Heat Transfer by Natural Convection in a Square Enclosure Containing PCM Suspensions

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Article Heat Transfer by Natural Convection in a Square Enclosure Containing PCM Suspensions

Ching-Jenq Ho 1,*, Chau-Yang Huang 2 and Chi-Ming Lai 3,*

1 Department of Mechanical Engineering, National Cheng-Kung University, Tainan City 701, Taiwan 2 Green Energy and Environment Research Laboratories, Industrial Technology Research Institute, Hsinchu 310, Taiwan; [email protected] 3 Department of Civil Engineering, National Cheng-Kung University, Tainan City 701, Taiwan * Correspondence: [email protected] (C.-J.H.); [email protected] (C.-M.L.); Tel.: +886-6-2757575 (ext. 62146) (C.-J.H.); +886-6-2757575 (ext. 63136) (C.-M.L.)

Abstract: Research on using phase change material (PCM) suspension to improve the heat transfer and energy storage capabilities of thermal systems is booming; however, there are limited studies on the application of PCM suspension in transient natural convection. In this paper, the implicit ﬁnite difference method was used to numerically investigate the transient and steady-state natural convection heat transfer in a square enclosure containing a PCM suspension. The following parameters were included in the simulation: aspect ratio of the physical model = 1, ratio of the buoyancies caused 3 5 by temperature and concentration gradients = 1, Raleigh number (RaT) = 10 –10 , Stefan number (Ste) = 0.005–0.1, subcooling factor (Sb) = 0–1.0, and initial mass fraction (or concentration) of PCM

particles (ci) = 0–0.1. The results showed that the use of a PCM suspension can effectively enhance 3 heat transfer by natural convection. For example, when RaT = 10 , Ste = 0.01, ci = 0.1, and Sb = 1, the steady-state natural convection heat transfer rate inside the square enclosure can be improved by 70% compared with that of pure water. With increasing Sb, the Nusselt number can change nonlinearly, Citation: Ho, C.-J.; Huang, C.-Y.; Lai, resulting in a local optimal value. C.-M. Heat Transfer by Natural Convection in a Square Enclosure Keywords: natural convection; heat transfer; phase change materials; numerical analysis Containing PCM Suspensions. Energies 2021, 14, 2857. https:// doi.org/10.3390/en14102857 1. Introduction Academic Editor: Phillip Ligrani Heat transfer is widely used in industry and daily life, and changes in the latent heat or sensible heat of heat transfer media are used to transfer heat to the external environment. Received: 5 April 2021 Performance improvements of a thermal system can be achieved not only by modifying Accepted: 13 May 2021 the system main body itself [1], but also by either enhancing the accessories [2,3] or by Published: 15 May 2021 changing the working fluid [4,5]. Eisapour et al. [1] numerically simulated the exergy and energy efficiencies of photovoltaic-thermal (PV-T) systems with various cooling tube Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in configurations, including straight tubes, wavy tubes, and variable wave-amplitude wavy published maps and institutional affil- tubes. The influence of the geometrical parameters, the sensitivity of the operating condi- iations. tions, and the working fluid effects were studied as well. Wijayanta et al. [2] experimentally investigated a double-pipe heat exchanger equipped with double-sided delta-wing (T-W) tape inserts. Liu et al. [3] numerically studied vortex flow and heat transfer enhancement in pipes with a segmented twisted element under constant wall temperature and different Reynolds numbers, ranging from 10,000 to 35,000. Muruganandam et al. [4] conducted ex- Copyright: © 2021 by the authors. periments on the thermal performance enhancement of a heat exchanger by incorporating Licensee MDPI, Basel, Switzerland. This article is an open access article nano-particles of the natural Glay at the cold end. Ghalambaz et al. [5] aimed to improve distributed under the terms and the thermal charging performance of the conical shell-and-tube LHTES (latent heat thermal conditions of the Creative Commons energy storage) unit by optimizing the shell shape and fin inclination angles. Attribution (CC BY) license (https:// Regarding the choice of working fluid, after the addition of suspended particles of creativecommons.org/licenses/by/ the phase change material (PCM) to a working fluid, fine solid–liquid PCM particles are 4.0/). dispersed in a carrier or suspending working fluid (PCM slurries [6] or PCM suspensions),

Energies 2021, 14, 2857. https://doi.org/10.3390/en14102857 https://www.mdpi.com/journal/energies Energies 2021, 14, 2857 2 of 17

and the absorption and release of latent heat by PCM particles at different positions can be utilized to reduce the maximum temperature of the target facility or improve the heat transfer efficiency of the system. The manufacturing methods of PCM particles can be divided into the categories of microencapsulation technology [7] and emulsion technology [8]. In microencapsulation technology, PCM is encapsulated by polymer materials to separate the dispersed phase (PCM) and the dispersion medium (such as water and glycerin), so that the PCM particles can be suspended in the dispersion medium. Emulsion technology reduces the surface tension between the dispersed phase and the dispersion medium by adding appropriate emulsifiers, so that the PCM is uniformly suspended in the dispersion medium to form a stable emulsion. Using a PCM suspension as a working fluid to improve heat transfer performance in forced convection in tubes, ducts, micro-channels, etc. [9,10], has been extensively displayed. Previously, studies applying the PCM suspension to natural convection were limited, and could be traced back to [11], who performed a natural convection experiment on an enclosure containing PCM particles. The heating surface was at the bottom of the enclosure, and the PCM particles were cylinders with a diameter of 3.2 mm and a height of 1.6 mm. The heat transfer coefficient obtained in this experiment was 1.9 times higher than that of pure water. Datta et al. [12] conducted a follow-up study on the experiment of [11]. The experimental model was a 150 mm × 150 mm × 150 mm cubic enclosure. The difference between these two experiments is that [12] used PCM microcapsules to replace the PCM particles used in [11]. The results showed that under low PCM concentrations (0.5% and 1%), the Nusselt number increased by approximately 80% compared with that of pure water; however, with a further increase in PCM concentration, the heat transfer efficiency gradually decreased, possibly because with increasing the PCM concentration, the uniform dispersion of the microcapsules in the fluid becomes progressively worse, and the microcapsules easily adhere to the high- or low-temperature walls, thus reducing the heat transfer efficiency. Harhira et al. [13] conducted a numerical simulation on the natural convection heat transfer from a vertical flat plate with PCM suspensions, and the results showed that the use of a working fluid containing PCM suspensions increased the Nusselt number to approximately eight times that of pure water. Recently, PCM and PCM suspensions in natural convection have been the subject of various engineering applications, such as thermal energy storage [14,15], thermal management of buildings [16,17], heat pump and air-conditioning systems [18], electronic cooling [19–21], and food industry [22], and research on the use of PCM suspensions in natural convection is booming. Ho [20] investigated the heat transfer of a rectangular thermosyphon loop containing a PCM suspension using numerical simulations. The results showed that a flow regime exists where the latent heat absorption or release of the PCM suspension can effectively enhance the heat transfer, which should be defined through more extensive parametric simulations. Therefore, Ho et al. [21] investigated the effects of the loop size, the heating section proportion, and the cooling section height on the heat transfer characteristics of a rectangular thermosyphon containing a PCM suspension. The results showed that under some conditions (the aspect ratio of the rectangular loop, length of the heated section, and relative elevation of the cooled section compared to the heated section), the best heating efficiency and the cooling efficiency can be obtained. Ghalambaz et al. [23] numerically investigated the natural convective behavior of nano encapsulated PCMs (NEPCMs) in a cavity. They showed that heat transfer enhancement is strongly related to the non-dimensional melting point. Zadeh et al. [24] numerically investigated the conjugated natural convection heat transfer and entropy generation in a square cavity composed of a porous matrix and PCM suspension with two solid blocks in it. The results illustrate that the heat transfer rates and the average Bejan number are maximized and the generated entropy is minimized when the PCM melting point is 0.5. Hajjar et al. [25] numerically studied the natural convection of a NEPCM suspension in a cavity with a hot wall with a time-periodic temperature. They indicated that the average Energies 2021, 14, 2857 3 of 18

number are maximized and the generated entropy is minimized when the PCM melting point is 0.5. Hajjar et al. [25] numerically studied the natural convection of a NEPCM suspension in a cavity with a hot wall with a time-periodic temperature. They indicated

Energies 2021, 14, 2857 that the average Nusselt number in the enclosure follows a periodic variation3 of 17 with the same frequency as for the hot wall temperature. Investigations on natural convection heat transfer with a PCM suspension in rectangularNusselt number enclosures in the enclosure are gradually follows aincreasing periodic variation, but most with of the them same deal frequency with assteady-state behavior.for the hot In wall this temperature. context, the objective of this study is to provide insights into details of the transientInvestigations buoyant on natural flow convectionand heat heat transfer transfer in with a square a PCM suspension enclosure in containing rectangu- a PCM suspension,lar enclosures for are which gradually little increasing, information but most is currently of them deal available. with steady-state Of particular behavior. emphasis in thisIn thisstudy context, is the the effects objective of of the this subcooling study is to provide factor insights (Sb), initial into details concentration of the transient ( c ), Raleigh buoyant flow and heat transfer in a square enclosure containing a PCM suspension, for i numberwhich little( RaT information), and Stefan is currently number available. (Ste). Of particular emphasis in this study is the effects of the subcooling factor (Sb), initial concentration (ci), Raleigh number (RaT), and 2. StefanProblem number Formulation (Ste). 2. ProblemA mathematical Formulation model for the heat transfer of a square enclosure containing a PCM suspensionA mathematical is established, model forwhich the heat is transferdiscretized of a square via numerical enclosure containing methods a PCMand simulated usingsuspension the iterative is established, method. which The is discretized physical viamodel numerical and methodsboundary and conditions simulated using are shown in Figurethe iterative 1. The method. square Theenclosure physical (aspect model and ratio boundary AR = H/W conditions = 1) is are filled shown with in Figure an aqueous1. fluid The square enclosure (aspect ratio AR = H/W = 1) is filled with an aqueous fluid containing containing PCM particles. The upper and lower walls of the enclosure are adiabatic, the PCM particles. The upper and lower walls of the enclosure are adiabatic, the left side is a left side is a hot isothermal wall (Th), and the right side is a cold isothermal wall (Tc). The hot isothermal wall (Th), and the right side is a cold isothermal wall (Tc). The buoyancy buoyancygenerated generated by the differences by the in thedifferences wall temperatures in the andwall PCM temperatures particle concentrations and PCM particle concentrationsdrive the fluid todrive flow the in the fluid enclosure, to flow resulting in the inenclosure, heat convection. resulting in heat convection.

(a) physical model (b) grid

FigureFigure 1. 1. SchematicSchematic diagramsdiagrams of of (a) ( thea) the physical physical model model and (b )and grid. (b) grid.

InIn the the mixedmixed continuum continuum formula formula for the for buoyancy-driven the buoyancy-driven circulation flow circulation of the PCM flow of the suspension, the simplified assumptions used are as follows. PCM suspension, the simplified assumptions used are as follows. 1. Suspended PCM particles are rigid spheres of a uniform size, whose size is much 1. Suspendedsmaller than PCM the characteristic particles are dimension rigid spheres of flow. of a uniform size, whose size is much 2. smallerPCM suspension than the characteristic is a dilute solution dimension and is treated of flow. as a Newtonian fluid. 2. 3. PCMThe dispersedsuspension PCM is particlesa dilute are solution neutrally and buoyant is treated in the as mixture. a Newtonian fluid. 3. 4. TheThe dispersed flow of the PCM PCM suspensionparticles are is a neutrally two-dimensional buoyant laminar in the flow. mixture. 5. Except for the phase change process of particles, the hypothesis of local thermal 4. The flow of the PCM suspension is a two-dimensional laminar flow. equilibrium between the particles and base fluid is established. 5. 6. ExceptThe melting for the and phase freezing change of the PCM process particles of occurparticles, at a fixed the temperature.hypothesis of local thermal 7. equilibriumThere are no between chemical orthe nuclear particles reactions and betweenbase fluid the is particles established. and the fluid. 6. 8. TheIn themelting initial stage,and freezing the PCM particlesof the PCM are uniformly particles suspended occur at ina thefixed fluid, temperature. which is a 7. Theretwo-phase are no fluid. chemical or nuclear reactions between the particles and the fluid. 8. 9. InThe the interface initial stage, effect ofthe the PCM surfactants particles coated are on uniformly the PCM is suspended ignored. in the fluid, which is 10. In addition to the buoyancy caused by the density differences, the thermal properties a two-phaseof the fluid can fluid. be regarded as constants. 9. 11. TheThe interface density change effect associatedof the surfacta with thents solid–liquid coated on phase the PCM change is inignored. the particles is 10. Innegligible. addition to the buoyancy caused by the density differences, the thermal properties 12. ofThe the thermalfluid can radiation be regarded and viscous as constants. dissipation in the flow are neglected.

Energies 2021, 14, 2857 4 of 17

13. The relative velocity between the particles and the ﬂuid is quite low compared with the overall solution velocity, and can be ignored. Based on the above assumptions and dimensionless variables and parameters, the governing equations can be expressed as follows: Continuity equation ∂U ∂V + = 0 (1) ∂X ∂Y Momentum equations

∂U ∂U ∂U ∂P ∂2U ∂2U + U + V = − + Pr + (2) ∂Fo ∂X ∂Y ∂X ∂X2 ∂Y2

∂V ∂V ∂V ∂P ∂2V ∂2V + U + V = − + Pr + + Ra Pr[θ + N(C − 1)] (3) ∂Fo ∂X ∂Y ∂Y ∂X2 ∂Y2 T Energy equation

2 2 ! ∂θ ∂θ ∂θ ∂ θ ∂ θ c Cp f ∂(Cξ ) ∂(Cξ ) ∂(Cξ ) + + = + − i , l + l + l U V 2 2 U V (4) ∂Fo ∂X ∂Y ∂X ∂Y Ste (1 + Sb) Cp,b ∂Fo ∂X ∂Y Concentration equation

∂C ∂C ∂C 1 ∂2C ∂2C + U + V = + (5) ∂Fo ∂X ∂Y Le ∂X2 ∂Y2

The dimensional forms and the process of derivation of the above equations can be obtained in Ho [20]. The liquid-phase volume fraction of the PCM particles can be obtained from the following equation [26]:

∂ξ ∂ξ ∂ξ αp Cp p Bip 1 Cp b ∂θ l + l + l = , ( + )( + ) − , ( + ) U V 12 2 Ste 1 Sb θ ε Bip Ste 1 Sb τ (6) ∂Fo ∂X ∂Y αb Cp,b Dp 3 Cp, f ∂Fo

hprp k 2(1 − c ) Bi = = e v p 1/3 (7) kp kp (2 − 3cv + cv)

The effective thermal conductivity is a function of the particle concentration (cv) and the Peclet number (Pep), which varies as the local shear rate is varying. Charunkyakorn et al. (1991) suggests the correlation that accounts for different values of particle Peclet number as follows: ke m = 1 + b cv Pep (8) kp where 2 edp Pep = (9) α f ∂u ∂v e = + (10) ∂y ∂x The constants (b, m) that depend on the Peclet number are as follows:

when Pep ≤ 0.67; b = 3, m = 1.5; when 0.67 < Pep < 250; b = 1.8, m = 0.183; when Pep ≥ 250; b = 3, m = 0.09091. The dimensionless continuity and momentum equations can be further arranged as follows: [27] Stream function ∂2ψ ∂2ψ + = −ω (11) ∂X2 ∂Y2 Energies 2021, 14, 2857 5 of 17

Vorticity function

∂ω ∂ω ∂ω ∂2ω ∂2ω ∂θ ∂C + U + V = Pr + + Ra Pr + N (12) ∂Fo ∂X ∂Y ∂X2 ∂Y2 T ∂X ∂X

The initial conditions are set as follows: Fo = 0 −Sb ψ = 0, ω= 0, θ = , C = 1, ξ = 0 (13) 1 + Sb l The boundary conditions are set as follows: Fo > 0 ∂θ ∂C ∂ξ 0 < X < 1, Y = 0, ψ= 1, = 0, = 0, l = 0 (14) ∂Y ∂Y ∂Y 1 ∂C 0 < Y < 1, X = 0, ψ= 1, θ = , = 0, ξ = 1 (15) 1 + Sb ∂X l ∂θ ∂C 0 < X < 1, Y = AR, ψ= 1, = 0, = 0 (16) ∂Y ∂X −Sb ∂C 0 < Y < 1, X = 1, ψ= 1, θ = , = 0 (17) 1 + Sb ∂X The local and average Nusselt numbers on the hot and cold walls are deﬁned as follows: ∂θ Nuh = − (18) ∂X X = 0

∂θ Nuc = − (19) ∂X X = 1 1 Z AR Nuh = NuhdY (20) AR 0 1 Z AR Nuc = NucdY (21) AR 0

3. Numerical Method In the grid system (as illustrated in Figure1b), non-uniform orthogonal grid points are used for a fine resolution near the hot and cold walls. On both sides of the x-axis, a tight grid is used, and in the middle of the enclosure, a sparse grid is used; on the y-axis, the equally spaced grid is used. In this paper, the finite difference method is used to discretize the differential equations, and for the terms of time, space, and convection, the first-order backward difference, second-order central difference, and QUICK for two dimensions (QUICK-2D, [28]) are used, respectively. The vorticity on the boundaries of the walls is solved with Thom’s formula. The discretization of the concentration equation on the wall uses the finite volume method. The time discretization method used is implicit discretization, and the problem-solving process is divided into two iterative loops, i.e., a time forward loop and a space iterative loop. At each time step, a spatial calculation is carried out first and must reach iterative convergence (Equation (22)), then the temporal calculation is conducted, using Equation (23) to check. If it also converges, then the iteration can advance to the next time step. The convergence criteria are defined as follows:

n,m+1 − n,m φi,j φi,j Iterative convergence error = n,m ≤ εi (22) φi,j

n+1,m − n,m φi,j φi,j Temporal convergence error = n,m ≤ εi (23) φi,j Energies 2021, 14, 2857 6 of 17

where m represents the number of iterations and n represents the number of time steps (the iteration advances). The spatial and temporal flow field, temperature, concentration, liquid volume ratio, and vorticity are first compared with the iterative convergence criteria, which are 10−8 for temperature and 10−5 for other parameters. In addition, the equilibrium of the inlet and outlet heats (Nuh and Nuc) of the high- and low-temperature walls is used to determine whether steady-state convergence is reached. The appropriate time step will depend on the parameter range. After testing, when time step ∆Fo is less than 5 × 10−6, the simulation results of different time steps have very little difference. Therefore, the time step used in this study is set to be 10−6. The spatial grid point tests are carried out for the steady

states, and the average discrepancy of ENu and EΨmax are calculated using the changes in the average Nusselt number and the maximum value of the stream function, respectively, due to different grid-point numbers:

Num1 − Num2 Ψmax m1 − Ψmax m2 ENu = ; EΨmax = (24) Num1 Ψmax m1 where m1 and m2 are the different grid points. Based on the grid test results (shown in Table1), this study decided to use a 71 × 71 grid for the calculation.

Table 1. Grid test results.

3 RaT = 10 , Ste = 0.05, Sb = 1, ci = 10%

Grid Nuh ENuh Nuc ENuc Ψmax EΨmax 71 × 71 1.3535 – 1.3535 – 0.9775 – 81 × 81 1.3515 0.152 1.3514 0.155 0.9770 0.05 91 × 91 1.3507 0.058 1.3506 0.061 0.9766 0.045 4 RaT = 10 , Ste = 0.05, Sb = 1, ci = 10%

Grid Nuh ENuh Nuc ENuc Ψmax EΨmax 71 × 71 2.7407 – 2.7405 – 3.4018 – 81 × 81 2.7368 0.142 2.7367 0.139 3.4073 0.16 91 × 91 2.7329 0.143 2.7328 0.144 3.4191 0.34 5 RaT = 10 , Ste = 0.05, Sb = 1, ci = 10%

Grid Nuh ENuh Nuc ENuc Ψmax EΨmax 71 × 71 5.1342 – 5.1335 – 9.6979 – 81 × 81 5.1021 0.629 5.1045 0.568 9.6788 0.200 91 × 91 5.0821 0.394 5.0837 0.409 9.6579 0.412

To validate the formulation and numerical method developed in this study, the steady- state forced convection of the PCM suspension in a circular tube, as shown in Figure2, which was reported in [29,30], is analyzed. The obtained results are qualitatively and quantitatively compared with the results from the literature. Energies 2021, 14, 2857 7 of 18

81 × 81 5.1021 0.629 5.1045 0.568 9.6788 0.200 91 × 91 5.0821 0.394 5.0837 0.409 9.6579 0.412

To validate the formulation and numerical method developed in this study, the Energies 2021, 14, 2857 7 of 17 steady-state forced convection of the PCM suspension in a circular tube, as shown in Figure 2, which was reported in [29,30], is analyzed. The obtained results are qualitatively and quantitatively compared with the results from the literature.

Figure 2. 2. PhysicalPhysical model model used used for the for model the model validation. validation.

TheThe basic basic assumptions assumptions of this of thistest are test as are follows: as follows:

(1) TheThe inlet inlet temperature temperature is uniform. is uniform. (2) The flow field of the fluid is a two-dimensional, hydrodynamically fully developed (2) The flow field of the fluid is a two-dimensional, hydrodynamically fully developed laminar flow. (3) Thelaminar fluid was flow. regarded as a Newtonian fluid. (3)(4) ExceptThe fluid for the was heat regarded conductivity, as a which Newtonian varies with fluid. the flow field, the bulk properties (4) ofExcept the fluid for are the homogeneous heat conductivity, and fixed. which varies with the flow field, the bulk properties (5) Theof theconcentration fluid are homogeneousdistribution is uniform. and fixed. (5)(6) TheThe buoyancy concentration caused distributionby the temperature is uniform. differences is negligible. (6)(7) TheThe temperature buoyancy field caused is axially by the symmetric. temperature differences is negligible. (7) BasedThe temperatureon the above fieldassumptions, is axially the symmetric. dimensionless governing equations of this problemBased using on the the methods above in assumptions, this paper can be the expressed dimensionless as follows: governing equations of this Energy equation: problem using the methods in this paper can be expressed as follows: ∂∂∂∂θθ** θ *∂∂ξξ Energy equation:2 11kCCell()2 () +−(1ηη ) = ( ) − +− (1 η ) (25) ∂∂Fo** Xηη∂∂k η Ste *** ∂ Fo ∂ X b b " # ∂θ∗ 2 ∂θ∗ 1 ∂ ke ∂θ∗ 1 ∂ C ξl) 2 ∂ C ξl) Liquid-phase∗ + ( volume1 − η ) fraction∗ = of the( ηPCM particles:)− ∗ ∗ + (1 − η ) ∗ (25) ∂Fo ∂X η ∂η kb ∂η Steb ∂Fo ∂X ∂∂ξξα CBi C ∂θ * ll+−η 2 =11pppp,,**θε+− * τ pb * (1 ) 12Ste ( ) Bip Ste (26) ∂∂Fo** X 1(/)1Liquid-phase+∀∀−Bi  volume1/3 fractionα CD of the2* PCM particles:3 C∂ Fo pppcmbpb, ppf,

 The dimensionless velocity of the fluid is completely developed: " C ∗ # ∂ξ ∂ξ  1  αp C*2p,p Bip 1 p,b ∂θ l + (1 − η2) l = 12 U =−2(1ηSte ) ∗(θ∗ + ε∗) − Bi τ Ste∗ (27) (26) ∂Fo∗ ∂X∗ h 1/3 i α C D2 3 p C ∂Fo∗  1 + Bi (∀p/∀pcm) − 1  b p,b p p, f Thep initial and boundary conditions for this problem are: ==θξ The dimensionless velocity ofFo the0, ﬂuid=0, isl completely 0 developed: (28) Fo > 0 U∗ = 2(1 − η2) (27) ∂θ 1 = At the end, η = 1 X> 0 (29) The initial and boundary∂Γη conditions for this problem are:

Fo = 0, θ= 0, ξl = 0 (28)

Fo > 0 ∂θ 1 = At the end, η = 1 X > 0 (29) ∂η Γ ∂θ = 0 At the end, η = 0 X > 0 (30) ∂η

θ = θi At the end, η > 0 ξl = 0 X = 0 (31) . Figure3a shows the wall temperature distribution of the circular tube with pure water (ci = 0). The numerical simulation results of this validation model are in good agreement with the analytical solution [29] and are roughly consistent with the experimental data [29], but due to the systematic error caused by the experimental equipment, some temperatures Energies 2021, 14, 2857 8 of 18

∂θ = 0 At the end, η = 0 X> 0 ∂η (30)

θθ= ηξ>= i At the end, 00 l X = 0. (31) Figure 3a shows the wall temperature distribution of the circular tube with pure = Energies 2021, 14, 2857 water ( ci 0 ). The numerical simulation results of this validation model are in 8good of 17 agreement with the analytical solution [29] and are roughly consistent with the experimental data [29], but due to the systematic error caused by the experimental equipment,are decreased some slightly. temperatures The maximum are decrease deviationd slightly. is about 20%.The maximum Figure3b shows deviation the inﬂuence is about 20%.of the FigureSte on 3b the shows wall the temperature influence distributionof the Ste on whenthe wallc temperature= 0.1. Qualitatively, distribution the when wall c = 0.1 i temperaturei . Qualitatively, increases with the increasingwall temperSteature. The increases wall temperatures with increasing obtained Ste in. thisThe paper wall temperaturesare slightly different obtained from in thethis experimentalpaper are slightly results different [29], but from are closer the experimental to the experimental results [29],results but than are thecloser analysis to the results experimental of [30]. Basedresults on than the the above analysis analysis, results the of predicted [30]. Based values on theof the above model analysis, in this the study predicted are qualitatively values of consistentthe model with in this the resultsstudy are of thequalitatively literature. consistentHowever, with quantitatively, the resultsthe of the differences literature are. However, still signiﬁcant, quantitatively, which maythe differences be due to theare stillneglect significant, of the axial which heat may transfer be due orto otherthe neglect control of variables,the axial heat which transfer is worthy or other of furthercontrol variables,discussion. which is worthy of further discussion.

Figure 3.3. ComparisonComparison of of the the predicted predicted wall wall temperature temperature with with the experimentalthe experimental and analyticaland analytical results results in the literaturein the literature [29,30] shows[29,30] thatshows there that was there no was signiﬁcant no significant difference. difference.

4. Results Results and and Discussion Discussion

The effects of Sb, ci, RaT, and Ste on the transient and steady-state natural convection The effects of Sb, ci , RaT , and Ste on the transient and steady-state natural heat transfer in a square enclosure containing PCM suspension are analyzed. The govern- convection heat transfer in a square enclosure containing PCM suspension are analyzed. ing equations used include continuity equation, momentum equations, energy equation, The governing equations used include continuity equation, momentum equations, energy concentration equation, and the liquid-phase volume fraction of the PCM particles. The equation, concentration equation, and the liquid-phase volume fraction of the PCM parameter ranges are as follows: Sb = 0–1.0, c = 0–0.1, Ra = 103 − 105, and Ste = 0.005–0.1. i T =−3 5 particles. The parameter ranges are as follows: Sb− =4 0–1.0, ci = 0–0.1, Ra 10 106 , and Other parameters are set as follows: Dp = 5 × 10 , AR = 1, Pr = 8.5, Le =T 3.3 × 10 , N = 1, =× −4 Steε = = 0.0050 (the– effect0.1. Other of PCM parameters supercooling are set is as not follows: considered), Dp 510 and τ ,= AR1. = 1, Pr = 8.5, Le = 3.3 × 106, N = 1, ε = 0 (the effect of PCM supercooling is not considered), and τ =1. 4.1. Transient Evolution 4.1. TransientThe transient Evolution evolution of the natural convection of a PCM suspension in an enclosure is affectedThe transient not only evolution by the temperature of the natural and co concentrationnvection of gradients,a PCM suspension but also byin thean enclosurerelease or is absorption affected not of only latent by heat the temperat of the containedure and concentration PCM particles. gradients, Figure4 showsbut also the by transient evolution of the flow field, temperature distribution, and PCM solid–liquid phase the release or absorption of latent heat of the contained PCM particles. Figure 4 shows the distribution with time when Ra = 105, Ste = 0.05, Sb = 1, and c = 0.05. The dotted transient evolution of the flow Tfield, temperature distribution, andi PCM solid–liquid line represents the temperature distribution,5 the solid line represents the flow field, the phase distribution with time when Ra = 10 , Ste = 0.05, Sb = 1, and c = 0.05. The dotted bold solid line represents the meltingT point, the shaded area and whitei area in the right figures represent the solid-phase and the liquid-phase PCM particles respectively, and the grayscale between the shaded and white areas is the phase change region. In the initial stage, the heat of the hot wall mainly transfers to the cold wall by the mechanism of heat conduction, as shown in Figure4a. Energies 2021, 14, 2857 9 of 18

line represents the temperature distribution, the solid line represents the flow field, the bold solid line represents the melting point, the shaded area and white area in the right figures represent the solid-phase and the liquid-phase PCM particles respectively, and the grayscale between the shaded and white areas is the phase change region. In the initial Energies 2021, 14, 2857 9 of 17 stage, the heat of the hot wall mainly transfers to the cold wall by the mechanism of heat conduction, as shown in Figure 4a.

Figure 4. Cont.

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Energies 2021, 14, 2857 10 of 18

Figure 4. Transient evolution of the flow field (solid line), temperature distribution (dashed line), Figure 4. Transient evolution of the flow field (solid line), temperature distribution (dashed line), and PCM solid–liquid phase distribution (shaded area is solid-phase PCM particles; FigureFigure 4. 4.TransientTransient evolution evolution of of the the flow ﬂow field ﬁeld (solid (solid line), line), temperature temperature distribution distribution==5 (dashed (dashed line), and PCM solid–liquid phase distribution (shaded area is solid-phase PCM particles; RaTi10 , Ste=0.05, Sb=1, c 5% ). ==5 andline), PCM and solid PCM–liquid solid–liquid phase distribution phase distribution (shaded (shadedarea is solid-phase area is solid-phase PCM particles; PCM particles; RaTi10 , Ste=0.05, Sb=1, c 5% ). 5 5 RaT=== 10 , Ste = 0.05, Sb = 1, ci = 5%). RaTi10 , Ste=0.05, Sb=1, c 5% ). With increasing time, the distance between the melting interface and the hot wall With increasing time,With the distance increasing between time, the the distance melting betweeninterfacethe and melting the hot interface wallincreases, andthe and hot the wall heat transfer mechanism changes from heat conduction to heat increases, and the heatincreases,With transfer andincreasing themechanism heat time, transfer changes the mechanism distance from changes heatbetween conduction from the heat melting conductionto heat interfaceconvection. to heat and convection. At thisthe moment,hot wall the velocity of PCM suspension gradually increases (Figure convection. At this moment, the velocity of PCM suspension gradually increases (Figure Ψ increases,At this moment, and the velocityheat transfer of PCM suspensionmechanism gradually changes increases from (Figure 4b,heat max4conductionb, =Ψ max24.6).= 24.6).After to theheat convection is enhanced, the PCM particles cross the melting 4b, Ψ max = 24.6). Afterconvection.After thethe convection convection At this is moment, isenha enhanced,nced, the the the velocity PCM PCM particles particles of PCM cross cross suspension the the melting melting graduallyinterface interface ( θincreases (θ = 0;  in (Figure Figure 4c) and expand the melting range to the right (). From the interface (θ = 0;  in Figure 4c) and expand the melting range to the right (). From thefigures on the right panel, the PCM inside the particles at both  is liquid. The flow 4b,Figure Ψ max4c) = and24.6). expand After the the melting convection range is to enha the rightnced, ( the). From PCM the particles figures oncross the the right melting   figures on the rightinterface panel, the (θ PCM = 0; inside  in Figurethe particles 4c) and at bothexpand the is liquid. melting The range flowturns to the downward right (). as From it approaches the the cold wall on the right side ( ). During the flow turns downward as it approaches the cold wall on the right side (). During the flowprocess, the PCM inside the particles is gradually cooled to solid. The downward flowing figures on the right panel, the PCM inside the particles at both  is liquid. The flow process, the PCM inside the particles is gradually cooled to solid. The downward flowingPCM suspension turns to the left under the influence of the wall below (), then it is  PCM suspension turnsturns to downward the left unde asr theit approaches influence of ththee coldwall belowwall on ( the), then right it isside ( ). During the flow process, the PCM inside the particles is gradually cooled to solid. The downward flowing PCM suspension turns to the left under the influence of the wall below (), then it is

EnergiesEnergies 2021, 14 2021, 2857, 14 , 2857 10 of 1810 of 18

Energies 2021, 14, 2857 Energies 2021, 14, 2857 10 of 18 10 of 18

Figure Figure4. Transient 4. Transient evolution evolution of the flowof the field flow (solid field li(solidne), temperature line), temperature distribution distribution (dashed (dashed line), line), andFigure PCMand 4. Transientsolid PCM–liquid solid evolution– phaseliquid distributionphase of the distribution flow (shadedfield (solid (shaded area line), is areasolid-phase temperature isFigure solid-phase PCM 4. distribution Transient particles; PCM particles; evolution (dashed line),of the flow field (solid line), temperature distribution (dashed line), ==5 ==5 andRaTi PCM10RaTi ,solid Ste=0.05,10–liquid , Ste=0.05, Sb=1, phase cSb=1,distribution 5% c ). 5% (shaded). area is solid-phaseand PCM PCM solid particles;–liquid phase distribution (shaded area is solid-phase PCM particles; ==5 ==5 RaTi10 , Ste=0.05, Sb=1, c 5% ). RaTi10 , Ste=0.05, Sb=1, c 5% ). With increasingWith increasing time, thetime, distance the distance between between the melting the melting interface interface and the and hot the wall hot wall increases,Withincreases, andincreasing theand heat thetime, transferheat the distancetransfer mechanism betweenmechanism changes the changesmelting fromWith interfacefromheat increasing conductionheat and conduction the time, hotto the heatwall todistance heat between the melting interface and the hot wall convection.increases,Energiesconvection.2021 ,and At14, 2857 thisthe At moment, heatthis moment,transfer the velocity themechanism velocity of PCM ofchanges suspensionPCM suspension increases,from gradually heat andgradually conduction increasesthe heat increases (Figuretotransfer heat (Figure mechanism changes 11from of 17 heat conduction to heat

4b,convection. Ψ max4b, =Ψ 24.6). maxAt this= After24.6). moment, theAfter convection thethe velocityconvection is enhaof PCM isnced, enha suspension nced,the PCMconvection. the graduallyparticlesPCM particlesAt crossthis increases moment, thecross melting (Figure the the melting velocity of PCM suspension gradually increases (Figure θ interface4b, Ψinterfacemax (= 24.6). = 0;(θ  After =in 0; Figure  the in convectionFigure 4c) and 4c) expand andis enha expand thenced, melting the the melting 4b, PCMrange Ψ maxparticles rangeto the= 24.6). toright crossthe After ( right the). Fromthe (melting ).convection theFrom the is enhanced, the PCM particles cross the melting θ Energies 2021θ, 14, 2857 11 of 18 figuresinterfacefigures on ( the =on right0; the in panel,right Figure panel, the 4c)panel, PCM andthe insidethePCMexpand PCM inside the the inside particles melting the theparticlesinterface particles atrange both atto ( both attheboth =isright 0; liquid.  ( isin ).liquid.isFigure TheFrom liquid. flow The4c)the The and flow flow expand turns the downward melting range as it to the right (). From the Energies 2021, 14, 2857 11 of 18 turnsfigures turnsdownward on thedownward right as itpanel, approaches as it the approachesapproaches PCM th einside cold th thee wallthe cold cold particles on wall wall the on onfiguresright at the theboth sideright righton  (the side ).is right During liquid.( ). panel, During Thethe theflowflow thethe PCM flowflow inside process, the theparticles PCM insideat both  is liquid. The flow process,turns process,downward the PCM the insidePCMas it approachesinside the particles the particles th ise graduallycold is wall gradually cooledon the cooledcooled turnstoright solid. todownwardtoside solid.Thesolid. ( downward). TheThe During as downwarddownward it approaches theflowing flow flowingflowing the PCM cold suspensionwall on the turns right to side (). During the flow PCMprocess, suspensionPCM the suspension PCM turnsinside turnsto the the particles totheleft the leftunde left is under graduallyr undethe the influencer the influence cooled influence ofprocess, to of the solid. the ofwall wall the theThe below belowwallPCM downward belowinside( ), thenthen (the flowing), itparticles itthen isdirected is directed it isis upwardgradually upward by by cooled the the thermal thermal to solid. buoyancy The downward near the flowing hot wall on the left (). Finally, it directedPCM upwardsuspension by theturns thermal to the buoyancy buoyancyleft unde rnear near the the theinfluence hot hot wall wall ofPCM on onthe the suspensionwall left below (  ). turns( Finally, ), thento the itit it returns leftis unde to r the the top influence region ( of the) to wall form below the main (), circulation. then it is returns to the top region () to form the main circulation. When Fo = 0.05 (Figure 4c), there is an inner circulation inside the above-mentioned When Fo = 0.05 (Figure 4c), there Whenis an innerFo = circulation 0.05 (Figure inside4c), there the above-mentioned is an inner circulation main inside circulation the above-mentioned (symbol (A)), and the right figure of Figure 4c shows that this small main circulation (symbol (A)), andmain the circulation right figure (symbol of Figure (A)), 4c andshows the that right this figure small of circulation Figure4c shows is formed that by this solid small PCM particles. As time passes, when Fo = 0.1 (Figure 4d), circulation is formed by solid PCMcirculation particles. is As formed time passes, by solid when PCM Fo particles. = 0.1 (Figure As time 4d), passes, this internal when smallFo =0.1 circulation (Figure4 d),becomes two small circulations (symbols (B) and (C)), and this internal small circulation becomesthis internal two small small circulations circulation (symbols becomes (B) two and small (C)), circulations and the right (symbols figure (B)of Figure and (C)), 4d andshows that the left circulation (B) is formed by solid PCM the right figure of Figure 4d showsthe that right the figure left circulation of Figure4 d(B) shows is formed that theby leftsolid circulation PCM particles, (B) is while formed the by right solid circulation PCM (C) is formed by liquid PCM particles, and there is a particles, while the right circulationparticles, (C) is formed while theby rightliquid circulation PCM particles, (C) is and formed there by is liquid a solid PCM PCM particles, region inside and there the small is a circulation (C). When Fo = 0.2, as shown in the right solid PCM region inside the smallsolid circulation PCM region (C). When inside Fo the = small0.2, as circulation shown in the (C). right When figureFo = of 0.2, Figure as shown 4e, the in aforementioned the right solid PCM region is melted, and a small circulation figure of Figure 4e, the aforementionedfigure solid of Figure PCM4 regione, the aforementioned is melted, and a solidsmall PCM circulation region (D) is melted, is fully and formed a small by circulation liquid PCM particles. When the circulation crosses the melting (D) is fully formed by liquid PCM(D) particle is fullys. formed When the by liquidcirculation PCM crosses particles. the Whenmelting the interface, circulation the crossesPCM particles the melting in the working fluid could be in different phases before and interface, the PCM particles in theinterface, working thefluid PCM could particles be in different in the working phases fluidbefore could and after be in the different crossing. phases Figure before 4f (steady-state) and shows that the internal circulation at (E) is full after the crossing. Figure 4f (steady-state)after the shows crossing. that Figure the internal4f (steady-state) circulation shows at (E) that is thefull internalof liquid circulation PCM particles. at (E) is fullWhen of the fluid flows to the right and contacts the low- of liquid PCM particles. When liquidthe fluid PCM flows particles. to the When right the and fluid contacts flows to thethe rightlow- andtemperature contacts theregion, low-temperature the particles dissipate heat. When the fluid flows back to the left temperature region, the particlesregion, dissipate the he particlesat. When dissipate the fluid heat. flows When back the to fluid the flowsleft region back toat the(F), leftthe regionPCM particles at (F), the have solidified. PCM particles have solidified. region at (F), the PCM particles have solidified. Figure 5 illustrates the parameter effects (Sb, c , Ra , and Ste) on the average Figure5 illustrates the parameter effects ( Sb, c , Ra , and Ste) on the average Nusselt i T Figure 5 illustrates the parameter effects (Sb, c , Ra , and Ste) on the averagei T numbers of hot and coldi walls.T Figure5a shows the transientNusselt development numbers of under hot and different cold walls. Figure 5a shows the transient development under Nusselt numbers of hot and cold walls. Figure 5a shows the transient development under different Sb values. When Sb = 0.5–1, compared with other Sb cases, the transient values Sb values. When Sb = 0.5–1, compared with other Sb cases, the transient values in Nuh different Sb values. When Sb = 0.5–1, compared with other Sb cases, the transient values and Nuc are large; however, the differences in Nu betweenin differentNuh andSb sNu arec notare significant.large; however, the differences in Nu between different Sbs are in Nuh and Nuc are large; however,Both Nutheh differencesand Nuc are in 1.4Nu after between reaching different steady-state, Sbs are which is higher than the Nu (1.3) not significant. Both Nuh and Nuc are 1.4 after reaching steady-state, which is higher Nu Nuwhen Sb = 0.2 and the Nu (1.13) of pure water. The effect of Sb on Nu is quite special, as not significant. Both h and c are 1.4 after reaching steady-state, which is higher than the Nu (1.3) when Sb = 0.2 and the Nu (1.13) of pure water. The effect of Sb on detailed in the next section. High ci (ci = 0.1) accompanies transient high values in Nu, than the Nu (1.3) when Sb = 0.2 and the Nu (1.13) of pure water. The effect of Sb on Nu c c = 0.1 as shown in Figure5b, and the 1.4 of the final steady-state Nu isis quite higher special, than those as detailed under in the next section. High i ( i ) accompanies Nu is quite special, as detailed in the next section. High c (c = 0.1 ) accompanies other ci values. i i transient high values in Nu , as shown in Figure 5b, and the 1.4 of the final steady-state transient high values in Nu , as shown in Figure 5b, and the 1.4 of theNu final steady-state Ra Ra Figure5c shows the changes in under differentNu T iss. higher The increases than those in underT can other ci values.

Nu is higher than those under otherresult ci in values. a strong convection effect, which further increases the heat transfer rate. As shown Figure 5c shows the changes in Nu under different RaT s. The increases in RaT can in the figure, the time required to reach equilibrium under a large RaT is shorter than Figure 5c shows the changes in Nu under different RaT s. The increases in RaT can result in a strong convection effect, which further increases the heat transfer rate. As that in other cases, and the higher the RaT, the higher the heat transfer rate is. When RaT result in a strong convection effect, which further increases the heat transfer rate. As Ra is increased from 103 to 105, the steady-state Nu is increasedshown fromin the 1.05 figure, to 5.0,the showingtime required to reach equilibrium under a large T is shorter shown in the figure, the time required to reach equilibrium under a large RaT is shorter Ra substantial growth. However, cases with high RaT are morethan susceptible that in other to thecases, influence and the of higher the T , the higher the heat transfer rate is. When than that in other cases, and the higher the RaT , the higher the heat transfer rate is. When 3 5 other parameters regarding the generation of oscillatory naturalRaT is convection.increased from It can 10 be to seen 10 , the steady-state Nu is increased from 1.05 to 5.0, Ra is increased from 103 to 10from5, the Figure steady-state5d that withNu decreasingis increasedSte from, Nu increases,1.05 to 5.0, and when Ste = 0.01, the steady-state T showing substantial growth. However, cases with high RaT are more susceptible to the Nu is 1.9. According to above analysis, under the condition of high ci, high RaT, and low showing substantial growth. However, cases with high RaT are more susceptible to the influence of other parameters regarding the generation of oscillatory natural convection. Ste, the transient and the steady-state values of Nu are high, and the time required to reach influence of other parameters regarding the generation of oscillatory natural convection. Nu steady-state is shortened. It can be seen from Figure 5d that with decreasing Ste, increases, and when Ste = It can be seen from Figure 5d that with decreasing Ste, Nu increases, and when Ste = 0.01, the steady-state Nu is 1.9. According to above analysis, under the condition of high 0.01, the steady-state Nu is 1.9. According to above analysis, under the condition of high 4.2. Steady-State Results ci , high RaT , and low Ste, the transient and the steady-state values of Nu are high, and ci , high RaT , and low Ste, the transientIn and this the section, steady-state analyses values are performedof Nu are onhigh, the and effects the time of Sb required, ci, RaT, to and reachSte steady-stateon the is shortened. the time required to reach steady-statesteady-state is shortened. results. It can be seen from Figure6 that with increasing ci or RaT, Nuh increases; however, with increasing Sb, Nuh increases first4.2. thenSteady-State decreases, Results and there is a 3 4.2. Steady-State Results local optimum Nu. When RaT = 10 (Figure6a), regardless ofInc ithis, the section, optimal analyses heat transfer are performed on the effects of Sb, c , Ra , and Ste on the 4 i T In this section, analyses are performed(Nuh) always on the occurs effects at Sb of= Sb 0.8;, c when, Ra Ra, andT = Ste10 on(Figure the 6b, the optimum heat transfer i T 5 steady-state results. It can be seen from Figure 6 that with increasing c or Ra , Nu characteristics occur at Sb = 0.5; and when RaT = 10 (Figure6c), the optimal heat transfer i T h steady-state results. It can be seen from Figure 6 that with increasing ci or RaT , Nuh characteristics occur at Sb = 0.5. In summary, within the discussedincreases; parameter however, ranges,with increasing when Sb, Nuh increases first then decreases, and there is a 3 =Nu = 3 increases; however, with increasingRa SbT , 10h , increases the maximum first thenNu hdecreases,is 1.20, which and there occurs is whena localc ioptimum0.1 and NuSb. =When 0.8; when Ra = 10 (Figure 6a), regardless of c , the optimal heat 4 T i Ra 3 = 10 , the maximum Nu is 2.89, which occurs when c = 0.1 and Sb = 0.5; and local optimum Nu . When Ra = 10T (Figure 6a), regardless ofh c , the optimal heat i 4 T 5 i transfer ( Nu ) always occurs at Sb = 0.8; when Ra = 10 (Figure 6b, the optimum heat when RaT = 10 , the maximum Nuh is 5.13, which occurs when ci =h 0.1 and Sb = 0.5. T transfer ( Nu ) always occurs at Sb = 0.8; when Ra = 104 (Figure 6b, the optimum heat = 5 h T transfer characteristics occur at Sb = 0.5; and when RaT 10 (Figure 6c), the optimal heat = 5 transfer characteristics occur at Sb = 0.5; and when RaT 10 (Figure 6c), the optimal heat transfer characteristics occur at Sb = 0.5. In summary, within the discussed parameter transfer characteristics occur at Sb = 0.5. In summary, within the discussed parameter = 3 = ranges, when RaT 10 , the maximum Nuh is 1.20, which occurs when ci 0.1 and Sb = 3 = ranges, when RaT 10 , the maximum Nuh is 1.20, which occurs when ci 0.1 and Sb = 4 = = 0.8; when RaT 10 , the maximum Nuh is 2.89, which occurs when ci 0.1 and Sb = = 4 = = 0.8; when RaT 10 , the maximum Nuh is 2.89, which occurs when ci 0.1 and Sb =

Energies 2021, 14, 2857 12 of 18

Energies 2021, 14, 2857 12 of 17 = 5 = 0.5; and when RaT 10 , the maximum Nuh is 5.13, which occurs when ci 0.1 and Sb = 0.5.

FigureFigure 5. 5.Transient Transient development development of of the the average average Nusselt Nusselt numbers numbers of of the the hot hot and and cold cold walls. walls.

Energies 2021, 14, 2857 13 of 17 Energies 2021, 14, 2857 13 of 18

3 FigureFigure 6. 6.The The effect effect of Sbof onSb theon averagethe average Nusselt Nusselt numbers numbers of the hot of walls.the hot ((a walls.): RaT ((=a):10 , Ste = 0.05, = 3 4 = 4 5 = 5 (bR):aTRaT10= , Ste=0.0510 , Ste =, (b0.05,): Ra (Tc): 10RaT , Ste=0.05= 10 , Ste, (c):=Ra0.05T 10. , Ste=0.05 )

Table2 shows the effects of Ra and Ste. According to the deﬁnition of Ste, a low Ste Table 2 shows the effects of RaT and Ste. According to the definition of Ste, a low Ste means that more latent heat than sensibleT heat is contained in the ﬂuid, and with increasing latentmeans heat, that the more overall latent heat transferheat than characteristics sensible heat can beis improved.contained According in the fluid, to the and data with 3 inincreasing the table, latent the lower heat, the theSte overall, the higher heat the transfer heat transfer characteristics characteristics. can be Whenimproved.RaT =According10 andto theSte data= 0.01, in the maximumtable, the lower increase the in Ste the, the Nusselt higher number the heat of thetransfer high-temperature characteristics. wall When = 3 canRaT be approximately10 and Ste = 70% 0.01, higher the maximum than that of increase pure water in (theNu hNusselt/Nuw = 1.699,number the of highest the high- value within the discussed parameter ranges). As RaT increases, the Nusselt number also temperature wall can be approximately 70% higher than that of pure water ( Nuhw/ Nu = decreases, which is consistent with the observations by Katz [11]. An increase in RaT means Ra that1.699, the the convection highest value effect ofwithin the PCM the particlesdiscussed in parameter the system ranges). is increased. As AsT aincreases, result of the Nusselt number also decreases, which is consistent with the observations by Katz [11]. An

increase in RaT means that the convection effect of the PCM particles in the system is increased. As a result of the enhanced convection, the time for the PCM particles to

Energies 2021, 14, 2857 14 of 17

the enhanced convection, the time for the PCM particles to experience a phase change in the phase change region decreases, which reduces the release and absorption of latent heat, thereby reducing the overall heat transfer capability. As RaT increases, Ste increases, which can induce oscillatory natural convection; in other words, low latent heat can induce oscillatory natural convection under high RaT. This phenomenon is not the focus of this study, but it is worthy of subsequent in-depth investigation.

Table 2. Ratio of the average Nusselt number and the Nusselt number of pure water under different RaT and Ste values.

Nuh/Nuw (Nuh or Nuc)

ci Ste Sb RaT 103 104 105 0.1 0.1 1 1.103 (1.243) 1.111 (2.556) 1.084 (5.188) 0.05 1.201 (1.354) 1.192 (2.741) 1.097 (5.250) 0.01 1.699 (1.915) 1.696 (3.900) 1.587 (7.600) 0.005 2.046 (2.305) (The bold numbers indicate the occurrence of oscillatory natural convection).

5. Conclusions Research on natural convection heat transfer with PCM suspension in rectangular enclosures are gradually increasing; however, there are limited studies on the transient behavior. In this context, the objective of this study focused on the transient and steady-state natural convection of PCM suspensions in a square enclosure, mainly inﬂuenced by the Raleigh number (RaT), Stefan number (Ste), subcooling factor (Sb), and initial concentration (ci). The following parameters were included: aspect ratio of the physical model = 1, ratio 3 5 of the buoyancies caused by temperature and concentration gradients = 1, RaT = 10 –10 , Ste = 0.005–0.1, Sb = 0–1.0, and ci = 0–0.1. The results can be summarized as follows:

1. Under the condition of high ci, high RaT, and low Ste, the transient and the steady- state values of the Nusselt number (Nu) are high, and the time required to reach steady-state is decreased. 2. With increasing Sb, Nu increases ﬁrst then decreases, and there is a local optimum Nu. 3 Within the discussed parameter ranges, when RaT = 10 , the maximum Nuh is 1.20, 4 which occurs when ci = 0.1 and Sb = 0.8; when RaT = 10 , the maximum Nuh is 5 2.89, which occurs when ci = 0.1 and Sb = 0.5; and when RaT = 10 , the maximum Nuh is 5.13, which occurs when ci = 0.1 and Sb = 0.5. 3. As RaT increases, Nuh decreases; moreover, as RaT increases, oscillatory natural convection is induced by higher Stes; in other words, the low latent heat of the PCM can induce oscillatory natural convection under high RaT. This phenomenon of oscillatory natural convection has not yet been mentioned in the literature. The modeling and simulation results in this study are limited to the PCM used, the geometric conﬁguration, and the chosen parameter ranges. An in-depth investigation on the behavior of PCM slurries or suspensions in practical applications is noticeable. This is not what we are exploring in this study, but it is worthy of further consideration.

Author Contributions: C.-J.H. and C.-Y.H. conceived of and designed the model; C.-J.H. and C.-Y.H. performed the simulation works; C.-J.H., C.-Y.H., and C.-M.L. analyzed the data; and C.-J.H. and C.-M.L. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: This study is supported by the Ministry of Sciences and Technologies (MOST) of ROC in Taiwan through Projects Nos. NSC 89-2212-E006-160. Conﬂicts of Interest: The authors declare no conﬂict of interest. Energies 2021, 14, 2857 15 of 17

Nomenclature A Surface area of the PCM particle AR Aspect ratio of the rectangular enclosure, H/W h f rp Bip Biot number of a PCM particle, kp c Mass fraction of PCM particles ca Concentration of species a ci Initial mass fraction of PCM particles c Volumetric fraction of PCM particles, c = ρb c v v ρp C Dimensionless mass fraction of PCM particles, c/ci Cp Specific heat dp Average diameter of PCM particles D Mass diffusivity or diffusion coefficient Dp Dimensionless average particle size of PCM particles, Dp/W E Error 2 Fo Fourier number, αbt/W ∗ 2 Fo Modified Fourier number, αbt/Rd g Acceleration due to gravity h Convection heat transfer coefficient hls Latent heat of fusion H Height of the rectangular enclosure i Grid point at the x-axis j Grid point at the y-axis k Thermal conductivity Le Lewis number, αb/Db N Buoyancy ratio of mass-to-heat transfer, βc ci βT (T h−Tc ) Nu Local Nusselt number Nu Average Nusselt number p Pressure 2 P Dimensionless pressure, p/ρbUc 2Rdum Pe Peclet number, α Pr Prandtl number, νb/αb q00 Heating flux R Radical direction ( − ) 3 Ra Thermal Raleigh number, gβT Th Tc W T νb αb Rd Tube radius rp Radius of a PCM particle Sb Subcooling factor, (Tm − Tc)/(Th − Tm) Ste Stefan number, Cp,b(Th − Tc)/hls C (q00 R /k ) Ste∗ Modified Stefan number, p,b d b b ci hls t Time T Temperature u Velocity in the X-direction um Average flow velocity of the fluid in the tube U Dimensionless velocity in the X-direction, u/Uc ∗ U Modified dimensionless velocity in the X-direction, u/um Uc Characteristic velocity, αb/W v Velocity in the Y-direction V Dimensionless velocity in the Y-direction, v/Uc W Width of a rectangular enclosure X Dimensionless x-coordinate, x/W X∗ Modified dimensionless x-coordinate, x Rd Peb Y Dimensionless y-coordinate, y/W ∆Tls Difference between the melting and freezing points of the PCM Energies 2021, 14, 2857 16 of 17

Greeksymbols α Thermal diffusivity 1 ∂ρ βc Concentration expansion coefficient (− ) ρ ∂ca T,P βT Thermal expansion coefficient ν Kinematic viscosity φ Variable under investigation θ Dimensionless temperature, (T − Tm)/(Th − Tc) (T−Tm) θ∗ Modified dimensionless temperature, 00 q Rd/kb ρ Density ω Vorticity τt Heat transfer time lag of the PCM particles 2 ∀p Cp,b τ Dimensionless heat transfer time lag of the PCM particles, τ /[( ) /αp]( ) t Ap Cp,p ξ` Liquid-phase volume fraction of the PCM particles Ψ Stream function η Dimensionless tube diameter, R/Rd ∀ Volume of a PCM particle ε Supercooling factor, ∆Tls Th−Tc ∗ ∆Tls ε Modified supercooling factor, 00 q Rd/kb εi Comparison criteria Subscripts b Bulk fluid c Cooled wall f Base fluid (water) h Heated wall i Inside ` Liquid phase of PCM m Melting point m1, m2 Different grid points p PCM particle pcm Phase change materials T Properties based on temperature s Solid phase of PCM w Water, wall Superscripts Averaged quantity m Number of iterations n Number of time steps (iteration advances)

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