Lecture 11 – Performance of simple cycles

Chalmers University of Technology

Lecture 11 – Performance of simple cycles

• The off-design problem

• Off-design operation for:– the single shaft engine– free turbine engine– the jet engine

• Design Task 3 description


The off design problem• Chapter 1-3 describes

the (on) design problem– Downstream engine

components are adapted to upstream. For instance:

• Turbine pressure ratio is selected to deliver power required by compressor

• Exhaust nozzle is sized to swallow flow exiting from turbine

Every point correspondsto new engine design - newturbine/compressor blading nozzle areas etc

Design = rubberengines


The off design problem• What happens when control

signals are changed such as:– Fuel flow– Nozzle exit area – Compressor variable

geometry• Conservation of mass flow,

energy and turbine/compressor rotational speed compatibility

Operating space reduced to an equilibrium runningline

Shows proximity to surge line


The off design simulation - component models• Component performance

– Semi-empirical models• Models with some constants set by

measurements or design experience. Ex:

– Scaled maps• Existing performance maps are scaled to new design point.

– Data from component rig tests– Higher order models (2D or 3D simulation)

(6.1) 11

221

T

TKKPLF

Obtained fromexperiments


The off design simulation - component modelsEngine system model is built by its component models• Iteration is frequently required

to determine the running line

• Some engine specific algorithms are found in chapter 8 and 9.


Part load importance• Aircraft– High. Taxiing and landing.

• Power generation– Low (except for ambient conditions).

However, surge free starting and shut down as well as time to max. power is important.

• Naval– High. Poor gas turbine part load

performance has given rise to a number of combined cycles:

• CODOG, COSAG, COGAG

• Vehicular gas turbine– High.

1% fuel efficiency idle

WR21 better fuel efficiency

than simple cycle


Layout types to be studied off design

• Single shaft engine:

• Free turbine engine:

• Jet engine:

Poses the same restriction

on upstream components


Off-design of single-shaft engine• Select a constant speed line on compressor

characteristic. Reading of point gives:

0101

02

01

01 ,,,T

N

p

p

P

RTmc

!Calculated

01

03

pressuredelivery compressor

of percentage Fixed

03

02

02

01

01

01

ratio pressure turbine

ofFunction

03

03

T

T

P

P

P

P

P

Tm

P

Tm

• By approximating the fuel flow as equal to bleeds, compatibility of flow gives:

• Turbine pressure ratio is obtained from (neglect inlet and exhaust losses):

01

02

02

03

04

03

P

P

P

P

P

P


Off-design of single-shaft engine• The turbine rotational speed

is now obtained:

itycompatibilflow from

Calculated

01

03

03

reading mapcompressor From

01 T

T

T

N

T

N

• Rotational speed and corrected mass flow gives turbine efficiency from turbine map.

• The power output is then:

1

1

1output power net

1

01

0201012

1

03

0403034

012034

a

a

g

g

p

pTT

p

pTT

TmcTmc

c

t

pam

pg


Off-design of single-shaft engine• We have now determined the

power output corresponding to the selected point in the compressor map.

• Does it match the load?

Performance problemexam 2003


Gas generator performance

Poses the same restriction

on upstream components

Jet engine

Free turbine engineDerive commonprocedure for bothengines - GASGENERATORmatching!


Off-design of gas generator• Select a constant speed line on compressor

characteristic. Reading of point gives:

0101

02

01

01 ,,,T

N

p

p

P

RTmc

!Calculated

01

03

pressuredelivery compressor

of percentage Fixed

03

02

02

01

01

01

ratio pressure turbine

ofFunction

03

03

T

T

P

P

P

P

P

Tm

P

Tm

• By approximating the fuel flow as equal to bleeds, compatibility of flow gives:

• Turbine pressure ratio is relatedto (neglect inlet and exhaust losses):

0401

02

02

03

RCOMOPRESSO OFITYCOMPATIBILWORK

TO COUPLED!!KNOWN! NOT

04

03

P

P

P

P

P

P

P

P a


Off-design of gas generator

• Guess turbine pressure ratio and proceed as usual:

012

???

034

1

01

0201012

1

03

0403034

1

1

TcTc

p

pTT

p

pTT

papgm

c

t

a

a

g

g

Verify assumption withpower balance


Off-design of gas generator and load• Every point on compressor rotational speed has a matching point,

but only one of these will match the exhaust nozzle/free turbine!!!

• A simple nested iteration will do:

match_load: DOmatch_gas_generator: DO

! gas generator simulation code

END DO match_gas_generator ! load check simulation code

END DO match_load


Off-design of free turbine engine - load match

• For the free turbine we obtain acorrected mass flow as input:

8.7 Eq.From

03

04

04

03

03

03

04

04

T

T

P

P

P

Tm

P

Tm

where:(8.7) 11

03

04

1

03

04

03

034

T

T

p

p

T

T g

g

t

• The free turbine pressure ratio is obtained from (power turbine exit pressure is approximately pa):

04

04

sticcharacteri turbinefrom

04

03

04

02

03

01

0204

P

Tm

P

P

P

P

P

P

P

P

P

P

a

a


Off-design of jet engine• The characteristics of the turbine nozzles are the same as the

exhaust nozzle => we have already solved the problem

• Use same procedure but check with exhaust nozzle characteristic instead of turbine characteristic!


Design Task 3

2 4 6 8 10 12 141

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Whittle compressor characteristic

m /

r c

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1x 10

-3 Whittle turbine characteristic

m

T0

3/ P

03

rt

Klassens, Wood, Schuman “Experimental Performance of a ….Centrifugal Compressor Designed

for a 6:1 Pressure Ratio”NASA TMX-3552 1977

2

*

*

3

03

03

1

11)(

t

t

grr

r

XAm

PT

)(3

03

03gXA

m

PT

You receive a Start Kit which contains characteristicsStart by solving warm up task


Design Task 3 – two nested iterations

Gasgen.Gasgen.m

Turbojet.m

Odp.m (off-design performance)

• Start with inner loop – gas generator

• Check that you get (T3/T1)work=(T3/T1)flow

when you run with Design Task 1 data• Then solve inner loop with fminbnd and

continue with outer


Design Task 3• Predict off-design T5,Thrust

and SFC

• Determine engine conditions at 500 mph and 30000 feet

2 4 6 8 10 12 141

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Whittle compressor characteristic

m /

r c


Use of fminbnd

Minimize a function of one variable on a fixed interval. Syntax:

x = fminbnd(fun,x1,x2)

x = fminbnd(fun,x1,x2,options)

x = fminbnd(fun,x1,x2,options,P1,P2,...)

Start by solving test example Fig217_test, that is make sure that you obtain fa = 0.01452 for t02=482.0 and t03=1046.0. (faair = 0.0, fastoch = 0.06760, Qnet=43200000 ) Make sure you understand the manual for fminbnd.

% Solve non-linear 1D equation by minimizationfa = fminbnd('fa_err',faair,fastoch,[],t02,t03,Qnet);


function mfp4_err = Turbojet(rc,i,pa,P01,T01,eta_t,eta_m,eta_j,deltaP_b,A3,A5, … gamma_a,gamma_g,cp_a,cp_g,R)

[mcorr1,eta_c,ncorr1] = CompChar(i,rc);

%p02 = rc*P01;%p03 = p02*(1.0-deltaP_b);r_b = p03/p02;

theta = T01/288.15;delta = P01/101325.0;

m = (mcorr1*delta)/sqrt(theta);mfp1 = m*sqrt(T01)/P01;…..

Use of non-dimensional numbers


Approximation for two turbines in series• For the gas generator exit we have:

03

04

1

03

04

03

034

03

04

04

03

03

03

04

04

11T

T

p

p

T

T

T

T

P

P

P

Tm

P

Tm

g

g

t

we can plot outflow and inflow

in same turbine map!

),(

]neglible is in changesby on effect Assume[

11

04

03

03

03

04

04

1

03

04

04

03

03

03

04

04

P

P

P

Tmf

P

Tm

p

p

P

P

P

Tm

P

Tm

t

t

g

g

Typically, variation in turbine efficiency will be limitedSame effect with nozzle

downstream of gas-generator turbine!!!


The compressible continuity function (x-function):

tT

T

TTT

PrrPP

P

P

PP

MXAP

RTm

t

cb

0101

030103

010101

02

02

0303

303

03 ),(

t

rC

t

rMXrAr

MXrrt

A

P

RTm

c

ccb

cb

1

3

301

01

choked nozzle Turbine

),(

),(

Theory 11.1 – Simplified turbojet running line



Assume that both exhaust nozzle and turbine operate choked:

If the exhaust nozzle operates choked,

the turbine will remain in the same

non-dimensional point! Assuming

a fixed efficiency => temperature

ratio will then remain constant.

Exhaustnozzle

03

04

constant

1

03

04 11T

T

p

p g

g

t

Nozzle choked andefficiency approx. const. => temperatureand pressure ratio isconstant over turbine



Finally, a work balance will be introduced:

03

04

01

03

01

0204030102 11

T

T

T

T

c

c

T

TTTcTTc

t

pa

pgmpgmpa

The compressor pressure ratio is obtained from: 1above From

01

021

010102 1 1 1

a

a

a

a

T

Trr

TTT ccc

c

Combining yields:

11

t

1 1 1

1

2

12

1constant

03

04

a

a

a

a

a

a

c

pa

pgmcc

rC

tCT

Tt

c

cr


Combining the two equations yield:

t

rC

P

RTm

rC

c

c

a

a

101

01

2

11

t1

11

11

321

01

01

a

a

a

a

c

c

c

c

r

Cr

r

CCr

P

RTm

We have derived an explicit expression for the running line!!!



Learning goals

• Master algorithms for calculating performance for: – Single shaft engine– Jet engine – Free turbine engine

• Know how to derive an expression for the running line as well as to state the requirements for this expression to hold

Documents

Lecture 11 – Performance of simple cycles