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Thermoeconomic optimization of a shell and tube condenser using both genetic algorithm and particle swarm Hassan Hajabdollahi a, Pouria Ahmadi b,*, Ibrahim Dincer b a

Mechanical Engineering Department, Valiasr University of Rafsanjan, Rafsanjan, Iran Department of Mechanical Engineering, Faculty of Engineering and Applied Science, University of Ontario Institute of Technology (UOIT) , 2000 Simcoe St. North, Oshawa ON L1H 7K4, Canada

b

article info

abstract

Article history:

This paper presents a thermoeconomic optimization of a shell and tube condenser, based

Received 31 March 2010

on two new optimization methods, namely genetic and particle swarm (PS) algorithms.

Received in revised form

The procedure is selected to find the optimal total cost including investment and operation

29 September 2010

cost of the condenser. Initial cost includes condenser surface area and operational cost

Accepted 18 February 2011

includes pump output power to overcome the pressure loss. Design parameters are tube

Available online xxx

number, number of tube pass, inlet and outlet tube diameters, tube pitch ratio and tube arrangements (30, 45, 60 and 90 ). Therefore, shell diameter should be selected less than

Keywords:

7 m, tube length should be less than 15 m, and ratio of diameter to tube length should be in

Shell-and-tube condenser

the range of 1/12 to 1/3. In addition, it is found that GA provides better results for computer

Refrigeration

CPU running time, compared to PS algorithm. Finally, a sensitivity analysis of design

Optimization

parameters at the optimal point is conducted. Results show that an increase in the tube

Effectiveness

number leads to decrease in the objective function first then it leads to a considerable

Cost

increment in objective function. ª 2011 Elsevier Ltd and IIR. All rights reserved.

Optimisation thermoećonomique d’un condenseur multitubulaire a` l’aide d’un algorithme e´volutif Mots cle´s : Condenseur multitubulaire ; Froid ; Optimisation ; Efficacite´ ; Couˆt

1.

Introduction

Condensers are considered to be a significant part of both refrigeration and heat pump systems, as well as other systems

and applications where the heat exchangers are employed. Their performance directly affects the performance of the system. In this regard, they have been receiving attention for research from various researchers.

* Corresponding author. Tel.: þ1 289600 9505. E-mail address: [email protected] (P. Ahmadi). 0140-7007/$ e see front matter ª 2011 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2011.02.014

Please cite this article in press as: Hajabdollahi, H., et al., Thermoeconomic optimization of a shell and tube condenser using both genetic algorithm and particle swarm, International Journal of Refrigeration (2011), doi:10.1016/j.ijrefrig.2011.02.014

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Nomenclature At,o Cmin Cin Cop Co Ctotal Cl di do Ds f hi ho i kel k m n ny NTU Nu N pt

2

total tube outside heat transfer area (m ) min((mcp)s,(mcp)t) cost of investment ($) total cost of operation ($) annual operating cost ($ yr1) total cost ($) tube layout constant (e) tube side inside diameter (m) tube side outside diameter (m) shell diameter (m) friction factor (e) tube-side heat transfer coefficient (W m2 K1) shell-side heat transfer coefficient (W m2 K1) annual discount rate (%) price of electrical energy ($ k1 W1 h1) thermal conductivity (W m1 K1) mass flow rate (kg s1) number of tube in a column (e) equipment life (yr) number of transfer units (e) Nusselt number (e) number of tube (e) tube pitch (m)

Shell and tube condensers are composed of with circular pipes which are installed in cylindrical shells. It is a well known heat exchanger which is a key component in refrigeration and heat pump systems, thermal system plants, petrochemical plants and refrigeration/air-conditioning systems (Zhao and Zhang, 2010; Haseli et al., 2008). Shell and tube heat exchanger, as shown in Fig. 1, is widely used in many industrial power generation plants as well as chemical, petrochemical, and petroleum industries. They are used to transfer heat between two or more fluids, between a solid surface and a fluid, or between solid particulates and a fluid, at different temperatures and in thermal contact. In heat exchangers, there are usually no external heat and work interactions. There are some effective parameters in shell and tube heat exchanger design such as tube numbers, tube length, tube arrangement, baffle spacing (Haseli et al., 2008). Therefore, optimization of this kind of heat exchanges is quite

P Pr Q Rf Rt Re tw T Um DTlm Dpt

pumping power (W) Prandtl number (e) rate of heat transfer (W) fouling factor (m2 K W1) the total thermal resistances (m2 k W1) Reynolds number (e) tube wall thickness (m) temperature ( C) mean overall heat transfer coefficient (m2 K1) logarithmic mean temperature difference ( C) pressure drop in tube side (Pa)

Greek letters 3 effectiveness m viscosity (pa s) h pump efficiency (e) s hours of operation per year (h yr1) r density (kg m3) Subscripts s shell-side t tube side w tube wall i inner or inlet o outer or outlet

interesting. Bejan (1978) demonstrated the use of irreversibility as a criterion for evaluation of the efficiency of a heat exchanger. The purpose was to minimize the wasted energy by optimum design of fluid passages in a heat exchanger. Since shell and tube type condensers are widely used in refrigeration and heat pump, as well as air-conditioning systems, there has been increasing interest by various researchers. Moreover, finned-tube condensers and evaporators are the predominant types of refrigerant-to-air heat exchangers. In fact, their performance is affected by a multitude of design parameters, some of which are limited by the application or available manufacturing capabilities (Dincer, 2003). On the other hand, according to the literature (Dincer and Rosen, 2007) refrigeration condenser is the most exergy destructor in comparison with other component used in refrigeration cycle. Thus, the optimization and exergy analysis of such devices are of the greatest importance.

Fig. 1 e Plant view of the shell and tube condenser. Please cite this article in press as: Hajabdollahi, H., et al., Thermoeconomic optimization of a shell and tube condenser using both genetic algorithm and particle swarm, International Journal of Refrigeration (2011), doi:10.1016/j.ijrefrig.2011.02.014


Berhane et al. (2010) performed the optimum heat exchanger area estimation used in an absorption system. In this work the structural method was used to derive a simple correlation for the straight forward estimation of the optimum area of a heat exchanger in a thermal system. Misra et al. (2003) carried out the thermoeconomic optimization of a single effect vapor absorption refrigeration system. The average cost approach (AVCO) was used to optimize thermoeconomically. They also performed the exergoeconomic analysis for the absorption system. Johannessen et al. (2002) proved that the entropy production due to heat transfer in a heat exchanger is a minimum, when the local entropy production is constant throughout all parts of the system. A new design strategy, involving losses due to fluid and heat transfer irreversibilities that lead to production of entropy, has recently been presented by Lerou et al. (2005) and applied to the thermal design of a counterflow heat exchanger through minimization of entropy generation. Haseli et al. (2008) optimized the temperatures in a shell and tube condenser with respect to exergy. The optimization problem is defined subject to condensation of the entire vapor mass flow and it is solved based on the sequential quadratic programming (SQP) method. From that analysis it was concluded that optimization results revealed new characteristics for the cooling water, with respect to the minimization of exergy destruction of the condensation process. Haseli et al. (2010) performed the exergy analysis for two phase flow in a shell and tube condenser. This analysis was used to evaluate both local exergy efficiency of the system (along the condensation path) and for the entire condenser, i.e., overall exergy efficiency. Hence, from what have been carried out by researches about condenser the thermoeconomic optimization of this component using evolutionary algorithm is so necessary. Khalifeh Soltan et al. (2004) estimated the maximum load for a shell and tube condenser at some special conditions. Ahmadi et al. (2011) performed the Multi-objective exergetic optimization of shell and tube heat recovery heat exchangers using a genetic algorithm. They considered two key parameters, such as exergy efficiency and cost and optimized them using multi-objective genetic algorithm. They also performed a sensitivity analysis of the variation of design parameters on both objective functions. On the other hand, there are some researches which have been carried out by considering the economic point of view (Selbas et al., 2006; Andre et al., 2008; Eryener, 2006). In the present study which is the extended version of earlier researches performed by Haseli et al. (2008, 2010) the primary goal is to thermodynamically model and optimize a shell and tube condenser using evolutionary algorithm which has been shown that is the best candidate for optimization problems. In summary, the following specific objectives are direct contribution of this paper to the area: To model a shell and tube condenser thermoeconomically. To validate the model results with the literature data. To optimize the shell and tube condenser by using two potential methods, namely genetic algorithm and particle swarm method.

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To compare these two different optimization methods and their results for various cases. To perform a sensitivity analysis of the variation of design parameters at the optimal point.

2.

Thermal modeling

2.1.

Assumptions

Here, the following assumptions are made for analysis purposes:

Condensing flow is in the shell direction. Cooling fluid is considered in tube side. The shell pressure loss is negligible. The condenser changes the water state from saturated steam to liquid.

2.2.

Modeling

The heat transfer between hot and cold fluid is calculated based of the following relation: Q ¼ Um At;o DTlm

(1)

where DTlm is the logarithmic mean temperature difference which is defined as DTlm ¼

DT1 DT2 DT1 ln DT2

(2)

Here Um is the mean value of total heat transfer coefficient and DT1 ¼ Ts Tt,i and DT2 ¼ Ts Tt,o. Since the total heat transfer coefficient can considerably vary along with the heat exchanger, the mean value of this parameter is used as follows: 1 Um ¼ ðU1 þ U2 Þ 2

(3)

where U1 , U2 are the total heat transfer at the inlet and outlet parts of the condenser. By assuming that both 1=U and DT change linearly with Q, a useful relation for determining the mean value of total heat transfer is obtained as follows: Um ¼

1 DTlm DT2 1 DT1 DTlm þ U DT1 DT2 U2 DT1 DT2

(4)

In addition, U can be defined as follows: 1 1 ¼ Rt þ ho U

(5)

where ho is a convection heat transfer coefficient and Rt is the total inner thermal resistances which are determined as: Rt ¼ Rfo þ

1 d o tw d o þ Rfi þ hi di kw Dm

(6)


4


Dm ¼

do di hd i o ln di

where pr is the pitch ratio and cl is tube layout constant that is equal to 1 for 45 , 90 and is 0.87 for 30 , 60 (Dentice and Vanoli, 2004). The shell and tube condenser effectiveness is evaluated as follow (Kakac and Liu, 2000):

(7)

Here, Rf is fooling factor and do, di, kw, tw are tube outlet diameter, tube inlet diameter, tube conduction conductivity and tube thickness respectively. Having known the tube number as well as the mass flow rate in the pipe, the velocity in the pipe will be determined as: V¼

4mt 4mt / Re ¼ pdi mN rpd2i N

3 ¼ 1 eNTU

where NTU is the number of transfer units that is determined as:

(8) NTU ¼

where N is the tube number. Having known the Re number, Nu number at the tube side will be defined as a Re number function (Kakac and Liu, 2000): Nu ¼

Nu ¼

0:5f ðRe 1000ÞPr if 2300 < Re < 104 0:5 2=3 Pr 1

1 þ 12:7ðf =2Þ

0:5fRePr 0:5

1:07 þ 12:7ðf =2Þ

Pr2=3 1

if 104 < Re < 5 106

2

f ¼ ð1:58 lnðReÞ 3:28Þ

(10)

(12)

Heat transfer coefficient for the condensate flow (ho) is obtained based on the following relation (Kakac and Liu, 2000): ho ¼ 0:728

rl ghfg ko ml DTw do

0:25

Hot fluid temperature (Th ¼ 125 C). Inlet cold fluid temperature (Tc1 ¼ 12 C). _ c ¼ 400 kg=sÞ. Cold fluid mass flow rate ðm _ h ¼ 8:7 kg=sÞ. Hot fluid mass flow rate ðm

3.

1 n1=6

(18)

Here, tube number, number of tube pass, inlet and outlet diameters, tube pitch ratio and tube arrangements are considered as 5 design parameters for the optimization procedure. To have a real optimization program some relevant constraints should be applied to the optimization program. Shell diameter should be selected less than 7 m and tube length should be selected less than 15 m respectively. Moreover, performance conditions of the shell and tube condenser are as follows:

(9)

(11)

Nu$ki di

Um At;o Cmin

2.3. Design parameters, constraints and performance conditions

Therefore, knowing the Nu number, the convection heat transfer coefficient is determined as hi ¼

(17)

Optimization: objective function

(13)

Here, n is the tube number in a column which may be predicted for each tube arrangement as follows:

In this study, total cost is considered as an objective function. Total cost includes the cost of heat transfer area as well as the operating cost for the pumping power.

For arrangement 45 , 90

Ctotal ¼ Cin þ Cop

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4cl p2t N=p n¼ do þ pt

The investment cost for both shell and tube from stainless steel shell and tubes is (Taal et al., 2003):

(14)

Cin ¼ 8500 þ 409A0:85 t;o

For arrangement 30 , 60

n¼

(19)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4cl p2t N=p pffiffiffi do þ 3pt

where At is the total tube outside heat transfer area. The total operating cost related to pumping power to overcome friction losses was computed from the following equations:

(15)

pt ¼ do pr

(20)

Cop ¼

(16)

ny X

Co

(21)

k

k¼1 ð1 þ iÞ

Table 1 e The comparison between our simulation code and the literature. Case No 1

HT area (m2) Pressure drop (kPa) Condensed length (m) Condensed diameter (m)

Case No 2

Case No 3

Simulation

(Kakac and Liu, 2000)

Simulation


Simulation


12,963 55.34 15.573 4.556

13,178.8 56.6 15.8 4.6

13,433 31.915 12.91 5.09

13,696.6 32.5 13.2 5.1

14,159 15.873 10.21 5.88

14,525.5 16.1 10.5 5.9

Maximum error %

2.5 2.2 2.76 0.95


5


Co ¼ Pkel s P¼

(22)

1 mt DPt h rt

(23)

where ny is the equipment life time, i is annual discount rate, kel, s and h are price of electrical energy, hours of operation per year and pump efficiency, respectively.

4.

Model validation and optimization results

To validate the modeling results, the simulation output was compared with the corresponding reported results given in literature. The comparison of our modeling results and the corresponding values from reference Kakac and Liu (2000), for the same input values is shown in Table 1. Results show that the difference percentage points of two mentioned modeling output results are acceptable for three studied cases. As it is shown in this table, differences between simulation code and references are reasonable in the engineering problems. Therefore, the developed code is valid for the optimization program. Furthermore, the optimization results using procedure presented in this paper for the same case study with the

Fig. 3 e Depiction of the velocity and position updates in particle swarm optimization.

same input values as taken from Kakac and Liu (2000) are performed. The results show that our optimization results lead to decrease in the total cost about 40%, 62% and 75% for three cases. Therefore, our simulation code has better results in comparison with the literature which can be considered a noticeable improvement.

Start

Coding of parameter space

Random creation of initial population

Evaluation of population Finesses

Application of operators

New population (Replacement of the old)

No

Is Number of Generation exceeded?

Yes Stop Fig. 2 e Genetic algorithm flowchart. Please cite this article in press as: Hajabdollahi, H., et al., Thermoeconomic optimization of a shell and tube condenser using both genetic algorithm and particle swarm, International Journal of Refrigeration (2011), doi:10.1016/j.ijrefrig.2011.02.014

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Table 2 e The range of change in design parameters. Variable Tube number Number of tube pass Inner tube diameter (m) Outer tube diameter (m) Tube pitch ratio Tube arrangements

Lower bound

Upper bound

100 1 0.023 0.43 1.25 30

1000 3 0.0254 0.453 1.5 90

5.

Evolutionary algorithm

5.1.

Genetic algorithm

In recent years, optimization algorithms have received increasing attention by the research community as well as the industry. In the area of evolutionary computation (EC), such optimization algorithms simulate an evolutionary process where the goal is to evolve solutions by means of crossover, mutation, and selection based on fitness with respect to the optimization problem at hand (Ghaffarizadeh, 2006) Evolutionary algorithms (EAs) are highly relevant for industrial applications, because they are capable of handling problems with non-linear constraints, multiple objectives, and dynamic components properties that frequently appear in real problems (Goldberg, 1989). Uniform crossover and random uniform mutation are employed to obtain the offspring population. The integer-based uniform crossover operator takes two distinct parent individuals and interchanges each corresponding binary bits with a probability, 0 < pc 1. Following crossover, the mutation operator changes each of the binary bits with a mutation probability, 0 < pm < 0.5. Genetic algorithms (GAs) are an optimization technique based on natural genetics. GAs were developed by Holland (1975) in an attempt to simulate growth and decay of living organisms in a natural environment. Even though originally designed as simulators, GAs proved to be a robust optimization technique. The term robust denotes the ability of the GAs for finding the global optimum, or a near-optimal point, for any optimization problem. The basic idea behind GAs could be described in brief as follows. A set of points inside the optimization space is created by random selection of points. Then, each set of points is transformed into a new one. Moreover, this new set will contain more points that are closer to the global optimum. The transformation procedure is based only on the information of how optimal each point is in the set, consists of very

simple string manipulations, and is repeated several times. This simplicity in application and the fact that the only information necessary is a measure of how optimal each point is in the optimization space, make GAs attractive as optimizers. Nevertheless, the major advantages of the GAs are the following: Constraints of any type can be easily implemented. GAs usually finds more than one near-optimal point in the optimization space, thus permitting the use of the most applicable solution for the optimization problem at hand.

The basic steps for the application of a GA for an optimization problem are summarized in Fig. 2. A set of strings is created randomly. This set, which is transformed continuously in every step of the GA, is called population. This population, which is created randomly at the start, is called initial population. The size of this population may vary from several tens of strings to several thousands. The criterion applied in determining an upper bound for the size of the population is that further increase does not result in improvement of the near-optimal solution. This upper bound for each problem is determined after some test runs. Nevertheless, for most applications the best population size lies within the limits of 10e100 strings (Ghaffarizadeh, 2006). The “optimality” (measure of goodness) of each string in the population is calculated. Then on the basis of this value an objective function value, or fitness, is assigned to each string. This fitness is usually set as the amount of “optimality” of each string in the population divided by the average population “optimality”. It is possible that a certain string does not reflect an allowable condition. For such a string there is no “optimality”. In this case, the fitness of the string is penalized with a very low value, indicating in such a way to the GA that this is not a good string. Similarly, other constraints may be implemented in the GA. A set of “operators”, a kind of population transformation device, is applied to the population. These operators will be discussed. As a result of these operators, a new population is created, that will hopefully consist of more optimal strings. The old population is replaced by the new one. A predefined stopping criterion, usually a maximum number of generations to be performed by the GA, is checked. If this criterion is not satisfied a new generation is started, otherwise the GA terminates. It is now evident that when the GA terminates, a set of points (final population) has been defined, and in this population more than one equivalently good (optimal) point may exist. As discussed earlier, the selection is done by the

Table 3 e Inner and outer diameters of 42 standard tubes. Inner diameter (in) 0.824, 0.742, 1.049, 0.957, 1.380, 1.278, 1.610, 1.500, 2.067, 1.939, 2.469, 2.323, 3.068, 2.900, 3.548, 3.364, 4.026, 3.826, 5.295, 5.047, 4.813, 6.3570, 6.065, 5.761, 8.329, 8.071, 7.625, 10.420, 10.192, 9.750, 12.390, 12.090, 11.750, 13.500, 13.250, 13.000, 15.500, 15.250, 15.000, 17.624, 17.250, 17.000 Outer diameter (in) 1.050, 1.050, 1.315, 1.315, 1.660, 1.660, 1.900, 1.900, 2.375, 2.375, 2.875, 2.875, 3.500, 3.500, 4.000, 4.000, 4.500, 4.500, 5.563, 5.563, 5.563, 6.6250, 6.625, 6.6250, 8.625, 8.625, 8.625, 10.750, 10.750, 10.750, 12.750, 12.750, 12.750, 14.000, 14.000, 14.000, 16.000, 16.000, 16.000, 18.000, 18.000, 18.000


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Table 4 e Constant parameters for the objective function based on Kakac and Liu (2000).

Hours of operation Price of electrical energy Pump efficiency Rate of annual discount Equipment life

Dimension

Value

h/year $/MWh e e hr

5000 20 0.6 0.1 5000

user. Thus, this advantage of the GAs permits the selection of the most appropriate solution for the optimization problem.

5.2.

Particle swarm optimization

Particle Swarm Optimization (PSO) was invented by Kennedy and Eberhart in the mid 1990s (Kennedy and Eberhart, 1995). The basic PSO algorithm consists of three steps, namely, generating particles’ positions and velocities, velocity update, and finally, position update. Here, a particle refers to a point in the design space that changes its position from one move (iteration) to another based on velocity updates. First, the positions, xki, and velocities, vki, of the initial swarm of particles are randomly generated using upper and lower bounds on the design variables values, xmin and xmax, as expressed in Eqs. (24) and (25). xi0 ¼ xmin þ randðxmax xmin Þ vi0

xmin þ randðxmax xmin Þ ¼ Dt

Table 5 e Optimal design parameters using GA. Design parameters Tube number Number of tube pass Inner and outer tube diameters (m) Tube pitch ratio Tube arrangement

vikþ1 ¼ wvik þ c1 rand

Here, the positions and velocities are given in a vector format with the superscript and subscript denoting the ith particle at time k. In Eqs (24) and (25), rand is a uniformly distributed random variable that can take any value between 0 and 1. This initialization process allows the swarm particles to be randomly distributed across the design space. The second step is to update the velocities of all particles at time k þ 1

Fig. 4 e The convergence of the objective function for various numbers of generations using genetic algorithm.

203 1 0.0406e0.0479 1.2598 45 or 90

using the particles objective or fitness values which are functions of the particles current positions in the design space at time k. The fitness function value of a particle determines which particle has the best global value in the current swarm, pkg, and also determines the best position of each particle over time, pi, i.e. in current and all previous moves. The velocity update formula uses these two pieces of information for each particle in the swarm along with the effect of current motion, vki, to provide a search direction, vkþ1i, for the next iteration. The velocity update formula includes some random parameters, represented by the uniformly distributed variables, rand, to ensure good coverage of the design space and avoid entrapment in local optima. The three values that effect the new search direction, namely, current motion, particle own memory, and swarm influence, are incorporated via a summation approach as shown in Eq (26) with three weight factors, namely, inertia factor, w, self confidence factor, c1, and swarm confidence factor, c2, respectively.

(24)

(25)

Optimal value

g pi xik p xik þ c2 rand k Dt Dt

(26)

The original PSO algorithm uses the values of 1, 2 and 2 for w, c1, and c2 respectively, and suggests upper and lower bounds on these values as shown in Eq (26) above. Position update is the last step in each iteration. The Position of each particle is updated using its velocity vector as shown in Eq (27) and depicted in Fig. 3. xikþ1 ¼ xik þ vikþ1 Dt

(27)

Fig. 5 e The convergence of the objective function for various numbers of iterations using Particle swarm algorithm.


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Table 6 e Optimal design parameters using PSO. Design parameters

Optimal value

Tube number Number of tube pass Inner and outer tube diameters (m) Tube pitch ratio Tube arrangement

204 1 0.0406e0.0479 1.2527 45 or 90

The three steps of velocity update, position update, and fitness calculations are repeated until a desired convergence criterion is met.

6.

Results and discussion

6.1.

Decision parameters and limitations

As mentioned earlier, 5 decision parameters with relevant constraints were considered to have a real optimization program. As known, for each optimization program there should be a reasonable range selected for design parameters. Therefore, the range of change in design parameters is shown in Table 2.

Fig. 7 e The variation of the objective function versus changes in number of tube passes at the optimum point.

the condenser total cost is about 29,122.13 $/yr based on GA. The optimal design parameters are listed in Table 5.

6.4. 6.2.

Case study

For the optimization program, 42 standard tubes are used. Table 3 shows the inner and outer diameters of each tube. Moreover, constant values for the objective function formula are listed in Table 4.

6.3.

Genetic algorithm optimization results

The genetic algorithm optimization was performed for 150 generations, using a search population size of M ¼ 100 individuals, crossover probability of pc ¼ 0.9, gene mutation probability of pm ¼ 0.035. The results for optimum total cost versus generation are shown in Fig. 4. The optimum value for

Fig. 6 e The variation of the objective function versus changes in the tube number at the optimum point.

Particle swarm optimization result

Particle swarm algorithm with 35 particles and c1 ¼ 1.7, c2 ¼ 2 with 200 iterations was performed. The variation of the objective function with number of iterations is shown in Fig. 5. In this case the optimum value for the objective function is about 29,112.66 $/yr which a poor difference in comparison with results obtained using a genetic algorithm. The optimal design parameters are listed in Table 6.

6.5.

Comparison of two optimization algorithms

By comparison of two mentioned optimization algorithm, it can be concluded that Particle Swarm algorithm has more

Fig. 8 e The variation of the objective function versus changes in tube diameter for the 42 considered tubes at the optimum point.



Fig. 9 e The variation of the objective function versus tube arrangement at the optimum point.

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Fig. 11 e The variation of the effectiveness versus rate of cooling fluid at the optimum point.

6.6.

Sensitivity analysis

convergence rate in comparison with Genetic Algorithm. It is due to the fact that unlike GA it does not have the decode part in the program and it does not need decoding in each iteration. It should be noted that results obtained by PSO have various results for each run. Hence, the results from PSO in this paper are the best results after 20 runs. However, GA provides better accuracy in finding the search domain. As a result, it needs less running time in comparison with PSO. This might be due to the mutation and crossover operators. Thus, this can result in a better search domain. On the other hand, GA has the same value for the objective function for each run. In conclusion, although GA has less convergence rate in comparison with PSO, the power of searching in GA is more powerful than PSO. Thus, GA is strongly suggested when the run time is significant. So, GA is also suggested for optimization problems with more design variables as well as numerous constraints.

In this section, the variation of design parameters at the design point in the reasonable range is shown. In this part, 4 design parameters are fixed at the optimal point and the tube number varies between 100 and 1000. Fig. 6 shows the variation of the objective function with change in the tube number. As it is shown in this figure, variation of tube number at the defined number results in optimizing the total cost. This number is about 204 tube number which is in a good manner of a number of tubes obtained from the optimization program. Therefore, it indicates the correct optimization results as well. Moreover, from this figure it is concluded that increasing the tube number above the optimal number leads to significant jump in the total cost. In this case, tube pass varies from 1 to 3 while other design parameters are fixed at the optimal point. Fig. 7 shows the variation of the objective function versus number of tube pass. It is shown that by increase in the number of tube pass, total

Fig. 10 e The variation of the objective function versus tube pitch ratio at the optimum point.

Fig. 12 e The variation of the effectiveness versus rate of condensation fluid at the optimum point.


10


a good insight into the study, the sensitivity analysis was conducted at the optimal point. The results showed that increase in the tube number leads to decrease in the objective function first then it leads to a considerable increment in the objective function. Also, results show that any increases in the number of tube pass led to increase in the total cost of the condenser. Moreover, it was concluded that although GA has less speed in comparison with PSC algorithm, it has better results as well as less CPU running time. The sensitivity analysis of the tube pitch ratio reveled that variation of this parameter had no effect of the variation of objective function because the pressure drop was in the tube side.

references Fig. 13 e The variation of the effectiveness versus cooling fluid temperature at the optimum point.

cost of the condenser increases and the optimal value for this number is 1 which is the obtained result by Genetic Algorithm and Particle Swarm method. In this case, the effect of changes in the tube diameters on the total cost of the condenser is shown in Fig. 8. It is concluded that total cost has the lower amount when the tube number 8 is selected. This indicates the results obtained by optimization procedure. The variation of the objective function versus tube arrangements is shown in Fig. 9. It is concluded that selecting the tube arrangement 45 or 90 leads to the optimal value for the total cost of the condenser. This result was obtained in the optimization part. The variation of the objective function at the optimal point versus tube pitch ration is shown in Fig. 10. Since, in this study the pressure drop in the tube side is considered negligible, one can predict that the variation of this parameters has no effect on the pressure drop and total cost of the condenser. Thus, there is no change in the objective function in Fig. 10. Finally, the effects of cold and hot flow mass flow rate as well as cold flow temperature in the optimal point on the condenser performance are obtained. The results are shown in Figs. 11e13. It is seen that by decrease in the mass flow rate of cold stream, and increase in the hot flow mass flow rate and increasing the cold flow temperature, condenser performance is increased.

7.

Conclusions

In the present study, thermoeconomic optimization of the shell and tube condenser has been performed using both genetic algorithm and Particle Swarm method. The objective function was introduced as a total cost of the condenser. Moreover, five design parameters were considered for the optimization program. To have a real analysis, the simulation code was compared with literature and the results showed the reasonable difference between the data. In addition, to have

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