Heat flow equation
| (1) | \frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z} |
Initial Conditions
| (2) | T(t=0, z) = T_0 |
Boundary conditions
| (3) | T(t, z=0) = T_f, \quad T(t, z= \infty) = T_0 |
The exact solution is given by following formula:
| (4) | T(t,z) = T_f + (T_0 - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi |
A fair approximation id given by following equation:
| (5) | T(t,z) = T_f + (T_0 - 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] |
where
| (6) | \zeta = \frac{z}{4 a t} |
The temperature gain from over- (under-) lying flow is going to be:
| (7) | dT(t, z) = T(t,z) -T_0= - (T_0 - T_f) \cdot \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) |