[Heat Conduction Problem Solving 7.11] A semi-infinite medium, x > 0, is initially at zero temperature. For times t > 0, the boundary surface at x = 0 is subjected to a periodically varying temperature f (t) as illustrated in Figure 7-5. Develop an expression for the temperature distribution in the medium at times (i) 0 < t < t, (ii) t < t < 2t, and (iii) 6t < t < 7t

Let’s solve a heat conduction problem.A semi-infinite medium, x > 0, is initially at zero temperature. For times t > 0, the boundary surface at x = 0 is subjected to a periodically varying temperature f (t) as illustrated in Figure 7-5. Develop an expression for the temperature distribution in the medium at times (i) 0 < t < t, (ii) t < t < 2t, and (iii) 6t < t < 7t

Heat Conduction problem solving

I will solve the below problem:

A semi-infinite medium, x > 0, is initially at zero temperature. For times t > 0, the boundary surface at x = 0 is subjected to a periodically varying temperature f (t) as illustrated in Figure 7-5. Develop an expression for the temperature distribution in the medium at times (i) 0 < t < t, (ii) t < t < 2t, and (iii) 6t < t < 7t

(sol)
By Example 7-2,
$$ \frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial x} \text{ in } 0<x<\infty, t>0$$
$$BC1: T(x=0,t) = f(t), BC2: T( x\to \infty, t) = 0, IC: T(x,t=0)=0$$
Auxiliary problem is below as:
$$ \frac{\partial^2 \Phi}{\partial x^2} = \frac{1}{\alpha} \frac{\partial \Phi}{\partial x} \text{ in } 0<x<\infty, t>0$$
$$BC1: \Phi(x=0,t) = 1, BC2: T( x\to \infty, t) = 0, IC: T(x,t=0)=0$$
Let $\Phi(x,t) = \Phi_H(x,t) + \Phi_{SS}(x)$. Here, $\Phi_{SS}(x)=1$.
$$ \frac{\partial^2 \Phi_H (x,t)}{\partial x^2} = \frac{1}{\alpha} \frac{\partial \Phi_H(x,t)}{\partial x} \text{ in } 0<x<\infty, t>0$$
$$BC1: \Phi_H(x=0,t) = 0, BC2: \Phi_H( x\to \infty, t) = -1, IC: T(x,t=0)=-1$$
$$\Phi_H(x,t) = -erf\left( \frac{x}{\sqrt{4\alpha t}}\right), \Phi_(x,t) = erfc\left(\frac{x}{\sqrt{4\alpha t}}\right)$$
By (7-18),
$$T(x,t) = \sum_{i=0}^7 (-1)^i \Delta f_0 U(t-i\Delta t) \Phi (x, t-i \Delta t)$$
$ (i).T(x,t)=\Delta f_0 \Phi(x,t)$
$(ii).T(x,t) =\sum_{i=0}^1 (-1)^i \Delta f_0 \Phi (x, t-i \Delta t) $
$(iii) T(x,t) =\sum_{i=0}^6 (-1)^i \Delta f_0 \Phi (x, t-i \Delta t) $

Reference:

Problem 7.11 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

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