- Thread starter
- #1

With proper non-dimensionalization and assuming for convenience that the ratio of convection and conduction parameters $h/k = 1$, the cooling process is described by the following equations:

\begin{alignat*}{4}

T_t & = & T_{xx} & \\

T_x(0,t) & = & 0 & \\

T_x + T & = & 0 & \quad \text{at} \ x = 1\\

T(x,0) & = & f(x) &

\end{alignat*}

Here, $x$ and $t$ are dimensionless positions and times wherein the original interval has been mapped from $[0,L]$ to $[0,1]$ and a time-scale based on the diffusivity $\alpha$ has been used; also, the temperature definition is such that the ambient temperature has a reference value of zero.