基于拓展分离变量法的层合材料瞬态传热分析
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摘要
层合材料各层热物理参数不同,难以用常规的分离变量法求解.针对此问题对常规分离变量法进行了拓展,将层合材料受热时的温度场在时间域上分成微小时间段,在每个微小时间段内层合材料交界处的温度可认为随时间正比变化,并假设比例系数,此时在微小时间段内对各层分别利用分离变量法单独求得解析解,根据交界处温度相等能量连续的关系可求出比例系数,进而求出该微小时间段内的温度场,通过循环求解可得整个时间段内的温度场.之后,利用拓展的分离变量法对常用层合隔热材料瞬态传热进行了分析,通过与有限元方法计算的结果比较,验证了本文方法的正确性,分析了隔热材料类型、厚度,材料表面对流换热系数,空气温度等参数对隔热效果的影响.拓展分离变量法利用解析的方式求解了层合材料瞬态传热问题,物理意义比常规的数值方法明确,计算效率也较高.
In general, when the one-dimensional heat conduction equation is solved by the method of separation of variables,we need to know the governing equations, two boundary conditions and initial condition. Because the thermophysical parameters in different layers of laminated materials are different, the heat conduction model cannot be expressed by the same governing equation. For each layer of laminated material, the boundary condition is unknown. That equation can-not be solved directly by the general separation variable method. In this work the separation of variable method is extended. The temperature field of laminated material's heat transfer is divided into many minute time intervals on the time axis. Based on differential conception, in a minimum time interval, the temperature at the junction of laminated materials can be considered to be proportional to time. Assume that the slope coefficient makes the boundary condition known, then for each layer of laminated materials, the general separation of variables method will be used to solve the temperature field. According to the same temperature and the energy continuity at the junction of laminated materials,one can solve the slope coefficient. The temperature field in the whole time domain can be obtained through cycling.Then the three-layer insulation materials are analyzed by the extended separation variable method. The correctness of the method is verified by comparing the calculated results with those from the finite element method. The influences of the type and thickness of heat insulation layer, heat transfer coefficient, air temperature on the heat insulation are studied. It is found that the thermal conductivity of the thermal insulation layer has a great influence on the insulation.The material with low heat conduction coefficient can enhance the heat insulation effect. The thicker the thickness of the insulation layer, the more slowly the surface temperature of the heat insulation material rises, and the lower the final temperature, the better the insulation effect is. The thicker the thickness of the insulation layer, the smaller the heat flux density of the heat insulation material shell is, and the better the heat insulation effect when the heat transfer reaches a stable state. All calculation results are consistent with physical phenomena. In this work, the analytical method is used to solve the heat transfer problem of laminated materials. Compared with the general numerical methods, the analytical method presents clear physical meaning and high efficiency of operation as well.
In general, when the one-dimensional heat conduction equation is solved by the method of separation of variables,we need to know the governing equations, two boundary conditions and initial condition. Because the thermophysical parameters in different layers of laminated materials are different, the heat conduction model cannot be expressed by the same governing equation. For each layer of laminated material, the boundary condition is unknown. That equation can-not be solved directly by the general separation variable method. In this work the separation of variable method is extended. The temperature field of laminated material's heat transfer is divided into many minute time intervals on the time axis. Based on differential conception, in a minimum time interval, the temperature at the junction of laminated materials can be considered to be proportional to time. Assume that the slope coefficient makes the boundary condition known, then for each layer of laminated materials, the general separation of variables method will be used to solve the temperature field. According to the same temperature and the energy continuity at the junction of laminated materials,one can solve the slope coefficient. The temperature field in the whole time domain can be obtained through cycling.Then the three-layer insulation materials are analyzed by the extended separation variable method. The correctness of the method is verified by comparing the calculated results with those from the finite element method. The influences of the type and thickness of heat insulation layer, heat transfer coefficient, air temperature on the heat insulation are studied. It is found that the thermal conductivity of the thermal insulation layer has a great influence on the insulation.The material with low heat conduction coefficient can enhance the heat insulation effect. The thicker the thickness of the insulation layer, the more slowly the surface temperature of the heat insulation material rises, and the lower the final temperature, the better the insulation effect is. The thicker the thickness of the insulation layer, the smaller the heat flux density of the heat insulation material shell is, and the better the heat insulation effect when the heat transfer reaches a stable state. All calculation results are consistent with physical phenomena. In this work, the analytical method is used to solve the heat transfer problem of laminated materials. Compared with the general numerical methods, the analytical method presents clear physical meaning and high efficiency of operation as well.
引文
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[8] Li J E, Wang B L, Chang D M 2011 Acta Mech. Solida.Sin. 32 248(in Chinese)[李金娥,王保林,常冬梅2011固体力学学报32 248]
[9] Rahideh H, Malekzadeh P, Haghighi M G 2012 Energ.Convers. Manage. 55 14
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[13] Li L, Zhou L, Yang M 2016 Int. J. Heat Mass Transfer93 834
[14] Wu Z K, Li F L, Kwak D Y 2016 Chin. J. Comput. Phys.33 49(in Chinese)[吴自库,李福乐, Kwak D Y 2016计算物理33 49]
[15] Liu F, Shi W P 2015 Appl. Math. Mech. 36 1158(in Chinese)[刘芳,施卫平2015应用数学和力学36 1158]
[16] Hu J X, Gao X W 2016 Acta Phys. Sin. 65 014701(in Chinese)[胡金秀,高效伟2016物理学报65 014701]
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[25] He C H, Feng X 2001 Chemical Principle(Beijing:Science Press)pp190–193(in Chinese)[何潮洪,冯霄2001化工原理(北京:科学出版社)第190—193页]