-
fo
ndustrIran
Heat exchangerOptimizationGenetic algorithmFirey algorithmCuckoo
search
y inchanple
tion is to maximize its thermal efciency. Obtained results for
several simulation optimizations indicatethat GA is unable to nd
permissible and optimal solutions in the majority of cases. In
contrast, design
(STHXincludifar, th
to the limited capability of the design engineers in
considerationand evaluation of all admissible designs. Budget
constrains duringthe design phase even worsen this. So it is not
surprising to see realworld STHX that their designs is far away
from being optimal.
Fig. 1 displays the layout and uid ows of a typical STHX. Baf-es
placed along the tube bundle force the uid to ow throughtubes [2].
Bafes simply intensify the turbulent level and improve
vel of calculationables make algorithmthe massi
of variable search space. Evolutionary algorithms, in contraable
to efciently explore the search space and nd approoptimal solutions
in a short time. They are also global optimmethods and can avoid
local optima using different mechanismsand operations. Therefore,
using evolutionary algorithms hasbecome a standard practice for
design of heat exchangers in thelast decade [8,9].
Despite many breakthroughs in the eld of evolutionary
optimi-zation (mainly reported in publications handled by IEEE
Computa-tional Intelligence Society), genetic algorithm is the most
used Corresponding author.
Energy Conversion and Management 89 (2015) 281288
Contents lists availab
n
lsesatises the trade-off between pressure drops and
thermalexchange transfers. No doubt, this process is massively
time-con-suming and expert expensive. Furthermore, there is no
guaranteethat the nal design is optimal in terms of considered
criteria due
optimal design of STHX. This is due to a high lecomplexity and
discrete nature of decision variobjective function
nondifferentiable. Also theshighly likely to be trapped in local
optima due
tohttp://dx.doi.org/10.1016/j.enconman.2014.09.0390196-8904/ 2014
Elsevier Ltd. All rights reserved.ing thes are
venessst, areximateizationused type of heat exchanger used in
different industries. Optimaldesign of STHX is a challenging
engineering task. Several criteriasuch as efciency and capital,
operating, and energy costs can beconsidered in the design. As
mentioned in [1], the design processhas an iterative nature and
includes several trials for obtaining areasonable conguration that
fullls the design specications and
has been considered in some studies as well [4,5]. Two
approachesare often used for design optimization. Some authors
focus onsimultaneous optimization of several variables [1,4], while
othersx some less important variables and try to nd the optimal
valuesfor the most important design variables [6,7].
Gradient descent optimization methods cannot be applied for1.
Introduction
Shell and tube heat exchangersoperation of many industrial
plantsstations, and manufacturing sites. Byvariables found by FA
and CS always lead to maximum STHX efciency. Also computational
requirementsof CS method are signicantly less than FA method. As
per optimization results, maximum efciency(83.8%) can be achieved
using several design congurations. However, these designs are
bearing differentdollar costs. Also it is found that the behavior
of the majority of decision variables remains consistent
indifferent runs of the FA and CS optimization processes.
2014 Elsevier Ltd. All rights reserved.
) play a critical role inng oil reneries, powerey are the most
widely
the shell lm coefcient of heat transfer. Detailed
informationabout components of a STHX can be found in [3]. The
existing lit-erature on design optimization of STHX greatly deal
with ndingthe optimal values for bafes (spacing and ratio) and the
number,length, diameter, and arrangement of tubes. Also tube pitch
ratioKeywords:
the STHX model. -NTU method and Bell-Delaware procedure are used
for thermal modeling of STHXand calculation of shell side heat
transfer coefcient and pressure drop. The purpose of STHX
optimiza-Effectiveness of evolutionary algorithmsexchangers
Rihanna Khosravi a,, Abbas Khosravi a, Saeid NahavaaCentre for
Intelligent Systems Research (CISR), Deakin University, Geelong,
VIC 3217, AbDepartment of Mechanical Engineering, Vali-e-Asr
University of Rafsanjan, Rafsanjan,
a r t i c l e i n f o
Article history:Received 26 June 2014Accepted 12 September
2014Available online 17 October 2014
a b s t r a c t
This paper comprehensivelof shell and tube heat exsearch (CS)
method are im
Energy Conversio
journal homepage: www.er optimization of heat
i a, Hassan Hajabdollahi b
alia
vestigates performance of evolutionary algorithms for design
optimizationgers (STHX). Genetic algorithm (GA), rey algorithm
(FA), and cuckoomented for nding the optimal values for seven key
design variables of
le at ScienceDirect
and Management
vier .com/ locate /enconman
-
method by process engineering researchers for design
optimiza-tion of heat exchangers [814]. Several optimization
methods havebeen introduced in recent years that outperform genetic
algorithmin term of optimization results. Also some of these
methods areeven computationally less demanding. Examples of these
methodsare particle swarm optimization [15,16], cuckoo search [17],
impe-rialist competitive algorithm [18], bee colony optimization
[19],and rey algorithm [20]. These methods show different
perfor-mances in different engineering applications. A conceptual
com-parison of these methods for several case studies can be found
in[21]. A few of these algorithms have been recently employed
fordesign and optimization of heat exchangers [2228].
The purpose of this paper is to comprehensively compare
per-formance of the genetic algorithm, rey algorithm, and
cuckoosearch method for the design of STHXs. To the best of our
knowl-edge, this is the rst study where rey algorithm and
cuckoosearch method are employed for optimal design of STHXs.
Seven
transfer surface area (A ) is dened as,
282 R. Khosravi et al. / Energy Conversion an2. Modelling shell
and tube heat exchanger
The efciency of the TEMA E-type STHX is calculated as,
2 1 C 1 C2
q1 eNTU
1C2
p
1 eNTU1C2
p !1
1
where the heat capacity ratio (C) is calculated as,
C CminCmax
minCs;CtmaxCs;Ct
min _mcps; _mcpt
max _mcps; _mcpt 2
where subscripts s and t stand for shell and tube respectively.
Thenumber of transfer units is dened as,design variables are
considered as part of the optimization process.These are tube
arrangement, pitch ratio, diameter, length, quantity,bafe spacing
ratio, and bafe cut ratio. Optimization is purelydone for
maximizing the efciency. Cost implications of this opti-mization
approach are then analyzed and discussed. Performanceof
optimization algorithms is compared on their ability to nd
per-missible and optimal congurations. The behavior of the
sevendesign variables are also studied in detail. Simulation
experimentsare done for an approximate thermal model of a real
world STHX.
The rest of this paper is organized as follows. Section 2
brieyintroduces the STHX model used in this study. Optimization
algo-rithms investigated in this study are briey described in
Section 3.Section 4 represents simulations results. Finally,
conclusions areprovided in Section 5.Fig. 1. The layout of a STHX
with shell and tube uid ows [2].NTU Uo AtCmin
3
where Cmin is,
Cmin minCh;Cc 4where Ch and Cc are the hot and cold uid heat
capacity rates, i.e.,Ch _mcph and Cc _mcpc . _m is the uid mass ow
rate. Specicheats cp are assumed to be constant.
The overall heat transfer coefcient (Uo) in (3) is then
computedas,
Uo 1ho Ro;f do lndo=di
2kw Ri;f dodi
dohidi
15
where L;Nt;di; do;Ri;f ;Ro;f , and kw are the tube length,
number, insideand outside diameter, tube and shell side fouling
resistances andthermal conductivity of tube wall respectively. hi
and ho are heattransfer coefcients for inside and outside ows,
respectively.
The total tube outside heat transfer area is calculated as.
At p L do Nt 6where L and do are the tube length and outside
diameter.
The tube side heat transfer coefcient (hi) is calculated as,
hi 0:024 ktdi Re0:8t Pr
0:4t 7
for 2500 < Ret < 124;000. kt and Prt are tube side uid
thermalconductivity and Prandtl number respectively. The tube ow
Rey-nold number (Ret) is also dened as,
Ret mt dilt Ao;t8
where mt is the tube mass ow rate and Ao;t is the tube side
owcross section area per pass,
Ao;t 0:25pd2iNtnp
9
where np is the number of passes.The average shell side heat
transfer coefcient is calculated
using the BellDelaware method correlation,
hs hk Jc Jl Jb Js Jr 10where hk is the heat transfer coefcient
for an ideal tube bank,
hk ji cp;s_msAs
ks
cp;sls
23 lsls;w
!0:1411
where ji is the Colburn j-factor for an ideal tube bank. As is
also thecross ow area at the centerline of the shell for one cross
owbetween two bafes. lsls;w is the viscosity ratio at bulk to wall
temper-
ature in the shell side. Jc; Jl; Jb; Js, and Jr in (10) are the
correction fac-tors for bafe conguration (cut and spacing), bafe
leakage, bundleand pass partition bypass streams, bigger bafe
spacing at the shellinlet and outlet sections, and the adverse
temperature gradient inlaminar ows.
The STHX total cost is made up of capital investment (Cinv )
andoperating (Copr) costs [1],
Ctotal Cinv Copr 12There are several methods for determining the
price of STHX.
Here we use the Halls method for estimation of the
investmentcost as detailed in [29] (alternative cost estimation
methods canbe found in [30]). Cinv as a function of the total tube
outside heat
d Management 89 (2015) 281288t
Cinv 8500 409 A0:85t 13
-
and their evolutions through generations. GA generates
candidate
GA is stochastic and gradient-free, so it can be easily applied
forminimization or maximization of discontinuous and
nondifferen-tiable objective functions. Theoretical literature of
GA is quite richand numerous applications of GA for real world
optimization prob-lems have been reported in the last two decades.
Detailed discus-sion about GA and its operators can be found in
basic readingsources such as [3335].
3.2. Firey algorithm
n and Management 89 (2015) 281288 283solutions from the space of
all possible solutions and examinestheir performance as per the
considered objective function. It hasbeen proven that GA performs
strongly well in both constrainedand unconstrained search problems
where the number of goodsolutions is very limited compared to the
size of the search space.
GA converges towards more competitive solutions by
applyingelitism, crossover, and mutation mechanisms. GA rst creates
apopulation (often randomly) of potential solutions (also
calledchromosomes) for the optimization problem. This population
isthen assessed using the objective function of the interest.
ThenGA uses its three operators to create the new population for
thenext generation. The best performing chromosome(s) is copied
tothe next generation unchanged. This process is called elitism
andmakes sure that the best solution(s) is not lost as the
optimizationproceeds.
Crossover operator is used for combing good parents and
gener-ating offspring. This operator is applied with the hope of
retainingthe spirit of good chromosomes. In its simplest form,
i.e., singlepoint, a random point (crossover point) is randomly
selected. Thenthe operator swaps portions of a pair chromosomes at
the cross-over point. Alternative crossover methods are
multi-points, uni-form, and arithmetic. Regardless of the type of
applied crossoveroperator, its generated offspring only include
information held bythe current population. A new operator is
required to introduceand bring new information (solutions) to the
population. Mutationoperator creates a new offspring by randomly
changing the valueswhere the construction materials are carbon and
stainless steel.The total discounted operating cost associated to
pumping
power is computed as follows [1],
Copr XNyk1
C01 ik
14
where i and Ny are the annual discount rate (%) and the STHX
lifetime in year. C0 is the annual operating cost and is calculated
asfollows,
C0 je P hopt 15where je and hopt are the price of electricity
($/kW h) and annualoperating hours. The pumping power (P) is also
calculated in watts(W),
P 1g
mtqtDpt
msqsDps
16
where g is pump efciency. qs and qt are uid density shell
andtube side respectively. Dps and Dpt are also total pressure drop
inshellside and tubeside, respectively.
More details about the STHX model used in this study can befound
in [4]. Calculations of shell and tube side heat transfer
coef-cients as well as pressure drops can be found in basic
heatexchanger design books [31,32].
3. Optimization algorithms
3.1. Genetic algorithm
Genetic algorithm (GA) is highly likely the most widely usedand
researched evolutionary optimization method in the scienticworld.
It is a guided stochastic search technique inspired from
theprinciples of natural ttest selection and population genetics.
Ingeneral terms, it is based on the parent and offspring
iterations
R. Khosravi et al. / Energy Conversioof genes at one or more
positions of a selected chromosome. Thepseudo code for GA including
three genetic operators is displayedin Fig. 2.Similar to
evolutionary optimization methods, rey algorithm(FA) is an
approximate rather than complete optimization algo-rithm. In the
family of approximate methods, the guarantee ofnding optimal and
perfect solutions is compromised for the sakeof obtaining
reasonably good solutions in a fraction of time andeffort required
by complete algorithms [36]. FA was originallydeveloped and
engineered by Prof. Yang in late 2007 and 2008 atCambridge
University [20]. The algorithm is inspired by the ash-ing behavior
and movement of reies. The method assumes thatthe attractiveness
between two reies is proportional to theirbrightness and the less
brighter one will move towards thebrighter one. Movement will be
random if there is no brighter adja-cent rey.
As attractiveness is proportional to the light intensity, the
vari-ation of attractiveness b with the distance r can be dened
as,
b b0 ec r2 17
where b0 is attractiveness at r 0. c is also the medium
lightabsorption coefcient. The distance between any two reies iand
j at spatial coordinates xi and xj is the Cartesian distance
calcu-lated as,
r kxi xjk Xd
k1xi;k xj;k2
r18
where xi;k is the kth component of the coordinate xi of ith rey.
Innormal 2D space, (18) is as follows,
r xi xj2 yi yj2
q19
Assuming the jth rey is brighter than ith rey, the move-ment of
xi towards xj is dened as,
xi xi b0 ec r2i;j xi xj a i 20
where the second and the third term in right are due to the
attrac-tion and randomization. a is a parameter multiplied in the
vector ofrandom numbers i. This vector is generated through drawing
num-bers from a normal or uniform distribution. As mentioned in
[20],often b0 1 and a 2 0;1 satisfy most of FA implementations.
Notethat (20) is a pure random walk search if b 0. Also other
distribu-tions such as Levy ights can be considered for the
randomizationterms () in (20).Fig. 2. Pseudo code for GA.
-
c is the key parameter of FA. It characterizes the variation of
theattractiveness between different reies. Its value has a
directimpact on the convergence speed of the algorithm and how
thespatial coordinates of reies change. While in theory c can
takeany value in 0;1, it is usually set to a value in 0;10.
In this paper, we use a modied version of FA algorithm
intro-duced in [37]. Two proposed modications aim to minimize
thechance of algorithm being trapped in local optima and to
eliminatethe effects of initialization process on the algorithm
performance.
3.3. Cuckoo search
The CS method is a nature-inspired metaheuristic
optimizationmethod which was proposed by Yand and Deb in 2009 [17].
The
tance between the current and best solution is applied as a
transi-
284 R. Khosravi et al. / Energy Conversion anreproduction
strategy of cuckoos is the core idea behind the CSmethod. The CS
method has been developed based on three ideal-ized assumptions:
(i) each cuckoo lays one egg at a time anddeposits it at a random
chosen nest, (ii) the best nests with thehighest quality eggs are
carried to the next generations, and (iii)the number of host nests
for depositing eggs are xed. Eggs laidby a cuckoo are discovered by
the host bird with a pre-set fractionprobability, pa 2 0;1. In case
of discovering alien eggs, the hostbird may simply through away
them or abandon the nest and builda completely new one.
In terms of optimization implementation, eggs in nests
repre-sent solutions. The idea is to replace not-so-good solutions
in thenests with new and potentially better solutions. Based on the
threeidealized assumption, Fig. 3 shows the pseudo code for
implemen-tation of the CS method. The method applies two
explorationmethods. Some solutions are generated in the
neighborhood ofthe current best solution (a Lvy walk). This speeds
up the localsearch. At the same time, a major fraction of new
solutions are gen-erated by far eld randomization and whose
locations are far awayfrom the current best solution location. This
is done to make surethe method is not trapped in a local optimum.
Fig. 3 presents thepseudo code for CS method including Lvy ights.
Note that CSmethod is in general population-based, elitist, and
single objective.
A Levy ight is considered when generating new solutions xt1
for the ith cuckoo,
xt1i xti a Levy 21
where a is the step size which depends on the scales of the
problemof interest. Often, a OL=100 satises the search requirements
formost optimization problems. L represents the difference
betweenthe maximum and minimum valid value of the problem of
interest.The product means entry-wise multiplication.Fig. 3. Pseudo
code for CS method including Lvy ights.tion probability to move
from the current location to the nextlocation. As per this, (21)
can be rewritten as,
xt1i xti as xti xbesti r 26
where xbesti is the current best solution and r is a random
numberdrawn from a normal distribution with zero mean and unit
vari-ance. The step length s is also calculated using (22). Further
discus-sion about CS method and its details can be found in
[20,17].
4. Simulation results
This section describes the simulation results for optimizing
thedesign of STHX using GA, FA, and CS method. STHX model used
insimulations is identical to one described and analyzed in
[4].Table 1 summarizes the list of decision variables (STHX
parame-ters) and their range. It is important to note that all
these 7 vari-ables are discontinuous due to practical construction
constraints.For instance, tube internal diameter is determined
according to rel-evant standards and suppliers catalogs.
For the three optimization methods, we set the number of
iter-ations (generations) to 30 and 60. The population size is also
set to10, 20, 30, and 50. Accordingly, 8 different sets of
experiments areperformed for each optimization method (combination
of differentpopulation sizes and iteration numbers). Each
experiment (e.g., GAwith 30 iterations and 10 populations) is
repeated 50 times andthen statistics of experiments are reported.
In total, 400 runs aresimulated and completed for each optimization
method. This isdone to make sure conclusions are made based on
general andextensive optimization scenarios rather than a few
tailored ones.Therefore, obtained results and driven conclusions
are statisticallymeaningful and believable. Simulations are
performed using aLenovo Thinkpad T420s laptop computer with Intel
Core i7-2640 M CPU @2.8G Hz and 8 GB memory, running Windows
7Professional.
The purpose of optimization is to maximize the efciencyThe Lvy
ight provides a random walk where its step is drawnfrom a Lvy
distribution. There are several ways to generate thisrandom step
[20]. The Mantegnas algorithm is one of the mostefcient algorithms
for generating symmetric (positive or nega-tive) Lvy distributed
steps. In this method, the step length in(21) is calculated as
s ujv j1=b
22
where u and v are drawn from normal distributions,
u N0;ru; v N0;rv 23where rv 1 and,
ru C1 b sinpb=2C1 b=2b2b1=2
( )1b
24
where Cz is the gamma function,
Cz Z 10
tz1 et dt 25
Two normal distributions are used by Mantegnas algorithm
togenerate a third random variable which has the same behavior of
aLvy distribution. In the CS method proposed by Yang and Deb[20],
the entry-wise multiplication of the random number and dis-
d Management 89 (2015) 281288through nding the best values for
seven design parameters listedin Table 1. For each run, the seven
decision variables are randomlyinitialized within their range (see
Table 1).
-
Fig. 4 shows the efciency of the optimized STHX using GA, FAand
CS method for 50 runs. According to these results, FA and CSshow a
much more consistent behavior in terms of maximizingthe efciency of
the heat exchanger. The maximum efciency ()is 83.80%. The efciency
of CS optimized heat exchanger (CS) isequal to this value almost in
all 400 simulations. There is onlyone case (#iter = 30, #pop = 10)
where CS is less than 80%. FA alsoshows a similar performance
although there are 5 out of 400 caseswhere FA cannot nd an
admissible solution. The efciency of GA-optimized heat exchangers
(GA) is equal to 83.80% in only a fewcases out of 400 simulations.
More interestingly, GA is less than80% in more than 75% of
simulations. GA cannot nd admissiblesolutions in 284 out of 400
simulations (71%). This indicates theinability of the GA operators
in nding permissible solutionswithin the search space. As per
demonstrated results in Fig. 4, thisis not something to be easily
solved by simply increasing the num-ber of iterations or the
population size. GA performance is highly
tion. It can nd globally optimal solutions if the initial
valuesselected for seven design parameters are proper (at least
being
Table 1The list of design variables (STHX parameters) and their
range.
Variable Minimum Maximum Increment Number ofsolutions
Tube arrangement (30, 45,90)
3
Tube insidediameter (m)
0.0112 0.0153 20 (as per standardtubes)
pt/do 1.25 3 0.001 1750Tube length (m) 3 8 0.001 5000Tube number
100 600 1 500Bafe cut ratio 0.19 0.32 0.001 130Bafe spacing ratio
0.2 1.4 0.001 1200
0 5 10 15 20 25 30 35 40 45 500
50
Effic
ienc
y (%GA
FACS
0 5 10 15 20 25 30 35 40 45 500
50
100
Effic
ienc
y (%
)
#iter=30, #pop=50
#iter=60, #pop=20
Fig. 5. The convergence behavior of FA for maximizing
efciency.
Table 2The mean of computation time for each optimization
run.
Simulation GA FA CS
#iter = 30, #pop = 10 0.90 3.77 0.97#iter = 30, #pop = 20 1.03
9.79 1.01#iter = 30, #pop = 30 1.53 21.39 1.49#iter = 30, #pop = 50
2.53 61.39 2.53#iter = 60, #pop = 10 0.55 4.97 1.01#iter = 60, #pop
= 20 1.01 19.15 1.97#iter = 60, #pop = 30 1.56 43.73 2.97#iter =
60, #pop = 50 2.54 122.94 5.02
R. Khosravi et al. / Energy Conversion and Management 89 (2015)
281288 285dependent on the initialization process for STHX design
optimiza-
0 5 10 15 20 25 30 35 40 45 500
50
100
Effic
ienc
y (%
)
#iter=30, #pop=10
0 5 10 15 20 25 30 35 40 45 500
50
100
Effic
ienc
y (%
)
#iter=30, #pop=30
#iter=60, #pop=100 5 10 15 20 25 30 35 40 45 500
50
100
Effic
ienc
y (%
)
0 5 10 15 20 25 30 35 40 45 500
50
100
Replicate
Effic
ienc
y (%
)
#iter=60, #pop=30
Fig. 4. STHX efciency optimization usingadmissible). Otherwise
it fails to nd generate optimal solutions
100)
#iter=30, #pop=200 5 10 15 20 25 30 35 40 45 500
50
100
Effic
ienc
y (%
)
0 5 10 15 20 25 30 35 40 45 500
50
100
Replicate
Effic
ienc
y (%
)
#iter=60, #pop=50
GA, FA, and CS method for 50 runs.
-
an82 82.5 83 83.5 8415,000
20,000
25,000
30,000
35,000
40,000
45,000
50,000
Efficiency (%)
Tota
l Cos
t ($)
286 R. Khosravi et al. / Energy Conversionusing its two
operators (crossover and mutation). In contrast, bothFA and CS
method always nd permissible solutions and maximizethe efciency
through appropriate exploration of the search space.They both
generate best results even with a small number of iter-ations and
populations (top plots in Fig. 4).
Fig. 5 displays the prole of efciency as the objective
functionalong optimization iterations. Here the optimal solution is
found inthe eighth generation. There is no need to continue
optimizationafter this. Similar patterns are also observed in other
runs of FA.Therefore, the effective and efcient required time for
FA is around2.6 s. Also note that all these computations and
optimizations aredone ofine. Therefore, computational burden is the
least impor-tant thing for the optimal design of STHX.
The computational cost of GA, FA, and CS method are also
com-pared in this section. Comparison is made based on the time
Fig. 6. The scatter plot of efciency and dollar cost
0 10 20 30 40 5020
40
60
80
100
Tube
Arra
ngem
ent
0 10 20 30 40 50
0
1000
2000
Pitc
h R
atio
0 10 20 30 40 50
0
200
400
600
Tube
Num
ber
Replicate
0 10 20
0
500
1000
1500
Cut
Rat
io
Rep
Fig. 7. Optimal values of STHX design para82 82.5 83 83.5
8415,000
20,000
25,000
30,000
35,000
40,000
45,000
50,000
Efficiency (%)
Tota
l Cos
t ($)
d Management 89 (2015) 281288required to nalize one optimization
run and return the optimizeddesign parameters. The mean values of
elapsed time for runningGA, FA, and CS method are shown in Table 2
for 8 experiments.Optimization times increase as the number of
iterations and pop-ulation size increase. GA and CS have almost the
same computa-tional burden. However, FA is much more demanding in
thisrespect. This is in particular more evident for simulations
with alarger population size (e.g., 30 and 50). For these cases,
tFA tGAand tFA tCS. However, we should note that GA is not able to
ndadmissible solutions in the majority of simulations. According
toall these, CS is the best in terms of global and fast
optimizationof STHX considering random initialization.
As the performance of GA for optimal design of STHX is
inferior,we hereafter just report the optimization results for the
FA and CSmethods. It is important to note that the week performance
of the
for solutions found by FA (left) and CS (right).
0 10 20 30 40 50
0
10
20
Tube
Dia
met
er
0 10 20 30 40 500
2000
4000
Leng
th
0 10 20 30 40 50
0
50
100
150
Spac
ing
Rat
io
Replicate
30 40 50
licate
meters in 50 runs of FA optimization.
-
100ent
20
Rep
R. Khosravi et al. / Energy Conversion an0 10 20 30 40 5020
40
60
80
Tube
Arra
ngem
0 10 20 30 40 50
0
1000
2000
Pitc
h R
atio
0 10 20 30 40 50
0
200
400
600
Tube
Num
ber
Replicate
0 10
0
500
1000
1500
Cut
Rat
ioGA is not something to be rectied purely by increasing the
num-ber of optimization generations or the population size. Even if
theperformance is improved, the computational burden for
ndingglobally optimal solutions will be massive.1
Fig. 6 displays the scatter plot of efciency and dollar cost
forSTHX optimized using FA (left) and CS (right) methods.
Theseresults are from the eighth experiment (#iter = 50, #pop =
60). Itis easy to see that while efciency is almost the same for in
themajority of experiments (83.80%), there is a huge difference
interms of the dollar cost. The total cost for the majority of
solutionsfound by FA and CS methods is around $45,000. Also the
plotclearly shows that the total cost increases as the
efciencyincreases. This is consistent with ndings in [4]. As per
results inthis gure, designs identical in terms of efciency can
have com-pletely different total costs.
The optimal values for seven design variables obtained using
FAoptimization are shown in Fig. 7. These are plotted for fty runs
ofFA simulation (#8) to see how their values change from one
simu-lation to another. The followings are observed:
The optimal values for tube arrangement are 30 and 90.
Theinteresting point is that 45 arrangement is not returned as
asolution for maximizing the efciency.
Tube diameter and pitch ratio often take a value between
thereminimum and median in 50 runs. This tendency is in
particularmore obvious for the pitch ratio.
Tube lengths between 3 m and 8 m are returned in
differentoptimization runs. However, there is a tendency towards
smal-ler values.
Fig. 8. Optimal values of STHX design para
1 Note that this does not mean that GA is not a suitable tool
for STHX designoptimization. GA can generate optimal results if
initialization is performed properly(admissible values are rst
picked and assigned to design parameters).0 10 20 30 40 50
0
10
20
Tube
Dia
met
er
0 10 20 30 40 500
2000
4000
Leng
th
0 10 20 30 40 50
0
50
100
150
Spac
ing
Rat
io
Replicate
30 40 50
licate
d Management 89 (2015) 281288 287 In contrast to the tube
length, the number of tube is oftenreturned close to the upper
bound (600). From a practical pointof view this makes sense. The
effect of short tube length is com-pensated by increasing the
number of tubes.
There is no obvious pattern in the bafe spacing ratio in the
50runs of the optimization process.
Bafe cut ratio is set to its minimum value in 42 out of 50
opti-mization runs. This is a strong indication of the optimality
of theminimum values of bafe cut ratios for optimal design of
heatexchangers.
Now, we look at the same experiment and results obtainedusing CS
method (see Fig. 8):
The optimal values for tube arrangement are 45 and 90.
Incontrast to FA method, 30 arrangement is not selected by
CSmethod.
Often middle values are returned for the tube diameter. Thepitch
ratio has lower value tendency. These patterns are similarto those
found by FA method.
There is no clear preference for the tube length. CS method
always picks the maximum tube number is the opti-mal value.
Similar to FA method, there is no consistent pattern for the
baf-e spacing ratio in the 50 runs of the CS method.
Bafe cut ratio is always set to its minimum value (similar to
FAresults).
According to these ndings, we may conclude that the tubenumber
is positively correlated with the STHX efciency. Thegreater the
number of tubes, the greater the efciency. Also, thecorrelation
coefcient between the bafe cut ratio and efciencyis negative. So,
it is reasonable to select the smallest allowable baf-
meters in 50 runs of CS optimization.
-
e cut ration to obtain maximum efciency. Selection of
middlevalues for tube diameter is the best in terms of efciency.
The pitchratio also should be set to values less than the median
value. Thesendings can smartly be used by engineers as rules of
thumb foroptimal design of STHXs. The design can then be revised as
per pro-ject requirements.
5. Conclusion
The optimization performances of genetic algorithm,
reyalgorithm, and cuckoo search method are comprehensively
exam-
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288 R. Khosravi et al. / Energy Conversion and Management 89
(2015) 281288ined for the design of shell and tube heat exchangers.
It is foundthat genetic algorithm cannot nd permissible design
congura-tions in the majority of simulation replicates. In
contrast, reyalgorithm nds permissible and optimal values for seven
designvariables regardless of search starting point. It is also
observed thatthere are several design congurations for STHX with
identical ef-ciency. However, these designs have greatly different
dollar costimplications. Different patterns are found for seven
design vari-ables in pure efciency-based design and optimization of
STHX.While the values of the bafe spacing ratio signicantly differ
fromone replicate to another, others such as the length, the number
oftubes, and the bafe cut ratio demonstrate consistent
patterns.These ndings can be used by STHX design engineers and
expertsto signicantly shorten the optimal design process.
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Effectiveness of evolutionary algorithms for optimization of
heat exchangers1 Introduction2 Modelling shell and tube heat
exchanger3 Optimization algorithms3.1 Genetic algorithm3.2 Firefly
algorithm3.3 Cuckoo search
4 Simulation results5 ConclusionReferences
