ASME-ATI-UIT 2010 Conference on Thermal and Environmental Issues in Energy Systems
16 – 19 May, 2010, Sorrento, Italy

Model for optimizing thermal efficiency of open indirect cycle of gas
turbine based on the mathematical programming
H.Farzaneh
Assistant professor, Graduate school of the environment and energy, Science and research branch, Islamic
Azad University, Tehran1415-755,Iran
Email: h.farzaneh@srbiau.ac.ir

ABSTRACTOpen indirect cycle of gas turbine is almost similar to a simple Brayton cycle but it can be used for heatrecovery means in different industries, specially, in nuclear power plants. In this investigation, a model of optimal
thermal efficiency of open indirect cycle of gas turbine has been developed and it has been applied for heatrecovery from a nuclear reactor. The model may be categorized as a nonlinear mathematical programming modeland the objective function is maximization of thermal efficiency in a given site conditions. Total compression
ratio, cycle temperature ratio and cycle pressure ratio are identified as three dimensionless numbers and they havebeen considered as basic decision variables. The model has been applied to identify the optimal design conditionof the heat recovery system.

1. INTRODUCTION:
Fuel consumption in industries contributes to the emissionof greenhouse gases in general and CO2 in particular.
Emission of CO2 is a function of technical operation of theindustries and the waste energy flow. Improvement in energy
efficiency and the design of technologies has reduced wastingenergy of industries considerably over last three decades. But
rapid economic growth in many developing countries andincreased welfare of population in industrialized nations has
led to expansion of industrial factories world wide. Numberof factories has increased considerably and air pollution isobserved in major cities in the world, including bothdeveloping and industrialized countries.
Energy recovery is considered as a sufficient method for
reducing air pollution due to converting waste energy touseful work in different industries. It may be categorized aspreheating of combustion air or fuel in industrial furnace,
preparing hot water in ventilation systems, producing
superheated steam in boilers or generating additional
electrical power in industries [1].
Open indirect cycle of gas turbine is one of the generalforms of heat recovery systems which may be used torecovery waste heat from cooling circuit of nuclear reactors[2]. Its cycle is similar to a simple Brayton cycle wherecombustion chamber is replaced with a recuperator in order toheat recovery from coolant fluid in nuclear reactor cooling
circuit. Absorbed heat is then used for increasing turbine inlet
temperature and generating of additional power in nuclear
power plant [3].

Many analytical tools have been developed for modelingopen indirect cycle of gas turbine and its contribution to theimplementation of the objectives of heat recovery. Somekinds of methods that have been suggested are thethermodynamically orientated simulators [4-12]. They study
operation of the system based on the predefined design
conditions.
Identification of optimal thermal efficiency of the open
direct cycle of gas turbine has been the subject of the present
work. A model of nonlinear mathematical programming has
been developed and it has been applied for identifying theappropriate means of achieving the objectives of optimal
design of the cycle. Theoretical concept of the model shall bepresented in the next section and it shall be followed bydemonstration of the application of model for defined casestudy.
2. THEORETICAL CONCEPT:The model has been found on the thermodynamics theory
and it has been tailored to identify the optimal design
condition of specified heat recovery technology. The
objective of the model has been to maximize the total thermal
efficiency of an open indirect cycle of gas turbine which isused for heat recovery from coolant flow of cooling circuit ofa nuclear reactor. Implementation of this concept has help toidentify optimal design condition while the technology is
considered as fixed.
Figure (1) shows the system of heat recovery. It contains
two stages compressors with one intercooler between them,
one recuperator for absorbing waste heat from coolant flow,

one gas turbine which is coupled with a generator for
producing additional electricity in nuclear power plant.

η C1 =

Cc1 − 1
1/η p

(8)

−1

Cc1

Substituting equations (8) and (7) into equation (5) yields:

T1 − T0 =T0(Cc1 − 1)

ηc

(9)

1

And it will be obtained for second stage compressor
similar to first stage compressor:

T2 − T4 =Figure 1- Heat recovery system Schematic
Total thermal efficiency in the Brayton cycle ( η th ) may be
defined as the ratio of net output power ( Wnet ) to total input
or absorbed heat ( Qadd ) as follows:η th =

Wnet
Qadd

(1)

According to figure (1), net output power and total
absorbed heat can be calculated by the help of following
equations:T2

ηc(Cc 2 − 1)

(10)

2

With following stream flow from inlet of first stagecompressor to outlet of second stage compressor,
dimensionless total compression ratio can be defined asfollows:

P1 = ( P0 − β1 )rc1

(11)

P2 = ( P1 − ( β 2 + ∆Pinc ))rc 2

(12)

γ

γ

( P − β1 )Cc1 γ −1 − (∆Pint + β 2 )Cc 2 γ −1
CcT = 0P0(13)

Wnet = m 0 C p {(T3 − T5 ) − (T2 − T4 ) − (T1 − T0 )}(2)

When compressed flow enters to the turbine, its pressure
will be reduced. It will be clear that the outlet pressure of theturbine should never reach to atmospheric pressure. A
dimensionless parameter is introduced to control this ratio:

Qadd = m 0 C p (T3 − T2 )

(3)

L=

If the heat capacity of working fluid is nearly considered
constant through the cycle,
thermal efficiency of cycle
should be rearranged as follows:η th =(T3 − T5 ) − (T2 − T4 ) − (T1 − T0 )(T3 − T2 )(4)

When the compressor unit is considered as an adiabatic
system, dimensionless compression ratio for each ofcompressors may be represented by equation (7) throughusing equations (5) and (6):T
T1 − T0 = T0 ( 1 − 1)
T0
T1  P1 = 
T0  P0 

 γ −1  γη p 



γ −1
γ
C1

Cc1 = r

(5)

P5
P0

(14)

According to equation (14), the dimensionless expansion
ratio of gas turbine can be represented by following
equations:γ −1P  γC t =  3 
 P0 

(15)γ −1

 P γC t =  3 
 P5 L 

(16)

γ −1

C t = (rt L) γ

(17)

γ −1

(6)

r γ
Ct = t
x

(18)
γ −1

T
1/η
⇒ 1 = Cc1 p
T0(7)

Efficiency of first stage compressor may be defined by thehelp of equation (8):1 γWhere, x = ( )Land rt =P3.
P5

Turbine stage efficiency then can be calculated by equation
(19):

1−

1

t=

ηp

Ct x
ηt =
1
1−
1
Ct x

(19)

According to the principles, difference between P5 and P0
should be negligible and also, air side pressure drop ofrecuperator should not be allowed to be high, therefore,
following relationship may be considered between outlet andinlet pressures ratio through recuperator and turbine:

P3 P0
=
P2 P5

(20)

According to the above assumption, x can be rewrite as
below:γ −1P  γP3x =  3  = 
γ −1 
 P Cc γ  P2 
 0 T 

γ −1
γ

=

Ct
CcT

(21)

T3
T0

(23)

Dimensionless cycle temperature ratio may be formed as afunction of technical indicators of the cycle if T3 is replacedby following equations:Thin − TCin
Tcout − TCin

(24)

T3 = (1 − ε H )T2 + ε H T8

(25)

T
T2
+εH 8
T0
T0

(26)

T4 = (1 − ε inc )T1 + ε inc T7

(27)

εH =

t = (1 − ε H )T2 = T4 (1 +Cc 2 − 1

ηc2

)

(28)

t = (1 − ε H )(1 − ε inc )Cc1 (1 +According to the principles, difference between P5 and P0
should be negligible and also, air side pressure drop ofrecuperator should not be allowed to be high, therefore,
following relationship may be considered between outlet andinlet pressures ratio through recuperator and turbine:

P3 P0
=
P2 P5

(20)

It can be concluded that equation (21) manifests therelationship between variations of turbine expansion ratio
based on the compressors compression ratio. Therefore, x is
named dimensionless cycle pressure ratio. It is also evident
that x should be set at the right level if the thermal efficiency
is to be maximized. Hence, identification of therelationship between Ct and CcT is an important factorof the aforementioned system design.

((1 − ε inc )ε inc (1 +

1
)
Ct.x

(22)

According to the principles of gas turbine design,
maximum power which may be obtained from the turbine isstrictly affected by the turbine inlet temperature (T3) [13].
This parameter will be influenced by the amount ofrecoverable heat through using the recuperator. Therefore,
dimensionless cycle temperature ratio ( t ) is introduced in themodel on the basis of the Carnot efficiency definition forcontrolling ratio of the maximum turbine inlet temperature toambient temperature:

ηC2

ηC 2

)+
(29)

T
) +εH 8T0

The total net output power can be determined fromequation (30):

Wnet = m 0 CpT0 DSW(30)
Where, DSW is defined as dimensionless specific work ofcycle. It will be calculated by substituting equations (9), (10),(22), (25), (27) and (28) into equation (2) as follow:
Cc − 1
Cc − 11) − (1 − ε int )Cc1 ( 2 ) − ε int ( 2 ) (31)
C t .x
ηc2
ηc2
Finally, thermal efficiency of open indirect cycle of gas
turbine may be represented by means of equation (32):
DSW = tη t (1 −Respectively, temperature difference between inlet andoutlet streams of the turbine may be represented by following
equation:

T3 − T5 = T3η t (1 −Cc 2 − 1

Cc 2 − 1η th =

Cc − 1
Cc − 1 Cc − 1
1) − (1 − ε int )Cc1 ( 2 ) − ε int ( 2 ) − 1(32)
C t .x
ηc2
ηc2
η c1

Cc − 1  
Cc − 1 t − (1 − ε int )Cc1 (1 + 2 ) − ε int (1 + 2 )
ηc2  ηc2 
tη t (1 −Equation (32) reveals that dimensionless numbers such as
cycle temperature ratio, total compression ratio and cycle
pressure ratio and also efficiency of compressors andeffectiveness coefficients of intercooler and recuperator
influence the thermal efficiency of open indirect cycle of gas
turbine through using it as a heat recovery system.
3. MATHEMATICAL MODELING APPROACH:
The model is formulated as a constrained optimization
programming with nonlinear objective function andconstraints (NLP)1. GAMS/CONOPT is used to solve the
NLP problem though following format [14]:1- Nonlinear Programming

Max Z = f ( x1 ,...., xn )(33)

Subject to:g i ( x1 ,..., x n ) ≤ bi

(34)

xj ≥ 0(35)Andi = 1,2,3,..., m

j = 1,2,3,..., n
Where, Z is defined as objective function and x and b
are represented as decision variables and upper limits ofconstraints.
According to this mathematical approach, variables,
parameters and constraints are categorized as follows:

3.1. Decision variables:
1) Cc1 , Cc 2 and x are defined as independed variables.

2) t , Ct , η t , η C1 and η C 2 are defined as first orderdepended variables.
3) Temperatures, Pressures and flow rate of working fluid
are defined as second order depended variables.
3.2. Exogenous Parameters
Temperature, Pressure and flow rate of coolant fluid which
its heat must be recovered by working fluid and also ambient
temperature and pressure are defined as exogenous
parameters of the model.
3.3. Technical Constraints:
Set of equations (8), (9), (10), (13), (14), (18), (19), (21),(22), (25), (27), (28), (29), and (31) are considered astechnical constraints of the model. The limitations of Cc
and x are included according the design condition:

1.1 < Cc < 2

(36)

x <1

(37)

3.4. Objective function
Objective function ( Z ) describes degree of utilization of
the system. In this model, thermal efficiency is defined as
objective function by the help of equation (32) which should
be maximized subject to satisfying technical constraints.
4. RESULTS AND DISCUSSION:
The model that has been developed in the present work has
been applied for analysis of open indirect cycle of gas turbine
as a heat recovery system. A case study has been designed on
the basis of heat recovery from the outlet coolant flow ofcooling circuit of a nuclear reactor.
Parameters of the abovementioned case study are describedin table (1). The optimum results of the model can beobserved in table (2).

4.1. Impact of dimensionless total compression ratio
Figure (2) shows variation of the thermal efficiency withthe dimensionless total compression ratio. It is clear that
thermal efficiency will be increased in simple Brayton cyclewith increasing of compression ratio of compressor. It can beobserved in this figure that the thermal efficiency is increased
to its maximum point at each of selected values for x from0.97 to 0.99 and after that, it will be decreased. Thermal
efficiency reaches to the optimum point of 0.268 at
CcT =2.020. It can be observed that the main difference
between simple Brayton cycle and open indirect cycle is theregressive trend of thermal efficiency after its optimum pointdue to replacing combustion chamber with recperator. Also,
the decision variable ( x ) has direct effect on improving
thermal efficiency due to increasing dimensionless specificwork of cycle.
Table 1- parameters of the case study
Parameter

Uni

Value

C
Bar
C
Bar

800
1.1
25
1

t
Outlet Coolant flow Temperature
Outlet Coolant flow pressure
Ambient temperature
Ambient Pressure

Table 2- Optimum results of the model
Item

η th
ηC
ηT

Unit
%

Value
26.8

%

80.2

%

93.5

Cc

1.49

Ct
t
T1
T2
T3

C

2
3.401
184.396C

208.862C

740.402

T4
T5

C

60.940C

271.273

P1
P2
P3

bar

3.489

bar

11.791bar

11.314

bar

3.4

bar

1.306

P4
P5
rc

3.489

rt

10.923

CcT
x

2.020

εH
ε inc

0.99
0.9
0.9

4.2. Impact of dimensionless cycle temperature ratio
Variation of thermal efficiency with dimensionless total
temperate ratio is depicted in figure (3). The figure represents
that the thermal efficiency increases smoothly until its

maximum point at t =3.401 and x =0.99 and then decreases
gradually. Unlike simple Brayton cycle, maximum thermal
efficiency will not be obtained at maximum turbine inlet
temperature. It may be explained by the fact that the turbine
inlet temperature will be limited by the amount of waste heat
and ability of the recperator for recovering it.
Whereas, the potential of heat source is constant, increasein turbine inlet temperature ( T3 ) necessitates increase in inlet
temperature of working fluid into recuperator ( T2 ) andconsequently increase in power consumption for two stages
compressors. Therefore, after the optimum point, thermal
efficiency will be reduced when the turbine inlet temperatureor the dimensionless cycle temperature ratio is increased.

can be obtained with CcT = 2.020 ,of the systemt = 3.401 and x = 0.99 at 26.8%.

The results of the model indicate that compression ratio
and turbine inlet temperature increase thermal efficiency until
certain point and after that, increment of these two factors
reduces thermal efficiency through replacing combustion
chamber with recuperator.
Application of the model results would enable to identify
maximum thermal efficiency at maximum allowable value ofdimensionless cycle pressure ratio. It means that,
dimensionless expansion ratio of turbine should be estimatedas much as closest to dimensionless total compression ratio. It
providesanalternative
approachtopredefined
thermodynamical simulations with the objective of theeconomic designing for the similar systems.
NOMENCLATURE

Figure 2. Variation of thermal efficiency withdimensionless total compression ratio

SymbolQuantity

Cp

Heat capacity

Ct
Cc
CcT

Dimensionless expanssion ratio

m0
P
rc
rt
T

Figure 3. Variation of thermal efficiency withdimensionless total temperature ratio

γ
ηp
ηth
β

∆Pinc
4.3. Impact of dimensionless cycle pressure ratio
It can be observed from figures (2) and (3) that maximum
thermal efficiency will be obtained at maximum
dimensionless cycle pressure ratio ( x = 0.99 ). The results ofthe model indicate that when dimensionless cycle pressure
ratio is increased, it will be possible to obtain maximum pointof thermal efficiency at lower value of dimensionless total
compression ratio ( CcT ) and cycle temperature ratio ( t ). It
means that a correct estimation of x is preferred which iscompatible with the objective of the economic designing of
the heat recovery system.
5. CONCLUSION
The objective of this research work has been to introduce
an optimal designing approach of open indirect cycle of gas
turbine as a heat recovery system based on the maximization
thermal efficiency. The results indicate that implementationof the optimal designing approach based on coordination ofdimensionless parameters such as total compression ratio,
cycle temperature ratio and cycle pressure ratio throughoperation mode would lead to maximum of thermal
efficiency. The results show that maximum thermal efficiency

Dimensionless compression ratio
Dimensionless total compression
ratio
Flow rate of working fluid
Pressure of working flow
Compression ratio of each stage

SI
Unit
kJ/kgC

Kg/sbar

Expansion ratio of turbine
Temperature
Heat capacity ratio
Polytrophic efficiency of compressor

C

Thermal effeicinecy of cycle
Pressure drop at the inlet section ofcompressor
Pressure drop of intercoolerεEffectiveness coefficient

Subscripts
hin
cout
cin
inc
H

Quantity
Inlet hot stream into recuperator
Outlet cold stream from recuperator
Inlet cold stream into recuperator
Intercooler
Recuperator

bar
bar

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