10 DOI: 10.5545/sv-jme.2023.887
Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 1-2   ▪  Y 2025 © The Authors. CC-BY 4.0 Int. Licencee: SV-JME
Thermal Design and Constrained Optimization  
of a Fin and Tube Heat Exchanger Using  
Differential Evolution Algorithm
Nader Afsharzadeh – Mohammad Eftekhari Yazdi  – Arash Mirabdolah Lavasani
Islamic Azad University, Department of Mechanical Engineering, Iran 
  moh.eftekhari_yazdi@iauctb.ac.ir
Abstract Fin and tube heat exchangers (FTHEs) are utilized for gas-liquid applications frequently. In the current study, a differential evolution (DE) algorithm 
and JDE as its variant, with α-level constraint-handling technique, are effectively applied to optimize an FTHE. Total weight and total annual cost are selected as 
objective functions. Seven design variables are taken into consideration: outside tube diameter, transverse pitch, longitudinal pitch, fin pitch, number of tube 
rows, height, and width of shape. Meanwhile, the logarithmic mean temperature difference (LMTD) method is used for heat transfer analysis under identical 
conditions such as mass flow rate, inlet and outlet temperatures, heat duty, and other thermal properties. The research findings indicate that the implementation 
of the DE algorithm coupled with α-level comparison method on optimization problems leads to better solutions for both objective functions compared with those 
achieved by other approaches such as the genetic algorithm (GA) and heat transfer search (HTS) algorithm.  In addition, a parametric analysis is performed 
for design parameters at the optimum points to show the effects on the objective functions and to identify the feasible design space. The proposed method is 
straightforward and can generally be employed for thermal design and optimization of FTHEs as well as any other type of compact heat exchangers (CHEs) under 
different specified duties.
Keywords Fin and tube heat exchanger, thermal design, constrained optimization, differential evolution (DE) algorithm, total weight, total annual cost
Highlights
 ▪ Compact heat exchangers aim to minimize weight and annual cost in industrial use.
 ▪ Metaheuristic algorithms outperform trial-and-error methods in optimization.
 ▪ Differential evolution with α-level constraints achieves superior optimization results.
 ▪ Proposed method cuts weight by 9.33% and cost by 6.87 % from previous best results. 
1  INTRODUCTION
The process of heat exchange between two fluids that are at different 
temperatures and separated by a solid wall occurs in many engineering 
applications. The device used to implement this exchange is termed a 
heat exchanger (HE), and specific applications may be found in space 
heating and air-conditioning, power production, waste heat recovery, 
and chemical processing [1]. The area density is the ratio of heat 
transfer surface to HE volume. A compact heat exchanger (CHE) has 
a high area density compared to traditional HEs. CHEs are highly 
effective and low in weight and cost. Fins are used on the gas side of 
CHEs to compensate for high thermal resistance and enhance the heat 
transfer coefficient. Plate-fin heat exchangers (PFHEs), fin and tube 
heat exchangers (FTHEs), and rotary regenerators are examples of 
CHEs for gas flow on one or both fluid sides [2].
The most common criteria for the optimization of CHEs are 
the minimum initial cost, minimum operation cost, maximum 
effectiveness, minimum pressure drop, minimum heat transfer area, 
minimum weight, or material. The optimization of a CHE can be 
transformed into a constrained optimization problem and then solved 
by modern optimization algorithms [3].
The following could be highlighted after looking into the studies 
accomplished on the thermal design and optimization of PFHEs: 
Rao and Patel [4] applied the particle swarm optimization (PSO) 
algorithm for the thermodynamic optimization of a PFHE based 
on three individual objective functions: total number of entropy 
generation units, total volume, and total annual cost. Ahmadi et al. 
[5] used the ε-NTU method and a nondominated sorting genetic 
algorithm (GA) for the thermal modeling of a PFHE to minimize 
cost and entropy generation. Wang and Li [6] introduced and carried 
out an improved multi-objective cuckoo search (IMOCS) algorithm 
for multi-objective optimization, including efficiency maximization, 
minimization of pumping power, and total annual cost. Hajabdollahi 
[7] compared two cases of similar and non-similar fins on each 
side of the PFHE by using a PSO algorithm for thermo economic 
optimization. 
If one fluid is a liquid, different fin and tube configurations are 
frequently used; the liquid passes through the tubes while the gas 
flows across the bank of finned tubes. Various tube shapes might 
be used such as circular tubes, ovals, rectangles, and any other 
complex type. Compressor inter-coolers, air-coolers, and fan coils 
are examples of power engineering and chemical applications that 
employ FTHEs. Xie et al. [8] and Raja et al. [9] were seeking to 
achieve the optimal design of an FTHE based on total weight and 
total annual cost by employing GA and heat transfer search (HTS) 
algorithms, respectively.
Compared to most other evolutionary algorithms (EAs), 
differential evolution (DE) is much simpler and more straightforward 
to implement. The main body of the algorithm takes a few lines to 
code in any programming language. Also, the performance of DE 
and its variants is largely better than other optimization algorithms 
such as PSO, PCX, ALEP, etc. [10]. Babu et al. [11] applied DE 
algorithm and its various strategies for the optimal design of shell 
and tube heat exchangers. Ayala et al. [12] proposed a novel multi 

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Process and Thermal Engineering
objective free search (FS) approach combined with DE (MOFSDE) 
algorithm for heat exchanger optimization. Segundo et al. [13] by 
considering a shell-and-tube heat exchanger and the total annual 
cost as the objective function, employed DE algorithm, and Tsallis 
differential algorithm (TDE) to optimize it. Also, Yuan et al. [14] 
compared two helically coiled tubes’ heat transfer characteristics 
and hydrodynamics by implementing an adaptive multi-objective 
optimization DE algorithm.
Now, what is the best solution? Perhaps, this is the main question 
that arises in an engineering optimization problem. However, in 
most discussed thermal design and optimization studies about 
CHEs, penalty function-based methods are employed to handle the 
constraints and seldom can achieve a global solution that satisfies all 
constraints. Differential evolution with Level Comparison (DELC) 
for the first time is proposed by Wang and Li [15] and achieves 
superior searching quality on all the problems with fewer evaluation 
times than other algorithms. In this paper, DELC and JDE as a variant 
of standard DE with level comparison (JDELC) are applied to the 
thermal design and optimization problem of an FTHE. 
The remainder of this paper is organized as follows: In Section 
2, an FTHE is modeled by using a closed-form equation to predict 
the heat transfer coefficient. The next section introduces objective 
functions including total weight and total annual cost in addition to 
corresponding constraints. Section 4 is about the traditional design 
method used for FTHEs. Section 5 demonstrates the DE algorithm 
and employs DE and JDE based on the α-level constraint-handling 
technique. Section 6 illustrates a case study of FTHE and explains 
the results and discussion. In Section 7, a parametric analysis is 
carried out to obtain feasible design space. Finally, the conclusions 
are delivered in Section 8, followed by the list of symbols and the list 
of references.
2  THERMAL DESIGN
In the present work, an intercooler is considered as an FTHE 
with a plain-fin type in which hot air flows normally to a finned 
tube bundle while cold water flows inside tubes, as illustrated in Fig. 
1. However, it is common to use other types of fins, such as wavy, 
slotted, and louvered.
2.1  Heat Transfer
For the geometry calculations of staggered tube arrangement, the 
minimum free flow area on the airside is given by the following Eq. 
(1) [2]:
A
H
P
cP dt NW
min
t
to ff









  









11 , (1)
where 
ca ab
cb ba





22 2
22 2
if
if
. (2)
Values of 2a and b are determined as follows:
21 aP dt N
tO ff
  

, (3)
bP Pd PdtN
tl ot off
  

  05
2
2
05
..
.
 (4)
The total heat transfer area of the heat exchanger is calculated as: 
AA A
pf
 , (5)
where, A
p
 and A
f
 are the primary and secondary (i.e., fin) surface area 
of the heat exchanger, respectively, and obtained by,
Ad WN tN
po tf f


 1, (6)
AN WL Hd Nt NW H
ff ot ff







 2
4
2
2

, (7)
where N
t
 is the number of tubes and N
f
 is the number of fins per unit 
length and defined as follows:
N
H
P
N
t
t









1, (8)
N
W
F
W
f
p

1
.
 (9)
For the air side, when number of tube rows is greater than one, the 
Colburn factor (j
a
) correlation is suggested by Wang [16]:
jR eN
F
d
F
d
F
P
ad
jj p
c
j
p
h
j
p
t
c
=0086
34
56
.
























0 09 3 .
,
 (10)
where j
3
 to j
6
 is calculated by the following formulations:
Fig. 1.  Fin and tube heat exchanger with cross section view

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j
N
Re
N
F
d
dc
p
c
3
04 1
0 361
0 042
0 158  


















.
.
ln
.l n,
.
 (11)
j
P
d
Re
l
h
dc
4
14 2
1 224
0 076
 







.
.
ln
,
.
 (12)
j
N
Re
dc
5
0 083
0 058
 

.
.
ln
, (13)
j
Re
N
dc
6
5 735 12 1  






.. ln , (14)
then the heat transfer coefficient on the air side can be achieved by,
hj
c
Pr
aa
aapa
a


,
.
.
06 7
 (15)
For the water side, the heat transfer area can be computed by the 
following relationship:
Ad WN
ii t
 . (16)
The Nusselt number was approximated through the correlation 
given by Gnielinski [17]:
Nu
f
Re Pr
f
Pr
w
w
ww
w
w

 









8
1000
11 27
8
1
1
2
2
3
.
, (17)
where f
w
 is the friction factor and obtained by,
fl og
w
Re
w



18 21 64
10
2
.. . (18)
The heat transfer coefficient on the water side is as follows:
h
Nu
d
k
w
ww
i
= . (19)
The basic equation for the design of FTHE is developed in the Eq. 
(20) [18],
QU AT UA TU AT
mo om ii m
   , (20)
under dry cooling conditions,
11
2
1
UA hA
R
ln
d
d
kWN
R
hA
ii i
fi
o
i
tt
fo
oo o
 







 
, (21)
where, R
fi
 and R
fo
 are the fouling resistances of inside and outside 
tubes, respectively, and assumed negligible, η
i
 and η
o
 are the fin 
efficiencies of inside and outside tubes, respectively. The air side fin 
efficiency is calculated from the modified Schmidt equation that has 
been proposed by Hong and Webb [19],




f
mr
mr
cosm r 


tanh
, (22)
where
m
h
tk
ff
=
2
, (23)
 





















r
r
r
r
fe qf eq ,,
.l n. 1103 5 (24)
For rectangular fins, the equivalent radius is introduced by 
Schmidt in the following correlation [20],
r
r
p
r
P
P
fe q
tl
t
,
.
/
., 








12 8
2
02 (25)
where r is the radius of the tube based on the outside tube diameter. 
Then the air side surface efficiency can be obtained by,

o
f
O
f
A
A











11 . (26)
Here, the logarithmic mean temperature difference (LMTD) 
method is applied for heat exchanger analysis. 
QU AT
lm
 . (27)
LMTD is the maximum temperature potential through the heat 
transfer process that occurs in a counterflow heat exchanger and is 
described as the following expression:




T
TT
T
T
lm


12
1
2
ln
, (28)
where
TT T
hi co 1

,,
, (29)
TT T
ho ci 2

,,
. (30)
For a crossflow arrangement, Eq. (27) is modified with a 
correction factor F
c 
, 
QU AF T
cl m
 . (31)
The correction factor F
c
 is determined from charts or formulas 
based on two dimensionless parameters: temperature effectiveness P, 
and the ratio of heat capacity rates, R. Details of calculating these 
parameters can be found in fundamentals of heat exchanger design 
[2].
2.2  Pressure Drop
On the air side, the friction factor is obtained from the following 
correlation given by Wang [16],
fR e
P
P
F
d
ad
f t
l
f
p
c
f
c

















0 0267
1
23
., (32)
where
f
P
P
F
dN
t
l
p
c
1
0 764 0 739 0 177
0 00758
   .. .
.
, (33)
f
Re
dc
2
15 689
64 021
 

.
.
ln
,
 (34)
f
Re
dc
3
1 696
15 695


.
.
ln
,
 (35)
Re
Gd
dc
ac
a


,
 (36)
where G
a
 is the mass velocity of air regarding minimum free flow 
area. Then the pressure drop on air side can be obtained as follows 
[2]:
p
G
f
A
A
a
a
i
aa i
a
m
ai
ao




















2
2
2
1
11






min
,
,
,
 



























,
,
s.t. and
11 11
2
11
  
a
m
am a
m
io  






. (37)
In the above equation, σ is the ratio of A
min
 to A. Finally, the 
pressure drop on the water side could be found by the following 
equation,

SV-JME   ▪   VOL 71   ▪   NO 1-2   ▪   Y 2025   ▪   13
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P
fW
d
w
www
i


2
2
. (38)
3  OBJECTIVE FUNCTIONS AND CONSTRAINTS
Total weight and total annual cost are considered objective functions. 
Total weight includes weight of fins and weight of tubes,
TW At NW dd
ff ft tO i












4
22
. (39)
Furthermore, total annual cost consists of investment cost and 
operating cost which are given as:
TACC C
in OP
, (40)
CC A
in A
p
= , (41)
CK
PV
K
PV
op el
t
a
el
t
w



















. (42)
The subsequent set of constraints is applied to the mentioned 
objective functions:
 PP
aa

,max
, (43)
 PP
ww

,max
,
 (44)
11 2 <<
A
A
h
., (45)
W
d
o
≥ 60, (46)
300 21 0
4
  Re
a
, (47)
2300 21 0
6
  Re
w
, (48)
where, Re
a
 and Re
w
 are Reynolds numbers based on d
c
 and d
h
, 
respectively. The maximum allowable pressure drops on the air side 
and water side, respectively, are denoted by ∆P
a, max
, and ∆P
w, max
.
4  DESIGN METHOD AND PARAMETERS
The steps involved in heat exchanger design based on the LMTD 
method using a trial-and-error process are as follows [18]:
1. Calculate the outlet temperature according to the heat transfer rate 
(heat duty) and operating parameters using the steady flow energy 
equation.
2. Look up the correction factor F
c
 and calculate LMTD; Eq. (28).
3. Select the size and arrangement of tubes and fins, and calculate 
the heat transfer area A
first
; Eq. (5).
4. Calculate the convective heat transfer coefficients of the two sides 
and then, the overall heat transfer coefficient U; Eqs. (15), (19), 
and (21).
5. Determine the Calculated heat transfer area A
cal
; Eq. (20).
6. Compare A
cal
 with A
first
. If A
cal
 > A
first
, then go back to step 3, until 
1 < A
first
 / A
cal
 <1.2.
7. Calculate pressure drops and Reynolds numbers on both sides; 
Eqs. (43), (44), (47), and (48). If they are larger than the allowable 
pressure drops or are not in valid ranges of Reynolds numbers, 
then adjust the size and arrangement of tubes and fins until they 
satisfy specified allowable pressure drops, Reynolds numbers, and 
step 6.
8. Calculate objective function.
Note that complex factors exist here without consideration and 
this issue is covered by allowing an additional area of up to 20 %.
The outside tube diameter (d
o
), transverse pitch (P
t
), longitudinal 
pitch (P
l
), fin pitch (F
p
), number of tube rows (N), height of shape 
(H), and width of shape (W ) are assumed as seven design parameters. 
These parameters and the range of their variations are listed in Table 1 
[8]. Shape length, L, is not considered an independent variable, 
because it can be directly obtained from N and P
l
.
Constructional parameters except for the number of tube rows are 
considered continuous for optimization purposes, however, they are 
available in discrete quantities. If the precision of design parameters 
is set to 0.01 (N excluded, which takes values 2, 3, 4, 5, and 6), there 
are 600, 1000, 1900, 750, 5, 350, and 200 choices for the above tabled 
variables and therefore 600 × 1000 × 1900 × 750 × 5 × 350 × 200 ≈ 10
17
 
trial-and-error efforts are needed to find the optimal design which 
is impossible. In the next section, first DE is explained, then we 
implement DELC and JDELC algorithms instead of a trial-and-error 
method to attain minimum objective functions in the feasible design 
space. 
Table 1.  The upper and lower bounds of design variables
Design variable Search range
Outside tube diameter, [mm] 7 to 13
Transverse pitch, [mm] 20.5 to 30.5
Longitudinal pitch, [mm] 13 to 32
Fin pitch, [mm] 1 to 8.5
Number of tube rows 2 to 6
Height of shape, [m] 4.5 to 8
Width of shape, [m] 3 to 5
5  DIFFERENTIAL EVOLUTION ALGORITHM
A heuristic called an evolutionary algorithm (EA) was first inspired 
by biological evolution and employs mechanisms such as mutation, 
recombination, and selection. In other words, EA evolves an initial 
population after several generations. Therefore, the use of these 
algorithms has become popular in solving many problems, including 
engineering optimizations. In an optimization problem, the candidate 
vectors represent the individuals of a population.
DE is a simple, yet powerful algorithm proposed by Price et al. [21] 
and as a metaheuristic seeks to evolve an initial population toward 
the optimal solutions by iteratively improving them. This algorithm 
makes a few assumptions about the problem and can quickly reach 
the best solutions.
5.1  Standard DE
The DE in its standard form has three main parameters: population 
size NP, mutation factor F, and crossover rate CR. Attaining better 
solutions and convergence completely depends on the setting of 
these parameters. To adjust them, a few authors have suggested 
as following: Price et al. [21] proposed the setting NP = 10n, 
where, n is the number of design parameters, F ∊ [0.5, 1], and CR 
∊ [0.8, 1]. According to Rönkkönen et al. [22], a reasonable choice 
for population size is between 2n and 40n, F ∊ (0.4, 0.95], and CR 
∊ (0, 0.2) or CR ∊ (0. 9,1) for separable and non-separable objective 
functions, respectively. Note that here objective functions are non-
separable. Zielinski et al. [23] reported that, in many cases, the best 
results are obtained with the setting of F ≥ 0.6 and CR ≥ 0.6. 
The standard DE includes four principal operations during an 
optimization problem: 1) initialization 2) mutation 3) crossover 4) 
selection. The general structure of the algorithm is shown in Fig. 2.
5.1.1  Initialization
In a problem with n design variable, each candidate vector is defined 
as X = (x
1
, x
1
, …, x
n
).  The purpose is to optimize objective function 

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f (X). In the beginning, an initial population is generated including 
vectors as many as NP. Each member of the initial population is 
generated from Eq. (49). As a result, the initial population is an NP 
by n matrix.
xx rand xx
iN Pj
ij jj j , minm ax min
,,
,, ,,
  

 
  
01
12 1 and2 2, ,,  n
 (49)
where, x
j(max)
 and x
j(min)
 are the maximum and minimum values of 
each design variable, respectively. Furthermore, rand(0,1) is a 
uniformly distributed random number between 0 and 1.
Fig. 2.  Flowchart of standard DE algorithm
5.1.2  Mutation
In this step, a noisy population (donor vectors) is produced from 
the initial population as follows: for each vector from the current 
population, i
th
, the mutated vector is obtained by combining three 
randomly selected vectors according to the formulation below:
MX XX
ii in ca b
mm F 

  
,,
,, ,
1
 (50)
where, X
a
, X
b
, and X
c
 are three random vectors in the current 
population between 1 and NP except i
th
. The mutation factor, F, is 
a positive real number that controls the rate at which the population 
evolves. F has no upper bound, however effective values are rarely 
larger than 1 [21]. This mutation step is replicated for all original 
vectors of the current population to produce new population members 
that would improve the search space. This strategy, named DE/
rand/1, is the most popular and simplest DE variant, which uses one 
difference as a perturbation of the base vector. There are many other 
variants of the mutation mechanism that have been subsequently 
proposed by Das et al. [10] and Price et al. [21], such as DE/rand/2, 
DE/best/1, DE/best/2, etc. [24].
5.1.3  Crossover
After mutation, the i
th 
original vector from the current population 
is recombined with the corresponding vector from the mutated 
population to produce the trial vector U
i
 = (u
i,1
, …, u
i,n
). Each element 
of the trial vector is determined based on the following equation:
u
mr and CR jR
x
ij
ij
ij
,
,
,
,
, 
  




if or
otherwise
01
 (51)
where CR is between 0 and 1 which represents the probability of 
selecting a trial vector from the original vector and mutated vector, 
and R is a random integer number between 1 and n.
5.1.4  Selection
As the last step, just one of the vectors, X
i
 (original) and U
i
 (crossed) 
can survive. This selection is done based on the type of problem as 
follows:
1. for an unconstrained problem, objective function values of the 
two above vectors are the comparison criteria. If the goal is a 
minimization problem, the vector with a lower objective function 
value will be selected, and vice versa. Eq. (52) represents the 
selection step due to the minimization of an objective function. 
The following process is repeated for a certain number of 
generations or until convergence criteria are satisfied.
X
UU X
X
i
t i
t
i
t
i
t
i
t
ff










1
if
otherwise
. (52)
2. If the problem is constrained, like the present case, in addition to 
checking the objective function, the constraints’ fitness should 
also be checked. 
In the next section, we apply α-level comparison to handle the 
optimization problem constraints.
5.2  Differential Evolution with Level Comparison
The canonical versions of EAs, including DE, lack a mechanism 
to bias the search to the most feasible area since they were not 
designed inherently to solve constrained optimization problems  
[25]. Hence, this has triggered a significant amount of investigation, 
and during the last years, many different methods for incorporating 
constraints into the fitness function of an EA have been proposed  
[26]. Practically, adjusting control parameters such as F and CR 
and coupling them with suitable and effective constraint-handling 
strategies can considerably enhance the search capability of DE 
algorithms. Differential evolution with level comparison (DELC) 
performs initialization, mutation, and crossover operations similar 
to standard DE, but besides objective function values, a satisfaction 
level for the constraints is considered, which indicates how well a 
search point (candidate vector) satisfies the constraints. In other 
words, this method quantifies the constraint violation [27].
Below, f (X) is assumed to be a general function that should be 
minimized by the inequality constraints set g
k
 (X)
 
with k = 1, …, p, 
and equality constraints set h
s
 (X) with s = 1, …, q. 
min;
,, ,; ;, ,, .
X
fg
kp hs q
k
s
XX
X
  
   
s.t. 0
12 01 2 (53)
From the first generation (t = 1) to the end (t = Gen
max
), the 
selection between each original vector (X
i
) and its trial (U
i
) from the 
current population, will be done regarding DELC. Also, f
1
 and f
2
 are 
the objective function values of the mentioned vectors, respectively, 
and µ
1
 and µ
2
 are their related satisfaction levels. For instance, the 
resulting satisfaction level of vector X
i
 is determined by,
 
1
     
XX X
i
ks
gi hi
ks
min,,
,
 (54)
where all constraints are calculated from piecewise linear functions 
as follows:

gi
ki
ki
k
ki k
k
g
g
b
gb X
X
X
X  
 


  





10
10
0
if
if
otherwise
,
 

 (55)

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h
s
i
si
s
is
b
hb
X
hX
X
 


 





1
0
if
otherwise
s
, (56)
where b
k
 and b
s
 are two positive numbers. In this study, the median 
values of the constraint violations in the initial population are 
employed and these parameters are updated after each generation. 
Here, the selection between two sets of (  f(X
i
), µ(X
i
)) and (  f(U
i
), 
µ(U
i
)) based on DELC with α satisfaction level is according to:
X
UU UX X
X
i
t i
t
i
t
i
tt
i
t
i
t
ff


 

  





1
,,
.

 i
else
 (57)
The α-level comparison <  α and  ≤  α between (  f(X
i
), µ(X
i
)) =  
(  f
1
, µ
1
) and (  f(U
i
), µ(U
i
)) = (  f
2
, µ
2
) are defined in this way:
ff
ff
ff
11 22
12 12
12 12
12
,,
,

 



   






if
otherwise
 

, (58)
It means that due to ≤  α comparison, (  f
1, 
µ
1
) is a better individual 
compared to (  f
2
, µ
2
) when f
1
 ≤   f
2
 (in case of µ
1
, µ
2
 ≥ α or µ
1
 = µ
2
) or 
µ
1
 ≥ µ
2
. Similar consequences can be pointed for < α comparison. Fig. 
3 depicts the detailed implementation of the DELC algorithm. For 
those problems subjected to strong equality constraints, the α-level 
should be controlled to obtain high quality results. However, in 
this study, the α-level does not need to be controlled and like many 
constrained problems, can be solved when α is constantly 1 [27].
5.3  JDE
A standard DE algorithm contains a set of parameters that are 
remained constant throughout the optimization process. In order 
to achieve optimal performance, the tuning of these parameters for 
each optimization problem is necessary. Some researchers claim that 
manually setting the DE inputs is not too difficult [21]. However, 
some argue that this process may be effortful, especially for certain 
optimization problems [28]. The setting of the control parameters 
greatly affects the efficiency, effectiveness, and robustness of the 
DE algorithm. Hence, best parameter selection is a problem-specific 
question, because some may work well for some problems but not 
for others [29]. JDE has self-adaptive control parameter settings and 
shows acceptable performance in benchmark problems. JDE operates 
in such a way that uses a self-adaptive approach to adjust F and CR 
parameters for the best optimization results. The control parameters 
for JDE algorithm are: 
F
Fr and Fr and
F
i
t lu
i
t


 


1 12 1
,
,
,
if
otherwise

 (60)
CR
rand rand
CR
i
t
i
t


 


1 34 2
,
,
.
if
otherwise

 (61)
rand
j
 with j ∈ {1, 2, 3, 4} are uniformly distributed random 
numbers between 0 and 1 and τ
1
 and τ
2
 represent probabilities to set 
factors F and CR where τ
1
 = τ
2
 = 0.1 is recommended, while F
l
  = 0.1 
and F
u
 = 0.9, therefore the new range for F value will be [0.1, 1] [30].
As well as DELC, we use JDE with level comparison (JDELC) 
to obtain optimal structural parameters associated with an FTHE.  
Note that JDELC has a similar procedure to Fig. 3, except that it 
does not need to guess good values of F and CR in the initialization 
step and it has a self-adaption nature. The task of thermal design and 
optimization by DELC and JDELC algorithms is conducted in the 
Fortran programming language.
6  A CASE STUDY AND RESULTS
To demonstrate the described procedure, a case study is considered 
and the effectiveness of the proposed algorithm is assessed by 
analyzing an application example that was earlier investigated by 
GA [8] and HTS [9]. The model of FTHE is cross-flow and both 
fluids are unmixed. The material of the tubes is stainless steel with 
a thermal conductivity of 15 W/(m·K) and a density of 7820 kg/m
3
, 
while the fin material is aluminum with a thermal conductivity of  
Fig. 3.  Flowchart of DELC

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170 W/(m·K) and a density of 2790 kg/m
3
. Tables 2 and 3 show 
operating conditions and cost function constants, respectively.
Table 2.  Operating parameters
Variable Water side Air side
Flow rate, [kg/s] 39.2 58.2
Inlet temperature, [°C] 20 104
Outlet temperature, [°C] 51
Inlet pressure, [kPa] 174.5 174.5
Allowable pressure drop, [Pa] 5200 5200
Heat duty, [kW] 3115 3115
Table 3. Economic constants
Variable Value Variable Value
Cost per unit area, [$/m
2
] 100 Electricity price, [$/MWh] 30
Exponent for area 0.6 Pump efficiency 0.5
Hour of operation, [h/year] 6500
For consistent comparison with previous works, air side, and 
water side allowable pressure drops are considered as 30 Pa and 4500 
Pa, respectively. Also, the thermophysical properties of the fluids 
have been shown in Table 4. The subscript t, m returns to the mean 
temperature value. 
Table 4.  Thermophysical properties
Fluid Air Water
T
m
 = (T
1
+T
2
)/2, [℃] 77.5 29.5
ρ
i
, [kg/m
3
] 1.612 -
ρ
o
, [kg/m
3
] 1.875 -
ρ
t,m
, [kg/m
3
] 1.734 995.8
ρ
ave
 = ( ρ
i
+ ρ
o
)/2, [kg/m
3
] 1.7435 -
μ
t,m
, [Pa·s] 0.00002085 0.0008059
Cp
t,m
, [J/(kgK)] 1008 4180
Pr
t,m
0.7162 5.489
k
t,m
, [W/(mK)] - 0.6137
ΔT
1
 = T
h,i
 – T
c,o
 = 104 – 39 = 65 [℃]
ΔT
2
 = T
h,o
 – T
c,i
 = 51 – 20 = 31 [℃]
ΔT
lm
 = (ΔT
1
 – ΔT
2
) / (ln (ΔT
1
/ ΔT
2
)) = 45.916 [℃]
F
c
 = 0.946
6.1  Minimum Weight Optimization
Table 5, reports the statistical results for total weight minimization, 
including the best, median, mean, worst, and SD. The optimum 
results are marked with boldface.
Table 5. Algorithms’ statistical results for total weight optimization
G
max
 = 500, NP = 70, and α = 1
DELC
CR = 0.9
F Best Median Mean Worst SD
0.2 1080.00 1206.5 1206.5 1464.11 1.27e2
0.4 1013.17 1013.17 1014.97 1041.77 5.9855
0.6 1013.17 1013.17 1013.17 1013.17 1.16e-16
0.8 1013.17 1015.02 1015.40 1019.14 1.5978
1 1013.17 1014.62 1017.11 1039.78 5.9981
1.2 1026.69 1055.82 1056.01 1089.65 16.668
JDELC: 
τ
1
 = τ
2
 = 0.1, 
F
l
 = 0.1, 
F
u
 = 0.9
F  
(Eq. (60))
CR  
(Eq. (61))
1013.17 1013.17 1013.31 1016.5 0.665
Table 6.  Design results for minimum total weight consideration compared to GA [8] and HTS [9]
GA HTS DE
Outside diameter, [mm] 12.63 11.9 10.72
Transverse pitch, [mm] 28.94 30.8 20.5
Longitudinal pitch, [mm] 27.26 30.7 14.89
Fin pitch, [mm] 1.23 1.2 1
Height of shape, [m] 7.29 7.79 8
Width of shape, [m] 4.08 3 3.39
Length of shape, [m] 0.0545 0.0614 0.0298
Number of tubes 502 504 779
Number of fins 3318 2501 3391
Number of tube rows 2 2 2
Fin thickness, [mm] 0.12 0.12 0.12
Volume, [m
3
] 1.62 1.43 0.81
Heat transfer area, [m
2
] 2294.48 2165.3 1222.59
Pressure drop on air side, [Pa] 30 30 30
Pressure drop on water side, [Pa] 4464 4495.1 4471.96
Total annual cost, [$/year] 10873.7 10523 7580.1
Weight of tubes, [kg] 584.66 406.45 630.61
Weight of fins, [kg] 750.42 711.65 383.12
Total weight, [kg] 1335.08 1118.1 1013.73
Fig. 4.  Algorithms’ convergence performance for total weight minimization

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From the results provided in Table 5, it is observed that DELC with 
F = 0.4, 0.6, 0.8, and 1 and JDELC obtain the finest solution at least 
once, but only DELC with F = 0.6 and JDELC have a minimum of the 
median, mean, and worst. Moreover, the SD of these two cases is the 
lowest. So, these concluded as the best schemes. Fig. 4 demonstrates 
the evolution process related to total weight optimization. As seen 
from Fig. 4, DELC with F = 0.6 converges considerably earlier. 
Also, after about 150 generations, there are very slight variations 
in individuals, and finally, an acceptable level of optimal weight is 
reached. As a result, DELC with F = 0.6 provides better performance 
than others and has been chosen as the best manner.
Table 6 shows DELC algorithm’s optimum solution for minimum 
total weight and those obtained by the GA and HTS algorithms 
previously. When compared to the results of the HTS algorithm, 
which were reported before as the best in terms of total weight, a 
significant decrease (33.4 %) in transverse pitch leads obviously 
to an enormous increase (54.6 %) in the number of tubes and, 
consequently, the weight of tubes increases by 55.2 %. The number 
of fins increases (35.6 %) due to the rise in width of shape (13 %) and 
decrease in fin pitch (16.7 %). Despite an increase in the number of 
fins, a significant decrease in the length of shape (51.5 %) besides an 
increase in the number of tubes (As confirmed by Eqs. (5) and (7), 
resulted in a noticeable decrease (43.5 %) in the heat transfer area, 
and so weight of fins decreases by 46.2 %.
Overall, the combined effect of changes in the weight of tubes 
and fins results in a reduction of total weight of about 24.07 % and 
9.33 % employing DE as compared to GA and HTS, respectively.
6.2  Minimum Cost Optimization
Like the first objective function, shown in Table 5, outcomes for 
total annual cost optimization are calculated and listed in Table 7. 
Similarly, the optimal cases are mentioned in boldface.
From the results presented in Table 7, five items achieve the elite 
solution, but the median, mean, and worst are minimum for DELC 
only with F = 0.6 and F = 0.8 besides the SD of both choices is too 
small. Again, DELC with F = 0.6 has a higher speed of convergence 
compared to the other as illustrated in Fig. 5. Furthermore, the 
number of generations needed to gain a good approximation is about 
150. It may make sense that convergence curves have considerable 
overshoot in both weight and cost optimizations, as shown in Figs. 4 
and 5, however, it has been observed that the behavior is significantly 
influenced by design variable N. The optimum solution found by the 
DELC algorithm for minimum total annual cost is gathered in Table 
8.
Table 7.  Algorithms’ statistical results for total annual cost optimization
G
max
 = 500, NP = 70, and α = 1
DELC
CR = 0.9
F Best Median Mean Worst SD
0.2 6778.76 7377.44 7491.69 8605.47 508.364
0.4 6286.38 6287.96 6484.78 7002.65 261.387
0.6 6286.38 6286.38 6286.38 6286.38 1.85e-12
0.8 6286.38 6286.38 6286.39 6286.68 0.05968
1 6286.38 6286.38 6288.43 6298.81 3.86347
1.2 6333.65 6403.27 6396.35 6462.06 37.4456
JDELC: 
τ
1
 = τ
2
 = 0.1,
F
l
 = 0.1, 
F
u
 = 0.9
F 
(Eq. (60))  
CR 
(Eq. (61))
6286.38 6286.38 6286.58 6291.36 0.996
Table 8.  Design results for minimum total annual cost consideration compared to GA [8] and 
HTS [9]
GA HTS DE
Outside diameter, [mm] 13 13 13
Transverse pitch, [mm] 21.78 20.4 20.5
Longitudinal pitch, [mm] 27.26 24.6 16.89
Fin pitch, [mm] 2.87 2.37 1.5
Height of shape, [m] 7.59 5.8 5.54
Width of shape, [m] 4.95 5 5
Length of shape, [m] 0.0545 0.0492 0.03378
Number of tubes 454 567 539
Number of fins 2645 2111 3335
Number of tube rows 2 2 2
Fin thickness, [mm] 0.12 0.12 0.12
Volume, [m
3
] 2.05 1.43 0.9357
Heat transfer area, [m
2
] 1114.45 989.37 876.73
Pressure drop on air side, [Pa] 30 29.14 30
Pressure drop on water side, [Pa] 2654 1856.9 4151.17
Total weight, [kg] 1388.52 1137.46 1054.13
Capital cost, [$/year] 6733.4 6269.2 5830.69
Operating cost, [$/year] 689.1 483.6 457.97
Total annual cost, [$/year] 7422.5 6752.8 6288.66
Fig. 5.  Algorithms’ convergence performance for total annual cost minimization

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As before, when comparing the optimum solution obtained 
by the DELC method to HTS findings, as the best result for total 
annual cost, while the width of shape remains constant, a remarkable 
increase (58 %) in the number of fins occurred by a large decrease 
(36.7 %) in fin pitch. Despite an increase in the number of fins, a 
noticeable decrease (31.3 %) in the length of shape has resulted in a 
significant decrease (11.4 %) in the heat transfer area (As expected 
from Eqs. (5) and (7)) and a considerable decrease in capital cost by 
7 %. Furthermore, an increase in transverse pitch and a decrease in 
height of shape reduce the number of tubes by 4.9 % and operating 
cost by 5.3 %. As a result, when compared to GA and HTS, the 
DELC algorithm saves 15.28 % and 6.87 % of the total annual cost, 
respectively. It is important to mention that the design variables are 
rounded in Tables 6 and 8 and the objective functions are calculated 
accordingly.
The achievements for both objective functions show that the 
DE algorithm with α-level constraint control, provides significant 
improvements in optimal designs by finding the desirable variables 
with satisfied constraints, compared to GA and HTS.
7  PARAMETRIC ANALYSIS
In this section, sensitivity analysis is used to determine how objective 
functions are affected based on changes in structural parameters 
and to indicate the feasible design space. For this purpose, each 
variable is changed within the predefined ranges, while others are 
kept constant at their optimum obtained by DE. Figs. 6 and 7 show 
the influence of variation of outside tube diameter, transverse pitch, 
longitudinal pitch, fin pitch, number of tube rows, height of shape, 
and width of shape on total weight. In these figures, only areas shown 
with symbols are feasible.
Fig. 6.  Effect of outside tube diameter, transverse pitch, longitudinal pitch, 
 and fin pitch on total weight
As seen from these figures, increases in the outside tube diameter, 
longitudinal pitch, number of tube rows, height of shape, and width 
of shape lead to a linear increment in the total weight of the heat 
exchanger, furthermore, as expected when transverse pitch and fin 
pitch increase, the mentioned objective function values decrease. 
Figures show that parametric optimization results agree with the 
solutions acquired by DE algorithm because total weight is optimum 
when transverse pitch and fin pitch are minimum and height of shape 
is maximum. Accordingly, the optimum solution occurs when there 
are two rows of tubes and the outside tube diameter, longitudinal 
pitch, and width of shape are equal to 10.72 mm, 14.89 mm, and 
3.39 m, respectively. Moreover, the feasible space for the design 
parameters; outside tube diameter, longitudinal pitch, and width of 
shape is obtained in this way: 10.72 mm to 13.5 mm, 14.89 mm to 
23.95 mm, and 3.39 mm to 3.41 m. Beyond this range, constraint 
violation happens. As a result of parametric analysis, 24.45 % of 
the solutions are practicable around the obtained optimal point. 
Similarly, Figs. 8 and 9 illustrate the variation of those parameters in 
the solution space for the total annual cost objective function.
Fig. 7.  Effect of height of shape, width of shape, and number of tube rows on total weight
Fig. 8.  Effect of outside tube diameter, transverse pitch, longitudinal pitch,  
and fin pitch on total annual cost
Fig. 9.  Effect of height of shape, width of shape, and number of tube rows on total annual cost
The above figures indicate that an increase in the outside tube 
diameter and fin pitch leads to a decrease in the total annual cost of 
the heat exchanger, primarily due to the reduction in pressure drop, 
which subsequently lowers the operation cost. However, increases in 
transverse pitch, longitudinal pitch, number of tube rows, as well as 
the height and width of the shape, result in a higher total annual cost.
This cost is optimal when the outside tube diameter, transverse pitch, 

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and width of shape are at their maximum (13 mm), minimum (20.5 
mm), and maximum (5 m), respectively. Also, the optimum solution 
is obtained where the longitudinal pitch, number of tube rows and 
height of shape are equal to 16.89 mm, 2 m and 5.54 m, respectively. 
In addition, as shown, those parameters are feasible for outside tube 
diameter, longitudinal pitch, fin pitch and height of shape which are 
within 13 to 13.5 mm, 16.89 mm to 25.57 mm, 1.17 mm to 1.5 mm, 
and 5.54 m to 8 m, respectively. It should be noted that an outside 
tube diameter between 7 mm and 13 mm is selected as the search 
range in the DE routine, but up to 13.5 mm is in the valid range [16], 
and used in the parametric analysis. A violation of the constraint 
occurs outside of these ranges. As a whole, 24.68 % of the space in 
total annual cost parametric analysis is applicable.
8  CONCLUSION
In this paper, the DE algorithm and JDE as a variant of standard DE 
are successfully implemented in the thermal design of a plain fin and 
tube heat exchanger based on the minimization of the total weight 
and total annual cost as two individual objective functions. For this 
purpose, the α-level constraint-handling method is incorporated 
into the global search capability of DE. Improvements in the results 
are observed for both objective functions with those reported by 
GA and HTS approaches previously. Finally, 24.07 % and 9.33 % 
reduction in total weight, and 15.28 % and 6.87 % in total annual 
cost are achieved using the DELC algorithm as compared to the two 
mentioned procedures. Consequently, applying the DELC algorithm 
to optimize FTHEs could give better results than the others. The 
performance of the gas side in the present FTHE may be improved 
by replacing continuous plain fins with high-performance fins, i.e., 
strip fins, slit fins, and louver fins. Furthermore, this study can be 
extended to the optimization of other types of heat exchangers for 
different goals in future work.
NOMENCLATURES
A initial heat transfer area, [m
2
]
A
h
 heat transfer area based on overall heat transfer coefficient, 
[m
2
]
A
i
 water side heat transfer area, [m
2
]
A
min
 minimum free flow area, [m
2
]
A
o
 air side heat transfer area, [m
2
]
CR crossover rate,
C
A
 price per unit area, [$/m
2
]
C
in
 annual cost of investment, [$/year]
C
op
 annual cost of operation, [$/year]
C
p
 specific heat at constant pressure, [J/(kgK)]
d
c
 fin collar outside diameter, [mm]
d
h
 hydraulic diameter, [mm]
d
i
 inside tube diameter, [mm]
d
o
 outside tube diameter, [mm]
f friction factor,
F mutation factor,
F
c
 correction factor,
F
p
 fin pitch, [mm]
F
s
 fin space, [mm]
H height of shape, [m]
k thermal conductivity, [W/(m
2
K)]
N
f
 number of fins per unit length,
NP population size,
N
t
 number of tubes,
Nu Nusselt number,
m fin parameter defined by Eq. (23),
n number of design variables,
p exponent of area,
P
l
 longitudinal tube pitch, [mm]
Pr Prandtl number,
P
t
 transverse tube pitch, [mm]
R radius of tube based on d
o
, [mm]
r
f,eq
 equivalent circular fin radius, [mm]
Re Reynolds number based on d
h
,
Re
dc
 Reynolds number based on d
c
,
R
fi
 internal fouling resistance, [m
2
K/W]
R
fo
 external fouling resistance, [m
2
K/W]
t
f
 fin thickness, [mm]
TAC total annual cost, [$/year]
TW total weight, [kg]
U overall heat transfer coefficient, [W/(m
2
K)]
V volumetric flow rate, [m
3
/s]
W width of shape, [m]
x design variable,
X vector of design variables
Greek letters
η pump and compressor efficiencies,
η
f
 fin efficiency,
η
i
 internal surface efficiency,
η
o
 external surface efficiency,
υ minimum flow velocity, [m/s]
µ fluid dynamic viscosity, [Pa·s]
σ contraction ratio of cross-sectional area,
ρ density, [kg/m
3
]
τ hours of operation per year, [hours/year]
ΔP pressure drop, [Pa]
ΔT
m
 mean temperature difference, [K]
ΔT
lm
 logarithmic mean temperature difference, [K]
ϕ parameter defined by Eq. (24)
Subscripts
a air side
c cold side
f fin
h hot side
i inlet
m mean 
min minimum
max maximum
o outlet
t tube
w water side 
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Received 2023-11-29, revised 2024-08-28, accepted 2024-11-05 
Original Scientific Paper
Data Availability The data supporting the findings of this study are included 
in the article.
Author Contribution Afsharzadeh N. contributed to conceptualization, 
methodology, resources, validation, data curation, formal analysis, and 
programming. Yazdi, M.E. provided conceptualization, methodology, 
supervision.  Lavasani A.M. completed conceptualization, review and editing, 
and consultation.
Uporaba diferencialnega evolucijskega algoritma  
za termodinamično snovanje in omejena optimizacija  
cevno orebrenega prenosnika toplote
Povzetek Cevno orebreni prenosniki toplote (COPT) so vrsta kompaktnih 
prenosnikov toplote (KPT), ki je razširjena v aplikacijah prenosa toplote med 
plini in tekočinami. V pričujoči študiji se za optimizacijo COPT učinkovito 
uporabljata algoritem diferencialne evolucije (DE) in njegova različica JDE 
s tehniko obvladovanja omejitev na ravni α. Za ciljni funkciji sta bili izbrani 
skupna teža in letni strošek obratovanja prenosnika toplote. Zajetih je bilo 
sedem konstrukcijskih spremenljivk: zunanji premer cevi, prečni razmik, 
vzdolžni razmik, razmik reber, število vrst cevi, višina in širina oblike. Za 
analizo prenosa toplote pod enakimi pogoji, kot so masni pretok, vstopna in 
izstopna temperatura, toplotna obremenitev in druge toplotne lastnosti, se 
uporablja metoda srednje logaritemske temperaturne razlike. Za reševanje 
optimizacijskih problemov so bistveno primernejši metahevristični iskalni 
algoritmi. Pri obeh ciljnih funkcijah so se izboljšali rezultati v primerjavi z 
genetskim algoritmom (GA) in tako imenovanim algoritmom iskanja prenosa 
toplote. Parametrična analiza je potrdila, da sta vrednosti ciljnih funkcij 
minimalni pri najmanjši vrednosti prečnega razmika. Predlagana metoda 
je preprosta in v splošnem uporabna pri termodinamičnem snovanju in 
optimizaciji COPT in drugih vrst KPT za različne namene. 
Ključne besede cevno orebreni prenosnik toplote, termodinamična 
zasnova, optimizacija z omejitvami, diferencialni evolucijski (DE) algoritem, 
skupna teža, skupni letni strošek