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模型假设采用1070nm连续激光,辐照方向垂直于电池表面,横截面强度为高斯分布。由于光束为中心对称,所以模型建立为2维轴对称模型。图 1为电池物理模型和各层尺寸示意图。图 1中,z轴为对称轴,r方向为太阳电池径向,a为太阳电池厚度,b为电池半径,其值为1.5cm。
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柱坐标系下各向同性均匀材料的热传导偏微分方程为:
$ \begin{array}{*{20}{c}} {{\rho _n}{C_n}\frac{{\partial {T_n}\left( {r,z,t} \right)}}{{\partial t}} = \frac{1}{r}\frac{\partial }{{\partial r}}\left[ {r{\kappa _n}\frac{{\partial {T_n}\left( {r,z,t} \right)}}{{\partial r}}} \right] + }\\ {\frac{\partial }{{\partial z}}\left[ {{\kappa _n}\frac{{\partial {T_n}\left( {r,z,t} \right)}}{{\partial z}}} \right] + {Q_n}\left( {T,r,z,t} \right)} \end{array} $
(1) 式中, n=1,2,3;ρn,Cn,κn和Tn(r, z, t)分别表示在t时刻第n层的材料密度、热容、热导率和温度场分布; Qn为热源项, T为温度场。
如表 1所示,由于GaInP2层和GaAs层对1070nm激光的吸收系数相较于Ge层的吸收系数很小,故可以假设只有底电池吸收能量,并将其吸收的热能作为热源[14]。热源项写为:
$ \begin{array}{*{20}{c}} {Q\left( {T,r,z,t} \right) = {I_0}\left( {1 - R - \eta } \right)\alpha \left( T \right) \times }\\ {I\left( {r,z} \right)g\left( t \right)\exp \left[ { - \alpha \left( T \right)z} \right]} \end{array} $
(2) Table 1. Forbidden band width and absorbable wavelength range of each layer of battery
material forbidden bandwidth/eV absorption wavelength range/nm GaInP2 1.85 350~700 GaAs 1.42 700~880 Ge 0.67 880~1750 式中, I0为激光辐照功率密度,R为电池表面反射率,η为光电转换效率, I(r, z)与g(t)分别为入射激光能量的空间分布函数与时间分布函数,α(T)为底电池的激光吸收系数。
$ \begin{array}{*{20}{l}} {\alpha \left( T \right) = 1.4 \times {{10}^4}\exp \left[ {2.81 \times \left( {1.16 + } \right.} \right.}\\ {\left. {0.67 - 0.83 + 3.9 \times {{10}^{ - 4}}T - 1.17} \right)} \end{array} $
(3) 高斯光束的空间分布函数为:
$ I\left( {r,z} \right) = \left\{ {\begin{array}{*{20}{l}} {\exp \left( { - 2{r^2}/a_0^2} \right),\left( {0 < r \le {a_0}} \right)}\\ {0,\left( {r > {a_0}} \right)} \end{array}} \right. $
(4) 式中,a0为激光辐照半径。连续激光的时间分布函数为:
$ g\left( t \right) = 1,\left( {0 < t \le \infty } \right) $
(5) 初始温度场为:
$ {\left. {T\left( {r,z,t} \right)} \right|_{t = 0}} = {T_0} = 303{\rm{K}} $
(6) 除电池上表面外,不考虑其它表面的热辐射与热对流。电池上表面边界条件为:
$ \begin{array}{*{20}{c}} { - {{\left. {\kappa \frac{{\partial T\left( {r,z,t} \right)}}{{\partial r}}} \right|}_{z = 0}} = - h\left[ {T\left( {r,z,t} \right) - {T_0}} \right] - }\\ {\sigma \varepsilon \left[ {{T^4}\left( {r,z,t} \right) - T_0^4} \right]} \end{array} $
(7) 式中, κ为导热系数, φ为热辐射率, σ为Stefan常数, h为传热系数。从初始温度场开始,通过时间、空间的网格划分,可以逐步求得模型中各点的瞬态温度分布。表 2中给出了Ge的热学参量与公式中各常数的值。
Table 2. Thermal parameters of Ge
parameter value density ρ/(g·cm-3) 5.49 thermal conductivity κ/(W·cm·K-1) 0.59 heat capacity C/(J·K-1) 0.31 thermal radiation rate φ 0.1 Stefan constant σ/(W·cm-2·K-4) 5.67×10-14 coefficient of heat transfer h/(W·cm-2·K-1) 1×10-5 -
热应力场的计算同样采用2维轴对称模型,在得到激光辐照太阳电池的温度场后,根据施加于靶材的温度载荷求解靶材的热应力场。根据von-Mises硬化准则,在材料弹性阶段材料的应力与应变为线性关系,满足Hooke定律,卸载后材料的变形与内部应力能逐渐恢复至初始值,当等效应力σ超过初始屈服强度时材料产生塑性变形。研究太阳电池在工作状态的热力效应时,激光功率并不高,所以电池温升相对较小,且变形量也较小,故忽略塑性变形,可按照热弹性模型计算。
2维轴对称模型中平衡方程为:
$ \left\{ {\begin{array}{*{20}{l}} {\frac{{\partial {\sigma _r}}}{{\partial r}} + \frac{{\partial {\tau _{zr}}}}{{\partial z}} + \frac{{{\sigma _r} - {\sigma _\theta }}}{r} = 0}\\ {\frac{{\partial {\sigma _z}}}{{\partial z}} + \frac{{\partial {\tau _{rz}}}}{{\partial r}} + \frac{{{\tau _{rz}}}}{r} = 0} \end{array}} \right. $
(8) 式中, σr, σθ, σz, τrz与τzr分别为径向应力、环向应力、轴向应力、剪应力和切应力。假设材料为各向同性,其热弹性本构方程为:
$ \left\{ \sigma \right\} = \left\{ {\begin{array}{*{20}{l}} {{\sigma _r}}\\ {{\sigma _\theta }}\\ {{\sigma _z}}\\ {{\tau _{zr}}} \end{array}} \right\} = {\mathit{\boldsymbol{D}}_{\rm{e}}}\left\{ {\begin{array}{*{20}{l}} {{\varepsilon _r}}\\ {{\varepsilon _\theta }}\\ {{\varepsilon _z}}\\ {{\gamma _{rz}}} \end{array}} \right\} - \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{1 - 2\mu }}}\\ {\frac{1}{{1 - 2\mu }}}\\ {\frac{1}{{1 - 2\mu }}}\\ 0 \end{array}} \right\}E{\alpha _\text{t}}\Delta T $
(9) 式中, De为弹性矩阵,E为材料弹性模量,μ为材料泊松比, εr, εθ, εz, γrz分别表示径向应变、环向应变、轴向应变及剪应变。{}表示列向量。表 3中列出了Ge材料的部分力学参量。表中,f为假设屈服强度,αt为热膨胀系数。
Table 3. Mechanical parameters of Ge
parameter value Yang’s module E/(N·cm-2) 1.31×107 fracture module f/(N·cm-2) 9.3×103 thermal expansion coefficient αt/(K-1) 5.92×106 Poisson ration μ 0.21 当材料应力超过屈服强度时, 必须考虑材料的塑性变形,考虑材料的塑性变形后,一个单元中的总应变增量表达式为:
$ \left\{ {\Delta \varepsilon } \right\} = \left\{ {\Delta {\varepsilon _{\rm{t}}}} \right\} + \left\{ {\Delta {\varepsilon _{\rm{e}}}} \right\} + \left\{ {\Delta {\varepsilon _{\rm{p}}}} \right\} $
(10) 式中, {Δε}为总应变增量;{Δεt}为热应变增量;{Δεe}为弹性应变增量;{Δεp}为塑性应变增量。
弹性阶段中应力增量与应变增量关系可表示为:
$ \left\{ {\Delta \sigma } \right\} = {\mathit{\boldsymbol{D}}_{\rm{e}}}\left( {\left\{ {\Delta \varepsilon } \right\} - \left\{ {\Delta {\varepsilon _{\rm{t}}}} \right\} - \left\{ {\Delta {\varepsilon _0}} \right\}} \right) $
(11) 其中,
$ \left\{ {\Delta {\varepsilon _0}} \right\} = \frac{{\partial \mathit{\boldsymbol{D}}_2^{ - 1}}}{{\partial T}}\left\{ \sigma \right\}{\rm{d}}T $
(12) 当材料等效应力超过屈服强度时材料发生塑性屈服产生塑性应变增量,根据表 3中的参量假设Ge实际屈服强度为93MPa。再根据Prandtl-Reuss塑性流动增量理论,热弹塑性本构方程如下:
$ \begin{array}{*{20}{c}} {\left\{ {\Delta \sigma } \right\} = {\boldsymbol{D}_{{\rm{ep}}}}\left( {\left\{ {\Delta \varepsilon } \right\} - \left\{ {\Delta {\varepsilon _{\rm{t}}}} \right\} - } \right.}\\ {\left. {\left\{ {\Delta {\varepsilon _0}} \right\}} \right) + \left\{ {\Delta {\varepsilon _0}} \right\}} \end{array} $
(13) 式中,Dep为弹塑性矩阵。
连续激光辐照三结GaAs太阳电池热应力场研究
Study on three-junction GaAs solar cell thermal stress field by continuous wave laser irradiation
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摘要: 为了研究1070nm光纤连续激光对三结GaAs太阳电池的热力作用, 使用COMSOL软件建立物理模型进行了数值仿真, 得到了不同激光功率密度激光作用下的热应力场。为了验证了热应力计算方法的正确性, 利用栅线投影法测量了功率为16.7W、辐照半径为1mm、辐照时间为10s的激光作用下电池表面的形变, 实验结果与模拟结果基本吻合。结果表明, 辐照半径为1.5cm、功率密度为16.7W/cm2的激光辐照20s时, 底电池中心温度刚好超过电池使用温度; 底电池中心等效应力为96.6MPa, 刚好超过底电池材料的屈服极限; 根据这一结果可推测电池失效与热应力导致的结构损伤有关。该数值模拟结果与实验结果为激光辐照太阳电池的热力效应研究提供了一定的理论依据。Abstract: In order to study thermodynamic effect of 1070nm continwous wave fiber laser on three-junction GaAs solar cells, physical model was built by COMSOL software and numerical simulation was carried out. Thermal stress fields under different laser power densities were obtained. In order to verify the correctness of thermal stress calculation method, surface deformation of batteries under laser irradiation with power of 16.7W, irradiation radius of 1mm and irradiation time of 10s was measured by grating projection method. The simulation results show that, when irradiation radius is 1.5cm and power density is 16.7W/cm2, laser irradiation time is 20s, central temperature of bottom battery is just above service temperature of the battery. Equivalent stress in the center of bottom battery is 96.6MPa. It just exceeds the yield limit of bottom battery material. According to this result, it can be inferred that the failure of battery is related to the structural damage caused by thermal stress. The experimental results are in good agreement with the simulation results. The numerical simulation results and experimental results provide theoretical basis for the study of thermal effects of laser irradiated solar cells.
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Key words:
- laser technique /
- thermal stress /
- numerical simulation /
- solar cell
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Table 1. Forbidden band width and absorbable wavelength range of each layer of battery
material forbidden bandwidth/eV absorption wavelength range/nm GaInP2 1.85 350~700 GaAs 1.42 700~880 Ge 0.67 880~1750 Table 2. Thermal parameters of Ge
parameter value density ρ/(g·cm-3) 5.49 thermal conductivity κ/(W·cm·K-1) 0.59 heat capacity C/(J·K-1) 0.31 thermal radiation rate φ 0.1 Stefan constant σ/(W·cm-2·K-4) 5.67×10-14 coefficient of heat transfer h/(W·cm-2·K-1) 1×10-5 Table 3. Mechanical parameters of Ge
parameter value Yang’s module E/(N·cm-2) 1.31×107 fracture module f/(N·cm-2) 9.3×103 thermal expansion coefficient αt/(K-1) 5.92×106 Poisson ration μ 0.21 -
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