Question

# Starting with an energy balance on a volume element, obtain the two-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for T(x, y, t) for the case of constant thermal conductivity and no heat generation.

Solution

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The volume element is centered about the general interior node $(m,n)$ and it involves heat conduction from four sides (right, left, top, bottom).

Assuming all heat transfer is into the element, the energy balance can be expressed as:

\begin{align*} \left(\begin{matrix} \text{Heat transferred into}\\ \text{the volume element}\\ \text{from all of its surfaces}\\ \text{during \Delta t}\\ \end{matrix} \right) + \left( \begin{matrix} \text{Heat generated}\\ \text{within the}\\ \text{volume element}\\ \text{during \Delta t}\\ \end{matrix} \right) = \left( \begin{matrix} \text{The change in the}\\ \text{energy content of}\\ \text{the volume element}\\ \text{during \Delta t}\\ \end{matrix} \right) \end{align*}

When we put that in equation, we get:

\begin{align*} \Delta t \cdot \sum_{\text{all sides}} \overset{.}{Q} + \Delta t \cdot \overset{.}{E}_{\text{gen, element}} = \Delta E_{element}\\ \end{align*}

The rate of heat transfer $\overset{.}{Q}$ consists of conduction terms for interior nodes.

By dividing the last equation by $\Delta t$ and substituting $Delta E_{element}$ with its expression, we get the following:

\begin{align*} \Delta E_{element} = mc_p\Delta T = \rho V_{element}c_p\Delta T\\ \sum_{\text{all sides}} \overset{.}{Q} + \overset{.}{E}_{\text{gen, element}} = \rho V_{element}c_p \frac{\Delta T}{\Delta t}\\ \end{align*}

We can also write that equation for any node $m$ in the medium of its volume element.

\begin{align*} \sum_{\text{all sides}} \overset{.}{Q} + \overset{.}{E}_{\text{gen, element}} = \rho V_{element}c_p \frac{T_m^{i+1}-T_m^i}{\Delta t}\\ \end{align*}

For a two-dimensional case, we can write it as:

\begin{align*} \overset{.}{Q}_{left}+\overset{.}{Q}_{top}+\overset{.}{Q}_{right}+\overset{.}{Q}_{bottom}+\overset{.}{E}_{\text{gen, element}} = \rho V_{element}c_p \frac{T_m^{i+1}-T_m^i}{\Delta t}\\ \end{align*}

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