Transient Conduction into a Semi-infinite Solid: Screencasts
The equation for the transfer of heat into a semi-infinite solid is derived, and several related concepts are discussed.
We suggest that after watching this screencast, you list the important points as a way to increase retention.
This screencast shows simulations of six semi-infinite heat transfer scenarios. The boundary conditions and governing equations are discussed for each scenario.
We suggest that after watching this screencast, you list the important points as a way to increase retention.
Important Equations:
There are a number of ways to characterize a semi-infinite surface. A common approach is to assume the temperature at the surface is a constant. In this case, the temperature as a function of depth and time, \(T(x,t)\) is:
\[T(x,t) = (T_0\, -\, T_S)erf\left( \frac{x}{\sqrt{4\alpha t}} \right)\,+\,T_S\]
where \(T_0\) (sometimes known as \(T_i\)) = initial temperature of the solid (°C),
\(T_S\) = constant surface temperature (°C),
\(x\) = depth below the surface (m),
\(\alpha\) = materials thermal diffusivity (m2/s), and
\(t\) = elapsed time (s).
A second approach is to assume heat transfer at the surface arises from convection. In this case, \(T(x,t)\) is:
\[T = T_0 \,+\, (T_{\infty} \, – \, T_0) \left[1 \,-\, erf \left( \frac{x}{\sqrt{4\alpha t}} \right) \,- \,exp \left( \frac{hx}{k} \, + \, \frac{h^2 \alpha t}{k^2}\right) \left( 1\,-\, erf\left( \frac{x}{\sqrt{4 \alpha t}} \,+\,\frac{h}{k} \sqrt{\alpha t}\right) \right) \right]\]
where \(T_{\infty}\) = temperature of the fluid (°C),
\(h\) = fluid’s heat transfer coefficient (W/m2-K),
and \(k\) = material’s thermal conductivity (W/m-K)
A third approach is to assume a constant heat flux at the surface. Not that \(T(x,t)\) in this equation will eventually go to inifinity. This is not realistic for long times:
\[T = T_0 \,+\, \frac{q”}{k} \left( \sqrt{\frac{4 \alpha t}{\pi}} exp\left( – \frac{x^2}{4 \alpha t} \right) -x \left(1 \,-\, erf\left( \frac{x}{\sqrt{4 \alpha t}} \right) \right) \right)\]
where \(q”\) = constant heat flux at the surface (W/m2)
Note there are several other approaches, which will not be reproduced here.
Important Definitions:
Error function: \(erf(z)\) is defined as \[erf(z) =\frac{2}{\sqrt{\pi}}\, \int^z _0 e^{-t^2} dt\] It must be calculated using software or looked up in a table. Note that \(erf(0) = 0\) and \(erf(\infty) = 1\).