Airbreathing Engines

Antonio L. Sánchez

March 6, 2020

1 Introduction

Unlike rocket engines, airbreathing engines use the oxygen of the surrounding air for the combustionprocess, so that they do not need to carry the oxidant. Since the fuel-to-air mass ratio needed in stoi-chiometric combustion is typically very small, most of the mass flow through the engine is provided bythe air stream, which is taken from the surrounding atmosphere through a diffuser. The main types ofairbreathing engines, to be analyzed below, are shown schematically in Fig. 1.

4

a

ape

ue

1 2 35

6

DIFFUSER

SUBSONIC

DIFFUSERSUPERSONIC

FUEL

INJECTORS

FLAME

HOLDERSp

u

a3 4 5

uea

2 6

p

8

a

ue

uefpa

3 4 5

a6

u

2

7

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Figure 1: Schematic view of the different types of airbreathing engines.

2 The thrust equation

A control-volume analysis can be used to derive an expression for the thrust generated by the engine.We consider first a turbojet engine attached to an airplane at takeoff, when the engine is surroundedby stagnant ambient air, the case represented on the left-hand side of Fig. 2. In steady operation massconservation (

∫

Scρv · ndS = 0) provides

me = ma + mf , (1)

indicating that the mass flow rate at the exit section is the sum of the air entrained by the engine and thefuel injected into the combustion chamber. The air flow is brought from all around the airplane. Since

1

the air velocity is inversely proportional to the square of the distance to the engine inlet, as follows fromcontinuity, the momentum flux of the air entering the control volume is very small. If its contribution isneglected when writing the horizontal component (x) of the momentum equation

∫

Sc

ρvxv · ndS = −

∫

Sc

(p− pa)nxdS + T , (2)

it follows thatT = meue + (pe − pa)Ae = ma(1 + f)ue + (pe − pa)Ae, (3)

a result analogous to that obtained previously for rocket engines. Equation (3) has been written in termsof the fuel-air ratio f = mf/ma. As previously mentioned, this is a very small quantity, although itsupper theoretical limit is f ≃ 1/15, corresponding to stoichiometric hydrocarbon-air combustion, typicalvalues are significantly smaller, as needed to limit the peak temperature of the gas exiting the combustor.

TT

n

uepe

pa

pa

pa

TT

n

pe

pa

pa

u

u

ma

ma

me

ue

me

n

mf mf

Streamline

Figure 2: Schematic view of a single-exhaust airbreathing engine at take off (left) and in flight (right).

The above result is modified when the airplane is flying at speed u, the case represented on the right-hand side of Fig. 2. The control volume is bound laterally by a stream surface, where the pressure isassumed to be pa. Since the air entering the engine has a momentum flux relative to the airplane mau,the momentum balance yields in this case the so-called thrust equation

T = meue − mau+ (pe − pa)Ae = ma[(1 + f)ue − u] + (pe − pa)Ae. (4)

As in the case of rockets, the contribution of the pressure term is typically small and can be correspondinglyneglected for many purposes.

3 Performance parameters

3.1 Specific thrust and TSFC

The thrust is proportional to the size of the engine, which enters in (4) through the air mass flow rate. Arelative measure of the thrust that enables the comparison between different engines, regardless of theirsize, is the so-called specific thrust

T

ma= (1 + f)ue − u+ (pe − pa)Ae/ma

[

kN · s

kg

]

. (5)

2

Neglecting the pressure term and using the approximation f ≪ 1 yields the simplified result

T

ma≃ ue − u, (6)

which reveals the key role of the exhaust speed in jet-propulsion applications. Note that the result furtherreduces to T /ma ≃ ue for the specific thrust at takeoff, a result that can be directly compared with thespecific impulse of rocket engines.

Fuel consumption is probably the most important factor in the selection of an airbreathing engine forcivilian transport applications. A useful measure of this aspect of the engine performance is given by thethrust-specific fuel consumption

TSFC =mf

T=

f

T /ma

[

kg

kN · s

]

, (7)

defined as the mass of fuel needed to produce the unit of impulse.

3.2 Engine efficiencies

Both the specific thrust and the TSFC have dimensions. A dimensionless description of the performanceof the engine can be made in terms of normalized efficiencies, expressing ratios of power output to powerinput. The objective of the jet engine is to generate thrust, with an associated power that for an airplanewith flight speed u is given by T u. The energy is obtained by burning the fuel with a mass consumptionrate mf . Correspondingly, the energy released per unit time is mfQR, with QR denoting the energyreleased per unit mass of fuel consumed (remember that, for rockets, the heat of reaction QR was referredto the unit mass of propellant instead). An ideal propulsion system should be able to convert all ofthe thermal energy into propulsion energy. In real systems, however, T u < mfQR, as measured by theso-called overall efficiency

ηo =T u

mfQR< 1. (8)

Using the approximation (6) leads to

ηo ≃mau(ue − u)

mfQR=

u2efQR

u

ue

(

1−u

ue

)

. (9)

This expression can be used to show that, for a fixed ue, the maximum overall efficiency ηo = u2e/(4fQR)is reached for flight conditions such that u = ue/2.

Conceptually, the conversion of the thermal power into propulsion power can be thought to occur intwo sequential steps. In the first step, the thermal energy is used to increase the kinetic energy of thegas stream flowing through the engine. The efficiency of this first step is measured by comparing theincrement in the flux of kinetic energy meu

2e/2 − mau

2/2 to the thermal power through the so-calledthermal efficiency

ηth =meu

2e/2− mau

2/2

mfQR< 1. (10)

In the second step, the increase of kinetic energy is used to produce thrust, as measured by the so-calledpropulsion efficiency

ηp =T u

meu2e/2− mau2/2< 1. (11)

Clearly,ηo = ηthηp, (12)

3

as follows from the above definitions. It is worth pointing out that in the approximation (6) with me =ma(1 + f) ≃ ma the propulsion efficiency reduces to

ηp ≃mau(ue − u)

ma(ue − u)(ue + u)=

2u/ue1 + u/ue

. (13)

It can be seen that the maximum propulsion efficiency ηp = 1 is reached as u → ue. Although the systemwould be very efficient in this limit, as measured by ηp, the associated specific thrust would be very small,as follows from (6), with the result that one would need to increase ma (and, therefore, the size of theengine) to provide a nonnegligible amount of thrust.

3.3 Aircraft range

The importance of the overall efficiency ηo becomes clear when considering the range of an aircraft (i.e.the distance that an aircraft can travel with a given amount of fuel mf at takeoff). At cruise conditionsthe forces acting on the aircraft are in equilibrium, so that the thrust T equals the drag D and the liftL equals the weight mAg, where mA denotes the instantaneous mass of the aircraft. This equilibriumbalance can be expressed in the form

T = D =L

L/D=

mAg

L/D, (14)

which can be used to write the thrust power

T u =mAg

L/Du = ηomfQR, (15)

the last expression following from the definition (8). On the other hand, the simple differential equations

ds

dt= u (16)

anddmA

dt= −mf , (17)

relating the aircraft speed u with the distance travelled by the aircraft s and the total mass of the aircraftmA with the fuel consumption rate mf , can be combined to give

ds = −u

mfdmA = −ηo

(

L

D

)

QR

g

dmA

mA

, (18)

where the value of u/mf is evaluated from the second equation in (15). If the dimensionless factor ηoL/Dis assumed to remain constant during flight, a reasonable approximation, then the above equation can beintegrated with initial condition mA = mo to give

s = ηo

(

L

D

)

QR

gln

(

mo

mA

)

(19)

for the distance traveled as a function of the decreasing aircraft mass mA. The range

smax = ηo

(

L

D

)

QR

gln

(

mo

mo −mf

)

(20)

can be correspondingly computed as the distance traveled after consuming all of the available fuel. Equa-tion (20) reveals, in particular, that the range is linearly proportional to the overall efficiency.

4

4 Energy balance in propulsion systems

The analysis of the performance of a given jet engine involves separate consideration of its differentcomponents, including the diffuser, compressor, combustor, turbine, and nozzle. In all cases, the integralenergy balance provides valuable information regarding the associated changes of stagnation enthalpy.For the analysis, we begin by formulating the energy equation (i.e. the first law of thermodynamics) inthe general integral form

d

dt

[∫

Vc

ρ

(

e+|v|2

2

)

dV

]

+

∫

Sc

ρ

(

e+|v|2

2

)

(v − vc) · ndS = −

∫

Sc

pv · ndS +

∫

Sc

v · τ · ndS

+

∫

Vc

ρg · v dV +

∫

Sc

κ∇T · ndS +

∫

Vc

qR dV. (21)

corresponding to a control volume Vc bounded by the control surface Sc moving with velocity vc.

Si

n

n

n

n

qRvb

Se

Sl

Sb

Figure 3: Schematic of a generic turbojet component.

The above equation is applied to the generic component represented in Fig. 3, including possiblyinternal moving parts (i.e. the rotating blades of a compressor or turbine moving locally with velocity vb)as well as an internal region where combustion takes place, qR representing the heat-release rate per unitvolume. The control surface Sc includes a fixed outer boundary with vc = 0, defined by the inlet sectionSi, the lateral surface Sl (where v = 0), and the exit section Se, and an internal surface Sb moving withthe rotating blades, so that v = vc = vb on Sb. For a system in steady operation, the accumulation termis negligible in (21). Neglecting the work done by viscous forces on the fixed outer boundary, becauseeither v = 0 (at Sl) or the local Reynolds number is large (at Si and Se), along with the effect of buoyancyforces and heat conduction reduces (21) to

∫

Si+Se

ρ

(

e+|v|2

2

)

v · ndS = −

∫

Si+Se

pv · ndS −

∫

Sb

pvb · ndS +

∫

Sb

vb · τ · ndS +

∫

Vc

qR dV. (22)

The heat release rate is related to the fuel mass flow rate by

∫

Vc

qR dV = ηbmfQR, (23)

involving a burning efficiency ηb that accounts for incomplete combustion and heat losses. The twointegrals −

∫

Sbpvb · ndS +

∫

Sbvb · τ · ndS account for the work done on the fluid per unit time by the

rotating blades. This “shaft power”

Ps = −

∫

Sb

pvb · ndS +

∫

Sb

vb · τ · ndS (24)

5

is positive for compressors and negative for turbines. Rewriting (22) in terms of the enthalpy h = e+ p/ρgives

∫

Si+Se

ρ

(

h+|v|2

2

)

v · ndS = Ps + ηbmfQR, (25)

to be used in the following analysis. For diffusers and nozzles Ps = mf = 0 , so that the above equationreduces to the condition that the stagnation enthalpy h0 = h+ 1

2 |v|2 is conserved, i.e.

h0i = h0e (diffusers and nozzles). (26)

For compressors and turbines the solution yields

m(h0e − h0i) = Pc (compressor) (27)

andm(h0i − h0e) = Pt (turbine), (28)

where Pc and Pt are the compressor power and turbine power, respectively 1. Finally, for combustors wefind that (ma+ mf )h0e − mah0a,i − mfh0f,i = ηbmfQR, where h0f,i and h0a,i are the stagnation enthalpiesof the fuel and the air in their feed streams. In most cases, the relative contribution of the term mfh0f,iis negligibly small and can be correspondingly neglected, thereby reducing the energy balance to

(ma + mf )h0e − mah0a,i = ηbmfQR (combustors). (29)

Equations (26)–(29) will be useful in the following development.

5 Ramjets

The ramjet is the simplest air-breathing engine. As indicated in Fig. 4, it comprises three main compo-nents, namely, (i) a diffuser, designed to compress the air by decelerating the incoming stream to low Machnumbers, (ii) a combustion chamber, where the oxygen of the air reacts with fuel that is continuouslyinjected, and (iii) a convergent-divergent nozzle, designed to expand the hot stream, thereby convertingthe hot slow gas stream into a high velocity jet. Since there are no moving parts, the design and main-tenance of ramjets are particularly simple. Besides, because of the absence of turbine blades betweenthe combustor and the nozzle, cooling limitations are less stringent, thereby enabling higher combustiontemperatures. The main drawback of ramjets is their inability to provide thrust at takeoff. They aremainly suitable for supersonic-flight applications at moderately large Mach numbers M ∼ 3. Their per-formance is hindered both at low Mach numbers (i.e. subsonic flight), when the compression provided bythe diffuser becomes insufficient, and at high Mach numbers (i.e. hypersonic flight), when the gas heatingin the diffuser, associated with the air-stream deceleration from M ≫ 1 to M ≪ 1 (through obliqueshocks), limits the temperature increase in the combustor. For that reason, the scramjet -a variant of theramjet involving supersonic combustion- is currently being considered instead as a better suited optionfor hypersonic-flight applications.

5.1 Analysis of ramjet performance

In analyzing the performance of the ramjet we must consider separately the gas evolution across thediffuser (a → 2), the combustion chamber (2 → 4), and the nozzle (4 → 6). The objective is to obtain the

1Note that the change in stagnation enthalpy across turbines and compressors, where the flow is nearly isentropic, is due

to the intrinsic unsteady character of the flow, associated with the motion of the blades.

6

4

a

ape

ue

1 2 3 5 6

DIFFUSER

SUBSONICDIFFUSER

SUPERSONIC FUELINJECTORS

FLAMEHOLDERS p

Figure 4: Schematic view of a ramjet, with indication of the different components.

specific thrust T /ma, the thrust-specific fuel consumption TSFC, and the engine efficiencies ηo, ηth, andηp for given ambient air conditions (i.e. Ta, pa, and aa) and given flight speed u and associated flight Machnumber M = u/aa. The fuel-to-air ratio f will be taken as an unknown variable, to be calculated fromthe condition that the peak temperature, found at the entrance of the nozzle, cannot exceed a maximumvalue, determined by the materials and cooling requirements of the nozzle walls. For simplicity, we shallconsider that the nozzle is designed to provide

pe = pa, (30)

so that the thrust (4) will be computed using

T = ma[(1 + f)ue − u]. (31)

5.1.1 Simplifications to the combustor flow

The low-Mach-number conditions prevailing in the combustion chamber are to be accounted in simplifyingthe description, i.e. since M2 ∼ M4 ≪ 1, it follows that

T02 = T2

(

1 +γ − 1

2M2

2

)

≃ T2 and T04 = T4

(

1 +γ − 1

2M2

4

)

≃ T4 (32)

and also that

p02 = p2

(

1 +γ − 1

2M2

2

)γ

γ−1

≃ p2 and p04 = p4

(

1 +γ − 1

2M2

4

)γ

γ−1

≃ p4. (33)

Furthermore, an order-of-magnitude analysis of the momentum balance equation indicates that, for high-Reynolds number flow, the relative spatial pressure differences scale with the square of the existing Machnumber. Since M2 ∼ M4 ≪ 1 in the combustion chamber, it follows that (p2−p4)/p2 ∼ (p02−p04)/p02 ≪ 1.We shall see below that these simplifications are no longer valid across the supersonic-combustion chamberof scramjets, where the pressure increases by a relative amount of order unity (p4 − p2)/p2 ∼ 1 as a resultof the heat-release process.

5.1.2 Energy balance

The energetics of the diffuser can be evaluated by straightforward application of (26) to give

h02 = h0a = ha +u2

2→

h02ha

=T02

Ta= 1 +

γ − 1

2M2. (34)

7

Similarly, we find

h04 = h06 = he +u2e2

→h04ha

=T04

Ta=

Te

Ta

(

1 +γ − 1

2M2

e

)

(35)

across the nozzle. Equation (29) can be used to write

(ma + mf )h04 − mah02 = ηbmfQR → (1 + f)T04

Ta−

T02

Ta=

ηbfQR

cpTa, (36)

with T02/Ta = 1+ γ−12 M2, as follows from (34). Since the peak temperature at the end of the combustion

process T4 ≃ T04 is taken as a known limiting parameter, as previously mentioned, we can solve the aboveequation to determine the fuel-to-air ratio

f =

(

T04

Ta

)

−(

1 + γ−12 M2

)

ηb

(

QR

cpTa

)

−(

T04

Ta

) (37)

in terms of known quantities (i.e., the peak-to-ambient temperature ratio T04/Ta, the flight Mach numberM , the burning efficiency ηb, and the ratio QR/(cpTa) of the heat of reaction to the ambient enthalpy).

5.1.3 Performance parameters

It is possible to relate all of the different ramjet performance parameters to the (still unknown) value ofthe exit Mach number Me. To that end, we begin by using (35) to write

ue = Meae = aaMe

√

Te/Ta = aa(T04/Ta)

1/2 Me(

1 + γ−12 M2

e

)1/2, (38)

which can be used in (31) to provide

T

ma= aa

(1 + f)

(T04/Ta)1/2 Me

(

1 + γ−12 M2

e

)1/2−M

(39)

for the ramjet specific thrust, with the corresponding

TSFC =f

T /ma(40)

evaluated from (7) once the fuel-to-air ratio f is computed from (37). Similarly, one can derive from thedefinitions (8), (10), and (11) the expressions

ηo =(γ − 1)M

f(

QR

cpTa

)

(1 + f) (T04/Ta)1/2 Me

(

1 + γ−12 M2

e

)1/2−M

(41)

ηth =(γ − 1)/2

f(

QR

cpTa

)

(1 + f) (T04/Ta)M2e

(

1 + γ−12 M2

e

) −M2

(42)

ηp = 2M

(1 + f) (T04/Ta)1/2 Me

(

1 + γ−12 M2

e

)1/2−M

/

(1 + f) (T04/Ta)M2e

(

1 + γ−12 M2

e

) −M2

(43)

8

for the engine efficiencies. In order to evaluate (38)–(43) we need to find the exit Mach number Me,following the procedure indicated below.

5.2 Ideal ramjets

It is instructive to consider first the case of ideal ramjets, where we assume that the flow is isentropic inthe diffuser and nozzle, so that sa = s2 and s4 = se, and neglect pressure losses across the combustor, sothat

p2 = p02 = p4 = p04 . (44)

The corresponding gas evolution is represented in the h−s diagram shown in Fig. 5, which for illustrativepurposes includes the approximation h04 − h02 = h4 − h2 ≃ ηffQR, stemming from (65) when f ≪ 1.

h

s

a

6

h4 ≃ h04

u2e

2

p 2=p 4

2

4

h2 ≃ h02

he

ha

u2

2

ηbQRf

p asa

Figure 5: The enthalpy-entropy variation for the gas flow across the ideal ramjet.

Since the flow is both steady and isentropic in the diffuser and nozzle, it follows that p0a = p02 andthat p04 = p0e , which can be combined with (44) and the condition pe = pa to give

p0a = p0e → pa

(

1 +γ − 1

2M2

)γ/(γ−1)

= pa

(

1 +γ − 1

2M2

e

)γ/(γ−1)

, (45)

leading to the conclusion that Me = M !! This is of course a direct consequence of the conservation ofstagnation pressure across the ramjet: a deceleration/compression followed by an acceleration/expansionto the same initial pressure leads to a stream with the same Mach number. As inferred from (31) withf ≪ 1, the generation of thrust requires that ue = Meae > u = Maa. For the ideal ramjet Me = M andTe = T04/(1 +

γ−12 M2), the latter following from (35), so that positive thrust is generated provided that

M < ( 2γ−1)

1/2(T04/Ta − 1)1/2, thereby placing an upper limit to the range of flight Mach numbers whereramjets can operate successfully.

Using the result Me = M in (39)–(43) allows us to evaluate the performance of the ideal ramjet. Thevariation with flight Mach number of T /ma, TSFC, ηo, ηp, and ηth are shown as solid curves in Fig. 6for an ideal Ramjet with γ = 1.3, T04/Ta = 12, QR/(cpTa) = 200, ηb = 1, and aa = 300 m/s. Theresults indicate that the specific thrust, measuring the thrust per unit size of engine, reaches a maximumat M ≃ 2.9 and decreases for higher values, eventually vanishing at M ≃ 8.5. By way of contrast, theoverall efficiency, which determines the range, increases continuously for increasing values of M . Also of

9

interest is that the value of TSFC remains almost constant over a wide range of intermediate values ofM .

1 2 3 4 5 6 7

0.2

0.4

0.6

0.8

1

1.2

1.4

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6: Variation with the flight Mach number of the performance parameters of a ramjet. Results areevaluated with γ = 1.3, T04/Ta = 12, QR/(cpTa) = 200, ηb = 1, and aa = 300 m/s for an ideal ramjetwith no aerodynamic losses (solid curves) and for a real ramjet with adiabatic efficiencies ηd = 0.8 andηn = 0.98 and combustion pressure ratio rc = 0.9 (dashed curves).

5.3 Effects of aerodynamic losses

The above analysis, assuming isentropic flow across the diffuser and nozzle and uniform pressure acrossthe combustor, provides an idealized picture of the ramjet performance. In reality, however, there arenonnegligible losses of stagnation pressure (and associated increases of entropy) across the diffuser, wherethe deceleration of the air stream takes place through a series of oblique shocks, and also across the nozzle,related mostly to the action of the viscous forces in the wall boundary layer. Also, the flow across thecombustor involves a small but finite pressure loss, needed to overcome the drag forces on flame holdersand fuel injectors and the viscous forces on the walls, so that p4 < p2. Because of these cumulativeaerodynamic losses the exit velocity ue is smaller than that predicted by the ideal-ramjet model. Thiscan be seen clearly in Fig. 7, which compares the h− s evolution of the gas in the ideal ramjet of Fig. 5with that of a real ramjet with s2 > sa, p4 < p2, and se > s4.

The simplest way to quantify these aerodynamic losses is by defining the pressure ratios across thediffuser, combustor, and nozzle

rd =p02p0a

< 1, rc =p4p2

=p04p02

< 1, rn =p06p04

=p0ep04

< 1. (46)

Since

p06p0a

= rdrcrn =

(

1 + γ−12 M2

e

1 + γ−12 M2

)γ

γ−1

< 1 (47)

the resulting Mach number at the exit

Me =

(

2

γ − 1

)1/2 [

(rdrcrn)γ−1

γ

(

1 +γ − 1

2M2

)

− 1

]1/2

(48)

10

h

s

a

h4 ≃ h04

u2e

2

p22h2 ≃ h02

ha

u2

2

ηbQRf

pa

sa

p 4<p 2

u2e

2

4

Figure 7: Comparison between the enthalpy-entropy variation for the gas flow across an ideal ramjet(blue) with that of a real ramjet with s2 > sa, p4 < p2, and se > s4 (red).

is smaller than M .For diffusers and nozzles, and also for compressors and turbines, to be discussed below, the departures

from isentropic behavior are usually measured in terms of the so-called adiabatic efficiencies. For a diffuser,the adiabatic efficiency

ηd =h2s − hah2 − ha

. (49)

is defined as the ratio of the enthalpy change involved in an isentropic compression from pa to p2 to theactual change occurring in the real system, with 2s denoting the downstream isentropic state, as indicatedon the left-hand side of Fig. 8. Since h2 − ha = u2/2, dividing by ha numerator and denominator of (49)and using the condition that the evolution a → 2s is isentropic with p2 = p02 provides

ηd =

(

p02pa

)γ−1

γ− 1

γ−12 M2

→p02pa

=p2pa

=

(

1 + ηdγ − 1

2M2

)γ

γ−1

. (50)

s

h

pa

a

2s

2

ha

u2

2

p2

sa

h

s

6s

4

6

p4 ≃ p04

pa

h4 ≃ h04

u2e

2

Figure 8: Comparison between the ideal and real gas evolution across diffusers and nozzles used in thedefinitions of their corresponding adiabatic efficiencies.

11

Similarly, for a nozzle the adiabatic efficiency

ηn =h4 − h6h4 − h6s

. (51)

is defined as the ratio of the enthalpy change h4 − h6 involved in the expansion from p4 = p04 to p6 = pato the change h4 − h6s that would be involved should the process be isentropic, with 6s denoting thedownstream isentropic state, as indicated on the right-hand side of Fig. 8. Using h4 −h6 = u2e/2 togetherwith the conditions T4 = T04 , (h6s/h4) = (T6s/T4) = (pa/p4)

(γ−1)/γ , and T04/T6 = T06/T6 = 1 + γ−12 M2

e

leads to

ηn =u2e/2

cpT04(1− h6s/h4)=

γ−12 M2

e(

1 + γ−12 M2

e

)

[

1−(

pap4

)(γ−1)/γ] . (52)

Using p4/pa = rc(p2/pa) with p2/pa evaluated from (50) as a function of M and solving the above equationfor Me yields

Me =

(

2

γ − 1

)1/2

r(γ−1)/γc

(

1 + ηdγ−12 M2

)

− 1

(1−ηn)ηn

r(γ−1)/γc

(

1 + ηdγ−12 M2

)

+ 1

1/2

, (53)

which reduces to Me = M when ηd = rc = ηn = 1. Equation (53) is used, together with (39)–(43),to generate the dashed curves of Fig. 6, representing the degraded performance of a real ramjet withηd = 0.8, ηn = 0.98, and rc = 0.9. As can be seen, although there exist quantitative differences betweenthe real and the ideal ramjet, the ideal model qualitatively captures the trends of the different performanceparameters, with the largest departures observed in the predictions of TSFC.

5.4 High-speed combustion

The previous analysis showed that the specific thrust reaches a maximum for a flight mach number M ≃ 2.9and decays for larger values of M . The reason for this decay is that for M ≫ 1 the compression across thediffuser places the system at temperatures T02 close to the maximum value allowed for T04 , thereby limitingthe enthalpy gain h04 − h02 ≃ h4 − h2 across the combustor and, therefore, the resulting exit velocityue. To overcome this limitation, one alternative for large M is to limit the deceleration/compressionof the air stream in the diffuser, so that the final Mach number M2 remains supersonic, as done insupersonic combustion ramjets (scramjets). The corresponding analysis of their performance parallelsthat presented above. The main differences are that the approximations (32) and (33) do not apply andthat the supersonic flow in the combustor involves large relative pressure differences of order unity thatneed to be accounted for, as indicated below.

The analysis of the combustion process when the Mach number is of order unity simplifies for acombustion chamber of constant section. As done earlier in section 4, we will use the subscripts i and e todenote the inlet and exit properties, which are related by the balances of mass, momentum, and energy.In particular, using the approximation f ≪ 1 when writing the energy equation (29)

(1 + f)cpT0e − cpT0i = ηbfQR (54)

provides the expressionT0e

T0i

= 1 +ηbfQR

cpT0i

(55)

for the jump in stagnation temperature. Neglecting friction along with the negligible momentum flux ofthe fuel stream reduces the integral momentum balance to pi + ρiu

2i = pe + ρeu

2e, which can be written in

12

the formpepi

=1 + γM2

i

1 + γM2e

(56)

relating the pressure jump with the inlet and exit Mach numbers (note, in particular, that if Mi ≪ 1 andMe ≪ 1, then pi ≃ pe, that being the case of ramjet combustors). If the combustor section is constant,then neglecting the mass of the fuel compared with the mass of air when writing the continuity equationme = ma(1 + f) ≃ ma provides

ρiui = ρeue →Te

Ti=

(

pepi

Me

Mi

)2

. (57)

Using (56) in this last expression along with the definition T0/T = 1 + γ−12 M2 leads to

T0e

T0i

=

[

Me(1 +γ−12 M2

e )1/2/(1 + γM2

e )

Mi(1 +γ−12 M2

i )1/2/(1 + γM2

i )

]2

=F 2(Me)

F 2(Mi), (58)

where the function

F =M(1 + γ−1

2 M2)1/2

1 + γM2(59)

is represented in Fig. 9.

0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 9: The function F defined in (59).

Equations (55) and (58) can be used together with Fig. 9 to illustrate some of the characteristics ofthe flow. Since T0e > T0i as a result of the heat-release process, as indicated by (55), then accordingto (58) F (Me) must be larger than F (Mi). Two cases present themselves:

13

• If the flow is initially subsonic (Mi < 1), then Fig. 9 reveals that the increase of F is associatedwith a larger value of Me > Mi and, according to (56), a smaller value of pe < pi, i.e., subsoniccombustion leads to an acceleration/expansion of the flow.

• Conversely, if the flow is initially supersonic (Mi > 1), then the increase of F is associated witha smaller value of Me < Mi and a larger value of pe > pi, i.e. supersonic combustion leads to adeceleration/compression of the flow.

In both cases, the function F is limited to a maximum value (e.g. F ≃ 0.456 for γ = 1.4), associated withthe maximum increase in stagnation temperature that can be achieved before the flow chokes.

6 Turbojet engines

To enable takeoff capabilities and extend the range of flight conditions, the aerodynamic compressor (i.e.the diffuser) of the ramjet can be supplemented with a mechanical compressor. The power needed tomove the compressor can be provided by a co-axial turbine, to be placed immediately downstream fromthe combustor. The resulting arrangement is the basis of the so-called turbojet, schematically representedin Fig. 10, with the corresponding enthalpy-entropy diagram shown in Fig. 11.

u

a3 4 5

uea

2 6

p1

Figure 10: Schematic view of a turbojet with indication of the numbering used in the analysis of the flow.

6.1 Performance analysis

In analyzing the performance of a turbojet separate consideration must be given to the gas evolutionacross the diffuser (a → 2), compressor (2 → 3), combustor (3 → 4), turbine (4 → 5), and nozzle(5 → 6). The expansion in the nozzle is assumed to be such that p6 = pe = pa, so that the thrustsimplifies to the expression given in (31). In typical systems the Mach number in the engine betweenthe diffuser and the nozzle is very small, with the result that T02/T2 = T03/T3 = T04/T4 = T05/T5 ≃ 1and p02/p2 = p03/p3 = p04/p4 = p05/p5 ≃ 1. The analysis assumes known ambient air conditions(i.e. Ta, pa, and aa), gas properties (γ and cp), combustion parameters (ηb and QR), flight speed uand associated flight Mach number M = u/aa, compressor pressure ratio Prc = p03/p02 , and maximumallowable temperature T04 . The values of the specific thrust T /ma, thrust-specific fuel consumptionTSFC, and engine efficiencies ηo, ηth, and ηp, depend on the unknown values of the fuel-to-air ratio fand exit speed ue, to be computed as indicated below. Formulas will be derived for a real turbojet withgiven values of the adiabatic efficiencies of the diffuser, compressor, turbine, and nozzle (ηd, ηc, ηt, ηn)and a given pressure ratio across the combustor rc = p4/p3 = p04/p03 , with formulas for the ideal turbojetobtained by substituting (ηd = ηc = ηt = ηn = rc = 1).

14

h

s

p 03=p 04

p =pa

3

2

a

4

5

6

Pcma

u2

2

Ptma+mf

u2e2

sa

ha

h

s

p 03

p =pa

3s

2s

a

4

5s

6

u2

2

Ptma+mf

u2e2

sa

ha

2

3

p 04=r cp 03

5

6s

Pcma

Ideal Turbojet Real Turbojet

Figure 11: The enthalpy-entropy variation for the gas flow across an ideal turbojet (left) and across a realturbojet (right).

6.1.1 Flow across the diffuser (a → 2)

The analysis of this component is identical to that presented earlier in relation to the ramjet. Thus energyconservation leads to h0a = h02 ≃ h2, which can be used to write

h2 − ha = u2/2 andT02

Ta=

T2

Ta= 1 +

γ − 1

2M2, (60)

while the pressure ratio reduces to

p02pa

=p2pa

=

(

1 + ηdγ − 1

2M2

)γ

γ−1

(61)

involving the diffuser adiabatic efficiency ηd.

6.1.2 Flow across the compressor (2 → 3)

The energy balance across the compressor (27) can be used to write

ma(h03 − h02) = Pc → h03 − h02 = h3 − h2 = Pc/ma, (62)

where Pc is the compressor power. Departures from isentropic behavior are measured through the com-pressor adiabatic efficiency

ηc =h3s − h2h3 − h2

(63)

defined as the enthalpy change involved in the isentropic compression from p2 to p3 divided by theactual enthalpy change (see the h − s evolution shown on the left-hand side of Fig. 12). Since h3s/h2 =

(p3/p2)γ−1

γ = Pγ−1

γrc , where Prc is the known compressor pressure ratio, it follows that

ηc =h3s − h2h3 − h2

=P

γ−1

γrc − 1

T3/T2 − 1→

T03

T02

=T3

T2= 1 +

Pγ−1

γrc − 1

ηc, (64)

which naturally reduces to T3/T2 = (p3/p2)γ−1

γ = Pγ−1

γrc for the ideal compressor (i.e. ηc = 1).

15

s

h

p2

2

3s

3p3

h

s

5s

4

5

p4

p5

Ptma+mf

Pcma

Figure 12: Comparison between the ideal and real gas evolution across compressors and turbines used inthe definitions of their corresponding adiabatic efficiencies.

6.1.3 Flow across the combustor (3 → 4)

The analysis parallels that presented earlier for the ramjet. Energy conservation provides

(ma + mf )h04 − mah03 = ηbmfQR → (1 + f)T04

Ta−

T03

T02

T02

Ta= ηbf

QR

cpTa, (65)

which can be combined with (60) and (64) to give

f =

(

T04

Ta

)

−

(

1 +P

γ−1

γrc −1ηc

)

(

1 + γ−12 M2

)

ηb

(

QR

cpTa

)

−(

T04

Ta

) (66)

for the fuel-air ratio.

6.1.4 Flow across the turbine (4 → 5)

The energy balance across the turbine (28) gives

(ma + mf )(h04 − h05) = Pt (67)

relating the enthalpy decrease across the turbine h04 −h05 = h4−h5 with the turbine power. The pressure

jump across the ideal turbine is given by the isentropic relation (p05/p04)γ−1

γ = T05/T04 . Departures fromthis isentropic behavior are characterized by the adiabatic efficiency

ηt =h4 − h5h4 − h5s

=1− T05/T04

1− (p05/p04)γ−1

γ

→

(

p05p04

)γ−1

γ

= 1−1− T05/T04

ηt(68)

associated with the expansion from p4 = p04 to p5 = p05 , as indicated on the right-hand side of Fig. 12.

6.1.5 Flow across the nozzle (5 → 6 = e)

Conservation of stagnation enthalpy (h0e = h05) with h0e = he + u2e/2 and h05 = h5 gives

h5 − he = u2e/2. (69)

16

On the other hand, from the definition of adiabatic efficiency,

ηn =h5 − h6h5 − h6s

=u2e/2

cpT05

[

1− (pe/p5)γ−1

γ

] , (70)

with pe = pa, p5 = p05 , and cp = γRg/(γ − 1) it follows that

u2e2

=ηna

2a

γ − 1

(

T05

T04

)(

T04

Ta

)

1−1

[(

p05p04

)(

p04p03

)(

p03p02

)(

p02pa

)]γ−1

γ

, (71)

where a2a = γRgTa. Substituting in the above equation (61), (68) and the known pressure jumps Prc =p03/p02 and rc = p04/p03 gives

u2e2

=ηna

2a

γ − 1

(

T05

T04

)(

T04

Ta

)

1−

1(

1−1−T05

/T04

ηt

)

rγ−1

γc P

γ−1

γrc

(

1 + ηdγ−12 M2

)

, (72)

which determines the gas speed at the exit ue in terms of the (still) unknown temperature jump T05/T04 .

6.1.6 The temperature jump across the turbine

Since the purpose of the turbine is to move the compressor, it follows that

Pt = Pc. (73)

Substituting (62) and (67) into the above equation provides

(ma + mf )(h04 − h05) = ma(h03 − h02) → (1 + f)(T04 − T05) = T03 − T02 , (74)

which can be rearranged with use made of (60) and (64) to give

T05

T04

= 1−

(

T03

T02

− 1)

T02

Ta

(1 + f)T04

Ta

= 1−

(

Pγ−1

γrc − 1

)

(

1 + γ−12 M2

)

ηc(1 + f)T04

Ta

(75)

for the temperature jump across the turbine. For many purposes, the above equation can be written inthe simplified form

T05

T04

= 1−

(

Pγ−1

γrc − 1

)

(

1 + γ−12 M2

)

ηc(T04/Ta)(76)

by using the approximation f ≪ 1 (the mass flow rate across the turbine is assumed to be equal to themass flow rate across the compressor). Equation (75) (or (76)) can be used in (72) to compute ue. Forthe ideal turbojet with f ≪ 1, the solution can be seen to simplify to

u2e2

=a2a

γ − 1

(

T04

Ta

)

1−(P

γ−1

γrc − 1)(1 + γ−1

2 M2)

T04/Ta−

1

Pγ−1

γrc (1 + γ−1

2 M2)

, (77)

which shows explicitly the dependence on the peak temperature T04 , compressor pressure ratio Prc , andflight Mach number M . The reader can verify that for Prc = 1 this last equation reduces to Me = M , asit should, since in the absence of mechanical compression the ideal turbojet reduces to an ideal ramjet.

17

6.2 Turbojet performance

Equations (66) and (72) (supplemented with (75) or (76) for the computation of T05/T04) can be used toevaluate the fuel-to-air ratio f and the exit gas speed ue, to be employed in evaluating the performanceof the turbojet through

T

ma= (1 + f)ue − u, TSFC =

f

(T /ma)(78)

and

ηo =(T /ma)u

fQR, ηth =

(1 + f)u2e/2− u2/2

fQR, ηp =

(T /ma)u

(1 + f)u2e/2− u2/2. (79)

The typical variation of the different performance parameters with the compressor pressure ratio is shownin Fig. 13 for a turbojet in supersonic flight at M = 1.5. Results are given for an ideal turbojet (solidcurves) and for a real turbojet (dashed curves). Note that the limiting values for Prc = 1 correspond tothose of a ramjet at the same flight conditions.

100

101

102

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

100

101

102

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 13: Variation with the compressor pressure ratio Prc of the performance parameters of a turbojetflying at M = 1.5. Results are evaluated with γ = 1.3, T04/Ta = 10, QR/(cpTa) = 200, ηb = 1, andaa = 300 m/s for an ideal turbojet with no aerodynamic losses (solid curves) and for a real turbojet withadiabatic efficiencies ηd = 0.9, ηc = 0.85, ηt = 0.85 and ηn = 0.95 and combustion pressure ratio rc = 0.97(dashed curves).

The curves for specific thrust in Fig. 13 reveal the existence of a maximum at an intermediate valueof Prc . To further explore this aspect of the problem, additional results are shown in Fig. 14 for an idealturbojet. The left plot shows the important effect of the peak temperature on the turbojet performance.The right-hand side plot shows that the performance degrades with increasing flight Mach number M . Itis of interest that the optimum Prc at which the specific thrust reaches its maximum value decreases withincreasing M . For M = 4 the maximum thrust is found at Prc = 1, indicating that for these high speedsa ramjet is more efficient than a turbojet.

7 Turbofan engines

The performance of the turbojet can be improved in a modified design where part of the air bypasses thecombustor, that being the idea underlying turbofan engines. The typical system is shown in Fig. 15. Thetotal air mass flow rate ingested by the engine is (1+B)ma, with ma representing the air flow rate through

18

100

101

102

0.4

0.6

0.8

1

1.2

1.4

100

101

102

-0.5

0

0.5

1

1.5

2

Figure 14: Variation of the specific thrust with the compressor pressure ratio Prc for a turbojet flying atM = 1.5 with different peak temperatures (left plot) and for a turbojet with T04/Ta

= 10 flying at differentMach numbers (right plot). Results are evaluated for an ideal turbojet with γ = 1.3, QR/(cpTa) = 200,ηb = 1, and aa = 300 m/s.

the combustor and Bma representing the air flow rate associated with the bypass stream. The bypassratio B is large in engines for commercial applications (B ∼ 10 − 12) and much smaller (B ∼ 2 − 3) incompact turbofans for military applications, the latter engines typically incorporating an afterburner. Thecore engine of a turbofan is basically a turbojet equipped with two turbines, one to move the compressorand one to move the fan that compresses the bypass stream, so that Eq. (73) must be replaced with

Pt = Pc + Pf , (80)

where Pt is the combined power of both turbines and Pf is the power needed to drive the fan.

a

2

7

4 56

8

uefpauef

uepaue

3

Combustor

High-Pressure

Turbine

Low-Pressure

Turbine

Com

pressor

Fan

Bma

ma

Figure 15: Schematic view of a turbofan with two separate exhaust streams.

There are designs where the hot and cold streams mix upstream from the nozzle, leading to a single-exhaust jet engine. In the design of Fig. 15, however, the cold and hot streams remain separate up to theexit section. Because of the presence of distinct exhaust streams with speeds ue and uef , the resultingthrust equation (assuming pe = pa) takes the form

T = (ma + mf )ue +Bmauef − ma(1 +B)u, (81)

19

with corresponding specific thrust and TSFC given by

T

ma= (1 + f)ue +Buef − (1 +B)u and TSFC =

f

(1 + f)ue +Buef − (1 +B)u. (82)

7.1 Performance analysis

As seen in (82), the turbofan performance depends on f , ue, and uef . The evaluation of these threequantities, involving the fan pressure ratio Prf and the bypass ratio B as additional inputs, parallels thatdeveloped above for the turbojet. In particular, since the analysis of the hot stream evolution across thecore engine is identical to that of the turbojet, the fuel-to-air mass ratio f and the exit speed for the hotjet ue are given by (66) and (72), respectively. However, the temperature jump across the turbine T05/T04 ,appearing in (72), is not given by the turbojet expression (75) stemming from the power balance (73).The value of T05/T04 for a turbofan is determined instead by the condition (80), as explained below.

The enthalpy-entropy evolution of the hot stream is that given on the right-hand side of Fig. 11. Thecorresponding evolution of the cold stream, including the effects of the diffuser, fan, and fan nozzle arerepresented in Fig. 16, with indication of the idealized states 7s and 8s used in the definitions of the fanand fan nozzle adiabatic efficiencies ηf and ηfn.

h

s

p07

p =pa

7s

2s

8

u2ef2

sa

ha

2

7

p02

a8s

Figure 16: Enthalpy-entropy variation for the bypass stream.

Across the fan, with shaft power Pf , energy conservation provides

Pf = Bma(h07 − h02). (83)

On the other hand, the temperature jump across the fan is related to the fan pressure ratio Prf = p07/p02through the fan adiabatic efficiency

ηf =h7s − h2h7 − h2

=P

γ−1

γrf − 1

T07/T02 − 1→

T07

T02

= 1 +P

γ−1

γrf − 1

ηf. (84)

Similarly, the analysis of the fan nozzle involves the energy equation

h7 = h07 = h08 = h8 +u2ef2

→ h7 − h8 =u2ef2

(85)

20

along with the definition of the corresponding adiabatic efficiency

ηfn =h7 − h8h7 − h8s

=u2ef/2

h7[1− (pa/p7)γ−1

γ ]→

u2ef2

=ηfna

2a

γ − 1

(

T07

T02

)(

T02

Ta

)

1−1

(

p07p02

p02pa

)γ−1

γ

, (86)

the latter written with use of h7 = h07 = cpT07 , p7 = p07 , and cpTa = a2a/(γ − 1). Substitution of (60),(61), (84), and Prf = p07/p02 finally yields

u2ef2

=ηfna

2a

γ − 1

1 +P

γ−1

γrf − 1

ηf

(

1 +γ − 1

2M2

)

1−

1

Pγ−1

γrf

(

1 + ηdγ−12 M2

)

(87)

for the exit speed of the bypass stream.To close the solution we need the temperature jump across the turbine T05/T04 , necessary in evaluating

ue from (72). Using (62), (67) and (83) to evaluate the power balance (80) gives

(1+ f)(T04 −T05) = (T03 −T02)+B(T07 −T02) →T05

T04

= 1−

(

T03

T02

− 1)

+B(

T07

T02

− 1)

(1 + f)T04/Ta

(

T02

Ta

)

(88)

finally yielding

T05

T04

= 1−

[

1

ηc

(

Pγ−1

γrc − 1

)

+B

ηf

(

Pγ−1

γrf − 1

)]

(

1 + γ−12 M2

)

(1 + f)(T04/Ta)(89)

upon substitution of (60), (64), and (84).

7.2 Turbofan performance

To evaluate the specific thurst and the TSFC from (82) we use (66) to compute f , (72) and (89) tocompute ue, and (87) to compute uef . The influence of the bypass ratio and the fan pressure ratio onthe performance of the system is shown in Fig. 17 for an ideal turbofan flying at M = 0.8 with Prc = 20(all other parameters in the evaluation are those of the turbojet analyzed in Fig. 13). As can be seen,for these subsonic conditions turbofans with a large bypass ratio are an attractive option with reducedfuel consumption and increased specific thrust. In reality, the value of B is limited by the structuralweight and aerodynamic drag of the nacelle that is holding the engine as well as by compressibility effectsaffecting the flow near the tip of the fan blades.

8 Turboprop and turboshaft engines

Figure 18 shows a schematic view of a turboprop. Just like the turbofan, the turboprop uses as coreengine a turbojet with two turbines, namely, a “compressor” turbine connected through a shaft to thecompressor of the gas generator and a “power” turbine, mechanically independent of the gas-generatorrotor elements, which is used to move a large external propeller. The amount of air that flows through thepropeller is 25-30 times larger than that flowing though the gas generator, so in some sense a turboprop isa high-bypass-ratio engine. Because of the absence of a surrounding duct, near-tip compressibility effectsin propellers are more pronounced than those found in turbofans. For that reason, turboprops are suitedfor subsonic flight at only moderately large Mach numbers M <

∼ 0.7.

21

0 1 2 3 4 5

1.6

1.8

2

2.2

2.4

2.6

0 1 2 3 4 5

0.02

0.022

0.024

0.026

0.028

0.03

0.032

Figure 17: Variation of the specific thrust and TSFC with the bypass ratio B for an ideal turbofan withPrc = 20, M = 0.8, γ = 1.3, QR/(cpTa) = 200, ηb = 1, and aa = 300 m/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

Figure 18: Schematic view of a turboprop with free-turbine configuration.

In general, the thrust of a turboprop can be expected to have comparable contributions coming fromthe propeller and from the hot-exhaust jet. A turboshaft is a turboprop in which the hot gases areexpanded through the power turbine to a near-ambient pressure, so that the exhaust velocity gives anegligible contribution to the thrust. Because of the extra weight of the power turbine, the gearbox(necessary to guarantee that the propeller and the engine operate at optimum rotational speeds), andthe large propeller, with its accompanying pitch-control mechanism, turboprops are typically about 50% heavier than conventional turbojets with the same gas-generator size. The increased performance attakeoff and low flight speeds often compensate for this additional weight.

8.1 Performance analysis

The evolution of the gas through the turboprop of Fig. 18 is represented in the h− s diagram of Fig. 19,including a detailed view of the last stages (5 → 6 → 7 = e), corresponding to the flow through the powerturbine and the nozzle. The description of the evolution of the gas upstream from the power turbine (i.e.a → 5) is identical to that described above in sections 6.1.1–6.1.4.

Assuming that the exit pressure pe is equal to the ambient pressure, the total thrust provided by the

22

turboprop isT = (ma + mf )ue − mau+ Tpr, (90)

where Tpr is the contribution of the propeller. Ideally, the corresponding propeller thrust power Tpru shouldbe equal to the power provided by the power turbine Ppt. In reality, however, Tpru < Ppt partly becauseof aerodynamic losses, measured by the propeller efficiency ηpr, and partly because of mechanical losses,measured by the gearbox efficiency ηg. These effects are incorporated in the equation Tpru = ηprηgPpt,which can be used to write (90) in the form

T

ma= (1 + f)ue − u+

ηprηgPpt

uma. (91)

The power-turbine power Ppt and the exhaust speed ue are not independent, as shown on the right-hand side of Fig. 19. As can be seen, the energy available in the gas stream, measured by the difference∆h = h5 −h∗ between the enthalpy h5 behind the compressor turbine and the enthalpy h∗ correspondingto an isentropic expansion from p5 to the ambient pressure pa, is partly employed to move the propellerand partly employed to accelerate the gas stream through the nozzle. In the following, α∆h will denotethe fraction of the available enthalpy used in the power turbine, with the remaining enthalpy (1 − α)∆hcorrespondingly used to accelerate the stream. The extreme cases α = 1 and α = 0 represent theturboshaft and the turbojet, respectively.

h

s

p03

p =pa

3s

2s

a

4

7u2

2

Ptma+mf

sa

ha

2

3

p 04

5

Pcma

6 6s

*

5

7s 7

6

∆h

α∆h

(1− α)∆h

Ppt

ma+mf

u2e2

p5

pa

p6

h

s

Figure 19: Enthalpy-entropy diagram for the gas evolution through the turboprop of Fig. 18.

It is of interest to begin by evaluating the value of ∆h in terms of known quantities. From the definitionof ∆h = h5 − h∗ if follows that

∆h

ha=

h5 − h∗

ha=

T05

Ta

[

1−

(

pap05

)γ−1

γ

]

=T05

T04

T04

Ta

1−1

(

p05p04

p04p03

p03p02

p02pa

)γ−1

γ

. (92)

Substituting (61), (68) and the known pressure ratios Prc = p03/p02 and rc = p04/p03 finally gives

∆h

ha=

T05

T04

T04

Ta

1−

1(

1−1−T05

/T04

ηt

)

rγ−1

γc P

γ−1

γrc

(

1 + ηdγ−12 M2

)

, (93)

with T05/T04 obtained from (75).

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The analysis continues by considering the flow across the power turbine, whose energy balance reducesto

Ppt = (ma + mf )(h05 − h06). (94)

The adiabatic efficiency of the power turbine can be written as

ηpt =h5 − h6h5 − h6s

. (95)

Since h5 − h6s = α∆h and h5 − h6 = h05 − h06 , it follows that h05 − h06 = ηptα∆h, leading to

Ppt

ma= (1 + f)ηptα∆h (96)

upon substitution into (94).Similarly, the energy balance across the nozzle gives

h06 = h07 = h7 +u2e2

→ h6 − h7 =u2e2. (97)

Since the enthalpy difference is h6 − h7 = ηn(1− α)∆h in terms of the nozzle adiabatic efficiency

ηn =h6 − h7h6 − h7s

≃h6 − h7

(1− α)∆h, (98)

it follows from the last equation in (97) that

ue =√

2ηn(1− α)∆h (99)

Substituting (96) and (99) into (91) provides

T

ma= (1 + f)

√

2ηn(1− α)∆h− u+ (1 + f)ηprηgηptα∆h

u(100)

for the turboprop specific thrust, with ∆h evaluated from (93) supplemented with (75).

8.2 Maximum specific thrust

For given values of the flight speed u, fuel-to-air ratio f , available enthalpy ∆h, and efficiencies ηn andηprηgηpt, the optimum value αOPT of α that results in the maximum specific thrust (T /ma)max can beobtained by straightforward differentiation of (100) to give

∂(T /ma)

∂α= 0 → αOPT = 1−

ηn(ηprηgηpt)2

u2/2

∆h, (101)

giving(

T

ma

)

max

= u

[

(1 + f)ηprηgηpt∆h

u2−

(

1−(1 + f)ηn2ηprηgηpt

)]

(102)

upon substitution of α = αOPT into (100). Values of α < αOPT (a faster exhaust stream and a propellerwith smaller thrust power) or α > αOPT (a slower exhaust stream and a propeller with higher thrustpower) result is smaller values of the thrust coefficient.

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