Mattingly J.D., Heiser W.H., Pratt D.T. Aircraft Engine Design

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APPENDIX K: TURBOPROP ENGINE CYCLE ANALYSIS 593

Development of an expression for the propeller work interaction coefficient

(Cprop)

starts with a power balance on the low-pressure spool, which gives

PTOL

tn4.5(ht4.5- ht5)~TmL =

Pprop +_

Og I~mPL

Solution for the power of the propeller

(Pprop)

in terms of enthalpy and mass flow

ratios and using Eq. (K.2) yields

Cprop = 17propOg

I rlmL

(1 +

fo -- t)rX'CmlrtH72m2

(1- Z'tL)- CTOL [

(K.8)

r~meL

The uninstalled specific power of the engine

(P/rho)

is given by

-- = CTOTAL ho

(K.9)

rh0

and the uninstalled power specific fuel consumption (Sp) is given by

Se = /n__.f_f _ fo

(K.10)

P CTOTALho

The uninstalled equivalent specific thrust of the turboprop engine

(F/mo)

given by

CTOTALh 0

-- -- (K.11)

rho Vo

and the uninstalled thrust specific fuel consumption (S) is given by

S- l~lf _

foVo

(K.12)

F CrorALho

The propulsive efficiency (0e) of the turboprop engine is defined as the ratio

of the total power interaction with the vehicle producing propulsive power to the

total energy available for producing propulsive power. Thus,

rnohoCToTAL

+ (m9V4-

moVg)/Zgc

/{Cprop ~--~I (Vg~2-Mg]}

(K.13)

rip = CrOrAL -- +

(1 +

fo -- t) --

rlprop

\ ao /

The thermal efficiency (~TH) of the turboprop engine is defined as the ratio of

the total power produced by the engine to the energy made available by the fuel.

Thus,

rhoho( CToTAL + CTOL -~- CTOH)

rlT H

rh f hpR

CrorAL + CrOL + Cron

O TH =

(K.14)

foheR/ho

594 AIRCRAFT ENGINE DESIGN

All of the equations needed for parametric cycle analysis of this turboprop

engine have now been identified. The following section presents these equations

in the order of solution.

Summary of Turboprop Engine Parametric Cycle Analysis Equations

This section presents the complete parametric cycle analysis equations for the

turboprop engine with bleed, turbine cooling, and power extraction. For conve-

nience in computer programming, the required computer inputs and principal com-

puter outputs are given. These are followed by the cycle equations in their order

of solution. Please note that iteration is required to calculatef.

Inputs

Flight parameters:

Aircraft system parameters:

Design limitations:

Fuel heating value:

Component figures

of merit:

Design choices:

Mo, To, eo

t, CroL, CrOH

hpR

El, ,~2

7r b, 71"dmax , 7( n

ec, etH, etL

rib, tiroL, rlmH, OmPL, OmPH, rlprop

2Tc, 7:t, Tt4

Outputs

Overall performance:

Component behavior:

Equations

F /rho, S, P /rho, Sp, fo, Cc,

Cprop, I"]p, OTH,

V9/ao

7"gtH , 7"gtL

"re, "~tH, "(tL , "gX

Oc, rhn, rhL

M9, Pt9 / P9, P9 / Po, T9 / To

FAIR (1, 0, To, h0,

Pro,

¢0,

Cpo,

R0, g0, a0)

Vo = Moao

hto =

ho + V2

2gc

FAIR (2, 0,

Tto, hto, Prto, (Pro, cpto, Rto,

Yt0,

ato)

rr = hto/ ho

7Or = Prto/ Pro

~d=rCamox forMo< 1

ht2 = hto

Prt2 = Prto

Prt3 = D 1/ee

rrt2Jtc

APPENDIX K: TURBOPROP ENGINE CYCLE ANALYSIS 595

FAIR (3, 0,

Tt3, ht3, Prt3, (bt3, Cpt3, Rt3, Yt3, at3)

-CcL= ht3/ ht2

Prt3i = ert27t'c

FAIR

(3, 0,

Tt3i,

ht3i, Prt3i, (bt3i, Cpt3i, Rt3i, )/t3i, at3i)

ht3i - ht2

rlc-

ht3 - ht2

Set initial value of fuel/air ratio at station 4 = f4i

FAIR(I,

f4i, Tt4, ht4, Prt4, (bt4, Cpt4, Rt4, Yt4, at4)

ht4 -- ht3

f--

rlbhpR -- ht4

If [f -

f4i[

> 0.0001, then

f4i = f

and go to A; else continue.

*:k = ht4/ ho

(1 -/3

- e I -- ez)(1

f) +

el'Cr'Cc/rZ

27ml =

(1 -/3

- E 1 --

82)(1 + f) +

E 1

"t'r('f c --

CroH/~mPH

"~tH = 1 --

rlmHZ'X{(1

-- /3 -- el -- e2)(1

f) +

elrrZc/ZX}

ht4.1 = ht4"Cml

f4.1 ~"

1 + f + el/(1 -/3 - el - e2)

FAIR

(2,

f4.1,

Tt4.1, ht4.1, Prt4.1, t~t4.1,

Cpt4.1,

gt4.1, ~/t4.1, at4.1)

ht4.4 = ht4.1ztH

FAIR

(2, f4.1,

Tt4.4,

ht4.4, Prt4.4, t~t4.4,

Cpt4.4,

Rt4.4, Yt4.4, at4.4)

Prt4.4i = 7rtH Prt4.1

FAIR

(3, f4.1,

Tt4.4i, ht4.4i, Prt4.4i,

q~t4.4i,

Cpt4.4i, Rt4.4i,

Yt4.4i,

at4.4i)

ht4.1 - ht4.4

l~t H --

ht4.1 --

ht4.4i

(1 -/3 - el

- e2)(1 +

f) + el +

S2{'Cr'Cc/(r~,.75ml'CtH)}

"Cm2 ~---

(1 -/5

- E 1 --

82)(1 + f) + El -~-

ht4.5 ~" ht4.4"Cm2

f4.5 ~---

1 + f +

(E 1 "{-

e2)/(1 -/5

- e 1 -

e2)

FAIR (2, f4.5,

T,4.5,

ht4.5,

Prt4.5, ~bt4.5,

Cpt4.5, Rt4.5, Yt4.5,

at4.5)

rtL --

"~m 1 "~'tH Tm2

ht5 = ht4.5"CtL

FAIR (2, f4.5,

Tt4.5, ht4.5, Prt4.5,

~t4.5,

Opt4.5, Rt4.5,

Yt4.5, at4.5)

= ( Prt5 ~ 1/etL

\ P ,4.5 /

596 AIRCRAFT ENGINE DESIGN

ert5i = 2"(tL ert4.5

FAIR

(3,

f4.5, Ttsi,

ht5i, Prt5i,

~bt5/,

cpt5i, et5i, Yt5i, at5i)

ht4.5 - ht5

OtL--

ht4.5 - ht5i

ht9 = ht5; Tt9 =

Tt5;

Prt9 = Prt5;

f9 = f4.5

M9=l

RGCOMPR (Tt9, Mg, f9, (Tt/T)9,

(Pt/P)9, MFP9)

Pt9

-- -~- ~r 7"(d 7gc Yfb ~tH ~tL 7gn

ifPt9

(Pt)

-- > then

Po- P9

r~9

r9-- --

(rt/T)9

FAIR(l, f9, T9, h9,

Pr9, qb9, Cp9,

R9,

Y9, a9)

Po (Pt/P)9

P9 Pt9 / Po

Else

Prt 9

Pr9----

P,9 / Po

FAIR (3, f9, T9, h9,

Pr9, ~b9, Cp9,

g9, Y9, a9)

--=1

End if

V9 = ~/2gc(ht9 -

h9)

M9 = V9/a9

= f(1 - fl - el - e2)

CTOTAL = fprop + CC

-- = CrorALho

rho

Sp--

CrorzL ho

F CroraLho

r'no Vo

foVo

N-- --

C TOTAL h o

APPENDIX K: TURBOPROP ENGINE CYCLE ANALYSIS 597

S--

F/r'no

rIp=CTOTAL/ICpr°p q-~---[(l-I-

\ao

CTOTAL + CrOL + CTOH

tiT H =

fohpR/ ho

Example Design Point Results

Consider a turboprop engine to be designed for a Mach of 0.8 at a standard day

altitude of 25 kft with the inputs listed here:

= 0.90; Zrd = 0.97; T~b

0.995;

Tt4 = 3200°R;

etH

= 0.89; Zrb = 0.96;

rlmL

= 0.99;

hpR

= 18,400 Btu/lbm; etL =

0.91; Zrn = 0.99; rlmH = 0.98;

gIpro p

0.82; Og =

0.99; el = e2 = 0.05; Yc = 1.4;

Cpc

= 0.240 Btu/lbm-°R; fl = 0;

CroH = CrOL

= 0; Yt = 1.4; and

Cpt

= 0.295

Btu/lbm-°R. Figures K.2 and K.3 present the parametric performance results for

variation in the two design variables, Zrc and ft. These results were obtained using

the methods of the preceding sections for values of 5 _< Zrc _< 35 and 0.45 _< rt _<

0.75. Of the three thermodynamic models available with the ONX program, we

have chosen to use for this exercise the modified specific heat model MSH.

Figures K.2 and K.3 reveal that for any rrc there is a turbine enthalpy ratio

(rt) for which

F/rho

is maximized and S is minimized. This is referred to as the

optimum turbine enthalpy ratio.

Example Design Point ResultsmOptimum

"rt

The design point performance of a family of optimum turboprop engines was

calculated for values of 5 _< nc _< 35 using the inputs from the preceding section.

The performance results are plotted as a dashed line in Figs. K.2 and K.3 vs the

compressor pressure ratio. The optimum turbine enthalpy ratio (zt*) for each com-

pressor pressure ratio is plotted in Fig. K.4.

Performance Analysis

The performance of a selected design point turboprop engine of the type shown

in Figs. K. 1 a and K. lb is desired at off-design flight conditions and throttle settings.

In this off-design problem, there are 10 dependent and four independent variables

as shown in Table K. 1.

The assumptions of Sec. 5.2.2 apply, except those referring to the exhaust mixer

and afterburner, and the exit nozzle has a fixed area. As a result of these assump-

tions, Eqs. (5.1) and (5.2) apply to this engine. Because the high-pressure turbine

drives the compressor, the power balance of the high-pressure spool yields the

expression for calculating the total temperature ratio of the compressor (rc) at

off-design conditions. The power balance of the high-pressure spool gives

PTOH

/~/4.1(ht4.1 -

ht4.4)rlmH = r'nc(ht3 - ht2) -~ --

llmPH

598

200

180

160

F / rh o

{lbf/(lbm/s) }

140

120

100

1.3

AIRCRAFT ENGINE DESIGN

Fig. K.2

' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '

..................... _ ....

------..2----<

I I I ' I , , , , I , , , , I , , , , I , , , , l , , , ,

5 10 15 20 25 30 35

7~" c

Design point specific thrust of turboprop engines.

(],'h)

1.2 -

1.1

1.0 "

o,9

0.8

0.7

0.6

0.5 ~

Fig. K.3

Tt = 0.75

0.70

0.65

0.60

, , , , I

0.55

0.50

0.45

10 15 20 25 30

7C c

Design point specific fuel consumption of turboprop engines.

APPENDIX K: TURBOPROP ENGINE CYCLE ANALYSIS

0.65

599

0.60

0.55

0.50

0.45

0.40

5 10 15 20 25 30 35

Fig. K.4 Optimum turbine enthalpy ratio.

Table K.1 Performance variables

Component

Variable

Independent Constant or known Dependent

Engine Mo, To, P0 fl hi0

Diffuser ~rd =

f(Mo)

Compressor r/c Zrc, rc

Burner

Tt4

T( b

Coolant mixer 1 sl "gml

High-pressure turbine

~hn ~tn, rtn

Coolant mixer 2 s2, 2"t'm2 "t'm2

Low-pressure turbine ~/tL ~rtL, rtL

Nozzle ~r. M9

Propeller r/prop =

f(Mo)

Total number 4 10

600 AIRCRAFT ENGINE DESIGN

which can be written as

rc = l _q_ rimH{(l _ fl _ el _ ez)(l + f) + el}'CX rml(1-- "CtH) CTOH

(K.15)

75r OmPH

The total pressure ratio of the compressor (zrc) is determined from Eq. (4.9c)

and the ideal exit state

t3i,

ht3i

= ht2{1 +

Oc(rc -

1)} and rc =

Prt3i/Prt2

(K.16)

An expression for the mass flow rate of air through the turboprop engine (rh0)

at any conditions can be obtained by using Eq. (5.17) for the case when ot = 0 and

rrc = rrcC ZrcH,

yielding

rn0 = (1 -- fl

- el --

e2)(1 +

f)Po rCr red rCc ZrbA4MFPn/~/-T~t4

(K.17)

The power produced by the low-pressure turbine at off-design is determined by

its total pressure ratio

(rrtr).

The total enthalpy ratio of the low-pressure turbine (rtL)

is related to its dependent variables by the uncooled turbine subroutine (TURB)

of Chapter 5 and Appendix F written as

TURB

[(Tt4.5,

f4.5,

A4.5/(A9Yrn),

M4.5, M8,

ritL, Tt5i, 7E'tL, rtL,

Tts)]

(K.18)

This subroutine includes the conservation of mass between stations 4.5 and 9,

constant turbine efficiency, and the variation of gas properties with temperature

and fuel-to-air ratio. For a calorically perfect gas, this relationship becomes

rtc = 1 - rill { 1

- (TgtL) (Yt-1/Y') }

(K.18-CPG)

The work interaction coefficient for the propeller

(Cprop)

is obtained by a power

balance of the low-pressure spool. For a constant value of

PrOL,

Cprop

is given by

(CToL)R /J'tOR hoR

-- (K.19)

Cpr°p=ripr°prig{rimL(l"~-f°--t~)7~)~TmlTtHlSm2(1--'gtL)}-- rimPL r'no

Determination of the value of

Cprop

thus depends mainly on Tta(rz) and rtL

because

riprop, Og,

and

rime, rml, 7:tH,

and rm2 are constant or essentially constant

in Eq. (K.19). Equation (K.18-CPG) shows the direct dependence of

rtL

ZrtL

for a constant

ritL.

The off-design

~rtL

depends on the total pressure ratio imposed

across the low-pressure turbine. Note that the flow is choked at the entrance to

the low-pressure turbine (station 4.5) and is normally unchoked at the engine exit

(station 9).

The total pressure at station 4.5 is related to the flight condition and throttle

setting by

Pt4.5 =

Nr Ygd 7[c 7gb TEtH

The total pressure at station 5 is related to the nozzle operation by

APPENDIX K: TURBOPROP ENGINE CYCLE ANALYSIS 601

and thus

and

et91Po

TEtL = (K.20)

TEr TEd TEc TEb TEtH TEn

The pressure ratio

Po/P9

is determined by the nozzle operation with

Pt9

(Pt) andPOPt9/P9

--~-9 = T M=I

P9 P,9/""'~O

for choked flow

P0 1 and Pt9

(Pt)

forunchokedflow

V99 = -if9 = 7M=,

Equating the mass flow rate of air at stations 4.5 and 9 yields the following

equation for the total pressure ratio of the low-pressure turbine in terms of the exit

Mach number and the total temperature ratio of the low-pressure turbine:

TEtL

A4.5

MFP(M4.5,

Tt4.5, f4.5)

-- -- (K.21)

~/Tt5/Tt4.5 TEnA9 MFP(M9, Tt5,

f4.5)

where M9 is determined by the conditions at the exit. Thus, M9 = 1 for choked

flow, and M9 is given by the compressible flow function RGCOMP for

Pt/P, Tt,

and f at station 9.

Determination of the conditions downstream of station 4.5 requires an iterative

solution of Eqs. (K. 18), (K.20), and (K.21). The method used is as follows:

1) Initially assume =

TEtL = TEtLe.

2) Calculate rtL using Eq. (K.18).

3) Calculate

Pt9/Po

and conditions at exit including M9.

4) Calculate new TEt~, using Eq. (K.21).

5) Compare new TEtL with earlier value. If the difference is greater than 0.0001,

then go to step 2 using the new zrtL.

The performance of the propeller can be simply modeled as a function of the

flight Mach number (see Appendix L); one such model, which is used in the

AEDsys computer program of this textbook, is shown in Fig. K.5 and expressed

algebraically as

l~pro p = lOMorlpropmax

17prop ~ 17propmax

(1 0 07)

~-- 0p rop max

3/0<0.1

0.1 < M0 _< 0.7

0.7 < M0 < 0.85

The equation for the Mach range of 0.7-0.85, just given, models the drop in

TIpro p

experienced in this flight regime as a result of transonic flow losses in the tip region

of the propeller.

The following section summarizes the performance equations in the order of

calculation.

602 AIRCRAFT ENGINE DESIGN

1 I I I I I I I I I

~Tpropmax

~T prop

0.5

0 0.2 0.4 0.6 0.8 1

Fig. K.5 Propeller efficiency vs Mo.

of Turboprop Engine Performance Cycle Analysis Summary

Equations

This section presents the complete performance cycle analysis equations for

the turboprop engine with bleed, turbine cooling, and power extraction. For con-

venience in computer programming, the required computer inputs and principal

computer outputs are given. These are followed by the cycle equations in their

order of solution. Note that iteration is required for solution.

Inputs

Performance choices:

Flight parameters:

Throttle setting:

Design constants:

~/:

Others:

Reference condition:

Flight parameters:

Component behavior:

Others:

Engine control limits:

Mo, To, Po

7rd max, Yrb, Yfn

~c, TItH, 17tL, Ob, OmL, T]mH, OmPL, OmPH, Opropmax

A4, A4.5,

/3, el, e2,

hpic, ProL, ProH

moR, Tog, PoR

7"(cR, 7ftHR, ~tLR

"~cR, 75tHR, 75tLR

Tt4R, frolic,

rmZic,

fic, Moic, Msic, CvoI~ic, CTOI-IR

FIC, ~:noic, Sic

7t'cmax, Tt3max, lPt3max,

%NL, %NH