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Heat Transfer Cheat Sheet

This page provides an “equation sheet” that students like to use on tests. Developed originally by Vanderbilt University’s heat transfer class (ME 248).

Notation Convention

name quantity units alternative
heat tansfer q W Q or \dot{q}
heat flux q'' W / m2 q
generation q''' W / m3 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 = TT), the base temperature (θb)is defined. Also, M = \sqrt{hPkA_c}\theta_b, and m2 = hP / kAc.

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

\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 dT / dx | x = L = 0:

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

q = MtanhmL

Convective tip kdθ / dx | x = L = hθ(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 = TT0. 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 θb 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 = TTmean and θm 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