Phase-change materials to improve solar panel's performance

doi:10.1016/j.enbuild.2013.02.059

Energy and Buildings 能源与建筑

Volume 62, July 2013, Pages 59-67
第 62 卷，2013 年 7 月，第 59-67 页

https://doi.org/10.1016/j.enbuild.2013.02.059 Get rights and content 获取权利和内容

Highlights 亮点

• -
A volume force was added to the buoyancy term in the Navier–Stokes’ momentum conservation equation.
在纳维-斯托克斯动量守恒方程的浮力项中加入了体积力。
• -
Using PCM kept panel temperature under 40 °C for 80 min under radiation of 1000 W/m².
在 1000 W/m ² 的辐射条件下，使用 PCM 使面板温度在 80 分钟内保持在 40 °C 以下。
• -
Adding cooling fins does not always help maintaining the panels’ temperature low.
增加散热片并不总是有助于保持电池板的低温。

Abstract 摘要

High operating temperatures induce a loss of efficiency in solar photovoltaic and thermal panels. This paper investigates the use of phase-change materials (PCM) to maintain the temperature of the panels close to ambient. The main focus of the study is the computational fluid dynamics (CFD) modeling of heat and mass transfers in a system composed of an impure phase change material situated in the back of a solar panel (SP). A variation of the enthalpy method allows simulating the thermo-physical change of the material properties. The buoyancy term in Navier–Stokes’ momentum conservation equation is modified through an additional term which forces the velocity field to be non-existent when the PCM is solid. For validation purposes, isotherms and velocity fields are calculated and compared to those from an experimental set-up. Results show that adding a PCM on the back of a solar panel can maintain the panel's operating temperature under 40 °C for 80 min under a constant solar radiation of 1000 W/m².
工作温度过高会导致太阳能光伏板和热能板的效率降低。本文研究了如何使用相变材料 (PCM) 将太阳能电池板的温度保持在接近环境温度的水平。研究的重点是对太阳能电池板（SP）背面的不纯相变材料组成的系统中的热量和质量传递进行计算流体动力学（CFD）建模。焓法的一种变体可以模拟材料特性的热物理变化。纳维-斯托克斯动量守恒方程中的浮力项通过一个附加项进行了修改，当 PCM 为固体时，该附加项迫使速度场不存在。为了验证，计算了等温线和速度场，并与实验装置中的等温线和速度场进行了比较。结果表明，在 1000 W/m ² 的恒定太阳辐射条件下，在太阳能电池板背面添加 PCM 可以将电池板的工作温度保持在 40 °C 以下 80 分钟。

Keywords 关键词

Solar panel

Operating temperature

Phase change material

太阳能电池板工作温度相变材料

Nomenclature 术语

thermal diffusivity, m² s⁻¹
热扩散率，m ² s ⁻¹

B₀

melt function 熔化功能

B₁

smoothed melt function 平滑熔融函数

C_p

specific heat, J kg⁻¹ K⁻¹
比热，J kg ⁻¹ K ⁻¹

smoothed delta Dirac function
平滑德尔塔狄拉克函数

F_a

added buoyancy force, N m⁻³
附加浮力，N m ⁻³

F_b

buoyancy force from Boussinesq approximation, N m⁻³
布森斯克近似浮力，N m ⁻³

gravitational constant, m s⁻²
引力常数，m s ⁻²

h_e

outdoor convection heat transfer coefficient, W m⁻² K⁻¹
室外对流传热系数，W m ⁻² K ⁻¹

h_i

indoor convection heat transfer coefficient, W m⁻² K⁻¹
室内对流传热系数，W m ⁻² K ⁻¹

PCM container height, m PCM 容器高度（米

thermal conductivity, W m⁻¹ K⁻¹
导热系数，W m ⁻¹ K ⁻¹

characteristic length, m 特征长度，米

L_F

latent heat of fusion, J kg⁻¹
聚变潜热，J kg ⁻¹

PCM container internal width, m
PCM 容器内部宽度，米

mass, kg 质量，公斤

pressure, Pa 压力，Pa

mass Peclet number 质量佩克莱特数

Rayleigh number 雷利数

temperature, K 温度，K

T_m

mean melt temperature, K 平均熔体温度，K

velocity, m/s 速度，米/秒

ΔT

half range of melt temperatures, K
熔体温度的一半范围，K

Δx

maximum length of mesh cells, m
网格单元的最大长度，米

Greek letters 希腊字母

coefficient of thermal expansion, K⁻¹
热膨胀系数，K ⁻¹

density, kg m⁻³ 密度，千克米 ⁻³

dynamic viscosity, Pa s
动态粘度，Pa s

kinematic viscosity, m² s⁻¹
运动粘度，m ² s ⁻¹

Subscripts 下标

liquid 液态

liquid phase 液相

solid 固体

solid phase 固相

1. Introduction 1.导言

The efficiency of solar panels depends on the three factors: the intensity of the solar radiation flux, the quality of the semiconductor in use, and the operating temperature of the semiconductor cell. The variations of solar radiation cannot be controlled. Therefore, the ongoing research focuses either on new material like copper, indium diselenium, cadmium tellurium and chalcopyrites, or on maintaining low operating temperatures. For photovoltaic (PV) panels, high operating temperatures create a drop in the conversion rate of about 0.5%/°C over the nominal cell operating temperature of 25 °C [1], as defined by the industry standard STC (Standard Test Conditions). In summer, panel's temperature typically ranges from 40 to 70 °C which makes a 7.5–22.5% drop in the conversion rate. In the same way, the efficiency of solar thermal panels decreases mainly because of radiation losses when the operating fluid temperature is above ambient.
太阳能电池板的效率取决于三个因素：太阳辐射强度、使用的半导体质量和半导体电池的工作温度。太阳辐射的变化是无法控制的。因此，正在进行的研究要么侧重于新材料，如铜、二硒铟、碲镉和黄铜矿，要么侧重于保持较低的工作温度。根据行业标准 STC（标准测试条件）的规定，对于光伏（PV）电池板而言，高工作温度会导致转换率下降，比标称电池工作温度 25°C 降低约 0.5%/°C[1]。在夏季，电池板的温度通常在 40 至 70 °C之间，这使得转换率下降了 7.5-22.5%。同样，当工作液体温度高于环境温度时，太阳能热板的效率也会下降，这主要是由于辐射损失造成的。

To lower the operating temperature, one can either improve the free cooling on the back of the panel using natural or forced convection, or try to absorb the excess heat by modifying the panel's architecture. The latter solution includes the use of PCMs situated on the back of solar panels. PCMs are materials that undergo reversible transition of phase depending on their temperature. They absorb or reject heat in the process. The numerical simulation of PCMs behavior has been extensively researched. Voller et al. [2] presented an enthalpy method for coupled convection and diffusion phase change. They later applied their technique to the melting of a pure metal [3]. Hu and Argylopoulos [4] made a review of the existing mathematical methods to model PCMs melting or solidification. Bertrand et al. [5] presented a model for PCM melting driven only by natural convection.
为了降低工作温度，我们可以利用自然对流或强制对流改善电池板背面的自由冷却，或者尝试通过修改电池板的结构来吸收多余的热量。后一种解决方案包括在太阳能电池板背面使用 PCM。PCM 是一种可根据温度进行可逆相变的材料。在此过程中，它们会吸收或排出热量。人们对 PCMs 行为的数值模拟进行了广泛研究。Voller 等人 [2] 提出了一种用于耦合对流和扩散相变的焓法。后来，他们将该技术应用于纯金属的熔化[3]。Hu 和 Argylopoulos [4] 综述了现有的 PCM 熔化或凝固建模数学方法。Bertrand 等人[5] 提出了一种仅由自然对流驱动的 PCM 熔化模型。

The specific use of PCMs for thermal energy storage (TES) was reviewed by Setterwall [6] and Zalba et al. [7]. In buildings, PCMs are being studied for free cooling or to enhance the building's thermal inertia. Research on this subject mainly deals with PCMs incorporated in wallboards [8], [9], roofs [10], windows [11], and ventilation heat exchangers [12]. A review on these applications can be found in Tyagi and Buddhi [13], Zhang et al. [14], and Raj and Velraj [15]. Regarding the materials used, paraffin waxes are very good candidates as PCMs for TES applications in buildings because of their low temperature of melting and their high heat of fusion. Rubitherm GmBH [16] technically graded the paraffin wax RT25 by the use of thermo sensors and differential scanning calorimetry analysis.
Setterwall [6] 和 Zalba 等人 [7] 综述了 PCM 在热能储存 (TES) 中的具体用途。在建筑物中，人们正在研究将 PCM 用于自由冷却或增强建筑物的热惯性。这方面的研究主要涉及在墙板 [8]、[9]、屋顶 [10]、窗户 [11] 和通风热交换器 [12] 中加入 PCM。Tyagi 和 Buddhi [13]、Zhang 等人 [14] 以及 Raj 和 Velraj [15]对这些应用进行了综述。在使用的材料方面，石蜡因其熔化温度低、熔融热高，非常适合作为 PCM 用于建筑物中的 TES。Rubitherm GmBH [16] 通过使用热传感器和差示扫描量热分析法对石蜡 RT25 进行了技术分级。

Only a few studies have been specifically devoted to passive cooling of solar panels by SP/PCM architectures. The hypothesis driving the research is simple: when the panel's temperature rises, the excess heat must be absorbed until the PCM has completely melted. When the panel's temperature decreases, the solidification of the PCM should provide additional heat for the operating liquid in solar thermal panels, provide heat to the building or act as an insulation material. The SP/PCM solution is expected to be very useful for roof or facade integrated panels where space for ventilation is limited.
只有少数研究专门针对利用 SP/PCM 结构对太阳能电池板进行被动冷却。推动这项研究的假设很简单：当太阳能电池板的温度升高时，多余的热量必须被吸收，直到 PCM 完全融化。当面板温度降低时，PCM 的凝固将为太阳能热板中的工作液体提供额外的热量，为建筑物提供热量或作为隔热材料。在通风空间有限的情况下，SP/PCM 解决方案预计将非常适用于屋顶或外墙集成面板。

Huang et al. [17] studied the melting of PCMs in an aluminum container submitted to a solar radiation of 750–1000 W/m². They used a finite volume model to analyze both the heat transfer diffusion and the Navier–Stokes equations. They later included cooling fins in the tank to improve the PCM bulk thermal conductivity [18], [19]. They found that the temperature rise in the system could be reduced by more than 30 °C for 130 min. Cellura et al. [20] analyzed the same architecture using a finite element partial differential equation (PDE) solver. However, they considered the PCM as pure, meaning that the PCM melting temperature is unique and does not change while the PCM is still melting. This property is not valid for most commercial PCMs which are generally mixtures of several different materials. By analyzing only the heat transfer diffusion equation, they showed that a PCM with a melting temperature between 28 °C and 32 °C can improve the energy conversion efficiency by around 20% in summer time. Jay et al. [21] experimentally studied a layout where PCM were contained in a honeycomb grid to improve conduction in the container. They showed that after 6 hours and 30 min of testing under an artificial insulation of 800 W/m² on real PV panels, the temperature of a PV/PCM system was still lower than that of a single panel, with a mean temperature difference of 24 °C. They also found that the panel's temperature drop using a PCM with a melt temperature at 27 °C was higher than using a PCM with a melt temperature at 45 °C.
Huang 等人[17] 研究了在太阳辐射为 750-1000 W/m ² 的铝制容器中 PCM 的熔化情况。他们使用有限体积模型分析了传热扩散和纳维-斯托克斯方程。后来，他们在水箱中加入了冷却鳍片，以提高 PCM 体积导热率 [18]、[19]。他们发现，在 130 分钟内，系统中的温升可降低 30 °C 以上。Cellura 等人 [20] 使用有限元偏微分方程 (PDE) 求解器分析了相同的结构。不过，他们认为 PCM 是纯的，这意味着 PCM 熔化温度是唯一的，在 PCM 熔化时不会发生变化。这一特性对大多数商用 PCM 都不适用，因为它们通常是几种不同材料的混合物。通过仅分析传热扩散方程，他们发现熔化温度介于 28 °C 和 32 °C 之间的 PCM 在夏季可将能量转换效率提高约 20%。Jay 等人[21] 对一种布局进行了实验研究，这种布局将 PCM 包含在蜂巢网格中，以改善容器中的传导。他们的研究表明，在 800 W/m ² 的人工隔热条件下，对真正的 PV 面板进行 6 小时 30 分钟的测试后，PV/PCM 系统的温度仍低于单个面板，平均温差为 24 °C。他们还发现，使用熔融温度为 27 °C 的 PCM 时，面板的温降高于使用熔融温度为 45 °C 的 PCM 时。

In this study, we consider the same geometry and same PCM as [17], except for the experimental validation. The transient conduction and convection heat transfer as well as the Navier–Stokes equations are simultaneously resolved in the PCM domain using a finite element model on a fixed grid. The buoyancy term in Navier–Stokes’ momentum conservation equation is modified through an additional term to force the velocity field to be zero when the PCM is solid. This scheme is validated using an experimental set-up. The model is then used for a parametric study of the SP/PCM architecture performances.
在本研究中，除实验验证外，我们考虑了与 [17] 相同的几何形状和相同的 PCM。在 PCM 域中，使用固定网格上的有限元模型同时求解瞬态传导和对流热传递以及纳维-斯托克斯方程。通过附加项修改 Navier-Stokes 动量守恒方程中的浮力项，以迫使 PCM 为固体时的速度场为零。通过实验装置对该方案进行了验证。然后利用该模型对 SP/PCM 结构的性能进行参数研究。

2. Methodology 2.方法论

2.1. Numerical case description
2.1.数值案例描述

The geometry of the model is presented in Fig. 1. The aluminum plate and fin widths are 4 mm.The thermo-physical properties of the simulated materials are presented in Table 1. The thermal boundary conditions are as follows: On the left plate, an inward heat flux E = 1000 W/m² is applied to mimic solar radiation. The same plate also receives a convection heat flux due to the difference between the plate temperature and the external air temperature which is imposed as T_e = 20 °C. The value of the outdoor convection heat transfer coefficient is h_e = 10 W/m² K. Similarly, a convection heat flux is applied on the right aluminum plate with T_i = T_e = 20 °C and h_i = 5 W/m² K. A symmetry boundary condition is applied on the top and bottom sides of the model.
模型的几何形状如图 1 所示。模拟材料的热物理性质见表 1。热边界条件如下：在左侧板上，施加内向热通量 E = 1000 W/m ² 以模拟太阳辐射。同一板块还接收到因板块温度与外部空气温度之差而产生的对流热通量，即 T _e = 20 °C。室外对流传热系数的值为 h _e = 10 W/m ² K。同样，在右侧铝板上也施加了对流热通量，T = T _e = 20 °C，h = 5 W/m ² K。

Table 1. Thermo-physical properties of RT25 [16] and aluminum.
表 1.RT25 [16] 和铝的热物理性质。

Empty Cell	C_p	k	ρ
Solid RT25 固体 RT25	1800	0.19	785
Liquid RT25 液体 RT25	2400	0.18	749
Aluminum 铝	903	211	2675
Constant properties of RT25: RT25 的恒定特性：
L_F: 232,000 J kg⁻¹
T_m: 26.6 °C
β: 1e−3 K⁻¹ β:1e-3 K ⁻¹
μ: 1.7976e−3 m² s⁻¹ μ:1.7976e-3 m ² s ⁻¹

A no slip boundary condition is applied on all internal surfaces of the PCM container.
PCM 容器的所有内表面都采用了无滑移边界条件。

2.2. Mathematical model 2.2.数学模型

2.2.1. Modeling heat transfers
2.2.1.传热建模

Over the front plate surface, we considered conduction, convection and radiation heat transfers as shown in Eq. (1). Long wave radiation with the sky was neglected in the model.(1)

ρ C_{p} \frac{\partial T}{\partial t} = - k \frac{\partial T}{\partial x} + h_{e} (T_{e} - T) + α E (t)

where α is the aluminum thermal absorptivity and E the solar radiation intensity. The heat transfers diffusion equation applies over the PCM, the air layer and the aluminum domains:(2)

ρ C_{p} \frac{\partial T}{\partial t} + \nabla \cdot (- k \nabla T) + ρ C_{p} \vec{u} \cdot \nabla T = 0

如公式 (1) 所示，我们考虑了前板表面的传导、对流和辐射传热。模型中忽略了对天空的长波辐射。 (1)

ρ C_{p} \frac{\partial T}{\partial t} = - k \frac{\partial T}{\partial x} + h_{e} (T_{e} - T) + α E (t)

其中，α 是铝的热吸收率，E 是太阳辐射强度。热量传递扩散方程适用于 PCM、空气层和铝域： (2)

ρ C_{p} \frac{\partial T}{\partial t} + \nabla \cdot (- k \nabla T) + ρ C_{p} \vec{u} \cdot \nabla T = 0

The velocity field u in Eq. (2) is given by Navier–Stokes equations for incompressible fluids. To model the changes in PCM RT25 thermo-physical properties occurring during the phase transition, we define function B₀ as the liquid fraction in the PCM domain. Let T_m be the mean melt temperature and ΔT the half range of melt temperatures:(3)

B_{0} (T) = \{\begin{array}{l} 0, & T < (T_{m} - Δ T) \\ \frac{T - T_{m} + Δ T}{2 Δ T}, & (T_{m} - Δ T) \leq T < (T_{m} + Δ T) \\ 1, & T > (T_{m} + Δ T) \end{array}

公式 (2) 中的速度场 u 由不可压缩流体的纳维-斯托克斯方程给出。为了模拟 PCM RT25 在相变过程中发生的热物理性质变化，我们将函数 B ₀ 定义为 PCM 域中的液体分数。假设 T _m 为平均熔体温度，ΔT 为熔体温度的半范围： (3)

B_{0} (T) = \{\begin{array}{l} 0, & T < (T_{m} - Δ T) \\ \frac{T - T_{m} + Δ T}{2 Δ T}, & (T_{m} - Δ T) \leq T < (T_{m} + Δ T) \\ 1, & T > (T_{m} + Δ T) \end{array}

Eq. (3) shows that B₀ is zero when the PCM is in solid and 1 when it is in liquid phase. B₀ linearly grows from zero to 1 between the two states. To ensure second order continuous differentiability of the liquid fraction over the temperature domain, B₀(T) is approximated by a second order differentiable function B₁. B₁(T) is the sixth-degree polynomial whose seven coefficients are calculated using the following conditions:(4)

\{\begin{array}{l} B_{1} (T_{m} - Δ T) = 0; {B^{'}}_{1} (T_{m} - Δ T) = 0; {B^{″}}_{1} (T_{m} - Δ T) = 0; B_{1} (T_{m}) = 0.5 \\ B_{1} (T_{m} - Δ T) = 1; {B^{'}}_{1} (T_{m} - Δ T) = 0; {B^{″}}_{1} (T_{m} - Δ T) = 0 \end{array}

where

{B^{'}}_{1}

and

{B^{″}}_{1}

are B₁(T) first and second derivatives. Function B₁(T) yields the same values as function B₀(T). The only difference is that B₁ is a second order continuously differentiable function. This scheme helps numerical convergence. B₁ is used to model the changes in the PCM thermo physical properties as follows:(5)

ρ (T) = ρ_{s o l i d} + (ρ_{l i q u i d} - ρ_{s o l i d}) \cdot B_{1} (T)

(6)

k (T) = k_{s o l i d} + (k_{l i q u i d} - k_{s o l i d}) \cdot B_{1} (T)

where ρ is the density and k the thermal conductivity of the PCM. The modeling of the specific heat includes an additional term representing the latent heat of fusion absorbed during the melting process:(7)

C_{p} (T) = C_{p s o l i d} + (C_{p l i q u i d} - C_{p s o l i d}) \cdot B_{1} (T) + L_{F} \cdot D (T)

where(8)

D (T) = \frac{e^{(- T {(T - T_{m})}^{2}) / (Δ T^{2})}}{\sqrt{π \cdot Δ T^{2}}}

公式 (3) 表明，当 PCM 处于固态时，B ₀ 为零，而当它处于液态时，B ₀ 为 1。在这两种状态之间，B ₀ 从 0 线性增长到 1。为确保液相分数在温度域内的二阶连续可微分性，B ₀ (T) 由二阶可微分函数 B ₁ 逼近。B ₁ (T) 是六度多项式，其七个系数通过以下条件计算得出： (4)

\{\begin{array}{l} B_{1} (T_{m} - Δ T) = 0; {B^{'}}_{1} (T_{m} - Δ T) = 0; {B^{″}}_{1} (T_{m} - Δ T) = 0; B_{1} (T_{m}) = 0.5 \\ B_{1} (T_{m} - Δ T) = 1; {B^{'}}_{1} (T_{m} - Δ T) = 0; {B^{″}}_{1} (T_{m} - Δ T) = 0 \end{array}

其中

{B^{'}}_{1}

和

{B^{″}}_{1}

是 B ₁ (T) 的一阶导数和二阶导数。函数 B ₁ (T) 与函数 B ₀ (T) 得到的值相同。唯一不同的是，函数 B ₁ 是二阶连续可微分函数。这种方案有助于数值收敛。B ₁ 用于模拟 PCM 热物理性质的变化，具体如下： (5)

ρ (T) = ρ_{s o l i d} + (ρ_{l i q u i d} - ρ_{s o l i d}) \cdot B_{1} (T)

(6)

k (T) = k_{s o l i d} + (k_{l i q u i d} - k_{s o l i d}) \cdot B_{1} (T)

其中，ρ 是 PCM 的密度，k 是 PCM 的导热系数。比热建模包括一个附加项，代表熔化过程中吸收的熔化潜热： (7)

C_{p} (T) = C_{p s o l i d} + (C_{p l i q u i d} - C_{p s o l i d}) \cdot B_{1} (T) + L_{F} \cdot D (T)

其中 (8)

D (T) = \frac{e^{(- T {(T - T_{m})}^{2}) / (Δ T^{2})}}{\sqrt{π \cdot Δ T^{2}}}

Function D is a smoothed Delta Dirac function which is zero everywhere except in interval [T_m − ΔT, T_m + ΔT]. It is centered on T_m and its integral is 1. Its main role is to distribute the latent heat equally around the mean melting point.
函数 D 是一个平滑的 Delta Dirac 函数，除区间 [T _m - ΔT, T _m + ΔT] 外，在其他地方均为零。它以 T _m 为中心，积分为 1。它的主要作用是将潜热平均分配到平均熔点附近。

2.2.2. Modeling of mass and momentum transfers for PCM domain
2.2.2.PCM 域的质量和动量传递模型

We assumed that the PCM in the liquid phase is a Newtonian fluid. The mass, momentum and energy conservation equations were resolved simultaneously with the heat transfer diffusion equation. However, to model the phase transition, the momentum conservation equation was modified as follows:(9)

ρ \frac{\partial \vec{u}}{\partial t} + ρ (\vec{u} \cdot \nabla) \vec{u} - μ \cdot \nabla^{2} \vec{u} = - \nabla P + {\vec{F}}_{b} + {\vec{F}}_{a}

where F_b is buoyancy force given by the Boussinesq approximation:(10)

{\vec{F}}_{b} = - ρ_{l i q u i d} (1 - β (T - T_{m})) \vec{g}

and(11)

{\vec{F}}_{a} = - A (T) \cdot \vec{u}

with the expression of A(T) inspired from the Carman–Koseny relation in a porous medium and the expression of

\nabla (P)

from the Darcy's law as presented in [23] and [24]:(12)

A (T) = \frac{C {(1 - B_{1} (T))}^{2}}{(B_{1} {(T)}^{3} + q)}

If we assume that the flow is laminar:(13)

\nabla (P) = \frac{- C {(1 - B_{1} (T))}^{2}}{B_{1} {(T)}^{3}} \cdot \vec{u}

我们假设液相中的 PCM 为牛顿流体。质量、动量和能量守恒方程与传热扩散方程同时求解。不过，为了模拟相变，动量守恒方程修改如下： (9)

ρ \frac{\partial \vec{u}}{\partial t} + ρ (\vec{u} \cdot \nabla) \vec{u} - μ \cdot \nabla^{2} \vec{u} = - \nabla P + {\vec{F}}_{b} + {\vec{F}}_{a}

其中 F _b 为浮力，由布西内斯克近似法给出： (10)

{\vec{F}}_{b} = - ρ_{l i q u i d} (1 - β (T - T_{m})) \vec{g}

和 (11)

{\vec{F}}_{a} = - A (T) \cdot \vec{u}

，A(T)的表达式来自多孔介质中的卡曼-科森关系，

\nabla (P)

的表达式来自达西定律，见文献[23]和[24]： (12)

A (T) = \frac{C {(1 - B_{1} (T))}^{2}}{(B_{1} {(T)}^{3} + q)}

如果我们假设流动为层流： (13)

\nabla (P) = \frac{- C {(1 - B_{1} (T))}^{2}}{B_{1} {(T)}^{3}} \cdot \vec{u}

The value of C depends on the morphology of the medium. In this study, C is given the constant value 10⁵. This value is chosen arbitrarily high. Constant q is chosen very low in order to make Eq. (12) valid even when B₁(T) is zero. The value of q was fixed at 10⁻³. When the temperature of the PCM is higher than T_m + ΔT, the PCM is completely liquid. Therefore, B₁ is 1 and consequently A and F_a are zero. In this case, the usual momentum conservation equation applies. During the transition state, 0 < B₁(T) < 1. A(T) increases along with the melting process until the added force F_a becomes greater than the convection and diffusion terms in Eq. (9). The momentum conservation equation becomes similar to the Darcy law for fluid flow in porous medium:(14)

\vec{u} = - \frac{K}{μ} \nabla (P)

where the permeability K is a function of B₁(T). When B₁(T) diminishes, the velocity field also diminishes until it reaches zero when the PCM becomes completely solid. At that point, the PCM temperature is lower than T_m − ΔT. Therefore, B₁ is 0. Eq. (12) shows that the value of A(T) becomes very high. Consequently, all the terms in the momentum conservation equation are dominated by the added force. The only solution of the Navier–Stokes equations is u = 0 which corresponds to a solid medium.
C 的值取决于介质的形态。在本研究中，C 的恒定值为 10 ⁵ 。这个值是任意选择的。常数 q 选得很低，以便在 B ₁ (T) 为零时，公式 (12) 仍然有效。q 的值固定为 10 ⁻³ 。当 PCM 的温度高于 T _m + ΔT 时，PCM 完全呈液态。因此，B ₁ 为 1，因此 A 和 F _a 为零。在这种情况下，通常的动量守恒方程适用。在过渡状态期间，0 < B ₁ (T) < 1。A(T) 随着熔化过程而增加，直到附加力 F _a 大于公式 (9) 中的对流和扩散项。动量守恒方程与多孔介质中流体流动的达西定律相似： (14)

\vec{u} = - \frac{K}{μ} \nabla (P)

其中渗透率 K 是 B ₁ (T) 的函数。当 B ₁ (T) 减小时，速度场也随之减小，直到 PCM 完全固化时速度场为零。此时，PCM 温度低于 T _m - ΔT。因此，B ₁ 为 0。公式 (12) 表明，A(T) 的值变得非常高。因此，动量守恒方程中的所有项都被附加力所支配。纳维-斯托克斯方程的唯一解是 u = 0，对应于固体介质。

2.2.3. Numerical method 2.2.3.数值方法

A 2D finite element model was used. The temperature and pressure field were approximated using 2D linear Lagrangian elements with three degrees of freedom:(15)

T = \sum_{i = 1}^{3} N_{i} T_{i}

(16)

P = \sum_{i = 1}^{3} N_{i} P_{i}

采用了二维有限元模型。温度场和压力场使用具有三个自由度的二维线性拉格朗日元素进行近似： (15)

T = \sum_{i = 1}^{3} N_{i} T_{i}

(16)

P = \sum_{i = 1}^{3} N_{i} P_{i}

To satisfy the Brezzi–Babuska condition [22] an additional degree of freedom was used by adding a node at the center of mass of each triangular element to approximate the velocity field as shown in Eq. (17) and Fig. 2.(17)

\vec{u} = \sum_{i = 1}^{3} N_{i} {\vec{u}}_{i} + N_{b} {\vec{u}}_{b}

为了满足布雷齐-巴布斯卡条件[22]，我们在每个三角形元素的质量中心添加了一个节点，以逼近速度场，如公式 (17) 和图 2 所示。 (17)

\vec{u} = \sum_{i = 1}^{3} N_{i} {\vec{u}}_{i} + N_{b} {\vec{u}}_{b}

Since the use of standard Galerkin mesh might create numerical instability when the convection term dominates the diffusion term in Eq. (9), the maximum mesh size was defined using a condition on the mass mesh Peclet number:(18)

P e = \frac{u Δ x}{ν} < 2

where Δx is the maximum length of mesh cell in a unit 2D surface. The maximum velocity in the PCM tank was experimentally determined at 10⁻² m/s. Therefore, Eq. (18) gives Δx ≈ 5e−4 m, Pe was kept lower than 1. Consequently, the maximum mesh size was 4.10e−4 m. The mesh was refined at the borders of the PCM domain (see Fig. 3). The final mesh had 135 240 elements and 465 857 degrees of freedom. No significant change of the results was observed when using a finer mesh. A Galerkin least-squares stabilization method was employed.
由于使用标准 Galerkin 网格可能会在公式 (9) 中对流项主导扩散项时产生数值不稳定性，因此使用质量网格佩克莱特数条件来定义最大网格尺寸： (18)

P e = \frac{u Δ x}{ν} < 2

，其中 Δx 是单位二维表面中网格单元的最大长度。经实验确定，PCM 罐中的最大速度为 10 ⁻² m/s。因此，公式 (18) 得出 Δx ≈ 5e-4 m，Pe 保持小于 1，因此最大网格尺寸为 4.10e-4 m。最终网格有 135 240 个元素和 465 857 个自由度。使用更细的网格后，结果没有明显变化。采用了 Galerkin 最小二乘稳定方法。

2.3. Elements of validation
2.3.验证的要素

As shown in Fig. 4, the experimental set-up consisted of a Plexiglas container without cooling fins and filled with a PCM whose thermo-physical properties are listed in Table 2. An air layer was left at the top of the tank to prevent it from breaking due to the PCM thermal expansion. A fixed temperature was imposed on each side of the tank using heating plates. The uncertainty in the applied temperature by heating plates was ±0.4 °C. The inner cavity containing the PCM was 167 mm × 167 mm × 30 mm-large. The transient two-dimensional velocity field in the cavity was measured through a particle image velocimetry (PIV) apparatus which included a Nd-Yag laser. The uncertainty of the PIV velocity measurement was ±2 mm/s.
如图 4 所示，实验装置由一个不带冷却鳍片的有机玻璃容器组成，容器中装满了 PCM，其热物理性质如表 2 所示。容器顶部留有一层空气，以防止容器因 PCM 热膨胀而破裂。利用加热板在罐体两侧施加固定温度。加热板施加温度的不确定性为 ±0.4 °C。含有 PCM 的内腔尺寸为 167 mm × 167 mm × 30 mm。空腔中的瞬态二维速度场是通过包含钕-钇钕石榴石激光器的粒子图像测速仪（PIV）测量的。PIV 速度测量的不确定性为 ±2 毫米/秒。

Table 2. Thermo-physical properties of the PCM used for validation.
表 2.用于验证的 PCM 的热物理性质。

ρ_solid = ρ_liquid	890 [kg m⁻³]
C_p solid = C_p liquid C _p solid = C _p liquid	2110 [J kg⁻¹ K⁻¹] 2110 [J kg ⁻¹ K ⁻¹ ]。
L_F	1.81 × 10⁵ [J kg⁻¹] 1.81 × 10 ⁵ [J kg ⁻¹ ]。
ν	1.1 × 10⁻⁵ [m² s⁻¹] 1.1 × 10 ⁻⁵ [m ² s ⁻¹ ]。
β	3.09 × 10⁻³ [K⁻¹] 3.09 × 10 ⁻³ [K ⁻¹ ]。
k_solid = k_liquid k _solid = k _liquid	0.182 [W m⁻¹ K⁻¹] 0.182 [W m ⁻¹ K ⁻¹ ]。
g	9.81 [m s⁻²]
T_m	25 [°C]

There were two different experimental comparisons: firstly, a transient comparison of the simulated and actual liquid–solid moving boundary location was carried out. On this first experiment, a fixed temperature of 40 °C was imposed on the left side of the PCM tank and 20 °C was imposed on the right side. A photograph of the boundary was taken every 20 min. This first comparison showed a good agreement between simulation and measurement except on the top of the PCM domain as visible in Fig. 5. The difference here is probably due to the air layer which was not simulated. On top of the simulated model, an adiabatic and a no slip boundary condition were used as presented in Fig. 1.
有两种不同的实验对比：首先，对模拟和实际的液固移动边界位置进行了瞬态对比。在第一次实验中，PCM 罐的左侧温度固定为 40 °C，右侧为 20 °C。每 20 分钟拍摄一次边界照片。第一次对比结果表明，除了图 5 所示的 PCM 域顶部外，模拟和测量结果非常吻合。这里的差异可能是由于未模拟的空气层造成的。在模拟模型的顶部，使用了绝热和无滑移边界条件，如图 1 所示。

Secondly, a stationary comparison of the simulated and measured two dimensional velocity fields in the completely melted PCM on the four cross-sections is shown in Fig. 6. For this second experiment, a fixed temperature of 38 °C was imposed on the left side of the PCM tank and 28 °C was imposed on the right side.
其次，图 6 显示了完全熔化的 PCM 在四个横截面上的模拟和测量二维速度场的静态对比。在第二次实验中，PCM 罐左侧的固定温度为 38 °C，右侧的固定温度为 28 °C。

As shown in Fig. 7, Fig. 8, the magnitude of velocity on the left side, upper side and bottom side of the system is of the order of 0.001 m/s to 0.002 m/s while on the right side, which is the colder side of the system, the velocity magnitude is around 0.005 m/s. This observed unbalanced velocity field was not reproduced by the numerical model which keeps a perfect balance due to the mass and momentum conservation equations. The simulated velocity always exceeded the experimental one. The discrepancy was of the order of 4 mm/s on the front plate and 2 mm/s on the back plate. It should be noted that the measured velocity was close to the uncertainty of the PIV measurement.
如图 7 和图 8 所示，系统左侧、上侧和下侧的速度大小约为 0.001 米/秒至 0.002 米/秒，而系统右侧（即较冷的一侧）的速度大小约为 0.005 米/秒。由于质量和动量守恒方程，数值模型保持了完美的平衡。模拟速度总是超过实验速度。前板上的差异为 4 毫米/秒，后板上的差异为 2 毫米/秒。值得注意的是，测量的速度接近 PIV 测量的不确定性。

3. Results and discussion
3.结果和讨论

The numerical model presented in Fig. 1 was used to carry out a parametric study. All the simulations were conducted using T_e = T_i = 20 °C, h_e = 10 W/m² K, h_i = 5 W/m² K and E = 1000 W/m² and ΔT = 0.5 K. The different cases simulated during the parametric study are presented in Table 3.
图 1 所示的数值模型用于进行参数研究。所有模拟都是在 T _e = T = 20 °C、h _e = 10 W/m ² K、h = 5 W/m ² K 和 E = 1000 W/m ² 以及 ΔT = 0.5 K 的条件下进行的。

Table 3. List of simulated cases.
表 3.模拟案例列表。

Empty Cell	(a)	(b)	(c)	(d)	(e)
L (mm) 长（毫米）	0	20	20	20	20
H (mm) 高（毫米）	132	132	132	40	132
Cooling fins 冷却片	No 没有	No 没有	Yes 是	No 没有	No 没有
PCM convection simulated 模拟 PCM 对流	No 没有	No 没有	Yes 是	Yes 是	Yes 是

3.1. SP/PCM system without cooling fins
3.1.无冷却片的 SP/PCM 系统

The simulated temperature and velocity fields in the SP/PCM system (e) without cooling fins at different instant times are presented in Fig. 9, Fig. 10. The melted PCM goes upward close to the heated aluminum plates, and downward along the solid PCM which is blue colored. The melting process gets accelerated after 40 min when the melted PCM touches the back plate because it gets warmer than the melt temperature. Then the remaining solid PCM is progressively separated from the back plate and another upward flow is observed from the bottom of the back plate to the top of the still solid PCM. The PCM is completely melted after 104 min. At this time, the mean temperature of the front plate is 48 °C. Afterwards its temperature rises more quickly. As shown in Fig. 10, the mean velocity of the liquid PCM is 7.3e−03 m/s on the front plate, 3.5e−03 m/s on the solid PCM surface, and 1.8e−03 m/s on the back plate.
图 9 和图 10 显示了不带冷却鳍片的 SP/PCM 系统 (e) 在不同瞬间的模拟温度场和速度场。熔化的 PCM 靠近加热的铝板向上移动，并沿着蓝色的固体 PCM 向下移动。40 分钟后，当熔化的 PCM 接触到背板时，熔化过程加速，因为它的温度高于熔化温度。然后，剩余的固体 PCM 逐渐与背板分离，并从背板底部向仍为固体的 PCM 顶部流动。104 分钟后，PCM 完全熔化。此时，前板的平均温度为 48 °C。之后，其温度上升得更快。如图 10 所示，液体 PCM 在前板上的平均速度为 7.3e-03 m/s，在固体 PCM 表面的平均速度为 3.5e-03 m/s，在后板上的平均速度为 1.8e-03 m/s。

The Rayleigh number in the liquid PCM was calculated as:

R a = \frac{g β (T_{p l a t e} - T_{l i q u i d}) \cdot H^{3}}{ν \cdot a}

where T_plate is the temperature of the left (heated) plate, T_liquid is the mean temperature of the liquid PCM and a its thermal diffusivity. The value of Ra was found equal to 9.5e06 at melt start and 4.8e07 at melt end. This result is coherent with our initial assumption of a laminar flow and confirms the observation of [4].
液体 PCM 中的雷利数计算公式为

R a = \frac{g β (T_{p l a t e} - T_{l i q u i d}) \cdot H^{3}}{ν \cdot a}

其中 T _plate 是左侧（加热）板的温度，T _liquid 是液体 PCM 的平均温度，a 是热扩散率。Ra 值在熔化开始时等于 9.5e06，在熔化结束时等于 4.8e07。这一结果与我们最初假设的层流相一致，并证实了 [4] 的观察结果。

3.2. SP/PCM system with cooling fins
3.2.带冷却片的 SP/PCM 系统

The simulated temperature and velocity fields in the SP/PCM system (c) with cooling fins at different instants of time are presented in Fig. 11, Fig. 12. There is a circulation of the liquid PCM through the three parts of the container. Convection of the liquid PCM is observed upward close to the heated panel, and downward close to the liquid–solid boundary through the space between the fins and the back plate. As shown in Fig. 12, the PCM velocity close to the front plate increases with time. The velocity is 40e−03 m/s when the PCM is completely melted with a mean value of 18e−03 m/s during the melting process.
图 11 和图 12 显示了带冷却鳍片的 SP/PCM 系统 (c) 在不同时刻的模拟温度场和速度场。液体 PCM 在容器的三个部分中循环。液体 PCM 在靠近加热板的地方向上对流，在靠近液固边界的地方向下对流，流经鳍片和背板之间的空间。如图 12 所示，靠近前板的 PCM 速度随着时间的推移而增加。PCM 完全熔化时的速度为 40e-03 m/s，熔化过程中的平均速度为 18e-03 m/s。

The Rayleigh number in the liquid PCM was calculated as:

R a = \frac{g β (T_{p l a t e} - T_{l i q u i d}) \cdot l^{3}}{ν \cdot a}

where the characteristic length l was taken as the height of each section of the PCM tank which is 0.04 m at the middle section and 0.042 m at the bottom and upper sections. The value of Ra was calculated as 5e06 at melt start and 5.3e07 at melt end. In spite of these medium Rayleigh numbers, the velocity streamlines plotted in Fig. 11 suggest that the flow in each section of the tank is weakly turbulent. Like [4], we also numerically observe a suspended solid PCM mass when the melting is nearly over. This is due to the mechanical resistance created by the fin and the liquid PCM under the solid mass.
液体 PCM 中的雷利数计算公式为

R a = \frac{g β (T_{p l a t e} - T_{l i q u i d}) \cdot l^{3}}{ν \cdot a}

其中，特征长度 l 取为 PCM 罐各部分的高度，中间部分为 0.04 米，底部和上部部分为 0.042 米。计算得出的 Ra 值为：熔体开始时为 5e06，熔体结束时为 5.3e07。尽管存在中等雷利数，但图 11 中绘制的速度流线表明，熔池各段的流动均为弱湍流。与文献[4]一样，我们也在数值上观察到，当熔化接近结束时，会出现悬浮的固体 PCM 质量。这是由于固体物质下的鳍片和液体 PCM 产生的机械阻力造成的。

3.3. Parametric study of the front plate temperature
3.3.前板温度参数研究

The transient temperature of the front plate was simulated for different sizes and shapes of the PCM container and compared to the temperature of a single aluminum plate – which is case (a) – under the same boundary conditions. The result is shown in Fig. 13.
模拟了不同尺寸和形状的 PCM 容器前板的瞬态温度，并与相同边界条件下的单块铝板（即情况 (a)）的温度进行了比较。结果如图 13 所示。

Three inflexions points can be observed on the transient operating temperature profile for cases (c), (d) and (e). The first one happens at the PCM melt temperature. After a steep increase, the panel temperature rises much more slowly from that point on because of the start of the melting process.
在情况 (c)、(d) 和 (e) 的瞬态工作温度曲线上可以观察到三个拐点。第一个拐点发生在 PCM 熔体温度处。在温度急剧上升之后，由于熔化过程的开始，面板温度从此时开始上升得更加缓慢。

Between the first and the second inflexion point, the PCM acts like an insulation material for the panel and heat transfer is dominated by conduction. The second inflexion point indicates the start of the convection heat transfer which balances conduction heat transfer in the PCM. The operating temperature remains more or less constant until the PCM has completely melted. The graph for case (b) shows that the simulated panel temperature may be overestimated by more than 20% after 3600 s when conduction only is considered in the numerical model.
在第一个拐点和第二个拐点之间，PCM 就像面板的隔热材料，热量主要通过传导传递。第二个拐点表示对流传热的开始，它平衡了 PCM 中的传导传热。在 PCM 完全融化之前，工作温度基本保持不变。情况 (b) 的曲线图显示，如果数值模型中只考虑传导，模拟面板温度在 3600 秒后可能会被高估 20% 以上。

The last inflexion point marks the end of the melting process. Heat transfer in the container is dominated by convection. Afterwards the temperature rises with a smaller slope than at the beginning of the process when the PCM was completely solid. This is due to the higher specific heat capacity of the liquid PCM in comparison with the specific heat capacity of the solid PCM. Nevertheless, the front plate temperature quickly rises toward its maximum since heat is no longer absorbed by the solid PCM.
最后一个拐点标志着熔化过程的结束。容器中的热量传递以对流为主。此后，温度上升的斜率小于过程开始时 PCM 完全固态时的上升斜率。这是由于液体 PCM 的比热容高于固体 PCM 的比热容。不过，由于固体 PCM 不再吸收热量，前板温度会迅速升至最高值。

Over the range of simulated sizes for the SP/PCM system, the temperature of the front plate always remains lower than 50 °C after 89 min under a constant radiation of 1000 W/m². The better performance is obtained with a 13.2 cm × 4.9 cm large PCM container featuring 3 cm long fins. In this case, the panel's temperature is 34.9 °C after 1 hour. The same temperature is reached after 5 min without PCM as shown in Fig. 13. These observations are in good agreement with [3] and [5]. The predicted percentages temperature drop of the front plate with respect to the PCM container shape and time are listed in Table 4.
在 SP/PCM 系统的模拟尺寸范围内，在 1000 W/m ² 的恒定辐射条件下，前板的温度在 89 分钟后始终低于 50 °C。13.2 cm × 4.9 cm 的大型 PCM 容器具有 3 cm 长的鳍片，性能更佳。在这种情况下，1 小时后面板的温度为 34.9 °C。如图 13 所示，在不使用 PCM 的情况下，5 分钟后达到相同的温度。这些观测结果与 [3] 和 [5] 非常吻合。表 4 列出了预测的面板温度下降百分比与 PCM 容器形状和时间的关系。

Table 4. Drop percentage of the panel's external temperature with respect to time and PCM layout by comparison with case (a).
表 4.与情况 (a) 相比，面板外部温度随时间和 PCM 布局变化的下降百分比。

Empty Cell	(c)	(d)	(e)
After 1 h 1 小时后	56.9%	57.2%	55%
After 2 h 2 小时后	21%	33.3%	36.6%
After 3 h 3 小时后	0%	12.6%	16.6%

As expected, the parametric study also showed that the operating temperature drops proportionally to the increase of the PCM width: after 3600 s, T = 34.9 °C when L = 0.049 m whereas T = 37 °C when L = 0.02 m. Comparing graphs (d) and (e) shows that the same trend is observed when the panel height is increased but only when the PCM has completely melted. In brief, it is better to increase the PCM width than its height to lower the panel's temperature. Adding cooling fins in the PCM tank provides a faster attenuation of the operating temperature because the PCM bulk conductivity is increased. But this layout accelerates the phase transition too, as shown in Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13. When the PCM has completely melted, the operating temperature rises faster than for all other SP/PCM architectures (Fig. 13). This fact mitigates the idea that adding cooling fins makes SP/PCM systems more efficient [4].
正如预期的那样，参数研究还表明，工作温度与 PCM 宽度的增加成正比：3600 秒后，当 L = 0.049 m 时，T = 34.9 °C，而当 L = 0.02 m 时，T = 37 °C。简而言之，要降低面板温度，增加 PCM 宽度比增加其高度更好。在 PCM 罐中添加冷却鳍片可以更快地降低工作温度，因为 PCM 的体传导性提高了。但这种布局也会加速相变，如图 9、图 10、图 11、图 12 和图 13 所示。当 PCM 完全熔化后，工作温度的上升速度要快于所有其他 SP/PCM 结构（图 13）。这一事实削弱了添加散热片可提高 SP/PCM 系统效率的观点 [4]。

4. Conclusion 4.结论

In this work, we proposed a detailed mathematical and numerical modeling of heat and mass transfers in coupled solar panel/phase change material architectures. A volume force was added to the buoyancy term in the Navier–Stokes’ momentum conservation equation in order to force the velocity field to be zero when the PCM is solid. In order to simulate both conduction and convection, the transient heat transfer diffusion equation was numerically solved simultaneously with the Navier–Stokes equations using a finite elements method.
在这项工作中，我们对太阳能电池板/相变材料耦合结构中的热量和质量传递进行了详细的数学和数值建模。在纳维-斯托克斯动量守恒方程的浮力项中加入了体积力，以迫使 PCM 为固体时的速度场为零。为了同时模拟传导和对流，使用有限元方法对瞬态传热扩散方程和纳维-斯托克斯方程同时进行了数值求解。

To validate the model, the experimental and simulated moving solid–liquid boundary was compared as well as the velocity field inside the PCM container through a PIV apparatus. Then a parametric study was performed on the temperature of the panel represented by a flat aluminum plate with the purpose of predicting the efficiency of SP/PCM architectures.
为了验证模型，通过 PIV 仪器比较了实验和模拟的移动固液边界以及 PCM 容器内的速度场。然后，对以铝制平板为代表的面板温度进行了参数研究，目的是预测 SP/PCM 结构的效率。

The simulated SP/PCM systems allowed maintaining the panel's temperature under 40 °C during 80 min of constant exposure to a radiation of 1000 W/m². The same temperature was reached by the panel after only 5 min without the PCM.
在持续暴露于 1000 W/m ² 辐射下 80 分钟期间，模拟 SP/PCM 系统可将面板温度保持在 40 °C 以下。在没有 PCM 的情况下，面板仅在 5 分钟后就达到了相同的温度。

One of the limitations of our study arises from the representation of the solar panel by an aluminum plate. This simplification fails to take into account the bulk specific capacity of real panels. The second limitation comes from the fact that the impact of sky temperature was not included in the numerical model, due to experimental validation difficulties. Despite these limitations, the observed results remain relevant. Future work should include experimental validations of this first numerical model using real solar panels under real climate conditions.
我们研究的局限性之一在于用铝板表示太阳能电池板。这种简化没有考虑到实际电池板的体积比容量。第二个局限是，由于实验验证困难，数值模型中没有包括天空温度的影响。尽管存在这些局限性，但观察到的结果仍然具有相关性。未来的工作应包括在真实气候条件下使用真实太阳能电池板对这一首个数值模型进行实验验证。

References

[1]
K. Emery, J. Burdick, Y. Caiyem, D. Dunlavy, H. Field, B. Kroposki, T. Moriatry
Temperature dependence of photovoltaic cells, modules and systems
Proceedings of the 25th IEEE PV Specialists Conference, Washington, DC, USA, May 13–19 (1996), pp. 1275-1278
View in Scopus Google Scholar
[2]
V.R. Voller, M. Cross, N.C. Markatos
An enthalpy method for convection/diffusion phase change
International Journal for Numerical Methods in Engineering, 24 (1) (1987), pp. 271-284
View at publisherCrossref View in Scopus Google Scholar
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A.D. Brent, V.R. Voller, K.J. Reid
Enthalpy–porosity technique for modelling convection–diffusion phase change: application to the melting of a pure metal
Numerical Heat Transfer, 3 (3) (1988), pp. 297-318
View in Scopus Google Scholar
[4]
H. Hu, S.A. Argylopoulos
Mathematical modelling of solidification and melting: a review
Modelling and Simulation in Materials Science and Engineering, 4 (1996), pp. 371-396
View in Scopus Google Scholar
[5]
O. Bertrand, B. Binet, H. Combeau, S. Couturier, Y. Delannoy, D. Gobin, M. Lacroix, P. Le Quéré, M. Médale, J. Mencinger, H. Sadat, G. Vieira
Melting driven by natural convection, a comparison exercise: first results
International Journal of Thermal Sciences, 38 (1999), pp. 5-26
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F. Setterwall
Phase change materials and chemical reactions for thermal energy storage state of the art
Proceedings of the Seventh International Conference on Thermal Energy Storage, Tokyo, Japan (1996), p. 1021
Google Scholar
[7]
B. Zalba, J.M. Marin, F. Cabeza
Review on thermal energy storage with phase change: materials, heat transfer analysis and applications
Applied Thermal Engineering, 23 (2003), pp. 251-283
View PDF View article View in Scopus Google Scholar
[8]
K. Peippo, P. Kauranen, P.D. Lund
A multi component PCM wall optimized for passive solar heating
Energy and Buildings, 17 (1991), pp. 259-270
View PDF View article View in Scopus Google Scholar
[9]
D.W. Hawes, D. Feldman, D. Banu
Latent heat storage in building materials
Energy and Buildings, 20 (1993), pp. 77-86
View PDF View article View in Scopus Google Scholar
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A. Pasupathy, R. Velraj, R.V. Seeniraj
Effect of double layered phase change materials in building roof for year round thermal management
Energy and Building, 40 (3) (2008), pp. 193-203
View PDF View article View in Scopus Google Scholar
[11]
K.A.R. Ismail, J.R. Henríquez
Thermally effective windows with moving phase change material curtains
Applied Thermal Engineering, 1 (18) (2001), pp. 1909-1923
View PDF View article View in Scopus Google Scholar
[12]
C. Arkar, S. Medved
Free cooling of building using PCM heat storage integrated into the ventilation system
Solar Energy, 81 (2007), pp. 1078-1087
View PDF View article View in Scopus Google Scholar
[13]
V.V. Tyagi, D. Buddhi
PCM thermal storage in buildings: a state of art
Renewable and Sustainable Energy Reviews, 11 (6) (2007), pp. 1146-1166
View PDF View article View in Scopus Google Scholar
[14]
Y. Zhang, G. Zhou, K. Lin, Q. Zhang, H. Di
Application of latent heat thermal energy storage in buildings: state-of-the-art and outlook
Building and Environment, 42 (2) (2007), pp. 197-209
Crossref Google Scholar
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V.A.A. Raj, R. Velraj
Review on free cooling of buildings using phase change materials
Renewable and Sustainable Energy Reviews, 14 (2010), pp. 2819-2829
View PDF View article Google Scholar
[16]
Rubitherm GmbH
Rubitherm Data Sheet
Rubitherm GmbH, Hamburg (2000)
Google Scholar
[17]
M.J. Huang, P.C. Eames, B. Norton
Thermal regulation of building integrated photovoltaics using phase change materials
International Journal of Heat and Mass Transfer, 47 (2004), pp. 275-2733
View in Scopus Google Scholar
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M.J. Huang, P.C. Eames, B. Norton
Phase change materials for limiting temperature rise in building integrated photovoltaics
Solar Energy, 0 (9) (2006), pp. 1121-1130
View PDF View article View in Scopus Google Scholar
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M.J. Huang, P.C. Eames, B. Norton
Comparison of a small-scale 3D PCM thermal control model with a validated 2D PCM thermal control model
Solar Energy Materials and Solar Cells, 0 (13) (2006), pp. 1961-1972
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M. Cellura, G. Ciulla, V. Lo Brano, A. Marvuglia, A. Orioli
Photovoltaic panel coupled with a phase changing material heat storage system in hot climates
Proceedings of the 25th Conference on Passive and Low Energy Architecture, Dublin, Ireland, October 22–24 (2008)
Google Scholar
[21]
A. Jay, S. Clerc, B. Boillot, A. Bontemps, F. Jay
Utilisation de matériaux à changement de phase pour réduire la température de panneaux PV intégrés au bâti
Proceedings of the International Building Performance Simulation Association conference, Moret sur Loing, France, November 9–10 (2010)
Google Scholar
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D.N. Arnold, F. Brezzi, M. Fortin, A Stable Element for the Stokes Equations, Calcolo ISSN: 0008-0624 (print) 1126–5434 (online), 21 (4), 1984, pp. 337–344.
Google Scholar
[23]
V.R. Voller, C. Prakash
A fixed grid numerical modeling methodology for convection–diffusion mushy region phase-change problems
International Journal of Heat and Mass transfer, 30 (1987), pp. 1709-1719
View PDF View article View in Scopus Google Scholar
[24]
A.D. Brent, V.R. Voller, K.J. Reid
Enthalpy–porosity technique for modeling convection–diffusion phase change: application to the melting of a pure metal
Numerical Heat Transfer, 13 (3) (1988), pp. 297-318
View in Scopus Google Scholar

Cited by (321)

Recent advancements in PV cooling and efficiency enhancement integrating phase change materials based systems – A comprehensive review
2020, Solar Energy
In recent years, researchers are fascinated to counter problem of PV-efficiency decline arising from high operating temperatures, especially in hot climates. This article conducts a comprehensive review of research activities performed in last 5 years, on cooling techniques with phase-change materials (PCMs), nanofluids and their combined use, leading to efficiency enhancement. By passive cooling approach with PCMs, it is found that maximum enhancement up to 20% in PV-efficiency can be achieved. Effectiveness of PCM for PV is more prominent in summer than in winter. Incorporations of fins inside PCM container at PV rear, results in much improved heat conduction within PCM. Now-a-days, researchers have grown interest in composite PCMs for PV cooling due to their enhanced thermal conductivity. Moreover, better heat regulation as well as PV-surface temperature uniformity can be achieved with two PCMs at a time having different melting points. Studies suggest that combination of passive & active cooling techniques helps in further lowering of PV-cell temperature, leading to enhancement in PV-efficiency with additional thermal power generation. PV-efficiency of water-based hybrid PV/T systems can be improved by 32% by integration with PCM. Although nanofluid-based PV/T systems have been proved to enhance PV-efficiency by more than 60%, but combined use of PCM & nanofluid is more effective approach for PV cooling than individual use of PCM or nanofluid. If combination is made between nanofluid & nano-PCM, electrical power & efficiency can further be enhanced. Nanofluids can also be considered a good spectral filter alternative as they require small thickness and are able to be tuned by varying nanoparticles conc. Finally, environmental impacts & economic viability of mentioned cooling techniques, were discussed. Studies show that PV/PCM systems become expensive & less feasible when operated in single junction due to long payback period up to 20 years. Economic feasibility can be increased by combining passive & active cooling techniques which can increase system compactness and lower its cost.
Electric vehicles batteries thermal management systems employing phase change materials
2018, Journal of Power Sources
Battery thermal management is necessary for electric vehicles (EVs), especially for Li-ion batteries, due to the heat dissipation effects on those batteries. Usually, air or coolant circuits are employed as thermal management systems in Li-ion batteries. However, those systems are expensive in terms of investment and operating costs. Phase change materials (PCMs) may represent an alternative which could be cheaper and easier to operate. In fact, PCMs can be used as passive or semi-passive systems, enabling the global system to sustain near-autonomous operations. This article presents the previous developments introducing PCMs for EVs battery cooling. Different systems are reviewed and solutions are proposed to enhance PCMs efficiency in those systems.
Increasing the efficiency of PV panel with the use of PCM
2016, Renewable Energy
The article presents how to increase electrical efficiency and power output of photovoltaic (PV) panel with the use of a phase change material (PCM). The focus of the work is in experimental setup and simulation heat extraction from the PV panel with the use of TRNSYS software. A modification of PV panel Canadian Solar CS6P-M was made with a phase change material RT28HC. The actual data of cell temperature of a PV panel with and without PCM were given and compared. A simulation of both PV panels in TRNSYS software was performed, followed by the comparison of results with the simulation and experimental actual data. The experimental results show that the maximum temperature difference on the surface of PV panel without PCM was 35.6 °C higher than on a panel with PCM in a period of one day. Referring to experimental results the calculation of the maximum and average increase of electrical efficiency was made for PV-PCM panel with TRNSYS software. Final results of simulation shows that the electricity production of PV-PCM panel for a city of Ljubljana was higher for 7.3% in a period of one year.
Uniform cooling of photovoltaic panels: A review
2016, Renewable and Sustainable Energy Reviews
Cooling of PV panels is a critical issue in the design and operation of concentrated photovoltaic (CPV) technology. Due to high cell temperature and non-uniform temperature distribution, current mismatching problem and hot spot occurs on the cell resulting in either reduction of efficiency or permanent structural damage due to thermal stresses. Temperature non-uniformity on the surface of PV panel has a major impact on the performance of CPV systems and directly increases cell temperature and series resistance. This review paper highlights the importance of uniform PV cooling by exploring the possible causes and effects of non-uniformity. Cooling techniques with low average cell temperature and uniform temperature distribution are analyzed. Economic and environmental impact on the importance of cooling of PV systems are discussed and an experimental case study is presented for comparison between uniform and non-uniform cooling methods. Immersion cooling is a promising solution for uniform cooling and has been reported to reduce the cell temperature to 20–45 °C for CPV systems. Heat pipes reduced the temperature down to 32 °C with the best case temperature non-uniformity of 3 °C. Passive cooling by heat sinks was found to reduce the cell temperature as low as 37 °C for high concentrations but with an expense of large heat sink area. Active cooling by microchannels, impingement cooling and hybrid microchannel-impingement cooling were found to be most effective in dissipating high heat flux from PV surface. Cell temperature was reported to decrease to 30 °C for 200× CPV using impingement cooling. For hybrid cooling, deviation of 0.46 °C surface temperature was obtained. Using PCM materials temperature of panel was controlled within 28–65 °C whereas optimization of heat exchanger designs also showed low and uniform temperature across surface. The impact of non-uniformity was found to be significant for all PV systems however the effect is more pronounced in CPV systems.
Using phase change materials in photovoltaic systems for thermal regulation and electrical efficiency improvement: A review and outlook
2015, Renewable and Sustainable Energy Reviews
The study presented in this paper is based on a detailed review of the literature focused on the use of phase change materials (PCM) for photovoltaic (PV) module thermal regulation and electrical efficiency improvement. The influence of high temperature on PV power generation has been examined and the findings have highlighted the importance of effective thermal regulation for PV models. Various PV cooling methods employed to maintain better PV performance are discussed and the recently emerging PV–PCM system concept for thermal regulation is introduced. A comprehensive literature review of the state-of-the-art aspects of this technology, such as system development, performance evaluation, materials selection, heat transfer improvement, numerical models, simulation, and application in practice is given. The PV–PCM system, however, might not be economically feasible if the enhancement of PV efficiency only is sought. The PV–ST–PCM system, i.e. integrated with a solar thermal (ST) system, has therefore been investigated as the stored heat can be extracted for other thermal applications. The dual PCM roles demonstrate significant application prospects for the combined technology. However, both PV–PCM and PV–ST–PCM systems are still mainly in the research and laboratory test stages, with obvious scope for practical applications but with attendant challenges. Suggestions for the future work are presented.
Review on thermal management systems using phase change materials for electronic components, Li-ion batteries and photovoltaic modules
2014, Renewable and Sustainable Energy Reviews
Improper operating temperature will degrade the performances of electronic components, Li-ion batteries and photovoltaic (PV) cells, which calls for a good thermal management system. In this paper, specific attention is paid to the thermal management systems based on the phase change materials (PCMs). Performances of the PCM-based thermal management systems for each kind of these three devices along with the type of PCM used, thermal properties of that kind of PCM, like phase change temperature, enthalpy of phase change and thermal conductivity are discussed. Discussion in detail on techniques to improve the thermal conductivity of PCMs is made because of its crucial influence. Advanced-structure heatsinks with multi-layer PCMs and hybrid passive heatsinks combined with active cooling are also introduced. The PCM-based thermal management system is powerful in ensuring electronic devices, Li-ion batteries and photovoltaic cells working safely and efficiently.

View all citing articles on Scopus

View Abstract

[1] [1]
K. Emery, J. Burdick, Y. Caiyem, D. Dunlavy, H. Field, B. Kroposki, T. Moriatry
Temperature dependence of photovoltaic cells, modules and systems
Proceedings of the 25th IEEE PV Specialists Conference, Washington, DC, USA, May 13–19 (1996), pp. 1275-1278
View in Scopus Google Scholar

[2] [2]
V.R. Voller, M. Cross, N.C. Markatos
An enthalpy method for convection/diffusion phase change
International Journal for Numerical Methods in Engineering, 24 (1) (1987), pp. 271-284
View at publisherCrossref View in Scopus Google Scholar

[3] [3]
A.D. Brent, V.R. Voller, K.J. Reid
Enthalpy–porosity technique for modelling convection–diffusion phase change: application to the melting of a pure metal
Numerical Heat Transfer, 3 (3) (1988), pp. 297-318
View in Scopus Google Scholar

[4] [4]
H. Hu, S.A. Argylopoulos
Mathematical modelling of solidification and melting: a review
Modelling and Simulation in Materials Science and Engineering, 4 (1996), pp. 371-396
View in Scopus Google Scholar

[5] [5]
O. Bertrand, B. Binet, H. Combeau, S. Couturier, Y. Delannoy, D. Gobin, M. Lacroix, P. Le Quéré, M. Médale, J. Mencinger, H. Sadat, G. Vieira
Melting driven by natural convection, a comparison exercise: first results
International Journal of Thermal Sciences, 38 (1999), pp. 5-26
View PDF View article View in Scopus Google Scholar

[6] [6]
F. Setterwall
Phase change materials and chemical reactions for thermal energy storage state of the art
Proceedings of the Seventh International Conference on Thermal Energy Storage, Tokyo, Japan (1996), p. 1021
Google Scholar

[7] [7]
B. Zalba, J.M. Marin, F. Cabeza
Review on thermal energy storage with phase change: materials, heat transfer analysis and applications
Applied Thermal Engineering, 23 (2003), pp. 251-283
View PDF View article View in Scopus Google Scholar

[8] [8]
K. Peippo, P. Kauranen, P.D. Lund
A multi component PCM wall optimized for passive solar heating
Energy and Buildings, 17 (1991), pp. 259-270
View PDF View article View in Scopus Google Scholar

[9] [9]
D.W. Hawes, D. Feldman, D. Banu
Latent heat storage in building materials
Energy and Buildings, 20 (1993), pp. 77-86
View PDF View article View in Scopus Google Scholar

[10] [10]
A. Pasupathy, R. Velraj, R.V. Seeniraj
Effect of double layered phase change materials in building roof for year round thermal management
Energy and Building, 40 (3) (2008), pp. 193-203
View PDF View article View in Scopus Google Scholar

[11] [11]
K.A.R. Ismail, J.R. Henríquez
Thermally effective windows with moving phase change material curtains
Applied Thermal Engineering, 1 (18) (2001), pp. 1909-1923
View PDF View article View in Scopus Google Scholar

[12] [12]
C. Arkar, S. Medved
Free cooling of building using PCM heat storage integrated into the ventilation system
Solar Energy, 81 (2007), pp. 1078-1087
View PDF View article View in Scopus Google Scholar

[13] [13]
V.V. Tyagi, D. Buddhi
PCM thermal storage in buildings: a state of art
Renewable and Sustainable Energy Reviews, 11 (6) (2007), pp. 1146-1166
View PDF View article View in Scopus Google Scholar

[14] [14]
Y. Zhang, G. Zhou, K. Lin, Q. Zhang, H. Di
Application of latent heat thermal energy storage in buildings: state-of-the-art and outlook
Building and Environment, 42 (2) (2007), pp. 197-209
Crossref Google Scholar

[15] [15]
V.A.A. Raj, R. Velraj
Review on free cooling of buildings using phase change materials
Renewable and Sustainable Energy Reviews, 14 (2010), pp. 2819-2829
View PDF View article Google Scholar

[16] [16]
Rubitherm GmbH
Rubitherm Data Sheet
Rubitherm GmbH, Hamburg (2000)
Google Scholar

[17] [17]
M.J. Huang, P.C. Eames, B. Norton
Thermal regulation of building integrated photovoltaics using phase change materials
International Journal of Heat and Mass Transfer, 47 (2004), pp. 275-2733
View in Scopus Google Scholar

[18] [18]
M.J. Huang, P.C. Eames, B. Norton
Phase change materials for limiting temperature rise in building integrated photovoltaics
Solar Energy, 0 (9) (2006), pp. 1121-1130
View PDF View article View in Scopus Google Scholar

[19] [19]
M.J. Huang, P.C. Eames, B. Norton
Comparison of a small-scale 3D PCM thermal control model with a validated 2D PCM thermal control model
Solar Energy Materials and Solar Cells, 0 (13) (2006), pp. 1961-1972
View PDF View article View in Scopus Google Scholar

[20] [20]
M. Cellura, G. Ciulla, V. Lo Brano, A. Marvuglia, A. Orioli
Photovoltaic panel coupled with a phase changing material heat storage system in hot climates
Proceedings of the 25th Conference on Passive and Low Energy Architecture, Dublin, Ireland, October 22–24 (2008)
Google Scholar

[21] [21]
A. Jay, S. Clerc, B. Boillot, A. Bontemps, F. Jay
Utilisation de matériaux à changement de phase pour réduire la température de panneaux PV intégrés au bâti
Proceedings of the International Building Performance Simulation Association conference, Moret sur Loing, France, November 9–10 (2010)
Google Scholar

[22] [22]
D.N. Arnold, F. Brezzi, M. Fortin, A Stable Element for the Stokes Equations, Calcolo ISSN: 0008-0624 (print) 1126–5434 (online), 21 (4), 1984, pp. 337–344.
Google Scholar

[23] [23]
V.R. Voller, C. Prakash
A fixed grid numerical modeling methodology for convection–diffusion mushy region phase-change problems
International Journal of Heat and Mass transfer, 30 (1987), pp. 1709-1719
View PDF View article View in Scopus Google Scholar

[24] [24]
A.D. Brent, V.R. Voller, K.J. Reid
Enthalpy–porosity technique for modeling convection–diffusion phase change: application to the melting of a pure metal
Numerical Heat Transfer, 13 (3) (1988), pp. 297-318
View in Scopus Google Scholar

Outline 概要

Cited by (321) 引用 (321)

Figures (13) 数字 (13)

Tables (4) 表格 (4)

Energy and Buildings 能源与建筑

Highlights 亮点

Abstract 摘要

Keywords 关键词

Nomenclature 术语

Greek letters 希腊字母

Subscripts 下标

1. Introduction 1.导言

2. Methodology 2.方法论

2.1. Numerical case description
2.1.数值案例描述

2.2. Mathematical model 2.2.数学模型

2.2.1. Modeling heat transfers
2.2.1.传热建模

2.2.2. Modeling of mass and momentum transfers for PCM domain
2.2.2.PCM 域的质量和动量传递模型

2.2.3. Numerical method 2.2.3.数值方法

2.3. Elements of validation
2.3.验证的要素

3. Results and discussion
3.结果和讨论

3.1. SP/PCM system without cooling fins
3.1.无冷却片的 SP/PCM 系统

3.2. SP/PCM system with cooling fins
3.2.带冷却片的 SP/PCM 系统

3.3. Parametric study of the front plate temperature
3.3.前板温度参数研究

4. Conclusion 4.结论

References

Cited by (321)

Recent advancements in PV cooling and efficiency enhancement integrating phase change materials based systems – A comprehensive review

Electric vehicles batteries thermal management systems employing phase change materials

Increasing the efficiency of PV panel with the use of PCM

Uniform cooling of photovoltaic panels: A review

Using phase change materials in photovoltaic systems for thermal regulation and electrical efficiency improvement: A review and outlook

Review on thermal management systems using phase change materials for electronic components, Li-ion batteries and photovoltaic modules

Phase-change materials to improve solar panel's performance改善太阳能电池板性能的相变材料

Highlights 亮点

Abstract 摘要

Keywords 关键词

Nomenclature 术语

Greek letters 希腊字母

Subscripts 下标

1. Introduction 1.导言

2. Methodology 2.方法论

2.1. Numerical case description2.1.数值案例描述

2.2. Mathematical model 2.2.数学模型

2.2.1. Modeling heat transfers2.2.1.传热建模

2.2.2. Modeling of mass and momentum transfers for PCM domain2.2.2.PCM 域的质量和动量传递模型

2.2.3. Numerical method 2.2.3.数值方法

2.3. Elements of validation2.3.验证的要素

3. Results and discussion3.结果和讨论

3.1. SP/PCM system without cooling fins3.1.无冷却片的 SP/PCM 系统

3.2. SP/PCM system with cooling fins3.2.带冷却片的 SP/PCM 系统

3.3. Parametric study of the front plate temperature3.3.前板温度参数研究

4. Conclusion 4.结论

References

Phase-change materials to improve solar panel's performance
改善太阳能电池板性能的相变材料

2.1. Numerical case description
2.1.数值案例描述

2.2.1. Modeling heat transfers
2.2.1.传热建模

2.2.2. Modeling of mass and momentum transfers for PCM domain
2.2.2.PCM 域的质量和动量传递模型

2.3. Elements of validation
2.3.验证的要素

3. Results and discussion
3.结果和讨论

3.1. SP/PCM system without cooling fins
3.1.无冷却片的 SP/PCM 系统

3.2. SP/PCM system with cooling fins
3.2.带冷却片的 SP/PCM 系统

3.3. Parametric study of the front plate temperature
3.3.前板温度参数研究