Melting of cyclohexane–Cu nano-phase change material (nano-PCM) in porous medium under magnetic field

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Melting of cyclohexane–Cu nano-phase change material (nano-PCM) in porous medium under magnetic field M. Tahmasebi Kohyani a, B. Ghasemi a,∗, A. Raisi a,∗, S.M. Aminossadati b a b

Engineering Faculty, Shahrekord University, Shahrekord, PO Box 115, Iran School of Mechanical and Mining Engineering, The University of Queensland, Brisbane, QLD 4072, Australia

a r t i c l e

i n f o

Article history: Received 24 July 2016 Revised 7 April 2017 Accepted 25 April 2017 Available online xxx Keywords: Melting Nano-PCM Porous medium Magnetic field Cavity

a b s t r a c t This paper presents a numerical study on the phase changing of cyclohexane–copper nano-PCM in a square porous cavity in the presence of a magnetic field. The horizontal walls of the cavity are insulated and its initial temperature is Ti . In the beginning, the temperature of the left vertical wall changes to a temperature higher than the melting point of nano-PCM. Four pertinent dimensionless parameters of porosity (ε ), solid volume fraction (ϕ ), Hartmann number (Ha) and Rayleigh number (Ra) are considered. The effects of these parameters on the flow and temperature fields, the heat transfer rate, and the melting time examined. The results show that changing the porosity has greater influence than changing the solid volume fraction on the melting time. The presence of an intensive magnetic field leads to a decrease in the melting time and prevail of the conductive heat transfer in the cavity. It changes the curvature of the melting line and converts it to a straight line being parallel to vertical walls of the cavity. As the Rayleigh number increases, the convection heat transfer becomes stronger and consequently, the melting speed in the cavity increases. © 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1. Introduction The heat transfer in porous media has increasingly received attention in various modern technologies and industrial applications. In particular, the thermal performance of heat exchangers, reactors, electronic components, heat insulators and geothermal energy systems have extensively been studied [1–4]. Most of these applications have considered the heat transfer in porous media under different boundary conditions [5–9]. As an example, Saeid [10] investigated the heat transfer in a porous medium surrounded by two walls with a limited thickness. He identified an increase in the recirculation flow intensity inside the porous medium as a result of a decrease of the thickness and the conductivity coefficient of the walls. The Nusselt number and horizontal temperature gradient increased as the wall conductivity increased. Another example is the study by Chamkha et al. [11], who studied a porous enclosure exposed to radiation. They examined the effect of radiation intensity on the Nusselt number and vortex power. Some researchers studied cases where the heat energy was either applied to the cavity walls or was placed inside the cavity. Pop and Saeid [12] studied the heat transfer in a vertical porous ∗

Corresponding authors. E-mail addresses: [email protected] (M.T. Kohyani), [email protected], [email protected] (B. Ghasemi), [email protected] (A. Raisi), [email protected] (S.M. Aminossadati).

medium with two heat sources. The variation of thermal characteristics of the porous medium as a function of Rayleigh and Peklet numbers and the distance between thermal sources was studied. Grosan et al. [13] investigated the effect of internal heat generation and magnetic field in a porous medium and showed that the amount of heat transfer was a function of the Rayleigh and Hartmann numbers. They also indicated that the local Nusselt number along the horizontal walls was significantly affected by changing the magnetic field angle from horizontal to vertical. More investigations in this field of research are reported in references [14–16]. Vynnycky and Kimura [17] investigated conjugated convection and conduction heat transfer in a semi-infinite porous medium in the presence of a solid plate. They presented their results for different Rayleigh numbers and unsteady heat transfer regimes. They showed that the steady-state condition was achieved very quickly at large Rayleigh numbers. Islam and Nandakumar [18] studied the effect of generating uniform internal energy on unsteady heat transfer in porous media. Their study highlighted the importance of heat transfer in porous media and its application in radioactive waste storage and geothermal science. They studied the effect of energy generation on flow regime and steady solution of problem. Reported studies in literature have used both single-phase and two-phase models to examine the behavior of nanofluids. The single-phase model has obviously been more popular due to its simplicity in the analysis; however, some researchers have used the two-phase model to ensure more realistic simulation results.

http://dx.doi.org/10.1016/j.jtice.2017.04.037 1876-1070/© 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Please cite this article as: M.T. Kohyani et al., Melting of cyclohexane–Cu nano-phase change material (nano-PCM) in porous medium under magnetic field, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.04.037

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Nomenclature B0 C Fo Fo g H,L hls I Ha K k Nu Nu P Ra S s Ste T t u v U V x, y X, Y

magnitude of magnetic field, kg/Am specific heat, J/kg ◦C Fourier number modified Fourier number gravitational acceleration, m/s2 height and width of the cavity, m enthalpy of fusion of nano-PCM, J/kg electric current, A Hartmann number permeability of the porous medium, m2 thermal conductivity, W/m ◦C mean Nusselt number Nusselt number pressure, Pa Rayleigh number dimensionless horizontal distance from the heated wall to the melting front, s/H horizontal distance from the heated wall to the melting front, m Stefan number temperature, ◦C time, s velocity component in x-direction, m/s velocity component in y-direction, m/s dimensionless horizontal velocity dimensionless vertical velocity horizontal and vertical coordinate, m dimensionless horizontal and vertical coordinate, x/H, y/H

Greek symbols α thermal diffusivity, m2 /s ϕ volume fraction (=volume of the nano-particle/total volume of nano-PCM) ε porosity of the porous medium (=void volume/total volume of porous medium) θ dimensionless temperature difference β volumetric thermal expansion coefficient, 1/K μ = ρ ·ν viscosity of the nano-PCM, N s/m2 ψ stream function, m2 /s

dimensionless stream function σ electric conductivity, 1/ m , effect of nanoparticles’ coefficient in momentum equation  symbol in convergence criteria equation

T temperature difference ρ density, kg/m3  mean specific heat ratio Subscripts c cold wall f fluid h hot wall i initial max maximum nf nano-fluid nm nanomaterial (solid or liquid phase) p nano-particle pm porous medium pnm mean value between porous medium and nanomaterial Sheikholeslami et al. [19,20] reviewed both single-phase and twophase models for simulating nanofluids. In most natural convec-

tion studies, the nanofluids have been assumed a single-phase and homogenous fluid. In this study, the suspended nanoparticles are assumed to have a low concentration, to be in the local thermal equilibrium and to move with the same velocity as the fluid. Chamkha and Muneer [21] studied conjugated heat transfer in a rectangular cavity that consisted of a porous medium and nanofluid. The cavity was heated by a solid triangular wall. They considered adding copper, aluminum-oxide and titaniumOxide nanoparticles to pure water; and examined the influence of Rayleigh number, solid volume fraction and solid wall thickness on the heat and fluid characteristics. They concluded that the heat transfer at low Rayleigh numbers was intensified as the solid volume fraction increased. They also showed that the heat transfer variation of the cavity depended on the solid wall thickness and the solid volume fraction. Sun and Pop [22] considered the convection heat transfer in a right triangle cavity that consisted of porous medium in the presence of nanofluid. They studied the temperature of a part of the cavity’s vertical and inclined walls. Their governing equations were based on Darcy theory. The nanofluid was simulated based on Tiwari and Das [23] model. The effects of different parameters such as Rayleigh number, heater size, and solid volume fraction were investigated. The results showed that the heat transfer from the cavity was enhanced as the solid volume fraction increased at low Rayleigh numbers. An opposite behavior was observed at large Rayleigh numbers. The study of porous media saturated with materials has attracted the attention of researchers in the field of energy efficiency of thermal systems. There are mainly two methods to save energy in storage materials: 1—sensible thermal energy storage; and 2—insensible thermal energy storage [24]. In the first method, the temperature of storage materials such as water, soil and concrete is changed by increasing or decreasing the energy amount. In the second method, the thermal energy is added to or subtracted from material (PCM) by freezing or melting latent insensible heat. The temperature of material is kept constant in this condition. The second method has many advantages compared to the first one and has attracted more attention. Examples of this method are the solar heating systems, low energy buildings and low temperature energy storage units [24–29]. Beckermann and Viskanta [30] numerically investigated the melting of Gallium in a porous medium consisting of glass particles. They found that melting process was affected by the fluid natural convection and the solid conduction heat transfer. They examined the streamlines and isotherms (for different time steps) at various melting stages. The effects of the Rayleigh and Stephane numbers on the total melting time in the cavity and other thermal or fluid flow characteristics were analyzed. Chang and Yang [31] studied the ice-melting process in a rectangle porous medium. They analyzed melting process using a non-Darcy model and considered the buoyancy, friction, and convection effects. The results indicated improvements of the heat transfer, and an increase of the melting speed, gradient and curvature of melting line by increasing the Darcy number. They found that the heat transfer increased over time at the cold boundary and decreased at the hot boundary. Zhang and Nguyen [32] investigated the ice-melting process in a porous medium and analyzed the effect of parameters such as the Rayleigh and Stephane numbers and density changes on the melting process. They noted a special mode of transition in the beginning of melting. They argued that changing the effective parameters on free convection could lead to intensive changes in the heat transfer and melting line progress speed in the cavity. The energy storage method by melting or freezing, described earlier, had advantages such as low conductivity coefficient of phase-change material heat transfer, resulting in an increase of melting or freezing time.

Please cite this article as: M.T. Kohyani et al., Melting of cyclohexane–Cu nano-phase change material (nano-PCM) in porous medium under magnetic field, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.04.037

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Researchers have conducted a number of studies in the field of fast melting/freezing due to their various emerging applications [33–35]. These include the studies of adding the nano-PCM with high conductivity coefficient to phase-change materials, use of porous metal structure, simultaneously use of several materials and placing metal rings to phase-change materials. Hossain et al. [36] studied nano-PCM melting in porous medium for the first time. The melting process was initiated from the upper boundary of square cavity while its other boundaries were insulated. The melting line moved from the top to bottom, being parallel to horizontal wall of the cavity. The results indicated a percentage of nano-PCM in different nanoparticles volume fractions at different time. The required energy for nano-PCM melting was decreased at high nanoparticles volume fractions. Tasnim et al. [37] investigated the effect of adding nanoparticles to phase-change materials in porous medium, and illustrated the effect of changing parameters such as Rayleigh number and nanoparticles volume fraction on melting process of these materials. Their study showed a delay of melting process and a decrease in the convection heat transfer and conductivity as the solid volume fraction increased. The isotherms and streamlines at different times were also presented. The change of heat transfer and the melting and freezing processes were affected by the fluid movement mechanism in the medium. Many studies on melting and freezing in porous media have been reported in literature; however, to the best knowledge of authors, the study of melting process of nano-PCM in porous cavities and in the presence of a magnetic field has not been conducted. An effective thermal control system is highly desired due to the increased heat generated from the tight integration of electrical components or atomic reactors. It is increasingly difficult when the systems are operating at a high temperature with a narrow space and strong magnetic field and/or nuclear radiation. For this problem, the thermal storage system can be simulated as a two-dimensional square cavity. One of the mechanisms that affects the melting process is the volume force due to the magnetic field, which in turn controls the melting or solidification processes. In this study, the effect of magnetic field intensity (by changing the Hartmann number) on melting processes is investigated. The effect of magnetic field on “weak-magnetic” materials is small; however, it must be considered for polymeric materials and PCMs. They are too small to be detected easily under the field strengths as low as generated by permanent magnets and electromagnets. The advent of liquidhelium free superconducting magnets has facilitated the use of high magnetic fields, which have resulted in a number of new findings that would have been impossible under low magnetic fields. In this context, the phase change in a porous cavity and in the existence of magnetic field has been studied by several researchers because of the high magnetic force acting on diamagnetic materials. In this case, the phase change material (PCM) experiences a Lorentz force. This force suppresses the convective flow, which in turn affects the growth rate of melting process [38–40]. It should be indicated that the ferro-fluid (base fluid with magnetic nanoparticle such as Fe3 O4 ) [41] is the most common issue in recent investigations. In this type of fluid, in addition to the Hartmann number, the other magnetic field effects appear in the governing equation. This issue has been considered to be outside of the scope of this study. Hence, the present study aims to investigate the effect of magnetic field and the presence of nanoparticles in the base nano-PCM on the heat transfer and melting process of a porous medium.

3

(i.e. Ti = Tm ). Initially, the temperature of the right vertical wall (Tc ) is equal to the initial temperature of the system (i.e. Tc = Ti ) and the temperature of the left vertical wall is set at a temperature higher than the melting point. The cavity is influenced by a steady magnetic field (intensity = Bo ) being parallel to the horizontal sides of the cavity. It is considered that the porous medium is homogeneous, isotropic and fully-saturated. Nanoparticles are distributed homogeneously in the phase-change material. The nano-PCM and the porous medium are in local thermodynamic equilibrium. The frictional waste effect and the inertia term in the energy and momentum equations are neglected. The thermodynamic properties of the porous medium and the fluid, except the fluid density in the buoyancy term, are constant. The fluid density is changed according to the Boussinesq theory and the fluid in porous medium follows the Darcy model. The effect of magnetic field on fluid movement in the porous medium is based on the models reported by Revnic et al. [42], Garandet et al. [43] and Alchaar et al. [44]. The electric current is a scalar:

∇ ·I =0

(1)

The electric current is related to the electromagnetic field based on the following equation:

I = σ (−∇φ + V × B )

(2)

where σ is the electric conductivity, V is the fluid velocity vector, B is the external magnetic field, and ∇φ depends on the electric field. Garandet et al. [43] showed that the Eqs. (1) and (2) can be converted to ∇φ 2 = 0 (special solution is ∇φ = 0). According to the study by Alchaar et al. [44], since the cavity boundaries are electrically insulated, the electric field within it is omitted. Hence, the two-dimensional equations of continuity, momentum and energy of the fluid are as follow [42]:

∂ u ∂v + =0 ∂x ∂y     = K −∇ P + ρ g + I × B V μ pnm

(3) (4)

 2  ∂ Tnm ∂ Tnm ∂ Tnm ∂ Tnm ∂ 2 Tnm +u +v = αpnm + ∂t ∂x ∂y ∂ x2 ∂ y2

(5)

where u and v are the velocity components in x and y directions, respectively, Tnm is the temperature of nano-PCM in solid or liquid phase and K is the permeability of porous medium. After simplification, the momentum equation is given by:

  g (1 − ϕ )(ρβ )f + ϕ (ρβ )p K ∂ Tnf σnf KB20 ∂ u ∂ u ∂v − =− − ∂y ∂x μnf ∂x μnf ∂ y

(6) The melting line equation can also be written as [37,45]:



−kpnm

∂T ∂s ∂T − ∂x ∂y ∂y



= [(1 − ε )(ρ h )pm + ε (ρ h )nf ] ≈ ε (ρ h )nf

∂s ∂t

∂s , x = s(y, t ) ∂t

(7)

The definitions of the parameters applied in the above equations are described in Table 1. The above equations can be rewritten based on the following stream function:

∂ψ ∂ψ and v = − . ∂y ∂x

2. Methodology

u=

A schematic diagram of the geometry studied in this study is illustrated in Fig. 1. The initial temperature of the system (Ti ) is assumed to be equal to the nano-PCM melting point (Tm )

According to Table 2 and by introducing the dimensionless parameters, the governing equations can be rewritten as dimensionless equations:

Please cite this article as: M.T. Kohyani et al., Melting of cyclohexane–Cu nano-phase change material (nano-PCM) in porous medium under magnetic field, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.04.037

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Fig. 1. A schematic diagram of the physical model.

Table 1 Models used in the study.

In the abovementioned equations, Ra is the Rayleigh number given by:

Model

Parameter [reference]

α nf = knf /(ρ c)nf (ρ c)nf = (1 − ϕ )(ρ c)f + ϕ (ρ c)p

Nanofluid thermal diffusivity [19,46] Nanofluid thermal capacity [19,47] Nanofluid thermal conductivity [19,21]

(k +2k )−2ϕ (k −k )

knf = (kpp +2kf )+ϕ (k f−kpp) kf f f μnf = μf /(1 − ϕ )2.5 3( (σp /σf )−1)ϕ σnf σ = 1 + ( (σp /σ )+2)−( (σp /σ )−1)ϕ f

f

f

(ρβ )nf = (1 − ϕ )(ρβ )f + ϕ (ρβ )p ϕ = ∀p /(∀p + ∀f ) (ρ c)pnm = (1 − ε )(ρ c)pm + ε (ρ c)nm kpnm = (1 − ε )kpm + ε knm (ρ c )

pm pnm = ε + (1 − ε ) (ρ c)nm α pnm = kpnm /(ρ c)nm

Ra = (gρf

Nanofluid viscosity [19,48] Nanofluid electrical conductivity coefficient [19,49] Nanofluid thermal expansion [19,21] Solid volume fraction Average heat capacity of nanomaterials and medium Average heat capacity of nanomaterials and medium

Table 2 Parameters used in the dimensionless equations. Symbol

=

ψ αpmf

θ=

T −Tc Th −Tc

x ; Y = Hy H tα = Hpnm 2 Fo

X= Fo

Fo = pnm

Location in horizontal and vertical directions Dimensionless time Modified dimensionless time

Dimensionless momentum

∂ 2 ∂ 2

∂θ ∂ 2

+ = −Ra · − Ha · ∂X ∂X2 ∂Y 2 ∂X2

(8)

∂θ ∂ ∂θ ∂ ∂θ αpnm ∂ θ ∂ θ + − = + αpmf ∂ X 2 ∂ Y 2 ∂ Fo ∂ Y ∂ X ∂ X ∂ Y 2

 (9)

Dimensionless melting line

 Ste  ∂θ ∂ S ∂θ  ∂S = −pnm × − ε ∂ X ∂Y ∂Y ∂ Fo

 σf KB20 /μf

(12)

The other parameters are given by:

  3( (σp /σf ) − 1 )ϕ = 1+ × (1 − ϕ )2.5 ( ( σ p / σ f ) + 2 ) − ( ( σ p / σ f ) − 1 )ϕ    ρp βp

= (1 − ϕ ) + ϕ × ( 1 − ϕ )2 . 5 ρf βf

(13)

As it can be seen from the definitions for and , the effect of presence of nanoparticles in the base fluid will be introduced to momentum equation by these parameters. This will be achieved in α energy equation by αpnm . The Stephane number in melting term is

Ste =

Temperature

2



pmf

Stream function



Ha =

defined by:

Dimensionless parameter

Dimensionless energy

(11)

where T = (Th – Tc ). Ha is the Hartmann number given by

Average thermal capacity ratio Average thermal diffusivity nano-PCM and medium Average thermal diffusivity of base fluid and medium

α pmf = kpmf /(ρ c)f

βf K T H )/(μf αpmf )

Cn (Th − Tm ) hls

(14)

The important point for introducing this parameter is to separate the porosity effect in the melting line equation (Ste/ε ). This provides an opportunity for investigating the effects of boundary conditions, nano-PCM properties and porosity individually. This issue has not been considered in previous works. The thermophysical properties of materials are presented in Table 3. The dimensionless initial and boundary condition are as follows: Boundary conditions

θ (0,Y ) = 1 , θ (1,Y ) = 0 ∂θ ∂θ (X , 0 ) = (X , 1 ) = 0 ∂Y ∂Y

(0, Y ) = (1, Y ) = (X, 0 ) = (X, 1 ) = 0

(15)

Initial conditions

(10)

θ0 (X, Y ) = 0 (X, Y ) = 0

(16)

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Table 3 Thermophysical properties of base material (cyclohexane), nanoparticles (copper) and porous medium (aluminum). Symbol

Liquid cyclohexane

Solid cyclohexane

Copper

Aluminum

Cp (J/kg/K) ρ (kg/m3 ) k(W/m/K) β × 105 (1/K) σ (1/ m) hls (J/kg)

1763 799 0.127 121 5 × 10−9 32,557

1600 856 0.136 –

385 8933 401 1.67 5.96 × 107 –

903 2702 237 – – –

In this study, the local and average Nusselt number (Nu and Nu) on the hot and cold walls are given by:

k ∂θ Nu = − nf , kf ∂ X

Nu =

1

NudY

(17)

0

Table 4 Grid independency ϕ = 0.01, ε = 0.3).

Number of grid points

3. Numerical solution method The finite volume method (FVM), used in this study, is a method for representing and evaluating partial differential equations in the form of algebraic equations [50,51]. In the last two decades, the Lattice Boltzmann method (LBM) has emerged as a promising tool for modeling the Navier–Stokes equations and simulating complex fluid flows. LBM is based on microscopic models and mesoscopic kinetic equations. The Lattice Boltzmann method [52–54] was originated from Ludwig Boltzmann’s kinetic theory of gases. LBM is very applicable to simulate multiphase/multicomponent flows and complex boundaries. The control volume finite element method (CVFEM) is another powerful method for solving fluid flow and heat transfer problems in complicated geometries and multi-physics problems [55,56]. It scores over finite volume methods using unstructured meshes by using element-based interpolation for variables at the control volume faces. Eqs. (8) and (9) are related to each other and should be solved simultaneously. One of the important parts of this problem is finding the time dependence boundary condition in melting line. This boundary condition can be determined by solving simultaneously the momentum (8), the energy (9) and the melting line (10) equations. The power-law method [57] is used for approximation of diffusion and convective terms. The central difference approximation is used for diffusion terms. An implicit computer code was developed in FORTRAN for the analysis. The achieved algebraic equations were solved by repeating line by line and using triple diametrical matrix algorithm. The convergence of repetition process was continued to achieve following limitation each time step:





Maxni,+1 − ni, j /ni, j  ≤ 10−4 j

study

46 × 46 56 × 56 76 × 76 101 × 101

in

steady

Ra = 100

condition

(Ha = 1,

Ra = 200

Nu

max

Nu

max

2.68 2.66 2.65 2.65

3.40 3.44 3.45 3.45

3.75 3.72 3.69 3.69

5.42 5.43 5.47 5.47

(18)

That  could be or θ and n is the repetition counter. The convergence criterion for the steady state is:

  nt  nt  nt Nuh − Nuc /Nuh ≤ 10−4

(19)

n is the time step, Nuh and Nuc represent the average Nusselt number for the hot and cold sides, respectively. For obtaining exact solutions, the proportional grid and time step should be used. The grid independence study was carried out by considering four grid sizes. The results are presented in Table 4, in terms of Num and max at steady state conditions for Ha = 1, ϕ = 0.01, ε = 0.3 and two Rayleigh numbers of Ra = 100 and 200. Fig. 2a presents the grid independence study in terms of changing Nuh with the dimensionless time (Ste.Fo). According to these results, a 76 × 76 uniformly-spaced grid was selected for the numerical scheme. The time step independence study is also presented in Fig. 2b for the melt fraction changing with dimensionless time.

Fig. 2. (a) Grid independency study, (b) Time step independency study (Ra = 100, Ha = 1, ϕ = 0.01, ε = 0.3).

Please cite this article as: M.T. Kohyani et al., Melting of cyclohexane–Cu nano-phase change material (nano-PCM) in porous medium under magnetic field, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.04.037

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Fig. 3. Validation of the present code for melting in the square porous cavity against other researchers (Ha = 0, Ra = 50). (a) ϕ = 0 and (b) ϕ = 0.1.

The results show that the dimensionless time step equal to 0.009 can be used in this study for achieving accurate and fast results. 4. Results and discussion The results of comparison with reference [37,58] for melting in the square porous cavity without effect of magnetic field are presented to ensure the accuracy of computer code developed in this study. First, the variation of Nuh in the cavity with dimensionless time at Ra = 50, Ha = 0, and ϕ = 0 are shown in Fig. 3a and validated against the work by Tasnim et al. [37] and Jany and Bejan [58]. The results are in good agreement. In addition, the variation of Nuh in the enclosure obtained from the present model is validated against Tasnim et al. [37] at Ra = 50, Ha = 0, ϕ = 0.1 in Fig. 3b. It is evident that the corresponding results from two studies match. 4.1. Effect of porosity As mentioned earlier, the use of metal structure with high conductivity coefficient as the porous medium is one of the melting mechanism control and improvement methods. Regarding the selection of aluminum as the medium solid structure, the effect of porosity change, which is saturated volume with nano-PCM to

Fig. 4. Streamlines (left) and isotherms (right) at different porosity (Ra = 100, Ha = 1, ϕ = 0.01, Ste.Fo = 1).

whole medium volume ratio, is investigated in this section. Hence, the investigations is carried out for constant values of Ra = 100, ϕ = 0.01 and Ha = 1, and for a range of 0.1 ≥ ε ≤ 0.6. The isotherms and streamlines for ε = 0.1, 0.3, 0.6 in the time of Ste.Fo = 1 is illustrated in Fig. 4. The results show that the vortexes are more evident at ε = 0.1 due to the presence of high percentage of solid part with high thermal conductivity in the cavity and more power heat transfer consequently. This leads to more melting in the cavity than bigger ε in Ste.Fo = 1. The amount of melting percent changes in the cavity for different values of medium porosity with time is shown in Fig. 5. The results show a higher melting speed for a smaller porosity due to easier permeate of heat through conductivity in solid part of aluminum with high thermal conductivity coefficient. In order to examine the increase of heat transfer due to the decrease of porosity, the variation of the average Nusselt number with time at the hot wall is presented in Fig. 6. The average Nusselt number decreases with time as a result of reduction of temperature difference between the wall and medium. In order to examine the variation of melting process in the cavity with time, the time history of isotherms and streamlines at constant Rayleigh and Hartmann numbers, nano-PCM volume ratio and medium porosity parameters and for three values of Ste.Fo are illustrated in Fig. 7. It is clear that the power of vortexes and the convection heat transfer increase with time due to the increase of

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Fig. 5. Nano-PCM melting percent in cavity over time (Ste.Fo) for different porosity (Ra = 100, Ha = 1, ϕ = 0.01).

Fig. 7. Streamlines (left) and isotherms (right) at different times (Ra = 100, Ha = 1, ϕ = 0.01, ε = 0.3). Fig. 6. Variation of the hot wall average Nusselt number with time (Ste.Fo) for different porosity (Ra = 100, Ha = 1, ϕ = 0.01).

the temperature difference between the up and bottom of the cavity. Fig. 8 shows the melting line form and increasing its curvature in the cavity over time. The straight melting line at the start of process is an indication of the dominant conduction heat transfer. The strength of convection heat transfer with time in the upper cavity and melting speed in this part is more than the bottom cavity. This leads to more advances of melting line in the upper cavity than in the lower cavity. 4.2. Effect of Rayleigh number The scope of this section is to examine the effect of Rayleigh number (50 ≤ Ra ≤ 500) on the melting process at constant values of ε = 0.3, ϕ = 0.01 and Ha = 1. Fig. 9 presents the isotherms and streamlines in the cavity for three values of Ra = 50, 250 and 500 and Ste.Fo = 1. It is clear that the convection heat transfer increases as the power of vortexes increases at higher Rayleigh numbers. This leads to more advanced melting in the upper cavity especially at large Rayleigh numbers. The change of isotherms from vertical to horizontal is seconded of overcoming the convection heat transfer as the Rayleigh number increases. Fig. 10 shows the variation of melting rate with Rayleigh number. It can be seen that at a fixed time, a higher amount of nano-PCM is melted at larger Rayleigh numbers. For example, at Ra = 500, the melting process is completed at Ste.Fo = 1.6; while

Fig. 8. Melting line advance in cavity over time (Ste.Fo = 0.1, 0.2, 0.5, 1, 1.5) (Ra = 100, Ha = 1, ϕ = 0.01, ε = 0.3).

at Ra = 50, the complete melting process occurs at Ste.Fo = 2. The effect of Rayleigh number is more evident in forming temperature field with required intensity differences for nano-PCM melting over time. Fig. 11 shows the variation of the local Nusselt number along the left and right walls for two values of Ste.Fo = 0.5 and 1.5. It can be seen that for Ste.Fo = 0.5, there is no heat transfer at the

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Fig. 11. Local Nusselt number along with hot and cold walls of the cavity for different times (Ra = 250, Ha = 1, ϕ = 0.01, ε = 0.3).

right wall due to the failure in achieving the melting line to this wall; however, after achieving the melting line at the right wall for Ste.Fo = 1.5, the Nusselt number becomes greater than zero. The local Nusselt number decreases from the bottom to the top of the hot wall due to the decrease of temperature difference between the hot wall and the adjacent material.

4.3. Effect of Hartmann number

Fig. 9. Streamlines (left) and isotherms (right) at different Rayleigh numbers (Ste.Fo = 1, Ha = 1, ϕ = 0.01, ε = 0.3).

Fig. 10. Nano-PCM melting percent in cavity over time (Ste.Fo) for different Rayleigh numbers (Ha = 1, ϕ = 0.01, ε = 0.3).

It was previously mentioned that a mechanism to effect on melting process is creating volume force like magnetic field which leads to control of melting process. In this section, the effect of magnetic field intensity change by changing the Hartmann number is investigated by considering Ra = 100, ϕ = 0.01, and ε = 0.3. First, the isotherms and streamlines are plotted in Fig. 12 in order to analyze the effect of magnetic field. It can be seen that by increasing the Hartmann number from zero to 100, the vortexes power is decreased. The change of prominent mechanism of heat transfer from convection to conduction can be seen from the results with isotherms be vertical by increasing the Hartmann number. On the other hand, the curvature of melting line is decreased by increasing the Hartmann number. For Ha = 100, the melting line becomes parallel to the vertical walls of cavity. By examining the melting percentage in the cavity over time for different Hartmann numbers (Fig. 13), it is obvious that the melting time is not changed significantly by a small change of the field intensity from zero to one. The power of vortexes in the melted nano-PCM part is, however, dramatically decreased and the heat transfer mechanism is changed to conductive heat transfer (Fig. 12) as the field intensity increases (Ha = 100). Although this phenomenon leads to a decrease in the advancement of melting process in the upper part of the cavity at a fixed time; however, the total amount of melted material is increased and the time duration of total nanoPCM melting is decreased as the melting in the bottom part of the cavity increases. The present of magnetic field leads to a change in the prominent heat transfer process to conduction heat transfer and an increase of melting speed by presence of aluminum with high conductivity coefficient as the medium solid part. On the other hand, it can be seen from Fig. 13 that in the half time of complete melting process, the melting percentage is approximately constant at three Hartmann numbers and in the second half time also has a little change. Thus, the melting line is kept vertical by applying the magnetic field without much changes in the melting time. This can be seen in some phenomena like controlled melt-

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9

Fig. 13. Nano-PCM melting percent in cavity over time for different Hartmann number (Ra = 100, ϕ = 0.01, ε = 0.3).

Fig. 12. Streamlines (left) and isotherms (right) at different Hartmann numbers (Ste.Fo = 1, Ra = 100, ϕ = 0.01, ε = 0.3).

ing and freezing in order to forming a regular crystalline structure intended for researchers.

4.4. Effect of nano-particle volume fraction

Fig. 14. Melting line advance in cavity over time (Ste.Fo = 0.2, 0.5, 1, 1.5) (ϕ = 0.01, 0.05, Ra = 100, Ha = 1, ε = 0.3). Table 5 Melt fraction (Mf) and the change of Melt fraction ( Mf). Dimensionless time

Volume fraction of nanoparticle

ϕ=0

ϕ = 0.01

ϕ = 0.05

Ste.Fo = 1.015

Mf

Mf Mf

Mf Mf

Mf

70.25 – 89.9 – 100 –

69.97 0.4% 89.7 0.22% 99.85 0.15%

69 1.8% 88.34 1.76% 98.3 1.72%

Ste.Fo = 1.48 Ste.Fo = 1.96

The effect of adding copper nanoparticles to the base material (cyclohexane) is investigated. Hence, the location of melting line for ϕ = 0, 0.05 and constant other effective parameters (ε = 0.3, Ha = 1, Ra = 100) are presented in Fig. 14 for different time steps. Adding nanoparticles causes a delay in the melting process of nano-PCM. It should be noted that the improvement effect of nano-PCM effective conductive coefficient is not significant by adding copper nanoparticles against negative effect of nanoparticles presence in the porous medium due to decreasing role of vortexes power and convection heat transfer mechanism in investigated condition and by regarding aluminum presence as porous medium part with its high thermal conductivity coefficient. This improvement does not have significant effect on acceleration of melting process and decreases the melting speed slightly. The amount of melt fraction (Mf) that changes in the cavity for different nanoparticle volume fraction and time step is presented in Table 5.

−Mf ].

Mf = [ Mfϕ=0 Mf

5. Conclusions This paper presented the results of a numerical investigation on the phase-change nano-PCM (cyclohexane–copper) melting in a square porous cavity and in the presence of a magnetic field. Four parameters of Ra, Ha, ϕ , and ε were considered in the analysis. The effects of these parameters on the temperature and stream fields, the heat transfer rate and the required nano-PCM melting time were examined. The results showed that the melting process was more affected under the influence of changing the porosity of the porous medium than changing the nanoparticles volume ratio. Faster nano-PCM melting was achieved at a low porosity while other parameters were kept constant. The presence

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Please cite this article as: M.T. Kohyani et al., Melting of cyclohexane–Cu nano-phase change material (nano-PCM) in porous medium under magnetic field, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.04.037