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Does nanoparticles dispersed in a phase change material improve melting characteristics?
International Communications in Heat and Mass Transfer 89 (2017) 219–229
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Does nanoparticles dispersed in a phase change material improve melting characteristics?
MARK
⁎
Rouhollah Yadollahi Farsania, Afrsiab Raisia, , Afshin Ahmadi Nadooshana, Srinivas Vanapallib a b
Shahrekord University, Engineering Faculty, PO Box 115, Shahrekord, Iran University of Twente, Faculty of Science and Technology, Post Bus 217, 7500 AE Enschede, The Netherlands
A R T I C L E I N F O
A B S T R A C T
Keywords: Melting Nanoparticles Heat storage Buoyancy driven convection
Nanoparticles dispersed in a phase change material alter the thermo-physical properties of the base material, such as thermal conductivity, viscosity, and specific heat capacity. These properties combined with the configuration of the cavity, and the location of the heat source, influence the melting characteristics of the phase change material. In this paper, an assessment of the influence of the nanoparticles in the base material subjected to a heat generating source located in the center of an insulated square cavity, which is a common configuration in thermal capacitors for temporal heat storage is investigated. The interplay between heat conduction enhanced due to an increase in thermal conduction and buoyancy driven heat convection damped by the increase in viscosity of nanoparticles dispersed in the phase change materials is studied with the calculated streamlines and isotherms. We observed three regimes during the melting process, first at an early time duration dominated by heat conduction, later by buoyancy driven convection till the melting front levels with the center of the cavity, and lastly once again heat conduction in the bottom portion of the cavity. During the first two regimes, addition of nanoparticles have no significant performance gain on the heat storage cavity, quantified by maximum temperature of the heat source and average Nusselt number at the faces of the heat source. In the late regime, nanoparticles provide a slight performance gain and this is attributed to the increase in the specific heat of the melt due to the nanoparticles.
1. Introduction Thermal capacitors consisting of materials with a solid to liquid phase change provides an excellent opportunity to bridge the cooling supply and demand needs, and therefore reduce the size of the thermal circuitry. Several researchers explored this idea in the context of heat storage, electronics cooling and many other applications [27,19,14,5]. These thermal management problems can be broadly divided into two categories based on the driving source, 1) temperature, 2) heat flux. An example of temperature driven application is the temporal storage of solar heat, where the temperature of heat generating part is fairly constant, as well as heat transport with heat pipes, where the condenser operates at a narrow temperature range depending on the working substance in the heat pipe [18,15,21]. The second case, is when the heat storage material has to suppress thermal runaway or dampen the temperature rise of a device, such as in electronics cooling. In case of a heat flux driven thermal storage, one key parameter of interest is the maximum temperature rise of the device [12,25]. Paraffin based materials are commonly used in terrestrial and space
⁎
applications to control the temperature of the devices around room temperature. These thermal capacitors consist of series of paraffin packed pockets with the heat generating part at the center, as shown in Fig. 1. Initially most designers have employed pure conduction models for design analyses, without considering convection during melting. The role of buoyancy driven convection was later identified to play a major role in the melt of a phase change material [9]. In a cavity with heat source at the center, the upper and the side regions surrounding the heat source have the right temperature boundary conditions for the onset of buoyancy driven flow (see Fig. 1), whereas at the bottom section below the heat source, buoyancy drive flow cannot develop. In order to increase the thermal diffusivity of the cavity, nanoparticles mixed with phase change materials (NePCM) have been recently proposed. The main hypothesis for proposing this mixture, is that the average thermal conductivity of the substance increase, thereby increasing the overall melting process, increasing the liquid fraction, and thus reduces the maximum device temperature for a certain heat generation rate. The increase in the viscosity in the liquid phase due to the presence of nanoparticles, dampen the buoyancy driven convective
Corresponding author at: Engineering Faculty, Shahrekord University, PO Box 115, Shahrekord, Iran. E-mail address: [email protected] (A. Raisi).
https://doi.org/10.1016/j.icheatmasstransfer.2017.10.006
0735-1933/ © 2017 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 89 (2017) 219–229
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Nomenclature b B Bz c f g h L,l k Nu p P Pr q‴ Ra Ste T Tm,Ts Th,Tc t u,v U,V x,y X,Y
Abbreviations
enthalpy-porosity coefficient, kg m− 3 s− 1 dimensionless enthalpy-porosity coefficient Boltzmann constant specific heat, J kg −1 K− 1 liquid fraction gravity, ms− 2 enthalpy, J kg− 1 K− 1 cavity and heat source dimension, m thermal conductivity, Wm− 1 K− 1 Nusselt number, − knf/kf(Ts − Tm)∂ T/∂ n pressure, Nm− 2 dimensionless pressure Prandtl number, Pr = νf/αf heat generation rate, W/m3 Rayleigh number, gβfq‴l5/νfαfks Stefan Number, cfq‴l2/hnfks temperature, K melting and solidification points, K hot and cold temperatures, K time, s velocity in the x,y direction, ms− 1 dimensionless velocity Cartesian coordinate, m dimensionless Cartesian coordinate
CLF PCM NePCM
cavity liquid fraction Phase Change Material Nano enhanced PCM
Greek symbol α β μ ν θ ρ ϕ σ τ
thermal diffusivity, m2 s− 1 expansion coefficient, K− 1 dynamic viscosity, N s m− 2 kinematic viscosity, m2 s− 1 dimensionless temperature density, kg m− 3 volume fraction electrical conduction, S m− 1 dimensionless time
Subscript f,s m nf ns np
fluid and solid melting point fluid PCM with nanoparticles solid PCM with nanoparticles nanoparticles
heating, compared to horizontal heating configuration. However, compared to the base PCM, the NePCM showed a lower melting rate for both the configurations. In water based nanofluid melting, Feng et al. [8] observed a completely different trend for a bottom heating configuration, in which the melting rate increased with the volume fraction of copper oxide nanoparticles. Similar findings were reported albeit for solidification (see Fig. 2(c)) by Khodadadi and Hosseinzadeh [13] for water and copper oxide NePCM. A concentric cylindrical configuration as shown in Fig. 2(d) is used by Dhaidan et al. [7] to investigate melting of n-octadecan mixed with copper oxide nanoparticles. The experimental results show that the melting rate decrease with an increase in volume fraction of nanoparticles. However, the maximum temperature of the wall during melting reduced with increasing volume fraction of nanoparticles. The authors suggested an eccentric configuration of the cylinders to utilize enhanced melting on the top section of the annulus. A similar configuration but a constant temperature boundary conditions as shown in Fig. 2(e) is investigated by Sebti et al. [20], with similar conclusions. In another work, Dhaidan et al. used a square enclosure with heat flux boundary conditions on one vertical side and the remain sides are insulated, as shown in the Fig. 2(f). Contrary to the findings of Arasu and Mujumdar [2] mentioned earlier, the results presented for this configuration showed that higher melting rate are possible as the volume fraction of the nanoparticles are increased. The above discussion of the previous work point that the merits of NePCM should be discussed in the context of a particular configuration combined with the appropriate heating boundary condition pertaining to an application. The geometry shown in Fig. 1, has combined features of configuration shown in Fig. 2(a) and (b), and immediate conclusions cannot be drawn by combining the results. In this paper, a systematic numerical approach is presented to assess the addition of nanoparticles in the phase change material for the geometry and boundary conditions shown in Fig. 1. The transient flow and temperature is evaluated and the corresponding macroscopic variables such as average melt temperature, temperature of the heat source, average Nusselt number and cavity liquid fraction (CLF) are determined.
currents, and therefore reduces the overall heat transfer. Therefore, when designing a nanoparticle enhanced phase change material thermal capacitor, the interplay between the increase in the thermal conductivity and decrease in convection should be carefully considered [18,15,21]. Arasu and Mujumdar [2] studied melting of paraffin wax mixed with alumina nanoparticles of a number of volume fractions, in a square cavity of side 25 mm, heated from a vertical side (see Fig. 2(a)) and in a different configuration where the cavity is heated from the bottom side (see Fig. 2(b)). The results suggest that the addition of nanoparticles decreases the melting rate of the PCM, and they conclude that the melting rate and the thermal energy storage is greater for vertical
Fig. 1. Physical model of the cross-section of a heat storage cavity with heat source at the center and insulated at the outside walls.
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Fig. 2. Literature review of configurations with boundary conditions of proposed cavities with nanoparticles mixed with phase change materials.
hnf ∂f ∂T ∂T ∂T +u +v = αnf ∇2 T − cnf ∂t ∂x ∂y ∂t
2. Problem definition and mathematical formulation Fig. 1 shows the mathematical domain of the physical system consisting of rectangular container with the heat source at its center and the outer walls are insulated. The phase change material is initially in the solid phase at its melting point temperature, Tm. The homogeneous heat generation rate of q‴ per unit volume is applied for t > 0 inside the heat source which is located at the center of the cavity and this region contains aluminium material with a thermal conductivity of ks = 202 Wm− 1 K− 1. Table 1 shows the thermophysical properties of the paraffin, alumina nanoparticles, and aluminium. Alumina nanoparticles are used in this study because the nanofluid formed with these particles have lower viscosity and higher thermal conductivity compared to other particles such as copper oxide. Moreover, alumina particles are cheap, safe and readily available [1]. As the PCM begins to melt from the side-walls of the heat source, the solid-liquid phase boundary develops inside the cavity. In the melt, both heat conduction and heat convection play a role, where as in the solid phase the heat transfer is only due to heat conduction. The flow is assumed to be two-dimensional, laminar and in-compressible. The governing equations describing natural convection and conduction within the phase change material are given below [2],
• Energy (solid phase): ∂T = αns ∇2 T ∂t
(5)
• Energy (heat source): q‴ ∂T = αs ∇2 T + (ρcp )s ∂t
(6)
The density and specific heat capacity of nanoparticles mixed in PCM is equal to the corresponding weight fraction of the constituents [10], as given below,
ρnf = (1 − ϕ) ρf + ϕρnp
(7)
(ρc )nf = (1 − ϕ)(ρc )f + ϕ (ρc )np
(8)
(ρβ )nf = (1 − ϕ)(ρβ )f + ϕ (ρβ )np
(9)
hnf =
• Continuity:
(1 − ϕ)(ρh)f ρnf
(10)
The empirical nanofluid dynamic viscosity is given by the following
∂u ∂v + =0 ∂x ∂y
(1)
Table 1 Thermophysical properties of the PCM [16] and Al2O3 [2] at T = 273.2 K.
• x momentum: ∂u ∂u ∂u 1 ⎛ ∂p + μnf ∇2 u + bu⎞ − +u +v = ∂t ∂x ∂y ρnf ⎝ ∂x ⎠
•
(4)
(2)
⎜
Paraffin (RT 44HC)
Al2O3
Al
Viscosity, μf Ns/m2
0.001 exp (−4.25 + 1790/T) 760 (780) 1.0 (10− 3)
–
–
3600 1.6 (10− 5)
2700 –
0.19 (0.21)
36
202
45 255,000 2000 (2000)
– – 765
60.32 – 903
Density, ρf (ρs) kg m− 3 Thermal expansion coefficient, β K− 1 Thermal conductivity, kf (ks) Wm− 1 K− 1 Melting point, Tm °C Latent heat fusion, hm J kg− 1 Specific heat, cf (cs) J kg− 1 K− 1
y momentum:
∂v ∂v ∂v 1 ⎛ ∂p + μnf ∇2 v + bv + ρnf g β(T − Tm) ⎞ +u +v = − ∂t ∂x ∂y ρnf ⎝ ∂y ⎠
Property
⎟
(3)
• Energy (liquid phase): 221
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significantly [11]. In this work, C1 = 1 × 108 kg·m− 3·s− 1 and C2 = 0.003 are used. The governing equations are reduced to the non-dimensional form by using the following variables including dimensionless parameters like, X = x/1, Y = y/1, U = ul/αf, V = vl/αf, θ = (T − Tm)/q‴l2/ks, P = pl2/ρnfαf, as well as dimensionless governing numbers like, τ = αft/l2 which is Fourier number, B = bl2/ρnfαf is the dimensionless form of b, Ra = gβfq‴l5/ksνfαf is the Rayleigh number, Ste = cfq‴l3/ hnfks is the Stefan number, Pr = νf/αf is the Prandtl number.
equation, [22].
μnf = 0.983e (12.959ϕ) μf
(11)
The effective thermal conductivity of the NePCM, taking into account particle size, particle volume fraction and temperature dependence as well as properties of the base PCM is proposed by Arasu et al. [2] and Ho and Gao [10] and is repeated below for completeness,
knf =
knp + 2kf − 2(kf − knp ) ϕ knp + 2kf + (kf − knp ) ϕ
kf + 5 × 10 4βk fϕρf cf
BzT ζ (T , ϕ) ρnp dnp
• Continuity:
(12) where Bz is Boltzmann constant, 1.381 × 10
βk =
− 23
8.4407(100ϕ)−1.07304
(13)
ζ (T , ϕ) = (2.8217 × 10−2ϕ + 3.917 × 10−3) + (−3.0669 ×
∂U ∂V + =0 ∂X ∂Y
J/K and
10−2ϕ
• X momentum:
T Tref
− 3.91123 ×
10−3)
μ ∂U ∂U ∂U ∂P +U +V =− + nf ∇2 U + BU ρnf α f ∂τ ∂X ∂Y ∂X
(14)
where Tref is the reference temperature equal to 273.15 K. The first part of Eq. (14) is obtained directly from the Maxwell model while the second part accounts for Brownian motion, which causes the temperature dependence of the effective thermal conductivity. Note that there is a correction factor, f in the Brownian motion term, since there should be no Brownian motion in the solid phase. The computed thermophysical properties of paraffin dispersed with 0, 0.02, and 0.05 volume fraction of Al2O3 nanoparticles using Eq. (11)–(14) are plotted as a function of temperature and volumetric concentration (see Fig. 3). The solid to liquid phase change process is mathematically described using enthalpy-porosity formulation proposed by Brent et al. [4]. In this approach, a parameter b is defined in the momentum equations (Eqs. (2) and (3)) to gradually reduce the velocities from a finite value in the liquid to zero in the solid, over the computational cell that undergoes the phase change. This is achieved by assuming that such cells behave like a porous media with a porosity equal to the liquid fraction. Based on the Carman-Kozeny relation the coefficient b is defined as,
−C1 (1 − f )2 b= f 3 + C2
(16)
(17)
• Y momentum: μ (ρβ)nf ∂V ∂V ∂V ∂P +V =− + nf ∇2 V + RaPr θ + BV +U ∂τ ∂X ∂Y ∂Y ρnf α f ρnf βf
(18)
• Energy (liquid phase): hnf ∂θ ∂θ ∂θ α ⎛ ∂f ⎞ +U +V = nf ∇2 θ − ∂τ ∂X ∂Y αf cnf ΔT ⎝ ∂τ ⎠
(19)
• Energy (solid phase): ∂θ α = ns ∇2 θ ∂τ αf
(20)
• Energy (heat source): ∂θ α α = s ∇2 θ + s ∂τ αf αf
(21)
The dimensionless initial and boundary conditions are:
(15)
• at initial time, τ = 0, U = V = 0 and θ = 0 • for the side walls, ∂ θ/∂ n = 0 • for melting front, U = V = θ = 0
In this equation, f = 1 and f = 0 in the liquid and solid region, respectively. f can take values between 0 and 1 in the mushy zone (liquid and solid). The constant C1 has a large value (107 − 1015) to suppress the velocity as the cell becomes solid and C2 is a small constant to avoid a division by zero when a cell is fully located in the solid region, namely f = 0. The choice of these constants is arbitrary. However, the constants should ensure enough depression of the velocity in the solid region and should not influence the numerical results
Where “n” is a normal vector to the side-walls. The absorption of latent heat during melting is included as a sink term in the last term in the right side of energy Eq. (19). The latent heat content of each control volume is evaluated after each energy equation iteration cycle. Based
Fig. 3. NePCM dynamic viscosity (a), thermal conductivity (b).
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time step of 0.005 yielded sufficient accuracy. The numerical results with the configuration shown in Fig. 1 are presented in two parts; First, the effect of heat generation rate in the heat source on the flow and phase change phenomena is studied by varying the Rayleigh number. Second, the effect of adding nanoparticles to the base PCM on the melting process is studied. In both these parts, the streamlines in the liquid melt, the isotherms in the melt, the liquid fraction in the cavity, the average Nusselt number on the heat source walls, and the maximum and the average temperature of the cavity are computed as a function of dimensionless time (Fourier number, τ).
on the latent heat content, a liquid fraction for each control volume is determined, which is elaborated in the next section. For control volumes containing a liquid phase of the substance, f is set to 1, and for control volumes containing solid phase, f is set to 0. The control volumes with values of f between 0 and 1 are treated as a mushy zone. The phase change is assumed to be non-isothermal, so the idea of the mushy zone is introduced to gradually switch off the velocities from liquid to solid at the solid-liquid interface [11]. 3. Numerical method of solution The governing equations (Eq. (16)–(21)) along with the boundary conditions are discretized using the finite-volume technique. This method appears to be suitable for numerical solution of the model equations describing phase change processes [23]. The power law scheme, which is a combination of the central difference and the upwind schemes, is used to discretize the convection terms. A staggered grid system, in which the velocity components are stored midway between control volumes is used (see Fig. 4). The SIMPLE algorithm proposed by Patankar [17] is used to solve the discretized and coupled continuity, momentum and energy equations and a line by line solver based on the tri-diagonal matrix algorithm is used to solve iteratively the algebraic discretized equations. Based on the finite difference method for the analysis of phase change problems proposed by Voller et al. [24], the enthalpy value is expressed as a function of temperature.
⎧ cns T h = cnf T + hnf (T − Ts )/(Tm − Ts ) ⎨ c T + hnf + cnf (Tm − Ts ) ⎩ nf
4.1. PCM melting The melting phenomena of the base PCM is studied for three values of Rayleigh numbers namely, 1.48 × 107, 2.72 × 106, and Ra = 5.74 × 105. These values are chosen based on the various values of volumetric heat generation in the source. It must be noted that Rayleigh number in this study essentially represents the rate of heat generation in the heat source. Considering the definition of Rayleigh number, Ra = gβq‴l5/(ksνfαf), for the specific thermo-physical properties of paraffin (Table 1), Rayleigh number is only dependent on the heat generation within the source. For example for a heat generation rate of 12 × 106 W/m3, the corresponding calculated Rayleigh number is equal to 2.72 × 106. Therefore a heat generation rate more than and less than this value (see Table 2) is considered in this study. For each of these Rayleigh numbers a constant value of Stefan number should be considered, which are shown in Table 2. Due to the change in melting region, the instantaneous value of Rayleigh number changes, however the reported Rayleigh and the associated Stefan numbers are for the completely melted situation. The streamlines in the melt region at various dimensionless times for all Rayleigh numbers are shown in the Fig. 6. The vortices in the liquid phase is due to the buoyancy driven convection induced by the temperature difference between the heat source surfaces and the solid PCM. The molten flow rises up along the side-surfaces of the source and then descends near the cold solid-liquid interface, resulting in two symmetric vortices in the opposite direction. The results show that at any given dimensionless time the strength of the vortices increase with the increase in Rayleigh number. The numbers labeled at the center of the streamlines in the Fig. 6 indicate the maximum strength of the circulation, Ψmax. An interesting trend is observed in the circulation strength as a function of dimensionless time. In case of Ra = 5.74 × 105, conduction
T < Ts Solid Ts ≤ T ≤ Tl Mushy T > Tl
Liquid
(22)
Assuming a linear temperature profile in the mushy zone, the enthalpy of the control volume as a function of liquid fraction, f, is given by,
h = hnf f + cnf (Te − Tm ) f − cns (Tm − Tw )(1 − f )
(23)
where Te and Tw are the left and right temperature of the control volume, respectively. Rearranging Eq. (12), the liquid fraction in the control volume is equal to,
f=
h − cns (Tm − Tw ) hnf + cnf (Te − Tm) + cns (Tm − Tw )
(24)
The location of the melting interface is determined by the liquid fraction contour f = 0.5. This formulation of the enthalpy method removes numerical oscillations in both temperature and moving boundary position, and produces a marked improvement in the accuracy of results, particularly for cases with a large ratio of the latent heat to sensible heat [6,3], which is the case for paraffin. 4. Results and discussion The accuracy of the numerical formulation is benchmarked with the results of Brent et al. [4], for rectangular cavity with gallium as PCM. They validated their numerical study with experimental findings of Gau and Viskanta [9]. Therefore, the boundary conditions and configuration of this study is adapted accordingly. The side-wall temperature is Tw = 38 °C, Tc = 28.3 °C, with an initial temperature of 28.3 °C in a domain size of 88.9 mm × 63.6 mm is considered. Fig. 5(a) shows the simulation results of the melting front inside the cavity at different times, and also includes the results of Brent et al. [19] and Gau and Viskanta [13]. A maximum error of 5% is observed between the data from the present model, and the results of Brent et al. [4]. A grid independency study is also performed for various grid sizes with a time step size in the range of 0.1 to 0.001 for a Ra = 5.75 × 105. The cavity liquid fraction (CLF) is selected as the monitoring variable. The results are shown in Fig. 5(b) and (c). A grid size of 180 × 180 grids with a
Fig. 4. Staggered mesh on the domain.
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Fig. 5. (a) Numerical code validation of the present study with the numerical results of Brent et al. [4] (b) Time step study (c) Grid size study at Ra = 5.74 × 105 and the experimental study of Gau and Viskanta [9].
interface is marked with zero value. As it is observed, the melt-front velocity increases with the increase of Rayleigh number. For Ra = 5.74 × 105 the isotherms have a ring-like shape indicating the primary mode of heat transfer is conduction. For the higher Rayleigh numbers, the vortices occupy larger area at the top section of the cavity and the melt-interface velocity increases due to the augmentation of convective heat transfer at this area. Therefore, for Ra = 1.48 × 107 and Ra = 2.72 × 106 the isotherms show a plump like shape above the heat source. The configuration of isotherms remarkable curve near the vertical surfaces of the heat source depicts the extreme temperature gradient and thereby conduction heat transfer at this area. Furthermore, they are perpendicular to the cavity walls due to the thermal insulation boundary condition. For Ra = 1.48 × 107, the isotherms show that the convection is the dominant heat transfer mechanism in the top section of cavity, while the conduction heat transfer controls melting process in the bottom section. The Fig. 7 also shows that for a given Rayleigh number, the maximum temperature (θmax) which occurs inside the heat source, increases as time elapses. For Ra = 2.72 × 106 the rate of increase in the θmax is lower in comparison with other Rayleigh numbers because in this case the heat generated inside the heat source transfers efficiently by conduction as well as conduction heat transfer to the solid PCM.
Table 2 The dimensionless time of NePCM when the cavity liquid fraction (CLF) approaches 1.0. q‴ (Wm− 3)
6
50 × 10 12 × 106 4.4 × 106
Ra
Ste
7
1.48 × 10 2.72 × 106 5.47 × 105
0.784 0.186 0.068
τ φ=0
φ = 0.02
φ = 0.05
0.155 0.38 1.38
0.140 0.365 1.32
0.165 0.41 1.42
is the dominant mechanism of heat transfer. Therefore, the small amount of the solid PCM in the cavity melts before the time of τ < 0.2. In this case, the strength of the vortices increases as time elapses due to the increase in the volume of the melt for buoyancy driven convective flow. Similar trend is observed for Ra = 2.72 × 106 case, up to τ = 0.15. At this time, almost all of the solid PCM in the upper half of the cavity has melted due to an increase in the heat generation rate inside the heat source as well as an increase of buoyancy force. However, after this time, the Ψmax decreases. The reason for this characteristic is that when the PCM melted thoroughly in the top section, the temperature difference as well as buoyancy driven force would decrease in the molten zone. In case of Ra = 1.48 × 107, due to high heat generation rate, this characteristic can be seen at earlier times between τ = 0.05 and 0.1. This reduction in the vortices strength causes the increase of heat source temperature which in turn causes an augmentation of the vortices strength once again after τ = 0.1 for the case of Ra = 1.48 × 107. Fig. 7 shows the isotherms in the cavity at several non-dimensional times from the start of the melting process. The solid-liquid melting
4.2. Nanoparticles mixed in PCM (NePCM) melting In this section, the effect of addition of alumina nanoparticles of diameter 50 nm dispersed in paraffin with a volume fraction of 0, 0.02 (NePCM-1), and 0.05 (NePCM-2) will be presented. The thermophysical 224
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Fig. 6. Streamlines in the base PCM, for various Rayleigh numbers at different dimensionless times.
Fig. 7. Isotherms in the base PCM, for various Rayleigh numbers at different dimensionless times.
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process for NePCM-2 compared to pure PCM as well as NePCM-1. However, it increases with increasing volume fraction of nanoparticles below the heat source due to the dominance of conduction in this rejoin. Fig. 10 shows the cavity liquid fraction (CLF), dimensionless maximum temperature, dimensionless average temperature, and the average Nusselt number as a function of dimensionless time for all the three PCM's. The cavity liquid fraction is lower for NePCM-2 compared to NePCM-1 and PCM. It is interesting to note that NePCM-1 and PCM have almost identical values. Referring to Figs. 8 and 9, one can observe that the possible effects of increase in thermal conductivity, and decrease in streaming are compensated. In the top section of the cavity above the heat source, the dominant mode of heat transfer at later time duration is convection, where as in the bottom section of cavity, the dominant mode of heat transfer is conduction as the temperature gradient is opposite to the gravitational force. Therefore, the slope of lines in the cavity liquid fraction plot, decreases after a certain duration of time, when the melt reaches the bottom section of the cavity. As a summary, the melting process inside the cavity can be separated into three distinguishing regions based on the dominant mode of heat transfer; first, at the initial time of melting, the conduction heat transfer is dominant (τ < 0.1). After that, convection plays a main role in the heat transfer (0.1 < τ < 0.2). Finally, after the PCM in the top section of cavity melts completely, conduction mode heat transfer is dominant (τ > 0.2). The maximum dimensionless temperature (see Fig. 10(b)) increases almost linearly at the early time of melting, then levels off and increases once again. The above mentioned three stages in the melting process are also recognizable in the maximum temperature trend. The results show that addition of nanoparticles to the base PCM reduces the
properties of the PCM and nanoparticles are listed in Table 1. As observed in the previous section, for a Rayleigh number equal to Ra = 2.72 × 106 both modes of heat transfer played a role in the melting dynamics, therefore this value of Rayleigh number will be used herewith. Xuan and Roetzel [26] showed that the dispersed nanoparticles and the base fluid behave as a homogeneous fluid, allowing the use of Eqs. (16) to (19) for NePCM. Figs. 8 and 9 shows the flow and temperature field, respectively for NePCM at several time duration's. Also the results without nanoparticles is repeated in the figures for convenience. The results show that at the early time duration, regarding the small vortices near the side-surfaces of the heat source, the dominant mode of heat transfer is conduction. As time elapses, the vortices get stronger and occupy a larger area mostly in the top section of the cavity. Basically, adding nanoparticles to the pure paraffin has different effects on the buoyancy-driven convection flow in the molten zone as well as on the thermal performance of the cavity. First, adding nanoparticles increases the effective thermal conductivity of the PCM (see Eq. (12)) and thereby conduction heat transfer rate from the heat source to the NePCM. Second, the addition of nanoparticles increases the dynamic viscosity of the PCM, (see Eq. (11)). As a result, the natural convective flow in the molten region is dampened by increasing the volume fraction of the nanoparticles. Nevertheless ψmax (see Fig. 8) generally reduces by increasing the volume fraction of the nanoparticles, θmax (see Fig. 9) does not have a definite trend versus the volume fraction of the nanoparticles at this period of time because of the two mentioned counter effects of nanoparticles on the thermal performance of the NePCM. The results also show that the melt-interface velocity decreases above the heat source where convection dominates the heat transfer
Fig. 8. Streamlines for various volume fraction of NePCM at different dimensionless times and for Ra = 2.72 × 106.
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Fig. 9. Isotherms for various volume fraction of NePCM at different dimensionless times and for Ra = 2.72 × 106.
sharply approaching a constant value, and then drops slightly in the last stage of melting. This trend confirms the assumption of the three stages of melting discussed before. The figures also show that for the large volume fraction of nanoparticles the average Nusselt number increases due to the increase of thermal conductivity and conduction heat transfer. Table 2 shows the dimensionless time when the end of melting is approached for various Rayleigh numbers and volume fractions. The results show that the end time of melting slightly decreases for low concentration of nanoparticles (ϕ = 0.02) in comparison with pure PCM. However, for NePCM-2 melting time is longer. The melting characteristics of NePCM with a heat generating source in the center of a square cavity is significantly different from the geometries considered in the previous literature (see Fig. 2). The location of the heat generating source in the center of the cavity does not allow buoyancy driven convection in the bottom side. The addition of nanoparticles did not have a significant influence on the melting rate of the PCM, which is contrary to the observations with other geometries in the literature. However, the maximum temperature at the heat source is lower with NePCM after a certain period of time. Given very limited benefits of the NePCM, they are possibly not a potential solution to improve the performance of thermal capacitors used to control temporal temperature rise in (power-) electronics thermal management.
maximum temperature, which is the heat source temperature. The average dimensionless temperature (see Fig. 10(c)), is calculated from the mean of the temperature in the PCM and heat source as wall. At the first stage, when conduction plays a main role of heat transfer, the average temperature increases linearly with time. After the melting front passes the center of the cavity, the average temperature rises once again till the end of melting. The trend shows that adding nanoparticles to the base PCM reduces the average temperature of the cavity especially for ϕ = 0.05. The reason is due to higher specific heat of NePCM-2 compared to NePCM-1 and the base PCM. The local Nusselt number on the side-walls of the heat source is a good indication of heat transfer rate from the heat source into the cavity.
Nu =
−knf q″l ∂T hl = = kf kf (TB − TI ) kf (TB − TI ) ∂n
(25)
where TB and TI are the temperature on the hot boundary wall and it's adjacent nod inside the PCM, respectively. This equation can be represented in dimensionless form as shown in the Eq. (26);
Nu = −
knf 1 ∂θ kf θB ∂n
(26)
The average Nusselt number is obtained by integrating the local Nusselt number along the surfaces of the cavity. The average Nusselt number on the hot surface as a function of dimensionless time is plotted in the Fig. 10(d). Eq. (26) shows that the Nusselt number depends on the surface temperature, and the temperature gradient on the side-walls of heat source. At the early time, the melt is relatively thin, leading to a large Nusselt number. As the thickness of the molten layer increases, the thermal resistance increases and the Nusselt number declines
5. Conclusions The effect of Rayleigh number and the volume fraction of nanoparticles on the heat transfer and melting process of a NePCM consisting of paraffin and alumina nanoparticles is investigated in a square cavity heated from the center with a constant heat source. The flow and 227
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Fig. 10. Macroscopic parameters during melting of the NePCM for Ra = 2.72 × 106: (a) liquid fraction, (b) average dimensionless temperature, (c) maximum dimensionless temperature, and (d) average Nusselt number.
temperature field in the melt is calculated, and the corresponding macroscopic variables, such as, the cavity liquid fraction, average and maximum temperature, Nusselt number during melting are determined. The following conclusions are drawn from this study:
[4]
• Melting is dominated by the conduction mode of heat transfer at the
[5]
• •
[3]
early stage, followed by buoyancy driven convection, resulting in early melting of the top side of the cavity. This is followed by the melting front advancing towards the bottom wall by heat conduction. These three phases of melting is a typical characteristic for this configuration. Addition of nanoparticles has no influence on the Nusselt number, average and maximum temperature of the cavity during the first two phases of melting. A marginal improvement in these quantities is observed in the last phase where the bottom section of the cavity melts. Due to very limited enhancement potential, addition of nanoparticles in phase change materials is not recommended.
[6] [7]
[8]
[9] [10]
[11]
Acknowledgement R. Yadollahi Farsani acknowledges the support of Shahrekord University for providing travel grant to perform part of the work described in this paper at University of Twente.
[12]
[13]
References
[14]
[1] Amirtham Arasu, Agus Sasmito, Arun Mujumdar, Thermal performance enhancement of paraffin wax with al2o3 and cuo nanoparticles—a numerical study, Front. Heat Mass Transf. 2 (4) (2012) 043005. [2] A. Valan Arasu, Arun S. Mujumdar, Numerical study on melting of paraffin wax
[15]
228
with Al2O3 in a square enclosure, Int. Commun. Heat Mass Transf. 39 (1) (2012) 8–16. Biswajit Basu, A.W. Date, Numerical modelling of melting and solidification problems a review, Sadhana 13 (3) (1988) 169–213. A.D. Brent, V.R. Voller, K. t J. Reid, Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal, Numer. Heat Transf. A Appl. 13 (3) (1988) 297–318. Tzai-Fu Cheng, Numerical analysis of nonlinear multiphase Stefan problems, Comput. Struct. 75 (2) (2000) 225–233. A.W. Date, A strong enthalpy formulation for the Stefan problem, Int. J. Heat Mass Transf. 34 (9) (1991) 2231–2235. Nabeel S. Dhaidan, J.M. Khodadadi, Tahseen A. Al-Hattab, Saad M. Al-Mashat, Experimental and numerical investigation of melting of nepcm inside an annular container under a constant heat flux including the effect of eccentricity, Int. J. Heat Mass Transf. 67 (2013) 455–468. Yongchang Feng, Huixiong Li, Liangxing Li, Bu Lin, Tai Wang, Numerical investigation on the melting of nanoparticle-enhanced phase change materials (nepcm) in a bottom-heated rectangular cavity using lattice boltzmann method, Int. J. Heat Mass Transf. 81 (2015) 415–425. C. Gau, R. Viskanta, Melting and solidification of a pure metal on a vertical wall, J. Heat Transf. 108 (1) (1986) 174–181. Ching Jenq Ho, J.Y. Gao, Preparation and thermophysical properties of nanoparticle-in-paraffin emulsion as phase change material, Int. Commun. Heat Mass Transf. 36 (5) (2009) 467–470. Annabelle Joulin, Zohir Younsi, Laurent Zalewski, Daniel R. Rousse, Stéphane Lassue, A numerical study of the melting of phase change material heated from a vertical wall of a rectangular enclosure, Int. J. Comput. Fluid Dyn. 23 (7) (2009) 553–566. Ravi Kandasamy, Xiang-Qi Wang, Arun S. Mujumdar, Transient cooling of electronics using phase change material (pcm)-based heat sinks, Appl. Ther. Eng. 28 (8) (2008) 1047–1057. J.M. Khodadadi, Y. Zhang, Effects of buoyancy-driven convection on melting within spherical containers, Int. J. Heat Mass Transf. 44 (8) (2001) 1605–1618. A. Kürklü, A. Wheldon, P. Hadley, Mathematical modelling of the thermal performance of a phase-change material (pcm) store: cooling cycle, Appl. Therm. Eng. 16 (7) (1996) 613–623. Lidia Navarro, Alvaro de Gracia, Shane Colclough, Maria Browne, Sarah J. McCormack, Philip Griffiths, Luisa F. Cabeza, Thermal energy storage in building integrated thermal systems: a review. Part 1. Active storage systems, Renew. Energy
International Communications in Heat and Mass Transfer 89 (2017) 219–229
R.Y. Farsani et al.
264–277. [22] Ravikanth S. Vajjha, Debendra K. Das, Devdatta P. Kulkarni, Development of new correlations for convective heat transfer and friction factor in turbulent regime for nanofluids, Int. J. Heat Mass Transf. 53 (21) (2010) 4607–4618. [23] R. Viskanta, Review of three-dimensional mathematical modeling of glass melting, J. Non-Cryst. Solids 177 (1994) 347–362. [24] V.R. Voller, Fast implicit finite-difference method for the analysis of phase change problems, Numer. Heat Transf. 17 (2) (1990) 155–169. [25] Xiang-Qi Wang, Arun S. Mujumdar, Christopher Yap, Effect of orientation for phase change material (pcm)-based heat sinks for transient thermal management of electric components, Int. Commun. Heat Mass Transf. 34 (7) (2007) 801–808. [26] Yimin Xuan, Wilfried Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transf. 43 (19) (2000) 3701–3707. [27] Zhang Zongqin, Adrian Bejan, The problem of time-dependent natural convection melting with conduction in the solid, Int. J. Heat Mass Transf. 32 (12) (1989) 2447–2457.
88 (2016) 526–547. [16] R. Pakrouh, M.J. Hosseini, A.A. Ranjbar, A parametric investigation of a pcm-based pin fin heat sink, Mech. Sci. 6 (1) (2015) 65–73. [17] Suhas Patankar, Numerical Heat Transfer and Fluid Flow, CRC press, 1980. [18] A. Safari, R. Saidur, F.A. Sulaiman, Yan Xu, Joe Dong, A review on supercooling of phase change materials in thermal energy storage systems, Renew. Sust. Energ. Rev. 70 (2016) 905–919. [19] K. Sasaguchi, R. Viskanta, Phase change heat transfer during melting and resolidification of melt around cylindrical heat source(s)/sink(s), J. Energy Resour. Technol. 111 (1) (1989) 43–49. [20] Seyed Sahand Sebti, Mohammad Mastiani, Hooshyar Mirzaei, Abdolrahman Dadvand, Sina Kashani, Seyed Amir Hosseini, Numerical study of the melting of nano-enhanced phase change material in a square cavity, J. Zhejiang Univ. Sci. A 14 (5) (2013) 307–316. [21] N.H.S. Tay, M. Liu, M. Belusko, F. Bruno, Review on transportable phase change material in thermal energy storage systems, Renew. Sustain. Energy 75 (2016)
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Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Does nanoparticles dispersed in a phase change material improve melting characteristics?
MARK
⁎
Rouhollah Yadollahi Farsania, Afrsiab Raisia, , Afshin Ahmadi Nadooshana, Srinivas Vanapallib a b
Shahrekord University, Engineering Faculty, PO Box 115, Shahrekord, Iran University of Twente, Faculty of Science and Technology, Post Bus 217, 7500 AE Enschede, The Netherlands
A R T I C L E I N F O
A B S T R A C T
Keywords: Melting Nanoparticles Heat storage Buoyancy driven convection
Nanoparticles dispersed in a phase change material alter the thermo-physical properties of the base material, such as thermal conductivity, viscosity, and specific heat capacity. These properties combined with the configuration of the cavity, and the location of the heat source, influence the melting characteristics of the phase change material. In this paper, an assessment of the influence of the nanoparticles in the base material subjected to a heat generating source located in the center of an insulated square cavity, which is a common configuration in thermal capacitors for temporal heat storage is investigated. The interplay between heat conduction enhanced due to an increase in thermal conduction and buoyancy driven heat convection damped by the increase in viscosity of nanoparticles dispersed in the phase change materials is studied with the calculated streamlines and isotherms. We observed three regimes during the melting process, first at an early time duration dominated by heat conduction, later by buoyancy driven convection till the melting front levels with the center of the cavity, and lastly once again heat conduction in the bottom portion of the cavity. During the first two regimes, addition of nanoparticles have no significant performance gain on the heat storage cavity, quantified by maximum temperature of the heat source and average Nusselt number at the faces of the heat source. In the late regime, nanoparticles provide a slight performance gain and this is attributed to the increase in the specific heat of the melt due to the nanoparticles.
1. Introduction Thermal capacitors consisting of materials with a solid to liquid phase change provides an excellent opportunity to bridge the cooling supply and demand needs, and therefore reduce the size of the thermal circuitry. Several researchers explored this idea in the context of heat storage, electronics cooling and many other applications [27,19,14,5]. These thermal management problems can be broadly divided into two categories based on the driving source, 1) temperature, 2) heat flux. An example of temperature driven application is the temporal storage of solar heat, where the temperature of heat generating part is fairly constant, as well as heat transport with heat pipes, where the condenser operates at a narrow temperature range depending on the working substance in the heat pipe [18,15,21]. The second case, is when the heat storage material has to suppress thermal runaway or dampen the temperature rise of a device, such as in electronics cooling. In case of a heat flux driven thermal storage, one key parameter of interest is the maximum temperature rise of the device [12,25]. Paraffin based materials are commonly used in terrestrial and space
⁎
applications to control the temperature of the devices around room temperature. These thermal capacitors consist of series of paraffin packed pockets with the heat generating part at the center, as shown in Fig. 1. Initially most designers have employed pure conduction models for design analyses, without considering convection during melting. The role of buoyancy driven convection was later identified to play a major role in the melt of a phase change material [9]. In a cavity with heat source at the center, the upper and the side regions surrounding the heat source have the right temperature boundary conditions for the onset of buoyancy driven flow (see Fig. 1), whereas at the bottom section below the heat source, buoyancy drive flow cannot develop. In order to increase the thermal diffusivity of the cavity, nanoparticles mixed with phase change materials (NePCM) have been recently proposed. The main hypothesis for proposing this mixture, is that the average thermal conductivity of the substance increase, thereby increasing the overall melting process, increasing the liquid fraction, and thus reduces the maximum device temperature for a certain heat generation rate. The increase in the viscosity in the liquid phase due to the presence of nanoparticles, dampen the buoyancy driven convective
Corresponding author at: Engineering Faculty, Shahrekord University, PO Box 115, Shahrekord, Iran. E-mail address: [email protected] (A. Raisi).
https://doi.org/10.1016/j.icheatmasstransfer.2017.10.006
0735-1933/ © 2017 Elsevier Ltd. All rights reserved.
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Nomenclature b B Bz c f g h L,l k Nu p P Pr q‴ Ra Ste T Tm,Ts Th,Tc t u,v U,V x,y X,Y
Abbreviations
enthalpy-porosity coefficient, kg m− 3 s− 1 dimensionless enthalpy-porosity coefficient Boltzmann constant specific heat, J kg −1 K− 1 liquid fraction gravity, ms− 2 enthalpy, J kg− 1 K− 1 cavity and heat source dimension, m thermal conductivity, Wm− 1 K− 1 Nusselt number, − knf/kf(Ts − Tm)∂ T/∂ n pressure, Nm− 2 dimensionless pressure Prandtl number, Pr = νf/αf heat generation rate, W/m3 Rayleigh number, gβfq‴l5/νfαfks Stefan Number, cfq‴l2/hnfks temperature, K melting and solidification points, K hot and cold temperatures, K time, s velocity in the x,y direction, ms− 1 dimensionless velocity Cartesian coordinate, m dimensionless Cartesian coordinate
CLF PCM NePCM
cavity liquid fraction Phase Change Material Nano enhanced PCM
Greek symbol α β μ ν θ ρ ϕ σ τ
thermal diffusivity, m2 s− 1 expansion coefficient, K− 1 dynamic viscosity, N s m− 2 kinematic viscosity, m2 s− 1 dimensionless temperature density, kg m− 3 volume fraction electrical conduction, S m− 1 dimensionless time
Subscript f,s m nf ns np
fluid and solid melting point fluid PCM with nanoparticles solid PCM with nanoparticles nanoparticles
heating, compared to horizontal heating configuration. However, compared to the base PCM, the NePCM showed a lower melting rate for both the configurations. In water based nanofluid melting, Feng et al. [8] observed a completely different trend for a bottom heating configuration, in which the melting rate increased with the volume fraction of copper oxide nanoparticles. Similar findings were reported albeit for solidification (see Fig. 2(c)) by Khodadadi and Hosseinzadeh [13] for water and copper oxide NePCM. A concentric cylindrical configuration as shown in Fig. 2(d) is used by Dhaidan et al. [7] to investigate melting of n-octadecan mixed with copper oxide nanoparticles. The experimental results show that the melting rate decrease with an increase in volume fraction of nanoparticles. However, the maximum temperature of the wall during melting reduced with increasing volume fraction of nanoparticles. The authors suggested an eccentric configuration of the cylinders to utilize enhanced melting on the top section of the annulus. A similar configuration but a constant temperature boundary conditions as shown in Fig. 2(e) is investigated by Sebti et al. [20], with similar conclusions. In another work, Dhaidan et al. used a square enclosure with heat flux boundary conditions on one vertical side and the remain sides are insulated, as shown in the Fig. 2(f). Contrary to the findings of Arasu and Mujumdar [2] mentioned earlier, the results presented for this configuration showed that higher melting rate are possible as the volume fraction of the nanoparticles are increased. The above discussion of the previous work point that the merits of NePCM should be discussed in the context of a particular configuration combined with the appropriate heating boundary condition pertaining to an application. The geometry shown in Fig. 1, has combined features of configuration shown in Fig. 2(a) and (b), and immediate conclusions cannot be drawn by combining the results. In this paper, a systematic numerical approach is presented to assess the addition of nanoparticles in the phase change material for the geometry and boundary conditions shown in Fig. 1. The transient flow and temperature is evaluated and the corresponding macroscopic variables such as average melt temperature, temperature of the heat source, average Nusselt number and cavity liquid fraction (CLF) are determined.
currents, and therefore reduces the overall heat transfer. Therefore, when designing a nanoparticle enhanced phase change material thermal capacitor, the interplay between the increase in the thermal conductivity and decrease in convection should be carefully considered [18,15,21]. Arasu and Mujumdar [2] studied melting of paraffin wax mixed with alumina nanoparticles of a number of volume fractions, in a square cavity of side 25 mm, heated from a vertical side (see Fig. 2(a)) and in a different configuration where the cavity is heated from the bottom side (see Fig. 2(b)). The results suggest that the addition of nanoparticles decreases the melting rate of the PCM, and they conclude that the melting rate and the thermal energy storage is greater for vertical
Fig. 1. Physical model of the cross-section of a heat storage cavity with heat source at the center and insulated at the outside walls.
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Fig. 2. Literature review of configurations with boundary conditions of proposed cavities with nanoparticles mixed with phase change materials.
hnf ∂f ∂T ∂T ∂T +u +v = αnf ∇2 T − cnf ∂t ∂x ∂y ∂t
2. Problem definition and mathematical formulation Fig. 1 shows the mathematical domain of the physical system consisting of rectangular container with the heat source at its center and the outer walls are insulated. The phase change material is initially in the solid phase at its melting point temperature, Tm. The homogeneous heat generation rate of q‴ per unit volume is applied for t > 0 inside the heat source which is located at the center of the cavity and this region contains aluminium material with a thermal conductivity of ks = 202 Wm− 1 K− 1. Table 1 shows the thermophysical properties of the paraffin, alumina nanoparticles, and aluminium. Alumina nanoparticles are used in this study because the nanofluid formed with these particles have lower viscosity and higher thermal conductivity compared to other particles such as copper oxide. Moreover, alumina particles are cheap, safe and readily available [1]. As the PCM begins to melt from the side-walls of the heat source, the solid-liquid phase boundary develops inside the cavity. In the melt, both heat conduction and heat convection play a role, where as in the solid phase the heat transfer is only due to heat conduction. The flow is assumed to be two-dimensional, laminar and in-compressible. The governing equations describing natural convection and conduction within the phase change material are given below [2],
• Energy (solid phase): ∂T = αns ∇2 T ∂t
(5)
• Energy (heat source): q‴ ∂T = αs ∇2 T + (ρcp )s ∂t
(6)
The density and specific heat capacity of nanoparticles mixed in PCM is equal to the corresponding weight fraction of the constituents [10], as given below,
ρnf = (1 − ϕ) ρf + ϕρnp
(7)
(ρc )nf = (1 − ϕ)(ρc )f + ϕ (ρc )np
(8)
(ρβ )nf = (1 − ϕ)(ρβ )f + ϕ (ρβ )np
(9)
hnf =
• Continuity:
(1 − ϕ)(ρh)f ρnf
(10)
The empirical nanofluid dynamic viscosity is given by the following
∂u ∂v + =0 ∂x ∂y
(1)
Table 1 Thermophysical properties of the PCM [16] and Al2O3 [2] at T = 273.2 K.
• x momentum: ∂u ∂u ∂u 1 ⎛ ∂p + μnf ∇2 u + bu⎞ − +u +v = ∂t ∂x ∂y ρnf ⎝ ∂x ⎠
•
(4)
(2)
⎜
Paraffin (RT 44HC)
Al2O3
Al
Viscosity, μf Ns/m2
0.001 exp (−4.25 + 1790/T) 760 (780) 1.0 (10− 3)
–
–
3600 1.6 (10− 5)
2700 –
0.19 (0.21)
36
202
45 255,000 2000 (2000)
– – 765
60.32 – 903
Density, ρf (ρs) kg m− 3 Thermal expansion coefficient, β K− 1 Thermal conductivity, kf (ks) Wm− 1 K− 1 Melting point, Tm °C Latent heat fusion, hm J kg− 1 Specific heat, cf (cs) J kg− 1 K− 1
y momentum:
∂v ∂v ∂v 1 ⎛ ∂p + μnf ∇2 v + bv + ρnf g β(T − Tm) ⎞ +u +v = − ∂t ∂x ∂y ρnf ⎝ ∂y ⎠
Property
⎟
(3)
• Energy (liquid phase): 221
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significantly [11]. In this work, C1 = 1 × 108 kg·m− 3·s− 1 and C2 = 0.003 are used. The governing equations are reduced to the non-dimensional form by using the following variables including dimensionless parameters like, X = x/1, Y = y/1, U = ul/αf, V = vl/αf, θ = (T − Tm)/q‴l2/ks, P = pl2/ρnfαf, as well as dimensionless governing numbers like, τ = αft/l2 which is Fourier number, B = bl2/ρnfαf is the dimensionless form of b, Ra = gβfq‴l5/ksνfαf is the Rayleigh number, Ste = cfq‴l3/ hnfks is the Stefan number, Pr = νf/αf is the Prandtl number.
equation, [22].
μnf = 0.983e (12.959ϕ) μf
(11)
The effective thermal conductivity of the NePCM, taking into account particle size, particle volume fraction and temperature dependence as well as properties of the base PCM is proposed by Arasu et al. [2] and Ho and Gao [10] and is repeated below for completeness,
knf =
knp + 2kf − 2(kf − knp ) ϕ knp + 2kf + (kf − knp ) ϕ
kf + 5 × 10 4βk fϕρf cf
BzT ζ (T , ϕ) ρnp dnp
• Continuity:
(12) where Bz is Boltzmann constant, 1.381 × 10
βk =
− 23
8.4407(100ϕ)−1.07304
(13)
ζ (T , ϕ) = (2.8217 × 10−2ϕ + 3.917 × 10−3) + (−3.0669 ×
∂U ∂V + =0 ∂X ∂Y
J/K and
10−2ϕ
• X momentum:
T Tref
− 3.91123 ×
10−3)
μ ∂U ∂U ∂U ∂P +U +V =− + nf ∇2 U + BU ρnf α f ∂τ ∂X ∂Y ∂X
(14)
where Tref is the reference temperature equal to 273.15 K. The first part of Eq. (14) is obtained directly from the Maxwell model while the second part accounts for Brownian motion, which causes the temperature dependence of the effective thermal conductivity. Note that there is a correction factor, f in the Brownian motion term, since there should be no Brownian motion in the solid phase. The computed thermophysical properties of paraffin dispersed with 0, 0.02, and 0.05 volume fraction of Al2O3 nanoparticles using Eq. (11)–(14) are plotted as a function of temperature and volumetric concentration (see Fig. 3). The solid to liquid phase change process is mathematically described using enthalpy-porosity formulation proposed by Brent et al. [4]. In this approach, a parameter b is defined in the momentum equations (Eqs. (2) and (3)) to gradually reduce the velocities from a finite value in the liquid to zero in the solid, over the computational cell that undergoes the phase change. This is achieved by assuming that such cells behave like a porous media with a porosity equal to the liquid fraction. Based on the Carman-Kozeny relation the coefficient b is defined as,
−C1 (1 − f )2 b= f 3 + C2
(16)
(17)
• Y momentum: μ (ρβ)nf ∂V ∂V ∂V ∂P +V =− + nf ∇2 V + RaPr θ + BV +U ∂τ ∂X ∂Y ∂Y ρnf α f ρnf βf
(18)
• Energy (liquid phase): hnf ∂θ ∂θ ∂θ α ⎛ ∂f ⎞ +U +V = nf ∇2 θ − ∂τ ∂X ∂Y αf cnf ΔT ⎝ ∂τ ⎠
(19)
• Energy (solid phase): ∂θ α = ns ∇2 θ ∂τ αf
(20)
• Energy (heat source): ∂θ α α = s ∇2 θ + s ∂τ αf αf
(21)
The dimensionless initial and boundary conditions are:
(15)
• at initial time, τ = 0, U = V = 0 and θ = 0 • for the side walls, ∂ θ/∂ n = 0 • for melting front, U = V = θ = 0
In this equation, f = 1 and f = 0 in the liquid and solid region, respectively. f can take values between 0 and 1 in the mushy zone (liquid and solid). The constant C1 has a large value (107 − 1015) to suppress the velocity as the cell becomes solid and C2 is a small constant to avoid a division by zero when a cell is fully located in the solid region, namely f = 0. The choice of these constants is arbitrary. However, the constants should ensure enough depression of the velocity in the solid region and should not influence the numerical results
Where “n” is a normal vector to the side-walls. The absorption of latent heat during melting is included as a sink term in the last term in the right side of energy Eq. (19). The latent heat content of each control volume is evaluated after each energy equation iteration cycle. Based
Fig. 3. NePCM dynamic viscosity (a), thermal conductivity (b).
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time step of 0.005 yielded sufficient accuracy. The numerical results with the configuration shown in Fig. 1 are presented in two parts; First, the effect of heat generation rate in the heat source on the flow and phase change phenomena is studied by varying the Rayleigh number. Second, the effect of adding nanoparticles to the base PCM on the melting process is studied. In both these parts, the streamlines in the liquid melt, the isotherms in the melt, the liquid fraction in the cavity, the average Nusselt number on the heat source walls, and the maximum and the average temperature of the cavity are computed as a function of dimensionless time (Fourier number, τ).
on the latent heat content, a liquid fraction for each control volume is determined, which is elaborated in the next section. For control volumes containing a liquid phase of the substance, f is set to 1, and for control volumes containing solid phase, f is set to 0. The control volumes with values of f between 0 and 1 are treated as a mushy zone. The phase change is assumed to be non-isothermal, so the idea of the mushy zone is introduced to gradually switch off the velocities from liquid to solid at the solid-liquid interface [11]. 3. Numerical method of solution The governing equations (Eq. (16)–(21)) along with the boundary conditions are discretized using the finite-volume technique. This method appears to be suitable for numerical solution of the model equations describing phase change processes [23]. The power law scheme, which is a combination of the central difference and the upwind schemes, is used to discretize the convection terms. A staggered grid system, in which the velocity components are stored midway between control volumes is used (see Fig. 4). The SIMPLE algorithm proposed by Patankar [17] is used to solve the discretized and coupled continuity, momentum and energy equations and a line by line solver based on the tri-diagonal matrix algorithm is used to solve iteratively the algebraic discretized equations. Based on the finite difference method for the analysis of phase change problems proposed by Voller et al. [24], the enthalpy value is expressed as a function of temperature.
⎧ cns T h = cnf T + hnf (T − Ts )/(Tm − Ts ) ⎨ c T + hnf + cnf (Tm − Ts ) ⎩ nf
4.1. PCM melting The melting phenomena of the base PCM is studied for three values of Rayleigh numbers namely, 1.48 × 107, 2.72 × 106, and Ra = 5.74 × 105. These values are chosen based on the various values of volumetric heat generation in the source. It must be noted that Rayleigh number in this study essentially represents the rate of heat generation in the heat source. Considering the definition of Rayleigh number, Ra = gβq‴l5/(ksνfαf), for the specific thermo-physical properties of paraffin (Table 1), Rayleigh number is only dependent on the heat generation within the source. For example for a heat generation rate of 12 × 106 W/m3, the corresponding calculated Rayleigh number is equal to 2.72 × 106. Therefore a heat generation rate more than and less than this value (see Table 2) is considered in this study. For each of these Rayleigh numbers a constant value of Stefan number should be considered, which are shown in Table 2. Due to the change in melting region, the instantaneous value of Rayleigh number changes, however the reported Rayleigh and the associated Stefan numbers are for the completely melted situation. The streamlines in the melt region at various dimensionless times for all Rayleigh numbers are shown in the Fig. 6. The vortices in the liquid phase is due to the buoyancy driven convection induced by the temperature difference between the heat source surfaces and the solid PCM. The molten flow rises up along the side-surfaces of the source and then descends near the cold solid-liquid interface, resulting in two symmetric vortices in the opposite direction. The results show that at any given dimensionless time the strength of the vortices increase with the increase in Rayleigh number. The numbers labeled at the center of the streamlines in the Fig. 6 indicate the maximum strength of the circulation, Ψmax. An interesting trend is observed in the circulation strength as a function of dimensionless time. In case of Ra = 5.74 × 105, conduction
T < Ts Solid Ts ≤ T ≤ Tl Mushy T > Tl
Liquid
(22)
Assuming a linear temperature profile in the mushy zone, the enthalpy of the control volume as a function of liquid fraction, f, is given by,
h = hnf f + cnf (Te − Tm ) f − cns (Tm − Tw )(1 − f )
(23)
where Te and Tw are the left and right temperature of the control volume, respectively. Rearranging Eq. (12), the liquid fraction in the control volume is equal to,
f=
h − cns (Tm − Tw ) hnf + cnf (Te − Tm) + cns (Tm − Tw )
(24)
The location of the melting interface is determined by the liquid fraction contour f = 0.5. This formulation of the enthalpy method removes numerical oscillations in both temperature and moving boundary position, and produces a marked improvement in the accuracy of results, particularly for cases with a large ratio of the latent heat to sensible heat [6,3], which is the case for paraffin. 4. Results and discussion The accuracy of the numerical formulation is benchmarked with the results of Brent et al. [4], for rectangular cavity with gallium as PCM. They validated their numerical study with experimental findings of Gau and Viskanta [9]. Therefore, the boundary conditions and configuration of this study is adapted accordingly. The side-wall temperature is Tw = 38 °C, Tc = 28.3 °C, with an initial temperature of 28.3 °C in a domain size of 88.9 mm × 63.6 mm is considered. Fig. 5(a) shows the simulation results of the melting front inside the cavity at different times, and also includes the results of Brent et al. [19] and Gau and Viskanta [13]. A maximum error of 5% is observed between the data from the present model, and the results of Brent et al. [4]. A grid independency study is also performed for various grid sizes with a time step size in the range of 0.1 to 0.001 for a Ra = 5.75 × 105. The cavity liquid fraction (CLF) is selected as the monitoring variable. The results are shown in Fig. 5(b) and (c). A grid size of 180 × 180 grids with a
Fig. 4. Staggered mesh on the domain.
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Fig. 5. (a) Numerical code validation of the present study with the numerical results of Brent et al. [4] (b) Time step study (c) Grid size study at Ra = 5.74 × 105 and the experimental study of Gau and Viskanta [9].
interface is marked with zero value. As it is observed, the melt-front velocity increases with the increase of Rayleigh number. For Ra = 5.74 × 105 the isotherms have a ring-like shape indicating the primary mode of heat transfer is conduction. For the higher Rayleigh numbers, the vortices occupy larger area at the top section of the cavity and the melt-interface velocity increases due to the augmentation of convective heat transfer at this area. Therefore, for Ra = 1.48 × 107 and Ra = 2.72 × 106 the isotherms show a plump like shape above the heat source. The configuration of isotherms remarkable curve near the vertical surfaces of the heat source depicts the extreme temperature gradient and thereby conduction heat transfer at this area. Furthermore, they are perpendicular to the cavity walls due to the thermal insulation boundary condition. For Ra = 1.48 × 107, the isotherms show that the convection is the dominant heat transfer mechanism in the top section of cavity, while the conduction heat transfer controls melting process in the bottom section. The Fig. 7 also shows that for a given Rayleigh number, the maximum temperature (θmax) which occurs inside the heat source, increases as time elapses. For Ra = 2.72 × 106 the rate of increase in the θmax is lower in comparison with other Rayleigh numbers because in this case the heat generated inside the heat source transfers efficiently by conduction as well as conduction heat transfer to the solid PCM.
Table 2 The dimensionless time of NePCM when the cavity liquid fraction (CLF) approaches 1.0. q‴ (Wm− 3)
6
50 × 10 12 × 106 4.4 × 106
Ra
Ste
7
1.48 × 10 2.72 × 106 5.47 × 105
0.784 0.186 0.068
τ φ=0
φ = 0.02
φ = 0.05
0.155 0.38 1.38
0.140 0.365 1.32
0.165 0.41 1.42
is the dominant mechanism of heat transfer. Therefore, the small amount of the solid PCM in the cavity melts before the time of τ < 0.2. In this case, the strength of the vortices increases as time elapses due to the increase in the volume of the melt for buoyancy driven convective flow. Similar trend is observed for Ra = 2.72 × 106 case, up to τ = 0.15. At this time, almost all of the solid PCM in the upper half of the cavity has melted due to an increase in the heat generation rate inside the heat source as well as an increase of buoyancy force. However, after this time, the Ψmax decreases. The reason for this characteristic is that when the PCM melted thoroughly in the top section, the temperature difference as well as buoyancy driven force would decrease in the molten zone. In case of Ra = 1.48 × 107, due to high heat generation rate, this characteristic can be seen at earlier times between τ = 0.05 and 0.1. This reduction in the vortices strength causes the increase of heat source temperature which in turn causes an augmentation of the vortices strength once again after τ = 0.1 for the case of Ra = 1.48 × 107. Fig. 7 shows the isotherms in the cavity at several non-dimensional times from the start of the melting process. The solid-liquid melting
4.2. Nanoparticles mixed in PCM (NePCM) melting In this section, the effect of addition of alumina nanoparticles of diameter 50 nm dispersed in paraffin with a volume fraction of 0, 0.02 (NePCM-1), and 0.05 (NePCM-2) will be presented. The thermophysical 224
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Fig. 6. Streamlines in the base PCM, for various Rayleigh numbers at different dimensionless times.
Fig. 7. Isotherms in the base PCM, for various Rayleigh numbers at different dimensionless times.
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process for NePCM-2 compared to pure PCM as well as NePCM-1. However, it increases with increasing volume fraction of nanoparticles below the heat source due to the dominance of conduction in this rejoin. Fig. 10 shows the cavity liquid fraction (CLF), dimensionless maximum temperature, dimensionless average temperature, and the average Nusselt number as a function of dimensionless time for all the three PCM's. The cavity liquid fraction is lower for NePCM-2 compared to NePCM-1 and PCM. It is interesting to note that NePCM-1 and PCM have almost identical values. Referring to Figs. 8 and 9, one can observe that the possible effects of increase in thermal conductivity, and decrease in streaming are compensated. In the top section of the cavity above the heat source, the dominant mode of heat transfer at later time duration is convection, where as in the bottom section of cavity, the dominant mode of heat transfer is conduction as the temperature gradient is opposite to the gravitational force. Therefore, the slope of lines in the cavity liquid fraction plot, decreases after a certain duration of time, when the melt reaches the bottom section of the cavity. As a summary, the melting process inside the cavity can be separated into three distinguishing regions based on the dominant mode of heat transfer; first, at the initial time of melting, the conduction heat transfer is dominant (τ < 0.1). After that, convection plays a main role in the heat transfer (0.1 < τ < 0.2). Finally, after the PCM in the top section of cavity melts completely, conduction mode heat transfer is dominant (τ > 0.2). The maximum dimensionless temperature (see Fig. 10(b)) increases almost linearly at the early time of melting, then levels off and increases once again. The above mentioned three stages in the melting process are also recognizable in the maximum temperature trend. The results show that addition of nanoparticles to the base PCM reduces the
properties of the PCM and nanoparticles are listed in Table 1. As observed in the previous section, for a Rayleigh number equal to Ra = 2.72 × 106 both modes of heat transfer played a role in the melting dynamics, therefore this value of Rayleigh number will be used herewith. Xuan and Roetzel [26] showed that the dispersed nanoparticles and the base fluid behave as a homogeneous fluid, allowing the use of Eqs. (16) to (19) for NePCM. Figs. 8 and 9 shows the flow and temperature field, respectively for NePCM at several time duration's. Also the results without nanoparticles is repeated in the figures for convenience. The results show that at the early time duration, regarding the small vortices near the side-surfaces of the heat source, the dominant mode of heat transfer is conduction. As time elapses, the vortices get stronger and occupy a larger area mostly in the top section of the cavity. Basically, adding nanoparticles to the pure paraffin has different effects on the buoyancy-driven convection flow in the molten zone as well as on the thermal performance of the cavity. First, adding nanoparticles increases the effective thermal conductivity of the PCM (see Eq. (12)) and thereby conduction heat transfer rate from the heat source to the NePCM. Second, the addition of nanoparticles increases the dynamic viscosity of the PCM, (see Eq. (11)). As a result, the natural convective flow in the molten region is dampened by increasing the volume fraction of the nanoparticles. Nevertheless ψmax (see Fig. 8) generally reduces by increasing the volume fraction of the nanoparticles, θmax (see Fig. 9) does not have a definite trend versus the volume fraction of the nanoparticles at this period of time because of the two mentioned counter effects of nanoparticles on the thermal performance of the NePCM. The results also show that the melt-interface velocity decreases above the heat source where convection dominates the heat transfer
Fig. 8. Streamlines for various volume fraction of NePCM at different dimensionless times and for Ra = 2.72 × 106.
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Fig. 9. Isotherms for various volume fraction of NePCM at different dimensionless times and for Ra = 2.72 × 106.
sharply approaching a constant value, and then drops slightly in the last stage of melting. This trend confirms the assumption of the three stages of melting discussed before. The figures also show that for the large volume fraction of nanoparticles the average Nusselt number increases due to the increase of thermal conductivity and conduction heat transfer. Table 2 shows the dimensionless time when the end of melting is approached for various Rayleigh numbers and volume fractions. The results show that the end time of melting slightly decreases for low concentration of nanoparticles (ϕ = 0.02) in comparison with pure PCM. However, for NePCM-2 melting time is longer. The melting characteristics of NePCM with a heat generating source in the center of a square cavity is significantly different from the geometries considered in the previous literature (see Fig. 2). The location of the heat generating source in the center of the cavity does not allow buoyancy driven convection in the bottom side. The addition of nanoparticles did not have a significant influence on the melting rate of the PCM, which is contrary to the observations with other geometries in the literature. However, the maximum temperature at the heat source is lower with NePCM after a certain period of time. Given very limited benefits of the NePCM, they are possibly not a potential solution to improve the performance of thermal capacitors used to control temporal temperature rise in (power-) electronics thermal management.
maximum temperature, which is the heat source temperature. The average dimensionless temperature (see Fig. 10(c)), is calculated from the mean of the temperature in the PCM and heat source as wall. At the first stage, when conduction plays a main role of heat transfer, the average temperature increases linearly with time. After the melting front passes the center of the cavity, the average temperature rises once again till the end of melting. The trend shows that adding nanoparticles to the base PCM reduces the average temperature of the cavity especially for ϕ = 0.05. The reason is due to higher specific heat of NePCM-2 compared to NePCM-1 and the base PCM. The local Nusselt number on the side-walls of the heat source is a good indication of heat transfer rate from the heat source into the cavity.
Nu =
−knf q″l ∂T hl = = kf kf (TB − TI ) kf (TB − TI ) ∂n
(25)
where TB and TI are the temperature on the hot boundary wall and it's adjacent nod inside the PCM, respectively. This equation can be represented in dimensionless form as shown in the Eq. (26);
Nu = −
knf 1 ∂θ kf θB ∂n
(26)
The average Nusselt number is obtained by integrating the local Nusselt number along the surfaces of the cavity. The average Nusselt number on the hot surface as a function of dimensionless time is plotted in the Fig. 10(d). Eq. (26) shows that the Nusselt number depends on the surface temperature, and the temperature gradient on the side-walls of heat source. At the early time, the melt is relatively thin, leading to a large Nusselt number. As the thickness of the molten layer increases, the thermal resistance increases and the Nusselt number declines
5. Conclusions The effect of Rayleigh number and the volume fraction of nanoparticles on the heat transfer and melting process of a NePCM consisting of paraffin and alumina nanoparticles is investigated in a square cavity heated from the center with a constant heat source. The flow and 227
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Fig. 10. Macroscopic parameters during melting of the NePCM for Ra = 2.72 × 106: (a) liquid fraction, (b) average dimensionless temperature, (c) maximum dimensionless temperature, and (d) average Nusselt number.
temperature field in the melt is calculated, and the corresponding macroscopic variables, such as, the cavity liquid fraction, average and maximum temperature, Nusselt number during melting are determined. The following conclusions are drawn from this study:
[4]
• Melting is dominated by the conduction mode of heat transfer at the
[5]
• •
[3]
early stage, followed by buoyancy driven convection, resulting in early melting of the top side of the cavity. This is followed by the melting front advancing towards the bottom wall by heat conduction. These three phases of melting is a typical characteristic for this configuration. Addition of nanoparticles has no influence on the Nusselt number, average and maximum temperature of the cavity during the first two phases of melting. A marginal improvement in these quantities is observed in the last phase where the bottom section of the cavity melts. Due to very limited enhancement potential, addition of nanoparticles in phase change materials is not recommended.
[6] [7]
[8]
[9] [10]
[11]
Acknowledgement R. Yadollahi Farsani acknowledges the support of Shahrekord University for providing travel grant to perform part of the work described in this paper at University of Twente.
[12]
[13]
References
[14]
[1] Amirtham Arasu, Agus Sasmito, Arun Mujumdar, Thermal performance enhancement of paraffin wax with al2o3 and cuo nanoparticles—a numerical study, Front. Heat Mass Transf. 2 (4) (2012) 043005. [2] A. Valan Arasu, Arun S. Mujumdar, Numerical study on melting of paraffin wax
[15]
228
with Al2O3 in a square enclosure, Int. Commun. Heat Mass Transf. 39 (1) (2012) 8–16. Biswajit Basu, A.W. Date, Numerical modelling of melting and solidification problems a review, Sadhana 13 (3) (1988) 169–213. A.D. Brent, V.R. Voller, K. t J. Reid, Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal, Numer. Heat Transf. A Appl. 13 (3) (1988) 297–318. Tzai-Fu Cheng, Numerical analysis of nonlinear multiphase Stefan problems, Comput. Struct. 75 (2) (2000) 225–233. A.W. Date, A strong enthalpy formulation for the Stefan problem, Int. J. Heat Mass Transf. 34 (9) (1991) 2231–2235. Nabeel S. Dhaidan, J.M. Khodadadi, Tahseen A. Al-Hattab, Saad M. Al-Mashat, Experimental and numerical investigation of melting of nepcm inside an annular container under a constant heat flux including the effect of eccentricity, Int. J. Heat Mass Transf. 67 (2013) 455–468. Yongchang Feng, Huixiong Li, Liangxing Li, Bu Lin, Tai Wang, Numerical investigation on the melting of nanoparticle-enhanced phase change materials (nepcm) in a bottom-heated rectangular cavity using lattice boltzmann method, Int. J. Heat Mass Transf. 81 (2015) 415–425. C. Gau, R. Viskanta, Melting and solidification of a pure metal on a vertical wall, J. Heat Transf. 108 (1) (1986) 174–181. Ching Jenq Ho, J.Y. Gao, Preparation and thermophysical properties of nanoparticle-in-paraffin emulsion as phase change material, Int. Commun. Heat Mass Transf. 36 (5) (2009) 467–470. Annabelle Joulin, Zohir Younsi, Laurent Zalewski, Daniel R. Rousse, Stéphane Lassue, A numerical study of the melting of phase change material heated from a vertical wall of a rectangular enclosure, Int. J. Comput. Fluid Dyn. 23 (7) (2009) 553–566. Ravi Kandasamy, Xiang-Qi Wang, Arun S. Mujumdar, Transient cooling of electronics using phase change material (pcm)-based heat sinks, Appl. Ther. Eng. 28 (8) (2008) 1047–1057. J.M. Khodadadi, Y. Zhang, Effects of buoyancy-driven convection on melting within spherical containers, Int. J. Heat Mass Transf. 44 (8) (2001) 1605–1618. A. Kürklü, A. Wheldon, P. Hadley, Mathematical modelling of the thermal performance of a phase-change material (pcm) store: cooling cycle, Appl. Therm. Eng. 16 (7) (1996) 613–623. Lidia Navarro, Alvaro de Gracia, Shane Colclough, Maria Browne, Sarah J. McCormack, Philip Griffiths, Luisa F. Cabeza, Thermal energy storage in building integrated thermal systems: a review. Part 1. Active storage systems, Renew. Energy
International Communications in Heat and Mass Transfer 89 (2017) 219–229
R.Y. Farsani et al.
264–277. [22] Ravikanth S. Vajjha, Debendra K. Das, Devdatta P. Kulkarni, Development of new correlations for convective heat transfer and friction factor in turbulent regime for nanofluids, Int. J. Heat Mass Transf. 53 (21) (2010) 4607–4618. [23] R. Viskanta, Review of three-dimensional mathematical modeling of glass melting, J. Non-Cryst. Solids 177 (1994) 347–362. [24] V.R. Voller, Fast implicit finite-difference method for the analysis of phase change problems, Numer. Heat Transf. 17 (2) (1990) 155–169. [25] Xiang-Qi Wang, Arun S. Mujumdar, Christopher Yap, Effect of orientation for phase change material (pcm)-based heat sinks for transient thermal management of electric components, Int. Commun. Heat Mass Transf. 34 (7) (2007) 801–808. [26] Yimin Xuan, Wilfried Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transf. 43 (19) (2000) 3701–3707. [27] Zhang Zongqin, Adrian Bejan, The problem of time-dependent natural convection melting with conduction in the solid, Int. J. Heat Mass Transf. 32 (12) (1989) 2447–2457.
88 (2016) 526–547. [16] R. Pakrouh, M.J. Hosseini, A.A. Ranjbar, A parametric investigation of a pcm-based pin fin heat sink, Mech. Sci. 6 (1) (2015) 65–73. [17] Suhas Patankar, Numerical Heat Transfer and Fluid Flow, CRC press, 1980. [18] A. Safari, R. Saidur, F.A. Sulaiman, Yan Xu, Joe Dong, A review on supercooling of phase change materials in thermal energy storage systems, Renew. Sust. Energ. Rev. 70 (2016) 905–919. [19] K. Sasaguchi, R. Viskanta, Phase change heat transfer during melting and resolidification of melt around cylindrical heat source(s)/sink(s), J. Energy Resour. Technol. 111 (1) (1989) 43–49. [20] Seyed Sahand Sebti, Mohammad Mastiani, Hooshyar Mirzaei, Abdolrahman Dadvand, Sina Kashani, Seyed Amir Hosseini, Numerical study of the melting of nano-enhanced phase change material in a square cavity, J. Zhejiang Univ. Sci. A 14 (5) (2013) 307–316. [21] N.H.S. Tay, M. Liu, M. Belusko, F. Bruno, Review on transportable phase change material in thermal energy storage systems, Renew. Sustain. Energy 75 (2016)
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