Temperature Diffusion

The unknown temperature $ T$ is a function of its location in the domain and time $ t>0$. The governing equation in the interior of the domain is given by

$\displaystyle \rho c\hackscore p T\hackscore{,t} - (\kappa T\hackscore{,i})\hackscore{,i} = q\hackscore H$ (14)

where $ \rho c\hackscore p$ and $ \kappa$ are given material constants. In case of a composite material the parameters depend on their location in the domain. $ q\hackscore H$ is a heat source (or sink) within the domain. We are using the Einstein summation convention as introduced in Chapter 1.2. In our case we assume $ q\hackscore H$ to be equal to a constant heat production rate $ q^{c}$ on a circle or sphere with center $ x^c$ and radius $ r$ and 0 elsewhere:

$\displaystyle q\hackscore H(x,t)= \left\{ \begin{array}{lcl} q^c & & \Vert x-x^c\Vert \le r \\ & \mbox{if} \\ 0 & & \mbox{else} \\ \end{array} \right.$ (15)

for all $ x$ in the domain and all time $ t>0$.

On the surface of the domain we are specifying a radiation condition which prescribes the normal component of the flux $ \kappa T\hackscore{,i}$ to be proportional to the difference of the current temperature to the surrounding temperature $ T\hackscore{ref}$:

$\displaystyle \kappa T\hackscore{,i} n\hackscore i = \eta (T\hackscore{ref}-T)$ (16)

$ \eta$ is a given material coefficient depending on the material of the block and the surrounding medium. $ n\hackscore i$ is the $ i$-th component of the outer normal field at the surface of the domain.

To solve the time-dependent Equation (1.15) the initial temperature at time $ t=0$ has to be given. Here we assume that the initial temperature is the surrounding temperature:

$\displaystyle T(x,0)=T\hackscore{ref}$ (17)

for all $ x$ in the domain. It is pointed out that the initial conditions satisfy the boundary condition defined by Equation (1.17).

The temperature is calculated at discrete time nodes $ t^{(n)}$ where $ t^{(0)}=0$ and $ t^{(n)}=t^{(n-1)}+h$ where $ h>0$ is the step size which is assumed to be constant. In the following the upper index $ {(n)}$ refers to a value at time $ t^{(n)}$. The simplest and most robust scheme to approximate the time derivative of the the temperature is the backward Euler scheme. The backward Euler scheme is based on the Taylor expansion of $ T$ at time $ t^{(n)}$:

$\displaystyle T^{(n)}\approx T^{(n-1)}+T\hackscore{,t}^{(n)}(t^{(n)}-t^{(n-1)}) =T^{(n-1)} + h \cdot T\hackscore{,t}^{(n)}$ (18)

This is inserted into Equation (1.15). By separating the terms at $ t^{(n)}$ and $ t^{(n-1)}$ one gets for $ n=1,2,3\ldots$

$\displaystyle \frac{\rho c\hackscore p}{h} T^{(n)} - (\kappa T^{(n)}\hackscore{,i})\hackscore{,i} = q\hackscore H + \frac{\rho c\hackscore p}{h} T^{(n-1)}$ (19)

where $ T^{(0)}=T\hackscore{ref}$ is taken form the initial condition given by Equation (1.18). Together with the natural boundary condition

$\displaystyle \kappa T\hackscore{,i}^{(n)} n\hackscore i = \eta (T\hackscore{ref}-T^{(n)})$ (20)

taken from Equation (1.17) this forms a boundary value problem that has to be solved for each time step. As a first step to implement a solver for the temperature diffusion problem we will first implement a solver for the boundary value problem that has to be solved at each time step.

esys@esscc.uq.edu.au