留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

一维传热方程瞬态问题解析解及其应用

仝睿 付浩 宋二祥

仝睿, 付浩, 宋二祥. 一维传热方程瞬态问题解析解及其应用[J]. 工程力学, 2022, 39(8): 9-18. doi: 10.6052/j.issn.1000-4750.2021.04.0304
引用本文: 仝睿, 付浩, 宋二祥. 一维传热方程瞬态问题解析解及其应用[J]. 工程力学, 2022, 39(8): 9-18. doi: 10.6052/j.issn.1000-4750.2021.04.0304
TONG Rui, FU Hao, SONG Er-xiang. ANALYTICAL SOLUTION OF ONE-DIMENSIONAL TRANSIENT HEAT CONDUCTION EQUATION AND ITS APPLICATION[J]. Engineering Mechanics, 2022, 39(8): 9-18. doi: 10.6052/j.issn.1000-4750.2021.04.0304
Citation: TONG Rui, FU Hao, SONG Er-xiang. ANALYTICAL SOLUTION OF ONE-DIMENSIONAL TRANSIENT HEAT CONDUCTION EQUATION AND ITS APPLICATION[J]. Engineering Mechanics, 2022, 39(8): 9-18. doi: 10.6052/j.issn.1000-4750.2021.04.0304

一维传热方程瞬态问题解析解及其应用

doi: 10.6052/j.issn.1000-4750.2021.04.0304
基金项目: 国家自然科学基金项目(51778339);国家重点基础研究发展计划(973计划)课题项目(2014CB047003)
详细信息
    作者简介:

    仝 睿(1996−),男,江苏人,博士生,主要从事岩土力学研究(E-mail: 2675870452@qq.com)

    付 浩(1996−),男,天津人,博士生,主要从事岩土力学研究(E-mail: 18622335312@163.com)

    通讯作者:

    宋二祥(1957−),男,河北人,教授,博士,主要从事岩土力学及工程方面的教学及科研(E-mail: songex@tsinghua.edu.cn)

  • 中图分类号: TU441

ANALYTICAL SOLUTION OF ONE-DIMENSIONAL TRANSIENT HEAT CONDUCTION EQUATION AND ITS APPLICATION

  • 摘要: 探究路基及浅层土体的温度随深度的分布规律,对冻土区域路基的合理设计有重要意义。该文根据实际浅层土体温度边界情况,建立了一维传热瞬态模型并给出其解析解,通过与数值模拟及现场测试结果的对比验证了解析解的正确性。仔细考察介质分层情况下的热传导过程发现,传热问题在一定程度上存在与波动问题类似的现象,即在分层界面处有热传导的反射与透射,结合对解析解的分析及数值模拟,对此种“温度波”现象进行了论证分析,发现此种波总是伴有明显的衰减。此外,还探讨了解析解在冻深估计、有关模型试验中几何相似关系的确定、“覆盖效应”数值模拟等方面的应用。
  • 图  1  叠加示意图

    Figure  1.  Composition of functions

    图  2  解析解和模拟结果对比

    Figure  2.  Comparison between analytical solution and simulation results

    图  3  路基土体温度

    Figure  3.  Temperature of subgrade soil

    图  4  边坡土体温度

    Figure  4.  Temperature of slope soil

    图  5  路基振幅-深度关系图

    Figure  5.  Amplitude vs depth diagram of subgrade

    图  6  边坡振幅-深度关系图

    Figure  6.  Amplitude vs depth diagram of slope

    图  7  路基最高温时间点-深度关系图

    Figure  7.  Maximum temperature time vs depth diagram of subgrade

    图  8  边坡最高温时间点-深度关系图

    Figure  8.  Maximum temperature time vs depth diagram of slope

    图  9  不同材料交界面反射示意图

    Figure  9.  Reflection in interface between different materials

    图  10  仿真结果与理论分析结果对照图

    Figure  10.  Comparison of simulation result and analytical result

    图  11  修正前、后温度对比图

    Figure  11.  Temperature comparison before and after modification

    图  12  修正前、后含水量对比图

    Figure  12.  Water volumetric content comparison before and after modification

  • [1] 夏锦红, 陈之祥, 夏元友, 等. 不同负温条件下冻土导热系数的理论模型和试验验证[J]. 工程力学, 2018, 35(5): 109 − 117. doi: 10.6052/j.issn.1000-4750.2017.01.0048

    Xia Jinhong, Chen Zhixiang, Xia Yuanyou, et al. Theoretical model and experimental verification of thermal conductivity of frozen soil under different negative temperatures [J]. Engineering Mechanics, 2018, 35(5): 109 − 117. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.01.0048
    [2] 陈佩佩, 白冰. 非饱和土中温度引起水分迁移的光滑粒子法数值模拟[J]. 工程力学, 2016(4): 150 − 156. doi: 10.6052/j.issn.1000-4750.2014.09.0772

    Chen Peipei, Bai Bing. Numerical simulation of temperature induced water transfer in unsaturated soil by smoothed particle method [J]. Engineering Mechanics, 2016(4): 150 − 156. (in Chinese) doi: 10.6052/j.issn.1000-4750.2014.09.0772
    [3] 左建平, 周宏伟, 谢和平. 不同温度影响下砂岩的断裂特性研究[J]. 工程力学, 2008(5): 124 − 130.

    Zuo Jianping, Zhou Hongwei, Xie Heping. Fracture characteristics of sandstone under different temperatures [J]. Engineering Mechanics, 2008(5): 124 − 130. (in Chinese)
    [4] 宋二祥, 仝睿, 罗爽, 李鹏. 路基土体“时变覆盖效应”的数值模拟分析[J]. 工程力学, 2019, 36(8): 30 − 39. doi: 10.6052/j.issn.1000-4750.2018.09.0505

    Song Erxiang, Tong Rui, Luo Shuang, Li Peng. Numerical simulation and analysis of "time-varying canopy effect" of moisture transport in subgrade soil [J]. Engineering Mechanics, 2019, 36(8): 30 − 39. (in Chinese) doi: 10.6052/j.issn.1000-4750.2018.09.0505
    [5] 仝睿, 宋二祥, 赵志宏, 于洪钦, 张爱涛. 某铁路路基冻胀过程实测及“时变覆盖效应”分析[J]. 铁道科学与工程学报, 2020, 17(8): 1949 − 1956.

    Tong Rui, Song Erxiang, Zhao Zhihong, Yu Hongqin, Zhang Aitao. Measurement of frost heave process of a railway subgrade and analysis of "time-varying canopy effect" [J]. Journal of Railway Science and Engineering, 2020, 17(8): 1949 − 1956. (in Chinese)
    [6] Zhang S, Gao F, X He, et al. Experimental study of particle migration under cyclic loading: Effects of load frequency and load magnitude [J]. Acta Geotechnica, 2021, 16(2): 367 − 380. doi: 10.1007/s11440-020-01137-x
    [7] Liu J, Teng J, Zhang S, et al. A frost heave model of unsaturated coarse-grained soil considering vapor transfer [J]. E3S Web of Conferences, 2020, 195(2): 02017.
    [8] 徐学祖, 王家澄, 张立新. 冻土物理学 [M]. 北京: 科学出版社, 2001.

    Xu Xuezu, Wang Jiacheng, Zhang Lixin. Permafrost physics [M]. Beijing: Science Press, 2001. (in Chinese)
    [9] 涂新斌, 戴福初. 土体一维传热方程解析解及热扩散系数测定[J]. 岩土工程学报, 2008, 5: 33 − 38.

    Tu Xinbin, Dai Fuchu. Analytical solution for one-dimensional heat transfer equation of soil and evaluation for thermal diffusivity [J]. Chinese Journal of Geotechnical Engineering, 2008, 5: 33 − 38. (in Chinese)
    [10] 李翊神. 关于波动方程与传热方程定解问题间的联系[J]. 北京大学学报(自然科学版), 1964(2): 3 − 10.

    Li Yishen. The relation between wave equation and heat transfer equation [J]. Journal of Peking University (Natural Science Edition), 1964(2): 3 − 10. (in Chinese)
    [11] 王兆瑞, 许鹏. 地面表层温度场及隧道排水沟埋置深度设计探讨[J]. 地下空间与工程学报, 2015, 11(1): 149 − 155.

    Wang Zhaorui, Xu Peng. Study on temperature field of ground surface and discussion of tunnel centre drains depth design in cold region [J]. Chinese Journal of Underground Space and Engineering, 2015, 11(1): 149 − 155. (in Chinese)
    [12] 林府标, 张千宏, 张俊, 等. 一维广义热传导方程的精确解[J]. 重庆师范大学学报 (自然科学版), 2018, 35(4): 88 − 92.

    Lin Fubiao, Zhang Qianhong, Zhang Jun, et al. Exact solutions of one-dimensional general heat conduction equation [J]. Journal of Chongqing Normal University (Natural Science Edition), 2018, 35(4): 88 − 92. (in Chinese)
    [13] 左冲, 姚鸿骁, 姚伟岸. 时域径向积分边界元法在平面单相凝固问题中的应用[J]. 工程力学, 2019, 36(3): 43 − 49. doi: 10.6052/j.issn.1000-4750.2018.01.0003

    Zuo Chong, Yao Hongxiao, Yao Weian. Application of time domain radial integral boundary element method to planar single phase solidification [J]. Engineering Mechanics, 2019, 36(3): 43 − 49. (in Chinese) doi: 10.6052/j.issn.1000-4750.2018.01.0003
    [14] 姜礼尚. 数学物理方程讲义[M]. 北京: 高等教育出版社, 1961.

    Jiang Lishang. Lecture notes on equations of mathematical physics [M]. Beijing: Higher Education Press, 1961. (in Chinese)
    [15] 郭鸿, 刘建强, 赵智强, 桂忠强, 罗向荣. 陕西西安幸福林带项目地源热泵工程开发利用[J]. 矿产勘查, 2020, 11(12): 2653 − 2659. doi: 10.3969/j.issn.1674-7801.2020.12.010

    Guo Hong, Liu Jianqiang, Zhao Zhiqiang, Gui Zhongqiang, Luo Xiangrong. Development and utilization of ground-source heat pump engineering in Xingfu Forest Belt, Xi’an, Shaanxi Province [J]. Mineral Exploration, 2020, 11(12): 2653 − 2659. (in Chinese) doi: 10.3969/j.issn.1674-7801.2020.12.010
    [16] 陈博. 中国冻土时空变化特征及其与东亚气候的关系[D]. 北京: 中国科学院研究生院 (大气物理研究所), 2007.

    Chen Bo. Characteristics of spatial and temporal variation of frozen soil in china and their association with the east asian climate [D]. Beijing: Chinese Academy of Sciences (Institute of Atmospheric Physics), 2007. (In Chinese)
    [17] 王琳. 双锅盖效应下土体中水气运移规律分析研究[D]. 北京: 北京航空航天大学, 2019.

    Wang Lin. Analysis and Research on water vapor migration law in soil under Double pot cover effect [D]. Beijing: Beijing University of Aeronautics and Astronautics, 2019. (in Chinese)
    [18] 贺佐跃, 张升, 滕继东, 姚仰平, 盛岱超. 冻土中气态水迁移及其对土体含水率的影响分析[J]. 岩土工程学报, 2018, 40(7): 1190 − 1197.

    He Zuoyue, Zhang Sheng, Teng Jidong, Yao Yangping, Sheng Daichao. Vapor transfer and its effects on water content in freezing soils [J]. Chinese Journal of Geotechnical Engineering, 2018, 40(7): 1190 − 1197. (in Chinese)
    [19] 宋二祥, 罗爽, 孔郁斐, 等. 路基土体"锅盖效应"的数值模拟分析[J]. 岩土力学, 2017, 38(6): 1781 − 1788.

    Song Erxiang, Luo Shuang, Kong Yufei, et al. Numerical simulation analysis of "pot cover effect" of subgrade soil [J]. Geotechnical Mechanics, 2017, 38(6): 1781 − 1788. (in Chinese)
  • 加载中
图(12)
计量
  • 文章访问数:  553
  • HTML全文浏览量:  212
  • PDF下载量:  163
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-04-21
  • 修回日期:  2021-07-15
  • 网络出版日期:  2021-07-28
  • 刊出日期:  2022-08-01

目录

    /

    返回文章
    返回