Thermodynamics Optimization of the Turbofan Cycle

8/12/2019 Thermodynamics Optimization of the Turbofan Cycle

1/14

Thermodynamic optimization of theturbofan cycle

Yousef S.H. Najjar

Mechanical Engineering Department, Jordan University of Science and Technology, Irbid, Jordan, and

Sharaf F. Al-Sharif

King Abdulaziz University, Jeddah, Saudi Arabia

AbstractPurpose To develop and find the effect of combination of four cycle design variables that minimizes the specific fuel consumption (sfc) of a turbofanengine.Design/methodology/approach After choosing the four variables, namely bypass ratio ( B), fan pressure ratio, overall pressure ratio, and turbineinlet temperature (T04), first the sfc was minimized without a minimum thrust constraint. Then, a minimum specific thrust constraint was introduced.Findings The unconstrained-specific thrust is a two-dimensional optimization problem, whereas the specific thrust constrained problem was foundto be a three-dimensional one.Research limitations/implications The variables Band are limiting factors to further improvement, as set by their maximum practical values,whereas the other two variables are to be optimized.Practical implications A very useful work, in which numerical optimization program was developed, for a turbofan cycle and could be extended to

other cycles.Originality/value This paper offers a great help to those intending to optimize certain cycles with a number of variables.

Keywords Optimization techniques, Fuel consumption

Paper type Research paper

NomenclatureA area, m2

As specific area (area/unit mass flowrate), m2 s/kg

alt. altitude, m

B bypass ratio

C velocity magnitude, m/s

cp constant pressure specific heat, J/kg KF thrust, N

Fs specific thrust (thrust/total air mass flow rate),

Ns/kg

f fuel air ratio of core flow (mass flow rate of fuel/

mass flow rate of core air flow)

Hc enthalpy of combustion of air, MJ/kg

M mach number

_m mass flow rate, kg/s

n polytropic index of compression

P pressure, bar, Pa

R ideal gas constant, J/kg K

sfc specific fuel consumption (mass flow rate of

fuel/unit thrust produced), g/kN s

T temperature, K

Greek symbols

g specific heat ratio

hi isentropic efficiency of the intake

hc polytropic efficiency of the compressor and fan

ht polytropic efficiency of the turbine

hm mechanical efficiency of the shaft

hb combustion efficiency of the burner

hj isentropic efficiency of the jet nozzles

pin intake pressure ratio

pf

fan pressure ratiopc overall pressure ratio

pb burner pressure ratio

pt turbine pressure ratio

tin intake temperature ratio

tf fan temperature ratio

tc overall compression temperature ratio

tt turbine temperature ratio

Subscripts

0 stagnation state

1, 2, . . . 8 station numbering

a air

b burner

c compressor, combustion, critical, cold

(bypass flow)f fan

g combustion gas

h hot (core flow)

i inlet

in inlet

ISA international standard atmosphere

j jet nozzle

m mechanical

p constant pressure

s specific

t turbine

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/1748-8842.htm

Aircraft Engineering and Aerospace Technology: An International Journal

78/6 (2006) 467480

q Emerald Group Publishing Limited [ISSN 1748-8842]

[DOI 10.1108/00022660610707139]

467
http://www.emeraldinsight.com/0002-2667.htmhttp://www.emeraldinsight.com/0002-2667.htm


2/14


3/14

It is now necessary to find expressions for the unknown

variables C7; C8; P7; P8;As7;As8 in terms of input variables.Starting with the core nozzle, the definition of stagnation

temperature gives:

cpgT07 cpgT7C27

2

Rearranging:C7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cpg T07 2 T7

qThe duct is assumed to be adiabatic so that T07 T06, and:

C7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2cpgT06 2 T7q

3

An expression for T06 is found by making an energy balance

over the gas-generator on a unit mass basis:1

B11 fcpgT04 2 T06

cpa

hm

B

B1T02 2 T01

1

B1

cpa

hmT03 2 T01

T042

T06

cpa

hmcpg

1

1 f T01 B

T02

T012

1

T03

T012

1

4

Making the following definitions:

tin ;T01

Ta; pin ;

P01

Pa

tf ;T02

T01; pf ;

P02

P01

tc ;T03

T01; pc ;

P03

P01

tt ;T06

T04; pt ;

P06

P04

equation (4) can be rewritten:

tt 1 2 cpa

hmcpg

1

1f

Ta

T04tinBtf2 1 tc 2 1; 5

which, except for the fuel air ratio f, is in terms of known or

calculable quantities. Specifically:

tin 1 C2a

2cpaTa;

tfpna21=naf ;

tc pna21=nac

The fuel air ratio will be dealt with shortly.

T06 is now given as:

T06 T04tt

Similarly, P06 is given by:

P06 Papinpcpbpt

in which:

pin 1hiC2a

2cpaTa

ga21=ga;

and:

pt tng=ng21t

An expression for T7is found from knowledge of the pressure

ratio and the definition of the isentropic efficiency of nozzle:

hj T06 2 T7

T06 2 T07

;

T7 T06 1 2 hj 1 2 T07T06

;

T7

T061 2 hj 1 2

P7

P06

gg21=gg" #

At this point the calculations will depend on whether the

nozzle is choked or unchoked. For the core nozzle, the critical

pressure ratio is given by:P06

Pc

1

1 2 1hj

gg21

gg 1

h igg=gg21IfP

06=P

a $ P

06=P

c; the nozzle is choked and:

P7 Pc P06

P06=Pc;

T7 Tc 2T06

gg 1

Substituting for T06 and T7 in equation (3):

C7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2CpgT04tt 1 2

2

gg 1

s choked

The density of exhaust gas is given by:

r7 P7

RT7;

knowledge of which is employed in the continuity equation tofind the specific area:

As7 A7

_mh

1

r7C7

Otherwise, if P06=Pa , P06=Pc the nozzle is unchoked,P7 Pa; and T7 is given by:

T7 T06 1 2 hj 1 2 P7

P06

gg21=gg" #( )

T7 T04tt 1 2 hj 1 2 1

pinpcpbpt

gg21=gg" #( )

Substitution into equation (3) yields:

C7 2cpgT04tthj 1 2 1pinpcpbpt

gg21=gg" #( )1=2unchoked

Similarly, for the bypass nozzle, the definition of stagnation

temperature gives:

cpaT08 cpaT8C28

2

from which:


2cpgT08 2 T8q

Againthe duct is assumed to be adiabatic so that T08 T02, and:

Thermodynamic optimization of the turbofan cycle

Yousef S.H. Najjar and Sharaf F. Al-Shar if


Volume 78 Number 6 2006 467 480

469


4/14


2cpaT02 2 T8q

6

T02 Tatintf

As before, the nozzle pressure ratio must be checked against the

critical pressure ratio, which is given by:P

02Pc

1

1 2 1hj

ga21ga1

h iga=ga21IfP02=Pa $ P02=Pc;the nozzle is choked:

P8 Pc P02

P02=Pc;

T8 Tc 2T02

ga1

C8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2CpaTatintf 1 2

2

ga1

s choked

r8 P8

RT8

;

As8A8

_mc

1

r8C8

IfP02=Pa , P02=Pc; the nozzle is unchoked, P8 Pa; andT8is given by:

T8 T02 1 2 hj 1 2 P8

P02

ga21=ga" #( )

T8 Tatintf 1 2 hj 1 2 Pa

Papinpf

ga21=ga" #( )

Substitution into equation (6) and some algebraic manipulation

yields:

C8 2cpaTatintfhj 1 2 1

pinpf

ga21=ga" #( )1=2unchoked

The expressionsfor the unknownvariablesin equation (2),which

we had set out to find, are now complete, and the specific thrust

may be evaluated. It remains now to address the sfc.

The sfc is given by:

sfc _mf

F

f _mh

Fs _ma

1

B1

f

Fs;

The fuel air ratio fis found by making an energy balance over

the combustor:

_mhcpa T03 2 298 f _mhhbHc _mh f _mh cpg T04 2 298

where Hcis the enthalpy of combustion at 258C. Solving for f

and employing some previously defined quantities, the above

equation becomes:

f cpgT04 2 298 2 cpaTatintc 2 298

hbHc 2 cpgT04 2 298

It is implemented in TK-Solver 3.0 in the rule function

SpecFuelCons and auxiliary rule functions Nozzle,

IsNozzleChoked and NozzleThrust. Having formulated the

objective function, we may now direct our attention to the

optimization method.

Optimization technique

The method of optimization that is used is called the

conjugate gradient method. In general, optimization

techniques can be grouped into two main categories:methods that use gradient information, and direct search

methods. As its name implies, the conjugate gradient method

falls into the first category.

In all gradient computing methods the optimization

problem is subdivided into two sub-problems:

1 determining a suitable search direction; and

2 taking the optimum step size in that direction.

The different members of the family vary in the way they

address the first sub-problem.

The simplest of these methods is the method of steepest

descents. In this method the local gradient is evaluated in

each step, and the search direction is taken (in the

minimization problem) as the negative of the gradient,

which is, by definition of the gradient, the direction of

steepest descent. This method is usually the least efficient of

the family, especially when the scales of the design variables

are not similar such that the objective function has a narrow,

stretched contour map. This results from two facts:

1 the local gradient of a function (i.e. its negative) does not

generally point to the minimum; and

2 at the minimum along some search direction the local

gradient is perpendicular to the search direction.

What this means is that the algorithm will zigzag its way along

small mutually perpendicular steps, even if it is relatively close

to the minimum (Arora, 1989).

The conjugate gradient method results from the idea of

searching along non-interfering directions (Arora, 1989).Simply put, minimization along one direction should not

interfere or ruin previous minimizations. Based on this idea a

sequence of arguments are made which lead to the derivation

of the method.

The two situations that are considered can be listed as

follows:

1 The sfc is to be minimized with no constraint on specific

thrust. The optimization will be run for a combination of

different operating conditions and maximum B

constraints given in the Table I.

2 sfc is to be minimized subject to a constraint of minimum

specific thrust.

The combination of operating conditions can be seen in

Table I.

Discussion of results

Minimizing sfc with no constraint on specific thrust

The results of the optimization runs for the previously

mentioned conditions are summarized in Table II. Each case

is given a number in the table for easy referral. A parametric

variation of T04 and pc around the optimum in each of the

cases was performed, and the results were plotted in the form

of carpet plots of sfc versus Fs in Figures 2-5. The effect ofpfis considered separately in Figures 6-8, where case 8 is taken




Volume 78 Number 6 2006 467 480

470


5/14

as the reference cycle. Finally, a sensitivity plot for the

reference case is shown in Figure 9.

Throughout this work, a constraint will be called active if

the value of the constrained variable in its respective optimumcycle is found to be equal to its limiting value (whether

maximum or minimum). Thus, if in an optimization run, B is

constrained to a maximum of five, for example, and the value

of optimum B found by the optimization process is also five,

then the constraint on B is called an active constraint. If, on

the other hand, the optimum value of the constrained variable

does not reach its limiting value, the constraint will be called

passive.

The concept of active and passive constraints helps in

determining the real limiting factors to further improvement

under the conditions considered.

General observations

A number of points are observed in Table II and thecorresponding carpet plots of Figures 2-5. These may be

listed as follows:. The constraints on bypass ratio and overall pressure ratio

are active constraints in all cases.. The optimum pf is a function of bypass ratio, and

decreases as B is increased. For example, in cases 7 and 8,

where B is doubled from 5 to 10 while all other variables

are kept constant, sfc decreases 9.3 percent from 1.93 to

1.75.. The optimum pfincreases as Mach number increases (all

other variables kept constant). This is seen, for example,

when cases 8 and 11 are compared, in which optimum pfincreases from 1.75 to 1.8 (2.8 percent) when M is

increased from 0.9 to 0.8 (12.5 percent)..

The optimum T04 increases as B is increased. This isexemplified by cases 8 and 9 where increasingB50 percent

from 10 to 15 increases optimumT04by 14.4 percent from

1,600 to 1,830 K.. The optimumT04increases as pcis increased. This is seen

in the carpet plots where the optimum T04 consistently

shifts to the right when pcis increased. This may be easier

to observe on the B 15 carpets because of their higher

curvature, but it is applicable to all cases.. Increasing B significantly lowers specific thrust, as can be

seen when comparing cases 8 and 9. Increasing B from 10

to 15 (50 percent) decreases Fs of the optimum cycle

from 137.1 to 124 N s/kg (210.6 percent).. The effect of increasing T04in increasing specific thrust is

stronger at lower bypass ratios. For example, in Figure 1,

when moving fromT04 1,400 to 1,550 K (an increase of10.7 percent) on the pc 45 line of the B 5 carpet, Fsincreases from 153.8 to 192.7 N s/kg. This represents a

25.3 percent increase in Fs. While a comparable

10.5 percent increase in T04 from 1,900 to 2,100 K on

the pc 45 line of the B 15 carpet produces a thrust

increase of only 18.3 percent (from 120 to 142N s/kg.)

The reason for that is that T04represents the energy input

to the core flow. As the bypass flow increases, the core

flow becomes relatively less important, and a larger

increase inT04is needed to obtain the same specific thrust

increase.

Impact of the bypass ratio constraint

An outstanding observation in Table II is that the B and pc

constraints are always active constraints. This means that (inthis situation, where no minimum Fsconstraint is placed) they

are limiting factors. That is to say, sfc can be further lowered

if these constraints were not present, or if they could be

extended. However, while it is true that the sfc calculated

from cycle analysis appears to improve continuously as B is

increased (provided T04 can also be increased), this does not

hold true when installation effects (inlet and nozzle drag) are

Table II Results of optimization at different operating conditions and maximum Bconstraint (with no constraint on Fs)

Operating cond. Constraints Optimum cycle sfc FsCase No. Altitude Mach B T04(K) pc B pf T04(K) pc (g/kNs) (N s/kg)

1 9,000 0.8 5 2,000 40 5 1.94 1,450 40 18.1 173.4

2 9,000 0.8 10 2,000 40 10 1.76 1,700 40 16.75 141.63 9,000 0.8 15 2,000 40 15 1.7 1,950 40 16.15 129

4 9,000 0.9 5 2,000 40 5 1.97 1,500 40 19.4 169.5

5 9,000 0.9 10 2,000 40 10 1.8 1,775 40 18.1 139.3

6 9,000 0.9 15 2,000 40 15 1.7 2,000 40 17.46 122.1

7 11,000 0.8 5 2,000 40 5 1.93 1,360 40 17.5 167.5

8 11,000 0.8 10 2,000 40 10 1.75 1,600 40 16.23 137.1

9 11,000 0.8 15 2,000 40 15 1.68 1,830 40 15.65 124

10 11,000 0.9 5 2,000 40 5 1.96 1,410 40 18.76 163.8

11 11,000 0.9 10 2,000 40 10 1.8 1,675 40 17.5 134.4

12 11,000 0.9 15 2,000 40 15 1.72 1,925 40 16.92 122.3

Table I Combinations of operating conditions and bypass ratioconstraints

Case Altitude Mach Bmax

1 9,000 0.8 5

2 9,000 0.8 10

3 9,000 0.8 15

4 9,000 0.9 55 9,000 0.9 10

6 9,000 0.9 15

7 11,000 0.8 5

8 11,000 0.8 10

9 11,000 0.8 15

10 11,000 0.9 5

11 11,000 0.9 10

12 11,000 0.9 15




Volume 78 Number 6 2006 467 480

471


6/14

Figure 2Parametric variation ofT04and pcaround optima at 9,000 m, 0.8 Mach





Volume 78 Number 6 2006 467 480

472


7/14






Volume 78 Number 6 2006 467 480

473


8/14

Figure 6sfc versus T04at differentpf

Figure 7sfc versus Bat different pf




Volume 78 Number 6 2006 467 480

474


9/14

Figure 8sfc versus pcat different pf

Figure 9Sensitivity of the sfc of the reference cycle to deviations from optimum values the design variables




Volume 78 Number 6 2006 467 480

475


10/14

accounted for. To account for this difference some references

such as Mattingly (1996) and Kerrebrock (1992) differentiate

between the two quantities installed sfc (tsfc), which includes

installation losses, and uninstalled sfc (sfc), from cycle

calculations. The installed sfc depends on how the engine is

installed in the nacelle, and is related to the uninstalled sfc

through the relation:

tsfc sfc

1 2 finlet 2 fnoz

where finlet;fnoz are the inlet and nozzle loss coefficients,respectively, and are dimensionless measures of their drag.

Wilson (1984) points out that due to increasing nozzle losses,

the installed specific fuel consumption (tsfc) reaches a

minimum at about B 20. This of course, cannot be

verified by cycle calculations, because cycle calculations do

not account for installation effects. The point of this

elaboration is to stress that the continuously improving

trend of sfc with increasing B, as predicted by cycle

calculations alone can be misleading if installation effects

are not kept in mind.

In addition to that, as the bypass ratio is increased, the

specific thrust of the engine decreases, as can be seen clearly

in Figures 1-4. This means that for a given thrust requirement

(not specific thrust), if B is increased the engine diameter

must be increased to increase the air mass flow rate.

Consequently, the weight and (external) aerodynamic drag

of the engine also increase. Thus, even if the sfc decreases, the

load on the engine, the thrust it must provide to overcome the

drag, will also increase. This means that if the actual amount

of fuel consumed (e.g. kg) is considered, instead of the sfc

(e.g. kg/kN), the optimum B is further lowered (Wilson,

1984). One study, mentioned in Wilson (1984), which

considered this point found that there was not much fuel

efficiency to be gained beyond a bypass ratio of about eight.

Ultimately, for any accurate conclusions to be made in thisregard, the engine must be studied with proper consideration

of installation effects and weight implications.

Impact of the pressure ratio constraint

It is seen in Figures 1-4 that, above a certain T04, increasing

pc improves sfc. For example, in Figure 1 in the B 5 carpet

at T04 1,400 K, there is little improvement in sfc when

moving from pc 40 to pc 45. Below this temperature sfc

actually deteriorates as can be seen on the 1,350 K line, where

the sfc increases from about 18.7 to slightly more than 19 g/

kN s when moving from pc 40 to 45. But above a certain

temperature, about 1,400 K in this case, sfc decreases with

increasing pc.

Is this trend continuous? Figure 8 suggests that if the othercycle design variables are held constant, there will be a value

ofpc after which sfc will start to increase. However, if T04 is

allowed to increase the trend is practically continuous.

The maximum overall pressure ratio is limited by the

temperature limit of the compressor materials, which is

currently 920 K (Presset al., 1988). If standard air at 288.2 K,

and a polytropic efficiency of 0.9 are assumed, this

corresponds to a pressure ratio of:

pc 920

288:2

0:93:538:7

The value ofpc,max has been assumed to be 40 throughout

this work.

An important point to make here is that, when pc is

increased beyond current limits, the value of polytropic

efficiency assumed in the model becomes questionable. It has

b ee n a ss um ed t ha t p ol yt ro pi c e ffi ci en cie s o f t he

turbomachinery are constant at 0.9, a value that reflects

current technology up to the current limit of pressure ratio(Press et al., 1988). Even if technological advances allow the

pressure ratio of the turbomachinery to increase, it becomes

harder to maintain high polytropic efficiencies. Therefore,

trying to draw conclusions from plots that go too far beyond

current limits is not reliable.

Several references such as Mattingly (1996, 1999),

Kerrebrock (1992) and Kurzke (1999) point out that the

validity of the results of parametric cycle analysis depends on

the realism with which the variation of efficiency with pressure

ratio is accounted for. Mattingly (1999) gives correlations for

turbine and compressor polytropic efficiency variation with

the respective pressure ratios that represent current

technology levels for industrial gas turbines. However, it is

mentioned that the correlations over estimate the losses for

multispool aircraft engines. According to Press et al. (1988),

polytropic efficiencies of 0.9 up to the current limit of

pressure ratio can be achieved for aircraft engines.

The observation in Table II that the B and pc constraints

are always active constraints, suggests that the problem of

minimizing sfc without considering a minimum specific thrust

is actually a two-dimensional optimization problem. After

setting B and pc to their maximum practical values, the

optimum T04 and pfcombination must be found.

Illustrating the two-dimensional nature of the problem

Figure 6 shows a plot of sfc versus T04for different pf, with B

and pc set to their maximum values for the reference case.

Two points are visible in this figure:

1 for eachpf there is an optimum T04 that minimizes sfc;and

2 of these pairs of (pf, T04,opt), there is one pair (pf,opt,

T04,opt) that gives the minimum sfc among the set, in this

case: 1.75,1600 K, respectively.

Since, it has been shown that increasing B or pc would

improve sfc, and since they are set to their maximum, it

follows that this pair (pf,opt, T04,opt) is the global minimum of

the problem as it is currently defined.

This is further shown in Figures 7 and 8. Figure 7 shows a

plot of sfc versus bypass ratio at different pf, while T04and pcare held at 1,600K and 40, respectively. The plot shows that

there is an optimum B for each pf, and that as pf decreases

the optimumB increases. When the maximumB is marked, it

can be seen that pf of 1.75 gives the lowest sfc. If pf isincreased or decreased sfc increases, but if B is increased

beyond ten, then sfc can be decreased.

The same concept is seen in Figure 8, which is a plot of sfc

versus pc at different pf, while holding B and T04 at 10,

1,600K, respectively. Again when the maximum pc is

marked, it becomes apparent that pf 1.75 gives the lowest

sfc, and that sfc can be lowered by increasing pc beyond its

current limit.

Sensitivity of sfc to deviations from optimum cycle

Figure 9 shows a sensitivity plot for the reference optimum

case. The plot was generated by holding the cycle design




Volume 78 Number 6 2006 467 480

476


11/14

variables at their optimum values (B 10, pc 40,

pf 1.75, T04 1,600K), and then systematically changing

one of the variables while holding the others constant.

The plot shows that the sfc of the cycle is most sensitive to

T04 and pf, with deviation in T04 being more detrimental

below the optimum and pfabove. As expected, sfc improves

when pc and B increase, but the plot shows sfc to improve

slightly and then to deteriorate when B is increased. This isbecause when B is increased the optimum pf changes

(decreases), but pf is held constant so sfc decreases slightly

until B reaches an optimum and then starts to increase. This

can also be seen in Figure 6. Table III is constructed from the

results in Figure 9 to summarize the effect of a 5 percent

decrease in each of the design variables on the sfc of the

optimum cycle.

Generally, the plot shows that if T04 and pf can be

controlled within ^5% of their optimum value, an sfc within

about 2.65 percent of the minimum can be achieved. It also

shows that B can be decreased to 80 percent of its maximum

value (220 percent) with only a 5 percent penalty in sfc. The

penalty may be even smaller if T04 and pf are optimized for

the new B. Similarly, pc

may be decreased 20 percent of its

maximum with a penalty of about 2.8 percent in sfc.

Minimizing sfc for a given specific thrust requirement

The problem becomes more meaningful when a minimum

specific thrust constraint is introduced. As mentioned

previously, if the specific thrust of an engine decreases the

engine must ingest more air to produce a given required

thrust. This means the engine diameter must be increased,

which introduces a number of penalties. Most importantly,

the weight and aerodynamic drag will increase. Other factors

include ground clearance and landing gear length, and

transportation difficulty for engines above 3 m in diameter

(Wilson, 1984).

A minimum specific thrust requirement for a given

application may be obtained by knowledge of the required

thrust and forward speed, and by specifying a maximum

allowed engine diameter.

Table IV summarizes the results of a series of optimization

runs for progressively increasingFs,min. The optimization runs

were carried out for an altitude of 11 km and a flight Mach

number of 0.8. The constraints were fixed at Bmax 10,

pc,max 40, T04,max 2,000 K, except for Fs,min which

progressively increases from 100 to 580N s/kg. The table

lists the optimum cycle design variables, and the status of the

constraints in each case. An A in a constraint status column

denotes an active constraint, while a P denotes an inactive

or passive constraint.

The table shows that, except in the first case, the minimum

Fs constraint becomes a limiting factor in minimizing sfc.Specifically, this begins above an Fs,min of 137N s/kg for these

conditions. To explain this figure, the table shows that in the

first case when Fs,min is set to 100 N s/kg, the constraint is

passive, and the specific thrust of the resulting optimum cycle

is found to be 137 N s/kg. This practically means that for

these conditions this minimum specific thrust is guaranteed

but ifFs,minis increased above this value, it becomes an active

constraint and a penalty on sfc is incurred. Another

outstanding feature is that B is no longer a limiting factorexcept at low Fs,min values. This means that with the

introduction of the minimum Fs constraint the problem has

become three-dimensional (pc is still an active constraint).

The table also shows that when B steps out as a limiting

factor (moving down the table), T04 steps in. After that, T04continues to be a limiting factor until the optimumB becomes

low enough to allow the required Fs,min to be achieved with a

lower T04. This occurs somewhere between 400 and 450 N s/

kg (between B 3 and 1.5).

Comparison with the graphical method

A graphical method for cycle optimization involving extensive

parametric variations is described in Cohen et al.(1987). This

method involves finding the pairs of (T04,pf,opt) at fixedB and

pc, plotting sfc versus Fs for these pairs, repeating for severalB, and finally repeating the whole process altogether at

different pc. The envelope curve for the family of different B

curves at constant pc gives the plot of optimum variation of

sfc with Fs at that pc.

Since, Table IV shows that pc is always a limiting factor,

this only needs to be done at the maximum pc. Figure 10

shows the result of this parametric variation performed at

pc 40, with B ranging from 0.1 to 10. Each constant B

curve was obtained by varying T04 from 900 to 2,000 K,

finding the optimum pf at each T04, and plotting the

corresponding sfc versus Fs. Superimposed on this plot is the

plot of optimum sfc versus Fs obtained from Table IV, shown

as a dashed line, which incidentally happens to be the

envelope curve that the graphical method seeks to find.

The obvious advantage of the numerical optimization

approach is the saving in calculation and plotting effort. But

more importantly, the identity of the cycle is difficult to

determine from the graph. The graph may outline the trend of

optimum variation, but the corresponding cycle design

variables B, T04 and pf cannot be read directly (unless

constant parameter lines are drawn, but that adds to the

effort).

Introducing additional constraints with: single stage fan

Another advantage of the numerical optimization approach

is the ease of incorporating practical design constraints.

Powel (1991) mentions that for a single stage fan pf, 1.9.

This constraint was added to generate Table V, which

shows that when this constraint is added pf

becomes the

limiting factor instead of T04, and the optimum B quickly

decreases as Fs,min increases. The impact of this constraint

on sfc can be visualized in Figure 11, in which the dashed

line is the plot of (constrained) optimum sfc versus Fsfrom Table V.

Conclusions

Consideration of the problem of minimizing sfc without a

constraint for minimum Fs has revealed a number of points.

Table I showed that the maximum B and pc are limiting

factors for all cases, which means that the problem is

Table III Sensitivity of optimum cycle to 5 percent decrease in designvariables

Design variable Dsfc (percent)

pf 1.10

pc 0.65

B 0.78

T04 2.65




Volume 78 Number 6 2006 467 480

477


12/14

essentially a two-dimensional optimization problem.

Generally, sfc continues to improve as B and pc are

increased, provided T04 and pf are optimized. Practicalconsiderations, however, limit the potential improvements.

Some general trends were observed, which might be

summarized as follows:. optimum T04 increases when either B or pc is increased;. optimum pfdecreases as B is increased; and. increasing B significantly decreases Fs.

The sensitivity analysis of the reference optimum cycle

showed that sfc is not very sensitive to small deviations from

optimum design values. It further revealed that the sfc of the

optimum cycle is relatively most sensitive to T04 and pf. A

5 percent decrease from optimum value in T04 and pf was

found to incur a penalty of 2.65 and 1.1 percent. This iscompared to a 0.652 and 0.776 percent penalty incurred by a

comparable decrease in pc and B, respectively.

The nature of the problem changes when a minimum Fsconstraint is introduced. B no longer becomes a limiting

factor, and the problem becomes three-dimensional (B, pf,

T04). The overall pressure ratio, however, remains to be

limiting factor in all the cases studied.

Using numerical optimization had a number of advantages.

First, it allowed a better (and quicker) understanding of the

problem by revealing key features such as trends and limiting

Figure 10Optimum variation of sfc with Fsat pc 40

Table IV Results of optimization with a progressively increasing minimum Fsconstraint

Constraints Optimum cycle Constraint status

Fs,min B pf T04 pc B pf T04 pc Fs sfc Fs,min B pf T04 pc

100 10 2,000 40 10 1.75 1,600 40 137 16.17 P A P A

150 10 2,000 40 10 1.85 1,683 40 150 16.2 A A P A

200 10 2,000 40 9.7 2.3 2,000 40 200 16.8 A P A A

250 10 2,000 40 7 2.84 2,000 40 250 18 A P A A300 10 2,000 40 5.2 3.54 2,000 40 300 19.3 A P A A

350 10 2,000 40 4 4.44 2,000 40 350 20.7 A P A A

400 10 2,000 40 3 5.6 2,000 40 400 22.14 A P A A

450 10 2,000 40 1.5 6.1 1,658 40 450 23.63 A P P A

500 10 2,000 40 0.5 6.08 1,407 40 500 25.12 A P P A

550 10 2,000 40 0.2 6 1,333 40 550 26.4 A P P A

580 10 2,000 40 0.1 6 1,333 40 580 27.15 A P P A

Notes:A, active constraint; P, passive constraint




Volume 78 Number 6 2006 467 480

478


13/14

criteria. Second, it significantly narrowed down the region of

interest for parametric study. Third, it allowed design

constraints to be easily incorporated in the study.

References

Arora, J.S. (1989), Introduction to Optimum Design, 1st ed.,

McGraw-Hill, New York, NY.

Cohen, H., Rogers, G.F.C. and Saravanamuttoo, H.I.H.

(1987), Gas Turbine Theory, 3rd ed., Longman, London.

Kerrebrock, J. (1992), Aircraft Engines and Gas Turbines, 2nd

ed., MIT Press, Cambridge, MA.

Kurzke, J. (1999), Gas turbine cycle design methodology: a

comparison of parameter variation with numerical

optimization, Transactions of the ASME, Vol. 121, p. 6.

Mattingly, J. (1996), Elements of Gas Turbine for Propulsion,

McGraw-Hill, Singapore, International edition.

Mattingly, J. (1999), Need info for BSc project, November23, 1999, Technical correspondence, E-mail: Jack@

aircraftenginedesign.com

Powel, D.T. (1991), Propulsion systems for twenty first

century commercial transports,Proc. Instn. Mech. Engrs, J.

of Eng. for Gas Turbine and Power, Vol. 205, p. 13.

Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling,

W.T. (1988), Numerical Recipes in C, 1st ed., Cambridge

University Press, Cambridge.

Wilson, D.G. (1984), The Design of High Efficiency

Turbomachinery and Gas Turbines, 1st ed., MIT Press,

Cambridge, MA.

Figure 11Optimum variation of sfc with Fswith a constraint ofpfmax 1.9

Table V Results of optimization with a progressively increasing minimum Fsconstraint and an additional constraint ofpf# 1.9

Constraints Optimum cycle Constraint status

Fs,min B pf T04 pc B pf T04 pc Fs sfc Fs,min B pf T04 pc

100 10 1.9 2,000 40 10 1.75 1,600 40 136.5 16.17 P A P P A

150 10 1.9 2,000 40 10 1.85 1,683 40 150 16.2 A A P P A

200 10 1.9 2,000 40 5.45 1.9 1,573 40 200 18.26 A P A P A

250 10 1.9 2,000 40 2.1 1.9 1,303 40 250 20.55 A P A P A300 10 1.9 2,000 40 1 1.9 1,213 40 300 22.13 A P A P A

350 10 1.9 2,000 40 0.5 1.9 1,167 40 350 23.28 A P A P A

400 10 1.9 2,000 40 0.185 1.9 1,142 40 400 24.14 A P A P A

420 10 1.9 2,000 40 0.1 1.9 1,137 40 420 24.4 A P A P A

430 10 1.9 2,000 40 0.063 1.9 1,134 40 430 24.56 A P A P A

Notes:A, active constraint; P, passive constraint




Volume 78 Number 6 2006 467 480

479


14/14

Further reading

Vanderplaats, G.N. (1984), Numerical Optimization Techniques

for Engineering Design, McGraw-Hill, New York, NY.

About the authors

Yousef S.H. Najjar, Founding Director of theEnergy Center, Fellow ASME (USA), Fellow

the Institute of Energy (UK), PE, C.Eng.

Professor of Mechanical Engineering. BSc,

Mech. Eng. (Power), Cairo University (1969);

MSc and PhD, Mech. Eng. (Thermal Power),

Cranfield Institute of Technology (UK) 1976

and 1979, respectively. Industrial experience: Chief Power

Engineer-Irbid District Electricity Company, Jordan (1969-

1975); Specialized industrial training with General Electric

(GEC) and related power industries (UK) (1973-1974).

Academic experience: Yarmouk University (1980-1986): The

founding chairman of the Mech. Eng. Dept., 1980-1982;

Member University Council (1985-1986). King Abdulaziz

University-Jeddah (1986-2001): participated effectively in

two funded research projects and ABET accreditation.

Published 123 papers in international refereed journals and

conferences; granted a patent by British Patent Office (1988);

two patent publications; lectured in 24 international

conferences; member of the Editorial Advisory Board for

the International Journals of: Energy and Environment, and

Applied Thermal Engineering. Awards: The 1995 Award for

excellence for an outstanding paper in J. Aircraft Eng. and

Aerospace Technology; Fellowship of ASME-USA (1999);Fellowship of Institute of Energy-UK (1990); Professional

Engineer; Chartered Engineer. Specialization: Energy-

Th er ma l Powe r i nc lu di ng G as Tu rb in es : Fu el s,

Combustion, Turbomachines and Advanced Energy

Systems; Internal Combustion Engines and Autotronics.

Initiated a course on Autotronics. Authored three books

and manuals. Latest research: Autotronics and Fuel cell

gas turbine hybrid power. Founded Pioneering Labs for:

automotive diagnosis energy audit and autotronics.

Yousef S.H. Najjar is the corresponding author and can be

contacted at: [email protected]

Sharaf F. Al-Sharif is a Researcher at King Abdulaziz

University, Jeddah, Saudi Arabia.




Volume 78 Number 6 2006 467 480

To purchase reprints of this article please e-mail: [email protected]

Or visit our web site for further details: www.emeraldinsight.com/reprints

Documents

Thermodynamics Optimization of the Turbofan Cycle