Thermal Science and Engineering Progress 36 (2022) 101520

Contents lists available at ScienceDirect

Thermal Science and Engineering Progress

journal homepage: www.sciencedirect.com/journal/thermal-science-and-engineering-progress

Design optimization of shell-and-tube heat exchanger using Rao algorithms

and their variants
Ravipudi Venkata Rao *, Meet Majethia
Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Surat 395007, Gujarat, India

A R T I C L E I N F O A B S T R A C T

Keywords: Shell-and-tube heat exchangers (STHEs) are widely used type of heat exchangers in the industrial applications.
Shell-and-tube heat exchanger Due to the high usage, it is essential to design the STHEs with minimum cost possible. The traditional method for
Multi-objective optimization achieving minimum cost involved manual calculations and re-iterating the same by changing few parameters
Bell-Delaware method
until obtained the minimum solution. However, this method would require lots of efforts and time but ultimately
Kern’s method
Rao algorithms
it would still not guarantee the optimum solution. In this work, the design of STHE is obtained with the help of
Rao Algorithms and their variants, which would bring the best solutions for the problem with a simple approach.
Rao algorithms collectively include Rao-1 algorithm, Rao-2 algorithm and Rao-3 algorithm. The salient feature of
these algorithms is that they do not require any algorithm-specific parameters and hence without working on the
algorithm for finding its best suitable parameters, one can immediately apply the algorithm easily. These al­
gorithms use the best and worst solutions for finding the optimal solution in the search space with the help of
random interactions between the alternative solutions. The self-adaptive multi-population Rao (SAMP Rao) al­
gorithms are the variants of Rao algorithms, and these algorithms adapt the population size automatically based
on the fitness value of the objective function, thus eliminating the tuning of population size. In this study, four
different case studies have been taken into consideration and the design optimization of these four shell-and-tube
heat exchangers for minimum cost is performed using Rao and SAMP Rao algorithms. The cost includes the fixed
and operating costs and is calculated from the pumping power. Discrete variables are also included in the study
to have the advantage of these in real-life applications. These studies were previously carried out using seven
different optimization algorithms and the results obtained now by Rao Algorithms and SAMP Rao algorithms are
compared with them. The results showed a significant improvement in minimizing the cost of STHE. For multi-
objective optimization, total cost and effectiveness are targeted and their solutions are obtained in the form of
Pareto front. It is found that the Rao algorithms and the SAMP Rao algorithms have provided improved results
compared to the other optimization algorithms.

cylinder called shell. One fluid is passed through the number of tubes,
and another through the shell. There are obstructing panels called baf­
Introduction fles attached to the shell to direct the flow of the shell fluid so as there is
more interaction between the two fluids and hence more efficiency is
A heat exchanger is a device used for transferring the heat energy achieved. Fig. 1 shows the basic shell-and-tube heat exchanger.
from one fluid at a higher temperature to another one at a lower tem­ The traditional design approach for heat exchangers involves rating
perature. The exchange is obtained via the solid surface between the many different exchanger geometries to identify those that satisfy a
fluids. Shell and tube heat exchangers are one such heat exchangers given heat duty and a set of geometric and operational constraints [2].
which finds their applications widely in the industries such as power However, this approach is time-consuming and does not assure an
generation, heating, ventilation, and air-conditioning (HVAC), marine optimal solution. Therefore, advanced optimization techniques are
applications, pharmaceuticals, metals and mining. Normally, these are preferred for the design of STHEs. Several advanced optimization al­
used where the heat exchange is between either two liquids or between gorithms have been developed so far for the optimum design of the
one liquid and one gas. Shell-and-tube heat exchanger (STHE) mainly STHEs in conjunction with Kern’s method or Bell-Delaware method. The
consists of a number of small tubes enclosed in a big capsule shaped

* Corresponding author.
E-mail address: rvr@med.svnit.ac.in (R. Venkata Rao).

https://doi.org/10.1016/j.tsep.2022.101520
Received 12 July 2022; Received in revised form 9 October 2022; Accepted 14 October 2022
Available online 22 October 2022
2451-9049/© 2022 Elsevier Ltd. All rights reserved.
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Nomenclature streams
Jl correction factor for tube-to-baffle leakages and baffle-to-
Symbol Description, unit shell leakages
Afrt window area occupied by tubes, m2 Jr correction factor for any adverse temperature gradient
Afrw gross (total) window area, m2 buildup in laminar flows
Aobp flow bypass area of one baffle, m2 Js correction factor for larger baffle spacing at the inlet and
Aocr flow area at or near the shell centerline for one crossflow outlet sections compared to the central baffle spacing
section in a shell-and tube exchanger, m2 K1 constant for determining number of tubes
Aosb shell-to-baffle leakage flow area, m2 Lbc baffle spacing at center, m
Aotb tube-to-baffle leakage flow area, m2 Lbi baffle spacing at inlet
Aow flow area through window zone, m2 Lbo baffle spacing at outlet
Cop operating cost, $ Lcalc calculated length of tube, m
Cph specific heat capacity of hot fluid, kJ/kgK Lassumed assumed length of tube, m
Cpt specific heat capacity of tube side fluid, kJ/kgK Li ratio of baffle spacing at inlet to baffle spacing at center
CE equipment cost, $ Lo ratio of baffle spacing at outlet to baffle spacing at center
Cfix fixed cost, $ NPass number of tube passes
Ci setup cost, $ Nb number of baffles
Cm cost correction factor for construction material Nrc Sum of Nrcc and Nrcw
Cp cost correction factor for operating pressure Nrcc The number of tube rows crossed during flow through one
Ct constant for number of tubes, depends on tube layout. crossflow section between baffle tips
CT cost correction factor for operating temperature Nrcw number of effective tube rows in crossflow in the window
Ctot total annual cost, $ section
Dctl diameter of the circle through the centers of the outermost Nss 1/Nrcc
tubes, m Nt number of tubes
De equivalent shell diameter, m Ntw the number of tubes in the window section
Dotl diameter of the outer tube limit, m Ppump pumping power, kW
Ds shell inside diameter, m Prt Prandtl number of tube side fluid
Fc fraction of the total number of tubes in the crossflow Pt tube pitch, m
section Res Reynold’s number of shell side fluid
Fw ratio of number of tubes in one window section to the area Ret Reynold’s number of tube side fluid
of the outer tube row Rfs shell side conductive fouling resistance
Gc cross-flow mass velocity Rft tube side conductive fouling resistance
TLMTD log mean temperature difference, ◦ C Th1 inlet temperature of hot fluid, ◦ C
Xl longitudinal tube pitch, m Th2 outlet temperature of hot fluid, ◦ C
Xt transverse tube pitch, m ηpump pumping efficiency
b1 − b4 constants for calculating pressure θb angle between two radii intersected at the inside shell wall
di tube inside diameter, m with the baffle cut, radians
f id ideal Fanning friction factor per tube row θctl angle between the baffle cut and two radii of a circle
fs friction factor at shell side through the centers of the outermost tubes, radians
ft friction factor at tube side μs viscosity of shell side fluid, Pa-s
hid Heat transfer coefficient for pure crossflow stream, W/m2K μt viscosity of tube side fluid, Pa-s
hs shell side heat transfer coefficient, W/m2K ρs density of shell side fluid, kg/m3
ht tube side heat transfer coefficient, W/m2K ρ density of tube side fluid, kg/m3
kt thermal conductivity of tube side fluid, W/mK ΔPbid ideal pressure drop in the central section, Pa
lc baffle cut, distance from the baffle tip to the shell inside ΔPs pressure drop at shell side, Pa
diameter, m ΔPt pressure drop at tube side, Pa
ṁs mass flow rate of fluid at shell side, kg/s ΔPwid ideal pressure drop in window section, Pa
ṁt mass flow rate of fluid at tube side, kg/s a constant power for calculating Colburn factor j, depends on
n1 constant for determining number of tubes a3, a4 and Res
tc1 inlet temperature for cold fluid, ◦ C a1 - a4 Colburn factor constants
tc2 outlet temperature for cold fluid, ◦ C j Colburn factor
vs shell side fluid velocity, m/s r rate of interest
vt tube side fluid velocity, m/s A heat transfer surface area, m2
δsb diametrical clearance of shell-to-baffle, m F temperature difference corrective factor
δtb tube-to-baffle hole diametral clearance, m Q heat transfer, W
ζb correction factor for bypass flow TL Tube layout [30◦ , 45◦ , 60◦ , 90◦ ]
ζl correction factor for tube-to-baffle and baffle-to-shell U overall heat transfer coefficient
leakage b constant power for calculating pressure drop, depends on
ζs correction factor for inlet and outlet sections having b3, b4 and Res
different baffle spacing from that of the central section d tube outer diameter, m
w width of pass divider lane f Friction factor
ρ tube side fluid density, kg/m3 ṁh mass flow rate of hot fluid, kg/s
Jc correction factor for baffle configuration
Jb correction factor for bundle and pass partition bypass

2
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

advanced optimization algorithms combined with the Kern’s or Bell- Delaware method. The total cost of the STHE including capital invest­
Delaware method are expected to yield optimal design of STHEs. ment and the sum of discounted annual energy expenditures was
Wildi-Tremblay and Gosselin [3] minimized the cost of an STHE considered as the objective for minimization. Tube length, tube outer
based on genetic algorithm (GA) and Bell-Delaware method. The diameter, pitch size and baffle spacing were considered as the design
objective was to minimize the total cost including the initial cost and the variables. It was reported that the ICA algorithm can also be successfully
operating cost. Eleven design variables such as tube pitch, tube layout applied for optimal design of STHE with higher accuracy involving less
patterns, number of tube passes, baffle spacing at the centre, baffle computational time.
spacing at the inlet and outlet, baffle cut, tube-to-baffle diametrical Asadi et al. [11] minimized the total annual costs of STHE by a
clearance, shell-to-baffle diametrical clearance, tube bundle outer cuckoo search (CS) algorithm. The performance of the CS algorithm was
diameter, shell diameter, and tube outer diameter were considered for compared with PSO and GA. Khosravi et al. [12] used GA, firefly algo­
optimization. Caputo et al. [4] applied the GA for economic optimiza­ rithm (FFA), and cuckoo search algorithm (CSA) for finding the optimal
tion of the STHE design. Three case studies were used to analyze the values for seven key design variables of the STHE. ∊ -NTU method and
performance of the GA in terms of cost. The results showed that the GA Bell-Delaware method were used for thermal modeling of STHE and
method significantly reduced the cost. calculation of shell side heat transfer coefficient and pressure drop.
Ponce-Ortega et al. [5] used Bell-Delaware method for the descrip­ Seven design variables considered were: tube arrangement, pitch ratio,
tion of the shell-side flow with no simplifications. The design variables diameter, length, quantity, baffle spacing ratio, and baffle cut ratio. The
included were the number of tube-passes, standard internal and external objective was to maximize the thermal efficiency. The CSA and FFA were
tube diameters, tube layout and pitch, type of head, fluids allocation, reported to perform better.
number of sealing strips, inlet and outlet baffle spacing, and shell-side Sadeghzadeh et al. [13] used GA and PSO algorithms for design
and tube-side pressure drops. GA was then used for obtaining the optimization of an STHE. A cost function including costs of the heat
optimal design of STHE. Guo et al. [6] developed an approach in which exchanger based on surface area and power consumption to overcome
the dimensionless entropy generation rate obtained by scaling the en­ pressure drops was treated as the objective function for minimization.
tropy generation on the ratio of the heat transfer rate to the inlet tem­ The tube diameter, central baffles spacing and shell diameter were
perature of cold fluid was considered as the objective function. The considered as the design variables and Bell-Delaware method was used
geometrical parameters of the STHE were taken as the design variables to calculate the heat transfer coefficient and the shell-side pressure drop.
and the Bell-Delaware method in conjunction with GA was applied to It was reported that the Bell-Delaware method accompanied by GA and
solve the optimization problem. PSO algorithms achieved optimal results. Hajabdollahi et al. [14]
Patel and Rao [7] explored the use of particle swarm optimization studied shell and tube and gasket-plate heat exchangers from economic
(PSO) algorithm for design optimization of STHE from economic view­ viewpoint. Seven design variables were used for gasket-plate while nine
point. Minimization of total annual cost was considered as the objective design variables were selected for STHE. Bell-Delaware method was
function with shell internal diameter, outer tube diameter and baffle used accompanied by GA to find the minimum cost.
spacing as the design variables. The results of PSO were reported to be Mohanty [15] used Kern method accompanied by firefly algorithm
better than that of GA. Sahin et al. [8] used artificial bee colony (ABC) (FFA) for design optimization of an STHE considering total annual cost
algorithm and Kern method for the economic optimization of the STHE as the objective function. It was reported that the FFA algorithm per­
design. The tube length, baffle spacing, shell diameter, and tube diam­ formed better than GA, PSO, BBO, CSA, and ICA. Tharakeshwar et al.
eter were considered as the design variables. The three case studies were [16] attempted to optimize the effectiveness and total cost. The two
used to analyze the performance of ABC algorithm. objective functions were plotted by considering various design param­
Hadidi and Nazari [9] developed an approach for design optimiza­ eters such as baffle cuts, baffle spacings, pitch, tube length, and tube
tion of STHE using biogeography-based optimization (BBO) algorithm layout pattern to get Pareto optimal solutions. This multi-objective
and Kern method to minimize the total cost of the equipment including optimization was performed using bat algorithm (BA) and GA in
capital investment and the sum of discounted annual energy expendi­ conjunction with Bell-Delaware method.
tures related to pumping of STHE. Design variables such as tube length, Rao and Saroj [17] used Bell-Delaware method accompanied by Jaya
tube outer diameter, pitch size, and baffle spacing were considered for algorithm for economic optimization of STHE design. Consistency and
optimization. Three different test cases were solved to demonstrate the maintenance because of fouling were also considered. The results were
effectiveness and accuracy of the BBO algorithm. In another work, compared with GA, PSO, and civilized swarm optimization (CSO) al­
Hadidi et al. [10] developed an approach for design optimization of gorithms. Eleven design parameters related with STHE configuration
STHE based on imperialist competitive algorithm (ICA) and Bell- were used as decision variables. Rao et al. [18] reviewed various

Fig. 1. A typical shell-and-tube heat exchanger [1].

3
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

advanced optimization algorithms used for design optimization of heat Thus, due importance needs to be considered to tune the algorithm-
exchangers including STHEs. specific parameters and population size in the process of design opti­
Segundo et al. [19] minimized the total annual cost of STHE by mization. Furthermore, these algorithms take comparatively longer time
Differential Evolution (DE) and a novel Differential Evolution variant to converge. Hence, there is a need to develop and apply new optimi­
named as Tsallis Differential Evolution (TDE). The shell internal diam­ zation algorithms which are simple and free from algorithm-specific
eter, outside tube diameter, and baffles spacing were considered as the control parameters to the design optimization of STHEs.
design variables. The results were reported better than those of GA, PSO, The motivation of the present work is to make an attempt to see if
BBO, and CSA. In their another work, Segundo et al. [20] proposed Owl any improvement is possible in the design of STHEs by using the recently
Optimization Algorithm (OOA) for multi-objective optimization of total developed Rao algorithms. Rao [27] introduced the three Rao algo­
cost and effectiveness of STHE. Dhavle et al. [21] used cohort intelli­ rithms: Rao-1, Rao-2, and Rao-3. The Rao algorithms are population-
gence (CI) algorithm in conjunction with Kern method for solving the based algorithms, simple, and easy to implement for optimization ap­
design and economic optimization of the STHEs. The associated design plications. These algorithms have no algorithm-specific parameters and
variables such as tube outside diameter, baffle spacing, pitch size, shell have no metaphorical explanation. The general control parameter, i.e.,
inside diameter and number of tube passes that decide the total cost of population size, is the only parameter that needs to be tuned once the
the heat exchanger were optimized. termination criterion is fixed. Rao and Pawar [28] developed self-
Iyer et al. [22] developed adaptive range genetic algorithm (ARGA) adaptive multi-population Rao (SAMP Rao) algorithms which adapts
by combining GA with cohort intelligence (CI) algorithm. The three case population size depending upon the solution strength, thus eliminating
studies of STHE were used to evaluate the efficiency of the ARGA. The the tuning of population size. Therefore, the implementation of Rao and
simulation results showed that the ARGA performed better than the GA, SAMP Rao algorithms for engineering applications becomes much
PSO, ABC, BBO, CI, and improved intelligent tuned harmony search (I- easier. During the iterative process, these algorithms use iteration best
ITHS) algorithms. Sai and Rao [23] used bacteria foraging optimization solution, iteration worst solution, and random interactions among the
(BFO) algorithm for increasing the efficiency and decreasing the cost of population to explore and exploit the search region. Therefore, the
STHE. The results of BFO algorithm were reported to be better compared implementation of Rao and SAMP Rao algorithms for engineering ap­
to the biogeography-based optimization (BBO) and adaptive range ge­ plications becomes much easier. The SAMP Rao algorithms do not need
netic algorithm (ARGA). Lara-Montaño et al. [24] reported the results of any special attention or importance as there are no algorithm-specific
optimization for the STHEs using PSO and GWO algorithms. A virtual parameters and population size in the process of design optimization.
environment was created using the Kern and the Bell-Delaware methods The basic objectives of this work are:
which were easily integrated with PSO and GWO algorithms. Total
annual cost of STHE was considered as the objective function. (i) to propose Rao and SAMP Rao algorithms for the single-objective
Lara-Montaño et al. [25] compared the performance of different design optimization of STHEs and comparing the same with the
optimization algorithms such as particle swarm optimization (PSO), other advanced optimization algorithms to see if any improve­
differential evolution (DE), grey wolf optimizer (GWO), teaching ment in design is possible.
learning based optimization (TLBO), cuckoo search (CS), whale opti­ (ii) To consider both discrete ad continuous design variables with
mization algorihm (WOA), and univariate marginal distribution algo­ their bounds in Kern method and Bell-Delaware method during
rithm (UMDA) for the minimization of total annual cost of the STHEs. the optimization process.
The Kern and Bell-Delaware methods were used with discrete and (iii) to check the performance of the proposed algorithms in solving
continuous diameters of tubes. The DE and GWO were reported to multi-objective design optimization problems of STHEs in terms
perform better for the four case studies considered. Caputo et al. [26] of coverage and hypervolume metrics.
presented a critical review and comparison of objective functions on a
consistent basis to show their effect on STHE configuration and assess The single- and multi-objective optimization case studies of STHEs
respective merits and demerits. It was shown that thermodynamic-based considered in the present work have shown that the proposed Rao and
objective functions might not lead to cost-effective design, and eco­ SAMP Rao algorithms are capable of producing highly competitive re­
nomic functions demonstrate their superiority. sults compared with other existing optimization algorithms.
It has been observed from the literature review on cost optimization The following section presents the design aspects of STHEs.
of STHEs [3–26], different advanced optimization algorithms such as
ABC, ARGA, BA, BBO, BFO, CI, CS, CSA, CSO, DE, FFA, GA, GWO, ICA, I- Design of STHEs
ITHS, Jaya, OOA, PSO, TDE, UMDA, and WOA. Either Kern’s method or
Bell-Delaware method was used accompanied by any of these advanced Among the various design methods found in the literature, the most
optimization algorithms. The algorithms have their merits and have popular and widely used are Kern’s and Bell-Delaware methods [1].
provided good results for design optimization of STHEs. However, These methods are used for calculating the shell side heat transfer co­
except Jaya algorithm, all other algorithms have their own algorithm- efficients and pressure drop analysis. Kern’s method was developed
specific control parameters to be tuned in addition to tuning of com­ based on experimental work on a commercial heat exchanger. It is
mon control parameter of population size. For example, GA requires the simple in calculation and gives satisfactory preliminary results for heat
tuning of crossover probability, mutation probability, selection oper­ transfer coefficients. However, in case of pressure drop analysis, Kern’s
ator, etc. The PSO algorithm requires tuning of its social and cognitive method does not take leakages into consideration. On the other hand,
parameters and the inertia weight. ABC algorithm needs number of bees Bell-Delaware method was developed at the University of Delaware for
(scout, onlooker and employed bees) and limit, BBO algorithm requires calculating the shell side heat transfer coefficient and pressure drop
immigration rate, emigration rate, etc. Similarly, the other optimization derivation. For an ideal STHE, the openings between the elements and
algorithms have their specific parameters to be tuned. The tuning is a the shell are neglected considering only a single stream of flow between
burden on the designer and finding the optimum algorithm-specific the baffles perpendicular to the tubes. While in actual design, the shell
control parameters of an algorithm to get optimal solution of the fluid is divided into multiple flows. This method uses empirical relations
given STHE design problem is itself a separate optimization problem! to calculate the friction factors and correction factors addressing these
The designer must make many trials to arrive at an optimum combina­ tube-baffle leakages and shell-baffle leakages. Many factors are intro­
tion of algorithm-specific control parameters to find an optimum solu­ duced to contain the complex flow of the fluid in the shell to obtain more
tion to the given design problem. Incorrect tuning of algorithm-specific accurate results. Due to this, Kakaç et al. [2] stated that the Bell-
control parameters may lead to sub-optimal or local optimum solution. Delaware method is the most reliable method for shell side analysis.

4
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

For space reasons, the general equations used for modeling the STHE ′ ( ) ⃒ ⃒
are given in Appendix-A. The equations related to tube side heat transfer Xj,k,i = Xj,k,i + r1j,k,i Xbest,k,i − Xworst,k,i + r2j,k,i (⃒Xj,k,i or XJ,k,i ⃒
⃒ ⃒)
coefficient, shell side heat transfer coefficient, overall heat transfer co­ − ⃒XJ,k,i or Xj,k,i ⃒ (2)
efficient, and pressure using Kern’s method and the corresponding
equations using Bell-Delaware method are given. The description of ′
Xj,k,i = Xj,k,i + r1j,k,i (Xbest,k,i
various terms used in the equations is given in the nomenclature table ⃒ ⃒) ⃒ ⃒
− ⃒Xworst,k,i ⃒ + r2 j,k,i (⃒Xj,k,i or XJ,k,i ⃒ − (XJ,k,i or Xj,k,i ) (3)
given above Appendix-A. The detailed calculations using Kern’s method
and Bell-Delaware method considering both continuous and discrete Let f be a fitness function to be minimized (or maximized). At any
design variables are also given in Appendix-A. The next section presents iteration i assuming there are ‘n’ number of populations (i.e. candidate
the details of the proposed Rao algorithms and SAMP Rao algorithms. solutions, j = 1,2,…,n). Let fbest be the most optimum value of the so­
lution obtained from the best candidate amongst the entire candidate
Rao algorithms and SAMP Rao algorithms solutions. Also let fworst be the least optimum value of the solution ob­
tained from the worst candidate amongst the entire population. Then Xj,
Rao algorithms are the new simple, metaphor-less metaheuristic al­ th th
k,i corresponds to the value of k variable for the j candidate during the
gorithms that require no algorithm specific parameters when compared th
i iteration and X’j,k,i is its new value obtained after the algorithm is
to other algorithms like GA, PSO, CS, and GWO. Thus, the designer’s applied. Therefore Xbest,k,i is the value of best candidate for the variable k
burden of fine-tuning the algorithm for best results is eliminated [27]. during the iteration i and Xworst,k,i is the value of the worst candidate for
The three variants of Rao algorithm: Rao-1, Rao-2 and Rao-3 use the the variable k during the iteration i. r1,j,k,i and r2,j,k,i are the two random
following equations (1), (2), and (3) respectively. values in the range [0,1] for the jth candidate of kth variable during ith
iteration.
(1)
′
Xj,k,i = Xj,k,i + r1j,k,i (Xbest,k,i − Xworst,k,i )
The term Xj,k,i or XJ,k,i in the equations (2) and (3) denote that any
randomly picked Jth candidate solution is being compared with the jth
candidate solution. This interaction is done on the basis of the fitness
value of the candidates. If the fitness value of jth candidate solution is

Fig. 2. Flowchart of Rao algorithms.

5
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

better than the fitness value of Jth candidate solution then the term Xj,k,i Apart from the above defined Rao algorithms, the modified versions
or XJ,k,i becomes Xj,k,i and the term XJ,k,i or Xj,k,i becomes XJ,k,i. Similarly, are used named self-adaptive multi-population Rao (SAMP Rao) algo­
If the fitness value of Jth candidate solution is better than the fitness rithms. Multi-population based algorithms are advanced algorithms that
value of jth candidate solution then the term Xj,k,i or XJ,k,i becomes XJ,k,i are based on the splitting the population into different groups for
and the term XJ,k,i or Xj,k,i becomes Xj,k,i. In simple terms for under­ searching the entire space and to detect the problem changes effectively
standing for a particular iteration, the equations above can be written as, as a result, the diversity of search is improved [28]. This is to change the
exploration and exploitation nature of the algorithm during the opti­
xnew = xold + r1 (xbest − xworst ) (4)
mization. After the initial population generation in the Rao algorithms,
the entire population is divided into m sub-populations on the basis of
xnew = xold + r1 (xbest − xworst ) + r2 (|xold orxrandom | − |xrandom orxold |) (5)
their quality and the process for each sub-population is performed
xnew = xold + r1 (xbest − |xworst |) + r2 (|xold orxrandom | − (xrandom orxold )) (6) separately. Let the best value of fitness function be fbest_before. The best
and worst solutions are identified from each sub-population. Then, Rao
The flow chart for Rao algorithms is shown in Fig. 2. algorithms are applied as per equations (1), (2), and (3) on each sub-

Fig. 3. Flowchart of SAMP Rao algorithms.

6
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

population. Then update the new solutions in the sub-population if they evaluating the operating costs. Finally, Rao algorithms are applied to the
give better fitness function value otherwise, retain the old solutions. cost objective function and the iteration is completed. It may be noted
Now, find the best solution and corresponding fitness value from the that the error limit used for this paper is 0.5 %. For example, for a length
entire population, let it be fbest_after. If fbest_after is better than fbest_before, m of 3 m of the tube, it might vary from 2.985 to 3.015. Since this paper
is increased by 1 (m = m + 1) indicating the exploration feature to be compares the results with the other optimization algorithms, the error
increased. Otherwise, decrease m by 1 so that the algorithm can be more percentage is kept the same as that in the reference studies. Moreover,
exploitative than explorative. This completes one iteration of SAMP Rao reducing this error percentage will affect the computation time, as the
algorithm. The flowchart for this algorithm is given in Fig. 3. error condition lies within the objective function calculation which is
The flowchart showing the steps for calculations in this study is complex and may even result in an infinite loop. Thus, considering the
shown in Fig. 4. The initial generation of design variables can be reference studies and the computation time, the error is set at 0.5 %.
modified for particular methods. The value of length of tube is initially
assumed and after calculating the surface area for heat transfer, it is Single- and multi-objective optimization aspects of STHEs
compared with initial assumption. If the error is<0.5 % then the
assumed value is taken to be correct. Otherwise, using the new obtained Rao algorithms are used in application of design optimization of
value of length as assumed length, the entire process is reiterated. Once STHE. The variants of Rao algorithms, named as SAMP Rao algorithms,
the design variables are finalized, the pressure analysis is performed for are also used to check the performance. This is accomplished using five

Fig. 4. Flowchart of STHE design process.

7
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

case studies of STHE. Case studies 1,2,3 and 4 are taken from Lara- Table 2
Montaño et al. [25] for reference and comparison. Lara-Montaño et al. Discrete design variables and their bounds for Kern’s method.
[25] presented four case studies and have applied 7 different algorithms Design variable to be optimized Discrete values
viz. PSO, GWO, TLBO, CS, WOA, UMDA and DE. In these case studies the
Tube pitch Pt [1.25d, 1.50d]
objective is to reach to a design with minimum cost of heat exchanger Tube layout angle TL [30◦ , 45◦ , 90◦ ]
which follows the given constraints. Case study 5 is taken from Segundo Number of tube passes Npass [1,2,4,6,8]
et al. [20] where the STHE problem for Sarchesmeh cupper production
power plant is evaluated.
Table 3
Single-objective optimization problem Continuous design variables and their bounds for Bell-Delaware method.
Design variable to be optimized Lower limit Upper limit
Different metaheuristic algorithms are applied to the four case
Diameter of shell Ds 0.3 m 1.5 m
studies for achieving minimum cost Lara-Montaño et al. [25]. In the
Outer diameter of tube d 6.35 mm 50.8 mm
present work, each of the four case studies is evaluated by both Kern’s Center Baffle spacing Lbc 0.2Ds 0.55Ds
method and Bell-Delaware method. Inlet and outlet Baffle spacing Lbo, Lbi Lbc 1.6Lbc
Tube-to-baffle hole diametral clearance δtb 0.01d 0.10d
Shell-to-baffle diametral clearance 0.01Ds 0.1Ds
Design variables and bounds
δsb
Outer diameter of tube bundle Dotl 0.8(Ds - dsb) 0.95(Ds - dsb)

For each case study, the design variables can be either continuous or
discrete. While few of them are fixed as discrete, tube diameter is taken
Table 4
both discrete and continuous. Thus, each case study will have two
Discrete design variables and their bounds for Bell-Delaware method.
methods with continuous and discrete design variables. For Kern’s
method, the set of design variables consists of 7 parameters. The Bell- Design variable to be optimized Discrete values

Delaware method takes up 11 design variables to solve the optimiza­ Tube pitch Pt [1.25d, 1.50d]
tion problem. The continuous and discrete design variables and their Tube layout angle TL [30◦ , 45◦ , 90◦ ]
Baffle cut Bc [0.25, 0.30, 0.40, 0.45]
bounds for Kern’s method and Bell-Delaware method are taken from
Number of tube passes Npass [1,2,4,6,8]
Lara-Montaño et al. [24,25] and Flynn et al. [30] and are given in Ta­
bles 1-4.
In metaheuristic optimizations, the continuous variables are The values of Ec, Hr, n and i are kept same for the comparison of
addressed directly and the discrete variables are treated differently. The results. For the first four case studies, cost of electricity is taken 0.1 US
discrete variables are converted into discrete via some arbitrary con­ $/kWh, operating hours are taken as 8000 h per year, projected life time
version calculation. For example, in the case of tube pitch, if the value of as 20 years and rate of interest as 5 %. The pumping efficiency required
design variable in any iteration is from 0 to 0.5, tube pitch is assigned as for calculating the pumping power is taken as 0.85.
1.25d and if the value is from 0.5 to 1, tube pitch is assigned as 1.50d.
Ci = Cm Cp CT CE (10)

Cost calculation ( )0.68

A
CE = 3.28x104 (11)
80
Cost minimization is the optimization criteria for all the case studies.
The total cost Ctotal is the sum of the fixed cost Cfix and operating cost Cop.
(Costindex2000) 607.5
The operating cost Cop is the product of the pumping power required by CEPCIratio = = = 1.54 (12)
(Costindex2019) 394
the equipment Ppump, the annual operating hours for which the equip­
ment is in use (Hr) and the energy cost per unit (Ec). The fixed cost deals Cm is the correction factor for the material, Cp is the correction factor
with the cost of setup of the heat exchanger which is a function of heat for the pressure and CT is the correction for temperature. For Cfix, r is the
transfer area. rate of interest and n is the projected life for the heat exchanger.
For estimating the operating cost, pumping power is calculated as Lastly, total cost is calculated as:
follows [1,25,31]:
Ctot = Cop + Cfix (13)
( )( )
1 mt ΔPt ms ΔPs
Ppump = + (7)
ηpump ρt ρs Constraints

Cop =
Ppump Ec Hr
(8) For any real application problem, there are always some constraints
1000 due to material limitations, manufacturing feasibilities, industrial re­
The fixed cost is estimated using the ‘Chemical Engineering Plant quirements and cost ranges. First constraint is about the maximum
Cost Index (CEPCI)’. It is given by: allowable pressure drop across the shell and tube shown by equation
(14). Second constraint deals with limiting the range of fluid velocity in
r(1 + r)n tube side (15). The final constraint is about the ratio of length of the
Cfix = Ci (CEPCIratio) (9)
(1 + r)n − 1 tubes and diameter of the shell which is as shown by equation (16).
ΔPs , ΔPt ≤ 70, 000Pa (14)
Table 1
Continuous design variables and their bounds for Kern’s method. 0.5 m/s ≤ vt ≤ 3 m/s
Design variable to be optimized Lower limit Upper limit
L/Ds ≤ 15 (16)
Diameter of shell Ds 0.3 m 1.5 m
Outer diameter of tube d 6.35 mm 50.8 mm Thus, the optimization problem can be defined as follows:
Center Baffle spacing Lbc 0.2Ds 0.55Ds
Outer diameter of tube bundle Dotl 0.8(Ds - δsb) 0.95(Ds - δsb)
Minimize : Ctot (17)

8
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Subject to: ΔPs ≤ 70, 000Pa (18). different best solutions addressing both cost and effectiveness. The
calculation of cost is performed as per the following [20]:
ΔPt ≤ 70, 000Pa (19)
Cfix = 8500 + 409A0.85
t
0.5m/s ≤ vt ≤ 3m/s (20)
The operating cost is calculated by equation (25).
L/Ds ≤ 15 (21) y
∑ Co
Cop = (25)
To deal with the constraints defined above, penalty approach is used k=1 (1 + i)k
to avoid the solutions which dissatisfy them. The formulation of the final
function considering the penalties is defined in the equation (22). Co = Ppump Ec Hr (26)
fobj = Ctot + g1 (max[ΔPs ≤ 70000, 0]) + g2 (max[ΔPt ≤ 70000, 0])
Where i is the annual discount rate, and y is the life period of equipment
in number of years.
g3 (max[0.5 − vt , 0]) + g4 (max[vt − 3, 0]) + g5 (max[L/Ds − 15, 0]) (22)
For calculating the effectiveness of the multi objective problem, the
Here g1 , g2 , g3 , g4 andg5 are functions to ensure that the penalty levied NTU-ε method is used [20].
is large to avoid the inferior solution and to get the optimum solution.
2
ε= ( ( ) (27)
2 )0.5 ( 2 )0.5
(1 + C* ) + 1+ C* coth NTU 1 + C*
Multi-objective optimization problem 2

In actual application of heat exchangers, although cost is considered Where, NTU is the number of transfer units, and C* is defined below.
to be important objective but single-objective design might not give the U o At
required results. Thus, in this study multi-objective optimization prob­ NTU = (28)
Cmin
lem is taken into consideration. Here we consider two objectives viz.
cost and effectiveness. The case study for this problem is taken from Cmin
C* = (29)
Segundo et al. [20] where the authors used Owl optimization algorithm Cmax
(OOA) to increase the thermal effectiveness and to minimize the cost.
The heat transfer co-efficient for tube side is given as [30].
Bell-Delaware method is used for this owing to its better accuracy.
( )
kt
ht = 0.024Re0.8
t Prt
0.4
(30)
Design variables and bounds di
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
The design variables for multi-objective problem are shown in (πNt )CL
Ds = 0.637pt (31)
Table 5. CTP

Where, CL is constant depending upon the tube layout and CTP is the
Combined objective function
constant that depends upon tube count.
{
The multi-objective problem is evaluated using a priori approach,
◦ ◦
1.00forTL45 and90
CL = (32)
where the problem is initially solved considering a single-objective
◦ ◦
0.87forTL30 and60
function separately for both the objectives and then the weighted ⎧
combined objective function shown by equation (23) is optimized. Here ⎨ 0.93forsingletubepass
costopt is the optimum cost obtained when cost minimization is only CTP = 0.90fortwopasses (33)
⎩
considered as the single-objective function, and effopt is the optimum 0.85forthreepasses
effectiveness obtained when effectiveness maximization is only consid­
ered as the single-objective function. Thus, the weighted combined Applications of Rao algorithms for the STHE case studies
objective function f is given by:
( ) ( ) Case study 1
eff cost
Minimizef = − we + wc (23)
ef fopt costopt Case study 1 consists of heat exchanger wherein methanol is used in
shell-side and sea water is used in the tubes. The flow rate of methanol is
Here, the we and wc are the weightages given for the effectiveness and 27.8 kg/s and that of sea water is 68.9 kg/s. The inlet temperature of
cost respectively. The values of we and wc are varied from 0 to 1 to obtain methanol is 95 ◦ C and outlet temperature of methanol is 40 ◦ C. The inlet
temperature of sea water is 25 ◦ C and outlet temperature of sea water is
Table 5 40 ◦ C. The materials used for construction of shell and tubes are carbon
Design variables and bounds for multi-objective optimization [20]. steel and stainless steel. The correction factors in this case study are
Design variable to be optimized Allowed values taken as Cm = 1.7, Ct = 1.0 and Cp = 1.0. The values of thermo-physical
Tube Layout TL (30◦ , 45◦ , 90◦ )
properties for first three case studies are taken from Caputo et al. [4].
Tube diameter D 0.0112 m 0.0153 m
Tube pitch Pt 1.25 2
Tube length L 3m 8m Case study 2
Number of tubes Nt 100 600
Baffle cut Bc 0.19 0.32 For case study 2, kerosene is used in shell side while crude oil is used
Baffle spacing ratio BSR 0.2 1.4
in tube side heat transfer. Kerosene flows at 5.52 kg/s with inlet tem­
Baffle spacing at inlet and outlet Lbo = Lbi 0.06 m 0.00064 m
Tube-to-baffle hole diametral δtb 0.01d 0.10d perature as 199 ◦ C and outlet temperature as 93.3 ◦ C. Crude oil is used
clearance with a flow rate of 18.8 kg/s with inlet temperature as 38.8 ◦ C and outlet
Shell-to-baffle diametral clearance δsb 0.01Ds 0.1Ds temperature as 76.7 ◦ C. Both shell and tubes are made with carbon steel.
Outer diameter of tube bundle Dotl 0.8(Ds - dsb) 0.95(Ds - dsb) The correction factors for calculating the fixed cost are Cm = 1.0, Ct =
Number of tube pass Npass (1, 2, 3)
1.6 and Cp = 1.0.

9
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Case study 3 Table 7

Results comparison of cost obtained by the proposed algorithms and seven other
Case study 3 involves distilled water placed in shell side and raw algorithms for case study 2.
water in tubes. The mass flow rate of distilled water is 22.07 kg/s with Algorithm BD C BD D Kern C Kern D
inlet temperature as 33.9 ◦ C and outlet temperature as 29.4 ◦ C. Raw CS* 6071.06 6103.24 6427.53 6057.82
water flows in tubes with mass flow rate of 35.31 kg/s with inlet tem­ DE* 6065.58 6093.28 6424.89 6055.89
perature as 23.9 ◦ C and outlet temperature as 26.7 ◦ C. The shell is made GWO* 6066.05 6094.22 6426.29 6055.96
up of carbon steel and tubes are made from stainless steel. The correc­ PSO* 6067.19 6113.65 6424.89 6055.89
TLBO* 6179.51 6341.27 6441.37 6062.44
tion factors for calculating the fixed cost are Cm = 1.7, Ct = 1.0 and Cp =
UMDA* 6083.93 6156.67 6436.81 6068.72
1.0. WOA* 6129.89 6116.24 6427.12 6069.21
Rao 1 4861.603 4861.918 5376.489 5405.627
Case study 4 Rao 2 4861.582 4861.582 5376.489 5405.589
Rao 3 4861.582 4861.658 5376.618 5405.589
SAMP Rao 1 4861.595 4861.597 5677.523 6016.223
In this fourth case study naphtha is placed in shell with cooling water SAMP Rao 2 4861.582 4861.582 5376.420 5405.589
in the tube side. The flow rate at shell side is 2.7 kg/s. The inlet tem­ SAMP Rao 3 4861.582 4861.582 5376.958 5405.589
perature at shell side is 114 ◦ C and outlet temperature is 40 ◦ C. Cooling
*Results are taken from Lara-Montaño et al. [25].
water flows with the rate of 30 kg/s with inlet temperature of 33 ◦ C and
outlet temperature of and 37.2 ◦ C. Shell and tubes are made up of carbon
steel and stainless steel, respectively. The correction factors in this case Table 8
study are taken as Cm = 1.7, Ct = 1.0 and Cp = 1.0. The set of values are Results comparison of cost obtained by the proposed algorithms and seven other
taken from Wildi-Tremblay [3] for this case study. algorithms for case study 3.
Algorithm BD C BD D Kern C Kern D
Case study 5 CS* 3682.78 3640.62 3517.75 3559.16
DE* 3682.78 3640.62 3517.74 3559.15
In the fifth case study oil flowing at 8.1 kg/s is passed through the GWO* 3682.78 3640.62 3517.74 3559.16
shell side with input temperature as 78.3 ◦ C. Cold water was allowed to PSO* 3682.78 3640.62 3517.74 3559.15
TLBO* 3682.78 3640.62 3517.91 3559.16
flow at 12.5 kg/s through the tube side with input temperature 30 ◦ C.
UMDA* 3861.87 3818.39 3535.18 3603.88
The life period of the STHE is taken as 10 years. The annual rate of WOA* 3682.78 3640.62 3517.79 3559.16
discount is taken as 10 %. The cost of electricity is taken as 0.15 US Rao 1 4858.347 4913.666 4725.215 4735.269
$/kWh, operating hours are taken as 7500 h per year. The pumping Rao 2 4834.708 4883.715 4723.748 4735.269
efficiency required for calculating the pumping power is taken as 0.60. Rao 3 4825.875 4913.622 4723.738 4735.269
SAMP Rao 1 4850.848 4911.716 4921.627 4924.313
SAMP Rao 2 4835.070 4883.576 4723.738 4735.269
Results and discussion SAMP Rao 3 4835.004 4913.625 4723.736 4735.269

*Results are taken from Lara-Montaño et al. [25].

The four defined case studies are applied with Rao-1, Rao-2, Rao-3,
SAMP Rao-1, SAMP Rao-2, and SAMP Rao-3 algorithms. Each problem
is solved by two methods: Kern and Bell-Delaware, and for continuous Table 9
and discrete variables. Thus, the total individual problems taken into Results comparison of cost obtained by the proposed algorithms and seven other
study are 96 (6 × 4 × 2 × 2). Since Kern’s method involves less design algorithms for case study 4.
variables and fewer calculations, it needed fewer function evaluations Algorithm BD C BD D Kern C Kern D
but as the Bell-Delaware method is more realistic and complex it
CS* 5122.41 5058.12 5915.35 5840.82
required a greater number of function evaluations. Finally, each of this
DE* 5122.16 5057.95 5915.33 5840.82
problem is solved using 10,000 function evaluations and 10 iterations GWO* 5122.18 5057.99 5915.33 5840.82
each of those evaluations. The algorithms are computed on Python 3.8.3 PSO* 5122.15 5057.98 5915.33 5840.82
version. The results of four case studies for these computations are TLBO* 5122.47 5058.78 5915.36 5847.19
UMDA* 5345.22 5328.84 5928.84 5856.8
shown in Tables 6-9 and Figs. 5-8. The results for seven other algorithms
WOA* 5122.6 5057.98 5920.72 5840.84
computed for single-objective optimization problem are taken from Rao 1 3326.817 3326.822 3242.083 3242.083
Lara-Montaño et al. [25]. Rao 2 3326.928 3326.944 3242.083 3242.083
Rao 3 3326.944 3326.944 3242.083 3242.083
SAMP Rao 1 3326.820 3326.826 3242.083 3242.083
Table 6
SAMP Rao 2 3326.817 3326.944 3242.083 3242.083
Results comparison of cost obtained by the proposed algorithms and seven other SAMP Rao 3 3326.817 3326.944 3242.083 3242.083
algorithms for case study 1.
*Results are taken from Lara-Montaño et al. [25].
Algorithm BD C BD D Kern C Kern D

CS* 12999.77 12866.71 14265.12 14125.95

The values of the total cost in US$ are given in Table 6 for case study
DE* 12997.48 12860.34 14265.09 14125.92
GWO* 12998.07 12860.64 14265.1 14125.92 1. The results are shown in Fig. 5 as well. It is evident that the Rao al­
PSO* 12997.41 12860.31 14265.09 14125.92 gorithms and their variants performed better than the listed seven al­
TLBO* 13000.32 12951.26 14265.21 14134.08 gorithms of Lara-Montaño et al. [25] for case study 1. The cost values
UMDA* 12999.37 12865.27 14298.87 14144.35 shown in Table 6 in boldface indicate the minimum cost in US$ in
WOA* 13022.66 12892.19 14265.1 14126.56
Rao 1 12471.007 12469.553 12663.849 12663.849
respective Bell-Delaware (BD) and Kern methods considering the design
Rao 2 12471.878 12471.850 12663.849 12663.849 variables as continuous (C) and discrete (D).
Rao 3 12471.841 12468.920 12663.849 12663.849 The values of the total cost in US$ are given in Table 7 for case study
SAMP Rao 1 12472.039 12468.879 13088.953 13088.953 2. The results are shown in Fig. 6 as well. It is evident that Rao algo­
SAMP Rao 2 12469.386 12469.021 12663.849 12663.849
rithms and their variants performed better than the listed seven algo­
SAMP Rao 3 12467.609 12467.606 12663.849 12663.849
rithms for case study 2.
*Results are taken from Lara-Montaño et al. [25].

10
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Case study 1
14500

14000

13500

13000
Cost

12500

12000

11500
CS DE GWO PSO TLBO UMDA WOA Rao 1 Rao 2 Rao 3 SAMP SAMP SAMP
Rao 1 Rao 2 Rao 3

CS1 BD C CS1 BD D CS1 Kern C CS1 Kern D

Fig. 5. Optimum cost in case study 1 for different algorithms.

Case study 2
7000

6500

6000

5500
Cost

5000

4500

4000
CS DE GWO PSO TLBO UMDA WOA Rao 1 Rao 2 Rao 3 SAMP SAMP SAMP
Rao 1 Rao 2 Rao 3

CS2 BD C CS2 BD D CS2 Kern C CS2 Kern D

Fig. 6. Optimum cost in case study 2 for different algorithms.

The values of the total cost in US$ are given in the Table 8 for case Fig. 9 shows the comparison of best cost values from the previous and
study 3. The results are shown in Fig. 7 as well. The results for case study present studies.
3 may give the impression that the proposed algorithms have not given The results of multi-objective optimization are compared with the
the good results. However, when the corresponding design variables Multi-Objective Owl Optimization Algorithm (MOOOA) presented by
from the previous study of Lara-Montaño et al. [25] are substituted in Segundo et al. [20]. The results are obtained in the form of a Pareto front
the equations for calculations, the constraints are found violated. Thus, shown in Fig. 10.
the results given by Rao and SAMP Rao algorithms are valid and reliable. Fig. 10 shows that SAMP Rao 1 has better spread at least efficiency
The values of the total cost in US$ are given in Table 9 for case study and minimum cost. Except the two points marked (point A and point B)
4. The results are shown in Fig. 8 as well. It is evident that Rao algo­ the algorithms give better or similar results in the middle region where
rithms and their variants performed better than the listed seven algo­ none of the objectives is given sole priority. But, both Rao 1 and SAMP
rithms for case study 4. Rao 1 do not reach the point of best efficiency with highest cost
It can be clearly seen from case studies 1, 2, and 4 that the minimum compared to the reference.
cost value obtained by Rao algorithms and their variants is less as Fig. 11 shows the Pareto front results obtained using the Rao 2 and its
compared to the minimum cost value obtained by seven other optimi­ variant SAMP Rao 2. It can be seen that both the algorithms have better
zation algorithms presented in the study of Lara-Montaño et al. [25]. For spread of the pareto front when compared with the reference, i.e.,
case study-3, as mentioned earlier, the minimum cost value presented by Segundo et al. [20]. Also, all the points give better results and the pareto
Lara-Montaño et al. [25] is not reliable as the substitution of the values obtained is below the reference. SAMP Rao 2 gives lower cost and higher
of the corresponding design variables violate the constraints. efficiency when the efficiency is the sole consideration.
Table 10 summarizes the best values of the previous and present It can be seen from Fig. 12 that both Rao 3 and SAMP Rao 3 give
studies and the percentage improvement obtained in the present study. better results when compared to reference pareto front, i.e., Segundo

11
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Case study 3
5500

5000

4500

4000
Cost

3500

3000

2500
CS DE GWO PSO TLBO UMDA WOA Rao 1 Rao 2 Rao 3 SAMP SAMP SAMP
Rao 1 Rao 2 Rao 3

CS3 BD C CS3 BD D CS3 Kern C CS3 Kern D

Fig. 7. Optimum cost in case study 3 for different algorithms.

Case study 4
6500
6000
5500
5000
4500
Cost

4000
3500
3000
2500
CS DE GWO PSO TLBO UMDA WOA Rao 1 Rao 2 Rao 3 SAMP SAMP SAMP
Rao 1 Rao 2 Rao 3

CS4 BD C CS4 BD D CS4 Kern C CS4 Kern D

Fig. 8. Optimum cost in case study 4 for different algorithms.

To get even better idea about the performance of the proposed al­
Table 10
gorithms and MOOOA of Sengundo et al. [20], a performance measure
Best values of cost and percentage improvement obtained by the present study.
known as ‘Coverage’ is evaluated for all the algorithms. Coverage
Case Optimization Best of Best of this Percentage measures the comparison of two set of solutions that are non-dominated
study problem previous study improvement
and evaluates the percent of solutions of one set that dominate the so­
study
lutions of another set. The calculated coverage values are shown in
CS1 BD C 12997.41 12467.609 4%
Table 11. Cov(A,B) represents the coverage of set A with respect to that
BD D 12860.31 12467.606 3%
Kern C 14265.09 12663.849 11 %
of set B. The calculation is done as per [29].
Kern D 14125.92 12663.849 10 % Table 11 clearly shows that all the algorithms used in this study
CS2 BD C 6065.58 4861.5816 20 % perform better than the reference algorithms presented by Sengundo
BD D 6093.28 4861.5816 20 % et al. [20]. Among these algorithms, SAMP Rao 2 and SAMP Rao 3 give
Kern C 6424.89 5376.4197 16 %
best results since maximum space of the pareto from these dominate the
Kern D 6055.89 5405.5888 11 %
CS4 BD C 5122.15 3326.8168 35 % solutions obtained by the reference study. To compare these results
BD D 5057.95 3326.8217 34 % better, another performance measure known as ‘Hypervolume’ is eval­
Kern C 5915.33 3242.0827 45 % uated which gives the volume of the search space of the dominated so­
Kern D 5840.82 3242.0827 44 % lutions from a given point in the space. The reference point that is taken
in this calculation is (x,y) = (0.074973661, 50369.37751). Table 12
et al. [20]. Also, when compared internally, SAMP Rao 3 gives even shows the comparison of hypervolumes.
better results than those obtained by Rao 3 algorithm as shown in the This comparison shows that SAMP Rao 3 gives the better solutions
highlighted points. and stands out as best algorithm for this study of multi-objective

12
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Fig. 9. Comparison of best cost values from previous and current studies.

Fig. 10. Pareto front comparison of MOOOA, Rao 1 and SAMP Rao 1 algorithms.

optimization of STHE design problem. The solutions of design variables SAMP Rao algorithms can find applications in complex solutions and
obtained from SAMP Rao 3 are given in Table 13. give better results than the other algorithms. Furthermore, it is observed
All the solutions obtained from the Rao algorithms and SAMP Rao that using variants to all algorithms may not guarantee better results as
algorithms are found to be feasible satisfying the constraints. It is further seen for SAMP Rao 1, where original Rao algorithm provided better
noted that Rao algorithms and their variants are easy and simple for hypervolume and coverage in this study.
application for any kind of optimization problem. From the study related
to multi-objective optimization, i.e., minimizing the total cost and Conclusions
maximizing the effectiveness (Ctot and ε), it is observed that perfor­
mance measures of solutions obtained from SAMP Rao 3 algorithm The design of heat exchangers is a complicated task which involves
showed best results. Thus, these easy-to-use straight forward Rao and many variables and calculations. Hence, advanced optimization

13
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Fig. 11. Pareto front comparison of MOOOA, Rao 2 and SAMP Rao 2 algorithms.

Fig. 12. Pareto front comparison of MOOOA, Rao 3 and SAMP Rao 3 algorithms.

Table 11 Table 12
Coverage performance measure. Comparison of hypervolume obtained using different
algorithms.
coverage sets Value
Algorithm Hypervolume
Cov(Rao 1, MOOOA) 0.6
Cov(Rao 2, MOOOA) 0.6 Ref. [20] 20024.62
Cov(Rao 3, MOOOA) 0.6 Rao 1 22766.51
Cov(SAMP Rao 1, MOOOA) 0.2 Rao 2 23565.51
Cov(SAMP Rao 2, MOOOA) 0.8 Rao 3 23100.84
Cov(SAMP Rao 3, MOOOA) 0.8 SAMP Rao 1 22539.45
Cov(MOOOA,Rao 1) 0.0 SAMP Rao 2 24182.72
Cov(MOOOA,Rao 2) 0.0 SAMP Rao 3 24242.55
Cov(MOOOA,Rao 3) 0.0
Cov(MOOOA,SAMP Rao 1) 0.0
Cov(MOOOA,SAMP Rao 2) 0.0 case studies (i.e., four single – objective case studies and one multi-
Cov(MOOOA,SAMP Rao 3) 0.0
objective case study). Two methods i.e., Kern’s method and Bell-
Delaware method are used in conjunction with the proposed Rao algo­
algorithms are advantageous in designing cheap and effective heat ex­ rithms and their variants for the design of STHE. The results of STHE
changers. This work presents the successful application of Rao algo­ design optimization problem are compared with the best results ob­
rithms and their variants known as SAMP Rao algorithms for STHE tained from CS, DE, GWO, PSO, TLBO, UMDA and WOA. The reduction
design. The performance of these algorithms is exhibited by taking 5 in the total annual cost achieved by SAMP Rao 3 for case study 1 is 3 % in

14
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Table 13
Design variables with solutions obtained from SAMP Rao 3.
TL d Pt L Nt Bc BSR Lbo δtb δsb Dotl Npass Ctot (€) eff

45 0.0112 2.0000 3.000 100 0.32 1.400 0.5874 0.0001 0.0026 0.2466 1 11586.79 0.0750
30 0.0112 1.8940 3.000 100 0.19 1.295 0.3 0.0001 0.0023 0.2179 1 11609.99 0.2468
30 0.0112 1.6352 3.000 100 0.19 1.364 0.2727 0.0001 0.002 0.1881 1 11636.18 0.2672
30 0.0112 1.6352 3.000 100 0.19 0.882 0.1765 0.0001 0.002 0.1881 1 11692.05 0.2871
45 0.0114 1.5025 3.000 100 0.21 0.750 0.15 0.0001 0.002 0.1881 1 11738.97 0.3026
45 0.0112 1.5252 3.000 100 0.20 0.708 0.2264 0.0001 0.002 0.1881 1 11703.07 0.2993
45 0.0115 1.5019 3.003 100 0.20 0.744 0.1502 0.0001 0.002 0.1898 1 11764.71 0.3051
45 0.0112 1.2500 3.000 153 0.19 1.056 0.2143 0.0001 0.002 0.1909 1 13007.51 0.4015
45 0.0112 1.2500 3.000 201 0.19 0.747 0.1765 0.0001 0.0024 0.2223 2 14210.49 0.4713
45 0.0112 1.2500 3.629 225 0.19 0.685 0.2739 0.0001 0.0025 0.2349 2 15810.71 0.5439
45 0.0112 1.2500 3.521 340 0.19 0.447 0.1413 0.0001 0.0032 0.2973 3 18587.72 0.6270
30 0.0112 1.2553 4.712 405 0.19 0.573 0.2878 0.0001 0.0031 0.2955 2 23284.86 0.7148
30 0.0112 1.6137 6.436 457 0.19 0.357 0.1534 0.0001 0.0043 0.4036 2 29793.01 0.7726
30 0.0112 1.5211 6.945 600 0.19 0.358 0.1656 0.0001 0.0046 0.4357 2 37035.15 0.8070
30 0.0144 1.3206 8.000 600 0.19 0.227 0.1212 0.0001 0.0053 0.5016 3 48360.15 0.8194

comparison to PSO for Bell Delaware method while for Kern’s method of Rao algorithms specifically in thermal system design problems of heat
Rao 3 and SAMP Rao 2 proved better by 10 % as compared to CS, DE, exchangers where the calculations are complex, and variables are in
GWO, and PSO. large numbers.
In case study 2 for STHE design by Bell Delware method, Rao 3,
SAMP Rao 2 and SAMP Rao 3 reduced 20 % cost in comparison to DE. CRediT authorship contribution statement
Also, Rao 2, Rao 3, SAMP Rao 2 and SAMP Rao 3 gave the same result
which is 11 % less cost than that obtained by DE. In case study 4, a Ravipudi Venkata Rao: Conceptualization, Methodology, Writing –
reduction of 34 % of the cost is achieved using Bell-Delware method by review & editing. Meet Majethia: Visualization, Software, Writing –
Rao 1, SAMP Rao2 and SAMP Rao 3 when compared to DE. Rao 3 al­ original draft, Validation.
gorithm has reduced the cost of STHE by 44 % using Kern’s method in
comparison to CS, DE, GWO and PSO. In case study 5, the coverage of all
the Rao algorithms and their variants presents their dominance over the Declaration of Competing Interest
MOOOA in the bi-objective optimization achieving minimum cost and
maximum effectiveness. Also, the hypervolume obtained by using SAMP The authors declare that they have no known competing financial
Rao 3 is 21 % more than that obtained by MOOOA. interests or personal relationships that could have appeared to influence
The ability of Rao algorithms and their variants is demonstrated in the work reported in this paper.
the present work and significant improvement in the performance is
observed when compared to those obtained by previous research. The Data availability
Rao algorithms do not have any algorithm-specific parameters and they
can be easily implemented. This characteristic enhances the application Data will be made available on request.

Appendix A

The nomenclature of different terms and parameters used in this Appendix are given in the ‘Nomenclature’ section.
Detailed calculations for design of STHE using Kern’s method
Initially, the 7 design variables Ds , d, Lbc , Dotl , Pt , TLandNPass are randomly generated within the bounds given. Given below are the steps followed to
calculate the Ctot using Kern’s method [30,31].
Based on the tube layout value generated, the constants k1 and n1 are selected from the Table A1 and tube length is assumed some value as Lassumed.
The heat transfer, logarithmic mean temperature difference and temperature correction factor are calculated using the below equations [30,31].
Q = ṁh Cph (Th1 − Th2 ) (A.1)

(Th1 − tc2 ) − (Th2 − tc1 )

TLMTD = ( ) (A.2)
ln TTh1h2 −− ttc2c1

√̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ( 1− P )
R2 + 1ln 1− RP
F= [ √̅̅̅̅̅̅̅̅ ] (A.3)
2− P(R+1− R2 +1 )
(R − 1)ln √̅̅̅̅̅̅̅̅
2− P(R+1+ 2 R +1 )

Where,
Table A1 Th1 − Th2
R= (A.4)
Values for constants k1 and n1 [20]. tc2 − tc1
No. of passes 1 2 4 6 8
tc2 − tc1
P= (A.5)
Triangular pitch K1 0.3190 0.2490 0.1750 0.0743 0.0365 Th1 − tc1
n1 2.1420 2.2070 2.2850 2.4990 2.6750
Square pitch K1 0.2125 0.1560 0.1580 0.0402 0.0331 Tube side heat transfer coefficient
n1 2.2070 2.2910 2.2630 2.6170 2.6430 For calculating tube side heat transfer coefficient, Reynold’s number

15
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

is required for which the number of tubes is found out by the following.
( )n1
Dotl
Nt = K1 (A.6)
d
Next, the velocity of the fluid in the tubes is found using:
ṁt Npass
vt = π 2
(A.7)
d
4 i
ρt 2 Nt

Where di = d/0.8. With the use of velocity, Reynold’s number is calculated and thus ultimately, friction factor is obtained.
ρvt di
Ret = (A.8)
μt

ft = (1.82log10 Ret − 1.64)− 2

(A.9)
Prandtl number is calculated and hence the tube side heat transfer coefficient is found.
μt Cpt
Prt = (A.10)
kt
[( ( ) )1/3 ]
kt Ret Prt di
ht = 1.86 forRet < 2300 (A.11)
di L
⎡ ⎤

kt ⎢
⎢
ft
Ret Prt ⎥
⎥
ht = ⎢ 2
⎥forRet < 10000 (A.12)
di ⎣1.07 + 12.7( ft )0.5 ( Prt 2/3 − 1) ⎦
2

kt 0.8 1/3
ht = Ret Prt forRet > 10000 (A.13)
di
Shell side heat transfer coefficient
For shell side calculation, the cross-sectional area normal to flow direction is calculated followed by calculation of fluid velocity at shell side
(Fig. A1).
( )
d
Acr = Ds Lbc 1 − (A.14)
Pt

ṁs
vs = (A.15)
ρs Acr
Further, the equivalent shell diameter for triangular and square layout is calculated from equations (A.16) and (A.17), respectively.
1.1 ( 2 )
De = Pt − 0.917do2 (A.16)
do

1.27 ( 2 )
De = Pt − 0.785do2 (A.17)
do
ms De
Res = (A.18)
μs Acr

μCps
Prs = (A.19)
ks
Finally, the equation below is used to compute the heat transfer coefficient of shell side.
( )
ks
hs = 0.36Re0.55 0.333
s Pr s (A.20)
De

Overall heat transfer coefficient

Finally, the overall heat transfer coefficient is calculated followed by
heat transfer area, calculated length of tubes and error.
[ ( ) ]− 1
1 do ln(do /di ) do 1
U= + Rfs + + Rft + (A.21)
hs 2kw di ht

Q
A= (A.22)
UFTLMTD

A
Lcalc = (A.23)
πdNt
Fig. A1. Types of tube pitch for equivalent diameter [30]

16
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

|Lcalc − Lassumed |
error = x100 (A.24)
Lassumed
If the error is <0.5 %, pressure drop evaluation is performed.
Pressure calculations
( )
ρvt 2 ft L
ΔPt = + 2.5 Np (A.25)
2 di
( )( )( )
ρs vs 2 L Ds
ΔPs = fs (A.26)
2 Lbc De

fs = 1.44Res − 0.15
(A.27)
Detailed calculations for design of STHE using Bell-Delaware method
Initially, the 11 design variables Ds , d, Lbc , Lbo , Lbi , δtb , δsb , Dotl , Pt , TL, Bc andNPass are randomly generated within the bounds given. Given below are
the steps followed to calculate the Ctot finally using Kern’s method [1,3,7,25].
Tube side heat transfer coefficient
Length of tube is initially assumed as Lassumed. The calculations for heat transfer, logarithmic mean temperature difference and temperature
correction factor are same. The estimation of number of tubes is as follow:
( π) 2 {
D Ct = 0.866forTL = 30, 60
Nt = 4 2ctl (A.28)
Ct Pt Ct = 1forTL = 45, 90

Where,
Dctl = Dotl − d (A.29)
With the help of number of tubes, velocity in the tubes is found followed by Reynold’s number and Prandtl number.
ṁt Npass
vt = π 2
(A.30)
d
4 i
ρt 2 Nt

ρvt di
Ret = (A.31)
μt

ft = (1.82log10 Ret − 1.64)− 2

(A.32)
[( ( ) )1/3 ]
kt Ret Prt di
ht = 1.86 forRet < 2300 (A.33)
di L
⎡ ⎤

kt ⎢
⎢
ft
Ret Prt ⎥
⎥
ht = ⎢ 2
( )0.5 ( ) ⎥forRet < 10000 (A.34)
di ⎣1.07 + 12.7 t f
Pr t
2/3
− 1 ⎦
2

( )0.14
kt 0.8 1/3 μt
ht = Ret Prt forRet > 10000 (A.35)
di μw
Shell side heat transfer coefficient
For shell side calculations, the heat transfer coefficient is given by:
hs = hid Jc Jl Jb Js Jr (A.36)

Table A2
Colburn factor constants.
Layout Angle log (Re) a1 a2 a3 a4 b1 b2 b3 b4

30 ◦
5–4 0.321 − 0.388 1.45 0.519 0.372 − 0.123 7 0.5
4–3 0.321 − 0.388 – – 0.486 − 0.152 – –
3–2 0.593 − 0.477 – – 4.57 − 0.476 – –
2–1 1.36 − 0.657 – – 45.1 − 0.973 – –
<10 1.4 − 0.667 – – 48 − 1 – –
45◦ 5–4 0.37 − 0.396 1.93 0.5 0.303 − 0.126 6.59 0.52
4–3 0.37 − 0.396 – – 0.333 − 0.136 – –
3–2 0.73 − 0.5 – – 3.5 − 0.476 – –
2–1 0.498 − 0.656 – – 26.2 − 0.913 – –
<10 1.55 − 0.667 – – 32 − 1 – –
90◦ 5–4 0.37 − 0.395 1.187 0.37 0.391 − 0.418 6.3 0.378
4–3 0.107 − 0.266 – – 0.082 0.022 – –
3–2 0.408 − 0.46 – – 6.09 − 0.602 – –
2–1 0.9 − 0.631 – – 32.1 − 0.963 – –
<10 0.97 − 0.667 – – 35 − 1 – –

17
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

2/3
Cps Prs −
hid = j (A.37)
Aocr

Where,
( )a
1.33
j = a1 Res a4 (A.38)
Pt /do
( )
a3
a= (A.39)
1 + 0.14Res a4
In the above equations, the constants a1, a2, a3 and a4 are taken from the Table A2.
For calculation of correction factors Jc , Jl , Jb , Js andJr following equations are used.
l c = Bc D s (A.40)
( )
Ds − 2lc
θctl = 2cos (A.41)
Dctl
( ) ( )
θctl θctl
Fc = 1 − + sin (A.42)
π π

Jc = 0.55 + 0.72Fc (A.43)

Fw = 0.5(1 − Fc ) (A.44)

Ntw = Nt Fw (A.45)
√̅̅̅
(A.46)
◦
Xt = Pt , Xl = 0.5 3Pt ifTL = 30

(A.47)
◦
Xt = Pt , Xl = Pt ifTL = 90
√̅̅̅
(A.48)
◦
Xt = 3Pt , Xl = 0.5Pt ifTL = 60

√̅̅̅ Pt
(A.49)
◦
Xt = 2Pt , Xl = √̅̅̅ ifTL = 45
2
( ( ) )
Dctl
(A.50)
◦ ◦
Aocr = Ds − Dotl + (Xt − d) Lbc ifTL = 30 , 90
Xt
( ( ) )
Dctl
(A.51)
◦ ◦
Aocr = Ds − Dotl + 2 (Pt − d) Lbc ifTL = 45 , 60
Xt

Aotb = 0.5πdδtb Nt (1 − Fw ) (A.52)

ms d
Res = (A.53)
Aocr μs
( ( ))
2lc
θb = 2cos− 1
1− (A.54)
Ds
( )
πDs δsb θb
Aosb = 1− (A.55)
2 2π

Aosb
rs = (A.56)
Aosb + Aotb

Aosb + Aotb
rlm = (A.57)
Aocr

Jl = 0.44(1 − rs ) + [1 − 0.44(1 − rs ) ](e− 2.2rlm

) (A.58)

Ds − 2lc
Nrcc = (A.59)
Xl
( )[
0.8
Nrcw = lc − 0.5(Ds − Dotl + d)] (A.60)
Xl

1
Nss = (A.61)
Nrcc
w = 0.05Ds (A.62)

18
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

⎧
⎨ Np = 1if Npass = 1
Aobp = Lbc (Ds − Dotl + 0.5Np w) Np = 1if Npass = 2 (A.63)
⎩
Np = 2if Npass ≥ 2

Aobp
rb = (A.64)
Aocr
{
1forNss ≥ 1/2
Jb = Crb [1− (2Nss + )1/3 ] (A.65)
e− forNss ≤ 1/2

Where,
{
1.35forRes ≤ 100
C= (A.66)
1.25forRes ≥ 100

Lbi Lbo
Li = , Lo = (A.67)
Lbc Lbc

L − Lbi − Lbo
Nb = +1 (A.68)
Lbc
{
n = 0.6and00, n2 = 1.0ifRes ≤ 4000
(A.69)
n = 0.333, n2 = 0.2ifRes > 4000

n)
Nb − 1 + L(1− + L(1− n)
Js = i o
(A.70)
Nb − 1 + L i + L o

Nrc = Nrcc + Nrcw (A.71)

⎧ ( )0.18
⎨ J = 10
⎪
ifRes ≤ 20
(A.72)
r
Nrc
⎪
⎩
Jr = 1ifRes ≥ 100

Overall heat transfer coefficient

Finally, the overall heat transfer coefficient is calculated as:
[ ( ) ]− 1
1 ln(do /di ) do 1
U= + Rfs + + Rft + (A.73)
hs 2kw di ht

Q
A= (A.74)
UFTLMTD

A
L= (A.75)
πdNt

|Lcalc − Lassumed |
error = x100 (A.76)
Lassumed
Pressure calculations
The pressure side calculations are as follows:
ṁs
Gc = (A.77)
Aocr

b3
b= ( ) (A.78)
1 + 0.14 Res b4
⎛ ⎞b
⎜1.33 ⎟
fid = b1 ⎝ pt ⎠ Res b2 (A.79)
d

Here, the constants b1, b2, b3 and b4 can be found from Table A.2.
4fid G2c Nrcc
ΔPbid = (A.80)
2ρs
( )( ( ( )) ( ))
D2s θb 2lc θb
Afrw = − 1− sin (A.81)
4 2 Ds 2
π
Afrt = d2 Fw Nt (A.82)
4

19
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

Aow = Afrw − Afrt (A.83)

4Aow
Dhw = (A.84)
πdNtw + πD360s θb
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
ṁ2
Gw = (A.85)
Aocr Aow
⎧ ( )
⎪
⎪ (2 + 0.6Nrcw ) m2s
⎪
⎪ forRes > 100
⎨ 2ρs Aocr Aow
ΔPwid = ( ) (A.86)
⎪
⎪ 26Gw μs Nrcw Lbc G2
⎪
⎪ + 2 + w forRes ≤ 100
⎩ ρ Pt − d Dhw ρs
s

{
D = 4.5ifRes ≤ 100
(A.87)
D = 3.7ifRes > 100

pc = − 0.15(1 + rs ) + 0.8 (A.88)

{ ( ( 1
))
Drb 1− (2Nss )3
ζb = e − if Nss < 0.5
(A.89)
ζb = 1if Nss ≥ 0.5

(A.90)
p
1.33(1+rs )(rlmc )
ζl = e(−
( )2− n2 ( )2− n2
Lbc Lbc
ζs = + (A.91)
Lbo Lbi
( )
Nrcw
ΔPs = [(Nb − 1)ΔPbid ζb + Nb ΔPwid ]ζl + 2ΔPbid 1 + ζ ζ (A.92)
Nrcc b s
( )
f = 0.046 Re0.2
t (A.93)
( )( )
4fL ρvt
ΔPt = + 2.5 Npass (A.94)
di 2

References [13] H. Sadeghzadeh, M.A. Ehyaei, M.A. Rosen, Techno-economic optimization of a

shell and tube heat exchanger by genetic and particle swarm algorithms, Energy
Convers. Manag. 93 (2015) 84–91, https://doi.org/10.1016/j.
[1] R.K. Shah, D.P. Sekulic, Fundamentals of Heat Exchanger Design, Jon Wiley, 2007.
enconman.2015.01.007.
https://doi.org/10.1002/9780470172605.ch10.
[14] H. Hajabdollahi, M. Naderi, S. Adimi, A comparative study on the shell and tube
[2] S. Kakaç, H. Liu, A. Pramuanjaroenkij, Heat Exchangers: Selection, Rating, and
and gasket-plate heat exchangers: the economic viewpoint, Appl. Therm. Eng. 92
Thermal Design, Third Edition, CRC Press, Taylor & Francis Group, 2012.
(2016) 271–282, https://doi.org/10.1016/j.applthermaleng.2015.08.110.
[3] P. Wildi-Tremblay, L. Gosselin, Minimizing heat exchanger cost with genetic
[15] D.K. Mohanty, Application of firefly algorithm for design optimization of a shell
algorithms and considering maintenance, Arch. Thermodyn. 31 (2006) 867–885,
and tube heat exchanger from economic point of view, Int. J. Therm. Sci. 102
https://doi.org/10.1002/er.
(2016) 228–238, https://doi.org/10.1016/j.ijthermalsci.2015.12.002.
[4] A.C. Caputo, P.M. Pelagagge, P. Salini, Heat exchanger design based on economic
[16] T.K. Tharakeshwar, K.N. Seetharamu, B. Durga Prasad, Multi-objective
optimisation, Appl. Therm. Eng. 28 (2008) 1151–1159, https://doi.org/10.1016/j.
optimization using bat algorithm for shell and tube heat exchangers, Appl. Therm.
applthermaleng.2007.08.010.
Eng. 110 (2017) 1029–1038, https://doi.org/10.1016/j.
[5] J.M. Ponce-Ortega, M. Serna-González, A. Jiménez-Gutiérrez, Use of genetic
applthermaleng.2016.09.031.
algorithms for the optimal design of heat exchangers, Appl. Therm. Eng. 29 (2009)
[17] R.V. Rao, A. Saroj, Economic optimization of heat exchanger using Jaya algorithm
203–209, https://doi.org/10.1016/j.applthermaleng.2007.06.040.
with maintenance consideration, Appl. Therm. Eng. 116 (2017) 473–487, https://
[6] J. Guo, L. Cheng, M. Xu, Optimization design of heat exchanger by entropy
doi.org/10.1016/j.applthermaleng.2017.01.071.
generation minimization and genetic algorithm, Appl. Therm. Eng. 29 (2009)
[18] R.V. Rao, A. Saroj, P. Oclon, J. Taler, Design optimization of heat exchangers with
2954–2960, https://doi.org/10.1016/j.applthermaleng.2009.03.011.
advanced optimization techniques: a review, Arch. Comput. Methods Eng. 27
[7] V.K. Patel, R.V. Rao, Design optimization of heat exchanger using particle swarm
(2020) 517–548.
optimization technique, Appl. Therm. Eng. 30 (2010) 1417–1425, https://doi.org/
[19] E.H.D.V. Segundo, A.L. Amoroso, V.C. Mariani, L.dos S. Coelho, Economic
10.1016/j.applthermaleng.2010.03.001.
optimization design for heat exchangers by a Tsallis differential evolution, Appl.
[8] A. Şencan Şahin, B. Kiliç, U. Kiliç, Design and economic optimization of shell and
Therm. Eng. 111 (2017) 143–151, https://doi.org/10.1016/j.
tube heat exchangers using Artificial Bee Colony (ABC) algorithm, Energy Convers,
applthermaleng.2016.09.032.
Manag. 52 (2011) 3356–3362, https://doi.org/10.1016/j.enconman.2011.07.003.
[20] E.H.D.V. Segundo, V.C. Mariani, L. dos S. Coelho,, Metaheuristic inspired on owls
[9] A. Hadidi, A. Nazari, Design and economic optimization of heat exchangers using
behavior applied to heat exchangers design, Therm. Sci. Eng. Prog. 14 (2019),
biogeography-based (BBO) algorithm, Appl. Therm. Eng. 51 (2013) 1263–1272,
100431, https://doi.org/10.1016/j.tsep.2019.100431.
https://doi.org/10.1016/j.applthermaleng.2012.12.002.
[21] S.V. Dhavle, A.J. Kulkarni, A. Shastri, I.R. Kale, Design and economic optimization
[10] A. Hadidi, M. Hadidi, A. Nazari, A new design approach for heat exchangers using
of heat exchanger using cohort intelligence algorithm, Neural Comput. Appl. 30
imperialist competitive algorithm (ICA) from economic point of view, Energy
(2018) 111–125, https://doi.org/10.1007/s00521-016-2683-z.
Convers. Manag. 67 (2013) 66–74, https://doi.org/10.1016/j.
[22] V.H. Iyer, S. Mahesh, R. Malpani, M. Sapre, A.J. Kulkarni, Adaptive Range Genetic
enconman.2012.11.017.
Algorithm: a hybrid optimization approach and its application in the design and
[11] M. Asadi, Y. Song, B. Sunden, G. Xie, Economic optimization design of heat
economic optimization of Heat Exchanger, Eng. Appl. Artif. Intell. 85 (2019)
exchangers by a cuckoo-search-algorithm, Appl. Therm. Eng. 73 (2014)
444–461, https://doi.org/10.1016/j.engappai.2019.07.001.
1032–1040, https://doi.org/10.1016/j.applthermaleng.2014.08.061.
[23] J.P. Sai, B.N. Rao, Efficiency and economic optimization of shell and tube heat
[12] R. Khosravi, A. Khosravi, S. Nahavandi, H. Hajabdollahi, Effectiveness of
exchanger using bacteria foraging algorithm, SN Appl. Sci. 2 (2020) 1–7, https://
evolutionary algorithms for optimization of heat exchangers, Energy Convers.
doi.org/10.1007/s42452-019-1798-0.
Manag. 89 (2015) 281–288, https://doi.org/10.1016/j.enconman.2014.09.039.

20
R. Venkata Rao and M. Majethia Thermal Science and Engineering Progress 36 (2022) 101520

[24] O.D. Lara-Montaño, F.I. Gómez-Castro, C. Gutiérrez-Antonio, Development of a [27] R.V. Rao, Rao algorithms: Three metaphor-less simple algorithms for solving
Virtual Environment for the Rigorous Design and Optimization of heat Exchangers, optimization problems, Int. J. Ind. Eng. Comput. 11 (2020) 107–130, https://doi.
Elsevier Masson SAS, 2020. https://doi.org/10.1016/B978-0-12-823377-1.50004- org/10.5267/j.ijiec.2019.6.002.
5. [28] R.V. Rao, R.B. Pawar, Self-adaptive multi-population rao algorithms for
[25] O.D. Lara-Montaño, F.I. Gómez-Castro, C. Gutiérrez-Antonio, Comparison of the engineering design optimization, Appl. Artif. Intell. 34 (2020) 187–250, https://
performance of different metaheuristic methods for the optimization of heat doi.org/10.1080/08839514.2020.1712789.
exchangers, Comput. Chem. Eng. 152 (2021), https://doi.org/10.1016/j. [29] R.V. Rao, Jaya: An advanced optimization algorithm and its engineering
compchemeng.2021.107403. applications, 2018. https://doi.org/10.1007/978-3-319-78922-4.
[26] A.C. Caputo, A. Federeci, P.M. Pelagagge, P. Salini, On the selection of design [30] A.M. Flynn, T. Akashige, L. Theodore, Kern’s Process Heat Transfer, Second
methodology for heat exchangers optimization problems, Therm. Sci. Eng. Prog. 34 Edition, Wley, 2019.
(2022), 101384, https://doi.org/10.1016/j.tsep.2022.101384. [31] G. Towler, R. Sinnott, Chemical Engineering Design, Sixth Edition, Elsevier, 2019.
https://doi.org/10.1021/ie50502a032.

21