ratio (rt) and low-pressure compressor temperature ratio (rcL). From the low-pressure spool power balance, we have
rtL = 1 - - rr{(rcC - 1) + u ( r f - 1)} + (1 + ot)CToL/rlmPL rlmnrXrta{(1 -- fl -- el -- 62)(1 + f ) + (el + F-.2/'EtH)'Er'~cLTEcH/~3.}
which can be rearranged to yield
r f = l +
(1--'CtL)rlmL{th~'4CC r:x'rtH+(61"~tH-I-62)'ccL~:cH } r r
(4.22a)
(1 + ot ) PTOL
~r OmPL rhoho {(rcL -- 1 ) / ( r f -- 1) + Or}
where, from Fig. 4.2, th4/thc = (1 - / ~ - el - 62)(1 + f )
Because the low-pressure compressor and the fan are on the same shaft, it is reasonable to assume that the ratio of the enthalpy rise across the fan to the enthalpy rise across the low-pressure compressor is constant. Using referencing, we can therefore write
htl3 --ht2 ~f -- 1 (Tf -- 1)R ht2.5 --ht2 "EcL- 1 (Z'cL -- 1)R Thus the fan enthalpy ratio can be written as
~:f = 1 +
{ r~t4 "E~"CtH _1¢_ (,Sl-Ct H ..1- e2)-CcLr;cH} (1 + or) PTOL ( 1 - - rtL)TlmL I'h C r''~ rrrlmPL fnoho
{(ZcL -- 1 ) R / ( r f -- 1)R + or}
(5.5a) and the low-pressure compressor enthalpy ratio can be written as
rcL = 1 + ( z f - 1)[(ZcL -- 1 ) g / ( r f -- 1)R] (5.5b) For a calorically perfect gas, h0 = cpcTo in Eq. (5.5a) and Eq. (5.5b) is unchanged.
Fan pressure ratio (~f) and low-pressure compressor pressure ratio
(~cL),
F r o m the definition of fan efficiency, Eq. (4.9a),
htl3i = ht2{1 q- rlf('gf -- 1)} (5.6a) Given h t l 3 i , the subroutine FAIR will give the reduced pressure Pr tl3i. Thus the fan pressure ratio is calculated using Eq. (4.9a) written as
ert 13i (5.6b)
~ f -- Pr t2
Likewise, from the definition of low-pressure compressor efficiency, Eq. (4.9b), ht2.fi = ht2{1 + 17cL(ZcL - - 1)} (5.6c)