Topics: Heat Transfer Cheat Sheet

Heat Transfer Cheat Sheet

Notation Convention

name	quantity	units	alternative
heat tansfer	$q$	$W$	$Q$ or $\dot{q}$
heat flux	$q''$	$W / m 2$	$q$
generation	$q'''$	$W / m 3$	$G$

Equivalent Resistance

Analogy of electricity to Fourier’s Law. (Why a thermal scientist would want to use an electrical analogy is beyond me.)

Assumptions

No generation
Constant properties
Steady state
One-dimensional conduction

$q=\frac{\Delta T}{R_{eq}}$

where $R_{eq}\,$ is the series and parallel combination of resistances as if they were resistors in units of K/W.

For Conduction:

$R_{\rm planar}= \frac{L}{kA}$

$R_{\rm cylindrical}= \frac{ln(r_{o}/r_{i})}{2 \pi l k}$

$R_{\rm spherical}= \frac{\frac{1}{r_{i}}-\frac{1}{r_{o}}}{\pi 4 k}$

For Convection:

$R_{conv}= \frac{1}{hA}$

where $A$ is the area available for heat transfer.

For Radiation:

$R_{rad} = \frac{1}{\epsilon \sigma A (T+T_{surr})(T^2 + T_{surr}^2)}$

Or if $\frac{(\delta T/ T_{m})^2}{4}<<1$ then $R_{rad} = \frac{1}{A h_{rad}}$ where $h_{rad}= 4 \sigma T_{m}^3 \epsilon$ and $T_{m} = T\,$ at the middle of two known temperatures.

For Contact Interfaces:

$R_{contact}= \frac{1}{h_{c}A}$

Extended Surfaces (1D Fins)

For these temperature distributions ( $θ = T - T \infty$ ), the base temperature ( $θ b$ )is defined. Also, $M = \sqrt{hPkA_c}\theta_b$ , and $m 2 = h P / k A c$ .

Specified tip temperature $θ(x = L) = θ L$ :

$\frac{\theta}{\theta_b} = \frac{(\theta_L/\theta_b) \sinh mx + \sinh m(L-x)}{\sinh mL}$

$q = M \frac{\cosh mL - \theta_L/\theta_b}{\sinh mL}$

Insulated tip $d T / d x | x = L = 0$ :

$\frac{\theta}{\theta_b} = \frac{ \cosh m(L-x) }{\cosh mL}$

$q = M tanh m L$

Convective tip $- k d θ / d x | x = L = h θ(x = L)$ :

$\frac{\theta}{\theta_b} = \frac{\cosh m(L-x) + (h/mk)\sinh m(L-x)}{\cosh mL + \sinh mL}$

$q = M\frac{\sinh mL + (h/mk) \cosh mL}{\cosh mL + (h/mk)\sinh mL}$

Infinitely long fin:

$\frac{\theta}{\theta_b} = \exp (-mx)$

$q = M$

Lumped Capacitance

Semi-Infinite Slab Solutions

All solutions expressed relative to the initial uniform temperature $θ = T - T 0$ . Assume constant properties.

For specified temperature at $x = 0$ :

$\frac{\theta(x,t)}{\theta_b} = {\rm erfc} \left ( \frac{x}{\sqrt{4\alpha t}} \right ),$

where $θ b$ is the specified boundary temperature. The corresponding heat flux is given as

$q''(x,t) = \frac{k}{\sqrt{\pi \alpha t}} \exp \left ( \frac{-x^2}{4\alpha t} \right )$

A consequence of the constant temperature boundary condition is that the interface temperature between two semi-infinite solids that suddenly come in contact can be expressed as

$T_i = \frac{\sqrt{(\rho c k)_A} T_A + \sqrt{(\rho c k)_B} T_B} {\sqrt{(\rho c k)_A} + \sqrt{(\rho c k)_B}}$

For specified constant heat flux at $x = 0$ :

$\theta(x,t) = \frac{q''}{2k} \left [ \sqrt{\frac{4\alpha t}{\pi}} \exp \left ( \frac{-x^2}{4\alpha t} \right ) + x {\rm erfc} \left ( \frac{x}{4 \alpha t} \right ) \right ]$

For convective condition at $x = 0$ :

$\frac{\theta(x,t)}{\theta_{\infty}} = {\rm erfc} \left ( \frac{x}{\sqrt{4\alpha t}} \right ) - \exp \left ( \frac{hx}{k} + \frac{h^2\alpha t}{k^2} \right ) {\rm erfc} \left ( \frac{x}{\sqrt{4\alpha t}} + \frac{h\sqrt{\alpha t}}{k} \right )$

For steady periodic heating at $x = 0$ :

$\frac{\theta(x,t)}{\theta_m} = \exp \left ( \frac{-x}{\sqrt{\alpha t/\pi}} \right ) \sin \left ( \frac{2\pi t}{\tau} - \frac{x}{\sqrt{\alpha t/\pi}} \right ),$

where $τ$ is the period in rad/s, $θ = T - T m e a n$ and $θ m$ is the maximum temperature. The heat flux is given as

$q''(0,t) = \sqrt{\frac{2\pi \rho c k}{\tau}} \theta_m \sin \left ( \frac{2\pi t}{\tau} + \frac{\pi}{4} \right )$

For an energy pulse at $x = 0$ and $t = 0$ :

$\theta(x,t) = \frac{E\exp \left ( \frac{-x^2}{4\alpha t} \right )}{\sqrt{4\pi k\rho c t}},$

where $E$ is in J/m^2^.

Last modified on 22 Jan, 2009

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