Multi-objective optimization of a cooling tower assisted vapor compression refrigeration system

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Multi-objective optimization of a cooling tower assisted vapor compression refrigeration system Hoseyn Sayyaadi*, Mostafa Nejatolahi Faculty 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

article info

abstract

Article history:

A cooling tower assisted vapor compression refrigeration machine has been considered for

Received 23 December 2009

optimization with multiple criteria. Two objective functions including the total exergy

Received in revised form

destruction of the system (as a thermodynamic criterion) and the total product cost of the

24 July 2010

system (as an economic criterion), have been considered simultaneously. A thermody-

Accepted 30 July 2010

namic model based on energy and exergy analyses and an economic model according to

Available online 6 August 2010

the Total Revenue Requirement (TRR) method have been developed. Three optimized systems including a single-objective thermodynamic optimized, a single-objective

Keywords:

economic optimized and a multi-objective optimized are obtained. In the case of multi-

Refrigeration system

objective optimization, an example of decision-making process for selection of the final

Cooling tower

solution from the Pareto frontier has been presented. The exergetic and economic results

Exergy

obtained for three optimized systems have been compared and discussed. The results have

Economy

shown that the multi-objective design more acceptably satisfies generalized engineering

Optimization

criteria than other two single-objective optimized designs. ª 2010 Elsevier Ltd and IIR. All rights reserved.

Optimisation d’un syste`me frigorifique a` compression de vapeur dote´ d’une tour de refroidissement mene´e avec plusieurs objectifs Mots cle´s : Syste`me frigorifique ; Tour de refroidissement ; Exergie ; E´conomie ; Optimisation

1.

Introduction

In selection, design and optimization of energy systems, several and commonly conflicting criteria might be considered. For example, for optimization of a vapor compression refrigeration system, a designer may consider one or more of the thermodynamic, economic and environmental criteria as

the objective function. If only the thermodynamic criterion is considered, the system will be an ideal system from thermodynamic point of view, but it might not be able to pass the economic criterion. On the other hand, by considering only the economic criterion, the system will be the cheapest one, but this system might not be a well designed system from thermodynamic and environmental points of view e say

* Corresponding author. Tel.: þ98 21 8867 4841-8x2212; fax.: þ98 21 8867 4748. E-mail addresses: [email protected] (H. Sayyaadi), [email protected] (M. Nejatolahi). 0140-7007/$ e see front matter ª 2010 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2010.07.026

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DHct

Nomenclature A a1ea6 BBY BD BL C CC CELF COP CRF Celect CP C_ P  C_ P

Cw E_ EV FC f g h H0 I_  I_ ieff j K k LD _ m n OMC P PEC Q_ R ROI rFC rW rOMC s T TCR TNI TRR U0 V_ _ W WC x x* Z_

Heat transfer area (m ), Approach ( C) Constants in computing the purchased equipment cost (PEC) of the cooling tower Balance at the beginning of the year Book depreciation Book life Constant coefficient Carrying charge The constant escalation levelization factor Coefficient of performance Capital-recovery factor Electricity price ($ kW1 h1) Heat Capacity (J kg1 K1) Total product cost ($ h1) Normalized form of C_ P 2

The cost of cooling water ($ m3) Rate of exergy (kW) Expansion valve Fuel cost ($ h1) Objective function Gravity acceleration (m s2) Enthalpy (kJ kg1) Height relative to the ground surface (m) Irreversibility rate (kW) Normalized form of I_ Interest rate (%) jth year of the system operation Number of objective functions Number of operating year; Component kth Tube length to shell diameter ratio Mass flow rate (kg s1) Number of years; Number of decision variables Operating and maintenance cost ($) Pressure (kPa) Purchase equipment cost ($) Heat transfer rate (kW) Temperature range in the cooling tower ( C) Return on investment Annual escalation rate for the cost of the electricity Annual escalation rate for the cost of the cooling water Annual escalation rate for the operating and maintenance cost Specific entropy (kJ kg1) Temperature ( C or K) Total capital-recovery Total net investment Total revenue requirement Velocity relative to the ground surface (m s1) Volumetric flow rate (m3 s1) Power (kW) Water cost ($ h1) A decision variable vector An optimum decision variable vector Cost rate associated with the capital investment and OMC ($ h1)

Height difference between the cooling tower inlet and outlet water conduits DP Pressure deference DT Temperature deference I, II States I, II on the vapor compression refrigeration system 1, 2, ., 11 States 1, 2, ., 11 on the vapor compression refrigeration system Greek Letters r Density (kg m3) h Efficiency (%) s Number of operating hours in a year (h) 3 Specific exergy (kJ kg1) Subscripts act Actual comp Compressor cond Condenser ct Cooling tower desired Desired elect Electrical evap Evaporator fan Fan i Inlet io Inlet/outlet difference isen Isentropic elect Electrical L Levelized Loss Loss net Net mech Mechanical OMC Operating and maintenance cost o Outlet pump Pump ref Refrigerant R Rational s Isentropic sub Subcooled sup Superheated tot Total used Used w Water wb Wet bulb 0 Dead state; Related to the first year of the system operation Superscripts CI Capital investment Ch Chemical i Inlet K Kinetic OMC Operating and maintenance cost o Outlet P Potential Ph Physical Q Heat W Work

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a system that consumes lots of energy or emits a lot of pollutants into the environment. Both of these systems are not acceptable from a comprehensive engineering point of view. Thus it seems that simultaneous consideration of all or some of these criteria might provide better option for engineers. This goal can be obtained by multi-objective optimization techniques. In this way, we will have a system that satisfies all of the optimization criteria as much as possible simultaneously. Thermodynamic criteria are usually the first law (energetic) and the second law (exergetic) criteria. In this paper the second law criterion (the total exergy destruction of the system) is considered as the thermodynamic objective function which has been proven that it better considers thermodynamic criterion than the first law optimization. The economic objective function is the total product cost of the system that is developed according to the total revenue requirement (TRR) method. These two criteria are considered in a multi-objective optimization of a cooling tower assisted vapor compression refrigeration system as an example of energy systems. As a powerful thermodynamic tool, the exergy analysis (availability or second law analysis) presented in this study is well suited for furthering the goal of more effective energy resource use, for it enables the location, cause, and true magnitude and waste and loss of exergy to be determined. Such information can be used in the design of new energyefficient systems and for increasing the efficiency of the existing system (Bejan et al., 1996). There have been several studies on the exergy analysis of different types of refrigeration and heat pump systems. Leidenfrost et al. (1980) used exergy analysis to investigate performance of a refrigeration cycle working with R-12 as the refrigerant. Dincer et al. (1996) investigated the thermal performance of a solar powered absorption refrigeration system. Meunier et al. (1997) studied the performance of adsorptive refrigeration cycles using the second law analysis. Nikolaidis and Probert (1998) utilized the exergy method in order to simulate the behavior of a twostage compound compression-cycle with flash inter-cooling running with R-22 as the refrigerant. The effects of temperature changes in the condenser and evaporator on the irreversibility rate of the cycle were determined. Bouronis et al. (2000) studied the thermodynamic performance of a singlestage absorption/compression heat pump using the ternary working fluid trifluoroethanolewateretetraethyleneglycol dimethylether for upgrading waste heat. Go¨ktun and Yavuz (2000) investigated the effects of thermal resistances and internal irreversibilities on the performance of combined cycles for cryogenic refrigeration. Chen et al. (2001) studied the optimization of a multistage endoreversible combined refrigeration system. Kanoglu (2002) presented a methodology for the exergy analysis of multistage cascade refrigeration cycle and obtained the minimum work relation for the liquefaction of natural gas. Yumrutas‚ et al. (2002) presented a computational model based on the exergy for the investigation of the effects of the evaporating and condensing temperatures on the pressure losses, exergy losses, second law efficiency, and the coefficient of performance (COP) of a vapor compression refrigeration cycle. Kanoglu et al. (2004) developed a procedure for the energy and exergy analyses of open-cycle desiccant cooling systems and applied it to an

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experimental unit operating in ventilation mode with natural zeolite as the desiccant. Kopac and Zemher (2006) presented a computational study based on the exergy analysis for the investigation of the effects of the saturated temperatures of the condenser and the evaporator on the efficiency defects in each of the components of the plant, the total efficiency defect of the plant, the second law efficiencies and the values of COP of a vapor compression refrigeration plant for NH3, HFC-134a, R-12 and R-22. Ozgener and Hepbasli (2007) reviewed the energy and exergy analyses of solar assisted heat pump systems that many of them were in the category of solar assisted ground source heat pump systems. On the other side, there have been several studies on the economic or thermoeconomic analysis of refrigeration and heat pump systems. Wall (1991) presented a pioneering work in application of thermoeconomic optimization of heat pump systems. In that study, the objective function was the total life cycle cost including the electricity and the capital costs. The Lagrange multipliers method was utilized for minimization of the objective function. Cammarata et al. (1997) presented an economic method for optimizing a thermodynamic cycle of an air conditioning system. Global optimum values were found using the direct mathematical method. d’Accadia and de Rossi (1998) investigated thermoeconomic optimization of a vapor compression refrigerator using the exergetic cost theory method. Dingec and Ileri (1999) carried out the optimization of a domestic R-12 refrigerator. The structural coefficient method was used in this optimization procedure. Their objective was to minimize the total life cycle cost, which includes both the electricity and capital costs, for a given cooling demand and system life. Tyagi et al. (2004) investigated the thermoeconomic optimization of an irreversible Stirling cryogenic refrigerator cycle. Al-Otaibi et al. (2004) used thermoeconomics to study a vapor compression refrigeration system. The efficiencies of the compressor, condenser, evaporator, and electric motor were also studied as the decision variables with cost parameters. Sanaye and Malekmohammadi (2004) presented a new method of thermal and economical optimum design of air conditioning units with vapor compression refrigeration system. So¨ylemez (2004) presented a thermoeconomic optimization analysis yielding a simple algebraic formula for estimating the optimum operating temperature for refrigeration systems, which utilizes energy recovery applications. In their work the method used is known as the P1P2 method, used with the usage factor and simplified wall gain load factors. Selbas‚ et al., 2006 applied an exergy-based thermoeconomic optimization application to a subcooled and superheated vapor compression refrigeration system. All calculations were made for three refrigerants: R-22, R-134a, and R-407c. Misra et al. (2003, 2005) investigated thermoeconomic optimization of single and double effect H2O/LiBr vapor-absorption refrigeration systems. Sanaye and Niroomand (2009) investigated the thermal-economic modeling and optimization of a vertical ground source heat pump. According to the above-mentioned paragraphs, there are comprehensive investigations in the field of exergy and economic analyses and optimization of refrigeration and heat pump systems, especially on vapor compression refrigeration systems. But as mentioned previously, by considering only

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one of the exergetic or economic criteria as the objective function of optimization, the systems would not satisfactory pass the other criteria. Thus it seems that a multi-objective optimization is required. Multi-objective optimization is developed to deal with different and often competing objectives is an optimization problem (e.g. see Fonseca and Fleming, 1997; Van Veldhuizen and Lamont, 2000; Deb, 2001 and Konak et al., 2006). Multi-objective optimization of energy systems has been paid attention by researchers nowadays (e.g. Toffolo and Lazzaretto, 2002, 2004). Moreover, Sayyaadi et al. (2009) and Sayyaadi and Amlashi (in press) performed multi-objective optimization for GSHP systems in cooling mode. Objective functions were the total product cost of the system and the total exergy destruction. They compared the results of exergy and thermoeconomic analyses of the base case, two single-objective optimized, and the multi-objective optimized systems. This work has been presented here as an attempt for multiobjective optimization of a cooling tower assisted vapor compression refrigeration system. Objectives are the total exergy destruction and the total product cost of the system. Product in the refrigeration system is defined as refrigeration effect of the evaporator, hence in our study, the cost of the system product is defined as the unit cost of refrigeration effect in the evaporator. Three optimization scenarios including the thermodynamic single-objective, the thermoeconomic single-objective and multi-objective optimizations are performed in this work. All optimization scenarios are conducted using an artificial intelligence technique known as evolutionary algorithm (EA). The output of the multi-objective optimization is a Pareto frontier that yields a set of optimal points. In the case of multi-objective optimization scenario, an example of decision-making process for selection of the final solution from the Pareto frontier is presented here. The

exergetic and economic results obtained for systems obtained in the three optimization scenarios are compared and discussed.

2.

System specification

A cooling tower assisted vapor compression refrigeration system with the cooling load of 352 kW (100 Ref. Ton) is considered as illustrated in Fig. 1. This configuration is similar with those systems utilized as chillers in HVAC systems. A scroll compressor is used to drive system with R-134a as a refrigerant. The condensers and evaporator are shell and tube heat exchanger in which refrigerant is placed in the shell side and water flows in the tube side. The water inlet and outlet temperatures in the evaporator are 12  C and 7  C respectively. The outdoor dry bulb and wet bulb temperatures at the site are 40  C and 24  C respectively. The ratio of the baffle spacing to the shell diameter, and the baffle cut are chosen as 1 and 0.45, respectively for all heat exchangers. The shell and tube heat exchangers (the evaporator and condenser) and cooling tower are designed based on the procedure given by Coulson and Richardson (1996) and Ludwing (1993) respectively.

3.

Energy modeling

3.1.

Governing equations

The following assumptions are considered for energy and exergy analyses: i. All processes are steady state and steady flow with negligible potential and kinetic energy effects. ii. The directions of heat transfer to the system and work done on the system are positive. iii. Heat transfer and refrigerant pressure drops in the pipeline are ignored. Under the aforementioned assumptions for a general steady state, steady flow process, the mass balance, energy balance (first law of thermodynamic) and the exergy balance equations are applied to find the rate of irreversibility.

3.2.

Energy balance

The general energy balance for a control volume in steady state conditions can be expressed as follows: X X _ net ¼ _ o ho  _ i hi Q_ net þ W m m (1) _ ref Þ is calcuTherefore, mass flow rate of the refrigerant ðm lated by applying the energy balance for evaporator: _ ref ¼ m

Fig. 1 e Schematic arrangement of the cooling tower assisted vapor compression refrigeration system.

Q_ evap h1  h4

(2)

Similarly by applying energy balance for the evaporator and condenser, the brine and cooling water mass flow rates are calculated as follows: Q_ _w¼ m (3) CP ðTi  To Þ

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The type of compressor considered in this study is Copeland scroll compressor. The compressor power is given according to the following equation:

Where 3 is the specific exergy of a steam of matter that includes kinetic (3K), potential (3P), physical (3Ph) and chemical (3Ch) exergies:

_ _ act;comp ¼ mref ðh2;s  h1 Þ W hisen;comp

3 ¼ 3K þ 3P þ 3Ch þ 3Ph

(13)

3K ¼ U20 =2

(14)

3 ¼ gH0

(15)

(4)

Where hisen is the compressor isentropic efficiency. By applying the energy balance for the entire cycle, heat load of the condenser is calculated as: _ comp Q_ cond ¼ Q_ evap þ W

(5)

The pumping power requirements for each pump in the system as per Fig. 1 are: _ pump;I ¼ V_ w ðrw gDHct þ DPcond Þ=hpump W

(6)

_ pump;II ¼ V_ w DPevap =hpump W

(7)

Where DHct is equal to the difference between the height of cooling tower water inlet and outlet connections of the cooling tower and V_ is volumetric flow rate (m3/s). The total consumed electrical power of the system is as follows: _ comp þ W _ pump;I þ W _ pump;II þ W _ fan _ tot ¼ W W

(8)

_ fan is the consumed power by the cooling tower fan Where W that is calculated according to procedure given by Ludwing (1993).

4.

Exergy analysis

An exergy analysis provides, among others, the exergy of each stream in a system as well as the real “energy waste” i.e., the thermodynamic inefficiencies (exergy destruction and exergy loss), and the exergetic efficiency for each system component (Bejan et al., 1996) Thermodynamic processes are governed by the laws of conservation of the mass and energy. However, exergy is not generally conserved but is destroyed by irreversibilities within a system. Furthermore, exergy is lost, in general, when the energy associated with a material or energy stream is rejected to the environment. The general form of the exergy balance for a control volume in steady state conditions is: I_ ¼ E_  E_ þ E_ þ E_ i

o

Q

W

(9)

P

The kinetic and potential exergies are ignored in this work. Further, since most material steams of the system are not associated with any kind of chemical reaction, therefore, the chemical exergy terms will be canceled out in the balance equation. Thus, the exergy of flow in this work (Eq. (13)) (except in the cooling tower that water changes phase and the chemical exergy terms are not canceled out) are comprised only from the physical component. The physical specific exergy is given by: 3Ph ¼ ðh  T0 sÞ  ðh0  T0 s0 Þ

(16)

Where the subscript 0 is referred to the environmental conditions (restricted equilibrium with the environmental). The specific chemical exergy for liquid and vapor forms of the water are equal to 2.4979 and 0 kJ kg1, respectively (Bejan et al., 1996). By applying the exergy balance for a control volume that encloses the entire system, the total exergy destruction is evaluated as follow:   _ pump;II þ W _ fan þ W _ comp þ E_ 8 _ pump;I þ W I_tot ¼ E_ 9  E_ 11 þ W

(17)

Different ways of formulating exergetic efficiency proposed in the literature have been given in detail elsewhere (Bejan et al., 1996). Among these, the rational efficiency or the overall rational efficiency is defined by as the ratio of the desired exergy output to the exergy used, namely hR ¼

E_ desired E_ used

(18)

Where E_ desired is the sum of all exergy transfer rates from the system, which must be regarded as constituting the desired output, plus any by product that is produced by the system, while E_ used is the required rate of input exergy for the process to be performed. For the vapor compression refrigeration system, these two terms are determined as follows: _ pump;I þ W _ pump;II þ W _ fan þ W _ comp þ E_ 8 E_ used ¼ W

(19)

  E_ desired ¼  E_ 9  E_ 11

(20)

Where I_ is the total exergy destruction or irreversibility. The E_ is the exergy flow associated with the heat transfer through the control volume boundaries and is calculated as follow: Q

Q _  T0 =TÞ E_ ¼ Qð1

(10)

Because work is an ordered energy, its associated exergy flow is equal to the amount of that work. Thus W _ E_ ¼ W

(11)

i o The E_ and E_ are the exergies of the control volume inlet and outlet streams of matter and are given by:

_ E_ ¼ m3

(12)

Table 1 e Exergy balance equations for each component of the cooling tower assisted vapor compassion refrigeration system (Fig. 1). _ pump;I I_pump;I ¼ ðE_ 6  E_ 7 Þ þ W _ pump;II I_pump;II ¼ ðE_ 9  E_ 10 Þ þ W I_cond ¼ ðE_ 7  E_ 5 Þ þ ðE_ 2  E_ 3 Þ I_evap ¼ ðE_ 10  E_ 11 Þ þ ðE_ 4  E_ 1 Þ I_EV ¼ ðE_ 3  E_ 4 Þ _ fan þ E_ 8 I_ct ¼ ðE_ 5  E_ 6 Þ þ W _ comp I_comp ¼ ðE_ 1  E_ 2 Þ þ W _ pump;I þ W _ pump;II þ W _ fan þ W _ comp þ E_ 8 I_tot ¼ ðE_ 9  E_ 11 Þ þ W

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From Eqs. (18)e(20), the rational efficiency is obtained as follows: hR ¼

ðE_ 9  E_ 11 Þ _ pump;II þ W _ fan þ W _ comp þ E_ 8 _ pump;I þ W W

(21)

Application of the exergy balance equation (Eq. (9)) for each component of the vapor compression refrigeration system (Fig. 1) leads to the balancing equations mentioned in Table 1.

rate (cost of money), and n denotes the system economic life expressed in years. In the case of the cooling tower assisted vapor compression refrigeration system, the annual total revenue requirement is equal to the sum of the following five annual amounts including the total capital-recovery (TCR); minimum return on investment (ROI ); fuel costs (FC ); water costs (WC ); and the operating and maintenance cost (OMC ): TRRj ¼ TCRj þ ROIj þ FCj þ WCj þ OMCj

5.

Economic models

The economic model takes into account the cost of the components, including amortization and maintenance, the cost of fuel consumption and the cost of water consumption. In order to define a cost function, which depends on the optimization parameters of interest, component costs have to be expressed as functions of thermodynamic variables. These relationships can be obtained by statistical correlations between costs and the main thermodynamic parameters of the component performed on the real data series. The following sections illustrate the total revenue requirement method (TRR method ) which is based on procedures adopted by the Electric Power Research Institute in their technical assessment guide (TAG, 1993). Based on the estimated total capital investment and assumptions for economic, financial, operating, and market input parameters, the total revenue requirement is calculated on a year-by-year basis. Finally, the non-uniform annual monetary values associated with the investment, operating (excluding fuel and water), maintenance, water and the fuel costs of the system being analyzed are levelized; that is, they are converted to an equivalent series of constant payments (annuities) (Bejan et al., 1996). The annual total revenue requirement (TRR, total product cost) for a system is the revenue that must be collected in a given year through the sale of all products to compensate the system operating company for all expenditures incurred in the same year and to ensure sound economic system operation (Bejan et al., 1996). The series of annual costs associated with the carrying charges CCj and expenses (FCj and OMCj) for the jth year of a system operation is not uniform. In general, carrying charges decrease while fuel costs increase with increasing years of operation (Bejan et al., 1996). A levelized value for the total annual revenue requirement, TRRL, can be computed by applying a discounting factor and the capital-recovery factor CRF: TRRL ¼ CRF

n X 1



TRRj 1 þ ieff

j

(22)

In applying Eq. (22), it is assumed that each monetary transaction occurs at the end of each year. The capitalrecovery factor CRF is given by:  n ieff 1 þ ieff CRF ¼  n 1 þ ieff 1

(23)

TRRj is the total revenue requirement in the jth year of system operation, ieff is the average annual effective discount

(24)

The calculation method for TCRj and ROIj is given by Bejan et al. (1996), the extension of TCRj and ROIj for a cooling system is developed by Sayyaadi et al. (2009) and Sayyaadi and Amlashi (in press). FCj, WCj and OMCj and their corresponding levelized values are obtained using the following procedure. If the series of payments for the annual fuel cost FCj is uniform over the time except for a constant escalation rFC (i.e., FCj ¼ FC0 (1 þ rFC)j), then the levelized value FCL of the series can be calculated by multiplying the fuel expenditure FC0 at the beginning of the first year by the constant escalation levelization factor CELF:   kFC 1  knFC FCL ¼ FC0 CELF ¼ FC0 CRF (25) ð1  kFC Þ Where, kFC ¼

1 þ rFC and rFC ¼ constant 1 þ ieff

(26)

The terms rFC and CRF denote the annual escalation rate for the fuel cost and the capital-recovery factor (Eq. (23)), respectively. The levelized value WCL is calculated the same as FCL. Accordingly, the levelized annual operating and maintenance costs (OMCL) are given as follows:   kOMC 1  knOMC (27) OMCL ¼ OMC0 CELF ¼ OMC0 ð1  kOMC Þ With kOMC ¼

1 þ rOMC and rOMC ¼ constant 1 þ ieff

(28)

The term rOMC is the nominal escalation rate for the operating and maintenance costs. Finally, the levelized carrying charges CCL are obtained from the following equation: CCL ¼ TRRL  FCL  WCL  OMCL

(29)

The annual carrying charges or capital investment (superscript CI ) and operating and maintenance costs (superscript OMC ) of the total system can be apportioned among the system components according to the contribution of the kth component to the purchased equipment cost P for the overall system ðPECtotal ¼ k PECk Þ: CCL PECk CI P Z_ k ¼ s k PECk

(30)

OMCL PECk OMC P Z_ k ¼ s PECk

(31)

k

Here, PECk and s denote the purchased equipment cost of the kth system component and the total annual time (in hours) of

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system operation at full load, respectively. PECk equations for various components of the cooling tower assisted vapor compression refrigeration system are given in Appendix A. The term Z_ k represents the cost rate associated with the capital investment and operating and maintenance expenses: Z_ k ¼

CI Z_ k

þ

OMC Z_ k

(32)

The annual fuel and cooling water costs for the first year of the system operation are given as follows respectively: _ tot FC0 ¼ Celect $s$W _ w;loss  WC0 ¼ Cw $s$m

(33) 3600 1000

(34)

In which, Celect is the electricity price per kWh, Cw is the water _ tot is price per m3, s is annual operating hours in cooling mode, W _ w;loss is the total power consumption of the system (Eq. (8)), and m the mass flow rate of make up water of the cooling tower (lit s1). The electricity and water prices are local prices in Iran that considered as 0.075 $ kW1 h1 and 0.0368 $ m3 respectively. The operating life of the system is assumed as 15 years. The total annual operating time of the system in cooling mode is considered as 1800 h. In this study, the magnitude of other economic constant such as rFC, rWC, ieff and rOMC are assumed 0.06, 0.06, 0.12 and 0.05, respectively (Bejan et al., 1996). The levelized cost rates of the expenditures for electricity and cooling water supplied to the system are respectively given as follows: FCL Z_ elect ¼ s

(35)

WCL Z_ w ¼ s

(36)

CI OMC Levelized costs, such as Z_ k ; Z_ k ; Z_ elect and Z_ w are used as input data for the economic analysis.

6. Objective function, decision variables and constraints Optimization problem usually involves with these elements: objective functions, decision variables and constraints. Following sections describe the element of optimization problem for the proposed cooling tower assisted vapor compression refrigeration system.

6.1.

Thermodynamic : I_tot ¼

X

I_k

economic : C_ P ¼ Z_ elect þ Z_ w þ

6.2.

X

249

(37) Z_ k

(38)

Decision variables

The following eight decision variables are chosen in this work: Tcond: the condenser saturation temperature Tevap: the evaporator saturation temperature Tw,i,ct: the water inlet temperature of the cooling tower Tw,o,ct: the water outlet temperature of the cooling tower LDcond: ratio of the tube length to the shell diameter for the condenser 6. LDevap: ratio of the tube length to the shell diameter of the evaporator 7. DTsub: the magnitude of sub-cooling in the condenser 8. DTsup: the magnitude of super-heating in the evaporator

1. 2. 3. 4. 5.

Fig. 2 shows a schematic for temperature profiles in the evaporator, and the condenser and cooling tower. This figure can be considered as a guideline to recognize some of abovementioned decision variables and their constraints.

6.3.

Constraints

In engineering application of the optimization problem, there are usually constraints on the trading-off of decision variables. In this case, some limitations are emanating from the technical view points. For example, the allowable water velocity in the tube sides of a shell and tube heat exchanger should be within the range of 1e3 m/s to prevent fouling and erosion, respectively. The recommended good practice value for LD (ratio of the tube length to the shell diameter) for the evaporator and condenser is a number between 5 and 15. The recommended values of DTsub and DTsup are something between 1  C and 10  C. Due to having several tube passes in the evaporator and condenser, in order to prevent the temperature cross problem in these exchangers, the maximum temperature of the cold stream is always lower than the minimum temperature of the hot stream. Using Fig. 2, the limitations on the maximum and minimum ranges of decision variables 1e4 can be obtained as follows: Tcond;max ¼ 65 C

(39)

Tcond;min ¼ Twb þ DTwb;min þ DTcond;min þ DTsup þ DTw;io;min

(40)

Tevap;max ¼ Tw;i  DTsup  DTevap;min

(41)

Tevap;min ¼ 5 C

(42)

 Tcond  DTcond;min  DTsub Tw;i;ct;max ¼ min Tct;max

(43)

Tw;o;ct;max ¼ Tw;i;ct;max  DTw;io;min

(44)

Tw;o;ct;min ¼ Twb þ DTwb;min

(45)

Tw;i;ct;min ¼ Tw;o;ct þ DTw;io;min

(46)

Objective functions

Objective functions for single-objective and multi-objective optimizations in this study are the thermodynamic and economic objective functions denoted by Eqs. (37) and (38), respectively. In the single-objective thermodynamic optimization, the aim is minimizing the total irreversibility of the cooling tower assisted vapor compression refrigeration system. In the single-objective economic optimization, the total product cost of the vapor compression refrigeration system is minimized. The product in this system is defines as a cooling load (refrigeration effect) that should be provided in the evaporator. The economic objective is denoted by Eq. (38).

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a

b

Fig. 2 e Schematic of temperature profile (a) in the evaporator; (b) in the condenser and the cooling tower. In which, DTwb,min is the minimum temperature difference between the cooling tower water outlet temperature and the ambient air wet bulb temperature. DTevap,min and DTcond,min are the minimum temperature differences between the maximum temperature of the cold stream and the minimum temperature of the hot stream in the evaporator and condenser, respectively. Tw,io,min is the minimum water inlet/ outlet temperatures difference in the cooling tower. Tw,i,ct,max is the maximum cooling tower inlet temperature. In this study, values of DTwb,min, DTevap,min, DTcond,min, Tw,io,min, and Tw,i,ct,max are taken as 2.0, 0.5, 2.0, 2.0, and 70  C, respectively.

7. Multi-objective optimization via evolutionary algorithms 7.1.

General concepts of multi-objective optimization

Consider a decision-maker who wishes to optimize K objectives such that the objectives are non-commensurable and the decision-maker has no clear preference of the objectives relative to each other. Without loss of generality, all objectives are of the minimization type. A minimization type objective can be converted to a maximization type by multiplying negative one. A minimization multi-objective decision problem with K objectives is defined as follows: Given an nondimensional decision variable vector x ¼ {x1,.,xn} in the solution space X, find a vector x* that minimizes a given set of K objective functions f(x*) ¼ {f1(x*),.,fK(x*)}. The solution space X is generally restricted by a series of constraints, such as gj(x*) ¼ bj for j ¼ 1,.,m, and bounds on the decision variables (Konak et al., 2006).In general, no solution vector X exists that

minimizes all the K objective functions simultaneously. In other word, in many real-life problems, objectives under consideration conflict each other. Hence optimizing X with respect to a single-objective often results in unacceptable results with respect to the other objectives. Thus a new concept, known as the “Pareto optimum solution”, is used in multi-objective optimization problems. A feasible solution X is called “Pareto optimal” if there exists no other feasible solution Y that dominates solution X. By definition, a feasible solution Y is said to dominate another feasible solution X, if and only if, fi(Y )  fi(X ) for i ¼ 1,.,K and fj(Y ) < fj(X ) for at least one objective function j. This means that a feasible vector X is called Pareto optimal if there is no other feasible solution Y that would reduce some objective function without causing a simultaneous increase in at least one other objective function. The set of all feasible non-dominated solutions in X is referred to as the “Pareto optimal set”, and for a given Pareto optimal set, the corresponding objective function values in the objective space are called the “Pareto optimal frontier” (Konak et al., 2006). For having a better insight into the concept of Pareto optimal consider Fig. 3. This figure may show the objective functions space for an optimization problem with two objective functions f1 and f2. The feasible and infeasible areas are demonstrated as the areas inside and outside of the curve respectively. Every point in the feasible area is a solution of the problem. In Fig. 3, the values of both functions f1 and f2 for point M are lower than the corresponding values of point J. Thus point M dominates point J or point M is better than point J. In the same way points L and N dominate M. But points like I and K neither dominate M nor are dominated by M. Thus only those points that are located in the left-down parts of M will

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 4 3 e2 5 6

2

251

final solution? For doing this a decision-making process is required.

infeasible area feasible area

7.2. J K P The Pareto Optimal Front

M L

A

I N

R E

f2,min

T

O

The Equilibrium (Ideal) Point f1,min

1

Fig. 3 e Schematic of the objectives space.

dominate M (like L and N). Now consider point R, there is no point in the feasible area located in the left-down part of it. Thus R is not dominated by any other feasible point. Therefore R is a Pareto optimal. This is true for points P, A, R, E, T, O and all of the other points located at the bold curve indicated as “The Pareto Optimal Frontier” in Fig. 3. Notice that, in the feasible area, the minimum values of f1 and f2 belong to points P and O respectively. Thus P and O are the solutions for singleobjective optimization problems that their objective functions are f1 and f2, respectively. Other points on the Pareto frontier are also optimal, but which of them should be selected as the

Multi-objective evolutionary algorithms

In this work, the Pareto frontier is found using Genetic Algorithm (GA) as a branch of evolutionary algorithm. Genetic Algorithms were developed by John Holland in the 1960s as a means of importing the mechanisms of natural adaptation into computer algorithms and numerical optimization (Holland, 1975). They are implemented as a computer simulation in which a population of abstract representations (called chromosomes or the genotype of the genome) of candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem evolves toward better solutions. The evolution usually starts from a population of randomly generated individuals and happens in generations. In each generation, the fitness of every individual in the population is evaluated; multiple individuals are stochastically selected from the current population (based on their fitness), and modified (recombined and possibly randomly mutated) to form a new population. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number of generations have been produced, or a satisfactory fitness level has been reached for the population. If the algorithm has terminated due to a maximum number of generations, a satisfactory solution may or may not have been reached. In genetic algorithms, a candidate solution to a problem is typically called a chromosome, and the evolutionary viability of each chromosome is given by a fitness function. This method is a powerful optimization tool for nonlinear problems

a

b

c

d

e

f

Fig. 4 e Convergence of the Pareto frontier toward the optimum solutions for the cooling tower assisted vapor compression refrigeration system problem in the evolutionary algorithm optimization process: (a) generation 1, (b) generation 10, (c) generation 20, (d) generation 50, (e) generation 100, (f) generation 200.

252

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 4 3 e2 5 6

Table 2 e The tuning parameters in MOEA optimization program. Tuning parameters

value

Population size Maximum No. of generations Probability of crossover Probability of mutation Selection process

300 200 0.7 0.01 Roulette wheel

(Holland, 1975; Goldberg, 1989). For more information about Multi-Objective Evolutionary Algorithms (MOEAs) see (Konak et al., 2006). Fig. 4 presents the trend of convergence for the Pareto frontier in this optimization process from the beginning to generation number 200. This figure reveals that between generations 1 and 10, the population suddenly converges to the left-down part of the objective functions space. Convergence continues in the next generations. After 50 generations, the trend of Pareto optimal solutions is converged to a curve namely as the Pareto frontier. Finally, the result of generation number 200 is assumed to be the final Pareto frontier in this study.

8.

Results and discussion

The proposed model for the cooling tower assisted vapor compression refrigeration system schematically shown in Fig. 1 including eight decision variables and their constraints (introduced in Section 6.2) is optimized using the evolutionary algorithm with the tuning parameters that are mentioned in Table 2. Three optimization scenarios including thermodynamic single-objective, economic single-objective and multiobjective optimizations are performed. Fig. 5a presented the normalized form of the Pareto frontier obtained in multi-objective optimization scenario. Commonly, it is better to work with the normalized data of the Pareto frontier instead of the real values. The horizontal and vertical

Table 3 e The values and normalized values of objective functions for three optimized designs.   C_ P ð$$hr1 Þ I_tot ðkWÞ C_ I_ P

Economic Optimized Multi-Objective Optimized Thermodynamic Optimized

49.86 55.03 69.85

81.32 54.60 44.570

0.0 0.26 1.0

tot

1.0 0.27 0.0

axes in Fig. 5a are the normalized form of the objective functions that defined as follow respectively:  C_ P ¼

C_ P  C_ P;min _CP;max  C_ P;min

(47)

I_tot  I_tot;min I_tot;max  I_tot;min

(48)

and  I_tot ¼

Where C_ P;min and C_ P;max are the minimum and maximum values of C_ P in the Pareto frontier. In the same way I_tot;min and I_tot;max are the minimum and maximum values of I_tot in the Pareto frontier. Obviously C_ P;min and I_tot;min belong to the economic optimized and the thermodynamic optimized designs, respectively. In this case, and commonly in most of the cases, the shape of the Pareto frontier is such that C_ P;mam and I_tot;max belong to the thermodynamic optimized and the economic optimized designs, respectively. Using Eqs. (47) and   (48), the values of C_ P and I_tot for the economic single-objective   optimized design are 0 and 1, respectively. Similarly, C_ P and I_tot in the thermodynamic single-objective optimized design are 1 and 0, respectively (see Fig. 5a). In multi-objective optimization scenario, selection of the final solution among optimum points exist on the Pareto frontier needs a process of decision-making. The process of decision-making performed using definition of an ideal point on Pareto frontier namely as the equilibrium point as shown on Figs. 3 and 5a. At this equilibrium point, both objective functions ðC_ P and I_tot Þ have their minimum value and thus

Fig. 5 e (a) Normalized form of Pareto frontier and schematic of the decision-making process, (b) the graph that shows the distance of each point on the Pareto frontier from the equilibrium point.

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Table 4 e The values of decision variables in the various optimization scenarios. Decision Base Economic Thermodynamic Multi-objective variables case optimized optimized optimized Tcond ( C) Tevap ( C) Tw,i,ct ( C) Tw,o,ct ( C) LDcond LDevap DTsub ( C) DTsup ( C)

50.25

43.45

32.33

36.27

0.10

2.78

5.50

5.07

30.80

32.84

29.33

32.13

27.48

26.97

26.00

26.02

10.00 10.00 5.50

12.85 12.54 8.56

5.00 5.00 1.00

13.05 11.68 1.83

3.00

1.00

1.00

1.01

  both of C_ P and I_tot are zero. It is clear that this point is not existing in real world and is not located on the Pareto frontier as is clear from Fig. 5a. Thus this point is not a possible design of a system and is only an ideal point. In this decision-making process, the point of Pareto frontier that has shortest distance from the equilibrium point is selected as a final optimum solution. This solution is not only located on the Pareto frontier but also it archives the minimum possible values for both objectives (Sayyaadi et al., 2009 and Sayyaadi and Amlashi, 2010). The presented data for the optimum solution of the multi-objective optimization scenario reveals the corresponding data for this selected solution as described in Fig. 5a,   hereinafter. Hence, values of C_ P and I_tot for the selected final optimal solution in the multi-objective optimized design are 0.26 and 0.27, respectively. Fig. 5a graphically shows the variation of the distance between the equilibrium point and points located on the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Pareto frontier in Fig. 5a. This distance is equal to C_ P þ I_tot . It can be seen from this figure that the minimum value of this distance belongs to the selected multi-objective optimized point. This distance is equal to its maximum value, 1, for both   of the single-objective economic optimized ðC_ P ¼ 0:0 and I_tot ¼  1:0Þ and the single-objective thermodynamic optimized ðC_ P ¼  _ 1:0 and Itot ¼ 0:0Þ designs. It should be mentioned that for

points close to the minimum point, the slope of the graph is near zero. Hence one of these points, instead of the selected point, could be selected as a final solution in the multiobjective optimization problem without significant increase in the distance to the equilibrium point. The selection of the final solution point depends on the decision-maker opinion. For example a decision-maker could select a point that has a value  of C_ P about 0.05 (19.23%) less than the corresponding values of  C_ P for selected multi-objective final optimal solution at the  cost of a 0.056 (20.74%) increasing in the value of I_tot . In this new selected point the distance to the equilibrium point is increased only 0.0147 (4%) from the corresponding value of selected final optimal solution in the multi-objective scenario. In Fig. 5b a part of the Pareto frontier that are located on the left hand side of the minimum of the curve have better economic sound however the right hand side points are better in thermodynamic point of view.   The values of C_ P ; I_tot ; C_ P and I_tot for three optimization scenarios are listed in Table 3. Table 4 indicates the magnitude of decision variables for the base case design and corresponding magnitudes obtained in the three optimization scenarios. Table 5 indicates the results of energy analysis for various designs including the base case, economic optimized, thermodynamic optimized and multi-objective optimized systems. Some useful data are listed in this table, such as flow rates, heat loads, electrical works, and COPs. Fig. 6 shows the results of exergy analysis for the base case design and three optimized designs. This figure indicates that the thermodynamic optimized design has minimum total exergy destruction equal to 44.57 kW. The next exergy destructive designs are the multi-objective optimized, the economic optimized and the base case designs that have exergy destructions equal to 54.60 kW, 81.32 kW and 147.30 kW. These values are 22.51%, 82.46% and 230.48% more than total exergy destruction for the thermodynamic design respectively. Fig. 6 also indicates that for all of the three optimized designs, the most exergy destruction occurs in the compressor, condenser and cooling tower. The results of economic analysis for three optimized systems and the base case system are given in Fig. 7. This figure indicates that the minimum purchased equipment

Table 5 e The results of energy analysis the various optimization scenarios.

Total refrigerant flow rate (kg s1) CT water flow rate (kg s1 s) CT make up water flow rate (kg s1) Evaporator water flow rate (kg s1) Compressor power (kW) CT pump power (kW) Evaporator pump power (kW) CT fan power (kW) Evaporator heat load (kW) Condenser heat load (kW) Thermodynamic cycle COP Heat pump COP

Base case

Economic optimized

Thermodynamic optimized

Multi-objective optimized

2.56 34.25 0.38 16.84 126.60 29.36 3.47 6.33 352.00 478.60 2.78 2.12

2.32 17.79 0.25 16.84 86.24 6.57 2.67 4.61 352.00 438.24 4.08 3.52

2.22 28.89 0.32 16.84 52.35 3.49 0.15 7.19 352.00 404.35 6.72 5.57

2.29 16.18 0.24 16.84 63.24 4.48 1.10 4.60 352.00 415.23 5.57 4.79

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Fig. 6 e Comparison of exergy destructions in the various sub-systems of the cooling tower assisted vapor compression refrigeration system.

belongs to the economic optimized system. The base case, multi-objective optimized and thermodynamic optimized designs are in the next ranks and have purchased equipments respectively 10.02%, 20.82%, and 61.79% more than the economic optimized system. The minimum electricity cost belongs to the thermodynamic optimized design. This is due to special attention paid to electricity use in the thermodynamic optimization. The multi-objective optimized, the economic optimized and the base case designs are in the next ranks and have electricity costs respectively 16.21%, 58.43%, and 162.37% more than the thermodynamic optimized system. The water costs for all systems are almost equal and are lower than 0.12% of the total product cost. Therefore, compared with the other system costs, the water cost may be neglected. Finally the economic optimized design has the minimum total product cost ðC_ P Þ and the multi-objective optimized, base case and thermodynamic optimized designs are in the next ranks and have total product costs respectively

40 30

1.60 1.46 2.36 1.76

20 10 0 Capital Investmaent

Operating and maintanance

Electricity

60.96 49.86 69.84 55.03

0.07 0.05 0.06 0.05

50

Base Case Economic Optimized Thermodynamic Optimized Multi-Objective Optimized

18.13 10.95 6.91 8.03

60 41.15 37.40

-1

Levelized Cost ($.hr )

70

45.19

60.52

80

10.37%, 40.09% and 22.26% more than the economic optimized design. The results in Figs. 6 and 7 have shown that the total product cost ðC_ P Þ of the thermodynamic optimized design is 82.46% more than this value for the economic optimized design, and the total exergy destruction ðI_tot Þ for the economic optimized design is 40.09% more than this value for the thermodynamic optimized design. In fact, I_tot of the thermodynamic optimized design and C_ P of the economic optimized design are the minimum possible values (or the ideal values) of I_tot and C_ P respectively. If any of the thermodynamic or economic criteria is selected for optimization, the design will poorly satisfy the other criterion. In this study when the thermodynamic optimized design is selected as the final system design, C_ P will be 82.46% more than their minimum possible values. On the other side, if the economic optimized design is selected, I_tot will be 40.09% more than their minimum possible values. Whereas if the

Water

Product

Fig. 7 e Comparison of the levelized costs including capital investment, operating and maintenance cost, electricity cost, water cost and product cost (the cost of cooling) for various designs of the cooling tower assisted vapor compression refrigeration system.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 4 ( 2 0 1 1 ) 2 4 3 e2 5 6

multi-objective design is selected as a final design of the system, I_tot and C_ P are 22.51% and 10.37% more than their minimum values, respectively. It can be said that these deviations form the minimum values in multi-objective optimized design are more acceptable than the other singleobjective designs.

_ ref is the refrigerant mass flow rate (kg s1). hisen, the Where m isentropic efficiency of a scroll compressor is fitted as follows:

hisen ¼ 0:85  0:046667

A.2.

9.

pcond pevap

! (A.2)

Evaporator and condenser

Conclusions

A new method for optimization of a cooling tower assisted vapor compression refrigeration system was presented. The proposed method covers both thermodynamic and economic aspects of the system design and the component selection. Irreversibility (exergy destruction) for the systems was determined. The economic model of the system was developed based on the total revenue requirement method (TRR method). The configuration of the optimization problem was built with eight decision variables and the appropriate feasibility and engineering constraints. The optimization process was carried out using a multi-objective evolutionary algorithm. Three optimization scenarios including the thermodynamic, economic and multi-objective optimizations were performed. It was concluded that the multi-objective optimization is a general form of single-objective optimization that considers two objectives of thermodynamic and economic, simultaneously. It was discussed that the final solution of the multiobjective optimization depends on decision-making process. However, its results were somewhere between corresponding results of thermodynamic and economic single-objective optimizations. The thermodynamic optimization is dedicated to consideration about limited source of energy whereas the economic single-objective optimization has respect only on economic resources. The multi-objective optimization focuses on limited energy and monetary resources, simultaneously. The results show that percentages of deviation from ideal values of thermodynamic and economic criteria for the thermodynamic optimized system were 0.00% and 40.09%, respectively. These percentages for the economic optimized system were 82.46% and 0.00%, respectively. Deviation values from minimum ideal point for the multi-objective optimized design were obtained 22.51% and 10.37% for thermodynamic and economic criteria, respectively. It was concluded that the multi-objective design satisfies the thermodynamic and economic criteria better than two single-objective thermodynamic and economic optimized designs.

Appendix A. Purchased equipment cost (PEC) Equations for calculating the purchased equipment costs (PEC) for the components of the vapor compression refrigeration system are as follow:

A.1.

255

Compressors

The purchase equipment cost for the scroll compressor is given by Valero (1994):      _ ref 573m Pcond Pcond ln (A.1) PECcomp ¼ 0:8996  hisen Pevap Pevap

The purchase equipment cost for the condenser and evaporator are as follows (Selbas‚ et al., 2006): PECcond ¼ 516:621Acond þ 268:45

(A.3)

PECevap ¼ 309:143Aevap þ 231:915

(A.4)

Where Acond and Aevap are the heat transfer areas of condenser and evaporator respectively.

A.3.

Pump

The purchase equipment cost for a pump is as follows (Sanaye and Niroomand, 2009):

_ PECpump ¼ 308:9W pump Cpump

(A.5)

_ pump is the pumping power in kW, Cpump is 0.25 for Where W pumping power in the range of 0.02e0.3 kW, 0.45 for pumping power in the range of 0.3e20 kW, and 0.84 for pumping power in the range of 20e200 kW.

A.4.

Cooling tower

The purchased equipment cost for a cooling tower can be calculated as follow (Peters and Timmerhaus, 1991): _ aw2  10a3 A$Rþa4 Aþa5 Rþa6 PECct ¼ a1 m

(A.6)

Where A is the difference between the water output temperature and the ambient air wet bulb temperatures ( C). R is the difference between input and output water temperatures ( C) _ w is the water mass flow rate (kg/s). The values of and finally m a1ea6 are 3950.9, 0.5872900, 0.0032091, 0.0267190, 0.0436540 and 0.1026000, respectively. It is required to mentioned that all costs are modified to the cost index of 2009 as follow (Bejan et al., 1996): PECnew ¼ PECref

  Inew Iref

(A.7)

PECnew and PECref are the renewed cost and cost at reference year for the proposed equipment. Inew and Iref are cost indexes at new and reference years, respectively. In this work marshal and swift index used for equipment as indicated in the Table A1 (Peters and Timmerhaus, 1991) (indexes of years after 1991 was obtained in this reference by forecasting).

256

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Table A1 e Marshal and swift index at various years (Peters and Timmerhaus, 1991). Year

Index

1990 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2009

915 1027.5 1039.2 1056.8 1061.9 1068.3 1089 1092 1100.2 1109 1115.6 1129.6 1143 1156.6 1170.2

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