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Energy 38 (2012) 362e375

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Energy

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Efficiency enhancement of a gas turbine cycle using an optimized tubularrecuperative heat exchanger

Hoseyn Sayyaadi*, Reza MehrabipourFaculty of Mechanical Engineering-Energy Division, K.N. Toosi University of Technology, P.O. Box: 19395-1999, No. 15-19, Pardis Str., Mollasadra Ave., Vanak Sq.,Tehran 1999 143344, Iran

a r t i c l e i n f o

Article history:Received 28 February 2011Received in revised form19 November 2011Accepted 22 November 2011Available online 30 December 2011

Keywords:Efficiency enhancementPayback time minimizationMulti-objective optimizationBellman-Zadeh decision-makingLINMAP decision-makingTOPSIS decision-making

* Corresponding author. Tel.: þ98 21 8867 4212; faE-mail addresses: sayyaadi@kntu.ac.ir,

(H. Sayyaadi).

0360-5442/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.energy.2011.11.048

a b s t r a c t

A simple gas turbine cycle namely as the Kraftwerk Union AG unit including a Siemens gas turbine modelV93.1 with 60 MW nominal power and 26.0% thermal efficiency utilized in the Fars power plant locatedis considered for the efficiency enhancement. A typical tubular vertical recuperative heat exchanger isdesigned in order to integrate into the cycle as an air pre-heater for thermal efficiency improvement.Thermal and geometric specifications of the recuperative heat exchanger are obtained in a multi-objective optimization process. The exergetic efficiency of the gas cycle is maximized while thepayback time for the capital investment of the recuperator is minimized. Combination of these objectivesand decision variables with suitable engineering and physical constraints makes a set of the MINLPoptimization problem. Optimization programming is performed using the NSGA-II algorithm and Paretooptimal frontiers are obtained in three cases including the minimum, average and maximum ambient airtemperatures. In each case, the final optimal solution has been selected using three decision-makingapproaches including the fuzzy Bellman-Zadeh, LINMAP and TOPSIS methods. It has been shown thatthe TOPSIS and LINMAP decision-makers when applied on the Pareto frontier which is obtained ataverage ambient air temperature yields best results in comparison to other cases.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper a simple gas turbine cycle namely as the KraftwerkUnion AG unit utilized in the Fars gas power plant located in theShiraz city of Iran is considered for the efficiency enhancement.This unit is a Siemens gas turbine model V93.1 with 60 MWnominal power and 26.0% thermal efficiency at ISO condition. Thereare two categories of methods for efficiency enhancement of gascycles. In the first category the efficiency of gas cycles is enhancedusing compressor inlet air cooling [1e3]. In this method, the inletair of the air compressor is cooled using evaporative coolers,absorption chillers, electric chillers and similar apparatuses inorder to increase volumetric efficiency of the compressor andreducing compression work which leads to an increasing in theefficiency of the cycle and power generation. Second category ofmethods dealing with increasing the combustion efficiency andtherefore the efficiency of the cycle is heat and gas recirculation[4e8]. In the mass recirculation system, a portion of flue gas after

x: þ98 21 8867 4748.hoseynsayyaadi@gmail.com

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leaving the turbine is recirculated and mixed with the compressedair coming from the air compressor or directly entered into thecombustion chamber. FGR (Flue gas recirculation) which some-times called as EGR (exhaust gas recirculation) leads to both heatrecirculation by preheating of the inlet air of the combustionchamber and dilution of the air-fuel mixture leading to less NOxformation. The FGR system systems offer the substantial reductionof the thermal NOx formation (There is three mechanism of NOxformation including thermal nitrogen oxidation, prompt NOx andfuel NOx [9]) due to reduction of N2 and O2 contents which aresubstituted with CO2 and H2O, shorter residence time of reactantsdue to preheating and dilution, and reduction of local peaktemperatures due to better intermixing [4]. In the heat recircula-tion systems the heat of combustion is recirculated to the inlet air ofthe combustion chamber through the use of recuperative heatexchanger or a regenerative one [4]. In heat recirculation thethermal energy is transferred from combustion products into coldsubstrates without mass transfer and thus without any dilution ofreactant [4]. Consequently, the total reactant enthalpy is increasedenabling sustained combustions. This leads to a self-sustained orauto-thermal combustion that are sometimes referred to assuperadiabatic or excess enthalpy combustion [4]. The ideal of heatrecirculation is typical in the combustion science [4,5] and several

Nomenclature

Ao Heat transfer area (calculated based on tubes outsidediameter)(m2)

BL Booked life (years)C Cost (US $)CI Capital investment (US $)CCL Levelized carrying charge (US $)c Unit Cost (US $ per unit of the proposed parameter)diþ Distance of point ith from the ideal pointdi� Distance of point ith from the non-ideal pointD Diameter (mm)_E The rate of exergy (kW)e Specific exergy (kJ kg�1)�e Specific molar exergy (kJ/kmol�1)F An objective functionh Heat transfer coefficient (W m�2 K�1)h Molar enthalpy (kJ kmol�1)ieff Interest rate (cost of money)j jth year of operationLHV Molar Lower Heat Value of fuel (kJ kmol�1)Ltp Tube pitch in tube bundle (mm)Nt Number of tubesNb Number of bafflesM Molecular weight (kg kmol�1)max Maximum operator in the fuzzy logicsmin Minimum operator in the fuzzy logicsMOEA Multi-objective evolutionary algorithm_m Flow rate (kg s�1)_n Molar flow rate (kmol.s�1)P Pressure (kPa)payback Payback time for the capital investment_Q Heat transfer rate (kW)rFC Annual escalation rate for the fuel costT Temperature (�C or K)TRRj Total revenue requirement for jth year of the system

operation (US $)s Molar specific entropy (kJ kmol�1 K�1)Uo Overall heat transfer coefficient (calculated based on

outside tube surface) (W m�2 K�1)_W Power (kW)W Fabrication weight (kg)DP Pressure difference or pressure drop (kPa)

Greek lettersr Density (kg m�3)h Efficiency

n Specific volume (m3 kg�1)m Membership functionε Exergetic efficiencyl fuel-air ratio (mass base)l Molar fuel-air ratiohsc Compressor isentropic efficiencyhsg Gas turbine isentropic efficiency

Subscripts0 Index for ambient condition of the

atmospheric air, Index for the first yearof the system operation

1,2,.,5,6 States 1,2,.,5,6 on regenerative gas cyclea Airac Air compressorbray Brayton gas cycle (simple gas cycle with

no air pre-heater)cc Combustion chamberduct Ducting for air and flu gas transfer

to/from the recuperatorfab Fabricationf Fuelhx Heat exchangerg Gas (flue gas)gt Gas turbinei ith elementit Tube insideitl Inner tube limit in tube bundlej jth element; jth year of the

system operationL Levelized valuelm Log mean temperature difference (LMTD)net Netot Tube outsideotl Outer tube limit in tube bundlerec recuperatorreg Regenerative gas cycles Isentropic, shell sidet Tube sidestack Stack

Superscriptsmax Maximum value for an objective

functionmin Minimum value for an objective

functionn Non-dimensional

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375 363

combustion technologies such as mild combustion, recirculatingcombustion have been developed based on this idea [6]. Moreover,such approach can be beneficial for a combustion process becausemain exergy losses in the combustion arise from the heat transferfrom hot products to cold reactants while this temperature differ-ence is beneficially minimized in the gas turbine system involvingthe air preheating [7]. Another technologies for heat recirculation isthe using a regenerative heat exchanger, however these equip-ments are not usual in gas cycles but widely used as the air pre-heatres of steam boilers namely as Ljungström. There are severalworks involving heat recirculation approach for efficiencyenhancement of gas turbines [5e8,10,11]. Ruixian et al. analyzedthe recuperative gas turbine cycle with a recuperator locatedbetween HP and LP turbines namely as ARC (alternative recupera-tion cycles) and compared it with the simple gas cycle and the CRC

(conventional recuperative gas cycle) [10]. They showed that theefficiency of this cycle might be higher than the CRC in the case ofsame temperature ratio [10]. Further they indicated that themaximum optimum efficiency of practical ARCs is always lowerthan that of CRCs and the optimum pressure ratio for efficiency ofARC is always higher than that of CRC.

Kim and Hwang analyzed part load performance of the recu-perated gas cycle and specified which part load control strategy issuitable for various configurations of the cycle [11].

In the currentwork, we consider integration of the air pre-heaterinto the gas cycle for efficiency enhancement of the proposedSiemens gas turbine model V93.1. In this regard a typical tubularvertical heat exchanger is designed as an air pre-heater of theproposed gas cycle. Further we employ optimization approach forintegrating of the new component (air pre-heater) into the existing

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375364

gas cycle. Optimization is usually employed when a system is indesign stage; however we will show that in improvement stage ofexisting systems optimization could be useful. Thermal andgeometric specifications of the recuperative heat exchanger areobtained in a multi-objective optimization process with the exer-getic efficiency and payback time of the recuperator investment asobjectives of optimization. Indeed multi-objective approach inoptimization of energy systems have been implemented byresearchers [12e18] to deal with energetic, economic and environ-mental criteria, simultaneously. Energetic, economic and environ-mental modeling of energy systems usually leads to a set of MINLP(mixed integer non-linear) optimization problem. Therefore, inorder to avoid obtaining a local optimum instead of the globaloptimum, meta-heuristic optimization approaches (e.g. geneticalgorithm or simulated annealing) have been utilized by researchersnowadays [13,15,18e20]. Chang and Hwang [13] developed a MINLPmulti-objective model for an energy system to reduce the cost andenvironmental impact. Roosen et al. [17] implemented multi-objective optimization to balance between the capital investmentand operating cost of a combined power cycle. Multi-objectiveoptimization of a benchmark cogeneration system namely asCGAM problemwas conducted by one of authors [13]. In that work,three objective functions including the exergetic, the total levelizedcost rate of the system product and the cost rate of the environ-mental impact were considered simultaneously.

In a most relevant work Sayyaadi et al. designed a particularnon-TEMA type recuperative heat exchanger utilized as a regener-ator of a regenerative gas turbine cycle, for a gas cycle [21]. Theyconsidered a tubular heat exchanger with a vertical annular tubebundle and disk and doughnut types baffles. The exergetic effi-ciency of the entire gas cycle together with the purchased equip-ment cost of its recuperative heat exchangers was considered asobjectives of optimization. It was assumed that the recuperator isdesigned for an existing gas turbine cycle to be retrofitted. Threescenarios for optimization of the system including the minimiza-tion of the recuperator cost, maximizing the cycle exergetic effi-ciency and simultaneous optimization of both objectives wereperformed. An example of decision-making was presented to selecta final optimal solution from the Pareto frontier. Finally the resultswhich were obtained by three optimization scenarios werecompared together and also compared with the base case system.

In this work, the cost of recuperator which was the secondobjective functionof the ref. [21] is substitutedwith thepayback timeof the recuperator investment. Therefore, the payback time of therecuperator is minimized instead of the recuperator cost. This newobjective function is comprised from the capital investment of therecuperator and the fuel cost of the gas cycle. Therefore, itmeans thatin the current research the three objectives including the exergeticefficiency, the cost of the recuperator and the saving of the fuel costare optimized. The last two objectives are integrated in the paybacktime objective in fact. Therefore, minimizing the payback time of therecuperator investment leads to either minimizing the recuperatorcost or maximizing the fuel cost saving through using of the recu-perative gas cycle. Hence, in this work one more objective besidethose objectives which were considered in ref. [21] is optimized.Further the economic model for evaluation of the carrying chargeand fuel cost is modified based on the value of interest rate and fuelescalation factor; hence levelized values for capital investment andfuel cost are employed in evaluation of the payback time. Combina-tion of aforementioned objectives and decision variables includingtubes length, tubes outside/inside diameters, tube pitch in the tubebundle, outer and inner tube limits of the tube bundle and the totalnumber of disc and doughnut baffles plus the air outlet temperaturefrom the recuperator with suitable engineering and physicalconstraints makes a set of the MINLP optimization problem.

Optimizationprogramming inMATLAB is performedusingone of themost powerful and robust multi-objective optimization algorithmsnamely as the NSGA-II. Since operation of the gas cycle is highlydependent to the ambient air temperature, the multi-objectiveoptimization is performed in three cases of ambient conditionincluding the minimum, average and maximum annual condition atthe site of gas cycle (Shiraz city in Iran). Further, in additionalimprovement on previous work [21], three decision-makingapproaches including the fuzzy Bellman-Zadeh [22], LINMAP[23,24] and TOPSIS [23,24] are utilized for selection of final optimalsolutions from the Pareto frontiers which obtained at the minimum,average andmaximumenvironmental temperature. Therefore, threeoptimal solutions are selected using aforementioned decision-makers at three ambient cases (minimum, average and maximumannual air temperature at the site). Since the performance of the gascycle is significantly affected by the ambient air condition, thequestion is which ambient condition should be considered as thebest design condition. In this paper we try to answer this questionand we will present a systematic method to specify which ambienttemperature should be taken account for designing of the regener-ative gas cycle.

2. Problem definition

As is previously mentioned in this paper, the proposed gasturbine cycle is a simple Brayton gas cycle. This gas cycle isa Siemens unit model V93.1 namely as the Kraftwerk Union AG unitwith 60 MW nominal power and 26.0% thermal efficiency at ISOcondition (15 �C ambient air temperature at sea level). One SiemensV93.1 gas turbine has been installed in the Fars gas power plantlocated in the Shiraz city, Iran and has been in operation since 1980.As the unit is an old unit with a relatively low efficiency incomparison to current technologies of gas turbines, the objective ofthis project is enhancement of the proposed V93.1 unit as much aspossible in order to operate it with a more reasonable efficiency.Table 1 show general specifications of the V93.1 Brayton gas cycle.

In this paper, the thermal efficiency of the proposed gas turbinewill be enhanced by integration of a recuperative heat exchanger asan air pre-heater. Fig. 1 shows a schematic arrangement of theproposed regenerative gas turbine cycle with a recuperative heatexchanger as an air pre-heater. Combustion chamber inlet air ispre-heated using the flue gas exhausts from the gas turbine.

The recuperative heat exchanger that will be integrated into thegas cycle as an air-preheater comprises of a vertical tubular shelland tube heat exchanger that directly is connected to a conical stackat the top of the heat transfer area. The compressed air at the outletof the air compressor enters to the shell side of the recuperatorfrom the top and is pre-heated by the flue gas that enters in tubesfrom the bottom of heat exchanger. The pre-heated air exits fromthe bottom of shell side and directed to the combustion chamber.Flue gas exits from tubes at the top and directed into the stacksection (conical section at the top of heat transfer area). Theproposed heat exchanger has an annular tube bundle with disk anddoughnut baffles. More detail on specifications of the proposedrecuperative heat exchanger can be found in [21].

The aim is finding the geometrical specifications of the recu-perative heat exchanger including tube length, outside/insidediameters of tubes, tube pitch in the tube bundle, outer and innertube limits of tube bundle and the number of baffles plus the pre-heated air outlet temperature from the recuperator so that theexergetic efficiency of the cycle is maximized and the payback timefor capital investment of the recuperator is minimized,simultaneously.

Table 1Specifications of the simple gas turbine.

Manufacturer Kraftwerkunion AG

Type Siemens V93.1Number of turbine stages 4Rotor speed (rpm) 3000Air flow rate at ISO condition (kg s-1) 343.4Flue Gas flow rate with gas oil at ISO condition (kg s-1) 348.6Flue Gas flow rate with natural gas at ISO condition (kg s-1) 347.8Turbine inlet temperature for the base load operation

at rated output (�C)850

Turbine inlet temperature for the peak load operationat rated output (�C)

870

Compressor type Single flow axialtype V 93.1

Number of compressor stages 16Compressor air flow rate at the ISO condition (kg s-1) 343.4Compression ratio of the compressor at the ISO condition 8.70Compression ratio of the turbine at the ISO condition 8.35Isentropic efficiency of the air compressor 0.84Isentropic efficiency of the turbine 0.85Combustion chamber type Vertical silo typeNumber of combustors 4Pressure loss in the combustion chamber 2% of the

inlet pressure

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375 365

3. System modeling

3.1. Thermodynamic modeling

Thermodynamic model of the entire regenerative gas turbinecycle is built based on the following basic assumptions

1. All processes are steady state.2. The principle of ideal-gas mixtures is applied for the air and

combustion products.3. The fuel is the natural gas and it is assumed to be 100%methane

and ideal gas.

Fig. 1. Schematic for a regene

4. Heat loss from the combustion chamber is considered to be 2%of the fuel lower heating value. All other components areassumed adiabatic.

5. Constant pressure loss ratios are considered in the systemcomponents except in the recuperator (pressure losses inrecuperator are calculated based on hydraulic calculations).

6. Molar fractions for composition of the inlet air are assumed tobe 0.7594N2, 0.2038 O2, 0.0003 CO2 and 0.0274H2O.

7. Isentropic efficiencies of the air compressor and gas turbine areassumed constant.

Therefore, thermodynamic equation of the cycles (simple andrecuperative gas cycles) are developed as follows,

3.1.1. Air compressorIn order to achieve the outlet isentropic temperature of the air

compressor we have:

T2sT1

¼ rk�1k

pc (1)

Where rpc is the compression ratio of the compressor (¼8.7). Thereal outlet temperature of the compressor is:

T2 ¼�T2s � T1

hSC

�þ T1 (2)

Isentropic efficiency of the compressor,hSC, is 0.84. Assuming anadiabatic compressor, the consumed power of the air compressoris,

_Wac ¼ _maðh2 � h1Þ (3)

rative gas turbine cycle.

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375366

3.1.2. RecuperatorDetail thermohydraulic formulas and thermodynamic correla-

tion for modeling of the recuperator have been presented in [21].

3.1.3. Combustion processChemical reaction equation of the reaction process can be

formulated as,

lCH4 þ�xN2;aN2 þ xO2;aO2 þ xCO2;aCO2 þ xHO2;aH2O

�/½1þ l�

� �xN2;gN2 þ xO2;gO2 þ xCO2;gCO2 þ xHO2;gH2O�

(4)

Where l is the molar fuel to air ratio and

xN2;g ¼ xN2 ;a

1þl;xO2 ;g ¼ xO2;B�2l

1þl;xCO2;g ¼ xCO2;Bþl

1þl;xHO2;g

¼ xHO2 ;Bþ2l

1þl(5)

In Eq. (5) subscripts ’a’ and ’g’ denote the property of species(molar composition) in the atmospheric air and flue gas, respec-tively. The energy balance for the combustion chamber is:

0 ¼ _QCV � _WCV þ _nf hf þ _naha � _nphp (6)

Where index ‘f’ presents the fuel, index ‘a’ points to the air andindex ‘p’ points to products of the combustion. In Eq. (6) we have_WCV ¼ 0 and since we assumed the heat loss from the combustionchamber to be 2% of LHV of the fuel,

_QCV ¼ �0:02� _nf � LHV ¼ �0:02� _na � l� LHV (7)

Therefore Eq. (6) is converted to the following form,

0 ¼ �0:02lLHVf þ ha þ lhf � ð1þ lÞhp (8)

LHVf for the methane as a fuel is 3124 kJ kmol�1. Further wehave:

ha ¼ �xN2;ahN2

þ xO2;ahO2þ xCO2;ahCO2

þ xHO2;ahH2O�at T3

(9a)

hp ¼ �xN2;ghN2

þ xO2;ghO2þ xCO2 ;ghCO2

þ xHO2;ghH2O�at T6

(9b)

Hence l is obtained from solution of Eq. (8), therefore, the massflow rate of the fuel is calculated as,

_nF ¼ l _na/ _mF ¼ l

�Mf

Ma

�_ma (10)

Where Mf and Ma are molecular weights of the fuel and air,respectively.

3.1.4. Gas turbineIn similar way to the air compressor, the isentropic outlet

temperature of the gas turbine is determined as,

T4T6s

¼ rk�1k

pg (11)

Where rpg is the expansion ratio of the gas turbine (¼8.35) and k ¼cp

cp � Rin which R ¼ R

Mt. Mt is the molecular weight of outlet gas.

Therefore,

T5 ¼ T4 � hsgðT4 � T5sÞ (12)

The isentropic efficiency,hsg, of the gas turbine is 0.85 here.

Considering the turbine as a control volume and an adiabaticturbine, from energy balance we have:

_Wgt ¼ ð1þ lÞ _naðh4 � h5Þ ¼ _mpðh4 � h5Þ (13)

3.1.5. Exergetic efficiency of the gas cycleThe exergetic efficiency of the gas cycle is determined as follows,

εtot ¼_Wnet

_mf echf¼

_Wgt � _Wac

_mf echf(14)

Where _Wnet is the net generated power and echf is the chemicalexergy of the fuel assumed as 53155.8 kJ kg�1 for methane.

3.2. Thermohydraulic modeling of the recuperative heat exchanger

Thermohydraulic model is used here in order to calculate therequired heat transfer area for the recuperative heat exchanger inthe one hand and calculating of the hot and cold streams pressuredrops which affect performance of the gas cycle on the other hand.An especial type of the shell and tube heat exchanger with a verticalannular tube bundle and disk and doughnut shape baffles is usedhere as the air pre-heater of the gas cycle. Complete thermohy-draulic model for this type of heat exchanger was presented bySayyaadi et al. in [21].

3.3. Economic modeling

As is previously mentioned, the payback time for the capitalinvestment of the recuperative heat exchanger is considered as thesecondary objective of this work. Total capital investment of therecuperator is comprised from the cost of heat transfer area plusthe cost of stack section and the piping cost. Therefore, the capitalinvestment of the gas cycle enhancement is,

CI ¼ Chx þ Cstack þ Cduct (15)

The capital investment of a heat exchanger (the heat transferarea) can be estimated using the following expression [25],

Chx ¼ 8500þ 409A0:85o (16a)

Where Ao is the total outside area of the tube bundle which iscalculated using the thermohydraulic model of the recuperativeheat exchanger [21]. The cost of stack section is obtained based onthe fabricated weight of the stack and the manufacturing cost perkg of the fabricated weight as follow,

Cstack ¼ cfabWstack (16b)

Wstack is the fabricated weight of the stack section obtained inmechanical design, cfab is the fabricated price of the stack per kg ofits weight taken as 3.0 $.kg�1 (in the Iran). In Eq. (18) the ductingcost, Cduct, based on the site investigation is considered as 38,500$in this case (ducts are used in order to transfer the air and flue gasesto/from the recuperator).

A levelized value for the total annual cost of the capitalinvestment, CCL, can be computed by applying a discountingfactor (the cost of money or interest rate) and the capital-recoveryfactor CRF:

CCL ¼ CRFXBL1

TRRj�1þ ieff

�j (17a)

Fig. 2. Comparison of the exhaust gas temperatures predicted by the model and realvalues.

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375 367

CRF ¼ieff�1þ ieff

�BL�1þ ieff

�n�1(17b)

Where TRRj is the total revenue requirement in the jth year of thesystem operation [13] obtained as,

TRRj ¼CIBL

(17c)

In Eqs (17aec) ieff is the average annual effective discount rate(cost of money), and BL denotes the system economic life (bookedlife) expressed in years. The operating life of the system is assumedto be 20 years and ieff in Iran is 17%.

Therefore, the payback time for the capital investment of therecuperator is calculated based on the levelized capital investment,CCL, (Eq. (17a)) and the annual saving on the fuel cost reduction asfollow,

payback¼ CCL�BL

cfL �86400�365� 1=rf

!��_mfSimple cycle

� _mfRecuperative cycle

�(18)

Where cfL is the levelized cost of each cubic meters of the naturalgas. If the series of payments for the annual fuel cost is uniform overtime except for a constant-escalation rFC then the levelized value forthe fuel cost per cubic meters, cfL , of the series can be calculated bymultiplying the unit cost of fuel at the beginning of the first year ofthe project cf0 by the constant-escalation levelization factor CELF asfollows

cfL ¼ cf0CELF ¼ cf0kFC�1� kBLFC

�ð1� kFCÞ

CRF (19a)

With

kFC ¼ 1þ rFC1þ ieff

and rFC ¼ constant: (19b)

The terms rFC and CRF denote the annual escalation rate for thefuel cost (assumed to be 5%) and the capital-recovery factor (Eq.(17b)), respectively.

In Eq. (19a) cf0 is 0.08 $ m�3 in Iran (the international price fornatural gas is approximately 0.30 $ m�3 about 3.75 times higherthan the local price in Iran).

Table 2Comparison of the performance of the simple gas cycle at the ISO condition pre-dicted by the thermodynamic model and catalogue reported values.

Specification Catalogue data Predicted by the code Error

Efficiency 0.26 0.266 2.3%Air flow rate (kg s�1) 343.4 335.3 2.4%Fuel flow rate (kg s�1) 4.4 4.49 2.0%

3.4. Model verification

Thermodynamic and thermohydraulic modeling of the cycle isperformed using the MATLAB programming. For verification of themodel, the registered data at the site of the proposed gas turbine forthe exhaust temperature of the gas is compared with correspond-ing data predicted by the thermodynamic model. The thermody-namic model has been performed for site condition with 84.7 kPaatmospheric pressure and the ambient registered temperature.Fig. 2 illustrates this comparison for 27 days selected during anoperating year 2009, randomly.

This figure indicates that the maximum error is 2.2%. Furtheroperation of the simple gas cycle at ISO condition predicted by thedeveloped model is compared with the system catalogue (reportedby the manufacturer) as indicated in Table 2.

As is clear, Table 2 indicates that in this case the maximum erroris also 2.4%. Therefore, our thermodynamic model is able to predict,

the gas cycle behavior with a maximum 2.4% error which isreasonable for our purpose.

4. Objective functions, decision variables and constraints

4.1. Definition of the objectives

As is already discussed, in this paper two objectives includingthe exergetic efficiency of the regenerative gas turbine and thepayback time for the capital investment of recuperator denoted byEqs. (14) and (18), respectively, are considered. The exergetic effi-ciency (Eq. (14)) is maximized while the recuperator investmentpayback time (Eq. (18)) is minimized. These objectives are consid-ered simultaneously in a multi-objective optimization process.

4.2. Choice of decision variables

Following geometrical and thermal specifications of the recu-perative heat exchanger are considered as decision variables,

� Lt: Tube length (m)� Dto: Tube outside diameter (m)� Dti: Tube inside diameter (m)� Ltp: Tube pitch in the tube bundle (center to center distance oftubes in m)

� Dotl: Outer tube limit in the tube bundle(m)� Ditl: Inner tube limit in the tube bundle (m)� Nb: Total number of baffles (including disk and doughnutbaffles)

� T3: The outlet temperature of the pre-heated air from therecuperator (K)

4.3. Constraints and limitations

Following limitations are considered for the regenerative gasturbine cycle:

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375368

3 � vair � 6m:s�1 (20)

Where vairis the air velocity in the circular cross section area ofdoughnut baffles and the annular area limited between the outertube limit of the tube bundle and shell inside area.

T3 � 1420 K (21)

T6 � 378:15 K (22)

6 � Lt � 12m (23)

LtpDto

˛f1:25;1:33;1:5g (24)

Nb˛f3;5;7;9;11;13;15g (25)

2 � Dsi � 4 m (26)

0:80 � Dotl

Dsi� 0:97 (27)

0:25 � Ditl

Dsi� 0:45 (28)

T3 < T4 < T5 (29)

DPs � 3:5kPa (30)

DPt � 5kPa (31)

Since implementing of the air pre-heater in some case maycause efficiency reduction of the gas cycle even lower than thesimple gas turbine with no air pre-heater, in order to avoid such unreasonable condition, the following constraint has been imposedon the optimization process,

εreg � εbray þ 0:01 (32)

Where εreg and εbray are exergetic efficiencies of the regenerativeand simple Brayton gas cycles, respectively.

5. Multi-objective optimization

Multi-objective optimization of objectives function expressedby Eqs. (14) and (17) is performed using the multi-objectiveevolutionary algorithm. A multi-objective optimization problemrequires the simultaneous satisfaction of a number of different andoften conflicting objectives. It is required to mention that nocombination of decision variables can optimize all objectives,simultaneously. Multi-objective optimization problems generallyshow a possibly uncountable set of solutions, whose evaluatedvectors represent the best possible trade-offs in the objectivefunction space. Pareto optimality is the key concept to establisha hierarchy among the solutions of a multi-objective optimizationproblem, in order to determine whether a solution is really one ofthe best possible trades-off [26]. Eq. (1) shows how a multi-objective optimization problem can be formulated mathematically.

min FjðXÞ cj˛f1;2;3;.:; kg subject to X˛L (33)

Where we have k � 2 objective functionsFj : Rn/R1. The feasibleobjective region Z is the image of the feasible region (i.e Z ¼ F (X)3Rk). The elements of Z are called objective vectors. The objectivevectors are denoted by F(X) or by Z¼ [z1,z2,z3.,zk]T, where zj¼ Fj (X)cj ˛{1,2,.,k} [27].

Classical search and optimization methods are not efficient infollowing the Pareto approach for multi-objective optimizations.The class of search algorithms that implement the Pareto approachfor multi-objective optimization in the most straightforward wayis the class of multi-objective evolutionary algorithms (MOEAs)[12]. In this paper, one of most powerful MOEA namely as Non-dominated sorting genetic algorithm, NSGA-II has beenemployed to find the Pareto optimal frontier for the proposedrecuperative gas cycle. This method was well described bySayyaadi et al. in [21].

6. Decision-making in the multi-objective optimization

In multi-objective optimization a process of decision-makingfor selection of the final optimal solution from available solutionsis required. There are several methods for decision-making processin decision problem. These methods can be employed for selectionof a final optimal solution from the Pareto frontier Since, dimen-sion of various objectives in a multi-objective optimizationproblem might be different (for example in our case the exergeticobjective has no dimension while the dimension of the paybacktime is in years), therefore, before any decision, dimension andscales of objectives space should be unified. In this regard, objec-tives vectors should be non-dimensionalized before decision-making. There are some methods of non-dimensionalizationutilized in decision making including linear non-dimensionalization, Euclidian non-dimensionalization, and fuzzynon-dimensionalization.

� Linear non-dimensionalization

Consider the matrix of objectives at various points of the Paretofrontier is denoted by Fij where i is the index for each point on thePareto frontier and j is the index for each objective in the objec-tives space. Therefore a non-dimensionalized objective, Fnij , isdefined as,

Fnij ¼ Fijmax

�Fij for maximizing objectives (34a)

Fnij ¼ 1=Fijmax

�1=Fij

for minimizing objectives (34b)

� Euclidian non-dimensionalization

In this method, a non-dimensionalized objective, Fnij , is definedas,

Fnij ¼ FijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPmi¼1�Fij22

q for minimizing and maximizing objectives

(35)

� Fuzzy non-dimensionalization

In this method, a non-dimensionalized objective, Fnij , is definedas,

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375 369

Fnij ¼ Fij �min�Fij

max�Fij�min

�Fij for maximizing objectives (36a)

Fnij ¼ max�Fij� Fij

max�Fij�min

�Fij for minimizing objectives (36b)

In this paper most famous and common type of decision-making processes including the fuzzy Bellman-Zadeh, LINMAPand TOPSIS method is used in parallel in order to specify the finaloptimal solution. The Bellman-Zadeh method utilizes the fuzzynon-dimensionalization while the other methods (LINMAP andTOPSIS) employ Euclidian non-dimensionalization. The followingsections are presented here in order to describe these decision-making algorithms.

6.1. Bellman-Zadeh decision-making method

When using the Bellman-Zadeh approach, each FjðXÞ of Eq. (33)is replaced by a fuzzy objective function or a fuzzy set

Aj ¼nX;mAj

ðXÞg X˛L; j ¼ 1;2;.:k (37)

Where mAjðXÞ is a membership function of Aj [28].

A final decision is defined by the Bellman and Zadeh model asthe intersection of all fuzzy criteria and constraints and is repre-sented by its membership function. A fuzzy solution D with settingup the fuzzy sets (38) is turned out as a result of the intersectionD ¼ Xk

j¼1Aj with a membership function

mDðXÞ ¼ Xkj¼1mAj

ðXÞ ¼ minj¼1;.:;k

mAjðxÞ X˛L (38)

Using Eq. (35), it is possible to obtain the solution proving themaximum degree as follows,

maxmDðXÞ ¼ maxX˛L

minj¼1;.:;k

mAjðxÞ (39)

X0 ¼ argmaxX˛L

minj¼1;.:;k

mAjðxÞ (40)

To obtain Eq. (39), it is necessary to build membershipfunctions mAjðXÞ, j ¼ 1,., k reflecting a degree of achieving “own”optima by the corresponding FjðXÞ;X˛L, j ¼ 1, . . . , k. This is satisfiedby the use of the membership functions [24]. The membershipfunction of objectives and constraints, linear or non-linear, can bechosen depending on the context of problem. One of possible fuzzyconvolution schemes is presented below [25].

� Initial approximation for X-vector is chosen. Maximum(minimum) values for each criterion Fj (X) are established viascalar maxi mization (minimization). Results are denoted as‘‘ideal’’ points fX0

j ; j ¼ 1;.;mg.� The matrix table {T}, where the diagonal elements are ‘‘ideal’’points, is defined as follows:

26 F1

�X01

�F2�X01

�. Fn

�X01

�� � � � � �

37

fTg ¼

666666664

F1 X02 F2 X0

2 . Fn X02

:::F1�X0n

�F2�X0n

�. Fn

�X0n

�

777777775(41)

� Maximum and minimum bounds for the criteria are defined:

Fmini ¼ min

jFj�X0j

�; i ¼ 1;.:;n

Fmaxi ¼ max

jFj�X0j

�; i ¼ 1;.:;n

(42a,b)

� The membership functions are assumed for all fuzzy goals asfollows. For minimized objective functions

8>>>< 0 if FiðxÞ>Fmaxi ;

Fmaxi � Fi min max

mFiðXÞ ¼ >>>: Fmax

i � Fmini

if Fi < Fi � Fi ;

1 if FiðxÞ � Fmini

(43a)

For maximized objective functions

mFiðXÞ ¼

8>>><>>>:

1 if FiðxÞ>Fmaxi ;

Fi � Fmini

Fmaxi � Fmin

i

if Fmini < Fi � Fmax

i ;

0 if FiðxÞ � Fmini

(43b)

� Fuzzy constraints are formulated:

GjðXÞ � Gmaxj þ dj; j ¼ 1;2;.:; k (44)

Where dj is a subjective parameter that denotes a distance ofadmissible displacement for the bound Gmax

j of the jth constraint.Corresponding membership functions are defined in followingmanner:

mGiðXÞ ¼

8>>><>>>:

0 if GiðxÞ>Gmaxi

1�GjðxÞ�Gmax

j

djif Gmax

i < GiðXÞ �Gmaxi þdj

1 if GiðxÞ � Gmaxi

(45)

� A final decision is determined as the intersection of all fuzzycriteria and constraints represented by its membership func-tions. This problem is reduced to the standard non-linearprogramming problems: to find the such values of X and kthat maximize k subject to

l � mFi ; i ¼ 1;2;.:;nl � m ; j ¼ 1;2;.:; k (46)

Gj

The solution of themulti-criteria problem discloses themeaningof the optimality operator and depends on the decision-makersexperience and problem understanding.

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375370

6.2. LINMAP decision-making method

An ideal point on the Pareto frontier is the point in which eachobjective is optimized regardless to satisfaction of other objectives.It is clear that in the multi-objective optimization it is impossible tohave each objective in its optimal condition obtained a single-objective optimization. Therefore, the ideal point is not locatedon the Pareto frontier. In the LINMAP method, after Euclidian non-dimensionalization of all objectives, the spacial distance of eachsolution on the Pareto frontier from the ideal point denoted by diþis determined as follow,

diþ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

j¼1

�Fij � FIdealj

�22

r(47)

Where n denotes the number of objective while i stand for eachsolution on the Pareto frontier (i ¼ 1,2,..,m). In Eq. (47),FIdealj is theideal value for jth objective obtained in a single-objective optimi-zation. In LINMAP method, the solution with a minimum distancefrom the ideal point is selected as a final desired optimal solution,hence, i index for a final solution, ifinal is,

ifinalhi˛minðdiþÞ i ¼ 1;2;.;m (48)

6.3. TOPSIS decision-making method

In this method beside the ideal point a non-ideal point isdefined. The non-ideal point is the ordinate in the objectives spacein which each objective has its worst value. Therefore, beside thesolution distance from ideal point, diþ, the solution distance fromthe non-ideal point denoted by di� is used as a criterion for selec-tion of the final solution. Hence,

di� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

j¼1

�Fij � FNon�ideal

j

�22

r(49)

In continuing the TOPSIS method a Cli parameter is defined asfollows,

Cli ¼di�

diþ þ di�(50)

In the TOPSIS method a solution with a maximum Cli is selectedas a desired final solution, therefore, if ifinal is index for the finalselected solution, we have,

ifinalhi˛maxðCliÞi ¼ 1;2;.;m (51)

7. Results and discussion

The present simple gas cycle is modeled using the thermody-namic model for the site minimum, average and maximum annual

Table 3Performance of the simple Brayton V93.1 gas cycle at site conditions (Fars gas powerplant located in Shiraz-Iran).

Ambienttemp. (�C)

Ambientpressure (kPa)

Net power(kW)

Exergeticefficiency (%)

Fuel mass flowrate (kg s�1)

Tmin ¼ �3.5 84.7 55683 26.36 3.9865Tave ¼ 18.8 84.7 48160 24.83 3.6610Tmax ¼ 40.6 84.7 40664 22.93 3.3472

temperatures. Table 3 indicates the simple gas cycle performance ataforementioned site conditions.

Now a recuperative heat exchanger is integrated to the V93.1gas cycle in order to convert it into a regenerative gas cycle thatschematically is shown in Fig. 1. In this regards, geometric andthermal specifications of the recuperative heat exchanger and theair outlet temperature from the recuperator are specified ina multi-objective optimization process with objective functionsexpressed By Eqs. (14) and (18) and constraints specified by Eqs.(21)e(32). Multi-objective optimization using NSGA-II algorithmis performed in three site cases including annual minimum,average and maximum ambient air temperatures and Paretofrontiers are obtained in these three cases as illustrated inFig. 3(aec), respectively. In each case, final optimal solutionshave been selected with fuzzy, LINMAP and TOPSIS decision-makers.

It is clear from Fig. 3b that at the average annual site tempera-ture, LINMAP and TOPSIS recommend the same final optimalsolution. Tables 4e6 indicates specifications of the regenerativecycles that recommended by fuzzy, LINMAP and TOPSIS decision-makers.

Now the question is that which decision-makingmethod shouldbe considered for a final selection. Table 7 and Fig. 3(aec) arepresented here to help us to decide between various decision-making methods. In the Table 7, it is assumed that the systemoptimized in each temperature (optimization base temperature) isoperated in other two ambient temperatures and the efficiencyimprovements in those conditions are calculated. From the lastcolumn of Table 7, we can see that all decision-makers lead toapproximately the same values for the average exergetic efficiency(þ1.63%, þ1.57% and 1.61% for fuzzy, LINMAP and TOPSIS decision-makers, respectively).

Fig. 4 shows that, in almost same exergetic efficiency improve-ment of three decision-makers, LINMAP and TOPSIS decision-makers lead to the lowest payback time for the recuperatorinvestment. Therefore, it seems that in this case, LINMAP andTOPSIS provide a more desirable final optimal solution.

It should be mentioning that in general there is no decision-making method having superiority over other methods in allcases. Indeed, various decision-makingmethods are applied to helpdecision-makers who select the final solution based on theirprofessional experience. In this case, we applied three decision-making methods and we found that in our case LINMAP and TOP-SIS decision-makers select a final optimal solution that more suitour engineering and economic criteria.

Another question is which ambient annual temperature shouldbe considered as a reference base temperature for optimization ofthe system. In this regard, once more we assumed that the systemoptimized at each ambient temperature is operated in other two-ambient temperatures. For example we assumed that the systemoptimized at the minimum temperature is utilized in the averageand maximum ambient temperatures. Now, the deviations of thereal system (obtained with �3.5 �C base temperature) from theoptimal solution at 18.8 �C and 40.6 �C are assessed. This procedureis more elucidated as follows,

d18:8+C

þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�p�3:5+Cn � p18:8+C

n2þ�ε�3:5+C

n � ε18:8+Cn

2q(52a)

d40:6+C

þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�p�3:5+Cn � p40:6+C

n2þ�ε�3:5+C

n � ε40:6+Cn

2q(52b)

Fig. 3. Pareto optimal frontiers at (a) Minimum annual temperature; (b) Average annual temperature; (c) Maximum annual temperature.

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375 371

Table 4Specifications of the recuperator and the regenerative gas cycle specified by the fuzzy decision-maker at various ambient temperatures.

Parameters T0 ¼ �3.5 �C T0 ¼ 18.8 �C T0 ¼ 40.6 �C

Tube arrangement Triangle(30e60) Triangle(30e60) Triangle(30e60)Tube outside/inside diameter di/do (m) 0.0173/0.01905 0.0173/0.01905 0.0173/0.01905Tube pitch ratio in the tube bundle 1.33 1.33 1.25Tube length (m) 7.683 7.674 7.688Number of tubes(Nt) 6805 6572 6348Number of baffles(Nb) 5 7 7Shell inside diameter Ds (m) 2.280 2.240 2.070Outer tube limit Dotl (m) 2.008 1.850 1.689Inner tube limit Ditl (m) 1.026 1.008 0.931Outlet temp. of the recuperator (�C) 397 412 427Effectiveness 0.5935 0.5930 0.5925Recuperator cost($) 470040 457240 445180Payback time(years) 2.23 2.73 2.84Exergetic efficiency (%) 28.62 26.64 24.65Improvement in the exergetic efficiency

compared to the simple cycle (%)þ2.26 þ1.81 þ1.72

Table 5Specifications of the recuperator and the regenerative gas cycle specified by the LINMAP decision-maker at various ambient temperatures.

Parameters T0 ¼ �3.5 �C T0 ¼ 18.8 �C T0 ¼ 40.6 �C

Tube arrangement Triangle(30e60) Triangle(30e60) Triangle(30e60)Tube outside/inside diameter di/do (m) 0.0173/0.01905 0.0173/0.01905 0.0173/0.01905Tube pitch ratio in the tube bundle 1.33 1.33 1.33Tube length(m) 7.665 7.659 7.702Number of tubes(Nt) 6118 5975 5642Number of baffles(Nb) 7 7 7Shell inside diameter Ds (m) 2.160 2.140 2.080Outer tube limit Dotl (m) 1.821 1.842 1.689Inner tube limit Ditl (m) 0.972 0.963 0.936Outlet temp. of the recuperator (�C) 395 411 425Effectiveness 0.5880 0.5873 0.5828Recuperator cost($) 431990 423850 405330Payback time(years) 2.13 2.63 2.70Exergetic efficiency (%) 28.52 26.58 24.57Improvement in the exergetic efficiency

compared to the simple cycle (%)þ2.16 þ1.75 þ1.64

Table 6Specifications of the recuperator and the regenerative gas cycle specified by the TOPSIS decision-maker at various ambient temperatures.

Parameters T0 ¼ �3.5 �C T0 ¼ 18.8 �C T0 ¼ 40.6 �C

Tube arrangement Triangle(30e60) Triangle(30e60) Triangle(30e60)Tube outside/inside diameter di/do (m) 0.0173/0.01905 0.0173/0.01905 0.0173/0.01905Tube pitch ratio in the tube bundle 1.33 1.33 1.33Tube length(m) 7.667 7.659 7.724Number of tubes(Nt) 6109 5975 5663Number of baffles(Nb) 7 7 7Shell inside diameter Ds (m) 2.160 2.140 2.080Outer tube limit Dotl (m) 1.909 1.842 1.666Inner tube limit Ditl (m) 0.969 0.963 0.936Outlet temp. of the recuperator (�C) 395 411 425Effectiveness 0.5879 0.5873 0.5834Recuperator cost($) 431520 423850 406850Payback time (years) 2.13 2.63 2.70Exergetic efficiency (%) 28.52 26.58 24.58Improvement in the exergetic efficiency

compared to the simple cycle(%)þ2.16 þ1.75 þ1.65

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375372

Where d18:8+Cþ and d40:6

+Cþ are deviations of the optimal solution

obtained at �3.5 �C from optimal solutions obtained at 18.8 �C and40.6 �C, respectively. pn and εn denote Euclidian non-dimensionalized payback and exergetic objective values. There-fore, the total deviation at �3.5 �C denoted by d�3:5+C

þ is,

d�3:5+Cþ ¼ d40:6

+Cþ þ d18:8

+Cþ (52c)

Similarly, for systems designed at optimization base tempera-tures of 18.8 �C and 40.6 �C, deviations d18:8

+Cþ and d40:6+C

þ are ob-tained as,

d40:6+C

þ ¼ �p18:8+Cn � p40:6+C

n2þ�ε18:8+C

n � ε40:6+Cn

2 (53a)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq

d�3:5+Cþ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�p18:8+Cn � p�3:5+C

n2þ�ε18:8+C

n � ε�3:5+Cn

2q(53b)

d18:8+C

þ ¼ d40:6+C

þ þ d18:8+C

þ (53c)

d18:8+C

þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�p40:6+Cn � p18:8+C

n2þ�ε40:6+C

n � ε18:8+Cn

2q(54a)

Table 7Exergetic efficiency improvement obtained by different decision-making methods at various ambient temperatures.

Decision-makingmethod

Optimization basetemperature (�C)

Ambienttemp. (�C)

Exergeticefficiency (%)

Exergetic efficiencyimprovement

Sum of efficiencyimprovementin other two cases (%)

Average efficiencyimprovement (%)

Fuzzy 18.8 40.6 24.66 þ1.73 þ5.49 þ1.63�3.5 28.31 þ1.95

40.6 18.8 26.23 þ1.40 þ4.56�3.5 27.80 þ1.44

�3.5 18.8 26.58 þ1.75 þ4.7440.6 23.66 þ0.73

TOPSIS 18.8 40.6 24.61 þ1.68 þ5.23 þ1.57�3.5 28.16 þ1.80

40.6 18.8 26.15 þ1.32 þ4.29�3.5 27.68 þ1.32

�3.5 18.8 26.58 þ1.75 þ4.6040.6 23.62 þ0.69

LINMAP 18.8 40.6 24.61 þ1.68 þ5.23 þ1.61�3.5 28.16 þ1.80

40.6 18.8 26.32 þ1.49 þ4.13�3.5 27.36 þ1.00

�3.5 18.8 26.12 þ1.29 þ5.1440.6 24.62 þ1.69

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375 373

d�3:5+Cþ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�p40:6+Cn � p�3:5+C

n2þ�ε40:6+C

n � ε�3:5+Cn

2q(54b)

d40:6+C

þ ¼ d�3:5+Cþ þ d18:8

+Cþ (54c)

A solution with a minimum deviation from optimal solution ofother two ambient temperatures is desired. In this regard, ourcalculation indicates,

d�3:5+Cþ ¼ 0:001936;d40:6

+Cþ ¼ 0:00247 and d18:8

+Cþ

¼ 0:001583 (55)

Therefore, as is clear, optimization at the average annualambient temperature leads to less deviation compared to othertemperatures. Therefore, we conclude that optimization at 18.8 �Cwith the TOPSIS decision-maker leads to more reasonable results.This fact can be found also by looking at results indicated on thesixth column Table 7. This column indicates that a more efficiencyimprovement is obtained at the average ambient temperature by all

Fig. 4. Pareto optimal frontiers at minimum, average and maximum a

decision-making methods. On the other hand, as we can see fromFigs. 3aec and 4, the TOPSIS and LINMAP recommend same solu-tion. As a summary, the final optimal solution selected by LINMAPand TOPSIS decision-makers at 18.8 �C is selected our desiredsolution for the regenerative gas cycle. It can be concluded that insimilar works on optimization of the gas cycle (which operate invariable ambient temperature), optimization might be performedat the average annual ambient temperature. Table 8 indicatesspecifications of the selected final optimal regenerative gas cycle.Further Figs. 5(a and b) compare the fuel consumption and exer-getic efficiency of the simple V93.1 gas cycle with improvedregenerative gas cycle in a more elucidating manner.

As is found from Table 8 and Figs. 5a, b the exergetic efficiencyand fuel consumption of the modified cycle are improved 1.75% and6.6% compared to the present Brayton V93.1 cycle. This improve-ment requires an investment of approximately 423,850 US $ whichits cost will be paid back within 2.63 years with the currentdomestic cost of the natural gas fuel in Iran. It is required tomention that since optimization was performed based on the local

nnual temperatures indicating various decision-making methods.

Table 8Specifications of the final optimal regenerative gas cycle obtained at the averageambient temperature of 18.8 �C and TOPSIS (and LINMAP) decision-makingmethods.

Parameters Value

Tube arrangement Triangle(30�)Tube inside/outside diameter (mm) 17.3/19.05Tube length (m) 7.659Number of tubes 5975Number of baffles 7Shell inside diameter (m) 2.14Tube pitch to tube outside diameter 1.33Outer tube limit Dotl (m) 1.842Inner tube limit Ditl (m) 0.963Shell side heat transfer coefficient(hs) (W m�2 K�1) 343.7Tube side heat transfer coefficient(ht) (W m�2 K�1) 129.38Total heat transfer coefficient Uo (W m�2 K�1) 168.3DTmð+CÞ 72.65Tube side Pressure drop(DPt)(kPa) 5.02Shell side Pressure drop(DPs)((kPa) 2.39Approximate pre-heater cost($) 423850Payback(year) 2.63Exergy efficiency improvement (%) þ1.75Fuel consumption reduction (%) þ6.6

Fig. 5. Comparison of the present Brayton V93.1 gas cycle with the modified cycle in(a) fuel consumption; (b) Exergetic efficiency.

H. Sayyaadi, R. Mehrabipour / Energy 38 (2012) 362e375374

natural gas price in Iran which is very cheap in comparison to theinternational market, the payback time for the optimized cycle isrelatively high. If optimization is performed with the internationalfuel pricewhich is 3.75 times higher than its corresponding value inIran, the payback time will be reduced to 17 months which isreasonable value for the payback time.

8. Conclusions

A simple Brayton gas cycle namely as the Kraftwerk V93.1Siemens gas turbine was improved by converting it into a regener-ative gas cycle having an optimized vertical tubular recuperativeheat exchanger. It was shown that optimization process can beemployed even for existing cycles which are not in design stage.Therefore the existing gas cycle was improved using an

optimization approach. Specifications of the recuperator and itsrole in the gas cycle were found in a multi-objective optimizationprocess using the NSGA-II algorithm. Optimization objective werethe exergetic efficiency of the gas cycle and the payback time for therecuperative investment. Multi-objective optimization of a gascycle as a benchmark for any energy system has been performedwith three famous decision-making methods. In this regard, thefuzzy Bellman-Zadeh, LINMAP and TOPSIS decision-makingmethods were employed for selection of a final optimal solutionfrom the Pareto frontier. Further, multi-objective optimization wasperformed at three ambient temperature cases includingminimum, average and maximum ambient air temperatures.However there is no decision-making method having superiorityover other methods, It was found that in our case optimization basetemperature equal to the average ambient air temperature and theTOPSIS decision-making leads to best results. Further it was foundthat at the average ambient air temperature both the TOPSIS andLINMAP decision-makers suggest the same final solution for theregenerative gas cycle. It was shown that optimization at theaverage ambient temperature leads to better results than any othersystems that are optimized at other ambient reference tempera-tures. It was discussed that in a very low price for the natural gas inIran the payback time for the optimized cycle is relatively highwhereas if the optimization performed at international for thenatural gas price, the payback time of the optimized cycle will be ina reasonable range.

Acknowledgment

This research work has been completed by the financial supportof Fars Regional Electric Company.

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