Let’s solve a heat conduction problem. In a one-dimensional infinite medium, −∞ <x< ∞, initially, the region a<x<b is at a constant temperature T0, and everywhere outside this region is at zero temperature. Obtain an expression for the temperature distribution T(x, t) in the medium for times t > 0.
Heat Conduction problem solving
I will solve the below problem:
In a one-dimensional infinite medium, −∞ <x< ∞, initially, the region a<x<b is at a constant temperature T0, and everywhere outside this region is at zero temperature. Obtain an expression for the temperature distribution T(x, t) in the medium for times t > 0.
(sol)
From (6-132) in Hahn, D. W., & Özisik, M. N. (2012). Heat conduction. John Wiley & Sons.
$$T(x,t) = \frac{1}{(4\pi \alpha t)^{1/2}} \int_{-\infty}^\infty F(x^\prime) \exp \left( – \frac{(x-x^\prime)^2}{4\alpha t}\right) dx^\prime$$
Where $F(x) = T_0$ for $a \leq x \leq b$, $F(x)=0$ otherwise.
$$T(x,t) = \frac{T_0}{(4\pi \alpha t)^{1/2}} \int_{a}^b \exp \left( – \frac{(x-x^\prime)^2}{4\alpha t}\right) dx^\prime $$
Let $\eta = \frac{x-x^\prime}{\sqrt{4\alpha t}}$. $dx^\prime = – \sqrt{4\alpha t} d \eta $
$$T(x,t) = \frac{T_0}{2} \left( \frac{2}{\sqrt{\pi}} \int_{\frac{x-b}{\sqrt{4\alpha t}}}^{\frac{x-a}{\sqrt{4\alpha t}}} \exp ( – \eta^2 ) d \eta \right)$$
$$T(x,t) = \frac{T_0}{2} \left( erf\left( \frac{x-a}{\sqrt{4\alpha t}}\right) +erf\left( \frac{-x+b}{\sqrt{4\alpha t}}\right) \right)$$
Reference: 6.3 in Hahn, D. W., & Özisik, M. N. (2012). Heat conduction. John Wiley & Sons.