Recap: Lecture 4, 13th January 2015, 0830-0925 hrs.

Ideal cycle for jet engines

Without and with afterburning

The generalised thrust equation

Momentum and pressure thrust

1

The thrust equation

(Reaction) Control surface

Thrust producer

u, Pa

u

u

Ai Ae

ue

2 1

x

y

Ae, Pe

amem

sm

fm

2

The thrust equation

The reaction to the thrust, , is transmitted to the support. The engine thrust is thus the vector summation of all forces on the internal and external surfaces of the engine.

Therefore,

Considering the components of force and the momentum flux in the x-direction only,

CS

dAnuuF ).(

CS

xx dAnuuF ).(

3

The thrust equation

The pressure and velocity can be assumed to be constant over the entire control surface, except over the exhaust area, Ae.

The net pressure force acting on this control volume is (PaPe)Ae.

The only other force acting on the control volume is the reaction to the thrust, .

Adding up the forces in the x-direction,

eeax APPF )(

4

The thrust equation

The mass flow that enters the capture area, Ai, is

Similarly, the mass flow crossing the exhaust area Ae, is,

Also,

Or,

Continuity equation for the CV gives,

ia uAm

eeee Aum

fae mmm

ieeef uAAum

)( is,Which

g,Rearrangin

0)(

ies

eeeefs

fseeee

AAum

AuuAmm

uAmmAAuAu

5

The thrust equation

From the momentum balance across the CV,

This is the net outward flux of x-momentum.

This equation reduces to

From the force balance equation, we have,

uAAuumuAAuumumdAnuu iaeseeCS

x )()().(

umumdAnuu aeeCS

x

).(

eaeaee APPumum )(

6

The thrust equation

If we define fuel-air ratio,

This is the generalised thrust equation for air-breathing engines.

The term (PePa)Ae is not zero only if the exhaust jet is supersonic and the nozzle does not expand the exhaust jet to ambient pressure.

However if Pa Pe, it can be substantial contribution.

eaeea APPuufm )()1(

7

Engine performance parameters

The engine performance is described by different efficiency definitions, thrust and the fuel consumption.

The efficiency definitions that we shall now be discussing are applicable to an engine with a single propellant stream (turbojets or ramjets).

For other types of jet engines (turbofan, turboprop) the equations need to be appropriately modified.

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Engine performance parameters

Propulsion efficiency: The ratio of thrust power to the rate of production of propellant kinetic energy.

If we assume that f1 and the pressure thrust term is negligible,

2/)2/)(1( 22 uufmu

ea

P

e

e

e

eP

uu

uu

uu

uuu

/1

/2

2/2/

)(22

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Engine performance parameters

Thermal efficiency: The ratio of the rate of production of propellant kinetic energy to the total energy consumption rate

For a turboprop or turboshaft engine, the output is largely shaft power. In this case,

fuel. theofreaction ofheat theis , where,

2/)2/)(1(2/)2/)(1( 2222

R

R

e

Rf

eath

Q

fQ

uuf

Qm

uufm

engine. theofoutput power shaft theis , where, sRf

sth P

Qm

P

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Engine performance parameters

Overall efficiency: The product of thermal efficiency and propulsion efficiency.

In the case of aircraft that generate thrust using propellers,

thpo

.efficiencypropeller theis Where, pr

thpro

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Engine performance parameters

Thrust specific fuel consumption, TSFC

For turbine engines that produce shaft power, brake specific fuel consumption, BSFC

For engine (like turboprop) that produce both, equivalent brake specific fuel consumption,

uufmmm

TSFCea

ff

)1(

s

f

P

mBSFC

uP

m

P

mEBSFC

s

f

es

f

12

PT6: Turboprop Engine

500 to 2,000 shaft horsepower class Multi- stage axial and single-stage centrifugal

compressor Reverse flow combustor Single-stage compressor turbine Independent free power turbine with shrouded

blades Forward facing output for fast hot section

refurbishment Epicyclic speed reduction gearbox

Source: http://www.pwc.ca/en/engines/pt6a 13

Ideal cycle for jet engines

s

T

2

4

5

3

a

7

Ideal turbojet cycle (without afterburning) on a T-s diagram

14

Ideal cycle for jet engines

For cycle analysis we shall take up each component and determine the exit conditions based on known inlet parameters.

Intake: Ambient pressure, temperature and Mach number are known, Pa, Ta and M

Intake exit stagnation temperature and pressure are determined from the isentropic relations:

)1/(

0202

2

022

11

a

a

a

T

TPP

MTT

15

Ideal cycle for jet engines

Compressor: Let the known compressor pressure ratio be denoted as

Combustion chamber: From energy balance,

Hence, we can determine the fuel-air ratio.

/)1(

0203

0203

c

c

TT

PP

030403

0304

0304

//

1/,

TTTcQ

TTfor

fQhh

pR

R

16

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Ideal cycle for jet engines

Turbine: Since the turbine produces work to drive the compressor, Wturbine = Wcompressor

0304

)1/(

04

050405

02030405

02030504

02030504

chamber, combustion idealan For

Hence,

)1/()(

)())(1(,

)()(

PP

T

TPP

fTTTT

TTTTfor

TTcmTTcm papt

17

Ideal cycle for jet engines

Nozzle: With no afterburner, T06=T05, P06=P05

Thrust, TSFC and efficiencies can now be determined using the formulae derived earlier.

/)1(06060607

707

2

/12

Since,

2

energy, kineticexit nozzle theTherefore,

PPTcu

hh

hhu

ape

e

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Ideal cycle for jet engines

Thrust,

Propulsion efficiency,

Thermal efficiency,

uufmAPP

APPuufm

ea

eae

eaeea

)1(

,negligible is )( If

)()1(

uufmmm

TSFCea

ff

)1(

2/)2/)(1( 22 uufmu

ea

P

R

eth

fQ

uuf 2/)2/)(1( 22

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