Modes of Heat Transfer
Conduction
| Description |
Equations |
| Heat rate |
$\mathbf{q} = \mathbf{q''}A$ |
| Fourier’s law (conduction) |
$\mathbf{q''} = -k \nabla T$ |
| Fourier’s law (conduction) |
$q_x'' = -k \dfrac{dT}{dx}$ |
Convection
| Description |
Equations |
| Newton’s law of cooling (convection) |
$q'' = h (T_s - T_\infty)$ |
Radiation
| Description |
Equations |
| Stefan-Boltzman’s law (radiation) |
$E_b = \varepsilon\sigma T_s^4$ |
| Gray body |
$\varepsilon = \alpha$ |
| Blackbody |
$\varepsilon = \alpha = 1$ |
| Irradiation from isothermal surrounding |
$G = \sigma T_{\text{surr}}^4$ |
| Absorbed irradiation |
$G_{\text{abs}} = \alpha G$ |
| Net heat flux from radiation |
$q_{\text{rad}}'' = \varepsilon\sigma T_s^4 - \alpha G$ |
Net heat flux from radiation
★ gray surface, isothermal surrounding $T_{\text{surr}}$ |
$q_{\text{rad}}'' = \varepsilon\sigma (T_s^4 - T_{\text{surr}}^4)$ |
| Net heat flux from radiation in rate law form |
$q_{\text{rad}}'' = h_{\text{rad}} (T_s - T_{\text{surr}}) \newline h_{\text{rad}} = \varepsilon\sigma (T_s + T_{\text{surr}})(T_s^2 + T_{\text{surr}}^2)$ |
Energy balance
| Description |
Equations |
| Total energy balance |
$\text{In - Out + Generation = Accumulation}$ |
| Total energy balance |
$E_{\text{in}} - E_{\text{out}} + E_{\text{gen}} = \Delta E_{\text{acc}}$ |
| Total energy balance |
$\dot{E}_{\text{in}} - \dot{E}_{\text{out}} + \dot{E}_{\text{gen}} = \dot{E}_{\text{acc}} = \dfrac{d E_{\text{acc}}}{dt}$ |
Surface energy balance
★ $\dot{E}_{\text{gen}} = \dot{E}_{\text{acc}} = 0$ |
$\dot{E}_{\text{in}} = \dot{E}_{\text{out}}$ |
Heat Diffusion Equation
| Description |
Equations |
| Heat Diffusion Equation |
$\nabla^2 T + \dfrac{\dot{q}}{k} = \dfrac{1}{\alpha}\dfrac{\partial T}{\partial t}$ |
| Heat Diffusion Equation |
$\nabla \cdot q'' = \dot{q} - \rho c_P\dfrac{\partial T}{\partial t}$ |
| Heat Diffusion Equation |
$\dfrac{\partial T}{\partial t} = \alpha\nabla^2 T + \dfrac{\dot{q}}{\rho c_P}$ |
| Volumetric heat generation |
$\dot{q}$ |
| Volumetric heat capacity (storage) |
$\rho c_P$ |
| Thermal diffusivity |
$\alpha = \dfrac{k}{\rho c_P}$ |
Cartesian coordinates
| Direction |
Heat flux |
Differential Area |
| $x \in (-\infty, \infty)$ |
$q_x'' = -k \frac{\partial T}{\partial x}$ |
$dA = dydz$ |
| $y \in (-\infty, \infty)$ |
$q_y'' = -k \frac{\partial T}{\partial y}$ |
$dA = dxdz$ |
| $z \in (-\infty, \infty)$ |
$q_z'' = -k \frac{\partial T}{\partial z}$ |
$dA = dxdy$ |
$$
\dfrac{\partial}{\partial x} \left( k \dfrac{\partial T}{\partial x} \right) +
\dfrac{\partial}{\partial y} \left( k \dfrac{\partial T}{\partial y} \right) +
\dfrac{\partial}{\partial z} \left( k \dfrac{\partial T}{\partial z} \right) +
\dot{q} = \rho c_P \dfrac{\partial T}{\partial t}
$$
Cylindrical coordinates
| Direction |
Heat flux |
Differential Area |
| $r \in [0, \infty)$ |
$q_r'' = -k \frac{\partial T}{\partial r}$ |
$dA = r \ d\phi dz$ |
| $\phi \in [0, 2\pi]$ |
$q_\phi'' = -k \frac{1}{r}\frac{\partial T}{\partial \phi}$ |
$dA = drdz$ |
| $z \in [-\infty, \infty)$ |
$q_z'' = -k \frac{\partial T}{\partial z}$ |
$dA = r \ drd\phi$ |
$$
\dfrac{1}{r}\dfrac{\partial}{\partial r} \left( k r\dfrac{\partial T}{\partial r} \right) +
\dfrac{1}{r^2}\dfrac{\partial}{\partial \phi} \left( k \dfrac{\partial T}{\partial \phi} \right) +
\dfrac{\partial}{\partial z} \left( k \dfrac{\partial T}{\partial z} \right) +
\dot{q} = \rho c_P \dfrac{\partial T}{\partial t}
$$
Spherical coordinates
| Direction |
Heat flux |
Differential Area |
| $r \in [0, \infty)$ |
$q_r'' = -k \frac{\partial T}{\partial r}$ |
$dA = r^2 \sin\theta \ d\theta d\phi$ |
| $\theta \in [0, 2\pi]$ |
$q_\theta'' = -k \frac{1}{r}\frac{\partial T}{\partial \theta}$ |
$dA = r\sin\theta drd\phi$ |
| $\phi \in [0, \pi]$ |
$q_\phi'' = -k \frac{1}{r \sin\theta}\frac{\partial T}{\partial \phi}$ |
$dA = r \ drd\theta$ |
$$
\dfrac{1}{r^2}\dfrac{\partial}{\partial r} \left( k r^2 \dfrac{\partial T}{\partial r} \right) +
\dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta} \left( k \sin\theta \dfrac{\partial T}{\partial \theta} \right) +
\dfrac{1}{r^2 \sin^2\theta}\dfrac{\partial}{\partial \phi} \left( k \dfrac{\partial T}{\partial \phi} \right) +
\dot{q} = \rho c_P \dfrac{\partial T}{\partial t}
$$