Problem AB.1.1 A landing aircraft rolling down the runway at u = 50 m/s has an idling turbofanthat consumes air at a rate ma = maC + maH = 100 kg/s and produces exhaust streams withvelocities (relative to the aircraft) ueC=100 m/s and ueH=150 m/s. For the analysis, consider thatthe bypass ratio of the turbofan is B = maC/maH = 6 and that the pressure at the exit equals theambient value.

a) What is the forward thrust of the turbofan during landing? Simplify the expression forf = mf/maH ≪ 1.

b) What is the magnitude and direction (i.e., forward or reverse) of the thrust if the bypassexhaust is deflected 90o without affecting either the air mass flows or the magnitudes of theexhaust velocities?

c) Compute the additional deflection α needed to get zero net thrust once the aircraft has cometo a stop. For the new flow conditions, assume that the bypass ratio B = 6 and the velocityratio u∗eC/u

∗

eH = 2/3 remain equal to those found while the aircraft is moving.

b)

Cma.

ma.

H

ueC

Cma.

ma.

H

ueC

ueC

ueH

Cma.

ma.

H

ueC

ueC

ueC

ueH

u u

ueH

α*

*

*

*

*

c)a)

Problem AB.1.2 Using the expression

T = maH(1 + f)ueH + maCueC − mau

for the thrust of a turbofan with ambient pressure at the exit, show that the associated thrustspecific fuel consumption is

TSFC =mf

T=

f

(1 + f)ueH + BueC − (1 + B)u,

where f = mf/maH , while the overall efficiency becomes

ηo =Tu

mfQR=

(1 + f)ueH + BueC − (1 + B)ufQR

u.

Introduce the thrust-averaged exhaust velocity

ue =maH(1 + f)ueH + maCueC

maH(1 + f) + maC

to derive the approximate result

ηo ≃ (1 + B)(ue − u)u

fQR

after using the condition f ≪ 1. For a given bypass ratio B and a given exhaust velocity ue showthat the overall efficiency is maximized for u = ue/2.

Problem AB.1.3 Use Brequet’s range equation

s =ηoQR(L/D)

gln

(

minit

mfinal

)

to show thatmpayload

mo= exp

(

− gsmax

ηoQR(L/D)

)

− mempty

mo,

where mo is the maximum aircraft mass at takeoff. Use the above equation to represent thepayload vs range diagram (mpayload/mo− smax with smax in km) for an aircraft with L/D =15 andmempty/mo = 0.7 whose jet engine has an overall efficiency ηo = 0.6. Use the value QR = 45, 000kJ/kg, typical of hydrocarbon fuels, in the evaluation.

Problem AB.1.4 A turbojet is flying at M = 1.3 at an altitude where Ta = 200 K. At the exhaustpe = pa, Me = 1.6, and Te = 1100 K. The fuel-to-air ratio is f = 0.06 and the heat of reactionis QR = 45, 000 kJ/kg. Assuming that the gas behaves as air at normal ambient conditions (i.e.,γ = 1.4 and Rg = 287 J/(kg K)), compute:

• The specific thrust T /ma. T /ma ≃ 0.759 (kN · s)/kg

• The TSFC. TSFC≃ 0.079 kg/(kN · s)

• The propulsion efficiency. ηp ≃ 0.526

• The thermal efficiency. ηth ≃ 0.1979

• The overall efficiency. ηo ≃ 0.1036

Problem AB.2.1 Consider the performance of a Ramjet. Using

Tma

= M√

γRTa

[

(1 + f)√

T04/Ta

(

1 +γ − 1

2M2

)

−1/2

− 1

]

together with

f =T04/Ta − T02/Ta

QR/(cpTa)− T04/Ta

and the definition ηo = T u/(QRmf ), derive explicit expressions for the thrust per unit mass flowof air T /ma, thrust specific fuel consumption TSFC= mf/T , and overall efficiency ηo of an idealramjet in terms of aa =

√γRTa, QR/(cpTa), T04/Ta, and the flight Mach number M , verifying that

the result can be written in the form

Tma

=M

√γRTa

QR/(cpTa)− T04/Ta

[

√

T04

Ta−

(

1 +γ − 1

2M2

)1/2]

√

T04

Ta+

QR/(cpTa)(

1 + γ−12 M2

)1/2

,

TSFC =

[

√

T04Ta

+(

1 + γ−12 M2

)1/2]

M√γRTa

[

√

T04Ta

+(

QR

cpTa

)

/(

1 + γ−12 M2

)1/2]

and

ηo =

(γ − 1)M2

[

√

T04Ta

+(

QR

cpTa

)

/(

1 + γ−12 M2

)1/2]

[QR/(cpTa)]

[

√

T04Ta

+(

1 + γ−12 M2

)1/2]

Use these results to determine an explicit expression for the value of M at which the thrust vanishesas well as the corresponding values of TSFC and ηo. For the case aa = 300 m/s, QR/(cpTa) = 200,and T04/Ta = 12, obtain numerically the value of M at which the T /ma reaches its peak value.Obtain also this peak value, giving the result in (kN s)/kg, as well as the accompanying values of ηoand TSFC, with the latter expressed in kg/(kN s). T /ma ≃ 1.07 (kN · s)/kg, ηo ≃ 0.3564, TSFC≃ 0.048 kg/(kN · s)

Problem AB.2.2 Ramjet diffusers typically employ pointy center bodies to slow down the incom-ing flow through a series of oblique shocks. As an alternative, investigate the use of a blunt centerbody, as shown in the figure. Assuming that the bow shock that emerges can be approximated as anormal shock wave, obtain the jump in stagnation pressure rd = p02/p0a as a function of the flightMach number M . Compare the solution with the expression rd = 1− 0.1(M − 1)3/2 that describesthe performance of typical ramjet diffusers (according to the Aircraft Industries Association). Doesthe blunt-body alternative appear to be an advantageous configuration?

Problem AB.2.3 Plot the variation with flight Mach number of the propulsion and thermalefficiencies of a Ramjet operating at high altitude (Ta = 200 K). The nozzle inlet temperature isT04 = 3000 K. In the evaluation assume that the Ramjet is ideal and that QR = 45 × 106 J/kg,γ = 1.3, and cp = 1000 J/(kg K).

Problem AB.2.4 The specific thrust of a Ramjet with pe = pa is given by

Tma

=

[

(1 + f)Me

(

T04

Ta

)1/2 (

1 +γ − 1

2M2

e

)

−1/2

−M

]

aa,

in terms of the flight Mach number M , ambient sound speed aa, peak-to-ambient temperature ratioT04/Ta, exhaust Mach number Me, and fuel-air ratio f . The latter can be evaluated from

f =(T04/Ta)− (1 + γ−1

2 M2)

ηb[QR/(cpTa)]− (T04/Ta),

as follows from the energy balance across the combustor, whereas the exhaust Mach number Me

corresponding to a given value of M depends on the losses of stagnation pressure in the diffuser,combustor, and nozzle.

• Derive an equation for Me for known values of the pressure ratio across the combustor rc =p04/p02 and of the adiabatic efficiencies of the diffuser and nozzle ηd and ηn.

• Determine the specific thrust of a Ramjet with M = 4, T04/Ta = 10, aa = 300 m/s, ηb = 1,QR/(cpTa) = 220, rc = 0.95, ηd = 0.8, and ηn = 0.95. In the calculations use γ = 1.2 for thespecific-heat ratio. T /ma ≃ 1.06 (kN · s)/kg

• Obtain the corresponding values of TSFC, ηp, ηth, and ηo. TSFC ≃ 0.0332 kg/(kN · s),ηp ≃ 0.728, ηth ≃ 0.501, and ηo ≃ 0.365

a

a

apue

1 2 3 5 6

DIFFUSER

SUBSONICDIFFUSER

SUPERSONIC FUELINJECTORS

FLAMEHOLDERS

4

p

Problem AB.2.5 The figure below represents schematically a ramjet flying at Mach numberM = 3. In the following analysis, assume that:

(i) The compression through the diffuser can be represented by a normal shock.

(ii) The combustion increases the stagnation temperature by a factor T03/T02 = 4, while keepingthe stagnation pressure constant (i.e., p03 = p02).

(iii) the fuel-to-air ratio f = mf/ma satisfies f ≪ 1.

(iv) the nozzle is ideal and has a throat-to-exit area ratio such that the gas stream is perfectlyexpanded (i.e. p4 = pa).

1. Calculate the Mach number M2 behind the shock. M2 ≃ 0.475

2. Obtain the stagnation pressure p02 behind the shock, giving the result in the form p02/pa.p02/pa ≃ 12.056

3. Compute the Mach number at the nozzle exit Me = M4. Me ≃ 2.2766

4. Compute the thrust, given the result in the form T /(maaa), where aa is the ambient value ofthe sound speed. T /(maaa) ≃ 2.3387

5. Determine the propulsion efficiency ηp. ηp ≃ 0.7195

4321Q.

pa

pa

M Me

m f.

Problem AB.2.6 In supersonic flight, most of the irreversibility associated with the flow throughthe diffuser comes from the presence of shock waves, while the effect of viscous dissipation in thediffuser-wall boundary layers is of lesser importance. Assuming that a single normal shock isstanding in front of the inlet and that the effect of the boundary layers is entirely negligible, obtainan explicit analytic expression for the adiabatic efficiency of the diffuser ηd in terms of the flightMach number M > 1 and the specific heat ratio γ.

Problem AB.2.7 The SCRAMJET (supersonic combustion RAMJET) has been considered asa viable propulsion device for future hypersonic planes. It includes an aerodynamic intake thatcompresses the flow through a series of oblique shocks. The stream, still supersonic, enters thena combustion chamber where chemical heat release takes places. Finally, isentropic accelerationproduces a high velocity jet, as needed for propulsion. A simple schematic representation is shownin the figure.

• For a flight Mach number M = 4, determine the Mach number M2 and the pressure p2/paat the burner entrance. Assume that the compression is equivalent to that provided by twooblique shocks with deflection δ = 18o. M2 ≃ 1.99, p2/pa ≃ 13.9

• Assuming the conditions at the burner exit to be sonic (M4 = 1), determine the pressure p4,giving the result in the form p4/pa. Use continuity to obtain T4/Ta (in the calculation, use theapproximation f ≪ 1). Compute the fuel-to-air ratio f for QR/(cpTa) = 200. p4/pa ≃ 37.95,T4/Ta ≃ 4.40

• If the burner cross section has a surface area Ab = 0.23A, where A is the exit area, obtain theMach numberMe and the relative pressure pe/pa downstream from the expansion. Me ≃ 3.02,pe/pa ≃ 1.90

• Calculate the thrust, giving the result in the dimensionless form T/(maaa).

• The gas expands downstream from the SCRAMJET exit to reach the external ambient pres-sure pa. Obtain the deflection angle θ for the resulting Prandtl-Meyer expansion. θ ≃ 7.1o

M M4

δ=18p

A

A

b

θ

M M1

2

e

Me

FUEL

pa

aa

Problem AB.2.8 One concept for hypersonic air-breathing propulsion is a detonative-combustionRAMJET, in which fuel added to the air burns in a detonation wave. As a simplified model foranalysis of such a device, assume that the standing detonation can be modeled as a normal shockwave followed by a region of constant-area heat addition that leaves the flow in sonic conditions(M4 = 1) and that the downstream divergent nozzle expands the stream to ambient pressure(pe = pa). In your calculations use γ = 1.4 for the specific heat ratio.

1. Determine the Mach number behind the shock wave M2 for M = 5. M2 ≃ 0.415

2. Obtain the associated pressure and temperature ratios p2/pa and T2/Ta. p2/pa ≃ 29, T2/Ta ≃5.8

3. Compute the pressure after combustion is completed, giving the result in the form p4/pa.p4/pa ≃ 15

4. Calculate the associated post-combustion temperature, giving the result in the form T4/Ta.T4/Ta ≃ 9.0

5. Compute the Mach number at the nozzle exit Me. Me ≃ 2.83

6. Find the exit-to-entrance temperature ratio Te/Ta. Te/Ta ≃ 4.15

7. Determine the exhaust speed ue, giving the result in the form ue/aa. ue/aa ≃ 5.76

8. Calculate the needed exit-to-combustor area ratio Ae/Ac. Ae/Ac ≃ 3.6

9. Obtain the specific thrust for this engine T /m = ue−u, giving the result in the form T /(maa),where aa is the ambient sound speed. T /(maa) ≃ 0.76

10. Find the propulsion efficiency ηp ≃ 0.929

ηp =T u

(m/2)(u2e − u2)

FUEL

MMMMp p p p2

2

a a

e

4

4Q

Problem AB.2.9 An ideal ramjet is flying at M = 3 at an altitude where Ta = 200 K. Themaximum temperature is T04 = 2, 600 K and the heat of reaction is QR = 45, 000 kJ/kg. Assumingthat the gas behaves as air at normal ambient conditions (i.e., γ = 1.4 and Rg ≃ 287 J/(kg K) withcp = Rgγ/(γ − 1)) and that the burning efficiency is ηb = 1, compute:

1. The fuel-to-air ratio f . f = 0.0483

2. The exit speed ue. ue = 1, 832.5 m/s

3. The specific thrust T /ma in kN s/kg. T /ma = 1.07 kN s/kg

4. The TSFC in kg/(kN s). TSFC=0.0452 kN s/kg

5. The overall efficiency ηo. ηo = 0.4185

Problem AB.3.1 Determine the compressor pressure ratio that maximizes the specific thrust ofan ideal turbojet for given values of the turbine inlet-to-ambient temperature ratio T04/Ta and theflight Mach number M . Use the approximation f ≪ 1 in the derivation. Obtain also an expressionfor the value of M above which a compressor is no longer needed to provide optimum specificthrust. Calculate this limiting value of M for the case T04 = 1700 K, Ta = 220 K, and γ =1.4.M ≃ 3.09

Problem AB.3.2 The Concorde Rolls-Royce Olympus 593 engines were designed to cruise atM = 2 at an altitude of 53,000 feet (Ta = 216 K). At the time, the maximum turbine inlettemperature was limited to T04 = 1350 K. The engine employed a compressor with pressure ratioprc = 11.3. In trying to understand the selection of the latter value it is of interest to determinethe variation of T /ma, TSFC, and ηo with prc for the Olympus 593 engine, assuming the followingvalues for the efficiencies and combustor pressure ratio: ηn = 0.97, ηd = 0.94, ηc = 0.80, ηt = 0.88,ηb = 1.0, and rc = 1.0. In the calculation, use γ = 1.4, R = 287 J/(kg K), and QR = 45×106 J/kg.

Problem AB.3.3 Two options are being considered for the propulsion device of a long-rangemissile designed to fly at a cruise Mach number M = 2.0 at an altitude where Ta = 220 K:

• A Ramjet with peak temperature T04 = 2700 K

• A Turbojet with compressor pressure ratio prc = 30 and peak temperature T04 = 1800 K

The nozzle in both designs will have the exit area required to match the ambient pressure at theexit section (pe = pa). Because weight is a key issue, the preferred option is the one that providesa higher specific thrust T /ma. In the calculation, assume that the two engines behave as ideal andthat the gas is a perfect gas with γ = 1.4 and R = 287 J/(kg K). In both cases, use QR = 45× 106

J/kg for the fuel heat of reaction.

1. Determine the values T /ma for both engines. Based on the result, select the propulsion devicefor the application at hand.

2. Obtain in both cases the associated thrust specific fuel consumption TSFC.

Problem AB.3.4 Two options are being considered for the propulsion device of a long-rangemissile designed to fly at a cruise Mach number M = 2.0 at an altitude where Ta = 220 K:

• A Ramjet with peak temperature T04 = 2900 K

• A Turbojet with compressor pressure ratio prc = 20 and peak temperature T04 = 1800 K

The nozzle in both designs will have the exit area required to match the ambient pressure at theexit section (pe = pa). Because weight is a key issue, the preferred option is the one that providesa higher specific thrust T /ma. In the calculation, assume that the two engines behave as ideal andthat the gas is a perfect gas with γ = 1.4 and R = 287 J/(kg K). In both cases, use QR = 45× 106

J/kg for the fuel heat of reaction.

1. Determine the values T /ma for both engines. Based on the result, select the propulsion devicefor the application at hand. T /ma ≃ (1.11, 0.73) (kN · s)/kg

2. Obtain in both cases the associated thrust specific fuel consumption TSFC. TSFC≃ (0.054, 0.027)kg/(kN · s)

Problem AB.3.5 Two options are being considered for the propulsion device of an experimentalhigh-speed aircraft designed to fly at a cruise Mach number M = 2.0 at an altitude where Ta = 200K

• A Ramjet.

• A Turbojet with compressor pressure ratio Prc = 20.

The nozzle in both designs will have the exit area required to match the ambient pressure at theexit section (pe = pa). Because weight is a key issue, the option preferred is the one that providesa higher specific thrust T /ma. In the calculation of the thrust use the approximation f ≪ 1 andassume that the two engines behave as ideal and that the gas properties are γ = 1.4 and R = 287J/(kg K).

1. Determine the values of T /ma for both engines using T04 = 1800 K for the peak temperature.Based on the result, select the propulsion device for the application at hand. T /ma ≃(0.7, 0.768) (kN · s)/kg

2. The maximum temperature in ramjets can be significantly higher (why?). Redo the compu-tation of the specific thrust of the ramjet with T04 = 2500 K and comment on the result.T /ma ≃ 0.927 (kN · s)/kg

Problem AB.3.6 The Concorde range was only sCON =4,000 miles, barely enough to cross theAtlantic (3,600 miles) but not enough for trans-Pacific operation, which is of primary interestnowadays. You are the chief engineer for the design of a new supersonic passenger aircraft, based onthe Concorde, which must be able to operate the San Francisco-Tokyo route (5,200 miles). The firstquestion you need to answer is whether the technical improvements in turbomachinery efficiencyand turbine-blade materials that have occurred over the last forty years are by themselves sufficientto increase the range by the necessary amount, maintaining the flight conditions of the original plane(M = 2 and Ta = 216 K). Note that, if one assumes that the aerodynamic characteristics and themass distribution of the new aircraft are identical to those of the Concorde (i.e., same values ofL/D and mfuel/(mpayload +mstruc)) then the new range will be given simply by

sNEW = (ηoNEW/ηoCON

)sCON

in terms of the overall engine efficiencies of the original Concorde Rolls-Royce Olympus 593 engine,ηoCON

, and of the new turbojet, ηoNEW. To compute these two overall efficiencies, use γ = 1.4,

R = 287 J/(kg K), and QR = 45×106 J/kg and assume that the burning efficiency is unity (ηb = 1)and that no pressure loss occurs in the combustor (rc = p04/p03 = 1). For the Olympus 593, thepeak turbine-inlet temperature was T04 =1350 K, whereas the combination of new metal alloys andadvanced blade cooling techniques enables the temperature of the new design to be increased toT04 =1750 K. Besides, the adiabatic efficiencies of the different engine elements have increased overtime from the values ηd = 0.94, ηc = 0.80, ηt = 0.88 and ηn = 0.97 of the Olympus 593 engineto the new high-performance values ηd = 0.97, ηc = 0.88, ηt = 0.93 and ηn = 0.99. To minimizefuel consumption, in the new design the compressor pressure ratio, which was prc = 11.3 for theConcorde, should be increased to prc = 40. Calculate the range of the new supersonic aircraft.Would it be able to fly nonstop the San Francisco-Tokyo route? sNEW ≃ 4, 724 miles

Problem AB.3.7 Consider the turbojet with afterburner shown in the figure below. Under normaloperation the afterburner is off and the specific thrust and TSFC are given by

Tma

= [(1 + f)(ue/aa)−M ] aa

and

TSFC =mf

T =f

T /ma,

where f = mf/ma. The afterburner is used to boost the thrust by burning an additional fuelmass-flow rate m∗

f downstream from the turbine, resulting in an increased exhaust speed u∗e. Theassociated specific thrust and TSFC are given by

Tma

= [(1 + f + f∗)(u∗e/aa)−M ] aa

and

TSFC =mf + m∗

f

T =f + f∗

T /ma,

where f∗ = m∗

f/ma.To investigate the effect of the afterburner, consider an ideal turbojet withM = 1.5, ηbQR/(cpTa) =

200, aa =300 m/s, Prc = 20, T04/Ta = 7, and γ = 1.3. Assume that the pressure losses in the maincombustion chamber and in the afterburner are negligible and that the nozzle area is variable, sothat the condition pe = pa is always satisfied.

• Determine the values of f and T05/T04 . f ≃ 0.0224, T05/T04 ≃ 0.81

• If the afterburner is off, obtain the values of ue/aa along with the corresponding values ofT /ma and TSFC. ue/aa ≃ 4.5, T /ma ≃ 0.930 (kN · s)/kg, TSFC≃ 0.024 kg/(kN · s)

• When the afterburner is in operation, apply conservation of energy to the associated combus-tion process to show that

f∗ = (1 + f)

(

T ∗

05Ta

)

−(

T05T04

)(

T04Ta

)

(

ηbQR

cpTa

)

−(

T ∗

05Ta

)

in terms of the peak downstream temperature T ∗

05/Ta.

• Obtain f∗ for T ∗

05/Ta = 10. f∗ ≃ 0.0233

• Analyze the isentropic flow in the nozzle to show that

γ − 1

2

(

u∗eaa

)2

=T ∗

05

Ta

1− 1T05T04

P(γ−1)/γrc (1 + γ−1

2 M2)

.

• Obtain the value of u∗e/aa along with the corresponding values of T /ma and TSFC. u∗e/aa ≃5.987, T /ma ≃ 1.428 (kN · s)/kg, TSFC≃ 0.032 kg/(kN · s)

*

am. a

m. f

5*

m. f

3 4 5

ue

a2 6

up

Problem AB.3.8 A turbojet engine has a compressor pressure ratio Prc = 30 and a maximumtemperature at the end of the combustor T04 = 1700 K. The adiabatic efficiencies of the diffuser,compressor, turbine, and nozzle are ηd = 0.9, ηc = 0.9, ηt = 0.8, and ηn = 0.9, respectively. Thepressure loss in the combustor is negligible and the combustion efficiency if ηb = 1. For a flightMach number M = 0.8 and ambient temperature Ta = 205 K, determine the specific thrust T /m,TSFC, and overall efficiency of the engine.

Problem AB.3.9 Show that for an ideal turbojet (ηd = ηc = rc = ηb = ηt = ηn = 1) at takeoff(M = 0) the fuel-to-air ratio f and the exhaust jet speed ue are given by

f =T04/Ta − P

(γ−1)/γrc

QR/(cpTa)− T04/Ta

and(

γ − 1

2

)(

ueaa

)2

=(P

(γ−1)/γrc − 1)(T04/Ta − P

(γ−1)/γrc )

P(γ−1)/γrc

,

respectively. Use this last equation to show that, in the limit f ≪ 1, the specific thrust at takeoffcan be expressed in the form

Tma

=

(

γ − 1

2

)

−1/2[

(P(γ−1)/γrc − 1)(T04/Ta − P

(γ−1)/γrc )

P(γ−1)/γrc

]1/2

aa.

Show that the specific thrust takes a maximum value

Tma

=

(

γ − 1

2

)

−1/2[

(

T04

Ta

)γ

2(γ−1)

− 1

]

aa

for a compressor pressure ratio Prc = (T04/Ta)γ

2(γ−1) .

Problem AB.3.10 A turbojet with compressor pressure ratio Prc = 12 is designed to fly atM = 1.5 at an altitude where the ambient temperature is Ta = 220 K. The maximum temperatureallowed at the entrance of the turbine is T04 = 1, 800 K. In the calculations use γ = 1.4, cp = 1, 004J/(kg K), Rg = 287 J/(kg K), and QR = 45× 106 J/kg for the properties of the gas and ηd = 0.97,ηc = 0.85, ηt = 0.92, and ηn = 0.98 for the adiabatic efficiencies of the diffuser, compressor, turbine,and nozzle, respectively.

1. Compute the ambient sound speed aa. aa ≃ 297.3 m/s

2. Calculate the fuel-to-air ratio f . In the calculation, assume that the burner efficiency isηb = 1.f ≃ 0.0254

3. Find the temperature jump across the turbine T05/T04 . T05/T04 ≃ 0.784

4. Determine the pressure jump across the turbine p05/p04 . p05/p04 ≃ 0.392

5. Assuming that the pressure drop across the combustor is negligible (i.e. rc = 1), computethe exhaust speed ue. ue ≃ 1, 239 m/s

6. Obtain the specific thrust T /ma, giving the result in (kN · s)/kg. T /ma ≃ 0.824 (kN · s)/kg

7. Compute the TSFC, giving the result in kg/(kN · s). TSFC≃ 0.0308 kg/(kN · s)

8. Determine the propulsion efficiency ηp. ηp ≃ 0.534

9. Calculate the thermal efficiency ηth. ηth ≃ 0.602

10. Obtain the overall efficiency ηo. ηo ≃ 0.321

ua

3 4 5

ue

a2 6

p

Problem AB.3.11 Consider a turbojet with compressor pressure ratio Prc = 20, nozzle exit areaAe = 0.5 m2, peak temperature T04 = 1800 K, and adiabatic efficiencies (diffuser, compressor,turbine, and nozzle) ηd = 0.85, ηc = 0.8, ηt = 0.9, and ηn = 1 (i.e., the nozzle is assumed to beideal). Follow the steps listed below to calculate the air mass flux

ma =ρeueAe

1 + f=

γ

1 + f

paAe

aaMe

(

pepa

)(

Te

Ta

)

−1/2

and the thrustT = maaa[(1 + f)ue/aa −M ] +Ae(pe − pa)

both at cruise conditions (36,000 feet) and at takeoff. In the calculations, assume that the effectiveheat of reaction is ηbQR = 45× 106 J/kg, that the pressure loss in the combustor is negligible (i.e.rc = p04/p03 = 1), and that γ = 1.4 and Rg = 287 J/(kg K), corresponding to cp = 1, 004.5 J/(kgK).

During flight at cruise conditions (M = 1.5, pa = 22, 600 Pa, and Ta = 217 K):

1. Determine the fuel-to-air mass ratio f . f ≃ 0.0222

2. Obtain the temperature jump across the turbine T05/T04 . T05/T04 ≃ 0.704

3. Assuming that the nozzle is dimensioned to give pe = pa at cruise, calculate the exit Machnumber Me, along with the exit temperature Te/Ta and the exit velocity ue/aa. Me ≃ 2.43,Te/Ta ≃ 2.68, ue/aa ≃ 3.98

4. Compute the nozzle throat-to-exit area ratio At/Ae. At/Ae ≃ 0.405

5. Determine the air mass flux ma (kg/s) and the thrust T (kN). ma ≃ 77.8 kg/s, T ≃ 59 kN

At takeoff (M = 0, pa = 101, 300 Pa, and Ta = 288 K):

6. Determine the fuel-to-air mass ratio f . f ≃ 0.0238

7. Obtain the temperature jump across the turbine T05/T04 and the pressure behind the turbine,giving the latter in the form p05/pa. T05/T04 ≃ 0.729, p05/pa ≃ 5.70

8. The nozzle has fixed geometry (i.e., the value of At/Ae is that computed above). Assumingthat the nozzle remains choked at takeoff, with an oblique shock forming right outside thenozzle exit, obtain the values of the Mach number Me, temperature Te/Ta, velocity ue/aa,and pressure pe/pa at the nozzle exit section. Me ≃ 2.42, Te/Ta ≃ 2.09, ue/aa ≃ 3.51,pe/pa ≃ 0.372

9. Obtain the deflection of the jet stream across the oblique shock near the nozzle rim δ, as wellas the corresponding post-shock Mach number Md. δ ≃ 17.5o, Md ≃ 1.75

10. Compute the takeoff values of the air mass flux ma (kg/s) and thrust T (kN). ma ≃ 127.17kg/s, T ≃ 123.8 kN

u

3 4 5

uea

2 6

pe

Problem AB.3.12 The performance of a jet engine is limited by the capability of turbine-bladematerials to withstand high temperatures. To investigate how a higher value of the postcombustiontemperature can significantly improve the performance, consider the case of a turbojet flying atM = 1.5. The compressor pressure ratio is Prc = 40 and the adiabatic efficiencies (diffuser,compressor, turbine, and nozzle) are ηd = 0.85, ηc = 0.8, ηt = 0.9, and ηn = 0.95. In thecalculations, assume that the nozzle is dimensioned to give pe = pa and that the pressure loss inthe combustor is negligible (i.e. rc = p04/p03 = 1). Evaluate the results using (ηbQR)/(cpTa) = 220,γ = 1.4, and aa = 300 m/s.

u

3 4 5

uea

2 6

pe

Consider first a turbojet designed to meet a peak temperature T04/Ta = 8.

1. Determine the fuel-to-air mass ratio f .

2. Obtain the temperature jump across the turbine T05/T04 .

3. Calculate the exit velocity ue.

4. Determine the specific thrust T /ma (in (kN · s)/kg) as well as the TSFC (in kg/(kN · s)).

5. Compute the propulsion efficiency ηp.

Consider now a turbojet with no limitation of peak temperature, so that one may use stoichiometricconditions in the combustor (i.e. f = 0.067).

6. Determine the peak temperature T04/Ta.

7. Obtain the temperature jump across the turbine T05/T04 .

8. Calculate the exit velocity ue.

9. Determine the specific thrust T /ma (in (kN · s)/kg) as well as the TSFC (in kg/(kN · s)).

10. Compute the propulsion efficiency ηp. ηp.

Comment on the results.

Problem AB.4.1 The figure below represents a schematic view of an ideal turbofan with com-pressor pressure ratio Prc = 15 and fan pressure ratio Prf = 2 designed to fly at M = 0.8. In thecalculations, assume pe = pa and use aa = 300 m/s for the ambient sound speed, γ = 1.4 for the ra-tio of specific heats, T04/Ta = 8 for the peak temperature at the turbine inlet, and QR/(cpTa) = 200for the heat of reaction.

1. Determine the fuel-to-air ratio f . f ≃ 0.02893

2. Obtain the exhaust speed of the bypass stream uef , expressing the result in the form uef /aa.uef /aa ≃ 1.369

3. Compute the temperature ratio across the turbine T05/T04 and the exhaust speed of the hotstream ue/aa for B = (4, 8, 12). T05/T04 ≃ (0.712, 0.588, 0.465), ue/aa ≃ (3.481, 2.67, 1.497)

4. For those three values of the bypass ratio calculate the specific thrust T /ma and the TSFC.T /ma ≃ (1.517, 1.95, 2.27) (kN · s)/kg, TSFC≃ (0.019, 0.0148, 0.0127) kg/(kN · s)

8

a

ue

uefpa

3 4 5a

6

u

2

7

p

Problem AB.4.2 Starting from

(

Tma

)

max

= u

[

(1 + f)ηprηgηpt∆h

u2−

(

1− (1 + f)

2

ηnηprηgηpt

)]

,

show that for a turboprop with optimized propeller-to-jet thrust ratio (optimum α), the thrust isgiven by

Tma

= M√

γRTa

[

(1 + f)ηprηgηpt(γ − 1)M2

T05

T04

T04

Ta

(

1− 1

(p05/pa)(γ−1)/γ

)

−(

1− (1 + f)ηn2ηprηgηpt

)]

Use the above expression, together with the approximation f ≪ 1, to determine the compressorpressure ratio prc that provides the maximum thrust for a turboprop with ideal compressor turbine(ηt = 1). The resulting value of prc should be a function of γ, M , T04/Ta, rc = p04/p03 , ηd and ηc.

Problem AB.4.3 The figure below represents a schematic view of a turbofan with small bypassratio B = 3, similar to the Pratt & Whitney F119 developed for the F22 Raptor, including anafterburner downstream from the turbine. To determine the associated specific thrust T /ma attake off assume that the diffuser is ideal, so that p02 = pa and T02 = Ta. In the calculations, useaa = 340 m/s for the ambient sound speed and γ = 1.4 for the ratio of specific heats.

1. For a fan with pressure ratio prf = 1.5 and adiabatic efficiency ηf = 0.85, obtain the temper-ature ratio T08/Ta. T08/Ta ≃ 1.145

2. Assuming that the bypass flow is expanded isentropically in the fan nozzle to reach theambient pressure at the exit, obtain the exit Mach number Mef and the associated exitvelocity uef . Mef ≃ .7836, uef ≃ 269 m/s

3. For a compressor with pressure ratio prc = 15 and adiabatic efficiency ηc = 0.85, obtain thetemperature ratio T03/Ta. T03/Ta ≃ 2.374

4. Using the condition Psc + Psf = Pst , compute the temperature ratio across the turbineT05/T04 . In the calculation, use T04/Ta = 5 for the peak temperature at the turbine inlet.T05/T04 ≃ 0.6385

5. If the turbine adiabatic efficiency is ηt = 0.95, determine the pressure ratio p05/pa. In thecalculation, neglect pressure losses across the combustion chamber. p05/pa ≃ 2.805

6. Consider that the combustion process in the afterburner increases the stagnation temperatureby a factor of three (i.e., T06 = 3T05) with a negligible pressure loss (i.e., p06 ≃ p05). If thenozzle is ideal, determine the Mach number at the exit Me and the associated jet speed ue.Me ≃ 1.31, ue ≃ 1, 190 m/s

7. Assuming a small fuel-to-air mass-flow ratio f ≪ 1, calculate the specific thrust T /ma of theturbofan at take off. T /ma ≃ 1, 997 (kN · s)/kg

ef

a

pa

3 4 5 6

ue

2

a7

8 9 u

p

Problem AB.4.4 The figure below represents schematically the engine of a VTOL aircraft, similarto the Pratt & Whitney F135-PW-600 turbofan engine used in the F-35B. Besides the compressorturbine, the engine has a separate power turbine. In flight mode, this power turbine is used to movethe fan, so that the engine operates like a normal turbofan with Prc = 20, Prf = 2, and B = 8. Fortakeoff and landing, however, the engine switches to a turboshaft mode, with the power turbineused to move an external lift fan. The pressure ratio across the combustor is rc = p04/p03 = 0.9,while the adiabatic efficiencies are ηd = 0.9, ηf = 0.85, ηc = 0.8, ηct = 0.9, ηpt = 0.9, ηn = 0.97,and ηfn = 0.97 for the diffuser, fan, compressor, compressor turbine, power turbine, nozzle, andfan nozzle, respectively. For the following analysis use T04/Ta = 7, (ηbQR)/(cpTa) = 200, aa = 300m/s, and γ = 1.3.

• We begin by analyzing the performance of the engine in flight mode at M = 0.8:

1. Obtain the fuel-to-mass ratio f . f ≃ 0.0235

2. Compute the temperature behind the compressor turbine, giving the result in the formT05a/Ta. In the calculation, use the condition Psc = Psct, stating that the compressorturbine is used to move the compressor. T05a/Ta ≃ 5.666

3. Calculate the pressure behind the compressor turbine, giving the result in the formp05a/pa. p05a/pa ≃ 9.2

4. Determine the temperature and the pressure behind the fan, giving the result in theform T07/Ta and p07/pa, respectively. T07/Ta ≃ 1.32, p07/pa ≃ 2.864

5. Assuming that p8 = pa, find the exhaust speed of the bypass stream uef , giving theresult in the form uef/aa. uef/aa ≃ 1.356

6. Use the condition Psf = Pspt to obtain the temperature behind the power turbine, givingthe result in the form T05b/Ta.T05b/Ta ≃ 3.92

7. Calculate the pressure jump across the power turbine, giving the result in the formp05b/pa.p05b/pa ≃ 1.49

8. Assuming that p6 = pa, obtain the exhaust speed of the hot stream ue, giving the resultin the form ue/aa.ue/aa ≃ 1.4943

9. Compute the specific thrust of the turbofan T /ma. T /ma ≃ 1.5542 (kN · s)/kg

• Consider now the performance of the engine when operating as a turboshaft during take-off/landing:

10. Compute f , T05a/Ta, and p05a/pa for M = 0. f ≃ 0.0246, T05a/Ta ≃ 5.785, p05a/pa ≃7.11

11. Assuming p05b = pa, determine the shaft power of the power turbine Pspt (which isused to move the lift fan), giving the result in the nondimensional form Pspt/(maa

2a).

Pspt/(maa2a) ≃ 6.4734

POWER

a

ue

uefpa

au

7 8

2 3 4 5a 5b

COMPRESSORCOMPRESSOR

TURBINE TURBINE

GEAR BOX

LIFT FAN

6

FAN

p

Problem AB.4.5 The figure below represents schematically a turbofan with bypass ratio B = 6and compressor pressure ratio Prc = 18 designed to fly at M = 0.7. The hot stream behind theturbine and the bypass stream behind the fan mix upstream from the nozzle, so that there is a singleexhaust stream with velocity ue. The pressure ratio across the combustor is rc = p04/p03 = 0.9,while the adiabatic efficiencies are ηd = 0.9, ηf = 0.85, ηc = 0.8, ηt = 0.9, and ηn = 0.97 forthe diffuser, fan, compressor, turbine, and nozzle, respectively. For the following analysis useT04/Ta = 7, (ηbQR)/(cpTa) = 200, aa = 300 m/s, and γ = 1.4.

1. Obtain the fuel-to-mass ratio f . f ≃ 0.0214

For the configuration selected, the condition that the stagnation pressure behind the turbine (p05)equals the stagnation pressure behind the fan (p07) must be used to determine the fan pressureratio Prf . In the computation, follow these steps:

2. Show that the condition p05 = p07 can be written in the form Prf = (p05/p04)rcPrc .

3. Combine this last equation with the definition of the turbine efficiency to show that

1− T05

T04

= ηt

[

1−(

Prf

rcPrc

)γ−1γ

]

.

4. Use the condition Psf + Psc = Pst together with the last equation to derive the equation

(T04/Ta)ηt

1 + γ−12 M2

[

1−(

Prf

rcPrc

)γ−1γ

]

=(P

γ−1γ

rc − 1)

ηc+B

(Pγ−1γ

rf − 1)

ηf,

which can be used to solve for Prf .

5. Determine Prf . Prf ≃ 1.68

6. Compute p05/pa = p07/pa. p05/pa ≃ 2.26

7. Calculate also T05/Ta and T07/Ta. T05/Ta ≃ 4.0, T07/Ta ≃ 1.304

Assuming that mixing takes place at constant pressure (i.e. p06 = p05 = p07) and with negligibleheat losses:

8. Obtain the temperature immediately upstream from the nozzle, giving the result in the formT06/Ta. T06/Ta ≃ 1.69

9. Find the exhaust speed ue. ue ≃ 391.53 m/s

10. Determine the specific thrust T /ma and the TSFC. T /ma ≃ 1.279 (kN · s)/kg, TSFC≃ 0.0167kg/(kN · s)

a

e

pa

pa

3 4 5au

2

7

6

7

MIX

ING

RE

GIO

N

u

Problem AB.4.6 The figure below represents a turboshaft that is used on a small airplane flyingat M = 0.5 at an altitude where Ta = 260 K. Since the gas is expanded through the power turbineto a pressure close to the ambient value (p6 ≃ pa), the resulting exhaust jet has a small speedue ≪ u that does not contribute significantly to the thrust. To obtain the specific thrust T /ma

and the TSFC of the turboshaft follow the steps suggested below. In the calculations, use γ = 1.4and R = 287 J/(kg K).

1. Assuming that the diffuser has an efficiency ηd = 0.98, determine the stagnation values of thetemperature and pressure at the compressor inlet, giving the results in the form p02/pa andT02/Ta. p02/pa ≃ 1.182, T02/Ta ≃ 1.05

2. For a compressor with pressure ratio prc = 15 and adiabatic efficiency ηc = 0.85, obtain thetemperature ratio T03/T02 . T03/T02 ≃ 2.374

3. Assuming that the maximum temperature at the turbine inlet is T04 = 1500 K, use the energybalance across the combustion chamber to determine the value of the fuel-to-air ratio f . Inthe calculation, employ QR = 45× 106 J/kg and ηb = 1. f ≃ 0.0196

4. Using the condition Psc = Pst , determine the value of the temperature ratio across thecompressor turbine T05/T04 . T05/T04 ≃ 0.75

5. If the compressor turbine adiabatic efficiency is ηt = 0.95, determine the pressure ratio p05/p04 .p05/p04 ≃ 0.343

6. Analyze the expansion across the power turbine to determine the associated shaft power Pspt ,giving the result in the dimensionless form Pspt/(mau

2). In the computation, neglect pressurelosses across the combustion chamber (i.e., p04 = p03) and use ηpt = 0.96 for the adiabaticefficiency of the power turbine. Pspt/(mau

2) ≃ 17.07

7. Taking into account that ue ≪ u, calculate the specific thrust T /ma and TSFC for theturboshaft. The efficiencies of the propeller and the gear box are ηpr = 0.8 and ηg = 0.93,respectively. T /ma ≃ 1.89 (kN · s)/kg, TSFC≃ 0.010366 kg/(kN · s)

e

2 3 4 5 6

au

u <<upa

Problem AB.4.7 A turboprop with peak temperature T04 = 1400 K and compressor pressureratio Prc = 12 is to be used for a large cargo airplane with cruise conditions M = 0.6 and Ta = 220K. For the calculations below use γ = 1.4, ηbQR/(cpTa) = 200, aa = 297 m/s, and pe = pa.

• Obtain the fuel-air ratio f assuming that the adiabatic efficiency of the compressor is ηc =0.85. f ≃ 0.0206

• Calculate the temperature ratio T05/T04 and the pressure ratio p05/p04 across the high-pressure (compressor) turbine, whose adiabatic efficiency is ηt = 0.85. T05/T04 ≃ 0.795,p05/p04 ≃ 0.38

• Obtain the ideal enthalpy drop available downstream from the turbine ∆h = h05 −h7s, givingthe result in the form ∆h/(cpTa) with

∆h

cpTa=

T05

Ta

(

1− 1

(p05/pa)(γ−1)/γ

)

.

Use rc = 1 and ηd = 0.9 for the pressure ratio across the combustor and for the adiabaticefficiency of the diffuser, respectively. ∆h/(cpTa) ≃ 1.98

• Using ηn = 0.98 and ηprηgηpt = 0.4 determine the optimal value αopt of the fraction α ofavailable enthalpy to be used in the power turbine. αopt ≃ 0.78

• For this value α = αopt calculate the specific thrust T /ma and the TSFC of the turboprop.T /ma ≃ 1.045 (kN · s)/kg, TSFC≃ 0.0197 kg/(kN · s)

• Compare the resulting values of T /ma and TSFC with those corresponding to a turbojetwith the same efficiencies and the same values of T04/Ta and Prc . Comment on the results.T /ma ≃ 0.766 (kN · s)/kg, TSFC≃ 0.027 kg/(kN · s)

NO

ZZ

LE

aau

3 4 5

ue

2 76

GE

AR

BO

X

DIF

FU

SE

R

CO

MP

RE

SS

OR

CO

MB

US

TO

R

TU

RB

INE

TU

RB

INE

CO

MP

RE

SS

OR

PO

WE

R

p

Problem AB.4.8 The Bristol Siddeley BS100 engine was a British turbofan with small bypassratio B = 1.2 designed and built over fifty years ago for the Hawker Siddeley P.1154, a supersonicvertical/short take-off and landing (V/STOL) fighter aircraft. For thrust augmentation at takeoffthe engine used “plenum chamber burning” (PCB), an alternative to the standard afterburner inwhich the reheat is applied in the bypass stream. To explore the advantages of this alternative,consider the two configurations represented below. In the analysis, assume that the nozzles havevariable geometry to give an exit pressure equal to the ambient pressure. The left-hand-side figurecorresponds to a standard afterburner, for which

T /ma = (1 + f + f∗)u∗e +Buef

at takeoff, whereas the figure on the right corresponds to the PCB engine, for which

T /ma = (1 + f)ue + (B + f∗)u∗ef

at takeoff. In the above expressions f∗ = m∗

f/ma, with m∗

f and ma representing, respectively, there-heat fuel burning rate and the air mass flux through the core engine. The compressor and fanpressure ratios are Prc = 16 and Prf = 2, and the adiabatic efficiencies are ηf = 0.85, ηc = 0.8,ηt = 0.9, ηn = 0.97, and ηfn = 0.97 for the fan, compressor, turbine, nozzle, and fan nozzle,respectively. For the following analysis use T04 = 1600 K, (ηbQR)/(cpTa) = 200, aa = 340 m/s,Ta = 288 K, and γ = 1.3 and neglect all pressure losses associated with combustion. The maximumcombustion temperature for the afterburner and the PCB is T05∗ = T07∗ = 2500 K.

8

a

ue

uefpa

3 4 5

a

62

7

p*5*

*

8

a

ue

uefpa

3 4 5

a62

7

p

7* **

1. Compute the fuel-to-air mass ratio f in the main combustor.

2. Determine the temperature ratio across the turbine T05/T04 .

3. Calculate the temperature behind the fan T07 , giving the result in the form T07/Ta.

For the afterburner engine:

4. Obtain the exhaust speed for the bypass stream uef .

5. Compute the value of f∗.

6. Determine the exhaust speed for the core engine u∗e.

For the PCB engine:

7. Obtain the exhaust speed for the core engine ue.

8. Compute the value of f∗ = m∗

f/ma.

9. Determine the exhaust speed for the bypass stream u∗ef .

Using the above values,

10. Calculate the specific thrust (in (kN · s)/kg) at takeoff for both designs and comment on theresults.