Let’s solve a heat conduction problem. A semi-infinite medium, 0 ≤ x < ∞, is initially at a uniform temperature T0. For times t > 0, the region is subjected to a prescribed heat flux at the boundary surface x = 0: −k ∂T ∂x = f0 = constant at x = 0 Obtain an expression for the temperature distribution T(x,t) in the medium for times t > 0 by using Laplace transformation.

## Heat Conduction problem solving

I will solve the below problem:

A semi-infinite medium, 0 ≤ x < ∞, is initially at a uniform temperature T0. For times t > 0, the region is subjected to a prescribed heat flux at the boundary surface x = 0: −k ∂T ∂x = f0 = constant at x = 0 Obtain an expression for the temperature distribution T(x,t) in the medium for times t > 0 by using Laplace transformation.

(sol)

$$\frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t} \text{ in } 0 <x<\infty, t>0$$

$$BC1: – k \frac{\partial T}{\partial x} = f_0 \text{ at } x=0, BC2: T(x \to \infty) = 0, IC: T(x,t=0) =T_0$$

By Laplace transform,

$$ \frac{\partial^2 \overline{T}}{\partial x^2} = \frac{s}{\alpha} \overline{T} – \frac{T_0}{\alpha} \text{ in } 0 <x < \infty, s>0$$

$$BC1: -k \frac{\partial \overline{T}}{\partial x} = \frac{f_0}{s}, BC2: \overline{T} ( x \to \infty) =0$$

Then, $\overline{T}(x,s) = C_1 e^{\sqrt{s/\alpha} x} + C_2 e^{- \sqrt{s/\alpha}x}+ \frac{T_0}{s}$

From BC2, $C_1=0$. From BC1, $C_2 = \frac{f_0 \sqrt{\alpha}}{k} s^{- \frac{3}{2}}$.

$$\overline{T}(x,s) =\frac{f_0 \sqrt{\alpha}}{k} s^{-\frac{3}{2}} e^{-\sqrt{s/\alpha}x}+ \frac{T_0}{s}$$

From integral table,

$$T(x,t) = T_0 + \frac{x}{\sqrt{\alpha}} \int_0^t \frac{\tau}{\sqrt{(t-\tau)^3}} \exp \left( – \frac{x^2}{4\alpha(t-\tau)}\right)$$

Reference:

Problem 9.2 in Hahn, D. W., & Özisik, M. N. (2012). Heat conduction. John Wiley & Sons.

HW problem in Advanced heat transfer lecture by Prof. Seol (설재훈 교수님) in GIST, Fall semester 2023

[Heat Conduction Problem Solving 6.3] In a one-dimensional infinite medium, −∞