Mathematical modelling and sensitivity analysis of solar photovoltaic panel integrated with phase change material

doi:10.1016/j.apenergy.2018.06.145

Applied Energy 应用能源

Volume 228, 15 October 2018, Pages 1147-1158
第 228 卷，2018 年 10 月 15 日，第 1147-1158 页

https://doi.org/10.1016/j.apenergy.2018.06.145 Get rights and content 获取权利和内容

Highlights 亮点

•
Enhanced conductivity method is adopted to incorporate PCM convective effect into 1-D thermal resistance model.
采用增强传导法将 PCM 对流效应纳入一维热阻模型。
•
An improved thermal resistance model, offering a good compromise between accuracy and simplicity, is developed.
我们开发了一种改进的热阻模型，在准确性和简便性之间取得了良好的平衡。
•
Neglecting PCM convective effect or radiative heat transfer in numerical simulation may result in significant errors.
在数值模拟中忽略 PCM 对流效应或辐射传热可能会导致重大误差。
•
A sensitivity analysis suggests that the optimal PCM melting temperature is about 5 °C higher than ambient.
灵敏度分析表明，最佳 PCM 熔化温度比环境温度高约 5 °C。

Abstract 摘要

It is reported that every degree rise in photovoltaic (PV) temperature could lead to a decrease in electricity output by 0.4–0.65%. Phase change material (PCM), which could absorb great amount of heat without raising the temperature of itself, is employed in this study to control PV module temperature and increase power generation. This kind of integrated system is the so-called PV-PCM system. In recent years, some work has already been conducted in using PCM for PV panel thermal regulation both numerically and experimentally, while some issues are still unsolved or unclear, for example, limited number of cases in simulation, difficulties in modelling PCM convective effect, the impact and uncertainty resulting from some common assumptions in numerical simulation. To examine these issues, an improved thermal resistance model through applying enhanced conductivity method is developed to incorporate PCM convective effect in 1-D model, offering a good compromise between accuracy and simplicity. The numerical simulation result illustrates that neglecting PCM convective and radiative heat transfer will cause significant errors. Finally, based on simulation of over 300 cases, the two-variable analysis demonstrates that every 100 W/m² increase in solar radiation can lead to about 5 °C increase in peak temperature, and an optimal performance can be achieved when the melting temperature of PCM is slightly higher, such as 5 °C, than the ambient temperature.
据报道，光伏（PV）温度每升高一度，发电量就会减少 0.4-0.65%。相变材料（PCM）可以吸收大量的热量而不会使自身温度升高，本研究采用这种材料来控制光伏组件的温度并增加发电量。这种集成系统就是所谓的 PV-PCM 系统。近年来，在利用 PCM 进行光伏面板热调节方面已经开展了一些数值和实验工作，但仍有一些问题尚未解决或不明确，例如模拟案例数量有限、PCM 对流效应建模困难、数值模拟中一些常见假设的影响和不确定性。为了研究这些问题，通过应用增强传导法开发了一种改进的热阻模型，将 PCM 对流效应纳入一维模型，在精确性和简便性之间实现了良好的折中。数值模拟结果表明，忽略 PCM 对流和辐射传热将导致重大误差。最后，基于 300 多个案例的模拟，双变量分析表明，太阳辐射每增加 100 W/m ² 可导致峰值温度增加约 5 °C，当 PCM 的熔化温度略高于环境温度（如 5 °C）时，可实现最佳性能。

Keywords 关键词

Solar photovoltaic (PV)

Phase change materials (PCM)

PV-PCM

Thermal regulation

Thermal resistance model

Enhanced conductivity method

太阳能光伏 (PV)相变材料 (PCM)PV-PCM热调节热阻模型增强传导法

Nomenclature 术语

ρ

density (kg/m³)
密度（千克/米 ³ )

μ

dynamic viscosity of phase change material (N s/m²)
相变材料的动态粘度（N s/m ² )

C_{p}

specific heat (J/(kg K)) 比热（焦耳/（千克 K）

g

gravity acceleration (m/s²)
重力加速度（米/秒 ² )

Δ t

time interval (s) 时间间隔（秒）

A

area of PV panel (m²)
光伏板面积 (m ² )

α

absorptivity of PV panel
光伏板的吸收率

G_{PV}

solar radiation on PV panel (W/m²)
光伏板上的太阳辐射 (W/m ² )

η

actual efficiency of PV panel (%)
光伏板的实际效率 (%)

ε_{glass}

emissivity of PV surface 光伏表面的发射率

F

view factor of PV surface
光伏表面视角系数

σ

the Stefan–Boltzmann constant
斯特凡-波兹曼常数

T_{surface}

surface temperature of PV-PCM system (°C)
PV-PCM 系统的表面温度（°C）

T_{sky}

sky temperature (°C) 天空温度（°C）

h_{free}

natural heat convection coefficient (W/(m² K))
自然热对流系数（W/（m ² K）

h_{forced}

forced heat convection coefficient (W/(m² K))
强制热对流系数（W/(m ² K)

ε_{Al}

emissivity of aluminum 铝的发射率

T_{back}

back surface temperature of PV-PCM system (°C)
PV-PCM 系统的背面温度（°C）

T_{ground}

ground temperature (°C) 地面温度（°C）

Q

net solar radiation input (W)
净太阳辐射输入（瓦）

q

total heat input (W) 总输入热量（瓦）

q_{sky}

radiative heat loss to sky (W)
天空辐射热损失（瓦）

T_{amb}

ambient temperature (°C) 环境温度（°C）

q_{ground}

radiative heat loss to ground (W)
地面辐射热损失（瓦）

k

thermal conductivity (W/(m K))
导热系数（瓦/（米 K）

L

height of PV panel (m)
光伏板高度（米）

Ra

Raleigh number 罗利号码

\Pr

Prandtl number 普朗特数

Re

Reynolds number 雷诺数

ρ_{s}

density of PCM in solid state (kg/m³)
固态 PCM 的密度（千克/米 ³ )

ρ_{l}

density of PCM in liquid state (kg/m³)
液态 PCM 的密度（千克/米 ³ )

c_{s}

specific heat of PCM in solid state (J/(kg K))
固态 PCM 的比热（焦耳/（千克 K）

c_{l}

specific heat of PCM in liquid state (J/(kg K))
液体状态下 PCM 的比热（焦耳/（千克 K）

T_{l}

liquidus temperature of phase change material (°C)
相变材料的液相温度（°C）

T_{s}

solidus temperature of phase change material (°C)
相变材料的固相温度（°C）

E

energy absorbed by phase change material (J/kg)
相变材料吸收的能量（焦耳/千克）

E_{0}

energy required to rise to the solidus temperature of phase change material (J/kg)
相变材料上升到固相温度所需的能量（焦耳/千克）

E_{l}

latent heat capacity of phase change material (J/kg)
相变材料的潜热容量（焦耳/千克）

P_{out}

output power of PV panel (W)
光伏电池板的输出功率（瓦）

η_{ref}

reference efficiency of PV panel
光伏板的参考效率

β_{ref}

coefficient between PV temperature and efficiency
光伏温度与效率之间的系数

k_{PCM}

thermal conductivity of PCM layer (W/(m K))
PCM 层的导热系数（瓦/（米 K）

1. Introduction 1.导言

Nowadays, solar energy harnessed by photovoltaic (PV) panels is regarded as one of the most promising energy sources to deal with world energy crisis and global warming [1]. For the purpose to generate more electricity from the same amount of solar energy, scientists relentlessly pursue higher and higher PV conversion efficiency [2]. However, not only the characteristics of PV panel itself determine the PV conversion efficiency, the ambient environment also has impact on it [3]. While under laboratory conditions, the efficiency could reach to over 30% [4], only about 5–20% solar energy can be converted to electricity under real conditions [5]. Various environmental factors, e.g., the intensity of solar radiation and the dust accumulation on the PV surface, can affect the efficiency. Moreover, the high PV temperature has negative impact, −0.4%/°C to −0.65%/°C, on PV conversion efficiency, [6], which has been extensively reported in academic circles, for example [7], [8]. As the PV temperature could rise up to 80 °C and even higher than 100 °C in desert area, a significant reduction in PV power generation was observed [9].
如今，光伏（PV）电池板所利用的太阳能被认为是应对世界能源危机和全球变暖最有前途的能源之一[1]。为了用相同数量的太阳能产生更多的电能，科学家们不断追求更高的光伏转换效率[2]。然而，决定光伏转换效率的不仅是光伏板本身的特性，周围环境也会对其产生影响 [3]。虽然在实验室条件下，光电转换效率可达 30% 以上 [4]，但在实际条件下，只有约 5-20% 的太阳能可以转化为电能 [5]。各种环境因素，如太阳辐射强度和光伏表面的积尘，都会影响效率。此外，光伏温度过高会对光伏转换效率产生-0.4%/°C 至-0.65%/°C 的负面影响[6]，学术界对此已有大量报道，例如[7]、[8]。在沙漠地区，由于光伏温度可升至 80 °C，甚至高于 100 °C，光伏发电量显著减少 [9]。

In this context, researchers have adopted various thermal management techniques to mitigate and control the PV surface temperature [10], e.g., natural/forced air cooling [11], water cooling [12] and heat pipe [13]. Generally, those cooling techniques could be classified into two categories: active cooling and passive cooling [14]. Active system consumes extra electricity to drive cooling water or other coolants to cool the PV panel which could achieve better cooling effect than passive systems but usually costs more. In 1978, Stultz et al. [15] proposed to use phase change material (PCM) to cool the PV panel since PCM has the latent heat capacity which can absorb substantial amount of heat from PV. From that on, some research has been carried out in the field of PCM for PV temperature regulation, that is so-called PV-PCM system. Huang et al. has studied the performance of PV-PCM system comprehensively in earlier years, for example, the 2-D computational fluid dynamics (CFD) model was firstly employed to analyze the micro-level operational variables in the PV-PCM system [16], i.e., temperature, velocity fields and vortex formation, and it was found that the existence of PCM could enhance PV conversion efficiency significantly [17]. After that, more papers were published to investigate different issues, such as impact of different types of PCM, impact of fins attached to the PCM chamber, melting behavior of PCM [18] and comparison between 3-D and 2-D CFD model [19].
在这种情况下，研究人员采用了各种热管理技术来缓解和控制光伏表面温度[10]，例如自然/强制空气冷却[11]、水冷却[12]和热管[13]。一般来说，这些冷却技术可分为两类：主动冷却和被动冷却 [14]。主动冷却系统需要消耗额外的电力驱动冷却水或其他冷却剂来冷却光伏板，与被动冷却系统相比，主动冷却系统能达到更好的冷却效果，但通常成本较高。1978 年，Stultz 等人[15] 提出使用相变材料（PCM）冷却光伏板，因为 PCM 具有潜热能力，可以从光伏板吸收大量热量。从那时起，在 PCM 用于光伏温度调节领域，即所谓的 PV-PCM 系统方面，已经开展了一些研究。Huang 等人早年对 PV-PCM 系统的性能进行了全面研究，例如首次采用二维计算流体动力学（CFD）模型分析了 PV-PCM 系统的微观运行变量[16]，即温度、速度场和涡流形成，发现 PCM 的存在可显著提高光伏转换效率[17]。此后，又有更多论文对不同问题进行了研究，如不同类型 PCM 的影响、附着在 PCM 腔上的翅片的影响、PCM 的熔化行为 [18] 以及三维和二维 CFD 模型的比较 [19]。

At the same time, more in-depth research has been conducted by other scholars to broaden the scope of study in PV-PCM. For example, Hasan et al. [20] performed a techno-economic study about PV-PCM system under two locations with different climates, i.e., Dublin and Vehari. The simulation result illustrates that the peak temperature of PV-only system and PV-PCM system is 63 °C and 42 °C respectively in a Vehari’s summer day, and the average efficiency can be improved by about 10%. It also concluded that the PV-PCM system was financially viable in Vehari’s hot climate but not in Dublin’s mild climate. A year-round numerical study conducted in United Arab Emirates suggests that in extremely hot climate, the PV-PCM system can raise the annual electricity yield by 5.9% [21]. One more study was conducted about the building-integrated photovoltaic panel (BIPV) with PCM, in which the indoor thermal effect of PV-PCM system was examined and compared with PV-only system, revealing that PV-PCM system not only could improve the PV efficiency but meanwhile decrease the heat flux going into indoor environment [22]. Besides, a multi-location study suggests that the BIPV-PCM system successfully reduces the indoor cooling load by 20–30% in Venice, Helsinki and Abu Dhabi, and meanwhile, the peak value of electricity generation is improved by 5–8% [23]. Another study also proves that the PV-PCM system could improve the electricity production by 1–1.5% annually in South Korea [24]. On the contrary, an experimental study from Japs et al. [25] indicates that from July to August in Germany, both the electricity generation and economic profit improvement are mostly negative compared to traditional PV module, which is mainly attributed to the low thermal conductivity of PCM. However, a recent experimental study of BIPV-PCM system completed in India demonstrates again that the efficiency can be improved by 10% daily compared to the reference BIPV system [26]. Moreover, a global analysis was presented by Smith et al. [27]. It reveals that by choosing the optimal PCM melting temperature, over 6% more electricity generation could be obtained in Mexico and eastern Africa. In many regions, e.g. Central and South America, Africa and Southern Asia, 5% enhancement is possible, while Europe might be the unfavorable region due to only about 2–5% enhancement. Similar to Hasan’s indoor study [22], a field study of a ventilated solar roof with PCM suggests that the cooling load is significantly decreased by adopting the PV-PCM roof in East Tennessee, USA [28]. Furthermore, various numerical modelling methods, including CFD model [29], thermal resistance model [30] and finite element model [31], are also introduced to simulate PV-PCM performance under various conditions.
与此同时，其他学者也进行了更深入的研究，以拓宽 PV-PCM 的研究范围。例如，Hasan 等人[20]在都柏林和维哈里两个气候不同的地点对 PV-PCM 系统进行了技术经济研究。模拟结果表明，在 Vehari 的夏日，纯 PV 系统和 PV-PCM 系统的峰值温度分别为 63 °C 和 42 °C，平均效率可提高约 10%。研究还得出结论，PV-PCM 系统在气候炎热的维哈里是经济可行的，但在气候温和的都柏林则不可行。在阿拉伯联合酋长国进行的一项全年数值研究表明，在极端炎热的气候条件下，PV-PCM 系统可将年发电量提高 5.9%[21]。还有一项研究是关于带有 PCM 的建筑一体化光伏板 (BIPV)，其中考察了 PV-PCM 系统的室内热效应，并与纯光伏系统进行了比较，结果表明 PV-PCM 系统不仅能提高光伏效率，还能降低进入室内环境的热通量[22]。此外，一项多地研究表明，在威尼斯、赫尔辛基和阿布扎比，BIPV-PCM 系统成功地将室内制冷负荷降低了 20-30%，同时，发电峰值提高了 5-8%[23]。另一项研究也证明，在韩国，PV-PCM 系统每年可将发电量提高 1-1.5%[24]。相反，Japs 等人的一项实验研究[25]表明，与传统光伏组件相比，德国 7-8 月间的发电量和经济效益改善大多为负值，这主要归因于 PCM 的低导热性。不过，最近在印度完成的一项关于 BIPV-PCM 系统的实验研究再次证明，与参考 BIPV 系统相比，其效率每天可提高 10%[26]。此外，Smith 等人[27] 提出了一项全球分析。该分析表明，通过选择最佳的 PCM 熔化温度，墨西哥和非洲东部的发电量可提高 6% 以上。在许多地区，如中美洲和南美洲、非洲和南亚，可以提高 5%的发电量，而欧洲可能是不利地区，因为只能提高约 2-5% 的发电量。与 Hasan 的室内研究[22]类似，在美国东田纳西州进行的一项关于带有 PCM 的通风太阳能屋顶的实地研究表明，采用 PV-PCM 屋顶可显著降低制冷负荷[28]。此外，还引入了各种数值建模方法，包括 CFD 模型[29]、热阻模型[30]和有限元模型[31]，以模拟 PV-PCM 在各种条件下的性能。

Another field of PV-PCM research is to utilize PCM for other kinds of PV system, e.g., concentrated PV (CPV) system, PV/thermal (PV/T) system and photovoltaic-thermal electric (PV/TE) system. Substantial amount of studies has been conducted on the CPV-PCM system since CPV technology can obtain very high temperature. It is reported that a V-trough CPV installed with phase change material matrix obtains 55% more electricity output than the reference system [32]. Another study demonstrates that CPV-PCM system with 45° tilted angle has optimal performance and uniform temperature distribution on the PV surface [33]. It is believed that CPV integrated with PCM could be a promising and practical application of PV-PCM system in the near future [34]. Besides, some studies attempted to combine the traditional PV/T system with PCM [35]. One numerical study of PV/T-PCM system indicates that 9% increase in electricity output and 20 °C increase in water temperature could be achieved at a high solar radiation and mild ambient temperature day [36]. The PV/TE-PCM system, a relatively novel concept, was investigated by Cui et al. [37], suggesting that it seems not a feasible option under current technologies since the PV module contributes to 98% electricity output while the thermal electric module only accounts for 2%. Moreover, some novel improvement approaches in PCM have been examined, e.g., microencapsulated PCM [38], [39] and using pork fat as a cheap source of PCM [40].
PV-PCM 研究的另一个领域是将 PCM 用于其他类型的光伏系统，如聚光光伏 (CPV) 系统、光伏/热 (PV/T) 系统和光伏-热电 (PV/TE) 系统。由于 CPV 技术可以获得非常高的温度，因此对 CPV-PCM 系统进行了大量研究。据报道，安装了相变材料矩阵的 V 型槽式 CPV 比参考系统多发 55% 的电 [32]。另一项研究表明，具有 45° 倾斜角的 CPV-PCM 系统性能最佳，光伏表面温度分布均匀 [33]。相信在不久的将来，CPV 与 PCM 集成可能会成为 PV-PCM 系统的一个前景广阔的实际应用[34]。此外，一些研究尝试将传统的 PV/T 系统与 PCM 系统相结合 [35]。一项关于 PV/T-PCM 系统的数值研究表明，在太阳辐射较强、环境温度较低的情况下，该系统的发电量可增加 9%，水温可提高 20 °C[36]。Cui 等人[37]对 PV/TE-PCM 系统这一相对新颖的概念进行了研究，认为在现有技术条件下，该系统似乎并不可行，因为 PV 模块的发电量占 98%，而热电模块仅占 2%。此外，还研究了一些新的 PCM 改进方法，如微胶囊 PCM [38]、[39] 和使用猪脂肪作为 PCM 的廉价来源 [40]。

Despite that obvious progress has been made by many scholars in this field, several issues are still unclear and have not been identified in those studies. For instance, numerical studies usually have some assumptions in their mathematical modeling [41], e.g., neglecting radiative heat transfer [42] or neglecting convective heat transfer within PCM [43], whereas the impact of those assumptions on simulation results are poorly discussed in the literature. Besides, to achieve an optimal system configuration, limited cases, usually in the range of 2–20, were compared through changing a single design parameter or boundary condition in numerical [44] and experimental studies [45]. These comparisons are a good start but such investigation cannot achieve a real optimal PV-PCM system configuration for practical implementation. One major reason for these problems is the limitation in mathematical models and numerical methods. As convection occurs in the liquid PCM practically, Navier-Stokes equation is required to be numerically solved. Consequently, the computation time becomes quite long, making it is nearly impossible to conduct substantial number of simulations in one study. To avoid this problem, some researchers assumed that the convective effect can be abandoned and only conductive heat transfer was considered [41], [43]. However, such assumption has not been well justified. Fortunately, the research on PCM heat transfer provides an effective solution to them [46]. Based on scaling theory and enhanced conductivity method, the convective effect in PCM can be taken into account through increasing conduction rate [47]. By applying this method, the calculation of Navier-Stokes equation can be avoided. For example, Kahraman et al. applied an enhanced conductivity method in modelling the melting of ice in rectangle enclosure, which shows excellent agreement with numerical solution [48], [49]. A similar method was used by Costa et al. to simulate a PCM energy storage system [50]. In recent years, several studies, for example, Refs. [51], [52], were carried out to apply the enhanced conductivity method to effectively simulate the movement of melting front of PCM. It is suggested that the computation time can be reduced significantly by applying this method [51].
尽管许多学者在这一领域取得了明显进展，但仍有几个问题尚不清楚，这些研究也没有发现。例如，数值研究在数学建模时通常会有一些假设[41]，如忽略辐射传热[42]或忽略 PCM 内部的对流传热[43]，而这些假设对模拟结果的影响在文献中讨论不多。此外，为了获得最佳系统配置，在数值[44]和实验研究[45]中，通过改变单个设计参数或边界条件，对有限的情况（通常在 2-20 个范围内）进行了比较。这些比较是一个良好的开端，但这种研究无法为实际应用实现真正的最佳 PV-PCM 系统配置。造成这些问题的一个主要原因是数学模型和数值方法的局限性。由于液体 PCM 中实际上存在对流，因此需要对 Navier-Stokes 方程进行数值求解。因此，计算时间变得相当长，几乎不可能在一次研究中进行大量模拟。为了避免这个问题，一些研究人员假设可以放弃对流效应，只考虑传导传热[41]、[43]。然而，这种假设并没有得到充分证明。幸运的是，有关 PCM 传热的研究为其提供了有效的解决方案 [46]。基于缩放理论和增强传导法，可以通过增加传导率来考虑 PCM 中的对流效应 [47]。通过应用这种方法，可以避免纳维-斯托克斯方程的计算。例如，Kahraman 等人的研究就采用了这种方法。应用增强传导性方法模拟矩形外壳中冰的融化，结果与数值解非常吻合[48]，[49]。Costa 等人也使用类似方法模拟了 PCM 储能系统[50]。近年来，一些研究（如参考文献 [51]、[52]）应用增强传导性方法有效地模拟了 PCM 的熔化前沿运动。有研究表明，应用这种方法可以大大缩短计算时间 [51]。

Therefore, after reviewing previous research in this field and relevant studies in PCM heat transfer, a comprehensive study about PV-PCM system is conducted to overcome the problems as described above. The CFD model and 1-D thermal resistance model are developed for numerical simulation, and a comparison is performed to analyze the advantages and disadvantages between two methods. Based on the comparison, an improved 1-D thermal resistance model through applying the enhanced conductivity method is then developed to highlight the convective heat transfer process taking place within PCM. Moreover, several commonly used assumptions in PV-PCM system modelling are examined in MATLAB to study their impact and uncertainty, aiming to bridge the research gap and offer meaningful insights for similar numerical studies in the future. Finally, over 300 cases of PV-PCM system with different design parameters and boundary conditions are simulated, and the result is innovatively investigated by a two-variable analysis method to obtain a clear understanding and provide reference for PV-PCM optimization.
因此，在回顾了该领域的前人研究和 PCM 传热的相关研究后，对 PV-PCM 系统进行了全面研究，以克服上述问题。建立了 CFD 模型和一维热阻模型进行数值模拟，并对两种方法的优缺点进行了比较分析。在比较的基础上，通过应用增强传导法开发了改进的一维热阻模型，以突出 PCM 内部发生的对流传热过程。此外，还在 MATLAB 中检查了 PV-PCM 系统建模中的几个常用假设，以研究其影响和不确定性，旨在弥补研究差距，并为未来类似的数值研究提供有意义的见解。最后，模拟了 300 多个具有不同设计参数和边界条件的 PV-PCM 系统案例，并通过双变量分析方法对结果进行了创新性研究，以获得清晰的认识，为 PV-PCM 优化提供参考。

2. System configuration and mathematical models
2.系统配置和数学模型

2.1. System configuration
2.1.系统配置

Basically, PV-PCM system has two major parts, i.e. PV panel and PCM chamber. Usually, crystalline silicon PV is used in the system due to its wide utilization and high temperature coefficient. There are several shapes of PCM chamber, e.g., rectangle chamber, semi-circle chamber, triangle chamber and finned chamber in the literature [10], and generally rectangle chamber is selected.
PV-PCM 系统基本上由两大部分组成，即 PV 面板和 PCM 室。由于晶体硅光伏的广泛应用和较高的温度系数，该系统通常使用晶体硅光伏。文献[10]中记载了几种形状的 PCM 腔体，如矩形腔体、半圆形腔体、三角形腔体和翅片腔体，一般选择矩形腔体。

The PV-PCM system in this study is assembled by a PV module and an aluminum chamber filled with PCM. Fig. 1(a) illustrates the three-dimensional graphic of the PV-PCM system and Fig. 1(b) shows the sectional view. Fins in the aluminum chamber is optional and in the 1-D thermal resistance model of this study, they are not applied. Table 1 summaries the basic properties of PV cell, glass cover, EVA, and chamber used in this study. RT35 from RUBITHERM Technologies is adopted as the PCM in this study since it is commercially available and widely used in the literature [14], [25] for the same purpose. Besides, RT35 is non-combustible and non-corrosive which is safe to be filled into the aluminum chamber.
本研究中的 PV-PCM 系统由一个 PV 模块和一个充满 PCM 的铝制腔体组装而成。图 1(a) 是 PV-PCM 系统的三维图形，图 1(b) 是剖面图。铝室中的鳍片是可选项，在本研究的一维热阻模型中没有应用。表 1 总结了本研究中使用的光伏电池、玻璃盖板、EVA 和腔体的基本属性。本研究采用 RUBITHERM Technologies 公司生产的 RT35 作为 PCM，因为它在市场上可以买到，并且在文献 [14] 和 [25] 中被广泛用于相同的目的。此外，RT35 不可燃、无腐蚀性，可以安全地填充到铝制腔体中。

Table 1. The properties of PV module and PCM chamber.
表 1.光伏组件和 PCM 室的特性。

Property 物业	Glass 玻璃	EVA	Silicon cells 硅电池	Tedlar	Polyester 聚酯纤维	RT35 (PCM)	Aluminum 铝
Density (kg/m³) 密度（千克/米 ³ )	2500	935	2330	1500	1390	800	2791
Specific heat (J/(kg K)) 比热（焦耳/（千克 K）	750	2500	700	1090	1172	2000	871
Thickness (mm) 厚度（毫米）	3.2	5.0	0.2	0.0375	0.25	20–50 20-50	5
Thermal conductivity (W/(m K)) 导热系数（瓦/（米 K）	1.04	0.29	150	0.35	0.155	0.2	202.4
Emissivity 发射率	0.95	/	/	/	/	/	0.095
Solidus temperature (°C) 凝固温度（°C）	/	/	/	/	/	29	/
Liquidus temperature (°C) 液相温度（°C）	/	/	/	/	/	36	/
Latent heat capacity (kJ/kg) 潜热容量（千焦/千克）	/	/	/	/	/	130	/

In recent years, the PCM based method for PV thermal regulation has attracted much attention from scholars, and now there are also some demonstration projects in the world. However, the major problem for promoting this technology is the economic issue, it will have a relatively long payback period taking into account the initial cost of PV panels and the additional cost of the PCM. However, if the thermal energy stored in PCMs can be used for the buildings involved such as space heating, ventilation and hot water, and this its economic performance can be improved and show a promising future [10].
近年来，基于 PCM 的光伏热调节方法引起了学者们的广泛关注，目前世界上也有一些示范项目。然而，推广这项技术的主要问题是经济性问题，考虑到光伏板的初始成本和 PCM 的额外成本，其投资回收期相对较长。不过，如果 PCM 中储存的热能可用于建筑物的相关用途，如空间供暖、通风和热水，那么其经济效益就会得到改善，并显示出广阔的前景[10]。

2.2. Mathematical models 2.2.数学模型

The mathematical models of the PV-PCM system should be developed first to study the system performance numerically, the models can also be used to investigate how design parameters and boundary conditions impact its performance. Various modelling methods are used in literature, including CFD model [29], thermal resistance model [30] and finite element model [31]. In this study, two numerical modelling methods were employed, i.e., CFD simulation and 1-D thermal resistance model. For a typical PV-PCM system, the mathematical model could be generally divided into four main parts, i.e. fluid model, heat transfer model, phase change model and power generation model.
应首先建立 PV-PCM 系统的数学模型，以便对系统性能进行数值研究，这些模型还可用于研究设计参数和边界条件对系统性能的影响。文献中使用了多种建模方法，包括 CFD 模型 [29]、热阻模型 [30] 和有限元模型 [31]。本研究采用了两种数值建模方法，即 CFD 模拟和一维热阻模型。对于典型的 PV-PCM 系统，数学模型一般可分为四个主要部分，即流体模型、传热模型、相变模型和发电模型。

2.2.1. Fluid model 2.2.1.流体模型

Based on the two-dimensional coordinate system in Fig. 1(b), the momentum equation, i.e., Navier–Stokes equation, is shown below:(1)

ρ \frac{\partial u}{\partial t} + ρ u \frac{\partial u}{\partial x} + ρ v \frac{\partial u}{\partial y} = \nabla (μ \nabla u) - \frac{\partial P}{\partial x}

(2)

ρ \frac{\partial v}{\partial t} + ρ u \frac{\partial v}{\partial x} + ρ v \frac{\partial v}{\partial y} = \nabla (μ \nabla v) - \frac{\partial P}{\partial y} - ρ g

where

ρ

is the fluid density (kg/m³);

u

and

v

is the x-velocity and y-velocity (m/s);

g

is the gravity acceleration (kg m/s²);

μ_{l}

and

μ_{s}

is the viscosity of liquid PCM and solid PCM respectively (N s/m²). The same denotation is used for density

ρ

.
根据图 1(b) 中的二维坐标系，动量方程（即纳维-斯托克斯方程）如下所示： (1)

ρ \frac{\partial u}{\partial t} + ρ u \frac{\partial u}{\partial x} + ρ v \frac{\partial u}{\partial y} = \nabla (μ \nabla u) - \frac{\partial P}{\partial x}

(2)

ρ \frac{\partial v}{\partial t} + ρ u \frac{\partial v}{\partial x} + ρ v \frac{\partial v}{\partial y} = \nabla (μ \nabla v) - \frac{\partial P}{\partial y} - ρ g

其中

ρ

是流体密度（千克/米 ³ ）；

u

和

v

是 x-速度和 y-速度（米/秒）；

g

是重力加速度（kg m/s ² ）；

μ_{l}

和

μ_{s}

分别是液体 PCM 和固体 PCM 的粘度（N s/m ² ）。密度

ρ

也使用了相同的表示方法。

In Eqs. (1), (2), when PCM is in solid state:

μ = μ_{s} = \infty, ρ = ρ_{s}

; when PCM is in liquid state:

μ = μ_{l}, ρ = ρ_{l}

.
在公式 (1)、(2) 中，当 PCM 处于固态时，

μ = μ_{s} = \infty, ρ = ρ_{s}

；当 PCM 处于液态时，

μ = μ_{s} = \infty, ρ = ρ_{s}

：

μ = μ_{s} = \infty, ρ = ρ_{s}

；当 PCM 为液态时：

μ = μ_{l}, ρ = ρ_{l}

。

2.2.2. Heat transfer model
2.2.2.传热模型

Heat transfer in a typical PV-PCM system is depicted in Fig. 1(b). Besides electricity production, solar energy absorbed by the PV panel is mostly converted into heat, which is subsequently transferred to the PCM. It is assumed that there is no heat loss at the surrounding of PV-PCM system as the area is relatively small. The heat conduction within the solid part of the system, i.e., PV module, aluminum plate and the solid PCM, can be expressed as:(3)

ρ C_{p} \frac{\partial T}{\partial t} = k \frac{\partial T}{\partial x}

where

C_{p}

is the specific heat of each solid part (J/(kg K)) and

k

is the thermal conductivity of each part (W/(m K)).
图 1(b) 描述了典型 PV-PCM 系统中的热传递情况。除发电外，光伏板吸收的太阳能大部分转化为热量，随后传递给 PCM。由于 PV-PCM 系统的面积相对较小，因此假定其周围没有热量损失。系统固体部分（即光伏组件、铝板和固体 PCM）内的热传导可表示为 (3)

ρ C_{p} \frac{\partial T}{\partial t} = k \frac{\partial T}{\partial x}

其中

C_{p}

为各固体部分的比热（J/(kg K)），

k

为各部分的导热系数（W/(m K)）。

At the front and back surface of PV-PCM system, the heat exchange with the ambient environment is dominated by the convective and radiative heat transfer. The energy balance equations are presented by Eqs. (4), (5):(4)

ρ C_{p} dx \frac{\partial T}{\partial t} = α G_{PV} (1 - η) - ε_{glass} F σ (T_{surface}^{4} - T_{sky}^{4}) - h (T_{surface} - T_{amb})

(5)

ρ C_{p} dx \frac{\partial T}{\partial t} = ε_{Al} F σ (T_{back}^{4} - T_{ground}^{4}) + h (T_{back} - T_{amb})

where the absorptivity

α

is a function of absorptive and reflective properties of the encapsulating glass, laminating material and the absorptivity characteristics of the PV cell material, and it is set as 0.9 in this study [44];

G_{PV}

is the solar radiation on the PV module (W/m³);

ε_{glass}

and

ε_{Al}

is the emissivity of the glass and the aluminum plate respectively;

T_{surface}

T_{sky}

T_{back}

and

T_{ground}

is temperature of the front surface, the sky, the back surface and the ground respectively (°C);

h

is the convective heat transfer rate (W/(m² K)), which could be further defined into free convection and forced convection in Eqs. (11), (12). The net solar radiation casts on PV front surface is given as:(6)

Q = α G_{pv} A

where

A

is the surface area of PV-PCM system (m²).
在 PV-PCM 系统的前后表面，与周围环境的热交换以对流和辐射传热为主。能量平衡方程见式 (4)、(5)： (4)

ρ C_{p} dx \frac{\partial T}{\partial t} = α G_{PV} (1 - η) - ε_{glass} F σ (T_{surface}^{4} - T_{sky}^{4}) - h (T_{surface} - T_{amb})

(5)

ρ C_{p} dx \frac{\partial T}{\partial t} = ε_{Al} F σ (T_{back}^{4} - T_{ground}^{4}) + h (T_{back} - T_{amb})

其中，吸收率

α

是封装玻璃、层压材料的吸收和反射特性以及光伏电池材料的吸收特性的函数，在本研究中设为 0.9 [44]；

G_{PV}

是光伏组件上的太阳辐射（W/m ³ ）；

ε_{glass}

和

ε_{Al}

分别是玻璃和铝板的发射率；

T_{surface}

、

T_{sky}

、

T_{back}

和

T_{ground}

分别为前表面、天空、后表面和地面的温度（°C）；

h

为对流换热率（W/(m ² K)），可进一步定义为自由对流和强制对流，见式（11）和（12）。(11), (12).投射到光伏前表面的净太阳辐射为 (6)

Q = α G_{pv} A

其中

A

是 PV-PCM 系统的表面积（m ² ）。

As part of the energy is converted into electricity by photovoltaic effect, the net heat input

q

is governed by Eq. (7).(7)

q = α G_{pv} (1 - η) A

where

η

is the real efficiency of PV module. The radiative heat transfer in the front and back surface is governed by Eqs. (8), (9), (10).(8)

q_{sky} = ε_{glass} F σ (T_{surface}^{4} - T_{sky}^{4}) A

(9)

T_{sky} = 0.0375 T_{amb}^{1.5} + 0.32 T_{amb}

(10)

q_{r} = ε_{Al} F σ (T_{back}^{4} - T_{ground}^{4}) A

where the radiative heat transfer with the sky and the ground is represented by

q_{sky}

and

q_{groud}

respectively (W). The sky temperature is estimated based on the study of Biwole et al. [53], and the ground temperature is assumed to be equal to the ambient temperature. The convective heat loss at the front and back surface of the PV-PCM system is also considered. The estimation of natural convective rate

h_{free}

and forced convective rate

h_{force}

is presented in Eqs. (11), (12) [54].(11)

h_{free} = \frac{k}{L} {\{0.825 + \frac{0.387 {Ra}_{L}^{1 / 6}}{{[1 + {(0.492 / \Pr)}^{9 / 16}]}^{8 / 27}}\}}^{2}

(12)

h_{forced} = \frac{k}{L} (0.664 {Re}_{L}^{1 / 2} \Pr^{1 / 3})

where

L

is the height of PV Panel (m);

{Ra}_{L}

is the Raleigh number;

Re

is the Reynolds number and

\Pr

is the Prandtl number.
由于部分能量通过光伏效应转化为电能，因此净输入热量

q

受公式 (7) 控制。 (7)

q = α G_{pv} (1 - η) A

其中

η

是光伏组件的实际效率。前后表面的辐射传热由式 (8)、(9) 和 (10) 控制。 (8)

q_{sky} = ε_{glass} F σ (T_{surface}^{4} - T_{sky}^{4}) A

(9)

T_{sky} = 0.0375 T_{amb}^{1.5} + 0.32 T_{amb}

(10)

q_{r} = ε_{Al} F σ (T_{back}^{4} - T_{ground}^{4}) A

其中，与天空和地面的辐射传热分别用

q_{sky}

和

q_{groud}

表示（W）。天空温度是根据 Biwole 等人的研究[53]估算的，地面温度假定等于环境温度。还考虑了 PV-PCM 系统前后表面的对流热损失。自然对流率

h_{free}

和强制对流率

h_{force}

的估算见式 (11)、(12)[54]。 (11)

h_{free} = \frac{k}{L} {\{0.825 + \frac{0.387 {Ra}_{L}^{1 / 6}}{{[1 + {(0.492 / \Pr)}^{9 / 16}]}^{8 / 27}}\}}^{2}

(12)

h_{forced} = \frac{k}{L} (0.664 {Re}_{L}^{1 / 2} \Pr^{1 / 3})

其中，

L

为光伏板高度（m）；

{Ra}_{L}

为雷利数；

Re

为雷诺数；

\Pr

为普朗特数。

2.2.3. Phase change model
2.2.3.相变模型

For PCM, the heat diffusion equation in liquid state and solid state is given as:(13)

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

对于 PCM，液态和固态的热扩散方程为 (13)

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

As PCM has both latent heat capacity and sensible heat capacity, the temperature during phase change process can be divided into three stages, as shown in Eq. (14).(14)

T = \{\begin{matrix} E / c_{S} + T_{0} & E < E_{0}, T < T_{l} \\ T_{s} + ((E - E_{0}) / E_{l}) (T_{l} - T_{s}) & E_{0} < E < E_{0} + E_{l}, T_{s} < T < T_{l} \\ T_{l} + (E - E_{0} - E_{l}) / c_{L} & E ⩾ E_{0} + E_{l}, T > T_{l} \end{matrix}

where

E

E_{0}

and

E_{l}

denotes the heat already absorbed by the PCM, the minimum heat required to begin the phase change process and the latent heat capacity of the PCM respectively (J);

T

T_{s}

and

T_{l}

represents the current PCM temperature, the solidus and liquidus temperature of the PCM respectively (°C).

c_{S}

and

c_{L}

is specific heat of solid PCM and liquid PCM respectively.
由于 PCM 同时具有潜热容量和显热容量，相变过程中的温度可分为三个阶段，如公式 (14) 所示。 (14)

T = \{\begin{matrix} E / c_{S} + T_{0} & E < E_{0}, T < T_{l} \\ T_{s} + ((E - E_{0}) / E_{l}) (T_{l} - T_{s}) & E_{0} < E < E_{0} + E_{l}, T_{s} < T < T_{l} \\ T_{l} + (E - E_{0} - E_{l}) / c_{L} & E ⩾ E_{0} + E_{l}, T > T_{l} \end{matrix}

其中，

E

、

E_{0}

和

E_{l}

分别表示 PCM 已吸收的热量、开始相变过程所需的最小热量和 PCM 的潜热容量（J）；

T

、

T_{s}

和

T_{l}

分别表示当前的 PCM 温度、PCM 的固相温度和液相温度（°C）。

c_{S}

和

c_{L}

分别是固体 PCM 和液体 PCM 的比热。

2.2.4. Power generation model
2.2.4.发电模型

The power generation equation of PV cells is simulated by Eq. (15).(15)

P_{out} = η \cdot A \cdot G_{PV}

where

P_{out}

represents the energy output of PV panel (W);

η

is the real PV conversion efficiency.
光伏电池的发电方程由式 (15) 模拟。 (15)

P_{out} = η \cdot A \cdot G_{PV}

其中，

P_{out}

表示光伏电池板的能量输出（瓦）；

η

为实际光伏转换效率。

The correlation between temperature rise and the decline of PV efficiency is presented in Eq. (16).(16)

η = η_{ref} \cdot {1 - β_{ref} (T_{surface} - T_{ref})}

where the reference PV efficiency,

η_{ref}

, is 17.1% based on the information provided by the manufacturer;

β_{ref}

is the temperature coefficient of PV efficiency (0.5%/°C);

T_{surface}

and

T_{ref}

represents the real PV temperature and the reference temperature at the standard test condition respectively (°C).
温度上升与光伏效率下降之间的相关性见式 (16)。 (16)

η = η_{ref} \cdot {1 - β_{ref} (T_{surface} - T_{ref})}

其中，根据制造商提供的信息，参考光伏效率

η_{ref}

为 17.1%；

β_{ref}

是光伏效率的温度系数（0.5%/°C）；

T_{surface}

和

T_{ref}

分别代表标准测试条件下的实际光伏温度和参考温度（°C）。

3. CFD simulation and 1-D thermal resistance modelling
3.CFD 模拟和一维热阻建模

After investigating the mathematical models of PV-PCM system above, two numerical methods, i.e., CFD simulation and 1-D thermal resistance modelling, are discussed in this section, and their merits and demerits are analyzed. Finally, the improved 1-D thermal resistance model which requires less computation time is selected for the further study. The analysis below can also provide reference in choosing modelling method.
在研究了上述 PV-PCM 系统的数学模型后，本节讨论了两种数值方法，即 CFD 仿真和一维热阻建模，并分析了它们的优缺点。最后，选择了计算时间更短的改进型一维热阻模型进行进一步研究。以下分析也可为选择建模方法提供参考。

3.1. CFD simulation 3.1.CFD 模拟

The CFD model was constructed in ANSYS Fluent 17.0 in our previous study [55]. In that study, mesh independence test was conducted, and a comparison with experimental and numerical from other studies were completed, demonstrating a good agreement in patterns of PV temperature profile. A sample mesh of the PV-PCM system constructed in ANSYS is illustrated in Fig. 2. To avoid the mesh dependency program, a grid sensitivity test was conducted and the result is presented in Fig. 3. The very slight temperature difference between the three simulations demonstrates that it is reasonable to set the cell size at 1 mm ∗ 1 mm for the subsequent numerical simulations.
我们之前的研究[55]使用 ANSYS Fluent 17.0 建立了 CFD 模型。在该研究中，我们进行了网格独立性测试，并完成了与其他研究的实验和数值结果的比较，结果表明光伏温度曲线的模式与实验结果非常吻合。在 ANSYS 中构建的 PV-PCM 系统网格示例如图 2 所示。为避免网格依赖性程序，进行了网格灵敏度测试，结果如图 3 所示。三次模拟之间的温度差异非常小，这表明在随后的数值模拟中将电池尺寸设置为 1 mm ∗ 1 mm 是合理的。

Fig. 4 shows a sample result of CFD modelling. The temperature distribution and flow process of liquid PCM in the system can be observed clearly. Moreover, micro-analysis, e.g., the temperature variation and fluid behavior at each small grid, could also be easily performed by CFD modelling. However, the computation time of CFD simulation is very long, and it often consumes several days to conduct a complete case simulation. That is the major reason why only 2–20 different cases of PV-PCM system were simulated in the literature.
图 4 显示了 CFD 建模的样本结果。可以清楚地观察到系统中液体 PCM 的温度分布和流动过程。此外，通过 CFD 建模还可以轻松进行微观分析，例如每个小网格的温度变化和流体行为。然而，CFD 模拟的计算时间很长，进行一次完整的案例模拟往往需要数天时间。这也是文献中只模拟了 2-20 个不同 PV-PCM 系统案例的主要原因。

3.2. 1-D thermal resistance modelling
3.2.一维热阻建模

The optimization of a PV-PCM system should compare substantial cases, the CFD simulation seems too complex and not an ideal method for such research. Therefore, another modelling method, i.e., thermal resistance model using finite difference method, is investigated in this study. The schematic diagram of 1-D thermal resistance model is shown in Fig. 5. As indicated, PV-PCM system can be divided into several layers, each layer is a node of the thermal resistance model representing a certain material, e.g., glass, EVA, PV cells, Tedlar and so on. Particularly, the PCM is separated into layers which have a thickness of 1 mm each node to obtain a better accuracy of simulation in PCM. The selection of this node thickness is decided by consideration of avoiding mesh dependence problem and the convenience of the study. Solar radiation, radiative heat transfer, natural air convection and forced air convection are considered at the front surface of the system and the conductive heat transfer is included within the system. At the back of the system, radiative heat transfer and convective transfer with the ambient are also considered. However, it is a challenging work to investigate the convective heat transfer process within PCM because the momentum equations should be considered while the thermal resistance model cannot do that. The solution of this problem is discussed in the next section.
PV-PCM 系统的优化需要对大量案例进行比较，而 CFD 模拟似乎过于复杂，不是此类研究的理想方法。因此，本研究采用了另一种建模方法，即使用有限差分法的热阻模型。一维热阻模型示意图如图 5 所示。如图所示，PV-PCM 系统可分为若干层，每一层都是热阻模型的一个节点，代表某种材料，如玻璃、EVA、PV 电池、Tedlar 等。特别是将 PCM 分成若干层，每个节点的厚度为 1 毫米，以获得更好的 PCM 模拟精度。节点厚度的选择主要是考虑到避免网格依赖问题和研究的便利性。系统的前表面考虑了太阳辐射、辐射传热、自然空气对流和强制空气对流，系统内还包括传导传热。在系统背面，还考虑了辐射传热和与环境的对流传热。然而，研究 PCM 内部的对流传热过程是一项具有挑战性的工作，因为需要考虑动量方程，而热阻模型无法做到这一点。下一节将讨论这一问题的解决方案。

3.3. A new model for PCM heat transfer rate
3.3.PCM 传热率的新模型

To solve the problems mentioned above, an enhanced conductivity model of PCM is proposed in this study. Generally, during the phase change process from solid to liquid, the molten PCM flows within the chamber, caused by the temperature gradient. The liquid PCM flow contributes to the convective heat transfer within PCM, resulting in an increase in the total heat transfer rate. In order to take the convective effect into account, the heat transfer rate of PCM should be enhanced when the PCM is melting. Moreover, when PCM is fully melted the temperature gradient between each PCM layer would continue to drive the convective heat transfer. Therefore, two assumptions are made to include the above convective heat transfer effect: (1) the heat transfer rate within PCM chamber during melting is positively related to the liquid fraction by using a logistic function; (2) the heat transfer rate between completely molten PCM nodes is positively related to their temperature gradient. Based on the assumptions, the heat transfer rate

k_{PCM}

inside the PCM can be expressed as:(17)

k_{PCM} = \{\begin{matrix} k_{solidus} & if T < T_{s} \\ k_{solidus} + \frac{k_{liquidus}}{1 + e^{\frac{ξ (T - \frac{T_{l} + T_{s}}{2})}{T_{l} - T_{s}}}} & if T_{s} < T < T_{l} \\ (k_{solidus} + k_{liquidus}) {(T_{n} - T_{n + 1})}^{ω} & if T ⩾ T_{l} \end{matrix}

where

k_{solidus}

is the heat conductivity of solid PCM (W/(m K));

k_{liquidus}

is the heat transfer rate rise at the moment when the PCM node is completely transferred into liquid state (W/(m K));

ξ

and

ω

are constants determined by the property of PCM;

T_{n}

and

T_{n + 1}

is the temperature of two adjacent PCM layers (°C).
为了解决上述问题，本研究提出了 PCM 的增强传导模型。一般来说，在从固态到液态的相变过程中，由于温度梯度的作用，熔融 PCM 会在腔体内流动。液态 PCM 流动会促进 PCM 内部的对流传热，从而提高总传热率。为了将对流效应考虑在内，PCM 的传热速率应在 PCM 熔化时得到提高。此外，当 PCM 完全融化时，每层 PCM 之间的温度梯度将继续推动对流传热。因此，为了包含上述对流换热效应，我们做了两个假设：（1）通过使用对数函数，熔化过程中 PCM 腔内的换热率与液体分数呈正相关；（2）完全熔化的 PCM 节点之间的换热率与它们的温度梯度呈正相关。基于上述假设，PCM 内部的传热速率

k_{PCM}

可表示为 (17)

k_{PCM} = \{\begin{matrix} k_{solidus} & if T < T_{s} \\ k_{solidus} + \frac{k_{liquidus}}{1 + e^{\frac{ξ (T - \frac{T_{l} + T_{s}}{2})}{T_{l} - T_{s}}}} & if T_{s} < T < T_{l} \\ (k_{solidus} + k_{liquidus}) {(T_{n} - T_{n + 1})}^{ω} & if T ⩾ T_{l} \end{matrix}

其中，

k_{solidus}

为固态 PCM 的导热系数（W/(m K)）；

k_{liquidus}

为 PCM 节点完全转为液态时的传热速率升高值（W/(m K)）；

ξ

和

ω

为由 PCM 特性决定的常数；

T_{n}

和

T_{n + 1}

为相邻两层 PCM 的温度（℃）。

Based on Eq. (17), the profile of RT35 heat transfer rate during its melting process is simulated and shown in Fig. 6.
根据公式 (17)，模拟了 RT35 熔化过程中的传热率曲线，如图 6 所示。

To calibrate the thermal resistance model, meaning determining the value of two constants

ξ

and

ω

in Eq. (17), the result from CFD simulation is used to compare with the result from the proposed new model as the CFD simulation method is widely used in the research of PV-PCM system. The value of

k_{liquidus}

is the averaged heat transfer rate rise derived from CFD model at the moment when all PCM grids are completely molten and is determined as 4.82. To determine the value of

ξ

and

ω

, melting processes and convection processes simulated by CFD model were compared with results of 1-D model according to the averaged discrepancy caused by varying these two values. Finally,

ξ

and

ω

is determined as 10 and 2 respectively. After calibration, a PV-PCM system, with 30 mm RT35 PCM under 27 °C ambient temperature and 900 W/m² solar radiation, was simulated for validation. To be noted that in simulation, the PCM chamber was not finned due to the difficulties in incorporating fins into 1-D thermal resistance model and the effect of fins would be equivalent to certain changes in thermal conductivity. Finally, according to the simulations, the root-mean-square deviation (RMSD) between two methods is 0.173 and the normalized RMSD calculated by the range of data is 0.5%, presenting a high degree of agreement as shown in Fig. 7. Moreover, the improved 1-D model shows great advantage in shortening computation time, i.e., over 300 times faster than the CFD model in this case. Therefore, it is indicated that the proposed new model is reliable and feasible for further investigation.
为了校准热阻模型，即确定式 (17) 中的两个常数

ξ

和

ω

的值，我们使用了 CFD 仿真的结果与所提出的新模型的结果进行比较，因为 CFD 仿真方法在 PV-PCM 系统的研究中得到了广泛应用。

k_{liquidus}

值是所有 PCM 网格完全熔化时从 CFD 模型得出的平均传热速率上升值，确定为 4.82。为了确定

ξ

和

ω

的值，根据这两个值变化所造成的平均差异，将 CFD 模型模拟的熔化过程和对流过程与一维模型的结果进行了比较。最后，确定

ξ

和

ω

分别为 10 和 2。校准后，模拟了一个 PV-PCM 系统，在 27 °C 环境温度和 900 W/m ² 太阳辐射条件下使用 30 mm RT35 PCM 进行验证。需要注意的是，在模拟过程中，PCM 室没有安装鳍片，这是因为在一维热阻模型中安装鳍片存在困难，而且鳍片的影响相当于热导率的某些变化。最后，根据模拟结果，两种方法的均方根偏差（RMSD）为 0.173，按数据范围计算的归一化均方根偏差（RMSD）为 0.5%，呈现出高度一致，如图 7 所示。此外，改进后的一维模型在缩短计算时间方面具有很大优势，在本例中比 CFD 模型快 300 多倍。因此，所提出的新模型是可靠和可行的，值得进一步研究。

4. Study on different assumptions in mathematical modelling
4.数学建模中不同假设的研究

In the literature, various assumptions have been made in their mathematical models but the impact of those assumptions on result has not been sufficiently reported. Therefore, as a first attempt, the significance of different assumptions on simulation result is examined to ensure its accuracy for the further analysis. Three commonly used assumptions are summarized as follows:
在文献中，人们在数学模型中做出了各种假设，但这些假设对结果的影响尚未得到充分报道。因此，作为首次尝试，我们研究了不同假设对模拟结果的影响，以确保进一步分析的准确性。现将三种常用假设总结如下：

•
Assumption 1: only conductive heat transfer is considered within PCM and the convective heat transfer within liquid PCM is neglected for simplification [41], [43];
假设 1：只考虑 PCM 内部的传导传热，为简化起见，忽略液体 PCM 内部的对流传热 [41], [43]；
•
Assumption 2: only natural air convection or forced air convection is considered at the surrounding of system, while the radiative heat transfer is neglected [56], [57];
假设 2：只考虑系统周围的自然空气对流或强制空气对流，而忽略辐射传热 [56], [57]；
•
Assumption 3: The PV module is treated as a layer of aluminum plate for simplification [31].
假设 3：为简化起见，将光伏组件视为一层铝板 [31]。

To examine the above assumptions, a typical PV-PCM system is simulated under ambient temperature at 25 °C, solar radiation at 800 W/m², thickness of RT35 PCM at 30 mm, and wind speed at zero.
为了验证上述假设，模拟了一个典型的 PV-PCM 系统，环境温度为 25 °C，太阳辐射为 800 W/m ² ，RT35 PCM 厚度为 30 mm，风速为零。

4.1. Assumption 1 – neglect of convective heat transfer within liquid PCM
4.1.假设 1 - 忽视液体 PCM 内部的对流传热

As mentioned earlier, due to the inherent weakness of 1-D thermal resistance model, the convective heat transfer in liquid PCM is usually neglected and only conductive heat transfer is considered. The convective effect, however, could be significant as the temperature gradient induces considerable flow in liquid PCM when the PCM is melting, and the heat transfer rate is thus enhanced. Based on the newly developed model for PCM heat transfer rate in Eq. (17), the impact of such convective heat transfer on the PV temperature can be simulated.
如前所述，由于一维热阻模型的固有缺陷，液体 PCM 中的对流传热通常被忽略，只考虑传导传热。然而，对流效应可能非常重要，因为当 PCM 溶化时，温度梯度会在液体 PCM 中产生大量流动，从而提高传热速率。根据公式 (17) 中新开发的 PCM 传热率模型，可以模拟这种对流传热对光伏温度的影响。

As presented in Fig. 8, a significant gap between the two temperature profiles can be observed. When the convective heat transfer in PCM is considered, the PV temperature is 10 °C lower than the one excluded. Therefore, the conclusion is that Assumption 1 can lead to some significant error and the convective heat transfer should be considered in the simulation.
如图 8 所示，两种温度曲线之间存在明显差距。当考虑 PCM 中的对流传热时，PV 温度比不考虑对流传热的温度低 10 °C。因此，结论是假设 1 可能会导致一些重大误差，模拟中应考虑对流换热。

4.2. Assumption 2 – neglect of radiative heat transfer with ambient
4.2.假设 2 - 忽略与周围环境的辐射热传递

It is worthwhile to examine the impact of neglect of radiative heat transfer with ambient on system performance. The result in Fig. 9 demonstrates that, if the radiative heat dissipation with the sky and the ground is not considered, the melting time of PCM in the system is reduced and the PV temperature increases more quickly. At the end of simulation, the temperature difference can be up to about 20 °C as the heat radiation plays a very important role in heat transfer under high temperature. Thus, it can be concluded that radiative heat transfer should be included.
值得研究的是忽略与环境的辐射传热对系统性能的影响。图 9 中的结果表明，如果不考虑与天空和地面的辐射散热，系统中 PCM 的熔化时间会缩短，光伏温度会上升得更快。在模拟结束时，由于热辐射在高温条件下的热传导中起着非常重要的作用，因此温差可达约 20 °C。因此，可以得出结论，辐射传热应包括在内。

4.3. Assumption 3 – simplification of PV panel
4.3.假设 3 - 简化光伏电池板

The PV module is simplified as an aluminum plate in Ref. [31]. However, as the PV module is usually composed of glass, PV cell, Ethylene-vinyl acetate (EVA) and Tedlar-Polyester-Tedlar (TPT), the simplification of PV module as an aluminum plate could lead to a quite different specific heat capacity and conductive heat transfer rate. The effect of such simplification is investigated in this section. The result in Fig. 10 illustrates that the difference between two temperature profiles is not very significant, with an average difference at about ±1.5 °C. Therefore, the conclusion from this section is that the simplification of PV module as an aluminum plate can bring some errors in result while it might be acceptable if some deviation is allowed.
参考文献 [31] 将光伏组件简化为铝板。[31].然而，由于光伏组件通常由玻璃、光伏电池、乙烯-醋酸乙烯（EVA）和 Tedlar-聚酯-Tedlar（TPT）组成，因此将光伏组件简化为铝板可能会导致不同的比热容和传导热传导率。本节将研究这种简化的影响。图 10 中的结果表明，两个温度曲线之间的差异并不大，平均差异约为±1.5 °C。因此，本节得出的结论是，将光伏组件简化为铝板可能会导致结果出现一些误差，但如果允许出现一些偏差，则是可以接受的。

5. Sensitivity analysis of the PV-PCM system design
5.PV-PCM 系统设计的敏感性分析

After taking both convective and radiative effect into account, the improved thermal resistance model is finally adopted for optimizing PV-PCM system design from over 300 cases with different configurations and different conditions, i.e. ambient temperature, wind speed and solar radiation.
在考虑了对流效应和辐射效应后，最终采用改进的热阻模型，从 300 多个不同配置和不同条件（即环境温度、风速和太阳辐射）的案例中优化 PV-PCM 系统的设计。

5.1. Investigation of the melting process
5.1.熔化过程调查

To understand heat transfer process and melting process in the system, a specific case was modeled as an attempt. The simulation result is presented in Fig. 11 by using the contour map. It takes more time to increase PV temperature from 35 °C to 40 °C in contrast to other temperature intervals. This is reasonable because the melting temperature range of RT35 is 29–36 °C, indicating that during the phase change period, huge amount of latent heat can be absorbed and stored in PCM. In total it takes about 120 min to melt the PCM completely, and then the PV surface temperature rises rapidly from 40 °C to over 60 °C in a short time. Specifically, it takes only about 15 min to increase from 40 °C to 45 °C while it needs about 100 min from 35 °C to 40 °C. This phenomenon demonstrates the role of the PCM in PV temperature control, particularly around the phase change temperature.
为了了解系统中的传热过程和熔化过程，我们尝试对一个具体案例进行了模拟。模拟结果如图 11 所示。与其他温度区间相比，光伏温度从 35 °C 升至 40 °C 所需的时间更长。这是有道理的，因为 RT35 的熔化温度范围为 29-36 °C，这表明在相变期间，PCM 可以吸收并储存大量潜热。PCM 完全熔化总共需要 120 分钟左右，然后光伏表面温度会在短时间内从 40 °C 迅速升至 60 °C 以上。具体来说，从 40 °C 升至 45 °C 只需约 15 分钟，而从 35 °C 升至 40 °C 则需要约 100 分钟。这一现象表明了 PCM 在光伏温度控制中的作用，尤其是在相变温度附近。

The melting process of PCM can be observed in Fig. 11(b), which presents the liquid fraction along the PCM depth during the melting process. It is worthy to point out that although it causes exactly same amount of energy to melt every 10% of PCM, it takes less time to melt the first several tenth of PCM than others afterward. This could be explained that the heat removal caused by natural air convection becomes intensive as the PV temperature rises. In this example, 143 min are required for PCM melting completely.
从图 11(b) 中可以观察到 PCM 的熔化过程，图 11(b) 显示了熔化过程中沿 PCM 深度的液体分量。值得注意的是，虽然熔化每 10%的 PCM 所需的能量完全相同，但熔化前十分之一的 PCM 所需的时间比熔化后其他 PCM 所需的时间要短。这可以解释为，随着光伏温度的升高，自然空气对流导致的热量带走变得越来越密集。在本例中，PCM 完全熔化需要 143 分钟。

5.2. Sensitivity analysis on a single variable
5.2.单一变量的敏感性分析

In this section, the effect of changing several key parameters on simulation result, i.e. ambient temperature, solar radiation, wind speed, PCM thickness and latent heat capacity, has been investigated. Specifically, results of 300 cases are graphically illustrated from Sections 5.2–5.3. Configurations and boundary conditions of the cases are also described in these sections. Such sensitivity analysis can help the designers understand the effect of uncertainty, and it can also help the modelers to determine the impact that variations in assumed inputs have on the behavior, feasibility, and economics of a particular system configuration. Sensitivity analysis can also check the robustness of a particular system configuration. That is whether it is optimal in other scenarios when initial conditions have been changed.
本节研究了改变几个关键参数（即环境温度、太阳辐射、风速、PCM 厚度和潜热容量）对模拟结果的影响。具体而言，第 5.2-5.3 节以图表说明了 300 个案例的结果。这些章节还描述了案例的配置和边界条件。这种敏感性分析可以帮助设计人员了解不确定性的影响，也可以帮助建模人员确定假定输入的变化对特定系统配置的行为、可行性和经济性的影响。敏感性分析还可以检查特定系统配置的稳健性。也就是说，当初始条件发生变化时，该系统配置在其他情况下是否最优。

5.2.1. Ambient temperature
5.2.1.环境温度

A sensitivity analysis on ambient temperature can help users to understand the performance of a particular system in different locations or seasons. Fig. 12 provides that as the ambient temperature increases, the simulated PV temperature increases as well and the melting time is reduced. The total PCM melting process only takes about 1 h for ambient temperature at 30 °C, while it takes more than 2 h when ambient temperature is 20 °C.
对环境温度的敏感性分析可以帮助用户了解特定系统在不同地点或季节的性能。图 12 显示，随着环境温度的升高，模拟光伏温度也随之升高，熔化时间缩短。环境温度为 30 °C 时，整个 PCM 熔化过程仅需 1 小时左右，而环境温度为 20 °C 时则需要 2 小时以上。

To achieve a better performance of PV-PCM system under high ambient temperature condition, like Shanghai, a high melting temperature or high latent heat capacity is recommended. Fig. 12 also suggests that 5 °C increase in ambient temperature can lead to almost the same increase in the PV temperature at the end of simulation.
为了使 PV-PCM 系统在高环境温度条件下（如上海）获得更好的性能，建议采用高熔融温度或高潜热容量。图 12 还表明，在模拟结束时，环境温度升高 5 °C 几乎会导致光伏温度的相同升高。

5.2.2. Solar radiation intensity
5.2.2.太阳辐射强度

This section examines how the solar radiation intensity affects the system performance. As shown in Fig. 13, solar radiation has significant impact on the system performance, a higher solar radiation results in a shorter melting period and consequently a higher PV temperature. At the end, all PCM is molten in these cases. On average, a 100 W/m² increase in solar radiation can result in a 5 °C increase in PV temperature when PCM is totally molten and PV temperature becomes stable. This relation could be applied to predict temperature variance in real applications.
本节将探讨太阳辐射强度如何影响系统性能。如图 13 所示，太阳辐射对系统性能有重大影响，太阳辐射越强，熔化时间越短，光伏温度越高。在这些情况下，所有 PCM 最终都会熔化。平均而言，当 PCM 完全熔化且 PV 温度趋于稳定时，太阳辐射每增加 100 W/m ² 会导致 PV 温度上升 5 °C。这一关系可用于预测实际应用中的温度变化。

5.2.3. Wind speed 5.2.3.风速

In this section, the effect of surrounding wind speed on system performance is discussed, this has not been sufficiently examined by other studies according to the knowledge of the authors. Fig. 14 illustrates that the heat dissipation effect of wind is significant when the wind speed increases from 0 m/s to 1 m/s, with PV temperature reduced by about 5 °C. However, such temperature reduction is not so obvious when the wind speed continues to rise. Fig. 14 also suggests that different wind speed has little impact on PCM melting time and PV temperature.
本节将讨论周围风速对系统性能的影响，据作者所知，其他研究尚未对此进行充分研究。图 14 显示，当风速从 0 m/s 增加到 1 m/s 时，风的散热效果显著，光伏温度降低了约 5 °C。然而，当风速继续上升时，这种温度降低就不那么明显了。图 14 还表明，不同风速对 PCM 熔化时间和 PV 温度的影响很小。

5.2.4. PCM thickness 5.2.4.PCM 厚度

Fig. 15 investigates the effect of the thickness of PCM, one of the key parameters of PV-PCM system. The result shows that PV temperature in PV-PCM system could be reduced by up to 35 °C when compared to traditional PV-only module. Correspondingly, the maximum output gap between two systems can be enlarged to 17.5% of PV reference output power. It is observed in Fig. 15 that every 5 mm increase in thickness prolongs the melting period by about 10 min on average.
图 15 研究了 PCM 厚度的影响，这是 PV-PCM 系统的关键参数之一。结果表明，与传统的纯光伏模块相比，PV-PCM 系统中的光伏温度最多可降低 35 °C。相应地，两个系统之间的最大输出差距可扩大到光伏参考输出功率的 17.5%。从图 15 中可以看出，厚度每增加 5 毫米，熔化时间平均延长约 10 分钟。

Another finding is that when the PCM is fully molten in the end, the PV temperature of PV-PCM system could be higher than that of traditional PV-only system. Thus, sometimes PCM could become a thermal resistance for heat dissipation at the back of PV panel, increasing PV temperature conversely. This phenomenon might also happen in winter when temperature of PCM would not reach to the melting point. It highlights that a proper selection of PCM thickness and melting point according to local climate is really important. Therefore, further research is suggested to deal with the real climate condition for a whole year.
另一个发现是，当 PCM 最终完全熔化时，PV-PCM 系统的光伏温度可能会高于传统的纯光伏系统。因此，有时 PCM 会成为 PV 面板背面散热的热阻，反而会增加 PV 温度。这种现象也可能发生在冬季，因为 PCM 的温度不会达到熔点。这突出表明，根据当地气候适当选择 PCM 厚度和熔点非常重要。因此，建议进一步研究全年的实际气候条件。

5.2.5. PCM latent heat capacity
5.2.5.PCM 潜热容量

Another impact factor is the latent heat capacity of PCM. Fig. 16 presents a sensitivity analysis on latent heat capacity of PCM RT35. The effect of latent heat capacity on simulation results is quite uniform, every 40 kJ/kg increase in latent heat capacity results in about 20 min’ increase in melting time. Similar to Fig. 15, Fig. 16 also indicates that the molten PCM could become a thermal resistance.
另一个影响因素是 PCM 的潜热容量。图 16 显示了对 PCM RT35 潜热容量的敏感性分析。潜热容量对模拟结果的影响相当一致，潜热容量每增加 40 kJ/kg，熔化时间就会增加约 20 分钟。与图 15 相似，图 16 也表明熔融 PCM 可能成为热阻。

5.3. Sensitivity analysis on two variables
5.3.两个变量的敏感性分析

A two-variable analysis method has been adopted in this study to optimize system configuration. The improved thermal resistance model is employed to simulate different system configurations by changing two key parameters simultaneously, including PCM thickness, PCM types, solar radiation, ambient temperature and wind speed. After simulation, a 2-D profile of PV temperature and power generation, which is also called contour map, based on two variables can be obtained. The two-variable sensitivity analysis method can demonstrate the system performance vividly, which is very helpful for researchers and engineers to optimize a system design if two variables should be considered simultaneously.
本研究采用了双变量分析方法来优化系统配置。通过同时改变两个关键参数，包括 PCM 厚度、PCM 类型、太阳辐射、环境温度和风速，采用改进的热阻模型模拟不同的系统配置。模拟后，可获得基于两个变量的光伏温度和发电量的二维剖面图，也称为等值线图。双变量灵敏度分析方法可以生动地展示系统性能，对于同时考虑两个变量的研究人员和工程师优化系统设计非常有帮助。

5.3.1. Effects of solar radiation and PCM thickness
5.3.1.太阳辐射和 PCM 厚度的影响

Figs. 17 and 18 present the system performance simulated at the ambient temperature of 25 °C. RT35 is used as PCM in this analysis. Based on Fig. 17, the optimal PCM thickness can be determined according to solar radiation intensity and the targeted PV temperature. For example, if PV temperature should be controlled under 60 °C when solar radiation is 800 W/m², the thickness should be at least 35 mm. Besides, according to the slope of isothermal line in Fig. 17, it would be very difficult to control the PV temperature to a low value if only PCM thickness is increased when solar radiation is at a high level. However, it does not mean that it is infeasible to apply PV-PCM system in high radiation area.
图 17 和 18 显示了在环境温度为 25 °C 时模拟的系统性能。本分析使用 RT35 作为 PCM。根据图 17，可以根据太阳辐射强度和目标光伏温度确定最佳 PCM 厚度。例如，当太阳辐射为 800 W/m ² 时，如果光伏温度应控制在 60 °C 以下，则 PCM 厚度至少应为 35 mm。此外，根据图 17 中等温线的斜率，当太阳辐射处于较高水平时，如果只增加 PCM 厚度，则很难将光伏温度控制在较低值。不过，这并不意味着在高辐射区域应用 PV-PCM 系统不可行。

Fig. 18 presents the electricity production enhancement of PV-PCM system compared to traditional PV-only module under the same condition. Indeed, it reveals that the PV-PCM system can perform better as the solar radiation rises. The latent heat capacity of PCM can control PV temperature around its melting temperature while the temperature of PV-only system rises rapidly at high solar radiation level. Therefore, the temperature difference between two systems is significant, resulting in a substantial increase in PV conversion efficiency when the PCM is added. As a whole, the improvement ranges from 5% to 12% after 5-h simulation.
图 18 显示了相同条件下 PV-PCM 系统与传统纯光伏组件相比的发电量提升情况。事实上，随着太阳辐射的增加，PV-PCM 系统的性能会更好。PCM 的潜热能力可以将光伏温度控制在其熔化温度附近，而纯光伏系统的温度在太阳辐射水平较高时迅速升高。因此，两个系统之间的温差很大，加入 PCM 后，光伏转换效率会大幅提高。总体而言，经过 5 小时的模拟后，光电转换效率提高了 5%至 12%。

5.3.2. Effects of ambient temperature and PCM thickness
5.3.2.环境温度和 PCM 厚度的影响

Figs. 19 and 20 discuss the effect of PCM thickness and ambient temperature. PCM RT35 is used and the solar radiation is set as 800 W/m². Fig. 19 shows that if the ambient temperature is higher than the melting temperature range of PCM, i.e. 29–36 °C, an increase in PCM thickness has no effects in PV temperature control. As illustrated in Fig. 20, when ambient temperature changes from 15 to 35 °C, the PV power production improvement increases in the beginning, peaking at ambient temperature of about 25 °C and then followed by an obvious descending tendency. This interesting phenomenon could be explained by the temperature gap between PV-only system and PV-PCM system during the 5-h simulation. In the beginning when ambient temperature rises from 15 to 25 °C, the latent heat storage capacity of PCM can successfully control the PV temperature at a lower value than the PV-only system, therefore, the PV power improvement is significant in this range. However, if ambient temperature continues to rise from 25 to 35 °C, the melting time of PCM has been shorten greatly due to reduced latent heat capacity since the ambient temperature is close to or over the low boundary of melting temperature range, which makes the PV temperature in PV-PCM system has a close or equal value with PV-only system. Therefore, the power enhancement effect of PCM is not obvious at high ambient temperatures. In this regard, as demonstrated in this case, it is recommended to use PCM in real applications with slightly higher melting temperature than the ambient temperature, for example 5 °C, in order to maximize the enhancement in PV conversion efficiency.
图 19 和 20 讨论了 PCM 厚度和环境温度的影响。使用的是 PCM RT35，太阳辐射设定为 800 W/m ² 。图 19 显示，如果环境温度高于 PCM 的熔化温度范围，即 29-36 °C，则增加 PCM 厚度对光伏温度控制没有影响。如图 20 所示，当环境温度从 15 °C变为 35 °C时，光伏发电量的提高幅度开始增大，在环境温度约为 25 °C时达到峰值，随后出现明显的下降趋势。这一有趣的现象可以用 5 小时模拟期间纯光伏系统和 PV-PCM 系统之间的温度差来解释。在环境温度从 15 °C上升到 25 °C的初期，PCM 的潜热存储能力可以成功地将光伏温度控制在比纯光伏系统更低的值，因此在此范围内光伏功率的提高非常显著。但是，如果环境温度在 25 至 35 ° C 之间持续上升，由于环境温度接近或超过了 PCM 熔化温度范围的低边界，PCM 的熔化时间因潜热容量降低而大大缩短，这使得 PV-PCM 系统中的 PV 温度与纯 PV 系统接近或相等。因此，在环境温度较高时，PCM 的功率增强效果并不明显。因此，如本案例所示，建议在实际应用中使用熔化温度略高于环境温度的 PCM，例如 5 °C，以最大限度地提高光伏转换效率。

6. Conclusions 6.结论

In this study, the mathematical model of the PV-PCM system is developed. The CFD model and 1-D thermal resistance model are compared for numerical study on the proposed PV-PCM system. It was found that the CFD simulation cannot perform large number of simulation cases while the traditional 1-D thermal resistance model cannot solve the momentum equations of the molten PCM. Therefore, an improved thermal resistant model is developed, which consumes less computation time than CFD method and can incorporate the convective heat transfer effect within melted PCM through applying the enhanced conductivity method. The newly developed equation for convective heat transfer was simulated in MATLAB software and validated through CFD method, presenting a high degree of agreement. The improved thermal resistance model was then further employed to examine the impacts of several common assumptions in mathematical modelling. The result shows that the neglect of convective heat transfer, neglect of radiative heat transfer, and simplification of PV module as an aluminate plate could result in about 20, 10 and ±1.5 °C difference in PV temperature respectively, indicating that the first two assumptions are unreasonable. Moreover, a study of over 300 cases with different configurations has been completed. It is observed that solar radiation is a dominant factor for determining the PV temperature. The results illustrate that, on average, every 100 W/m² increase in solar radiation can lead to 5 °C increase in PV temperature. Furthermore, a two-variable sensitivity analysis is performed, illustrating that the PV-PCM system has great potential for implementation in high solar radiation area, and around 5 °C higher than the ambient temperature is recommended for the PCM’s phase change temperature. It is believed that the findings from this study can provide some reference for optimizing PV-PCM system in real application. By integrating the two methods, i.e., the enhanced conductivity method and two-variable sensitivity analysis, future study would be carried out in simulation under real climate conditions during a year-round time period. The enhanced conductivity method can also be further developed by using dimensionless parameters to expand its application in different geometries and properties.
本研究建立了 PV-PCM 系统的数学模型。在对拟议的 PV-PCM 系统进行数值研究时，比较了 CFD 模型和一维热阻模型。结果发现，CFD 仿真无法执行大量的仿真案例，而传统的一维热阻模型无法求解熔融 PCM 的动量方程。因此，我们开发了一种改进的热阻模型，它比 CFD 方法消耗更少的计算时间，并能通过应用增强传导法将熔融 PCM 内的对流传热效应纳入其中。新开发的对流传热方程在 MATLAB 软件中进行了模拟，并通过 CFD 方法进行了验证，结果显示两者高度一致。改进后的热阻模型被进一步用于检验数学建模中几个常见假设的影响。结果表明，忽略对流传热、忽略辐射传热以及将光伏组件简化为铝酸盐板分别会导致光伏温度相差约 20 °C、10 °C 和 ±1.5 °C，这表明前两种假设是不合理的。此外，还完成了对 300 多个不同配置案例的研究。研究发现，太阳辐射是决定光伏温度的主要因素。结果表明，太阳辐射平均每增加 100 W/m ² 可导致光伏温度上升 5 °C。此外，还进行了双变量敏感性分析，结果表明 PV-PCM 系统在高太阳辐射地区的应用潜力巨大，建议 PCM 的相变温度比环境温度高 5 ℃ 左右。相信本研究的结果能为实际应用中的 PV-PCM 系统优化提供一些参考。通过整合两种方法，即增强电导率法和双变量敏感性分析法，未来的研究将在全年实际气候条件下进行模拟。增强传导性方法还可以通过使用无量纲参数进一步发展，以扩大其在不同几何形状和属性中的应用。

Acknowledgments 致谢

The authors would appreciate the financial supports provided by the National Natural Science Foundation of China (NSFC) through the Grant 51506183. Special thanks would be given to the colleagues from Renewable Energy Research Group in The Hong Kong Polytechnic University who provided great support for this project.
作者感谢国家自然科学基金委员会（NSFC）通过 51506183 号基金提供的资金支持。特别感谢香港理工大学可再生能源研究组的同事为本项目提供的大力支持。

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Mathematical modelling and sensitivity analysis of solar photovoltaic panel integrated with phase change material集成相变材料的太阳能光伏板的数学建模和敏感性分析

Highlights 亮点

Abstract 摘要

Keywords 关键词

Nomenclature 术语

1. Introduction 1.导言

2. System configuration and mathematical models2.系统配置和数学模型

2.1. System configuration2.1.系统配置

2.2. Mathematical models 2.2.数学模型

2.2.1. Fluid model 2.2.1.流体模型

2.2.2. Heat transfer model2.2.2.传热模型

2.2.3. Phase change model2.2.3.相变模型

2.2.4. Power generation model2.2.4.发电模型

3. CFD simulation and 1-D thermal resistance modelling3.CFD 模拟和一维热阻建模

3.1. CFD simulation 3.1.CFD 模拟

3.2. 1-D thermal resistance modelling3.2.一维热阻建模

3.3. A new model for PCM heat transfer rate3.3.PCM 传热率的新模型

4. Study on different assumptions in mathematical modelling4.数学建模中不同假设的研究

4.1. Assumption 1 – neglect of convective heat transfer within liquid PCM4.1.假设 1 - 忽视液体 PCM 内部的对流传热

4.2. Assumption 2 – neglect of radiative heat transfer with ambient4.2.假设 2 - 忽略与周围环境的辐射热传递

4.3. Assumption 3 – simplification of PV panel4.3.假设 3 - 简化光伏电池板

5. Sensitivity analysis of the PV-PCM system design5.PV-PCM 系统设计的敏感性分析

5.1. Investigation of the melting process5.1.熔化过程调查

5.2. Sensitivity analysis on a single variable5.2.单一变量的敏感性分析

5.2.1. Ambient temperature5.2.1.环境温度

5.2.2. Solar radiation intensity5.2.2.太阳辐射强度

5.2.3. Wind speed 5.2.3.风速

5.2.4. PCM thickness 5.2.4.PCM 厚度

5.2.5. PCM latent heat capacity5.2.5.PCM 潜热容量

5.3. Sensitivity analysis on two variables5.3.两个变量的敏感性分析

5.3.1. Effects of solar radiation and PCM thickness5.3.1.太阳辐射和 PCM 厚度的影响

5.3.2. Effects of ambient temperature and PCM thickness5.3.2.环境温度和 PCM 厚度的影响

6. Conclusions 6.结论

Acknowledgments 致谢

References

Mathematical modelling and sensitivity analysis of solar photovoltaic panel integrated with phase change material
集成相变材料的太阳能光伏板的数学建模和敏感性分析

2. System configuration and mathematical models
2.系统配置和数学模型

2.1. System configuration
2.1.系统配置

2.2.2. Heat transfer model
2.2.2.传热模型

2.2.3. Phase change model
2.2.3.相变模型

2.2.4. Power generation model
2.2.4.发电模型

3. CFD simulation and 1-D thermal resistance modelling
3.CFD 模拟和一维热阻建模

3.2. 1-D thermal resistance modelling
3.2.一维热阻建模

3.3. A new model for PCM heat transfer rate
3.3.PCM 传热率的新模型

4. Study on different assumptions in mathematical modelling
4.数学建模中不同假设的研究

4.1. Assumption 1 – neglect of convective heat transfer within liquid PCM
4.1.假设 1 - 忽视液体 PCM 内部的对流传热

4.2. Assumption 2 – neglect of radiative heat transfer with ambient
4.2.假设 2 - 忽略与周围环境的辐射热传递

4.3. Assumption 3 – simplification of PV panel
4.3.假设 3 - 简化光伏电池板

5. Sensitivity analysis of the PV-PCM system design
5.PV-PCM 系统设计的敏感性分析

5.1. Investigation of the melting process
5.1.熔化过程调查

5.2. Sensitivity analysis on a single variable
5.2.单一变量的敏感性分析

5.2.1. Ambient temperature
5.2.1.环境温度

5.2.2. Solar radiation intensity
5.2.2.太阳辐射强度

5.2.5. PCM latent heat capacity
5.2.5.PCM 潜热容量

5.3. Sensitivity analysis on two variables
5.3.两个变量的敏感性分析

5.3.1. Effects of solar radiation and PCM thickness
5.3.1.太阳辐射和 PCM 厚度的影响

5.3.2. Effects of ambient temperature and PCM thickness
5.3.2.环境温度和 PCM 厚度的影响