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Citation: Popescu, L.L.; Popescu, R.S.;
Catalina, T. Study of Heat Recovery
Equipment for Building Applications.
Buildings 2023,13, 3125. https://
doi.org/10.3390/buildings13123125
Academic Editor: Ricardo M. S.
F. Almeida
Received: 22 November 2023
Revised: 13 December 2023
Accepted: 14 December 2023
Published: 17 December 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
buildings
Article
Study of Heat Recovery Equipment for Building Applications
Lelia Letitia Popescu 1,* , Razvan Stefan Popescu 1and Tiberiu Catalina 1,2
1Buildings’ Services Faculty, Technical University of Civil Engineering of Bucharest,
020396 Bucharest, Romania; razvan.popescu@utcb.ro (R.S.P.); tiberiu.catalina@utcb.ro (T.C.)
2National Research and Development Institute URBAN-INCERC, 021652 Bucharest, Romania
*Correspondence: lelia.popescu@utcb.ro
Abstract: Nowadays, heat exchangers find widespread use across various applications in different
fields, particularly in the field of heat recovery. This paper provides a detailed explanation of a
plate heat exchanger counter-flow model developed in Simulink/Matlab. Water was employed in
simulations for both circuits, although the thermal properties of other fluids can be investigated by
modifying them. The “Tanks in series” method was used for simulation purposes. The developed
model enables users to explore the impact of various parameters on heat exchanger functionality,
such as altering the number of plates, the material or thickness of the plates, and the nature of
thermal agents (gaseous or liquid). These models play a crucial role not only in simulating and
sizing heat exchangers but also in achieving parametric optimization. Parameter variations can be
employed to examine the operation of existing equipment under conditions different from their design
specifications. The Simulink/Matlab proposed model, featuring a variable number of finite volumes
to ensure high accuracy, was compared to the classical design method for plate heat exchangers. The
results revealed good accuracy, with relative errors for heat transfer rate remaining below 2.6%. This
research also considered the study of the number of finite volumes necessary for achieving accurate
results. For the 40 finite volumes model, the relative error for heat transfer rate is less than 10%.
Dividing the mesh into 50 finite volumes along the fluid flow direction resulted in relative errors
ranging from 1.6% to 1.7%, indicating that a finer mesh was not necessary. To validate the conceived
model, experimental data from the literature were compared. The relative errors for heat transfer rate
between the Matlab/Simulink model’s results and experimental data ranged from 1.58% to 11.92%,
demonstrating a strong agreement between the conceived model and the experimental values.
Keywords: heat transfer; plate heat exchanger; “tank in tank” method; finite volumes; dynamical
simulation; Matlab/Simulink
1. Introduction
Heat exchangers are used for a wide array of applications, with various types and
sizes available on the market [
1
–
3
]. Almost every household is equipped with such devices,
operating with different fluids on the primary and secondary circuits. The progression of
these systems is characterized by the incorporation of novel technologies and materials,
along with the utilization of dynamic simulation models for the purpose of optimization
and performance evaluation. The advancement of these models, specifically for plate heat
exchangers, has been significantly aided by software tools such as Simulink and Matlab,
which allow for thorough examination and enhancement under different operational
circumstances [4–6].
The interest in developing dynamic simulation models lies in implementing heat
transfer equations in a simulation environment that facilitates the creation of a fast and user-
friendly model. The modeled heat exchanger in this study is of the counter-flow type, using
a plate surface and employing the “tanks in series” method for meshing [
7
,
8
]. This method
involves a mesh structure that can be optimized to ensure acceptable computing times.
Buildings 2023,13, 3125. https://doi.org/10.3390/buildings13123125 https://www.mdpi.com/journal/buildings
Buildings 2023,13, 3125 2 of 14
Literature studies have focused on modeling tube-in-tube heat exchangers using re-
frigerant cycles, particularly for the evaporator and condenser of mechanical compression
cooling machines [
9
]. Given that every mechanical compression machine incorporates at
least two heat exchangers, the significance of these devices and their dynamic simulations
becomes evident. Adsorption machines used for cooling purposes feature four heat ex-
changers, using different cooling agents and fluids at varying temperatures in either vapor
or liquid states [
10
,
11
]. Each aspect is crucial for the sizing and design of heat exchangers,
employing diverse approaches and dynamic simulation tools.
Other simulation models for heat exchangers address steady-state conditions, offering
faster simulation times. Optimizations of the number of channels, passes, and various
configurations are analyzed and refined [
12
]. Additionally, variable inlet parameters play a
key role in simulating real conditions encountered in applications with heat exchangers
(such as variable seawater temperature, outside air temperature, or well water temperature).
The purpose of these simulations is to explore the optimal configuration of the parameters
described above and to understand the influence of a constant or variable overall heat
transfer coefficient.
Papers addressing malfunctions in the operation of heat exchangers are particularly
interesting in cold climates [
13
]. These exchangers are commonly used for air cooling in
cooling batteries within air handling units, frequently found in numerous office buildings.
In marine applications, these heat exchangers utilize air-to-water fluids. Malfunctions may
occur due to freezing water, which can partially or completely block the heat exchanger.
Therefore, it is crucial to input the water temperature at the inlet of the heat exchanger into
the model, considering that seawater temperature can vary, sometimes reaching freezing
conditions depending on the depth of aspiration.
The Laplace transform was employed as a simulation tool in Matlab/Simulink soft-
ware R2015b, and validation results are also presented. The heat exchanger in question
is a cross-flow unit, and the discretization uses the Laplace transform to incorporate bi-
dimensional modeling.
A similar simulation model explored water-to-water heat exchangers developed in the
Matlab/Simulink environment, a user-friendly simulation platform with a useful library
featuring numerous examples of different component types that can be connected to create
the desired model for study [
14
]. The implementation involved differential finite equations,
and various parameter variations are presented. The mathematical model is iterative, with
variable simulation times for different solvers, offering the flexibility of using either fixed
or variable steps based on the equations implemented in the design.
Matlab/Simulink simulation software presents significant advantages, being a well-
known tool with ample online resources for assistance, including a comprehensive pre-
sentation of similar topics on the developer’s website. It comprises two distinct software
components, with Matlab serving as a classic computer programming tool, while Simulink
operates as block connection software featuring blocks with diverse functions, ranging
from simple to more complex.
The results presented [
14
] used a discretization with just five elementary cells due to
the required computing time, considering both summer and winter sea water temperatures
in steady-state conditions with water properties derived from the Black Sea. This research
provides numerical results at each node without optimizing the discretization or comparing
them with experimental data. Future work should involve the incorporation of variable
inlet parameters, such as water temperature and flow, along with a comparison with
experimental data and optimization of the model’s discretization.
Over the past few years, there has been an increasing use and modeling of a specific
heat exchanger: the vertical ground heat exchanger. Due to the prevalence of heat pumps,
particularly the ground-to-water type, optimizing and designing such a heat exchanger
becomes crucial for long-term usage. Dedicated software applications exist for this purpose,
as well as experimental tests that play a vital role in simulation and sizing [
15
–
17
]. For
buildings requiring a substantial number of wells and ground heat exchangers, their
Buildings 2023,13, 3125 3 of 14
quantity is of great significance for economic reasons [
18
–
20
]. The Thermal Response Test
(TRT) experimental method is a widely employed measurement in such applications. It
involves measuring the properties of the soil in a specific location to provide essential data
for simulations and design capacities of ground heat exchangers, especially for heat pumps
in large buildings.
In pursuit of energy efficiency and a higher fresh air rate, air-to-soil heat exchangers
are also used in large building applications with significant airflow requirements. These
systems involve placing a special type of plastic pipe, typically containing silver ions
for bacteriological reasons, into the ground. This setup heats and cools the indoor air
needed for the building’s Air Handling Units, recovering energy freely from the ground.
Commonly referred to as a Canadian well or Provençal well, such a heat exchanger can also
be designed for cooling purposes in warm climates [
21
–
23
]. The performance of this heat
exchanger depends on various factors, such as soil nature, airflow, humidity, meteorological
conditions, groundwater depth, and the depth of the grounded heat exchanger.
The diverse heat exchangers described above have broad applications in our lives and
are available in various types on the market. The present work focuses on the simulation
and optimization of a plate heat exchanger’s meshing into several elementary cells. As a
prospective step, besides comparing the fundamentals of heat transfer theory, an experi-
mental setup would be valuable. A comparison between modeling in Matlab/Simulink and
measured values is presented and discussed [
24
]. The model used consists of three partial
differential equations (PDEs) employing the finite volume method. Due to the short fluid
residence time and limited dynamics of the inlet temperatures in this work, only one cell
volume was used in the simulations. A good agreement was found with the experimental
results, validating the simulated model.
Another study presents dynamic simulations of a counter-flow heat exchanger [
25
].
The results of this study are presented and compared to experimental data found in the
literature to validate the results. Simulations include inlet temperatures for both fluids
and changes in the mass flow rate. The numerical method in this case is based on the
analytical solution of the energy equation, a model that, according to the authors, offers a
fast computing time.
An interesting and similar work are presented in [
26
] with a simplified plate heat
exchanger model, compared and validated with experimental data. The Orthogonal Collo-
cation Method (OCM) is used for discretization, and the accuracy and comparison with
experimental data are reported to be very good.
Studies on heat transfer involving 3D nanofluids (water-based) are presented in [
27
].
Another mathematical model concerning fluid flow through a porous and stretching sheet
is analytically simulated, including mass transfer, thermal radiation, and Hall current [
28
].
Detailed mathematical approaches concern hybrid nanofluid flow energy transfer through
a permeable vertically rotating surface [
29
] and other numerical models of heat transfer,
including convergence of the studied models [
30
,
31
]. These nanofluids have demonstrated
promising capabilities in enhancing thermal conductivity and heat transfer efficiency. The
investigation of Al
2
O
3
-water nanofluids has been conducted to assess their impact on
the efficiency of cross-flow microheat exchangers, revealing notable enhancements in
heat transfer capabilities [
32
–
36
]. Furthermore, the incorporation of heat exchangers into
intricate systems, such as solar thermal heat pump hybrid systems, has been subjected to
modeling in order to enhance energy preservation in buildings. This analysis underscores
the significance of heat exchangers in sustainable energy alternatives [37].
An interesting study presents an investigation on how ultrasonic excitation affects
heat transfer rates in a fin-and-flat tube heat exchanger [
38
]. Parameters such as ambient
temperature, flow rate, air passing velocity, Reynolds number, and Nusselt number were
varied. Results show that reducing flow rate, ambient temperature, and air passing velocity
enhances ultrasonic effects, with the highest heat transfer enhancement reaching 70.11%.
The findings offer valuable insights for optimizing the design of ultrasonic vibrating fin-
and-tube heat exchangers.
Buildings 2023,13, 3125 4 of 14
Studies investigate how nanofluid concentration influences heat transfer in an ultra-
sonic finned tube heat exchanger using a Multi-Walled Carbon Nanotube (MWCN) [
39
].
This research validates results and assesses uncertainties, exploring parameters like ambi-
ent temperature and MWCNT concentration. Findings indicate enhanced heat transfer with
increased nanofluid concentration, proposing a promising avenue for combining nanofluid
and ultrasonic benefits in optimal conditions.
Recent studies explore the heat performance of a Multi-Layered Oscillating Heat Pipe
Heat Exchanger (ML-OHPHE) for heat recovery in HVAC systems [
40
]. Experimental
tests under different conditions were conducted and compared with simulations using
Honeywell’s UniSim
®
Design Suite software. The results suggest that the ML-OHPHE
could effectively serve as a passive heat transfer device for HVAC heat recovery [40].
Metal additive manufacturing (AM) with SUS316L material to create oscillating heat
pipes was considered in recent studies [
41
]. Experiments on printing parameters and
thermal performance were conducted. Suitable laser parameters produced oscillating heat
pipes with good compactness and minimal dimensional error. This study also investigated
the impact of inter-channel spacing on thermal performance, revealing that reducing ther-
mal interaction can enhance the oscillation effect, resulting in improved equivalent thermal
conductivity. The experiments demonstrated higher equivalent thermal conductivity at
low power with reduced thermal interaction [41].
In summary, the ongoing investigation and progress in heat exchanger technology,
which encompasses a range of kinds and applications, plays a vital role in the progression
of energy systems. The incorporation of novel materials, computer modeling, and opti-
mization methods plays a crucial role in augmenting the effectiveness and sustainability of
heat exchangers. This, in turn, contributes to worldwide endeavors aimed at conserving
energy and safeguarding the environment.
The novelty of this paper lies in the detailed explanation of a counter-flow plate heat
exchanger model developed using Simulink/Matlab. Water was employed in simulations
for both circuits, although the thermal properties of other fluids can be investigated by
modifying them. The “Tanks in series” method was utilized for simulation purposes.
The developed model enables users to explore the impact of various parameters on heat
exchanger functionality, such as altering the number of plates, the material or thickness
of the plates, and the nature of thermal agents (gaseous or liquid). The results of the
proposed Simulink/Matlab model were compared to the classical design method for plate
heat exchangers and experimental data from the literature.
2. Materials and Methods
This study aim was to describe by numerical simulations heat transfer between two
fluids crossing a plate heat exchanger. Heat transfer equations were solved using Mat-
lab/Simulink mathematical modeling software. The Matlab/Simulink model was tested
under the same conditions as a heat exchanger designed with a classical algorithm from the
literature [
31
]. The input parameters were the same for the Matlab/Simulink model, and
for the classical calculated method, the output temperatures for both fluids obtained with
both methods were compared. Also, the heat flow values were compared. Relative errors
were calculated between the two methods, both for fluids’ outlet temperatures and heat
flow. Experimental data from the literature [
31
] were taken for the plate heat exchanger, and
the conceived Matlab/Simulink model was tested for the same conditions. The purpose
of both comparisons with the conceived model was to express the good agreement of our
model with two very different methods of validation.
The conceived model can be used by engineers to determine heat exchanger work-
ing parameters for other conditions than the ones presented in the technical report of an
equipment, being very useful and user-friendly. As well, changing the number of plates
or the type of plate can be of interest to a heat exchanger designer. Generally, the work-
ing parameters are given for testing conditions, but real-life situations differ from those
conditions. So, easily determining the correct value for heat flow for different inlet/outlet
Buildings 2023,13, 3125 5 of 14
temperatures is often very useful for thermal engineers. All parameters are introduced by
the model user in a panel, a “mask.”
2.1. Overall Heat Transfer Coefficient and Plate Heat Exchanger Calculation
The first step in our study was to calculate the overall heat transfer coefficient and
plate heat exchanger design according to [
31
], as described hereafter. The input data for
the heat exchanger were: both fluids were water; the primary fluid’s temperatures vary
between 90
C and 104
C; and the two fluids’ temperatures vary between 60
C and
80
C. The exchanger’s heat transfer rate was 168 kW. The heat exchanger was made of
stainless-steel plates, each one having 0.2 m
2
surface area. The plate’s thickness was 5.5 mm.
Plate thickness is not a usual one, but we considered it to test the model limitations under
very different conditions.
The heat balance equation written for a heat exchanger is as follows:
?=⌘⇢
a1Va1cpa1 (Ta1 in Ta1 out)=⇢a2Va2cpa2 (Ta2 in Ta2 out)=USDTlog (1)
where:
?
is the heat flow [W];
⌘
is the thermal isolation efficiency,
⇢a1
and
⇢a2
are the
fluids’ density at mean temperature [kg/m
3
];
Va1
and
Va2
are the volumetric flows for both
fluids [m
3
/s];
cpa1 and cpa2
are the fluids’ heat capacity at mean temperature [J/(kg
·
K)];
Ta1 in and Ta1 out
the inlet and outlet temperature for primary fluid [K];
Ta2 in and Ta2 out
the inlet and outlet temperature for secondary fluid [K]; U is the overall heat transfer
coefficient [W/(m
2·
K)]; S is the total surface of all thermally effective plates [m
2
];
DTlog
is
the logarithmic mean temperature difference [K], calculated for counter-flow arrangements.
According to the heat balance Equation (1) and fluids’ thermal properties, the mass
flow rate for each fluid can be calculated. From the third part of the same equation, the
total surface area of all thermally effective plates can be estimated as a first step. For this,
the logarithmic mean temperature difference can be calculated using the input data and
an overall heat transfer coefficient estimated in a custom range for a plate heat exchanger
(in our case, between 3000 and 7000 W/m
2·
K). The plate’s characteristics depend on the
manufacturer we considered the following for our study: stainless steel plates, each having
a 0.2 m2surface area and a 5.5 mm thickness.
The effective number of plates is:
N02=S0
s(2)
where N
0
is the preliminary total number of plates and S
0
is the total surface of all thermally
effective plates [m2].
The hydraulic diameter of the channel d
h
(Equation (3)), channel flow area, flow
rates (obtained from Equation (1)), and thermodynamic fluid properties at their mean
temperature are used to calculate the first flow speed and the heat transfer coefficient.
dh=4 channel flowarea area
wetted perimeter =4 gL
2(g+L')⇡2g
'(3)
where g is the channel spacing, L is the plate’s width, and
'
is the surface enlargement
factor, defined as the ratio of the actual effective area given by the manufacturer to the
projected plate area.
The mass flow rate per channel, mass velocity, and Reynolds number for cold and hot
fluids can be calculated afterwards. Heat transfer coefficients strongly depend on a large
number of parameters, like chevron inclination angle relative to flow direction, corrugation
profile, channel spacing, surface enlargement factor (
')
, thermodynamic temperature-
dependent fluids’ properties, etc. Parts of these parameters are not completely presented
by the manufacturer. The conventional approach for heat transfer coefficient employs a
correlation between the Nusselt number and Reynolds number based on the hydraulic
diameter of the channel (Equation (3)) [24].
Buildings 2023,13, 3125 6 of 14
The overall heat transfer coefficient, which also considers the heat exchanger’s fouling,
can be estimated as:
U=1
hhot fluid +1
hcold fluid +plate
plate +Rfouling cold side +Rfouling hot side (4)
where: h
hot fluid
and h
cold fluid
are the heat transfer coefficients for hot and cold fluid
(W/m
2
K),
plate
is the plate thickness (m),
plate
is the thermal conductivity of the plate
material (W/m K),
Rfouling cold side
and
Rfouling hot side
are the fouling resistance on cold and
hot sides (m2K/W). For our calculus, the fouling was neglected on both sides.
After the overall heat transfer is calculated, if the difference between the calculated
value and the estimated one used as a preliminary one is less than 3%, the design step
is finished. Otherwise, the next iteration is started using the last calculated overall heat
transfer coefficient, and the iterations are continued until a less than 3% error is obtained.
The process specifications and construction data for the plate heat exchanger are
presented in Table 1. Table 2presents the calculated parameters for the heat exchanger,
based on the described method and the value from the last iteration. These values are
further used in simulations in Matlab/Simulink.
Table 1. The process specification data for the plate heat exchanger.
Parameter Hot Fluid Cold Fluid
Fluids Wastewater Cooling water
Mass flow rates (kg/s) 2.88 2.01
Inlet temperature (C) 104 60
Outlet temperature (C) 90 80
Specific heat (J/g K) 4.21 4.19
Viscosity (s/m2) 0.284 ⇥1060.415 ⇥106
Thermal conductivity (W/m K) 0.6836 0.6676
Density (kg/m3)955.4 977.7
Nusselt (-) 195 168.7
Heat transfer coefficient (W/m2K) 12,090 10,238
Table 2. The constructional data for the plate heat exchanger.
Parameter Value U.M.
Plate material Stainless Steel (-)
Thermal conductivity of plate material 17 W/m K
Plate surface 0.2 m2
Plate thickness 5.5 mm
Plate height 989 mm
Plate width 242 mm
Plate material density 7850 kg/m3
Plate material-specific heat 0.49 J/g K
Chevron angle 60 degrees
Enlargement factor 1.19 -
Effective number of plates 10 -
Buildings 2023,13, 3125 7 of 14
2.2. Heat Exchanger Simulation in Matlab/Simulink
Each system’s element to be modeled is represented by a block containing either the
equations’ characteristics or blocks that represent the system’s sub-elements. Those blocks
can also be pre-programmed elements representing different operators, like integration,
sum, product, etc., or different software functionalities, like visualization of results, for
example. Whatever their nature, the blocks are connected to each other in the graphic
interface to assemble the system’s equations and reproduce its operating dynamics.
The heat transfer algorithm based on the heat transfer equations is presented in
Figure 1for two simulation elementary cells, cell “i” and cell “i 1”.
Buildings 2023, 13, x FOR PEER REVIEW 7 of 14
Each system’s element to be modeled is represented by a block containing either the
equationsʹ characteristics or blocks that represent the system’s sub-elements. Those
blocks can also be pre-programmed elements representing different operators, like inte-
gration, sum, product, etc., or different software functionalities, like visualization of re-
sults, for example. Whatever their nature, the blocks are connected to each other in the
graphic interface to assemble the system’s equations and reproduce its operating dy-
namics.
The heat transfer algorithm based on the heat transfer equations is presented in
Figure 1 for two simulation elementary cells, cell “i” and cell “i − 1”.
Figure 1. Heat transfer algorithms are considered in the Matlab/Simulink model.
In order to describe the heat transfer, five elementary cells were considered in the
heat transfer rate direction. For each cell, a heat transfer equation was considered based
on previous research experience [41,42], as follows:
● Cell i,1, which describes fluid 1 temperature variation, is noted with “Ta1”. Fluid 1 is
the one with the highest temperature. This cell is placed in the fluid 1 zone, with
forced convection heat transfer being considered.
ρ V c
dT
dt =h
A T
−T
)+w
c T −T
) (5)
where: ρ is the primary fluid’s density at mean temperature [kg/m3]; V is the pri-
mary fluid volumetric flow [m3/s]; c is the primary fluid’s heat capacity at mean
temperature [J/(kg・K)]; T is the celli,1 temperature for primary fluid [K]; T is the
celli,1 inlet temperature for primary fluid [K]; hT1 is the heat transfer coefficient for pri-
mary fluid [W/(m2・K)], A is cell i,1 heat surface [m2], T
is the cell i,2 temperature [K];
w is the primary fluid mass flow [kg/s].
● Cell i,2, which describes the plate temperature variation on the side of fluid 1, is
noted with “T1”. Half of this cell dimension in the heat transfer rate direction is
placed in the fluid 1 zone, and the other half is in the plate thickness. The plate
thickness is, as presented in Figure 1, “2δx”. Each elementary cell has a “ δx” thick-
ness. The thickness of the cell is its dimension in the heat transfer rate direction. Heat
Figure 1. Heat transfer algorithms are considered in the Matlab/Simulink model.
In order to describe the heat transfer, five elementary cells were considered in the heat
transfer rate direction. For each cell, a heat transfer equation was considered based on
previous research experience [41,42], as follows:
•
Cell
i,1
, which describes fluid 1 temperature variation, is noted with “T
a1
”. Fluid 1
is the one with the highest temperature. This cell is placed in the fluid 1 zone, with
forced convection heat transfer being considered.
⇢a1Va1cpa1
dTa1
dt =hT1As(T1Ta1)+wa1cpa1 (Ta1in Ta1)(5)
where:
⇢a1
is the primary fluid’s density at mean temperature [kg/m
3
];
Va1
is the primary
fluid volumetric flow [m
3
/s];
cpa1
is the primary fluid’s heat capacity at mean temperature
[J/(kg
·
K)];
Ta1 is
the cell i,1 temperature for primary fluid [K];
Ta1 in
is the cell
i,1
inlet
temperature for primary fluid [K]; h
T1
is the heat transfer coefficient for primary fluid
[W/(m
2·
K)],
As
is cell
i,1
heat surface [m
2
],
T1
is the cell
i,2
temperature [K];
wa1
is the
primary fluid mass flow [kg/s].
•
Cell i,2, which describes the plate temperature variation on the side of fluid 1, is noted
with “T
1
”. Half of this cell dimension in the heat transfer rate direction is placed
in the fluid 1 zone, and the other half is in the plate thickness. The plate thickness
is, as presented in Figure 1, “2
x
”. Each elementary cell has a “
x
” thickness. The
Buildings 2023,13, 3125 8 of 14
thickness of the cell is its dimension in the heat transfer rate direction. Heat transfer
inside this cell is obtained by forced convection in the fluid layer and conduction in
the metal layer.
⇢m
x
2cpm
dT1
dt =m
x(T2T1)+hT1(Ta1 T1)(6)
where:
⇢m
is the metal density [kg/m
3
]; “
x
” cell’s thickness [m];
cpm
is the metal’s heat
capacity [J/(kg
·
K)];
T1
is the cell
i,2
temperature [K];
m
is the metal’s thermal conductivity
[W/(m
·
K)];
T2
is the cell
i,3
temperature [K]; h
T1
is the heat transfer coefficient for primary
fluid [W/(m2K)]; Ta1 is the cell i,1 temperature for primary fluid [K].
•
Cell i,3, which describes the plate temperature variation at its half thickness, is noted
with “T
2
”. The entire cell is made of metal, so heat transfer by conduction is considered.
⇢mxcpm
dT2
dt =m
x(T1T2)m
x(T2T3)(7)
where:
⇢m
is the metal density [kg/m
3
]; “
x
” cell’s thickness [m];
cpm
is the metal’s heat
capacity [J/(kg
·
K)];
T2
is the cell
i,3
temperature [K];
m
is the metal’s thermal conductivity
[W/(m·K)]; T1is the cell i,2 temperature [K]; T3is the cell i,4 temperature [K].
•
Cell i,4, which describes the plate temperature variation on the side of fluid 2, is noted
with "T
3
." Half of this cell dimension in the heat transfer rate direction is placed in the
fluid 2 zone, and the other half is in the plate thickness. Heat transfer inside this cell is
obtained by forced convection in the fluid layer and conduction in the metal layer.
⇢m
x
2cpm
dT3
dt =m
x(T2T3)+hT2(Ta2 T3)(8)
where:
⇢m
is the metal density [kg/m
3
]; “
x
” cell’s thickness [m];
cpm
is the metal’s heat
capacity [J/(kg
·
K)];
T3
is the cell
i,4
temperature [K];
m
is the metal’s thermal conductivity
[W/(m
·
K)];
T2
is the cell
i,3
temperature [K]; h
T2
is the heat transfer coefficient for secondary
fluid [W/(m2K)]; Ta2 is the cell i,1 temperature for secondary fluid [K].
•
Cell i,5, which describes fluid 2 temperature variation, is noted with “T
a2
”. Fluid 2
is the one with the lowest temperature. This cell is placed in the fluid 2 zone; forced
convection heat transfer is being considered.
⇢a2Va2cpa2
dTa2
dt =hT2As(T3Ta2)+wa2cpa2 (Ta2in Ta1)(9)
where:
⇢a2
is the secondary fluid’s density at mean temperature [kg/m
3
];
Va2
is the sec-
ondary fluid volumetric flow [m
3
/s];
cpa2
is the secondary fluid’s heat capacity at mean
temperature [J/(kg
·
K)];
Ta2 is
the cell
i,5
temperature for secondary fluid [K]; h
T2
is the heat
transfer coefficient for secondary fluid [W/(m
2
K)];
As
is cell
i,5
heat surface [m
2
],
T3
is the
cell
i,4
temperature [K];
wa2
is the secondary fluid mass flow [kg/s];
Ta2in
is the cell
i,5
inlet
temperature for secondary fluid [K].
The heat exchanger “tanks in series” model was performed in five elementary cells
along the heat transfer direction and one or more cells along the fluid flow. For the first
model, only one cell along the fluid flow was considered. Thus, in order to verify the
model’s convergence, after this step, the number of elementary cells along the fluid flow
was increased in order to obtain its influence on the results, until little influence was
observed. Figure 2presents a block diagram corresponding to the second level of modeling
in Matlab/Simulink.
Buildings 2023,13, 3125 9 of 14
Buildings 2023, 13, x FOR PEER REVIEW 9 of 14
Figure 2. Block diagram corresponding to the 2nd level of modeling in Matlab/Simulink for a 1-cell
model.
For the model evaluation, data obtained following the heat transfer steps for the
plate heat exchanger were used. The primary fluid’s temperatures vary between 90 °C
and 104 °C, and the fluidʹs temperatures vary between 60 °C and 80 °C. Both fluids were
water. The exchanger’s heat transfer rate was 168 kW. The heat exchanger was made of 10
middle plates made of stainless steel, each one having 0.2 m2 surface area. The plate’s
thickness was 5.5 mm. According to the values presented, the heat transfer coefficient by
convection for fluid one was calculated to be equal to 12,090 W/m2K, and the heat transfer
coefficient by convection for fluid two was equal to 10,238 W/m2K. The calculated pa-
rameters were used in Matlab/Simulink in order to verify the differences between the two
calculation methods.
Figure 3 shows the second heat transfer equation, wrien for “T1” temperature from
the 3rd level of modeling in Matlab/Simulink.
Figure 3. Block diagram corresponding to the 3rd level of modeling in Matlab/Simulink, T1 equa-
tion.
3. Results and Discussions
In Figure 2, where the second level of modeling from Matlab/Simulink is presented,
the five temperatures can be observed. The outlet temperature of fluid one obtained with
Matlab/Simulink is 95.99 °C, and the outlet temperature of fluid two is 71.54 °C, with
relative errors of −6.66% and 10.58% between the two calculation variants for primary
Figure 2. Block diagram corresponding to the 2nd level of modeling in Matlab/Simulink for a
1-cell model.
For the model evaluation, data obtained following the heat transfer steps for the plate
heat exchanger were used. The primary fluid’s temperatures vary between 90
C and
104
C, and the fluid’s temperatures vary between 60
C and 80
C. Both fluids were water.
The exchanger’s heat transfer rate was 168 kW. The heat exchanger was made of 10 middle
plates made of stainless steel, each one having 0.2 m
2
surface area. The plate’s thickness was
5.5 mm. According to the values presented, the heat transfer coefficient by convection for
fluid one was calculated to be equal to 12,090 W/m
2
K, and the heat transfer coefficient by
convection for fluid two was equal to 10,238 W/m
2
K. The calculated parameters were used
in Matlab/Simulink in order to verify the differences between the two calculation methods.
Figure 3shows the second heat transfer equation, written for “T
1
” temperature from
the 3rd level of modeling in Matlab/Simulink.
Buildings 2023, 13, x FOR PEER REVIEW 9 of 14
Figure 2. Block diagram corresponding to the 2nd level of modeling in Matlab/Simulink for a 1-cell
model.
For the model evaluation, data obtained following the heat transfer steps for the
plate heat exchanger were used. The primary fluid’s temperatures vary between 90 °C
and 104 °C, and the fluidʹs temperatures vary between 60 °C and 80 °C. Both fluids were
water. The exchanger’s heat transfer rate was 168 kW. The heat exchanger was made of 10
middle plates made of stainless steel, each one having 0.2 m2 surface area. The plate’s
thickness was 5.5 mm. According to the values presented, the heat transfer coefficient by
convection for fluid one was calculated to be equal to 12,090 W/m2K, and the heat transfer
coefficient by convection for fluid two was equal to 10,238 W/m2K. The calculated pa-
rameters were used in Matlab/Simulink in order to verify the differences between the two
calculation methods.
Figure 3 shows the second heat transfer equation, wrien for “T1” temperature from
the 3rd level of modeling in Matlab/Simulink.
Figure 3. Block diagram corresponding to the 3rd level of modeling in Matlab/Simulink, T1 equa-
tion.
3. Results and Discussions
In Figure 2, where the second level of modeling from Matlab/Simulink is presented,
the five temperatures can be observed. The outlet temperature of fluid one obtained with
Matlab/Simulink is 95.99 °C, and the outlet temperature of fluid two is 71.54 °C, with
relative errors of −6.66% and 10.58% between the two calculation variants for primary
Figure 3. Block diagram corresponding to the 3rd level of modeling in Matlab/Simulink, T
1
equation.
3. Results and Discussion
In Figure 2, where the second level of modeling from Matlab/Simulink is presented,
the five temperatures can be observed. The outlet temperature of fluid one obtained with
Matlab/Simulink is 95.99
C, and the outlet temperature of fluid two is 71.54
C, with
relative errors of
6.66% and 10.58% between the two calculation variants for primary
fluid and secondary fluid outlet temperatures, respectively. Concerning the relative errors
calculated between the heat flow transferred from the primary fluid and received by the
secondary one, with respect to the design theme, 42.8% and 42.3% values were respectively
Buildings 2023,13, 3125 10 of 14
obtained. These errors are induced by the coarse model mesh along the fluid flow. In
order to observe the impact of model mesh along the fluid flow, results from models
having 1, 10, 20, 30, 40, and 50 cells along the fluid flow are compared in Table 3. Modeled
values and calculated errors with respect to the given values by the design theme for outlet
temperatures, temperature difference between inlet and outlet, and heat transfer rates for
both fluids are presented in Table 3. In Figure 4, the second level from Matlab/Simulink for
the 50-cell model is presented for three consecutive cells. Each of the 50 cells contains under
the mask the five equations presented in Figure 2and explained before (Equations (5)–(9)).
Table 3. Modeled values and calculated errors with respect to the values given by the design theme
for outlet temperatures of the two fluids and heat transfer rates.
Parameter 1 Cell 10 Cells 20 Cells 30 Cells 40 Cells 50 Cells Reference
ta1 out [C] 95.99 93.12 92.29 91.74 90.6 90.37 90
ta2 out [C] 71.54 75.68 76.87 77.66 79.31 79.64 80
Relative error for t
a1
out [%]
6.66 3.47 2.54 1.93 0.67 0.41
Relative error for t
a2
out [%]
10.58 5.40 3.91 2.93 0.86 0.45
(ta1 in–ta1 out) [C] 8.01 10.88 11.71 12.26 13.4 13.63 14
(ta2 out–ta2 in) [C] 11.54 15.68 16.87 17.66 19.31 19.64 20
ø1 [kW] 97.1 131.9 142.0 148.7 162.5 165.3 169.7
ø2 [kW] 97.1 131.9 141.9 148.6 162.5 165.2 168.3
Relative error for ø1 [%] 42.8 22.3 16.4 12.4 4.3 2.6
Relative error for ø2 [%] 42.3 21.6 15.7 11.7 3.4 1.8
Buildings 2023, 13, x FOR PEER REVIEW 10 of 14
fluid and secondary fluid outlet temperatures, respectively. Concerning the relative er-
rors calculated between the heat flow transferred from the primary fluid and received by
the secondary one, with respect to the design theme, 42.8% and 42.3% values were re-
spectively obtained. These errors are induced by the coarse model mesh along the fluid
flow. In order to observe the impact of model mesh along the fluid flow, results from
models having 1, 10, 20, 30, 40, and 50 cells along the fluid flow are compared in Table 3.
Modeled values and calculated errors with respect to the given values by the design
theme for outlet temperatures, temperature difference between inlet and outlet, and heat
transfer rates for both fluids are presented in Table 3. In Figure 4, the second level from
Matlab/Simulink for the 50-cell model is presented for three consecutive cells. Each of the
50 cells contains under the mask the five equations presented in Figure 2 and explained
before (Equations (5)–(9)).
Table 3. Modeled values and calculated errors with respect to the values given by the design theme
for outlet temperatures of the two fluids and heat transfer rates.
Parameter 1 Cell 10 Cells 20 Cells 30 Cells 40 Cells 50 Cells Reference
ta1 out [°C] 95.99 93.12 92.29 91.74 90.6 90.37 90
ta2 out [°C] 71.54 75.68 76.87 77.66 79.31 79.64 80
Relative error for ta1 out [%] −6.66 −3.47 −2.54 −1.93 −0.67 −0.41
Relative error for ta2 out [%] 10.58 5.40 3.91 2.93 0.86 0.45
(ta1 in–ta1 out) [°C] 8.01 10.88 11.71 12.26 13.4 13.63 14
(ta2 out–ta2 in) [°C] 11.54 15.68 16.87 17.66 19.31 19.64 20
ø1 [kW] 97.1 131.9 142.0 148.7 162.5 165.3 169.7
ø2 [kW] 97.1 131.9 141.9 148.6 162.5 165.2 168.3
Relative error for ø1 [%] 42.8 22.3 16.4 12.4 4.3 2.6
Relative error for ø2 [%] 42.3 21.6 15.7 11.7 3.4 1.8
Figure 4. Block diagram corresponding to the 2nd level of modeling in Matlab/Simulink, 3 consec-
utive cells from the 50-cell model.
The results presented in Table 3 show that there is a strong correlation between the
Matlab/Simulink obtained results and the calculated results by means of stationary heat
transfer equations for plate heat exchangers. The relative error calculated for primary
agent outlet temperature varies from −6.66% to −0.41% between the 1 cell model and the
50 cells model. For the secondary agent outlet temperature, the relative error varies from
10.58% to 0.45% between the 1 cell model and the 50 cells model. The relative error cal-
culated between the heat transfer rate obtained in Matlab/Simulink and the design theme
varies between 42.8% and 2.6% for the primary fluid and 42.3% and 1.8% for the sec-
ondary fluid. It can be noticed that with the 40-cell model, the relative error for heat
transfer rate is less than 10%, the relative error accepted in engineering. Dividing the
mesh into 50 cells along the flow direction, the relative errors vary by 1.6–1.7%, meaning
that a finer mesh is not necessary.
Figure 5 presents the transitory part of the temperature distribution of the extreme
cells for the first and 50th cells of the model. The constant of time for the system is more
than 15 s, when equilibrium is reached for all the elementary cells in the heat exchanger.
Figure 4. Block diagram corresponding to the 2nd level of modeling in Matlab/Simulink, 3 consecu-
tive cells from the 50-cell model.
The results presented in Table 3show that there is a strong correlation between the
Matlab/Simulink obtained results and the calculated results by means of stationary heat
transfer equations for plate heat exchangers. The relative error calculated for primary agent
outlet temperature varies from
6.66% to
0.41% between the 1 cell model and the 50 cells
model. For the secondary agent outlet temperature, the relative error varies from 10.58%
to 0.45% between the 1 cell model and the 50 cells model. The relative error calculated
between the heat transfer rate obtained in Matlab/Simulink and the design theme varies
between 42.8% and 2.6% for the primary fluid and 42.3% and 1.8% for the secondary fluid.
It can be noticed that with the 40-cell model, the relative error for heat transfer rate is
less than 10%, the relative error accepted in engineering. Dividing the mesh into 50 cells
along the flow direction, the relative errors vary by 1.6–1.7%, meaning that a finer mesh is
not necessary.
Figure 5presents the transitory part of the temperature distribution of the extreme
cells for the first and 50th cells of the model. The constant of time for the system is more
than 15 s, when equilibrium is reached for all the elementary cells in the heat exchanger.
In this figure, we can notice the primary fluid’s temperature for the first cell is 104 C and
around 90
C (90.37
C) for the last one. The counter-flow, the secondary fluid, has an
outlet temperature of around 80
C (79.64
C) after the first cell, at its outlet, and of 60
C
after the 50th cell, at its inlet. All temperature distributions represented have an initial
temperature of 15
C for the system, which is assumed to be the ambient temperature
Buildings 2023,13, 3125 11 of 14
around and inside the heat exchanger before being heated and can be modified according
to environmental conditions.
Buildings 2023, 13, x FOR PEER REVIEW 11 of 14
In this figure, we can notice the primary fluid’s temperature for the first cell is 104 °C and
around 90 °C (90.37 °C) for the last one. The counter-flow, the secondary fluid, has an
outlet temperature of around 80 °C (79.64 °C) after the first cell, at its outlet, and of 60 °C
after the 50th cell, at its inlet. All temperature distributions represented have an initial
temperature of 15 °C for the system, which is assumed to be the ambient temperature
around and inside the heat exchanger before being heated and can be modified according
to environmental conditions.
Figure 5. All temperature variation occurs after the 1st cell (left side) and after the 50th cell (right
side).
From a physical point of view, the temperatures’ variations presented in Figure 5 are
in good correlation with the heat transfer phenomena. If we compare the temperature
difference between the primary agent and the plate’s temperature on its side and the
temperature difference between the secondary agent and the plate’s temperature on its
side, we can see that on the primary agent side, the temperature difference is smaller as
the convective heat transfer coefficient is higher. If we compare the temperature differ-
ence between the primary fluid and the plate’s temperature on its side with the temper-
ature difference between the plate’s outside and its middle, we can notice that the first
one is smaller as the convective heat transfer coefficient is higher than the conductive one.
The plate thickness of 5.5 mm is not a common one for plate heat exchangers, being
chosen to verify the model’s limitations.
4. Model Validation with Experimental Data
Experimental data from the literature were used in order to validate the conceived
Simulink model for heat exchangers. [31] performed an experimental study on plate heat
exchangers for different outlet temperatures of secondary fluid, like 40 °C, 45 °C, and 50
°C, and different mass flow rates of the same fluid, like 3,4,5,6 and 7 lpm. Fluid heat
transfer properties and heat exchanger properties (as a plate’s surface, number of plates,
plate thickness, material, etc.) were used in the Simulink model to compare the simula-
tion results with the experimental ones taken from the literature [31]. Fluid heat transfer
coefficients found in the literature [31] were used in the Simulink model, and heat flow
rates obtained in Simulink were compared with those experimentally determined by [31].
In Figure 6, a comparison between the heat flow rate on the cold side of the fluid obtained
with the Simulink model and the heat flow rate experimentally determined by [31], as a
function of the cold fluid heat transfer coefficient, for three levels of outlet temperature
for the cold fluid is presented.
Figure 5. All temperature variation occurs after the 1st cell (left side) and after the 50th cell
(right side).
From a physical point of view, the temperatures’ variations presented in Figure 5are
in good correlation with the heat transfer phenomena. If we compare the temperature
difference between the primary agent and the plate’s temperature on its side and the
temperature difference between the secondary agent and the plate’s temperature on its
side, we can see that on the primary agent side, the temperature difference is smaller as
the convective heat transfer coefficient is higher. If we compare the temperature difference
between the primary fluid and the plate’s temperature on its side with the temperature
difference between the plate’s outside and its middle, we can notice that the first one is
smaller as the convective heat transfer coefficient is higher than the conductive one. The
plate thickness of 5.5 mm is not a common one for plate heat exchangers, being chosen to
verify the model’s limitations.
4. Model Validation with Experimental Data
Experimental data from the literature were used in order to validate the conceived
Simulink model for heat exchangers. [
31
] performed an experimental study on plate heat
exchangers for different outlet temperatures of secondary fluid, like 40
C, 45
C, and
50
C, and different mass flow rates of the same fluid, like 3,4,5,6 and 7 lpm. Fluid heat
transfer properties and heat exchanger properties (as a plate’s surface, number of plates,
plate thickness, material, etc.) were used in the Simulink model to compare the simulation
results with the experimental ones taken from the literature [
31
]. Fluid heat transfer
coefficients found in the literature [
31
] were used in the Simulink model, and heat flow
rates obtained in Simulink were compared with those experimentally determined by [
31
].
In Figure 6, a comparison between the heat flow rate on the cold side of the fluid obtained
with the Simulink model and the heat flow rate experimentally determined by [
31
], as a
function of the cold fluid heat transfer coefficient, for three levels of outlet temperature for
the cold fluid is presented.
Buildings 2023,13, 3125 12 of 14
Figure 6. Heat flow rate for a cold fluid as a function of its heat transfer coefficient and outlet
temperature of 40 C, 45 C, and 50 C.
Relative errors for heat flow rate were calculated between simulated results and
experimental ones with the help of Equation (10), and the values vary from 1.58% to 11.92%,
showing a good agreement between the conceived model and the experimental values [
31
].
Relative error[%]=|Simulation value Experimental value|
Experimental value ⇥100 (10)
The good agreement between the Simulink model and the experimental values demon-
strates that the model is valid and can be used by engineers to quickly evaluate one
parameter impact on heat transfer by using a model instead of the usual heat transfer
design calculus.
5. Conclusions
Unsteady heat transfer equations were used in order to describe temperatures’ vari-
ation inside a plate heat exchanger using Matlab/Simulink. The results obtained in Mat-
lab/Simulink were compared with calculated ones using steady state criterion equations of
heat transfer used by engineers to design plate heat exchangers. Heat transfer coefficients
for both fluids obtained with steady state equations were used in the Matlab/Simulink
model, together with heat exchange surface, fluids’, and metals’ thermal properties. Good
validation was obtained for the steady-state zone between the Matlab/Simulink 50 cell
model and the results obtained by using the heat transfer equations for the plate heat
exchanger. The relative error decreased from 6.66% to 0.41% between the 1-cell model and
the 50-cell model for the outlet temperature of the primary fluid. The relative error de-
creased from 10.58% to 0.45% between the 1-cell model and the 50-cell model for the outlet
temperature of secondary fluid. The relative error calculated between the heat transfer
rate obtained in Matlab/Simulink and the design theme varies between 42.8% and 2.6%
for the primary fluid and 42.3% and 1.8% for the secondary fluid. It can be noticed that
with the 40-cell model, the relative error for heat transfer rates is less than 10%, the relative
error accepted in engineering. Dividing the mesh into 50 cells along the flow direction, the
relative errors vary by 1.6–1.7%, meaning that a finer mesh is not necessary.
The developed model offers engineers a user-friendly tool to determine heat exchanger
working parameters under various conditions, extending beyond those specified in techni-
cal equipment reports. Unlike classical methods, our Simulink/Matlab-based approach
provides rapid numerical results with good convergence, enabling visualization of transient
zones and easy modification of variables like plate material or thickness. Experimental
validation [
31
] showed a 1.58% to 11.92% error range in heat flow rate, affirming the
model’s accuracy. This versatile Matlab/Simulink model simulates the impact of differ-
Buildings 2023,13, 3125 13 of 14
ent parameters in various heat exchanger applications, including geothermal probes and
air-water exchangers, with the capability to adjust for fluid and metal properties. Future
improvements aim to incorporate phase changes and mass transfer considerations into
the model.
Author Contributions: Conceptualization, L.L.P. and R.S.P.; methodology, T.C.; software, L.L.P., T.C.
and R.S.P.; writing—original draft preparation, L.L.P. and R.S.P.; visualization, T.C. All authors have
read and agreed to the published version of the manuscript.
Funding: This work was supported by a National Research Grants of the Technical University of
Civil Engineering of Bucharest, project number UTCB-31 and Cnfis-FDI-2023-F-0655.
Data Availability Statement: The data presented in this study are available in the article.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Zhang, F.; Liao, G.; E, J.; Chen, J.; Leng, E. Comparative study on the thermodynamic and economic performance of novel
absorption power cycles driven by the waste heat from a supercritical CO
2
cycle. Energy Convers. Manag. 2021,228, 113671.
[CrossRef]
2.
Zhao, X.; E, J.; Wu, G.; Deng, Y.; Han, D.; Zhang, B.; Zhang, Z. A review of studies using graphenes in energy conversion, energy
storage and heat transfer development. Energy Convers. Manag. 2019,184, 581–599. [CrossRef]
3.
Zhao, X.; E, J.; Zhang, Z.; Chen, J.; Liao, G.; Zhang, F.; Leng, E.; Han, D.; Hu, W. A Review on Heat Enhancement in Thermal
Energy Conversion and Management Using Field Synergy Principle. Appl. Energy 2020,257, 113995. [CrossRef]
4.
Lai, F.; Yang, Y. Influence of frequency of exchanges in quasi-dynamic methods for transient conjugate heat transfer problems. Int.
J. Therm. Sci. 2011,50, 1725–1737. [CrossRef]
5.
Ofualagba, G.; Oshevire, O.P. Model-based simulation of a solar thermal heat pump hybrid system. J. Eng. Sci. Technol. Rev. 2020,
13, 1–8. [CrossRef]
6.
Ionescu, V.; Neagu, A. Numerical simulation of Al
2
O
3
—Water nanofluid effects on the performance of a cross flow micro heat
exchanger. IOP Conf. Ser. Mater. Sci. Eng. 2019,400, 042032. [CrossRef]
7.
Axley, J. Modeling Sorption Transport in Rooms and Sorption Filtration Systems for Building Air Quality Analysis. Indoor Air
1993,3, 298–309. [CrossRef]
8.
Popescu, R.S. Activated Carbon Filter Modeling for Indoor Air Epuration. Ph.D. Thesis, Technical University of Civil Engineering
of Bucharest, Bucharest, Romania, 2008.
9.
Nadawi, A.; Khudhair, A. Distributed parameters modeling for heat exchangers using pure and zeotropic blend refrigerants. In
Advanced Analytic and Control Techniques for Thermal Systems with Heat Exchangers; Elsevier: Amsterdam, The Netherlands, 2020;
pp. 49–129.
10.
Abdulakdir, M.S.; Kulla, D.M.; Usman, K.M.; Ahmadu, T.; Kofa, A.M. Application of renewable Energy options—The role of solar
adsorption cooling technology. J. Power Energy Eng. 2022,10, 38–46. [CrossRef]
11. White, J. Litterature review on adsorption cooling systems. Lat. Am. Caribb. J. Eng. Educ. 2013,378, 28.
12.
Gut, J.A.W.; Pinto, J.M. Modeling of plate heat exchangers with generalized configurations. Int. J. Heat Mass Transf. 2003,46,
2571–2585. [CrossRef]
13.
Damiani, L.; Revetria, R.; Giribone, P. A Dynamic Simulation Model for a Heat Exchanger Malfunction Monitoring. Energies 2022,
15, 1862. [CrossRef]
14.
Preda, A.; Popescu, L.L.; Popescu, R.S. Numerical calculation of a sea water heat exchanger using Simulink software. IOP Conf.
Ser. Mater. Sci. Eng. 2017,227, 012104. [CrossRef]
15.
Carcel, P.M.M.; Acuna, J.; Fossa, M.; Palm, B. Numerical generation of the temperature response factors for a Borehole Heat
Exchangers field. In Proceedings of the European Geothermal Conference, Pisa, Italy, 3–7 June 2013.
16.
Zhang, F.; Fang, L.; Zhu, K.; Yu, M.; Cui, P.; Zhang, W.; Zhuang, Z.; Fang, Z. Long-term dynamic heat transfer analysis for the
borehole spacing planning of multiple deep borehole heat exchanger. Case Stud. Therm. Eng. 2022,38, 102373. [CrossRef]
17.
Fang, L.; Diao, N.; Shao, Z.; Cui, P.; Zhu, K.; Fang, Z. Thermal analysis models of deep borehole heat exchangers. In Proceedings
of the International Ground Source Heat Pump Association Conference, Stockholm, Sweden, 18–19 September 2018.
18.
Fujii, H.; Akibayashi, S. 2002 Analysis of thermal response test of heat exchange wells in ground-coupled heat pump systems
Shigen-to-Sozai. J. Min. Mater. Process. Inst. Jpn. 2002,118, 75–80.
19.
Minaei, A.; Maerefat, M. Thermal resistance capacity model for short-term borehole heat exchanger simulation with non-stiff
ordinary differential equations. Geothermics 2017,70, 260–270. [CrossRef]
20. Koohi-Fayegh, S.; Rosen, M.A. Modeling of vertical ground heat exchangers. Int. J. Green Energy 2021,18, 755–774. [CrossRef]
21.
Boukail, I.; Fenchouch, L.; Kharoua, N.; Semmari, H. Effects of soil properties on Canadian wells performance: Numerical
simulation. In Proceedings of the Convective Heat and Mass Transfer Conference, Izmir, Turkey, 5–10 June 2022.
Buildings 2023,13, 3125 14 of 14
22.
Kaboré, B.; Ouedraogo, G.W.P.; Moctar, O.; Zeghmati, V.; Zeghmati, B.; Chesneau, X.; Kam, S.; Dieudonné, J.B. Habitat Cooling by
a Canadian Well in Ouagadougou (Burkina Faso): Numerical Approach. Phys. Sci. Int. J. 2021,25, 21–28. [CrossRef]
23.
Kharoua, N.; Semmari, H.; Haroun, M.; Korichi, H.; Islam, M. Numerical Simulation of a Canadian Well with One Circumferential
Row of Internal Vortex Generators. In Proceedings of the ASME 2022 Fluids Engineering Division Summer Meeting, Toronto,
ON, Canada, 3–5 August 2022. [CrossRef]
24.
Michel, A.; Kugi, A. Accurate low-order dynamic model of a compact plate heat exchanger. Int. J. Heat Mass Transf. 2013,61,
323–331. [CrossRef]
25.
Ansari, M.R.; Mortazavi, V. Simulation of the dynamical response of a countercurrent heat exchanger to inlet temperature of mass
flow rate change. Appl. Therm. Eng. 2006,26, 2401–2408. [CrossRef]
26.
Ullah, A.; Ikramullah Selim, M.; Abdeljawad, T.; Ayaz, M.; Mlaiki, N.; Ghafoor, A. A magnetite water based nanofluid three-
dimensional thin film flow on an inclined rotating surface with non linear thermal radiations and coupled stress effects. Energies
2021,14, 5531. [CrossRef]
27.
Khan, S.; Selim, M.M.; Khan, A.; Ullad, A.; Abdeljawad, T.; Ikramullah Ayaz, M.; Mashwani, W.K. On the analysis of the
non-newtonian fluid flow past a stretching permeable surface with heat and mass transfer. Coat. J. 2021,11, 566. [CrossRef]
28.
Rizk, D.; Ullah, A.; Ikramullah Elattar, S.; Alharbi, K.; Sohail MKhan, R.; Khan, A.; Mlaiki, N. Impact of the KKL correlation model
on the activation of thermal energy for the hybrid nanofluid (GO + ZnO + water) flow through permeable vertically rotating
surface. Energies 2022,15, 2872. [CrossRef]
29.
Arifeen, S.; Haq, S.; Ghafoor, A.; Ullah, A.; Kumam, P. Chaipanya, Numerical solutions of higher order boundary value problems
via wavelet approach. Adv. Differ. Equ. 2021,2021, 347. [CrossRef]
30. Kakaç, S.; Liu, H. Heat Exchangers: Selection, Rating and Thermal Design, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2002; p. 501.
31.
Sözen, A.; Khanlari, A.; Çiftçi, E. Experimental and Numerical Investigation of Nanofluid Usage in a Plate Heat Exchanger for
Performance Improvement. Int. J. Renew. Energy Dev. 2019,8, 27–32. [CrossRef]
32.
Frühwirth, C.; Lorbeck, R.; Schutting, E.; Eichlseder, H. Co-Simulation of a BEV Thermal Management System with Focus on
Advanced Simulation Methodologies. SAE Technical Paper, 2023-01-1609, 2023. Available online: https://saemobilus.sae.org/
content/2023-01-1609 (accessed on 1 November 2023).
33.
Menni, Y.; Chamkha, A.J.; Ameur, H.; Ahmadi, M.H. Hydrodynamic Behavior in Solar Oil Heat Exchanger Ducts Fitted with
Staggered Baffles and Fins. J. Appl. Comput. Mech. 2020,6, 735–749. [CrossRef]
34.
Reagan, J.; Kurtz, S. Vertical Bifacial Solar Panels as a Candidate for Solar Canal Design. In Proceedings of the 2022 IEEE 49th
Photovoltaics Specialists Conference, Philadelphia, PA, USA, 5–10 June 2022. [CrossRef]
35.
Karwa, R.; Srivastava, V. Experimental investigation on the thermal performance of a solar air heater having absorber plate with
artificial roughness. Renew. Energy 2009,34, 2617–2623. [CrossRef]
36.
Topcu, M.S.; Sarac, I. Theoretical and experimental investigation of heat transfer and friction factor in a tube with twisted tape
insert under turbulent flow conditions. Int. J. Heat Mass Transf. 2010,53, 468–474. [CrossRef]
37.
Alim, M.A.; Garimella, S. Performance analysis of a solar-assisted heat pump system with phase change material storage. Energy
Convers. Manag. 2019,196, 1119–1131. [CrossRef]
38.
Delouei, A.A.; Sajjadi, H.; Atashafrooz, M.; Hesari, M.; Ben Hamida, M.B.; Arabkoohsar, A. Louvered Fin-and-Flat Tube Compact
Heat Exchanger under Ultrasonic Excitation. Fire 2023,6, 13. [CrossRef]
39.
Delouei, A.A.; Atashafrooz, M.; Sajjadi, H.; Karimnejad, S. The thermal effects of multi-walled carbon nanotube concentration on
an ultrasonic vibrating finned tube heat exchanger. Int. Commun. Heat Mass Transf. 2022,135, 106098. [CrossRef]
40.
Kabát, J.; Gužela, Š.; Peciar, P. HVAC Systems Heat Recovery with Multi-Layered Oscillating Heat Pipes. Stroj. ˇ
Casopis-J. Mech.
Eng. 2019,69, 51–60. [CrossRef]
41.
Chen, K.-L.; Luo, K.-Y.; Gupta, P.P.; Kang, S.-W. SLM Additive Manufacturing of Oscillating Heat Pipe. Sustainability 2023,
15, 7538. [CrossRef]
42.
Popescu, R.S.; Blondeau, P.; Jouandon, E.; Costes, J.; Fanlo, J. Elemental modeling of adsorption filter efficiency for indoor air
quality applications. Build. Environ. 2013,66, 11–22. [CrossRef]
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