The Turbojet Engine

4.3 Aircraft Gas Turbine Engines

4.3.1 The Turbojet Engine

An aircraft turbojet (TJ) engine is basically a gas generator fitted with an inlet and exhaust system. A schematic drawing of a TJ-engine is shown in Figure 4.2.

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1

0 2 3 4 5

Air flow

Burner

Compressor Turbine

9 F I G U R E 4 . 2 Fuel

Schematic drawing of a turbojet engine

The additional parameters that are needed to calculate the performance of a turbojet engine are the inlet and exhaust component efficiencies. These parameters will be defined in the next section. The station numbers in a turbojet engine are defined at the unperturbed flight condition, 0, inlet lip, 1, compressor face, 2, compressor exit, 3, burner exit, 4, turbine exit, 5, and the nozzle exit plane, 9.

4.3.1.1 The Inlet. The basic function of the inlet is to deliver the air to the compressor at therightMach numberM₂and therightquality, that is, low distortion. The subsonic compressors are designed for an axial Mach number ofM₂=0.5–0.6. Therefore, if the flight Mach number is higher than 0.5 or 0.6, which includes all commercial (fixed wing) transports and military (fixed wing) aircraft, then the inlet is required todecelerate the air efficiently. Therefore, the main function of an inlet is todiffuse or deceleratethe flow, and hence it is also called a diffuser. Flow deceleration is accompanied by the static pressure rise or what is known as theadverse pressure gradientin fluid dynamics. As one of the first principles of fluid mechanics, we learned that the boundary layers, being of a low-energy and momentum-deficit zone, facing an adverse pressure gradient environment tend to separate. Therefore, one of the challenges facing an inlet designer is to prevent inlet boundary layer separation. One can achieve this by tailoring the geometry of the inlet to avoid rapid diffusion or possibly through variable geometry inlet design. Now, it becomes obvious why an aircraft inlet designer faces a bigger challenge if the inlet has to decelerate a Mach 2 or 3 stream to the compressor face Mach number of 0.5 than an aircraft that flies at Mach 0.8 or 0.9. In the present section, we will examine thethermodynamicsof an aircraft inlet. This exciting area of propulsion, that is, inlet aerodynamics, will be treated in more detail in Chapter 6.

An ideal inlet is considered to provide areversible and adiabatic, that is, isentropic, compression of the captured flow to the engine. The adiabatic aspect is actually met in real inlets as well. This requirement says that there is no heat exchange between the captured stream and the ambient air, through the diffuser walls. We remember (from the Fourier law of heat conduction) that for heat transfer to take place through the inlet wall, we need to set up atemperature gradientacross the wall, such that

q_n≡ Q̇_n

A = −k𝜕T

𝜕n (4.1)

where n denotes the direction of heat conduction, k the thermal conductivity of the wall, andq_nthe heat flux, which is defined as the heat transfer rateQ̇_nper unit areaA.

0

t2 t2s

h

s

p⁰ p^t2

s

p^t0

t0

p^t

V₀²/2

V ²₂/2 h_t2 = h_t0

h_{t 2s}

h₀

s₀ s₂ 2

p²

d F I G U R E 4 . 3

h–sdiagram of an aircraft engine inlet flow under real and ideal conditions

Equation 4.1 signifies the importance of agradientto set up the heat transfer. The tem- perature gradient across the inlet wall (i.e., nacelle) is negligibly small, and consequently the inlet aerodynamics is considered to beadiabatic,even in a real flow. Therefore, it is only thereversibleaspect of our ideal inlet flow assumption that negates the realities of wall friction and any shocks, which are invariably present inrealsupersonic flows. The process of compression in a real inlet can be shown on theh-sdiagram of Figure 4.3.

Now, let us describe the information we observe in Figure 4.3. First, four isobars are shown on theh-sdiagram. From the lowest to the highest value, these arep₀, the static pressure at the flight altitude,p₂, the static pressure at the engine face,p_t2, the total pressure at the inlet discharge (or compressor face), andp_t0, the flight total pressure. We also note five thermodynamic states identified, as the flight static (0), the flight total (t0), the compressor face or inlet discharge static (2), and total (t2) and a stagnation state (t2s) that does not actually exist! Note, that the first four states, that is, 0, t0, 2, and t2, do all in reality exist and are measured. The states 0 and t0 are measured by a pitot-static tube on an aircraft, and the static and total pressures at the engine face are measured by a pressure rake or inlet pitot tubes. We note that the state (t2s) is at the intersection ofs₀and p_t2, which is arrived at isentropically from state (0) to a state that shares the same total pressure as that of a real inlet (t2). We note an entropy riseΔs_dacross the inlet as well.

The total enthalpy of the real inleth_t2is identified to be the same as the total enthalpy of flighth_t0. This is based on the earlier assertion that arealinlet flow may be considered to beadiabatic. It is to be noted that although the total energy of the fluid in a real inlet is not changing, there is anenergy conversionthat takes place in the inlet. The inlet kinetic energy is partially converted to the static pressure (rise) and partially it is dissipated into heat. It is the latter portion, that is, the dissipation, which renders the process irreversible and cause the entropy to rise. The entropy rise in an adiabatic process leads to a total pressure lossΔp_tfollowing the combined first and second law of thermodynamics, that is, Gibbs equation, according to

p_t2 p_t0 =e⁻

s2−s0

R =e^−ΔsR (4.2)

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From Figure 4.3, we also note that the states (t0) and (0) are separatedisentropicallyby the amount ofV₀²∕2, as expected from the definition of stagnation state and its relation to the static state. The states (t2s) and (0) are separated isentropically by a kinetic energy amount, which producesp_t2as its stagnation state and obviously the rest of the kinetic energy, that is, the gap between the states (t0) and (t2s), represents the amount of dissipated kinetic energy into heat. Therefore, the smaller the gap between the states (t2s) and (t0), the more efficient will the diffuser flow process be. Consequently, we use the fictitious state (t2s) in a definition of inlet efficiency (or a figure of merit) known as theinlet adiabatic efficiency.

Symbolically, the inlet adiabatic efficiency is defined as 𝜂d≡ h_t2s−h₀

h_t2−h₀ = (V²∕2)_ideal V²

0∕2 (4.3)

The practical form of the above definition is derived when we divide the numerator and denominator byh₀to get

𝜂d= h_t2s

h₀ −1 h_t2

h₀ −1

= T_t2s

T₀ −1 h_t0

h₀ −1

= (p_t2

p₀ )^𝛾−1

𝛾 −1

𝛾−1 2 M₀²

= (p_t2

p₀ )^𝛾−1

𝛾 −1

𝜏r−1 (4.4)

where we have used the isentropic relation between the states (t2s) and (0). Note that the only unknown in Equation 4.4 isp_t2for a given flight altitudep₀, flight Mach numberM₀, and an inlet adiabatic efficiency𝜂d. We can separate the unknown termp_t2and write the following expression:

p_t2 p₀ =

{ 1+𝜂d

𝛾−1 2 M₀²

} ^𝛾

𝛾−1

(4.5) It is interesting to note that Equation 4.5 recovers the isentropic relation for a 100%

efficient inlet or𝜂d=1.0. Another parameter, or afigure of merit, that describes the inlet performance is the total pressure ratio between the compressor face and the (total) flight condition. This is given a symbol𝜋d and is often referred to as theinlet total pressure recovery:

𝜋d≡ p_t2

p_t0 (4.6)

As expected, the two figures of merit for an inlet, that is,𝜂dor𝜋d, are not independent from each other and we can derive a relationship between𝜂dand𝜋dworking the left-hand side of Equation 4.5, as follows:

p_t2 p₀ = p_t2

p_t0 p_t0

p₀ = {

1+𝜂d

𝛾−1 2 M²₀

} ^𝛾

𝛾−1

(4.6a)

𝜋d= {

1+𝜂d

𝛾−1 2 M₀²

} ^𝛾

𝛾−1

p_t0 p₀

=

⎧⎪

⎨⎪

⎩ 1+𝜂d

𝛾−1 2 M₀² 1+𝛾−1

2 M²₀

⎫⎪

⎬⎪

⎭

𝛾−1𝛾

= [

1+𝜂d

𝛾−1 2 M₀

] ^𝛾

𝛾−1

𝜋r

(4.6b)

Stagnation point

Separation

Flow area A₂ Inner cowl

Outer cowl Capture

streamtube

2

0

Flow area A₀ F I G U R E 4 . 4

Schematic drawing of an inlet flow at low-speed and/or takeoff flight condition (note:A₀>A₂)

Therefore, Equation 4.6b relates the inlet total pressure recovery𝜋dto the inlet adiabatic efficiency𝜂dat any flight Mach numberM₀. We note that Equation 4.6b as𝜂d→1, then 𝜋d→1 as well, as expected. Figure 4.3 also shows the static state 2, which shares the same entropy as the total state t2 and lies below it by the amount of kinetic energy at 2, namely,V²

2∕2. Compare the kinetic energy at 2 and at 0, shown in Figure 4.3, and then justify the static pressure rise achieved in the inlet (diffuser),p₂−p₀.

So far, we have considered the cruise condition or the high-speed end of the flight envelope and have thought of inlets as diffusers. Under low-speed or takeoff conditions, the captured stream tube will instead undergo acceleration to the engine face and as such the inlet acts like a nozzle! In Figure 4.4, the schematics of a captured stream tube, under a low-speed flight condition, are shown. Note that the stagnation point is on the outer cowl and flow accelerates to the engine face (A₀>A₂). The aerodynamics of the outer nacelle geometry affects the drag divergence of the inlet and plays a major role in the propulsion system integration studies of engine installation. However, the cycle analysis phase usually disregards the external performance and concentrates on the internal evaluation of the propulsion system.

The nature of the flow path into the inlet at low forward speed or under static engine testing conditions requires the inlet to act as a nozzle; therefore, it explains the use of a bellmouth in static test rigs. Figure 4.5 shows the schematic drawing of an inlet configuration in a static test rig. In Figure 4.5, we note that the static pressure along the stream lines drop from the ambientp₀to the engine face pressure ofp₂. Also, we note

p₀

Engine face

Bellmouth inlet p

2

0

p₂ Ambient pressure

V V₂

V₀~0

x

x x

F I G U R E 4 . 5 Bellmouth inlet guides the flow smoothly into the engine on a static test rig

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157

that the gas speed starts at almost zero, since the captured flow areaA₀is fairly large, and continues to grow to the engine face speed ofV₂.

In summary, we have learned that

the inlet flow may be considered to be adiabatic, that is,h_t2=h_t0

the inlet flow is always irreversible, that is,p_t2<p_t0, with viscous dissipation in the boundary layer and in a shock as the sources of irreversibility (s₂>s₀)

there are two figures of merit that describe the extent oflossesin the inlet and these are𝜂_dand𝜋_d

the two figures of merit are related (via Equation 4.6b)

in cruise, A₀< A₂and hence a diffusing passage and at low speed or takeoff, A₀>A₂, that is, a nozzle

the outer nacelle geometry of an inlet dictates the drag divergence and high angle of attack characteristics of the inlet and is crucial for the installed performance.

E X A M P L E 4 . 1

An aircraft is flying at an altitude where the ambient static pressure is p₀ = 10 kPa and the flight Mach number is M₀ =0.85. The total pressure at the engine face is mea- sured to bep_t2=15.88 kPa. Assuming the inlet is adiabatic and𝛾=1.4, calculate

(a) the inlet total pressure recovery𝜋d

(b) the inlet adiabatic efficiency𝜂d

(c) the nondimensional entropy rise caused by the inlet Δs_d∕R

SOLUTION

We first calculate the flight total pressure p_t0, and from definition of𝜋d(i.e., Equation 4.6), the inlet total pressure recovery.

p_t0=p₀[1+(𝛾−1)M₀²∕2]^𝛾−1𝛾 =10 kPa[1+0.2(0.85)²]^3.5

=16.04 kPa

𝜋d≡p_t2∕p_t0=15.88∕16.04=0.990

Inlet adiabatic efficiency𝜂dis calculated from Equation 4.4

𝜂d = [(p_t2

p₀ )^𝛾−1_𝛾

−1

]/ (𝛾−1 2 M²₀

)

=[(15.88∕10)^0.2857−1]∕[0.2(0.85)²]≅0.9775 The entropy rise is linked to the inlet total pressure loss parameter𝜋dvia Equation 4.2,

Δs_d∕R= − ln( 𝜋d

)= − ln(0.99)≅0.010

4.3.1.2 The Compressor. The thermodynamic process in a gas generator begins with the mechanical compression of air in the compressor. As the compressor discharge contains higher energy gas, that is, the compressed air, it requires external power to operate. The power comes from the turbine via a shaft, as shown in Figure 4.1 in case of an operating gas turbine engine. Other sources of external power may be used tostartthe engine, which are in the form of electric motor, air turbine, and hydraulic starters. The flow of air in a compressor is considered to be an essentially adiabatic process, which suggests that only anegligibleamount of heat transfer takes place between the air inside and the ambient air outside the engine. Therefore, even in a real compressor analysis, we will

still treat the flow as adiabatic. Perhaps a more physical argument in favor of neglecting the heat transfer in a compressor can be made by examining the order of magnitude of the energy transfer sources in a compressor. The power delivered to the medium in a compressor is achieved by one or more rows of rotating blades (called rotors) attached to one or more spinning shafts (typically referred to asspools). Each rotor blade, which changes thespin(or swirl) of a medium, will experience a countertorque as a reaction to its own action on the fluid. If we denote the rotor torque as𝜏, then the power delivered to the medium by the rotor spinning at the angular speed𝜔follows Newtonian mechanics, that is,

℘=𝜏⋅𝜔 (4.7)

A typical axial-flow compressor contains hundreds of rotor blades (distributed over several stages), which interact with the medium according to the above equation. Therefore, the rate of mechanical energy transfer in a typical modern compressor is usually measured in mega-Watt (MW) and is several orders of magnitude larger than heat transfer through the compressor wall. Symbolically, we may present this as

℘c⋙Q̇_w (4.8)

Therefore, a real compressor flow is assumed to be adiabatic.

Q_w≡0

℘c

C

2 3

℘^c=m₀(h_t3−h_t2) πc≡(p_t3/p_t2) τc≡T_t3/T_t2

m₀

As a real process, however, the presence of wall friction acting on the medium through the boundary layer and shock waves caused by the relative supersonic flow through compressor blades will render the process irreversible. Therefore, the sources of irreversibility in a compressor are due to the viscosity of the medium and its consequences (boundary layer formation, wakes, vortex shedding) and the shock formation in relative supersonic passages.

The measure of irreversibility in a compressor may be thermodynamically defined through some form of compressor efficiency. There are two methods of compressor efficiency definitions. These are

compressor adiabatic efficiency𝜂c compressor polytropic efficiencye_c.

To define the compressor adiabatic efficiency 𝜂c, we depict a “real” compression process on an h–s diagram, as shown in Figure 4.6 and compare it to an ideal, that is, isentropic process. The state “t2” represents the total (or stagnation) state of the gas entering the compressor, typically designated by p_t2 andT_t2. An actual flow in a

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159

Δs_c s t3

t2

p_{t 2} p_{t 3}

Actual power consumed

Ideal power required

h

t3s F I G U R E 4 . 6

Enthalpy–entropy (h–s) diagram of an actual and ideal compression process (note𝚫sc)

compressor will follow the solid line from “t2” to “t3”, thereby experiencing an entropy rise in the process,Δs_c. The actual total state of the gas is designated by “t3” in Figure 4.6.

The ratiop_t3∕p_t2is known as the compressor pressure ratio, with a shorthand notation𝜋c. The compressor total temperature ratio is depicted by, the shorthand notation,𝜏c≡T_t3∕T_t2. Since the state “t3” is the actual state of the gas at the exit of the compressor and is not achieved via an isentropic process, we cannot expect the isentropic relation between𝜏_c and𝜋c, to hold, that is,

𝜏c≠𝜋^𝛾

−1

c𝛾 (4.9)

It can be seen from Figure 4.6 that the actual𝜏cis larger than the ideal, that is, isentropic 𝜏c, which is denoted by the end state T_t3s. This fact actually helps with the exponent memorization in a real compression process. The compressor adiabatic efficiency is the ratio of the ideal power required to the power consumed by the compressor, that is,

𝜂c≡ h_t3s−h_t2

h_t3−h_t2 = Δht,isentropic

Δh_t,actual (4.10)

The numerator in Equation 4.10 is the power-per-unit mass flow rate in anideal compressor and the denominator is the power-per-unit mass flow rate in the actual compressor. If we divide the numerator and denominator of Equation 4.10 byh_t2, we get

𝜂c= T_t3s∕T_t2−1

T_t3∕T_t2−1 (4.11)

Since the thermodynamic states “t3s” and “t2” are on the same isentrope, the temperature and pressure ratios are then related via the isentropic formula, that is,

T_t3s T_t2 =

(p_t3s p_t2

)^𝛾−1

𝛾 =

(p_t3 p_t2

)^𝛾−1

𝛾 =𝜋^𝛾

−1

c𝛾 (4.12)

Therefore, compressor adiabatic efficiency may be expressed in terms of compressor pressure and temperature ratios as

𝜂c= 𝜋

𝛾−1𝛾

c −1

𝜏c−1 (4.13)

Equation 4.13 involves three parameters,𝜂_c,𝜋_c, and𝜏_c. It can be used to calculate𝜏_cfor a given compressor pressure ratio and adiabatic efficiency. As compressor pressure ratio and efficiency are typically known and assumed quantities in a gas turbine cycle analysis, the only unknown in Equation 4.13 is𝜏c.

A second efficiency parameter in a compressor ispolytropic efficiency e_c. As might be expected, compressor adiabatic and polytropic efficiencies are related. The definition of compressor polytropic efficiency is

e_c≡ dh_ts

dh_t (4.14)

It is interesting to compare the definition of compressor adiabatic efficiency, involving finite jumps (Δh_t), and the polytropic efficiency, which takes infinitesimal steps (dh_t).

The conclusion can be reached that the polytropic efficiency is actually the adiabatic efficiency of a compressor withsmallpressure ratio. Consequently, compressor polytropic efficiency is also calledsmall stage efficiency. From the combined first and second law of thermodynamics, we have

T_tds=dh_t−dp_t 𝜌t

(4.15)

We deduce that for an isentropic process, that is,ds=0,dh_t=dh_tsand, therefore,

dh_ts= dp_t 𝜌t

(4.16)

If we substitute Equation 4.16 in 4.14 and replace density with pressure and temperature from the perfect gas law, we get

e_c= dp_t

p_t dh_t RT_t

= dp_t

p_t C_pdT_t

RT_t

= dp_t

p_t 𝛾 𝛾−1

dT_t T_t

(4.17)

dp_t p_t = 𝛾e_c

𝛾−1 dT_t

T_t (4.18)

which can now be integrated between the inlet and exit of the compressor to yield

p_t3 p_t2 =𝜋c=

(T_t3 T_t2

)^𝛾ec

𝛾−1 =(

𝜏c

)^𝛾ec

𝛾−1 (4.19)

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161

To express the compressor total temperature ratio in terms of compressor pressure ratio and polytropic efficiency, Equation 4.19 can be rewritten as

𝜏c=𝜋

𝛾−1 𝛾ec

c (4.20)

The presence ofe_cin the denominator of the above exponent (Equation 4.20) causes the exponent of𝜋cto be greater than its isentropic exponent (which is (𝛾−1)/𝛾), therefore,

𝜏c,real≻ 𝜏c,isentropic or T_t3>T_t3s (4.21) as noted earlier (Figure 4.6). The physical argument for higher actualT_tthan the isentropic T_t(to achieve the same compressor pressure ratio) can be made on the grounds thatlost workto overcome the irreversibility in the real process (friction, shock) is converted into heat, a higher exitT_t is reached in a real machine due to dissipation. On the contrary, for a given compressor pressure ratio𝜋c, an ideal compressor consumes less power than an actual compressor (the factor being 𝜂c, as defined earlier). Again, the absence of dissipative mechanisms, leading to lost work, is cited as the reason for a reversible flow machine to require less power to run.

Now, to relate the two types of compressor efficiency description,e_c and𝜂c, we may substitute Equation 4.20 into Equation 4.13, to get

𝜂c= 𝜋

𝛾−1𝛾

c −1

𝜏c−1 = 𝜋

𝛾−1𝛾

c −1

𝜋

𝛾−1 𝛾.ec

c −1

(4.22)

Equation 4.22 is plotted in Figure 4.7.

The compressor adiabatic efficiency𝜂cis a function of compressor pressure ratio, while the polytropic efficiency is independent of it. Consequently, in a cycle analysis, we usually assume the polytropic efficiencye_cas the figure of merit for a compressor (and turbine) and then we can maintaine_cas constant in our engine off-design analysis. Typical values for the polytropic efficiency in modern compressors are in the range 88–92%.

So far, we have studied the thermodynamics of the mechanical compression through the stagnation states of the working medium. It is also instructive to examine the static states of the gas in a compressor as well. The mental connection between these states will help us with the fluid dynamics of the compressors. Figure 4.8 is basically an elaborate version of the earlier Figure 4.6, to which we have added the static states of the gas at the inlet and exit of the compressor. These states, that is, the local stagnation and static states, when plotted on theh–sdiagram, are distanced from each other only by the amount of local (specific) kinetic energyV²/2 and on the same isentrope.

The flow velocity at the compressor inlet,V₂, is nearly the same as the fluid velocity at the compressor exit, V₃, by design. This explains why the vertical gap shown in Figure 4.8, between the static and stagnation states at 2 and 3 is nearly the same, that is,

V₂²∕2≈V₃²∕2 (4.23)

This design philosophy, that is,V₂=V₃, is a part of the so-called constant through flow assumption often used in compressor aerodynamics. We will return to this topic, in detail, in Chapter 8.