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Analysis of Transient Heat Conduction of Infinite Flat Plate Subjected to Heat Flow Boundary
DOI: 10.12677/IJM.2023.123011, PDF, HTML, XML, 下载: 242  浏览: 382

Abstract: Understanding and grasping the temperature condition around aircraft in real time is critical for its safe operation. In this paper, the problem of an aircraft subjected to heat flow is regarded as a one-dimensional infinite plate with a time-varying heat flow boundary condition, and is solved analytically by the method of separation of variables. Since this is an inhomogeneous boundary-value problem, we describe the problem as homogeneous by introducing auxiliary functions, and give a specific solving method. Finally, the theoretical analysis results are compared with the finite element simulation results, and the results show that the calculation results obtained by the two methods are basically consistent.

1. 引言

2. 问题的提出及模型建立

Figure 1. An analytical model for heat conduction problem with heat flux boundary condition

$\left\{\begin{array}{l}{u}_{t}=D{u}_{xx}\text{}\left(0\le x\le l，t\ge 0\right)\hfill \\ u\left(x,0\right)=\phi \left(x\right)\text{}\left(0\le x\le l\right)\hfill \\ {\beta }_{1}{u}_{x}\left(0,t\right)=g\left(t\right)\text{}\left(t\ge 0\right)\hfill \\ {\beta }_{1}{u}_{x}\left(l,t\right)=0\text{}\left(t\ge 0\right)\hfill \end{array}$ (1)

3. 求解过程

$u\left(x,t\right)=v\left(x,t\right)+w\left(x,t\right)$ (2)

$w\left(x,t\right)=-\frac{g\left(t\right)}{l}f\left(x\right)+g\left(t\right)$ (3)

$\left\{\begin{array}{l}{v}_{t}-D{v}_{xx}-\frac{{g}^{\prime }\left(t\right)}{l}f\left(x\right)+{g}^{\prime }\left(t\right)+\frac{D}{l}g\left(t\right){f}^{\text{'}\text{'}}\left(x\right)=0\hfill \\ {\beta }_{1}{v}_{x}\left(0,t\right)-{\beta }_{1}\frac{g\left(t\right)}{l}{f}^{\prime }\left(0\right)=g\left(t\right)\hfill \\ {v}_{x}\left(l,t\right)-\frac{g\left(t\right)}{l}{f}^{\prime }\left(l\right)=0\hfill \\ v\left(x,0\right)-\frac{g\left(0\right)}{l}f\left(x\right)+g\left(0\right)=\phi \left(x\right)\hfill \end{array}$ (4)

$\left\{\begin{array}{l}{v}_{t}-D{v}_{xx}=0\hfill \\ {\beta }_{1}{v}_{x}\left(0,t\right)=0\hfill \\ {v}_{x}\left(l,t\right)=0\hfill \\ v\left(x,0\right)=\phi \left(x\right)\hfill \end{array}$ (5)

$\left\{\begin{array}{l}{f}^{″}\left(x\right)-\frac{1}{D}\frac{{g}^{\prime }\left(t\right)}{g\left(t\right)}f\left(x\right)+\frac{l}{D}\frac{{g}^{\prime }\left(t\right)}{g\left(t\right)}=0\hfill \\ {f}^{\prime }\left(0\right)=-\frac{l}{{\beta }_{1}}\hfill \\ {f}^{\prime }\left(l\right)=0\hfill \\ -\frac{g\left(0\right)}{l}f\left(x\right)+g\left(0\right)=0\hfill \end{array}$ (6)

(a) (6)式的通解为：

$f\left(x\right)=\left\{\begin{array}{l}\frac{b{\text{e}}^{\sqrt{a}l}{\text{e}}^{-\sqrt{a}x}+b{\text{e}}^{-\sqrt{a}l}{\text{e}}^{\sqrt{a}x}}{\sqrt{a}{\text{e}}^{\sqrt{a}l}-\sqrt{a}{\text{e}}^{-\sqrt{a}l}}+l\text{}a>0\hfill \\ -\frac{b}{\sqrt{-a}}\left(\mathrm{cot}\sqrt{-a}l\mathrm{cos}\sqrt{-a}x+\mathrm{sin}\sqrt{-a}x\right)+l\text{}a<0\hfill \\ l\text{}a=0\hfill \end{array}$ (7)

(b) 利用分离变量法可得公式(5)的解为：

$v\left(x,t\right)={\sum }_{i=1}^{\infty }{v}_{n}\left(x,t\right)={E}_{0}+{\sum }_{n=1}^{\infty }{E}_{n}{\text{e}}^{-{\left(\frac{n\pi }{l}\right)}^{2}Dt}\mathrm{cos}\left(\frac{n\pi }{l}x\right)$ (8)

${E}_{0}=\frac{1}{l}{\int }_{0}^{l}\left[\phi \left(x\right)+\frac{g\left(0\right)}{l}f\left(x\right)-g\left(0\right)\right]\text{d}x$ (9)

${E}_{n}=\frac{2}{l}{\int }_{0}^{l}\left[\phi \left(x\right)+\frac{g\left(0\right)}{l}f\left(x\right)-g\left(0\right)\right]\mathrm{cos}\left(\frac{n\pi }{l}x\right)\text{d}x$ (10)

1) 当 $a>0$

$u\left(x,t\right)={E}_{0}+{\sum }_{n=1}^{\infty }{E}_{n}{\text{e}}^{-{\left(\frac{n\pi }{l}\right)}^{2}Dt}\mathrm{cos}\left(\frac{n\text{π}}{l}x\right)-\frac{g\left(t\right)}{l}\frac{b{\text{e}}^{\sqrt{a}l}{\text{e}}^{-\sqrt{a}x}+b{\text{e}}^{-\sqrt{a}l}{\text{e}}^{\sqrt{a}x}}{\sqrt{a}{\text{e}}^{\sqrt{a}l}-\sqrt{a}{\text{e}}^{-\sqrt{a}l}}$ (11)

${E}_{0}=\frac{1}{l}\left[{\int }_{0}^{l}\phi \left(x\right)\text{d}x+\frac{g\left(0\right)}{l}\left(\frac{b}{a}+{l}^{2}\right)-g\left(0\right)l\right]$ (12)

(13)

2) 当 $a<0$

$u\left(x,t\right)={E}_{0}+{\sum }_{n=1}^{\infty }{E}_{n}{\text{e}}^{-{\left(\frac{n\pi }{l}\right)}^{2}Dt}\mathrm{cos}\left(\frac{n\text{π}}{l}x\right)+\frac{g\left(t\right)}{l}\left(\frac{b}{\sqrt{-a}}\mathrm{cot}\sqrt{-a}l\mathrm{cos}\sqrt{-a}x+\frac{b}{\sqrt{-a}}\mathrm{sin}\sqrt{-a}x\right)$ (14)

${E}_{0}=\frac{1}{l}\left[{\int }_{0}^{l}\phi \left(x\right)\text{d}x+\frac{g\left(0\right)}{l}\left(\frac{b}{a}+{l}^{2}\right)-g\left(0\right)l\right]$ (15)

$\begin{array}{c}{E}_{n}=\frac{2}{l}\left[{\int }_{0}^{l}\phi \left(x\right)\mathrm{cos}\left(\frac{n\pi }{l}x\right)\text{d}x\stackrel{}{{}_{}^{}}\\ -\frac{g\left(0\right)}{l}\frac{b}{\sqrt{-a}}\left(\frac{l}{2}\mathrm{cot}\sqrt{-a}l\left(\frac{1}{\sqrt{-a}l+n\pi }\mathrm{sin}\left(\sqrt{-a}l+n\pi \right)+\frac{1}{\sqrt{-a}l-n\pi }\mathrm{sin}\left(\sqrt{-a}l-n\pi \right)\right)\\ +\frac{l}{2}\frac{1}{\sqrt{-a}l+n\pi }\left(1-\mathrm{cos}\left(\sqrt{-a}l+n\pi \right)\right)+\frac{1}{\sqrt{-a}l-n\pi }\left(1-\mathrm{cos}\left(\sqrt{-a}l-n\pi \right)\right)\right)\right]\end{array}$ (16)

3) 当 $a=0$ 时，

$u\left(x,t\right)=\phi \left(x\right)$ (17)

4. 结果分析

Table 1. Influencing factors of heat flow boundary conditions

Figure 2. Heat flow exponentially varies with time

Figure 3. Comparison of temperature change with time under different thermophysical property parameters: (a) specific heat capacity; (b) density; (c) thermal conductivity

Figure 4. Step change of heat flow with time

Figure 5. Temperature change with time under different thermophysical property parameters: (a) specific heat capacity; (b) density

5. 结论

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