## Related questions with answers

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

VerifiedThe 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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