bjc 5.1 1/4/10

C

HAPTER

5T

HE

TURBOFAN

CYCLE

5.1 T

URBOFAN

THRUST

The figure below illustrates two generic turbofan engine designs.

Figure 5.1 Engine numbering and component notation

The upper figure shows a modern high bypass ratio engine designed for long distance cruise atsubsonic Mach numbers around 0.83 typical of a commercial aircraft. The fan utilizes a singlestage composed of a large diameter fan (rotor) with wide chord blades followed by a single

1 2 13 18 1e

3 4

4.5

5

0

7 8 e

referencer

diffuserd

fan1c

fan nozzle1n

compressorc

burnerb

turbinet

nozzlen

0r

reference

1 2d

diffuser

2.5

1cfan

c3 4

compressor burnerb

5turbine

t 6 7 8 eafterburner

a nnozzle

4.5

2.5

Commercial Turbofan

Military Turbofan

1.5

1.5

ma fan

macore

macorem f+

m f

Turbofan thrust

1/4/10 5.2 bjc

nozzle stage (stator). The bypass ratio is 5.8 and the fan pressure ratio is 1.9. The lower figureshows a military turbofan designed for high performance at supersonic Mach numbers in therange of 1.1 to 1.5. The fan on this engine has three stages with an overall pressure ratio of about6 and a bypass ratio of only about 0.6. One of the goals of this chapter is to understand whythese engines look so different in terms the differences in flight condition for which they aredesigned. In this context we will begin to appreciate that the thermodynamic and gasdynamicanalysis of these engines defines a continuum of cycles as a function of Mach number. We hada glimpse of this when we determined that the maximum thrust turbojet is characterized by

. (5.1)

For fixed turbine inlet temperature and altitude, as the Mach number increases the optimumcompression decreases and at some point it becomes desirable to convert the turbojet to a ram-jet. We will see a similar kind of trend emerge for the turbofan where it replaces the turbojet asthe optimum cycle for lower Mach numbers. Superimposed on all this is a technology trendwhere, with better materials and cooling schemes, the allowable turbine inlet temperatureincreases. This tends to lead to an optimum cycle with higher compression and higher bypassratio at a given Mach number.

The thrust equation for the turbofan is similar to the usual relation except that it includes thethrust produced by the fan.

(5.2)

The total air mass flow is

. (5.3)

The fuel/air ratio is defined in terms of the total air mass flow.

. (5.4)

The bypass fraction is defined as

(5.5)

and the bypass ratio is

. (5.6)

Note that

!cmax thrust

!"!r

---------=

T macoreUe U0–( ) ma fan

U1e U0–( ) m f Ue Pe P0–( )Ae P1e P0–( )A1e+ + + +=

ma macorema fan

+=

fm fma-------=

Bma fan

ma fanmacore

+-------------------------------------=

#ma fan

macore

---------------=

The ideal turbofan cycle

bjc 5.3 1/4/10

. (5.7)

5.2 T

HE

IDEAL

TURBOFAN

CYCLE

In the ideal cycle we will make the usual assumption of isentropic flow in the inlet, fan, com-pressor, turbine and fan and core nozzles as well as the assumption of low Mach number heataddition in the burner. The fan and core nozzles are assumed to be fully expanded. The assump-tions are

(5.8)

(5.9)

. (5.10)

For a fully expanded exhaust the normalized thrust is

(5.11)

or, in terms of the bypass ratio with

. (5.12)

5.2.1 T

HE

FAN

BYPASS

STREAM

First work out the velocity ratio for the fan stream

. (5.13)

The exit Mach number is determined from the stagnation pressure.

. (5.14)

Since the nozzle is fully expanded and the fan is assumed to behave isentropically, we can write

(5.15)

#B

1 B–-------------= B #

1 #+-------------=

P1e P0= Pe P0=

$d 1= $b 1= $n 1= $n1 1=

$c !c

%% 1–------------

= $c1 !c1

%% 1–------------

= $t !t

%% 1–------------

=

Tmaa0------------- M0 1 B– f+( )

UeU0------- 1–& '( )* +

BU1eU0--------- 1–& '( )* +

f+ +, -. /0 1=

f 1«

Tmaa0------------- M0

11 #+-------------& '* +

UeU0------- 1–& '( )* + #

1 #+-------------& '* +

U1eU0--------- 1–& '( )* +

+, -. /0 1=

U1eU0---------

M1eM0----------

T 1eT 0---------=

Pt1e P0$r$c1 P1e 1 % 1–2

------------M1e2+& '

* +

%% 1–------------

= =

!r!c1 1 % 1–2

------------M1e2+=

The ideal turbofan cycle

1/4/10 5.4 bjc

therefore

. (5.16)

The exit temperature is determined from the stagnation temperature.

. (5.17)

Noting (5.15) we can conclude that for the ideal fan

. (5.18)

The exit static temperature is equal to the ambient static temperature. The velocity ratio of thefan stream is

. (5.19)

5.2.2 THE CORE STREAM

The velocity ratio across the core is

. (5.20)

The analysis of the stagnation pressure and temperature is exactly the same as for the idealturbojet.

. (5.21)

Since the nozzle is fully expanded and the compressor and turbine operate ideally the Machnumber ratio is

. (5.22)

The temperature ratio is also determined in the same way in terms of component temperatureparameters

. (5.23)

M1e2

M02

----------!r!c1 1–!r 1–

----------------------=

T te T 0!r!c1 T 1e 1 % 1–2

------------M1e2+& '

* += =

T 1e T 0=

U1eU0---------& '( )* + 2 !r!c1 1–

!r 1–----------------------=

UeU0-------& '( )* + 2 Me

M0--------& '( )* + 2 T e

T 0------=

Pte P0$r$c$t Pe 1 % 1–2

------------Me2+& '

* +

%% 1–------------

= =

Me2

M02

--------!r!c!t 1–!r 1–

------------------------& '( )* +

=

T te T 0!r!d!c!b!t!n=

The ideal turbofan cycle

bjc 5.5 1/4/10

In the ideal turbofan we assume that the diffuser and nozzle flows are adiabatic and so

(5.24)

from which is determined

. (5.25)

The velocity ratio across the core is

. (5.26)

5.2.3 TURBINE - COMPRESSOR - FAN MATCHING

The work taken out of the flow by the high and low pressure turbine is used to drive both thecompressor and the fan.

(5.27)

Divide (5.27) by and rearrange. The work matching condition for a turbofan is

. (5.28)

The approximation is generally a pretty good one for a turbofan. Using this approxima-tion the work matching condition becomes

(5.29)

where the bypass ratio appears for the first time. If the bypass ratio goes to zero the matchingcondition reduces to the usual turbojet formula.

5.2.4 FUEL/AIR RATIO

The fuel/air ratio is determined from the energy balance across the burner.

. (5.30)

Divide (5.30) by and rearrange. The result is

T te T 0!r!c!b!t T e 1 % 1–2

------------Me2+& '

* + T e!r!c!t= = =

T eT 0------

!"!r!c----------=

UeU0-------& '( )* + 2 !r!c!t 1–

!r 1–------------------------& '( )* + !"

!r!c----------=

macorem f+( ) ht4 ht5–( ) macore

ht3 ht2–( ) ma fanht13 ht2–( )+=

maC pT 0

!t 1!r!"----- 1 B–

1 B– f+------------------------ !c 1–( )

B1 B– f+------------------------ !c1 1–( )+

, -. /0 1–=

f 1«

!t 1!r!"----- !c 1–( ) # !c1 1–( )+{ }–=

#

m f h f ht4–( ) macoreht4 ht3–( )=

maC pT 0

Maximum specific impulse ideal turbofan

1/4/10 5.6 bjc

. (5.31)

5.3 MAXIMUM SPECIFIC IMPULSE IDEAL TURBOFANThe specific impulse is

. (5.32)

Substitute (5.12) and (5.31) into (5.32). The result is

. (5.33)

The question is: what value of maximizes the specific impulse? Differentiate (5.33) withrespect to and note that appears in (5.29). The result is

. (5.34)

We can write (5.34) as

(5.35)

or

. (5.36)

This becomes

. (5.37)

From (5.19),the expression in parentheses on the left side of (5.37) is

. (5.38)

f 11 #+-------------& '* +

!" !r!c–! f !"–

----------------------=

Ispga0

---------- Tm f g----------& '* + g

a0-----& '* + T

maa0-------------& '* + 1

f---& '* += =

Ispga0

---------- M0! f !"–!" !r!c–----------------------& '( )* + Ue

U0------- 1–& '( )* +

#U1eU0--------- 1–& '( )* +

+, -. /0 1=

#

# #

#22 Ispg

a0----------& '( )* +

#22 Ue

U0-------& '( )* + U1e

U0--------- 1–& '( )* +

+ 0= =

12 Ue U0( )---------------------------

#22 Ue

2

U02

-------& '( )( )* + U1e

U0--------- 1–& '( )* +

+ 0=

12 Ue U0( )---------------------------

!"!r 1–--------------& '( )* +

#2

2!t U1eU0--------- 1–& '( )* +

–=

12 Ue U0( )---------------------------

!r !c1 1–( )

!r 1–----------------------------& '( )* + U1e

U0--------- 1–& '( )* +

=

12 Ue U0( )---------------------------

U1eU0---------& '( )* + 2

1–& '( )* + U1e

U0--------- 1–& '( )* +

=

Maximum specific impulse ideal turbofan

bjc 5.7 1/4/10

Factor the left side of (5.38) and cancel common factors. The velocity condition for a maximumimpulse ideal turbofan is

. (5.39)

According to this result for an ideal turbofan one would want to design the turbine such that thevelocity increment across the fan was twice that across the core in order to achieve maximumspecific impulse. Recall that depends on through (5.29) (and weakly though (5.31)

which we neglect). The value of that produces the condition (5.39) corresponding to the max-imum impulse ideal turbofan is

. (5.40)

The figure below shows how the optimum bypass ratio (5.40) varies with flight Mach numberfor a given set of engine parameters.

Figure 5.2 Ideal turbofan bypass ratio for maximum specific impulse as a function of Mach number.

U1eU0--------- 1–& '( )* +

2UeU0------- 1–& '( )* +

=

Ue U0 #

#

#max impulse ideal turbofan1

!c1 1–( )---------------------- ×=

!"!r!c---------- 1–& '( )* +

!c 1–( )!"

!r2!c

---------- !r 1–( )14---!r 1–!r

--------------& '( )* + !r!c1 1–

!r 1–----------------------& '( )* + 1 2

1+& '( )* +

–+

, -3 3. /3 30 1

!r

#max impulse ideal turbofan

!c 2.51=!c1 1.2=

!" 8.4=

1.5 2 2.5 3 3.5 4 4.5 5

2.5

5

7.5

10

12.5

15

17.5

20

Turbofan thermal efficiency

1/4/10 5.8 bjc

It is clear from this figure that as the Mach number increases the optimum bypass ratiodecreases until a point is reached where one would like to get rid of the fan altogether and con-vert the engine to a turbojet. For the ideal cycle the turbojet limit occurs at an unrealisticallyhigh Mach number of approximately 3.9. Non-ideal component behavior reduces this Machnumber considerably.

The next figure provides another cut on this issue. Here the optimum bypass ratio is plottedversus the fan temperature (or pressure) ratio. Several curves are shown for increasing Machnumber.

Figure 5.3 Ideal turbofan bypass ratio for maximum specific impulse as a function of fan temperature ratio. Plot shown for several Mach numbers

It is clear that increasing the fan pressure ratio leads to an optimum at a lower bypass ratio. Thecurves all seem to allow for optimum systems at very low fan pressure ratios and high bypassratios. This is an artifact of the assumptions underlying the ideal turbofan. As soon as non-idealeffects are included the low fan pressure ratio solutions reduce to much lower bypass ratios. Tosee this we will compute several non-ideal cases.

5.4 TURBOFAN THERMAL EFFICIENCYRecall the definition of thermal efficiency from Chapter 2.

(5.41)

1.5 2 2.5 3 3.5 4 4.5 5

2

4

6

8

10

#max impulse ideal turbofan

!c1

!" 8.4=!c 2.51=

!r 1.02=

!r 1.2=

!r 1.6=

!r 2.5=

!r 3.5=

4th

Power to the vehicle 5 kinetic energy of air second

------------------------------------------------------- 5 kinetic energy of fuel second

---------------------------------------------------------+ +

m f h f---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

Turbofan thermal efficiency

bjc 5.9 1/4/10

For a turbofan with a core and bypass stream the thermal efficiency is

(5.42)

If both exhausts are fully expanded so that the thermal efficiencybecomes

(5.43)

which reduces to

(5.44)

We can recast (5.44) in terms of enthalpies using the following relations

(5.45)

where the fan and core nozzle streams are assumed to be adiabatic. Now

(5.46)

Rearrange (5.46) to read

4th

T U0

macoreUe U0–( )

2

2----------------------------------------------

ma fanU1e U0–( )

2

2----------------------------------------------+

m f Ue U0–( )2

2------------------------------------

m f U0( )2

2----------------------–+ +

m f h f-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

Pe P0 ; P1e P0= =

4th

macoreUe U0–( ) ma fan

U1e U0–( ) m f Ue+ +( )U0

m f h f--------------------------------------------------------------------------------------------------------------------------------- +=

macoreUe U0–( )

2

2----------------------------------------------

ma fanU1e U0–( )

2

2----------------------------------------------+

m f Ue U0–( )2

2------------------------------------

m f U0( )2

2----------------------–+

m f h f-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

4th

macore

Ue2

-------2 U0

2-------

2–& '

* + m fUe

2

2------- m+ a fan

U1e2

---------2 U0

2-------

2–& '

* ++

m f h f-----------------------------------------------------------------------------------------------------------------------------------=

m f h f ht4–( ) macoreht4 ht3–( )=

ht5 heUe

2

2-------+=

ht13 h1eU1e

2

2---------+=

4th

macoreht5 he–( ) ht0 h0–( )–( ) ma fan

ht13 h1e–( ) ht0 h0–( )–( ) m f ht5 he–( )+ +[ ]

m f macore+( )ht4 macore

ht3–-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

Turbofan thermal efficiency

1/4/10 5.10 bjc

(5.47)

Recall the turbofan work balance (5.27). This relation can be rearranged to read

(5.48)

where it has been assumed that the inlet is adiabatic . Now use (5.48) to replace thenumerator or denominator in the first term of (5.47). The thermal efficiency finally reads

(5.49)

The expression in (5.49) for the heat rejected during the cycle,

(5.50)

brings to mind the discussion of thermal efficiency in Chapter 2. The heat rejected comprisesheat conduction to the surrounding atmosphere from the fan and core mass flows plus physicalremoval from the thermally equilibrated nozzle flow of a portion equal to the added fuel massflow. From this perspective the added fuel mass carries its fuel enthalpy into the system and theexhausted fuel mass carries its ambient enthalpy out of the system and there is no net massincrease or decrease to the system.

The main assumptions underlying (5.49) are that the engine operates adiabatically, the shaftmechanical efficiency is one and the burner combustion efficiency is one. Engine componentsare not assumed to operate ideally - they are not assumed to be isentropic.

5.4.1 THERMAL EFFICIENCY OF THE IDEAL TURBOFAN

For the ideal cycle assuming constant equation (5.49) becomes, in terms of temperatures,

. (5.51)

Using (5.25) Equation (5.51) becomes

(5.52)

4th

macorem f+( )ht5 ma fan

ht13 macorema fan

+( )ht0–+

m f macore+( )ht4 macore

ht3–--------------------------------------------------------------------------------------------------------------------------------------- –=

macorehe h0–( ) ma fan

h1e h0–( ) m f he+ +

m f macore+( )ht4 macore

ht3–------------------------------------------------------------------------------------------------------------

m f macore+( )ht4 macore

ht3– macorem f+( )ht5 ma fan

ht13 macorema fan

+( )ht0–+=

ht2 ht0=

4th 1Qrejected during the cycleQinput during the cycle

----------------------------------------------------– 1macore

m f+( ) he h0–( ) ma fanh1e h0–( ) m f h0+ +

m f macore+( )ht4 macore

ht3–---------------------------------------------------------------------------------------------------------------------------------–= =

Qrejected during the cycle macorem f+( ) he h0–( ) ma fan

h1e h0–( ) m f h0+ +=

C p

4thideal turbofan1

1 1 #+( ) f+( )T e T 0–1 1 #+( ) f+( )T t4 T t3–

--------------------------------------------------------------– 1 1!r!c----------& '* +

1 1 #+( ) f+( )T eT 0------ 1–

1 1 #+( ) f+( )!"!r!c---------- 1–

-----------------------------------------------------------

& '( )( )( )( )* +

–= =

4thideal turbofan1 1

!r!c----------& '* +–=

The non-ideal turbofan

bjc 5.11 1/4/10

which is identical to the thermal efficiency of the ideal turbojet. Notice that for the ideal turbo-fan with the heat rejected by the fan stream is zero. Therefore the thermal efficiencyof the ideal turbofan is independent of the parameters of the fan stream.

5.5 THE NON-IDEAL TURBOFANThe fan, compressor and turbine polytropic relations are

(5.53)

where is the polytropic efficiency of the fan. The polytropic efficiencies , and

are all less than one. The inlet, burner and nozzles all operate with some stagnation pres-sure loss.

. (5.54)

5.5.1 NON-IDEAL FAN STREAM

The stagnation pressure ratio across the fan is

. (5.55)

The fan nozzle is still assumed to be fully expanded and so the Mach number ratio for the non-ideal turbofan is

. (5.56)

The stagnation temperature is (assuming the inlet and fan nozzle are adiabatic)

(5.57)

and

. (5.58)

h1e h0=

$c1 !c1

%4pc'% 1–-------------

= $c !c

%4pc% 1–------------

= $t !t

%% 1–( )4pe

---------------------------

=

4pc' 4pc 4pc'

4pe

$d 1< $n1 1< $n 1< $b 1<

Pt1e P0$r$d$c1$n1 P1e 1 % 1–2

------------M1e2+& '

* +

%% 1–------------

= =

M1e2

M02

----------!r!c1

4pc' $d$n1( )

% 1–%

------------1–

!r 1–----------------------------------------------------------=

T t1e T 0!r!c1 T 1e 1 % 1–2

------------M1e2+& '

* + T 1e!r!c14pc' $d$n1( )

% 1–%

------------= = =

T 1eT 0---------

!c11 4pc'–

$d$n1( )

% 1–%

-------------------------------------------=

The non-ideal turbofan

1/4/10 5.12 bjc

Now the velocity ratio across the non-ideal fan is

. (5.59)

5.5.2 NON-IDEAL CORE STREAM

The stagnation pressure across the core is.

. (5.60)

The core nozzle is fully expanded and so the Mach number ratio is

. (5.61)

In the non-ideal turbofan we continue to assume that the diffuser and nozzle flows are adiabaticand so

(5.62)

from which is determined

. (5.63)

The velocity ratio across the core is

. (5.64)

The work balance across the engine remains essentially the same as in the ideal cycle

U1e2

U02

--------- 1!r 1–-------------- !r!c1

!c11 4pc'–

$d$n1( )

% 1–%

-------------------------------------------–

& '( )( )( )( )* +

=

Pte P0$r$d$c$b$t$n Pe 1 % 1–2

------------Me2+& '

* +

%% 1–------------

= =

Me2

M02

--------!r!c

4pc!t

14pe---------

$d$b$n( )

% 1–%

------------1–

!r 1–------------------------------------------------------------------------=

T te T 0!r!c!b!t T e!r!c4pc!t

14pe---------

$d$b$n( )

% 1–%

------------= =

T eT 0------

!c1 4pc–

!t

1 14pe---------–& '

* +

!"

!r!c $d$b$n( )

% 1–%

------------------------------------------------------------=

UeU0-------& '( )* + 2

1!r 1–-------------- !"!t

!c1 4pc–

!t

1 14pe---------–& '

* +

!"

!r!c $d$b$n( )

% 1–%

------------------------------------------------------------–

& '( )( )( )( )* +

=

The non-ideal turbofan

bjc 5.13 1/4/10

(5.65)

where a shaft mechanical efficiency has been introduced defined as

. (5.66)

5.5.3 MAXIMUM SPECIFIC IMPULSE NON-IDEAL CYCLE

Equation (5.35) remains the same as for the ideal cycle.

. (5.67)

Figure 5.4 Turbofan bypass ratio for maximum specific impulse as a function of fan temperature ratio comparing the ideal with a non-ideal cycle. param-eters of the nonideal cycle are , , ,

, , , , .

!t 1!r

4m!"------------- !c 1–( ) # !c1 1–( )+{ }–=

4m

macoreht3 ht2–( ) ma fan

ht13 ht2–( )+

macorem f+( ) ht4 ht5–( )

---------------------------------------------------------------------------------------------=

12 Ue U0( )---------------------------

#22 Ue

2

U02

-------& '( )( )* + Ue1

U0--------- 1–& '( )* +

+ 0=

1.1 1.2 1.3 1.4 1.5

5

10

15

20

25

30

!c1

!" 8.4=

!r 1.162=

!c 2.51=

#max impulse turbofan

ideal cycle

non - ideal cycle

$d 0.95= 4pc1 0.86= $n1 0.96=

4pc 0.86= $b 0.95= 4m 0.98= 4pe 0.86= $n 0.96=

The non-ideal turbofan

1/4/10 5.14 bjc

The derivative is

. (5.68)

Equations (5.59), (5.64), (5.65) and (5.68) are inserted into (5.67) and the optimal bypass ratiofor a set of selected engine parameters is determined implicitly. A typical numerically deter-mined result is shown in Figure 5.4 and Figure 5.5.

Figure 5.5 Turbofan bypass ratio for maximum specific impulse as a function of Mach number comparing the ideal with a non-ideal cycle. Parameters of the nonideal cycle are , , ,

, , , , .

These figures illustrate the strong dependence of the optimum bypass ratio on the non-idealbehavior of the engine. In general as the losses increase, the bypass ratio optimizes at a lowervalue. But note that the optimum bypass ratio of the non-ideal engine is still somewhat higherthan the values generally used in real engines. The reason for this is that our analysis does notinclude the optimization issues connected to integrating the engine onto an aircraft where thereis a premium on designing to a low frontal area so as to reduce drag while maintaining a certainclearance between the engine and the runway.

#22 Ue

2

U02

-------& '( )( )* + !r !c1 1–( )–

4m !r 1–( )------------------------------- 1

1 14pe---------–& '

* + !c

1 4pc–!t

14pe---------–

!r!c $d$b$n( )

% 1–%

------------------------------------------------------------------------–

& '( )( )( )( )( )( )* +

=

1.2 1.4 1.6 1.8 2

2.5

5

7.5

10

12.5

15

17.5

20

#max impulse turbofan

!r

!" 8.4=

!c1 1.2=

!c 2.51=

ideal cycle

non - ideal cycle

$d 0.95= 4pc1 0.86= $n1 0.96=

4pc 0.86= $b 0.95= 4m 0.98= 4pe 0.86= $n 0.96=

Problems

bjc 5.15 1/4/10

Nevertheless our analysis helps us to understand the historical trend toward higher bypassengines as turbine and fan efficiencies have improved along with increases in the turbine inlettemperature.

5.6 PROBLEMS

Problem 1 - Assume , M2/(sec2-°K), M2/(sec2-°K). The

fuel heating value is J/Kg. Where appropriate assume . The ambient tem-

perature and pressure are and . Consider a turbofanwith the following characteristics.

. (5.69)

The compressor, fan and turbine polytropic efficiencies are

. (5.70)

Let the burner efficiency and pressure ratio be . Assume the shaftefficiency is one. Both the fan and core streams use ideal simple convergent nozzles. Determinethe dimensionless thrust , specific fuel consumption and overall efficiency. Supposethe engine is expected to deliver 8,000 pounds of thrust at cruise conditions. What must be thearea of the fan face, ?

Problem 2 - Use Matlab or Mathematica to develop a program that reproduces Figure 5.4and Figure 5.5.

Problem 3 - An ideal turbofan operates with a heat exchanger at its aft end.

% 1.4= R 287 = C p 1005 =

4.28 107× f 1«

T 0 216K= P0 2 104× N M2

=

M0 0.85 ; !" 8.0 ; $c 30 ; $c1 1.6 ; # 5= = = = =

4pc 0.9 ; 4pc1 0.9 ; 4pt 0.95= = =

4b 0.99 ; $b 0.97 = =

T P0 A0( )

A2

2.5

ma fan

macore Q

Q

Problems

1/4/10 5.16 bjc

The heat exchanger causes a certain amount of thermal energy (Joules/sec) to be transferredfrom the hot core stream to the cooler fan stream. Let the subscript refer to the heat exchanger.

Assume that the heat exchanger operates without any loss of stagnation pressure and that both nozzles are fully expanded. Let and

. The thrust is given by

(5.71)

where we have assumed .

1) Derive an expression for in terms of and .

2) Write down an energy balance between the core and fan streams. Suppose an amount of heat is exchanged. Let where , . Show that

. (5.72)

3) Consider an ideal turbofan with the following characteristics.

. (5.73)

Plot versus for where corresponds to the value of

such that the two streams are brought to the same stagnation temperature coming out of the heatexchanger.

Problem 4 - The figure below shows a turbojet engine supplying shaft power to a lift fan.Assume that there are no mechanical losses in the shaft but the clutch and gear box thattransfers power to the fan has an efficiency of 80%. That is, only 80% of the shaft poweris used to increase the enthalpy of the air flow through the lift fan. The air mass flow rate

Qx

$x $x' 1.0= = !x T te T t5=

!x' T te' T t5'=

T macorea0M0 U6 U0 1–( ) # U6' U0 1–( )+{ }=

f 1«

T macorea0( ) !" !r !c !c' #, , , , !x !x',

Q !x 1 6–= 6 Q macoreC pT t5( )= Q 0>

!x' 1!"!t#!r!c'---------------& '( )* +

6+=

T 0 216K ; M0 0.85 ; !" 7.5 ; $c 30 ; $c' 1.6 ; # 5= = = = = =

T macorea0( ) 6 0 6 6max< < 6max 6

Problems

bjc 5.17 1/4/10

through the lift fan is equal to twice the air mass flow rate through the engine. The polytropic efficiency of the lift fan is and the air

flow through the lift fan is all subsonic. The flight speed is zero.

The ambient temperature and pressure are and . The

turbine inlet temperature is and . Relevant area ratios are

and . Assume the compressor, burner and turbine alloperate ideally. The nozzle is a simple convergent design and stagnation pressure lossesdue to wall friction in the inlet and nozzle are negligible. Assume f<<1. Let the nozzle areabe set so that .

1) Is there a shock in the inlet of the turbojet?

2) Determine the stagnation temperature and pressure ratio across the lift fan

(74)

m Lift Fan 2ma= 4p Lift Fan 0.9=

1 3 4 52 e

A*4

A1throat M=1

P0 T 0

LiftFan

2f

3f

m Lift Fanma

gearbox clutch

T 0 300K= P0 1.01x105N M2=

T t4 1800K= $c 25=

A2 A4*

15= A1throat A2 0.5=

Pt5 P0 3=

!Lift FanT t3 LiftFanT t2 LiftFan-------------------------= $ Lift Fan

Pt3 LiftFanPt2 LiftFan------------------------=

Problems

1/4/10 5.18 bjc