Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling

2.1 Governing equations of fluid flow and solid-liquid phase change

To simulate the charging and discharging processes of PCMs, the following assumptions are usually made to simplify the mathematical models: (1) the fluid flow of liquid PCMs is Newtonian, laminar, and incompressible with negligible viscous dissipation and (2) the thermophysical properties of PCMs, nanoparticles, fins, and metal foams are constant, except the PCM density ρ, which is a linear function of temperature Tusing the Boussinesq approximation. Based on the above assumptions, the flow of liquid PCMs is governed by the continuity equation and the momentum equation expressed in the Cartesian coordinate as:

where ρis the density; pis the pressure; μis the dynamic viscosity; tis the time; βis the thermal expansion coefficient of PCMs; gis the magnitude of gravitational acceleration; Tis the temperature; Tris the reference temperature; xis the horizontal coordinate; yis the vertical coordinate; zis the coordinate, which is orthogonal with xand ycoordinates; and u, v, and ware the fluid velocities in the x, y, and zdirections, respectively.

The solid-liquid phase change process of PCMs is governed by the enthalpy equation as:

where cpand kare the specific heat and thermal conductivity of PCMs. The enthalpy Hof PCMs is defined as:

where flis the PCM liquid fraction, and Lis the latent heat of PCMs. By calculating the enthalpy Hof PCMs, the liquid fraction fland temperature Tcould be updated by the following equations:

where Hsis the enthalpy of solid state PCMs, Hlis the enthalpy of liquid state PCMs, and Tmis the melting/solidification temperature of PCMs.

2.2 Nanofluid models

For simulating the fluid flow and heat transfer of nanoparticle-enhanced PCMs (NEPCMs), the nanoparticle is assumed to be spherical in shape, so that the Brinkman model and the Maxwell model for nanofluid are valid. In addition, the NEPCMs are considered as continuous media with the thermal dispersion being neglected. Based on this, the effective viscosity of NEPCMs is computed using the Brinkman model as [108]:

where Φis the volume fraction of nanoparticles, μPCMis the dynamic viscosity of pure PCM, and μnfis the dynamic viscosity of NEPCMs. The thermal conductivity of NEPCMs is calculated according to the Maxwell model as [109]:

where kPCM, kp, and knfare the thermal conductivities of pure PCMs, nanoparticles, and NEPCMs, respectively. Furthermore, the density of nanofluid ρnfis computed using interpolation as:

where ρPCMand ρpare the densities of pure PCM and nanoparticles. The heat capacitance of NEPCMs ρcpnfis defined as:

where ρcpPCMis the heat capacitance of pure PCM, and ρcppis the heat capacitance of nanoparticles. The thermal expansion volume of NEPCMs ρβnfis given as:

where ρβPCMand ρβpare the thermal expansion volume of pure PCM and nanoparticles, respectively. The latent heat of NEPCMs ρLnfis computed as:

where ρLPCMis the latent heat of pure PCM. Then, the corresponding enthalpy of NEPCMs Hnfis given as:

2.3 Conjugate heat transfer between PCMs and fins or metal foams

When the extended fins or metal foams are used as the heat transfer enhancement techniques for LHTES unit, the conjugate heat transfer occurs between the PCMs and the fins or metal foams. The heat transfer inside the fins or metal foams is governed by the conduction equation as:

ρs, cps, and ksare the density, specific heat, and thermal conductivity of fins or metal foams, respectively. On the interface of PCMs and fins or metal foams, the coupled Dirichlet-Neumann boundary conditions for conjugate heat transfer should be satisfied:

TPCMand Tsare the temperature of PCMs and fins or metal foams, kPCMis the thermal conductivity of PCMs, and nis the normal unit of the interface. The above boundary conditions could be satisfied in the lattice Boltzmann method automatically by using the harmonic mean value of thermophysical properties of PCMs and fins or metal foams as discussed in the following section. To investigate the melting and solidification processes of PCMs filled with metal foams at pore scale, the morphology of the real metal foam structures could be reconstructed using the quartet structure generation set (QSGS) as shown in Figure 7 [110, 111], where εaveand dpare the porosity and the pore size of metal foams. Besides, the metal foams could also be reconstructed with scanning electron microscopy (SEM).

3.1 The development history of enthalpy-based LBM for solid-liquid phase change

As a mesoscopic numerical approach developed during the past more than two decades, the lattice Boltzmann method (LBM) has become a powerful tool for simulating complex heat transfer and fluid flow problems such as single-phase flows, multiphase flows, turbulence flows, flows in porous media, heat transfer, phase change, and transport in microfluidics [112, 113, 114, 115]. The mesoscopic nature of LBM makes it appropriate for tackling the evolvement of solid-liquid interface during the phase change process. The existing LBM for solid-liquid phase change problems could be generally categorized as follows: (1) the phase-field based method, (2) the enthalpy-based method, and (3) the immersed boundary method. For the phase-field method, an auxiliary parameter, which varies smoothly across the diffusive phase interface, is used to track the solid-liquid interface implicitly [116, 117]. Unfortunately, the extremely finer grids are indispensable in the interfacial region, which increases the computational effort. In addition, Huang and Wu developed the immersed boundary-thermal lattice Boltzmann method for modeling the solid-liquid phase change phenomenon [118]. The melting interface is explicitly tracked by the Lagrangian grids, and the temperature and fluid flow are solved on the Eulerian nodes. However, an interpolation between the Lagrangian nodes and the Eulerian nodes should be carried out using the Dirac delta function, which reduces the computation accuracy. Compared with other methods, the enthalpy-based method becomes attractive for solid-liquid phase change due to its high efficiency and robustness. Jiang et al. first used LBM with enthalpy formulation to investigate the phase change problems [119]. However, the convective heat transfer is not considered in this work. Huber et al. developed a LBM model using double distribution functions to simulate the solid-liquid phase change process with convective heat transfer [120], and the numerical results are analyzed and compared with scaling laws and previous numerical work. Eshraghi and Felicelli developed an implicit LBM scheme to investigate the conduction with phase change [121]. Different from the previous enthalpy-based LBM, the iteration of nonlinear latent heat source term in the energy equation is avoided by solving a linear system of equations. Feng et al. further extended this implicit model with a consideration of natural convection to investigate the melting process of nanoparticle-enhanced PCMs [122]. To improve the computational efficiency, Huang et al. modified the equilibrium distribution function of temperature for enthalpy, so that the iteration of heat source term or solving a linear system of equations is not indispensable [123]. Unfortunately, the stability range of relaxation time for this model is narrow, which limits its applications for real scientific and engineering problems. To handle this drawback, Huang and Wu improved their work by using a multiple-relaxation-time (MRT) collision scheme instead of the single-relaxation-time (SRT) one, so that the numerical stability is highly ameliorated [124]. In addition, the thermal conductivity and the specific heat capacity are decoupled from the relaxation time and equilibrium distribution function, which make this model appropriate for satisfying the Dirichlet-Neumann boundary conditions for conjugate heat transfer. As the model developed by Huang et al. is limited in two-dimensional cases, Li et al. recently developed the SRT and MRT models for axisymmetric and three-dimensional solid-liquid phase change problems [125, 126]. Besides, to speed up the computation, Su and Davidson worked out a mesoscopic scale timestep adjustable non-dimensional LBM [127], and the time steps can be adjusted independent of mesh size by changing the transient mesoscopic Mach number. Furthermore, although the key advantage of LBM is to carry out the pore-scale numerical modeling of heat transfer in porous media, a few lattice Boltzmann models for solid-liquid phase change in porous media at the representative elementary volume (REV) scale are also developed [128, 129, 130]. The classical enthalpy-based MRT lattice Boltzmann model developed by Huang and Wu is reviewed in this section because of its simplicity and wide application by researchers for latent heat thermal energy storage problems.

3.2 Multiple-relaxation-time (MRT) method

3.3 Classical examples for code validation

3.3.1 One-dimensional transient conjugate heat transfer

To validate the capability of MRT LBM for tackling the differences in thermophysical properties, the one-dimensional transient conjugate heat transfer in two regions without phase change is used to compare the numerical results with analytical solutions. Initially, T is equal to 1 in the region A at x>0, and T is equal to 0 in the region B at x<0. The analytical solution for this problem is given as [141]:

TAxt=11+ρCpBkB/ρCpAkA[1+ρCpBkB/ρCpAkAerfx2kAt/ρCpA]E40

TBxt=11+ρCpBkB/ρCpAkAerfc−x2kBt/ρCpBE41

When the aluminum is used for region A while the liquid water is chosen as the material in region B, the comparison between LBM results and analytical solutions is presented in Figure 8. A good agreement is observed, and the maximum L2 error with 200 × 200 lattice grids is only 6.7983×10−5[142].

3.3.2 One-dimensional melting by conduction

In order to verify the MRT LBM for solid-liquid phase change phenomenon, the one-dimensional melting by conduction at a constant phase change temperature Tmis simulated. Initially, the substance is uniformly solid at a temperature T0(T0<Tm). The melting process begins at time t=0when the temperature of the left wall is at a high temperature of Th(Th> Tm). Then, the analytical solution for the temperature in this problem is given as [143]:

Txt=Th−Th−Tmerfλerfx2klt/ρcp,l,0≤x≤Xit,liquidE42

Txt=T0+Tm−T0erfcλ/Racerfcx2kst/ρcp,s,x>Xit,solidE43

where kland ksare the thermal conductivities of liquid PCM and solid PCM, respectively. The parameter Rac=kst/ρcp,sklt/ρcp,lis the ratio of thermal diffusivity between the solid and liquid phases of PCM, and the location of phase interface Xiis calculated as:

Xi=2λklt/ρcp,lE44

The parameter λis the root of the transcendental equation:

Stelexpλ2erfλ−StesRacexpλ2/Racerfcλ/Rac=λπE45

where the Stefan numbers, Steland Stes, are defined as:

Stel=cp,lTh−TmL,Stes=cp,sTm−T0LE46

The comparison of temperature Tbetween the analytical solutions and the SRT or MRT LBM with different relaxation times is presented in Figure 9. It could be observed that the numerical diffusion exists for the small τlclose to0.5and large τl>2relaxation times when the SRT model is applied. However, the numerical diffusion is highly reduced once the MRT model with a “magic” parameter relation shown in Eq. (36) is used.

3.3.3 Two-dimensional melting by convection

The natural convection with melting in a square cavity heated from the side wall is usually used to validate the code for solid-liquid phase change. First, the solid-liquid phase change with convection in a cavity at the Rayleigh number Ra=25000, the Prandtl number Pr=0.02, and the Stefan number Ste=0.01is compared with the results by Mencinger [144]. The average Nusselt number Nuaveat the hot wall in terms of the Fourier numbers Fois plotted in Figure 10(a), and the melting interface positions at Fo=10and Fo=20are shown in Figure 10(b). In addition, the average melting fraction flis also presented in Figure 10(c). It is obvious that the current LBM results are consistent with the work of Mencinger [142]. Besides, for the cases of Pr>1, the MRT LBM for natural convection with melting in cavity could be calibrated by the scaling laws and correlations derived by Jany and Bejan [145]. The average Nusselt number Nuaveat hot wall is given as:

Nuave=2FoSte−1/2+0.33Ra1/4−2FoSte−1/21+0.0175Ra3/4FoSte3/2−2−1/2E47

As shown in Figure 10(d), the average Nusselt number Nuaveat Pr=6.1989and Ste=0.1agrees well with the results of scaling law correlations at different Rayleigh number Ra.

3.4 GPU acceleration

The characteristic of highly parallel nature is a significant advantage of LBM over other traditional macroscopic numerical methods. In the CUDA programming platform, the CPU and GPU work as the host and the devices, respectively, as shown in Figure 11 [146]. It means that the parallel tasks are executed on GPU, while the CPU is responsible for the initial conditions and all the sequential commands. First, the initial conditions are set up in the host memory, and then the data are moved to the memory of GPU. The threads are grouped into blocks, which are the component of grids as displayed in Figure 11. For instance, in the real simulation, if the lattice grid number is (Mx, My, Mz) and the block size is (Nx, Ny, Nz), the corresponding grid size is Mx/NxMy/NyMz/Nz. Furthermore, a kernel is a function, which is executed on the concurrent threads on GPU. The collision step, streaming step, tackling of boundary conditions, and computation of macroscopic variables are completed in different kernels. It is important to note that all the kernels should be synchronized between CPU and GPU. Finally, the data on the GPU should be copied to the host memory of CPU for printing at any specific time when the results are needed. During the recent years, LBM is demonstrated to be appropriate for GPU computing [147, 148, 149, 150, 151], and it has been applied to solve several different physical problems [152, 153, 154, 155].

4.1 Enhancement of PCM performance with fins

As presented in the previous sections, the extended fins could be used to increase the heat transfer depth in LHTES system and accelerate the energy storage efficiency. In the recent years, LBM is applied by several researches to study the solid-liquid phase change of PCMs with extended fins [142, 156, 157]. Jourabian et al. studied the melting process of PCM in a cavity with a horizontal fin heated from the sidewall by enthalpy-based D2Q5 LBM [156]. The results indicated that adding a fin enhances the melting rate for all positions and different lengths compared with the LHTES cavity without fin. They also found that although varying the position of the fin from the bottom to the middle has a negligible effect on melting rate, the melting time is increased once the fin is mounted on the top of LHTES cavity. Talati and Taghilou used an implicit LBM to study the PCM solidification in a rectangular finned container [157]. It was found that the optimum aspect ratio of container for solidification equals to 0.5, and changing the fin material from aluminum to copper has no significant influence on the solidification rate. Ren and Chan applied enthalpy-based MRT LBM to investigate the PCM melting performance in an enclosure with internal fins and finned thick walls with GPU acceleration [142]. The transient PCM melting process with different number of fins at the Fourier number Fo=0.15is shown in Figure 12 in terms of temperature contours, liquid fraction, and streamlines. It could be found that the PCM melting rate is obviously enhanced by adding more internal fins in the cavity. However, the energy storage capacity of LHTES system is reduced when the number of internal fins increases, so that the appropriate fin configuration and number should be designed for engineering applications. They found that using a less number of longer fins is more effective than applying shorter fins for enhancing the thermal performance of PCMs. Besides, compared with the LHTES cavity with horizontal fins heated from side walls, the LHTES enclosure using vertical fins heated from the bottom surface has a better charging rate. From the above researches, it should be concluded that the enthalpy-based LBM is successful for simulating the conjugate heat transfer with solid-liquid phase change for melting and solidification of PCMs accelerated with fins.

4.2 Nanoparticle-enhanced PCM

The nanoparticles are commonly used to ameliorate the low thermal conductivity of most PCMs. With some appropriate assumptions as presented in Section 2, the thermal performance of PCMs with nanoparticles could be modeled using enthalpy-based LBM. Jourabian et al. studied the convective melting process of Cu-nanoparticle enhanced water-ice in a cylindrical-horizontal annulus [158, 159]. It was demonstrated that the melting rate of NEPCM is accelerated due to the improvement of thermal conductivity and the reduced latent heat. When the heated cylinder located in the bottom section of the annulus, the increment effect on NEPCM melting rate by adding more nanoparticles decreases because of the augmentation of NEPCM viscosity, which weakens the convective heat transfer. Feng et al. investigated the melting of water (ice)-copper nanoparticle NEPCM in a bottom-heated rectangular cavity by treating the latent heat source term with an implicit scheme [122]. They also found that the heat transfer rate of NEPCM increases with respect to the increment of nanoparticle volume fractions. However, the energy storage rate is the most significant parameter for a LHTES unit. Although adding nanoparticles into PCMs could increase their thermal conductivity, it increases the viscosity of PCM, which weakens the convective heat transfer. Due to this reason, the energy storage rate of LHTES system may even decrease with the increasing nanoparticle volume fractions, especially for the case with large temperature gradient. On the other hand, the increment of nanoparticle volume fraction decreases the energy storage capacity of LHTES unit, so that the energy storage rate could also be affected. Under this circumstance, more future research attentions should be paid on the influences of nanoparticles on the energy storage rate of NEPCM. In addition, the solid-liquid phase change model of NEPCM using LBM could be extended to study the charging and discharging of NEPCM under hybrid heat transfer enhancement techniques such as nanoparticle-fin or nanoparticle-metal foam combinations.

4.3 PCM filled with metal foams

Due to its intersected and connected heat transfer channels, metal foam is inserted into the LHTES system for enhancing the melting and solidification rates of PCMs. The numerical modeling of PCM solid-liquid phase change phenomenon filled in metal foams using enthalpy-based LBM could be categorized as the representative elementary volume (REV) scale modeling [128, 129, 130, 160] and pore-scale modeling [111, 161]. Tao et al. investigated the performance of metal foams/paraffin composite PCM in a LHTES cavity using LBM at REV scale [160]. They found that increasing the metal foam PPI (number of pores per inch) could enhance the conduction heat transfer, while the convective heat transfer is weakened. In addition, although decreasing the metal foam porosity could accelerate the PCM melting rate, the energy storage density of LHTES unit is dramatically reduced. Due to the above two tradeoffs, the optimum metal foam structure with the porosity of 0.94 and PPI of 45 is highly recommended. However, the REV scale modeling requires the use of some empirical relations, and the influences of metal foam morphology on energy storage rate are difficult to be analyzed. As a comparison, with the advantage of LBM for tackling complex boundary conditions, pore-scale modeling without using any empirical equations achieves the detailed fluid flow and heat transfer information inside the metal foams, so that the energy storage efficiency of LHTES unit could be further optimized. Ren et al. investigated the effects of metal foam characteristics on the PCM melting rate through pore-scale LBM modeling [111]. The transient PCM melting process inside the metal foams of different pore sizes at the Fourier number Fo=0.04is plotted with respect to temperature field, melting interface, and fluid velocity vectors in Figure 13. It could be observed that the charging rate of PCM is accelerated when the pore size decreases from dp=1mmto dp=0.75mmat a porosity of εave=0.95. Furthermore, the temperature field in the melted region of LHTES with smaller pore size dp=0.75mmis found to be more uniform than that of LHTES with pore size dp=1.0mm, which means that the thermal conductivity and its corresponding conduction heat transfer in LHTES unit are improved with a decreasing pore size (increasing PPI). Besides, they also concluded that an appropriate metal foam porosity should be chosen in the real engineering applications in order to balance the PCM melting speed and the energy storage density of LHTES unit. By reconstructing the microstructure of metal foam with X-ray computed tomography, Li et al. investigated the solid-liquid phase change phenomenon of PCM inserted with metal foams under different gravitational acceleration conditions [161]. The results indicated that the transition of the dominant heat transfer mechanism from convection to conduction occurs when the gravity gradually decreases. Due to the attenuated convection effect with decreasing gravity, the PCM melting development is dramatically hindered. Besides, they also concluded that the decreasing metal foam porosity could enhance the effective thermal conductivity of LHTES unit because of the extended heat transfer area. However, the above pore-scale modeling of PCM charging and discharging processes enhanced by metal foams is limited in two-dimensional cases. The three-dimensional pore-scale modeling, which is indispensable for optimizing the complicated metal foam structures, should be carried out in the future work.

4.4 PCM with heat pipes

The heat pipes are used to transfer heat between PCMs and heat transfer fluid (HTF), so that the charging and discharging of LHTES system could be accelerated. The configuration, arrangement, and number of heat pipes in the LHTES unit are essential for the PCM melting and solidification speed. Luo et al. applied LBM to study the convection melting in complex LHTES system with heat tubes [162]. The effects of inner heat pipe arrangement on PCM melting process are illustrated in Figure 14 with respect to temperature, flow, and phase fields at the Fourier number Fo=3, the Prandtl number Pr=0.2, the Stefan number Ste=0.02, and the Rayleigh number Ra=5×104. For the case using centrosymmetric inner heat pipes, the conduction heat transfer is dominant because the inner heat pipe is surrounded by solid PCM at the melting temperature, so that its melting rate is faster than the LHTES system with inline or staggered heat tubes as displayed in Figure 15. In addition, there is no obvious difference between the melting rates of LHTES systems using inline or staggered heat pipes. Although using heat pipe individually could enhance the thermal performance of LHTES to some extent, other heat transfer enhancement techniques are usually coupled with heat pipes such as fins or nanoparticles to further improve the PCM charging and discharging rate.

4.5 PCM with hybrid heat transfer enhancement techniques

In this section, the melting and solidification of PCMs enhanced using combined heat transfer enhancement techniques modeled by LBM are discussed, and the effectiveness of different approaches for improving the heat transfer capability of LHTES unit is compared. Huo and Rao carried out an investigation of NEPCM solid-liquid phase change process in a cavity with a heat pipe and separate plates using LBM [163]. It was found that the NEPCM of the case with separate located in the middle of cavity melts fastest because of the weakened heat accumulation. The results also showed that when the location of separate is less than 0.3, the melting rate of NEPCM is even slowed down due to the heat accumulation around the separate plate. Gao et al. developed an enthalpy-based MRT LBM model with a free parameter in the equilibrium distribution function for solid-liquid phase change in porous media and conjugate heat transfer with high-computational efficiency and stability [164]. Then, they investigated the PCM melting process in porous media with a conducting fin, and the results indicated that the heat transfer rate could be further improved by adding a fin into the porous medium. It was also observed that the vertical position of the fin has no remarkable impact on the PCM melting speed when there exists porous media. Jourabian et al. investigated the constrained ice melting around a cylinder using three heat transfer enhancement techniques by LBM [165]. They pointed out that adding nanoparticles may increase the dynamic viscosity of the base PCM, which has a negative effect on natural convective heat transfer. Besides, the tradeoff between consolidating the conduction heat transfer and weakening the convective heat transfer through decreasing the metal foam porosity needs further attention and investigation. Ren et al. completed a pore-scale comparative study of nanoparticle-enhanced PCM melting in a heat pipe–assisted LHTES unit with metal foams [166]. The melting process of NEPCM in a LHTES cavity filled with metal foams using a heat pipe at the Fourier number Fo=0.06, pore size dp=1mm, and heat pipe radius R=2mmis shown in Figure 16. Although different combinations of nanoparticle volume fraction Φ and metal foam porosity εaveare used for the LHTES unit, it should be noted that the volume fraction of pure PCM is kept at 90%, so that the energy storage capacity is unchanged for a fair comparison. It could be observed that the NEPCM with less nanoparticle volume fraction and lower metal foam porosity melts faster than that with more nanoparticles in the higher porosity metal foam. This finding indicated that using metal foams is more effective than adding nanoparticles for enhancing the charging rate of NEPCMs. They also found that there exists an optimum heat pipe radius for achieving the best energy storage rate in LHTES unit. As discussed in the above sections, LBM is demonstrated to be appropriate for simulating the charging and discharging processes of PCMs in LHTES system with different kinds of heat transfer enhancement technologies. Under this circumstance, the enthalpy-based LBM will definitely play a more significant role in the future research of thermal energy storage using PCMs.