## Heat Transfer from Infinite Plate-Time Dependence

**Heat Conduction in a Finite Slab**

This simple example covers the heating of a finite slab and how the temperature varies with time. We will set up the problem in COMSOL Multiphysics after which we compare the solution to the analytical solution.

### Introduction

This simple example covers the heat conduction in a finite slab, modeling how the temperature varies with time. You first set up the problem in COMSOL Multiphysics and then compare it to the analytical solution given in Ref. 1. In addition, this example also shows how to avoid oscillations due to a jump between initial and boundary conditions by using a smoothed step function.

### Model Definition

The model domain is defined between x = −b and x = b. The initial temperature is constant, equal to T_{0}, over the whole domain; see the figure below. At time t = 0, the temperature at both boundaries is lowered to T_{1}.

To compare the modeling results to the literature (Ref. 1), introduce new dimensionless variables according to the following definitions:

$ \displaystyle \Theta =\frac{{{{T}_{1}}-T}}{{{{T}_{1}}-{{T}_{0}}}}$ ; $ \displaystyle \eta =\frac{x}{b}$ ; $ \displaystyle \tau =\frac{{\alpha t}}{{{{b}^{2}}}}$

The model equation then becomes

$ \displaystyle \frac{{\partial \Theta }}{{\partial \tau }}=\frac{{{{\partial }^{2}}\Theta }}{{\partial {{\eta }^{2}}}}$

with the associated initial condition

$ \displaystyle \tau =0$ ; $ \displaystyle \Theta =1$

and boundary conditions

$ \displaystyle \eta =\pm 1$ ; $ \displaystyle \Theta =0$

The analytical solution of this problem is (see Ref. 1, equation 12.1-31):

$ \displaystyle \Theta =2\sum{{\frac{{{{{\left( {-1} \right)}}^{n}}}}{{\left( {n+\frac{1}{2}} \right)\pi }}}}\exp \left[ {-{{{\left( {n+\frac{1}{2}} \right)}}^{2}}{{\pi }^{2}}\tau } \right]\cos \left( {\left( {n+\frac{1}{2}} \right)\pi \eta } \right)$ ; $ \displaystyle n=0,…,\infty $

To model the temperature, decrease at the boundaries use a smoothed step function of time f(τ).

$ \displaystyle \eta =\pm 1$ ; $ \displaystyle \Theta =f\left( \tau \right)$

This method is usually more realistic from a physical point of view than the sudden change in the temperature, and it is also better from a numerical point of view.

### Results and Discussion

Figure 2 shows the temperature as a function of position at the dimensionless times τ = 0.01, 0.04, 0.1, 0.2, 0.4, and 0.6. In this plot, the slab’s center is situated at x = 0 with its end faces located at x = −1 and x = 1. The temperature profiles shown in the graph are identical to the analytical solution given in Ref. 1.

The plot of the L2 error between the analytical and numerical solutions over time (see Figure 3) confirms this conclusion.

**Source:** **1.** R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, 2nd ed., John Wiley & Sons, 2007. / **2.** COMSOL Muliphysics