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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.
[1] Wang M, Feng J Z, Jiang Y G, Zhang Z M, Feng J 2016 Mater. Rev. 30 461 (in Chinese)[王苗, 冯军宗, 姜勇刚, 张忠明, 冯坚 2016 材料导报 30 461]
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[3] Song H Y, Tian M M, Wu Y Y 2016 Pack. Eng. 37 56 (in Chinese)[宋海燕, 田萌萌, 伍亚云 2016 包装工程 37 56]
[4] Daryabeigi K 2002 J. Thermophys. Heat Transfer 17 10
[5] Daryabeigi K, Cunnington G R, Knutson J R 2013 J. Thermophys. Heat Transfer 27 414
[6] Al-sanea S A, Zedan M F 2011 Appl. Energy 88 3113
[7] Xu F, Lu T J, Seffen K A 2008 J. Mech. Phys. Solids 56 1852
[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
[10] He K L, Chen Q, Dong E F, Ge W C, Hao J H, Xu F 2018 Appl. Therm. Eng. 129 1551
[11] Wang B L, Cui Y J 2017 Appl. Therm. Eng. 119 207
[12] Liu K C, Wang Y N, Chen Y S 2012 Int. J. Thermal Sci. 58 29
[13] Li L, Zhou L, Yang M 2016 Int. J. Heat Mass Transfer 93 834
[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]
[17] Wang H G, Wu D, Rao Z H 2015 Acta Phys. Sin. 64 244401 (in Chinese)[王焕光, 吴迪, 饶中浩 2015 物理学报 64 244401]
[18] Wang G, Xie Z H, Fan X D, Chen L G, Sun F R 2017 Acta Phys. Sin. 66 204401 (in Chinese)[王刚, 谢志辉, 范旭东, 陈林根, 孙丰瑞 2017 物理学报 66 204401]
[19] Ma J, Sun Y, Yang J 2017 Int. J. Heat Mass Transfer 115 606
[20] Lin S M, Li C Y 2016 Int. J. Thermal Sci. 110 146
[21] Wang W, Ding H L, Zhang Z K, Shen L 2013 Acta Mater. Compos. Sin. 30 14 (in Chinese)[汪文, 丁宏亮, 张子宽, 沈烈 2013 复合材料学报 30 14]
[22] Zhao Y P, Yan G J, Chen D M, Chen L, Dong Z Z, Fu W G 2013 J. Funct. Mater. 16 697 (in Chinese)[赵义平, 阎家建, 陈丁猛, 陈莉, 董知之, 付维贵 2013 功能材料 16 697]
[23] Liu C N, Zhang S X, Zhou H Q, Xu D M, Li M Q 2011 J. Jilin Inst. Chem. Tech. 28 29 (in Chinese)[刘翠娜, 张双喜, 周恒勤, 许冬梅, 李美芹 2011 吉林化工学院学报 28 29]
[24] Liu B Z, Wang D W 2011 J. Northeast Univ. 32 302 (in Chinese)[刘保政, 汪定伟 2011 东北大学学报 32 302]
[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]
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[1] Wang M, Feng J Z, Jiang Y G, Zhang Z M, Feng J 2016 Mater. Rev. 30 461 (in Chinese)[王苗, 冯军宗, 姜勇刚, 张忠明, 冯坚 2016 材料导报 30 461]
[2] Tang J J, Xu X M 2011 Pack. Eng. 32 34 (in Chinese)[唐静静, 徐雪萌 2011 包装工程 32 34]
[3] Song H Y, Tian M M, Wu Y Y 2016 Pack. Eng. 37 56 (in Chinese)[宋海燕, 田萌萌, 伍亚云 2016 包装工程 37 56]
[4] Daryabeigi K 2002 J. Thermophys. Heat Transfer 17 10
[5] Daryabeigi K, Cunnington G R, Knutson J R 2013 J. Thermophys. Heat Transfer 27 414
[6] Al-sanea S A, Zedan M F 2011 Appl. Energy 88 3113
[7] Xu F, Lu T J, Seffen K A 2008 J. Mech. Phys. Solids 56 1852
[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
[10] He K L, Chen Q, Dong E F, Ge W C, Hao J H, Xu F 2018 Appl. Therm. Eng. 129 1551
[11] Wang B L, Cui Y J 2017 Appl. Therm. Eng. 119 207
[12] Liu K C, Wang Y N, Chen Y S 2012 Int. J. Thermal Sci. 58 29
[13] Li L, Zhou L, Yang M 2016 Int. J. Heat Mass Transfer 93 834
[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]
[17] Wang H G, Wu D, Rao Z H 2015 Acta Phys. Sin. 64 244401 (in Chinese)[王焕光, 吴迪, 饶中浩 2015 物理学报 64 244401]
[18] Wang G, Xie Z H, Fan X D, Chen L G, Sun F R 2017 Acta Phys. Sin. 66 204401 (in Chinese)[王刚, 谢志辉, 范旭东, 陈林根, 孙丰瑞 2017 物理学报 66 204401]
[19] Ma J, Sun Y, Yang J 2017 Int. J. Heat Mass Transfer 115 606
[20] Lin S M, Li C Y 2016 Int. J. Thermal Sci. 110 146
[21] Wang W, Ding H L, Zhang Z K, Shen L 2013 Acta Mater. Compos. Sin. 30 14 (in Chinese)[汪文, 丁宏亮, 张子宽, 沈烈 2013 复合材料学报 30 14]
[22] Zhao Y P, Yan G J, Chen D M, Chen L, Dong Z Z, Fu W G 2013 J. Funct. Mater. 16 697 (in Chinese)[赵义平, 阎家建, 陈丁猛, 陈莉, 董知之, 付维贵 2013 功能材料 16 697]
[23] Liu C N, Zhang S X, Zhou H Q, Xu D M, Li M Q 2011 J. Jilin Inst. Chem. Tech. 28 29 (in Chinese)[刘翠娜, 张双喜, 周恒勤, 许冬梅, 李美芹 2011 吉林化工学院学报 28 29]
[24] Liu B Z, Wang D W 2011 J. Northeast Univ. 32 302 (in Chinese)[刘保政, 汪定伟 2011 东北大学学报 32 302]
[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]
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