Heat and mass transport of nano-encapsulated phase change materials in a complex cavity: An artificial neural network coupled with incompressible smoothed particle hydrodynamics simulations

Weaam Alhejaili; Sang-Wook Lee; Cao Quang Hat; Abdelraheem M. Aly; Weaam Alhejaili; Sang-Wook Lee; Cao Quang Hat; Abdelraheem M. Aly

doi:10.3934/math.2024271

AIMS Mathematics

2024, Volume 9, Issue 3: 5609-5632. doi: 10.3934/math.2024271

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Research article Special Issues

Heat and mass transport of nano-encapsulated phase change materials in a complex cavity: An artificial neural network coupled with incompressible smoothed particle hydrodynamics simulations

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
School of Mechanical Engineering, University of Ulsan, Ulsan, South Korea
3.
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

Received: 18 November 2023 Revised: 06 January 2024 Accepted: 15 January 2024 Published: 30 January 2024
MSC : 76

This work simulates thermo-diffusion and diffusion-thermo on heat, mass transfer, and fluid flow of nano-encapsulated phase change materials (NEPCM) within a complex cavity. It is a novel study in handling the heat/mass transfer inside a highly complicated shape saturated by a partial layer porous medium. In addition, an artificial neural network (ANN) model is used in conjunction with the incompressible smoothed particle hydrodynamics (ISPH) simulation to forecast the mean Nusselt and Sherwood numbers ( $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ ). Heat and mass transfer, as well as thermo-diffusion effects, are useful in a variety of applications, including chemical engineering, material processing, and multifunctional heat exchangers. The ISPH method is used to solve the system of governing equations for the heat and mass transfer inside a complex cavity. The scales of pertinent parameters are fusion temperature ${\theta }_{f} = 0.05-0.95$ , Rayleigh number $Ra = {10}^{3}-{10}^{6}$ , buoyancy ratio parameter $N = -2-1$ , Darcy number $Da = {10}^{-2}-{10}^{-5}$ , Lewis number $Le = 1-20$ , Dufour number $Du = 0-0.25$ , and Soret number $Sr = 0-0.8$ . Alterations of Rayleigh number are effective in enhancing the intensity of heat and mass transfer and velocity field of NEPCM within a complex cavity. The high complexity of a closed domain reduced the influences of Soret-Dufour numbers on heat and mass transfer especially at the steady state. The fusion temperature works well in adjusting the intensity and location of a heat capacity ratio inside a complex cavity. The presence of a porous layer in a cavity's center decreases the velocity field within a complex cavity at a reduction in Darcy number. The goal values of $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ for each data point are compared to those estimated by the ANN model. It is discovered that the ANN model's $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ values correspond completely with the target values. The exact harmony of the ANN model prediction values with the target values demonstrates that the developed ANN model can forecast the $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ values precisely.

Keywords:

Citation: Weaam Alhejaili, Sang-Wook Lee, Cao Quang Hat, Abdelraheem M. Aly. Heat and mass transport of nano-encapsulated phase change materials in a complex cavity: An artificial neural network coupled with incompressible smoothed particle hydrodynamics simulations[J]. AIMS Mathematics, 2024, 9(3): 5609-5632. doi: 10.3934/math.2024271

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Abstract

Nomenclature

Acronym	Full name	Acronym	Full name
ANN	Artificial neural network	$Pr$	Prandtl number
$C$	concentration	$U, V$	velocity components
${c}_{p}$	specific heat ( $J{\mathrm{k}\mathrm{g}}^{-1}{K}^{-1}$ )	MoD	margin of deviation (%)
MLP	multilayer perceptron	R	coefficient of determination
MSE	mean squared error	$X, Y$	Cartesian coordinates
$\mathrm{g}$	gravity ( $m{s}^{-2}$ )	$T$	dimensional temperature ( $K$ )
$Da$	Darcy number	$\beta$	thermal expansion coefficient, ( ${K}^{-1}$ )
$Du$	Dufour number	$\epsilon$	porosity
$Cr$	heat capacity ratio	$\delta$	temperature parameter ( $K$ )
$k$	thermal conductivity ( $W{m}^{-1}{K}^{-1}$ )	$\theta$	dimensionless temperature
$K$	permeability	${\rho }_{p}$	density of NEPCM particles
$N$	buoyancy ratio parameter	$\phi$	nanoparticle parameter
$p$	dimensional pressure ( $Pa$ )	$\tau$	dimensionless time
$P$	dimensionless pressure	$\rho$	density ( $\mathrm{k}\mathrm{g}{m}^{-3}$ )
$Sr$	Soret number	$\mu$	dynamic viscosity ( $\mathrm{k}\mathrm{g}{m}^{-1}{s}^{-1}$ )
$Ra$	Rayleigh number	$\mathrm{\Phi }$	dimensionless concentration

1. Introduction

The phase change materials (PCMs) are applied to thermal administration and evaluated as latent heat storage. PCMs can be used in textiles, reduction of energy consumption, and solar thermal utilization. The healed textiles by PCM support a cooling effect and can melt by absorbing heat from the human body ^[1]. The review studies on the uses of phase change materials and heat transfer for thermal energy storage are discussed by Cabeza and his coauthors ^[2,3,4,5,6]. Abu-Hamdeh et al. ^[7] introduced the utilization of PCM into the building envelope for saving energy. Arshad et al. ^[8] prepared mono and hybrid NEPCMs for the thermal management of microelectronics. The natural convection of NECPMs inside different closed domains has received several numerical studies ^{[9,10,11,12,13,14,15,16]}. Sri et al. ^[17] implemented finite element method (FEM) to study unsteady simulation of NEPCMs in a rotated enclosure. In their studies, the heat transfer efficiency was estimated by various factors such as porous length, temperature, and angular velocity. In ^[18,19], the authors studied the thermal convection in the cavity between circular cylinders filled with NEPCM and they investigated the bioconvection flow of NEPCM and oxytactic microorganisms inside a circular cavity. The thermosolutal convection within complex shapes can be applied to thermal buildings, power collection systems, and heat exchangers. Tayebi et al. ^[20] studied the magnetic natural convection of a nanofluid in an annular enclosure installed by fins. The double diffusion of a hybrid nanofluid in an H-shaped cavity including a baffle is discussed by Eshaghi et al. ^[21]. Dogonchi et al. ^[22] introduced numerical studies on magnetic impacts and the presence of cylinders in a porous enclosure filled with a nanoliquid.

Double diffusion is introduced widely in the research laboratory ^[23,24,25]. Thermosolutal convection in closed geometries is gaining interest because of its treatments in solar systems, food processing, material processing, heat exchangers, chemical equipment, and electronic devices ^[26,27,28]. Nanofluid is a suspension of nanoparticles with host fluid such as water. It is common knowledge that the addition of nanoparticles alters the transfer of possessions. Nanofluids gain higher thermal conductivity than traditional fluids as the added nanoparticles enhance the performance of heat transfer. There are large applications of nanofluids in solar energy systems ^[29,30,31], heat exchangers ^[32,33], nuclear reactors ^{[34,35,36,37,38]}, etc. Furthermore, the nanofluid flow within porous media is significant in fuel cells, drying processes, oil recovery, and geothermal energy ^[39,40]. The mesh methods are well known, such as finite volume method (FVM) or FEM for computational fluid dynamics ^{[41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56]}.

Recently, the meshfree methods have been commonly employed in computational fluid dynamics ^{[57,58,59,60,61,62,63,64,65,66,67]}. Long et al. ^[68] proposed an innovative smoothed finite element technique (ES-FEM) coupling strategy using the smoothed particle hydrodynamics (SPH) method established for thermal-fluid-structure interaction (TFSI) problems. Long and Su ^[69] created an edge-based ES-FEM with the ISPH approach for modeling TFSI issues, where the ISPH method simulates the fluid domain, and the ES-FEM method simulates the structure domain. Based on the SPH approach, Salehizadeh and Shafiei ^[70] created a completely Lagrangian methodology for numerical modeling of fluid-elastic structure interaction (FSI) issues. The ISPH approach is employed for the fluid phase, whereas the total Lagrangian SPH (TLSPH) method is used to solve the equations of motion for structure dynamics. Aly and El-Sapa ^[71] adopted the ISPH scheme to investigate the magnetic impacts on double diffusion from rotated circular cylinders within a cavity. During these investigations, the highly complex geometry of a closed domain was considered during heat and mass transfer. Hence, this study adopted the ISPH method to handle heat and mass transfer adjoined by the effects of thermo-diffusion and diffusion-thermo on NEPCM inside a highly complex geometry. The complexity of a closed domain during double diffusion is helpful in several industries of thermal management such as electronic devices, and solar collectors.

2. Mathematical analysis

The complete original setting of a physical problem is introduced in (). Here, the cavity's center walls are adiabatic, and the left/right and top/bottom walls are kept at a superior temperature ${T}_{h}$ and concentration ${C}_{h}$ . The other cavity's walls are fixed at ${T}_{c}$ and ${C}_{c}$ as shown in Figure 1(d). The reader can distinguish between each boundary by material identification as shown in . The initial setting of the porous layer is located at the cavity's center as shown in . The complex cavity's length and height are around $1.5$ m.

Figure 1. Initial setting of the current physical problem.

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For simplicity, the model is considered in two dimensions only because it represents the physical phenomena with less computations. Table 1 presents the initial simulation setting such as initial particle size, smoothing length, time step, number of total particles, and boundary treatment of the ISPH method. The dummy wall particles are adopted to treat the boundary domains. It is assumed that the flow is laminar, time-dependent, and incompressible. Table 2 represents the physical features of a mixture fluid ^[57].

Table 1. The initial simulation setting of the ISPH method.

Particle size $\left({d}_{o}\right)$	Smoothing length $\left(h\right)$	Time step $\left(\Delta \tau \right)$	Number of particles $\left(np\right)$	Boundary treatments
$0.008$	$0.0096$	$0.000001$	$15684$	Dummy boundary particles

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Table 2. The features of a mixture fluid ^[10].

Material	${C}_{p}\left[\frac{KJ}{kg.K}\right]$	$\rho [kg/{m}^{3}]$	$\beta \times {10}^{-5}$ [1/K]	$k[W/m.K]$
Base fluid	4179	997	21	0.613
Core	1317.7	786	50	0.19
Shell	2037	721	$17.28$	0.025
Porous matrix	840	2700	------	1.05

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The dimensionless governing equations ^[72,73] in Lagrangian description are:

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y} = 0,$

(1)

$\frac{1}{\epsilon }\frac{DU}{D\tau }+\frac{{\mu }_{b}}{{\mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Pr\frac{U}{Da}+\frac{1.75}{\sqrt{150}}\frac{1}{\sqrt{Da{\epsilon }^{3}}}\sqrt{{U}^{2}+{V}^{2}}U = -\frac{{\rho }_{f}}{{\rho }_{b}}\frac{\partial P}{\partial X}+\frac{{\mu }_{b}}{{\epsilon \mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Pr\left(\frac{{\partial }^{2}\mathrm{U}}{\partial {X}^{2}}+\frac{{\partial }^{2}\mathrm{U}}{\partial {Y}^{2}}\right),$

(2)

$\frac{1}{\epsilon }\frac{DV}{D\tau }+\frac{{\mu }_{b}}{{\mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Pr\frac{V}{Da}+\frac{1.75}{\sqrt{150}}\frac{1}{\sqrt{Da{\epsilon }^{3}}}\sqrt{{U}^{2}+{V}^{2}}V = \frac{{\left(\rho \beta \right)}_{b}}{{\left(\rho \beta \right)}_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Ra\mathit{Pr}\left(\theta +N\mathrm{\Phi }\right)-\frac{{\rho }_{f}}{{\rho }_{b}}\frac{\partial P}{\partial Y}+\frac{{\mu }_{b}}{{\epsilon \mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}P\left(\frac{{\partial }^{2}\mathrm{V}}{\partial {X}^{2}}+\frac{{\partial }^{2}\mathrm{V}}{\partial {Y}^{2}}\right),$

(3)

$\left(\epsilon Cr+(1-\epsilon )\frac{{\left(\rho C\right)}_{s}}{{\left(\rho C\right)}_{f}}\right)\frac{D\theta }{D\tau } = \frac{{k}_{m, b}}{{k}_{f}}\left(\frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)+Du\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {X}^{2}}+\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {Y}^{2}}\right) ,$

(4)

$\frac{D\mathrm{\Phi }}{D\tau } = \frac{1}{Le}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {X}^{2}}+\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {Y}^{2}}\right)+Sr\left(\frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right) .$

(5)

The dimensionless quantities ^[73] are:

$\tau = \frac{t{{\alpha }_{f}}_{}}{{L}^{2}}, X = \frac{x}{L}, Y = \frac{y}{L}, U = \frac{uL}{{{\alpha }_{f}}_{}}, V = \frac{vL}{{{\alpha }_{f}}_{}}, \theta = \frac{T-{T}_{c}}{{T}_{h}-{T}_{c}}, P = \frac{p{L}^{2}}{{\rho }_{f}{\alpha }_{f}^{2}}, \mathrm{\Phi } = \frac{C-{C}_{c}}{{C}_{h}-{C}_{c}} .$

(6)

The related boundary conditions are:

Left/right and top/bottom walls: $U = 0 = V, \theta = 1 = \mathrm{\Phi }$ .

$\mathrm{Inner\; walls}: U = 0 = V, \theta = 0 = \mathrm{\Phi } .$

(7)

Other walls: $U = 0 = V, \frac{\partial \theta }{\partial \mathbf{n}} = 0 = \frac{\partial \mathrm{\Phi }}{\partial \mathbf{n}}$ .

The heat capacity ratio is:

$Cr = \frac{{{(\rho C}_{p})}_{b}}{{{(\rho C}_{p})}_{f}} = \frac{\phi }{\delta Ste}\left[\frac{\pi }{2}\mathrm{\Gamma }\mathrm{sin}\left(\frac{\pi }{\delta }(\theta -{\theta }_{f}+\frac{\delta }{2}\right)\right]+1-\phi +\lambda \phi$

(8)

with

$\mathrm{\Gamma } = \left\{\begin{array}{cc}0& \theta < {\theta }_{f}-\frac{\delta }{2}\\ 1& \left({\theta }_{f}-\frac{\delta }{2}\right) < \theta < \left({\theta }_{f}+\frac{\delta }{2}\right)\\ 0& \theta > {\theta }_{f}+\frac{\delta }{2}\end{array}\right.$

(9)

where

$\lambda = \frac{({{(C}_{p})}_{c, l}+\chi {{(C}_{p})}_{s}){\rho }_{s}{\rho }_{c}}{({\rho }_{s}+\chi {\rho }_{c}){({\rho C}_{p)}}_{f}}, {\theta }_{f} = \frac{{T}_{f}-{T}_{c}}{\mathrm{\Delta }T}, \delta = \frac{{T}_{Mr}}{\mathrm{\Delta }T}, Ste = \frac{{({\rho C}_{p)}}_{f}\mathrm{\Delta }T{(\rho }_{s}+\chi {\rho }_{c})}{(1+\chi ){h}_{sf}{\rho }_{s}{\rho }_{c}} .$

The mean Nusselt/Sherwood numbers are:

$\stackrel{-}{Nu} = \frac{-1}{{L}_{h}}{\int }_{0}^{{L}_{h}}\frac{{k}_{m, b}}{{k}_{f}}\frac{\partial \theta }{\partial \mathit{n}}d\zeta$

(10)

$\stackrel{-}{Sh} = \frac{-1}{{L}_{h}}{\int }_{0}^{{L}_{h}}\frac{\partial \mathrm{\Phi }}{\partial \mathit{n}}d\zeta$

(11)

where ${L}_{h}$ is a whole length of hot/cold walls in a complex cavity.

3. The numerical approach

3.1. The SPH formulation

The basic concept of SPH description for any function is:

$f\left({\mathit{X}}_{i}\right) = \sum _{j = 1}^{n}\frac{{m}_{j}}{{\rho }_{j}}f\left({\mathit{X}}_{j}\right)W\left({\mathit{r}}_{ij}, h\right)$

(12)

$W\left(q, h\right) = \frac{7}{478\pi {h}^{2}}\left\{\begin{array}{c}{\left(3-q\right)}^{5}-{6\left(2-q\right)}^{5}+15{\left(1-q\right)}^{5}, 0\le q < 1\\ {\left(3-q\right)}^{5}-{6\left(2-q\right)}^{5}, 1\le q < 2\\ {\left(3-q\right)}^{5}, 2\le q < 3\\ 0, q\ge 3\end{array}\right. .$

(13)

Here, $W$ is a kernel function. The adjusted first derivative ^[74] in the SPH approach is:

$\stackrel{~}{\nabla }{W}_{ij} = \mathit{L}\left({\mathit{r}}_{ij}\right){\nabla W}_{ij}$

(14)

where

$\mathit{L}\left({\mathit{r}}_{ij}\right) = {\left(\begin{array}{cc}\sum _{j = 1}^{n}\frac{{m}_{j}}{{\rho }_{j}}\left({X}_{j}-{X}_{i}\right)\frac{\partial {W}_{ij}}{\partial {X}_{i}}& \sum _{j = 1}^{n}\frac{{m}_{j}}{{\rho }_{j}}\left({X}_{j}-{X}_{i}\right)\frac{\partial {W}_{ij}}{\partial {Y}_{i}}\\ \sum _{j = 1}^{n}\frac{{m}_{j}}{{\rho }_{j}}\left({Y}_{j}-{Y}_{i}\right)\frac{\partial {W}_{ij}}{\partial {X}_{i}}& \sum _{j = 1}^{n}\frac{{m}_{j}}{{\rho }_{j}}\left({Y}_{j}-{Y}_{i}\right)\frac{\partial {W}_{ij}}{\partial {Y}_{i}}\end{array}\right)}^{-1} .$

(15)

The velocity divergence and pressure gradient, as well as the approximation of a second derivative, are as follows:

$\nabla \cdot \mathbf{U}\left({\mathit{X}}_{i}\right) = \sum _{j = 1}^{n}\frac{{m}_{j}}{{\rho }_{j}}\left(\mathbf{U}\left({\mathit{X}}_{j}\right)-\mathbf{U}\left({\mathit{X}}_{i}\right)\right)\cdot \stackrel{~}{\nabla }{W}_{ij} ,$

(16)

$\nabla P\left({\mathit{X}}_{i}\right) = {\rho }_{i}\sum _{j = 1}^{n}{m}_{j}\left(\frac{{P}_{i}}{{\rho }_{i}^{2}}+\frac{{P}_{j}}{{\rho }_{j}^{2}}\right)\nabla {W}_{ij} ,$

(17)

${\nabla }^{2}f\left({\mathit{X}}_{i}\right) = \sum _{j = 1}^{n}{m}_{j}\left(\frac{{\rho }_{i}+{\rho }_{j}}{{\rho }_{i}{\rho }_{j}}\frac{\left({\mathit{r}}_{i}-{\mathit{r}}_{j}\right)\cdot {\nabla }_{i}{W}_{ij}}{{r}_{ij}^{2}+0.0001{h}^{2}}\right)\left(f\left({\mathit{X}}_{j}\right)-f\left({\mathit{X}}_{i}\right)\right) .$

(18)

3.2. The ISPH method solution procedures

The procedures are:

Predictor velocities:

${U}^{*} = \epsilon \left(\Delta \tau \right)\left(\frac{{\mu }_{b}}{{\epsilon \mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Pr{\left(\frac{{\partial }^{2}U}{\partial {X}^{2}}+\frac{{\partial }^{2}U}{\partial {Y}^{2}}\right)}^{n}-\frac{{\mu }_{b}}{{\mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Pr\frac{{U}^{n}}{Da}-{\left(\frac{C}{\sqrt{Da}}\frac{\sqrt{{U}^{2}+{V}^{2}}}{{\epsilon }^{\frac{3}{2}}}\right)}^{n}{U}^{n}\right)+{U}^{n} ,$

(19)

${V}^{*} = \epsilon \left(\Delta \tau \right)\left(\frac{{\mu }_{b}}{{\epsilon \mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Pr{\left(\frac{{\partial }^{2}V}{\partial {X}^{2}}+\frac{{\partial }^{2}V}{\partial {Y}^{2}}\right)}^{n}+\frac{{\left(\rho \beta \right)}_{\mathrm{n}\mathrm{f}}}{{\rho }_{\mathrm{n}\mathrm{f}}{\beta }_{\mathrm{f}}}Ra\mathrm{Pr}\left({\theta }^{n}+N{\mathrm{\Phi }}^{n}\right)-\frac{{\mu }_{b}}{{\mu }_{f}}\frac{{\rho }_{f}}{{\rho }_{b}}Pr\frac{{V}^{n}}{Da}-{\left(\frac{C}{\sqrt{Da}}\frac{\sqrt{{U}^{2}+{V}^{2}}}{{\epsilon }^{\frac{3}{2}}}\right)}^{n}{V}^{n}\right)+{V}^{n} .$

(20)

Pressure Poisson equation (PPE):

${\nabla }^{2}{P}^{n+1} = \mathrm{{\rm Y}}\frac{\left({\rho }_{\mathrm{f}}-{\rho }^{num}\right)}{{\rho }_{\mathrm{f}}{\left(\Delta \tau \right)}^{2}}+\frac{1}{\epsilon \Delta \tau }\frac{{\rho }_{b}}{{\rho }_{f}}\left(\frac{\partial {U}^{*}}{\partial X}+\frac{\partial {V}^{*}}{\partial Y}\right) .$

(21)

Corrected velocities:

${U}^{n+1} = {U}^{*}-\Delta \tau \frac{{\epsilon \rho }_{f}}{{\rho }_{b}}{\left(\frac{\partial P}{\partial X}\right)}^{n+1} ,$

(22)

${V}^{n+1} = {V}^{*}-\Delta \tau \frac{{\epsilon \rho }_{f}}{{\rho }_{b}}{\left(\frac{\partial P}{\partial Y}\right)}^{n+1} .$

(23)

The thermal equation:

${\theta }^{n+1} = {\theta }^{n}+\frac{\Delta \tau }{\left(\epsilon Cr+(1-\epsilon )\frac{{\left(\rho C\right)}_{s}}{{\left(\rho C\right)}_{f}}\right)}\left(\frac{{k}_{m, b}}{{k}_{f}}{\left(\frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)}^{n}+Du{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {X}^{2}}+\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {Y}^{2}}\right)}^{n}\right) .$

(24)

The mass equation:

${\mathrm{\Phi }}^{n+1} = {\mathrm{\Phi }}^{n}+\Delta \tau \left(\frac{1}{Le}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {X}^{2}}+\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {Y}^{2}}\right)}^{n}+Sr{\left(\frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)}^{n}\right) .$

(25)

The following positions have been updated:

${X}^{n+1} = {X}^{n}+\Delta \tau {U}^{n+1} ,$

(26)

${Y}^{n+1} = {Y}^{n}+\Delta \tau {V}^{n+1} .$

(27)

The shifting technique ^[60] is:

${\mathcal{F}}_{{i}^{\text{'}}} = {\left(\nabla \mathcal{F}\right)}_{i}\cdot \left(-\mathcal{D}\nabla {{C}^{\text{'}}}_{i}\right)+{\mathcal{F}}_{i}+\mathcal{O}\left(\delta {\left(-\mathcal{D}\nabla {{C}^{\text{'}}}_{i}\right)}^{2}\right) .$

(28)

The ISPH simulation computations in this study are based on the in-house code of the ISPH technique developed in FORTRAN-90. These computations are carried out in a cluster equipped with an Intel(R) Xeon(R) CPU E5-2690 v4 @ 2.60 GHz and 128 GB RAM. The simulation results are considered to converge when the error tolerances RTOL = 10^-6 are satisfied.

4. Validation tests

Here, the efficacy of the ISPH method in calculating $\stackrel{-}{Nu}$ during natural convection is tested two times without porous media ^[75] and with porous media ^[76,77]. Table 3 represents the comparison between the ISPH method and benchmark data ^[75]. The present results support the confidence in the ISPH method because of a good agreement with the benchmark data ^[75]. The second numerical test is represented in . This comparison introduces the values of $\stackrel{-}{Nu}$ among numerical data ^[76,77] and the ISPH method. The test compares the values of $\stackrel{-}{Nu}$ during natural convection flow in a porous cavity under the effects of $Da, Ra$ , and porosity parameter $\epsilon$ . This test presented a good agreement with numerical data of ^[76,77]. Several experimental/numerical tests are validated for the ISPH method in the previous studies ^{[13,14,15,78]}.

Table 3. Comparison among benchmark data of ^[75] and the ISPH method.

	$Ra={10}^{3}$	$Ra={10}^{4}$	$Ra={10}^{5}$	$Ra={10}^{6}$
De Vahl Davis ^[75]	1.118	2.243	4.519	8.798
The ISPH method	1.085	2.208	4.388	8.581

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Table 4. Comparison of

$\stackrel{-}{Nu}$ among numerical data of ^[76,77] and the ISPH method.

$\mathit{D}\mathit{a}$	$\mathit{R}\mathit{a}$	$\mathit{\epsilon }=0.4$			$\mathit{\epsilon }=0.6$			$\mathit{\epsilon }=0.9$
		^[76]	^[77]	ISPH method	^[76]	^[77]	ISPH method	^[76]	^[77]	ISPH method
${10}^{-2}$	${10}^{3}$	$1.01$	$1.008$	$1.006$	$1.015$	$1.012$	$0.994$	$1.023$	$--$	$1.013$
	${10}^{4}$	$1.408$	$1.359$	$1.404$	$1.530$	$1.489$	$1.533$	$1.64$	$--$	$1.667$
	${10}^{5}$	$2.983$	$2.986$	$3.15$ 9	$3.555$	$3.430$	3.602	3.91	$--$	4.125
	$5\times {10}^{5}$	$4.99$	$--$	$5.225$	5.740	$--$	6.031	6.70	$--$	6.778
${10}^{-4}$	${10}^{5}$	1.067	1.064	1.081	1.071	1.066	1.048	1.072	$--$	1.056
${10}^{-4}$	${10}^{6}$	2.55	2.580	2.611	2.725	2.686	2.708	2.74	$--$	2.609

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5. ANN modelling

The basic objective of using the ANN model in this research is to improve the predictability and efficiency of forecasting heat and mass transport characteristics within a complicated cavity filled with nano-encapsulated phase change materials (NEPCM). The following are the major aspects concerning the significance and benefits of the suggested ANN model:

Complexity handling: The complex geometry and intricate conditions of the closed cavity present a challenging environment for traditional analytical methods. The ANN model serves as a valuable tool in handling the complexity of the system, providing a robust and efficient means of predicting the heat and mass transfer characteristics.

General applicability: The ANN model developed in this study is not limited to the specific conditions investigated but can be adapted for a broader range of scenarios involving heat and mass transfer within complex geometries. This generalizability enhances the utility of the model in diverse engineering applications, including but not limited to chemical engineering, material processing, and multifunctional heat exchangers.

Reduced computational costs: The ANN model complements the ISPH simulation, offering a computationally efficient alternative for predicting heat and mass transfer characteristics. This reduction in computational costs is particularly valuable for simulations involving highly complex geometries and large datasets.

The Levenberg-Marquardt algorithm (LMA) is a well-known trust region approach for locating the minimum of a function (either linear or nonlinear) across a set of parameters. A trustworthy zone of the target function is essentially internally described using a function such as a quadratic. Kenneth Levenberg and Donald Marquardt ^[79,80], derived the Levenberg-Marquardt method, which gives a numerical solution to the issue of minimizing a nonlinear function. It is quick and steady in its convergence. This approach is appropriate for training small and medium-sized issues in the field of artificial neural networks. In this study, a neural network based on the LMA approach is used to predict the values of $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ . Due to its layered nature, the ANN employs the multilayer perceptron (MLP) and according to Çolak ^[81], MLP has a high learning ability. A dimensionless time $\tau$ and the Dufour number $Du$ are used as input parameters to create acceptable predictions, and there are enough data points with the number 990341. $\stackrel{-}{Nu}$ values are included in the output layer. A model with 14 neurons was discovered in the MLP network model's hidden layer ^[81,82,83]. The following are the transfer functions used by the MLP network's hidden and output layers:

$f\left(x\right) = \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{g}\left(x\right) = -1+\frac{2}{1+{e}^{\left(-2x\right)}}$

(29)

$purelin\left(x\right) = x .$

(30)

Figure 2 depicts the structure of the created MLP network. The input data consists of 990341 data points, with 15% set aside for testing, 15% for validation, and 70% used to train the model. Figure 3 depicts a simple neural network configuration. The performance parameters that are commonly used in the literature to assess the training and prediction performance of the ANN model were chosen. The chosen performance measures, namely the mean squared error (MSE), coefficient of determination (R), and margin of deviation (MoD), are calculated using the following mathematical expressions ^[82]:

$\mathrm{M}\mathrm{S}\mathrm{E} = \frac{1}{\mathrm{N}}\sum _{\mathrm{i} = 1}^{\mathrm{N}}{({\mathrm{X}}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\left(\mathrm{i}\right)}-{\mathrm{X}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\left(\mathrm{i}\right)})}^{2} ,$

(31)

$\mathrm{R} = \sqrt{1-\frac{\sum _{\mathrm{i} = 1}^{\mathrm{N}}{\left({\mathrm{X}}_{\mathrm{targ}\left(\mathrm{i}\right)}-{\mathrm{X}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\left(\mathrm{i}\right)}\right)}^{2}}{\sum _{\mathrm{i} = 1}^{\mathrm{N}}{\left({\mathrm{X}}_{\mathrm{targ}\left(\mathrm{i}\right)}\right)}^{2}}} ,$

(32)

$\mathrm{M}\mathrm{o}\mathrm{D} = \left[\frac{{\mathrm{X}}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}}-{\mathrm{X}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}}{{\mathrm{X}}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}}}\right].$

(33)

Figure 2. The built MLP network's structure.

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Figure 3. A basic neural network arrangement.

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The MSE performance of the proposed MLP model during training is shown in Figure 4. After starting the training phase with a large number, the graph illustrates how the MSE values reduced with each epoch. When each of the three data sets produced the optimum validation value, the MLP model's training phase was finished. The error histogram constructed using data from the training phase is shown in Figure 5. A close look at the error histogram indicates that the error values are mostly grouped around the zero-error line. Furthermore, it was discovered that the numerical values of the errors were not particularly high. The regression curves for the MLP model are shown in . The linear link (R) between inputs and targets is known as regression. All training, validation, and testing values for R = 1 in the current model have the correct linear connection. illustrates the gradient state of the MLP model. The graphs indicate that the proposed MLP model is convergent at a modest gradient value of 4.0054e-06 at epoch 100 and a step size of Mu = 1e-09 at epoch 100. The goal values for each data point, as well as the $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ values computed by the ANN model, are shown in . When the graphs are examined, the $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ values obtained from the ANN model agree exactly with the goal values. The perfect harmony of the ANN model prediction values with the target values shows that the created ANN model can accurately forecast the $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ values.

Figure 4. MSE performances for the proposed MLP model's operation during training.

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Figure 5. Histogram of errors for the MLP model.

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Figure 6. The MLP model's regression profile.

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Figure 7. MLP model gradient state.

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Figure 8. The ANN model's values for

$\stackrel{-}{Nu}$ and

$\stackrel{-}{Sh}$ , as well as the target values for each data point.

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6. Results and discussion

This section introduces the obtained simulations of the effects of Soret/Dufour numbers on the double diffusion of NEPCM inside a complex cavity. The ranges of physical parameters are Rayleigh number $Ra = {10}^{3}-{10}^{6}$ , buoyancy ratio parameter $N = -2-1$ , fusion temperature ${\theta }_{f} = 0.05-0.95$ , Darcy number $Da = {10}^{-2}-{10}^{-5}$ , Lewis number $Le = 1-20$ , Dufour number $Du = 0-0.25$ , and Soret number $Sr = 0-0.8$ . The complexity of a closed domain during thermosolutal convection may be used in a variety of engineering applications, including electronic device cooling, chemical engineering, multifunctional heat exchangers, and material processing. Coupled heat-mass transport inside a partial layer of porous medium can also be used for natural phenomena and a variety of industrial processes. The mean Nusselt number $\left(\stackrel{-}{Nu}\right)$ is a physical quantity that describes the ratio of convective to conductive heat transfer inside a fluid and provides information on the efficiency of heat exchange between a solid surface and the surrounding fluid. The mean Sherwood number $\left(\stackrel{-}{Sh}\right)$ is a dimensionless quantity that represents the ratio of convective to diffusive mass transfer in mass-transfer activities.

represents the effects of fusion temperature ${\theta }_{f}$ on a heat capacity ratio $Cr$ , temperature $\theta$ , concentration $\mathrm{\Phi }$ , and velocity field. Increasing ${\theta }_{f}$ shifts $Cr$ from the center's cavity toward the hot walls. This result returns to the relationship between the temperature, ${\theta }_{f}$ and heat capacity ratio $Cr$ in Eq 8 $\left(Cr = \frac{\phi }{\delta Ste}\left[\frac{\pi }{2}\mathrm{\Gamma }\mathrm{sin}\left(\frac{\pi }{\delta }(\theta -{\theta }_{f}+\frac{\delta }{2}\right)\right]+1-\phi +\lambda \phi \right)$ . There are almost marginal variations on the distributions of temperature and intensity of a velocity field under the differences of ${\theta }_{f}$ .

Figure 9. Effects of θ_f on

$Cr$ , θ, and

$\mathbf{V}$ at

$Ra$ = 10⁴,

$Da = {10}^{-3}, \phi$ = 0.05,

$\alpha = 0.99, N = 2, Sr = 0.2, Du = 0.05,$ and

$Le = 10$ .

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establishes the effects of Rayleigh number $Ra$ on $Cr$ , $\theta$ , $\mathrm{\Phi }$ , and $\mathit{V}$ . Physically, the Rayleigh number $Ra$ is a dimensionless number that signifies the relation between buoyancy and thermal diffusivity. Increasing $Ra$ varies the contours of $Cr$ due to the strong enhancement in heat/mass transfer within a complex cavity. The intensity of temperature and concentration is enhanced under an increase in $Ra$ . Also, the velocity field is effectively enhanced with an increase in $Ra$ . These results are relevant to power in buoyancy forces that enhance heat-mass transition and velocity of a nanofluid flow at an increase in $Ra$ .

Figure 10. Effects of

$Ra$ on

$Cr$ ,

$\theta$ ,

$\mathrm{\Phi }$ , and

$\mathbf{V}$ at

${\theta }_{f} = 0.2, Da = {10}^{-3}, \phi = 0.05, \alpha = 0.99, N = 2, Sr = 0.2, Du = 0.05,$ and

$Le = 10$ .

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signifies the effects of buoyancy ratio parameter $N$ on $Cr$ , $\theta$ , $\mathrm{\Phi }$ , and the velocity field. Since the buoyancy ratio parameter is working effectively in changing the directions of buoyancy forces, at $N = -2$ , the strength of temperature and concentration appears in the bottom area of a complex cavity. Whilst, at $N\ge 0$ , this strength appears in the top area of a complex cavity. As a result, the contour of $Cr$ is affected by the variations of $N$ . Increasing $N$ from $-2$ to $1$ enhances the velocity field's maximum by $117.94\%$ .

Figure 11. Effects of

$N$ on

$Cr$ ,

$\theta$ ,

$\mathrm{\Phi }$ , and

$\mathbf{V}$ at

${\theta }_{f} = 0.2, Da = {10}^{-3},$

$\phi = 0.05, \alpha = 0.99, Ra = {10}^{4}, Sr = 0.2, Du = 0.05,$ and

$Le = 10$ .

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depicts the effects of Darcy number $Da$ on $Cr$ , $\theta$ , $\mathrm{\Phi }$ , and $\mathit{V}$ . Physically, a decrease in Darcy number $Da$ is a consequence of a reduction in the porous medium's permeability, which raises the porous resistance of fluid flow. Hence, as $Da$ decreases from 0.01 to 0.00001, the velocity field reduces a long a porous layer in a complex cavity. Due to occurrence of a porous layer in a complex cavity's center and the high complexity of a cavity, the distributions of temperature, concentration and heat capacity ratio are affected slightly by a reduction in $Da$ .

Figure 12. Effects of

$Da$ on

$Cr$ ,

$\theta$ ,

$\mathrm{\Phi }$ , and

$\mathbf{V}$ at

${\theta }_{f} = 0.2, N = 2, \phi = 0.05, \alpha = 0.99, Ra = {10}^{4}, Sr = 0.2, Du = 0.05,$ and

$Le = 10$ .

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implies the effects of Lewis number $Le$ on $Cr$ , $\theta$ , $\mathrm{\Phi }$ , and velocity field $\mathit{V}$ . The dimensionless Lewis number $Le$ is a fraction of thermal to mass diffusivities. $Le$ is implemented in fluid flows during heat and mass transfer. There are little variations on the contours of $Cr$ , isotherms, and velocity field under the variations of $Le$ . The intensity of concentration is reduced according to an increase in $Le$ .

Figure 13. Effects of

$Le$ on

$Cr$ ,

$\theta$ ,

$\mathrm{\Phi }$ , and

$\mathbf{V}$ at

${\theta }_{f} = 0.2, N = 2, \phi = 0.05, \alpha = 0.99, Ra = {10}^{4}, Sr = 0.2, Du = 0.05,$ and

$Da = {10}^{-3}$ .

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The effects of Dufour number $Du$ on $Cr$ , $\theta$ , $\mathrm{\Phi }$ , and velocity field $\mathit{V}$ are shown in . Dufour number $Du$ defines the concentration gradients to the thermal energy equation inside a fluid flow. Accordingly, an increase in $Du$ may result in an augmentation in heat transfer. Here, the contributions of $Du$ in enhancement of temperature distributions are lesser because of the cavity's complexity. Also, the variations of $Du$ have minor influences on $Cr$ , $\mathrm{\Phi }$ , and velocity field within a complex cavity.

Figure 14. Effects of

$Du$ on

$Cr$ ,

$\theta$ ,

$\mathrm{\Phi }$ , and

$\mathbf{V}$ at

${\theta }_{f} = 0.2, N = 2, \phi = 0.05, \alpha = 0.99, Ra = {10}^{4}, Sr = 0.2, Le = 10,$ and

$Da = {10}^{-3}$ .

DownLoad: Full-Size Img PowerPoint

shows the effects of Soret number $Sr$ on $Cr$ , $\theta$ , $\mathrm{\Phi }$ , and velocity field $\mathit{V}$ . Soret number $Sr$ defines the contributions of temperature gradients to the mass equation inside a fluid flow. Increasing $Sr$ slightly enhances heat-mass transmission within a complex cavity. So, the distributions of heat capacity ratio, temperature, concentration, and velocity field are slightly changed with a raise in $Sr$ .

Figure 15. Effects of

$Sr$ on

$Cr$ ,

$\theta$ ,

$\mathrm{\Phi }$ , and

$\mathbf{V}$ at

${\theta }_{f} = 0.2, N = 2, \phi = 0.05, \alpha = 0.99, Ra = {10}^{4}, Le = 10, Du = 0.05,$ and

$Da = {10}^{-3}$ .

DownLoad: Full-Size Img PowerPoint

represents the effects of $N, Du, Sr, Le$ , and $\phi$ on $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ . It is seen that the values of $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ are decreased by $7.27\%$ and $4.76\mathrm{\%}$ according to an increase in $N$ from $-2$ to $1$ . At the initial time instants $\tau \le 0.15$ , an increase in $Du$ augments the values of $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ , whilst at the steady state $\tau \ge 0.15$ , the contributions of $Du$ on values of $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ are lesser at a steady state. Similar trends are appearing under the influences of $Sr$ . Initially, a growth in $Sr$ enhances $\stackrel{-}{Nu}$ and reduces $\stackrel{-}{Sh}$ . At the steady state, the contributions of $Sr$ on values of $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ are less. $Le$ works well in enhancing the values of $\stackrel{-}{Sh}$ . Adding more concentration of nanoparticles by an increase in $\phi$ to $0.1$ enhances $\stackrel{-}{Nu}$ and reduces $\stackrel{-}{Sh}$ .

Figure 16. Effects of

$N, Du, Sr, Le$ , and

$\phi$ on

$\stackrel{-}{Nu}$ and

$\stackrel{-}{Sh}$ at

${\theta }_{f} = 0.2, \alpha = 0.99, Ra = {10}^{4},$ and

$Da = {10}^{-3}$ .

DownLoad: Full-Size Img PowerPoint

7. Conclusions

The ISPH method is utilized to emulate the thermosolutal convection of NEPCM in a complex cavity filled by a partial layer porous medium. The effects of Soret-Dufour numbers are tested. The complexity of a closed domain with heat and mass transfer inside this domain can be applied in several environmental and engineering fields such as solar collectors, chemical engineering, food processing, material processing, heat exchangers, electronic devices, insulation, and climate control. The main results of the executed numerical simulations are:

● The fusion temperature is a chief factor in dominating the strength and location of a heat capacity ratio in a complex cavity.

● The high complexity of a closed domain reduced the influences of Soret-Dufour numbers on heat-mass transmission in a complex cavity.

● The Rayleigh number is effective in augmenting the strength of heat and mass transfer as well as the velocity field.

● Due to the occurrence of a porous layer in the cavity's center, a reduction in Darcy number reduces the velocity field inside this layer.

● The ISPH method is a excellent tool in handling thermosolutal convection within a highly complex cavity.

● The perfect concord of the ANN model prediction values with the target values reveals that the constructed ANN model can correctly foresee the $\stackrel{-}{Nu}$ and $\stackrel{-}{Sh}$ values. The current ANN model is still restricted in its ability to anticipate complete physical occurrences without relying on numerical simulations.

As future work, the current scheme of the ISPH method will be developed to handle double diffusion inside a three-dimensional complex cavity. An ANN model will be developed to handle the complete physical problem.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP. 2/42/44). This research was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R229), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that no conflicts of interest exists in this paper.

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Heat and mass transport of nano-encapsulated phase change materials in a complex cavity: An artificial neural network coupled with incompressible smoothed particle hydrodynamics simulations

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Heat and mass transport of nano-encapsulated phase change materials in a complex cavity: An artificial neural network coupled with incompressible smoothed particle hydrodynamics simulations

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1. Introduction

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4. Validation tests

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