传热-2
热量传递
1. 热传导
- 导热基本定律 (傅里叶定律):
- $\Phi = -\lambda A \frac{dT}{dx}$
- $q = -\lambda \frac{dT}{dx}$
2. 热对流
- $\Phi = hA\Delta T$
- $q = h(t_w - t_f)$
- 对流换热热阻: $R_h = \frac{1}{hA}$
- $q = \frac{\Delta T}{R_h}$
- $r_h = \frac{1}{h}$ (单位热阻)
3. 热辐射
- $\Phi = A\sigma T^4$
- $q = \sigma T^4$
- $\sigma = 5.67 \times 10^{-8} W/(m^2 \cdot K^4)$
- 实际物体: $\Phi = \epsilon A \sigma T^4$
- 辐射换热量: $\Phi = \epsilon_1 A_1 \sigma (T_1^4 - T_2^4)$ ($A_1 << A_2$)
传热过程
- 流体 - 固体壁面 - 流体
- 稳态下：$\Phi_1 = \Phi_2 = \Phi_3$
- 传热系数: $K = \frac{1}{\frac{1}{h_1} + \frac{\delta}{\lambda} + \frac{1}{h_2}} = \frac{1}{r_{h_1} + r_x + r_{h_2}}$
- 导热热阻: $R_x = \frac{\delta}{\lambda A}$
温度场
- 稳态: $\frac{\partial T}{\partial t} = 0$
- 等温面 and 等温线
- 非稳态
- 温度梯度: $\nabla T = \frac{\partial T}{\partial x}\overrightarrow{i} + \frac{\partial T}{\partial y}\overrightarrow{j} + \frac{\partial T}{\partial z}\overrightarrow{k}$ （指向温度增加方向）
- $\overrightarrow{q} = -\lambda \nabla T$ （传热方向指向温度减小方向）
- 常温常压下气体热导率: $\lambda = \frac{1}{3}\overline{u} \rho l C_v$ (分子量小的气体导热率大)
- 液体导热系数经验公式： $\lambda = A \rho P^3 / M^{\frac{2}{3}}$; $A_0 = const$
- 固体: $T \uparrow \rightarrow \lambda \uparrow$ (多孔材料) , $\rho$ 湿度 $\rightarrow \lambda \uparrow$
三维、非稳态，变物性、有内热源
- 直角坐标：$\rho c \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}(\lambda \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y}(\lambda \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z}(\lambda \frac{\partial T}{\partial z}) + \dot{\Phi}$
- 热扩散率 (导温系数): $a = \frac{\lambda}{\rho C}$
- $(\lambda = const)$: $\frac{\partial T}{\partial t} = \frac{\lambda}{\rho C} \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + \frac{\dot{\Phi}}{\rho c}$
- 稳态无内热源: $∇^2t = 0 = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}$
坐标系
- 圆柱坐标: $\rho c \frac{\partial T}{\partial t} = \frac{1}{r} \frac{\partial}{\partial r}(r \lambda \frac{\partial T}{\partial r}) + \frac{1}{r^2} \frac{\partial}{\partial \phi}(\lambda \frac{\partial T}{\partial \phi}) + \frac{\partial}{\partial z}(\lambda \frac{\partial T}{\partial z}) + \dot{\Phi}$
$q_r = -\lambda \frac{\partial T}{\partial r}$
$q_{\theta} = -\lambda \frac{1}{r} \frac{\partial T}{\partial \phi}$
$q_{\phi} = -\lambda \frac{\partial T}{\partial z}$ - 球坐标: $\rho c \frac{\partial T}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \lambda \frac{\partial T}{\partial r}) + \frac{1}{r^2 sin \theta} \frac{\partial}{\partial \theta}(sin \theta \lambda \frac{\partial T}{\partial \theta}) + \frac{1}{r^2 sin^2 \theta} \frac{\partial}{\partial \phi}(\lambda \frac{\partial T}{\partial \phi}) + \dot{\Phi}$
$q_r = -\lambda \frac{\partial T}{\partial r}$
$q_{\theta} = -\lambda \frac{1}{r} \frac{\partial T}{\partial \theta}$
$q_{\phi} = -\lambda \frac{1}{r sin \theta} \frac{\partial T}{\partial \phi}$