Fig. 1. Flow and temperature pattern for Semispace Linear Conduction model.

Heat flow equation for Semispcae Linear Conduction:

(1)	$\begin{array}{l}\displaystyle \frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z}\end{array}$

Initial Conditions

(2)	$\begin{array}{l}\displaystyle T(t=0, z) = T_G(z)\end{array}$

Boundary conditions

(3)	$\begin{array}{l}\displaystyle T(t, z=0) = T_f = {\rm const}, \quad T(t, z \rightarrow \infty) = T_G(z)\end{array}$

The exact solution is given by following formula:

(4)	$\begin{array}{l}\displaystyle T(t,z) = T_f + (T_G(z) - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi\end{array}$

A fair approximation id given by following equation:

(5)	$\begin{array}{l}\displaystyle T(t,z) = T_f + (T_G(z) - T_f) \cdot \Bigg[ 1- \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) \Bigg]\end{array}$

where

(6)	$\begin{array}{l}\displaystyle \zeta = \frac{z}{4 a t}\end{array}$

The temperature gain from over- (under-) lying flow is going to be:

(7)	$\begin{array}{l}\displaystyle dT(t, z) = T(t,z) - T_G(z)= - (T_G(z) - T_f) \cdot \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg)\end{array}$

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