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\section{Heat Transfer}


References: 

\begin{rawhtml} <a href=''http://en.wikipedia.org/Heat_equation''>Wikipedia</a> \end{rawhtml}

Numerical  Recipes, Third edition by W. Press et al,

A Survey of Computational Physics by R. Landau et al


\section{General}

The heat transfer problem seeks to find the temperature profile
in a given domain and over a time interval. This is to be done
while the boundary conditions are satisfied. The
model is summarized by the heat equation, 
accompanied by the specification of the physical
boundary conditions and possibly, heat sources.
This equation is derived in the 
<a href=''http://en.wikipedia.org/Heat\_equation''> Wikipedia webpages.

\section{Physical Intuition}

This equation is a parabolic partial differential equation (PDE).  
Its solution $T(x,y,z,t)$ models the temperature profile
and the heat transfer over a domain in $1-D$, $2-D$ or $3-D$.
It is an explicit function of the time, $t$.  
In other words, the solution of the
heat equation (PDE) models the evolution of the temperature
profile overtime, starting with a given distribution
$T(x,y,z,t=0)$ at $t = 0$.  The heat equation assumes 
independent space and time variables.

Another way to visualize the solution of this equation is 
though plotting a 
graph of the temperature profile generated at each $\Delta t$
on a discretized time grid. These graphs will make a {\em movie}
if displayed in order and at fixed time intervals.

\section{Derivation of the Heat Equation}

The heat equation equation is derived in the 
\begin{rawhtml}
<a href=''http://en.wikipedia.org/Heat\_equation''> Wikipedia site.</a>
\end{rawhtml}
It is instructive to read this derivation.

It is based on Fourier's law which says that {\em the heat exchanged
through a surface is proportional to minus the gradient of the
temperature across the surface} and on energy conservation.

The equation reads
\begin{equation}
\frac{ \partial T}{\partial t} = \alpha \Delta^2 T = 0
\end{equation}
where 
\begin{equation}
 \alpha = \frac{k}{C_p |rho}
\end{equation}
The constants are  $k$, the {\em thermal conductivity},  $C_p$, 
the {\em specific heat capacity} and $\rho$ the {\em density}.

$\alpha$ is called the thermal diffusivity.

\section{Analytical Solution}

Joseph Fourier published a significant text titled {\em “Theorie Analytique
de la Chaleur”} in 1822. He proposed the expansion of arbitrary
functions in infinite sums of trigonomic functions, sin() and cos(),
to precisely solve the heat equation. 

The solution of a $1-D$ heat equation for a simple Dirichlet boundary
condition and no heat source is reproduced in the
<a href=''http://en.wikipedia.org/Heat\_equation''> Wikipedia site.</a>

This solution is very similar to that for the elliptic PDE. It starts
with a separation of variables, $X(x)$ function of $x$ and $T(t)$ 
function of $t$, and imposes the Boundary conditions.  This
brings in the Fourier sums, and mixes the variables. The details
are found in yhe Wikepidia web site.

\section{Numerical Solution}

Various numerical methods have been devived to solve the heat equation
numerically.  The method based on 
Finite Difference start with
rewriting the heat equation in terms of finite differences on a
space grid (equally spaced, $\Delta_x$) and a time grid (equally spaced,
$\Delta_t$).

Picture a $1-D$ space and time grid as in the following figure;

\begin{rawhtml}
<img src=../Heat/grid1.GIF>
\end{rawhtml}


The initial temperature field $T(s,t=0)$ is on the  $t=0$ line.
It must satisfy the boundary conditions. The task is now to solve
for $T(x,t)$ for all the subsequent time lines under the BCs 
constraints.

Note that the grid does not close with a known field profile for large time.
This makes it impossible to use an algorithm based on a stencil
(averaging) as in the elliptic PDE scheme.  

An {\em explicit} scheme follows if we use a first
order time derivative, i.e., the forward derivative,  
\begin{equation}
\frac{\partial T(x, t)}{\partial t} = \frac{T(x,t+\Delta t) - T(x,t)}{\Delta t}
\end{equation}
The space derivative can make use of the finite difference second order form.
Therefore, the heat equation becomes
\begin{equation}
   \frac{T(x,t+\Delta t) - T(x,t)}{\Delta t} = 
     \frac{k}{C \rho} 
       \frac{ T(x+\Delta_x), t)  - 2 T(i,j) + T(x-\Delta_x, t)  } { \Delta_x^2 }
\end{equation}

Introducing the grid indices, $ x = i \Delta_x $ and $ t = j \Delta_t $, the
heat equation becomes
\begin{equation}
T_{i, j+1} = T_{i, j} + \kappa \left[ T_{i+1.j} - 2 T_{i, j} + T_{i-1, j} \right]
\end{equation} 
where $\kappa = \frac{k \Delta t}{c \rho {\Delta t}^2 }$.


This algorithm is {\em explicit}.  The $j$ index starts at $t=0$, 
the initial time, with $T_{ i, j=0}$ , and proceeds up to 
the desired time.  

\begin{rawhtml}
<img src=../Heat/grid2.GIF>
\end{rawhtml}

This yields the {\em finite difference equation for heat transfer}.





\section{Von Neumann Stability Analysis}

The method above is known as {\em Foward Time Centered Space (FTCS) }.
Unfortunately, contrary to the finite diffrence method used to solve Poisson and
Laplace equation, the FTCS is an unstable method.

The fact that the solution depends on
space and time introduces a criteria for the choice of the space grid
 and the time grid intervals. Assume that the coefficients
of the finite difference equation are roughly constant.  Then
the eigenmode have the following form 
\begin{equation}
  T_{n,j} = \xi^nj e^{ n i k / Delta_x} 
\end{equation} 
where $\xi(k)$ is some arbitry complex amplitude and k$k$ a constant.
Substitution in the finite difference heat equation
 leads to this factor
\begin{equation}
\xi = 1 - \frac{2 \kappa \Delta_t}{ \Delta_x^2 } 
\end{equation}
Requiring that $|\xi|<1$ implies stability criterion (the nodes
remain finite)
\begin{equation}
  \frac{2 \kappa \Delta_t}{ \Delta_x^2 } < 1
\end{equation}

 This condition derives from the
physical fact that the maximum allowed time interval $\Delta_t$ 
is, up to a numerical factor, the diffusion time across
a cell of width $\Delta_x$.

Note that diffusion expands proportional to $\sqrt{t}$.


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