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基于增量微分求积单元法的功能梯度材料夹层板非线性瞬态传热分析

张忠 许家婧 曹小建 王艳超 朱军 姚潞

张忠, 许家婧, 曹小建, 等. 基于增量微分求积单元法的功能梯度材料夹层板非线性瞬态传热分析[J]. 复合材料学报, 2024, 42(0): 1-13.
引用本文: 张忠, 许家婧, 曹小建, 等. 基于增量微分求积单元法的功能梯度材料夹层板非线性瞬态传热分析[J]. 复合材料学报, 2024, 42(0): 1-13.
ZHANG Zhong, XU Jiajing, CAO Xiaojian, et al. Nonlinear transient heat transfer analysis of functionally graded material sandwich slabs by incremental differential quadrature element method[J]. Acta Materiae Compositae Sinica.
Citation: ZHANG Zhong, XU Jiajing, CAO Xiaojian, et al. Nonlinear transient heat transfer analysis of functionally graded material sandwich slabs by incremental differential quadrature element method[J]. Acta Materiae Compositae Sinica.

基于增量微分求积单元法的功能梯度材料夹层板非线性瞬态传热分析

基金项目: 国家自然科学基金(11802145;52308260);江苏省高等学校基础科学(自然科学)研究项目(23KJB560021);南通市科技项目(JC12022058);南通市社会民生科技计划面上研究项目(MS22022103)
详细信息
    通讯作者:

    姚潞,博士,讲师,研究方向为复合材料结构 E-mail: yaolu@ntu.edu.cn

  • 中图分类号: TK124

Nonlinear transient heat transfer analysis of functionally graded material sandwich slabs by incremental differential quadrature element method

Funds: National Natural Science Foundation of China (11802145; 52308260); Natural Science Foundation of the Jiangsu Higher Education Institutions of China (23KJB560021); Science and Technology Project of Nantong City (JC12022058); Nantong City Social Livelihood Science and Technology Project (MS22022103)
  • 摘要: 作为首次尝试,采用增量微分求积单元法(IDQEM)开展了功能梯度材料(FGM)夹层板的一维非线性瞬态传热分析。夹层板组分材料的热工参数随空间位置变化,且具有温度依赖性。基于IDQEM,沿层界面将夹层板划分为三个空间子域,同时将整个受热过程划分为若干时间子域。采用微分求积技术对任一时间子域内的控制方程、初始条件、界面条件以及边界条件进行离散处理。由于所获得的离散方程建立在不同区域的节点上,因此对方程进行修改并将其表示为矩阵形式,以便它们可以建立在同一区域中。采用Kronecker积将联立的矩阵方程转化为一系列代数方程组,并采用Newton-Raphson迭代法近似求解,即可获得单个时间子域内的温度解。由于每个时间子域的初始条件可由上一个时间子域最终时刻的温度分布决定,因此从第一个时间子域逐渐递推到最后一个子域,即可获得整个受热过程的温度分布。数值算例验证了本方法的快速收敛性,与已有文献的解析和数值结果的对比验证了本方法的正确性。最后,讨论了热工参数温度依赖性、体积分数指数以及热边界条件对FGM夹层板温度分布的影响。

     

  • 图  1  功能梯度材料(FGM)夹层板示意图

    Figure  1.  Schematic view of the functionally graded material (FGM) sandwich slab

    x – Global coordinate for the sandwich slab; x(s) (s = 1, 2, 3) – Local coordinate for the sth layer; H – Total thickness of the sandwich slab;Hs – Thickness of the sth layer

    图  2  r个时间子域下的节点分布

    Figure  2.  Grid point distribution in the rth temporal sub-domain

    Mr and Ns – Grid point numbers in the rth temporal sub-domain and the sth spatial sub-domain, respectively; t – Time variable

    图  3  η = 0.2和2两种情况下SUS304/Si3N4夹层板x = 0.5H处温度变化

    Figure  3.  Temperature variations at x = 0.5H for the SUS304/Si3N4 sandwich slab with η = 0.2 and 2

    T – Temperature; η – volume fraction index

    图  4  η = 2时SUS304/Si3N4夹层板x = 0.5H处温度变化

    Figure  4.  Temperature variations at x = 0.5H for the SUS304/Si3N4 sandwich slab with η = 2

    TI – Temperature independent; TD – Temperature dependent

    图  5  不同时刻SUS304/Si3N4夹层板的温度分布

    Figure  5.  Temperature profiles at different times for the SUS304/Si3N4 sandwich slab

    图  6  不同上表面温度Tc作用下SUS304/Si3N4夹层板的温度分布

    Figure  6.  Temperature profiles of the SUS304/Si3N4 sandwich slab under different top surface temperatures Tc

    图  7  Tc = 600 K时SUS304/Si3N4夹层板的热扩散率α时空分布

    Figure  7.  Thermal diffusivity α contour of the SUS304/Si3N4 sandwich slab under the surface temperature Tc = 600 K

    图  8  SUS304/Si3N4夹层板x = 0.5H处的温度与Tc的关系

    Figure  8.  Temperature at x = 0.5H versus Tc for the SUS304/Si3N4 sandwich slab

    图  9  η = 0.2, 1, 5三种情况下Ti−6Al−4V/ZrO2夹层板的温度变化

    Figure  9.  Temperature variations of the Ti−6Al−4V/ZrO2 sandwich slab for η=0.2, 1, 5

    (a) x = 0 (b) x = H

    图  10  Ti−6Al−4V/ZrO2夹层板的热扩散率时空分布

    Figure  10.  Thermal diffusivity contour of the Ti−6Al−4V/ZrO2 sandwich slab

    (a) η = 0.2 (b) η = 5

    图  11  Ti−6Al−4V/ZrO2夹层板上表面的热流

    Figure  11.  Heat flux on the top surface of the Ti−6Al−4V/ZrO2 sandwich slab

    qtop – Heat flux generated on the top surface

    图  12  Ti−6Al−4V/ZrO2夹层板的温度时空分布

    Figure  12.  Temperature contour of the Ti−6Al−4V/ZrO2 sandwich slab

    表  1  组分材料热工参数的温度系数[2326]

    Table  1.   Temperature coefficients of thermophysical properties for the component materials[2326]

    Material Property P0 P−1 P1 P2 P3
    SUS304 λ/(W∙(m∙K)−1) 15.379 0 −1.264 × 10−3 2.092×10−6 −7.223×10−10
    c/(J∙(kg∙K)−1) 496.56 0 −1.151×10−3 1.636×10−6 −5.863×10−10
    ρ/(kg∙m−3) 8166.0 0 0 0 0
    Si3N4 λ/(W∙(m∙K)−1) 13.723 0 −1.032× 10−3 5.466 × 10−7 −7.876×10−11
    c/(J∙(kg∙K)−1) 555.11 0 1.016×10-3 2.920×10−7 −1.670×10−10
    ρ/(kg∙m−3) 2370.0 0 0 0 0
    Ti−6Al−4V λ/(W∙(m∙K)−1) 1.0000 0 1.704×10−2 0 0
    c/(J∙(kg∙K)−1) 625.30 0 −4.224×10−4 7.179×10−7 0
    ρ/(kg∙m−3) 4420.0 0 0 0 0
    ZrO2 λ/(W∙(m∙K)−1) 1.7000 0 1.276×10−4 6.648×10−8 0
    c/(J∙(kg∙K)−1) 487.34 0 3.049×10−4 −6.037×10−8 0
    ρ/(kg∙m−3) 5700.0 0 0 0 0
    Notes: λ, c, and ρ−Thermal conductivity, specific heat, and density, respectively; Pk (k = −1, 0, 1, 2, 3) − Temperature coefficients; SUS304, Si3N4, Ti−6Al−4V, and ZrO2 denote stainless steel, silicon nitride, titanium alloy, and zirconia, respectively.
    下载: 导出CSV

    表  2  η = 0.2时SUS304/Si3N4夹层板x = 0.75H处的温度结果(单位:K)

    Table  2.   Temperature results at x = 0.75H for the SUS304/Si3N4 sandwich slab with η = 0.2 (Unit: K)

    TimeTemporal grid point numberSpatial grid point number
    N = 12N = 20N = 28N = 36
    t = 0.01 sM = 4342.16 (0.0312 s)342.96 (0.0625 s)342.98 (0.0938 s)342.99 (0.1154 s)
    M = 12342.16 (0.1406 s)343.01 (0.1719 s)343.00 (0.2193 s)343.01 (0.4688 s)
    M = 20342.16 (0.1875 s)343.01 (0.3281 s)343.01 (0.5806 s)343.01 (1.0312 s)
    M = 28342.16 (0.2031 s)343.01 (0.5156 s)343.01 (1.1562 s)343.01 (2.1094 s)
    t = 0.03 sM = 4362.95 (0.0469 s)362.80 (0.0712 s)362.81 (0.1024 s)362.80 (0.1193 s)
    M = 12362.87 (0.1562 s)362.69 (0.1736 s)362.68 (0.2056 s)362.68 (0.5021 s)
    M = 20362.87 (0.1719 s)362.70 (0.3598 s)362.69 (0.6006 s)362.69 (1.0156 s)
    M = 28362.87 (0.2031 s)362.70 (0.5469 s)362.69 (1.0625 s)362.69 (2.2188 s)
    Notes: M and N – Total numbers of grid points in the temporal and spatial domains, respectively. The content in parentheses represents the CPU-time.
    下载: 导出CSV

    表  3  两种情况下数值结果和CPU计算时间的对比

    Table  3.   Comparison of numerical results and CPU-time in the two cases

    PositionTime/sCase 1/KCase 2/KErrorCPU-time of case 1/sCPU-time of case 2/s
    x = 0.25H0.01302.43301.400.34%0.60940.6310
    0.03314.28312.230.66%0.59380.5781
    x = 0.75H0.01349.05343.011.76%0.60940.6310
    0.03365.93362.690.89%0.59380.5781
    Notes: In Case 1, only the thermal conductivity is considered to be TD; in Case 2, all the thermophysical properties are considered to be TD.
    下载: 导出CSV

    表  4  时变热流的多项式拟合

    Table  4.   Polynomial fits for the time-dependent heat flux

    Time/sa0a1a2/10−3a3/10−6
    0-431.100.0031961.021−1.383
    431.1-660.2−228.71.595−2.6701.471
    660.2-1561253.4−0.5961−0.6477−0.2043
    1561-2200−12572.307−1.2120.1929
    Notes: ak (k = 0, 1, 2, 3) – Fitting coefficients.
    下载: 导出CSV
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