Derive the energy balance equation for a corner control volume in a three-dimensional steady conduction problem with heat generation in a rectangular coordinate system. Assume an adiabatic boundary condition and equal node spacing in all three dimensions.
GIVEN
- Three-dimensional steady conduction in a rectangular coordinate system, corner boundary control
volume with specified temperature boundary condition
FIND
(a) Energy balance equation for the control volume
SKETCH
First, define the nodal indices as follows
and for simplicity, let
where, as usual, m is the time index. Note that the volume of the control volume is
Referring to the sketch above, we see that there are three surfaces across which heat is transferred by
conduction. For these surfaces, the heat transferred into the control volume is
Heat generation in the control volume is
and the rate at which energy is stored in the control volume is
The resulting energy balance equation for the control volume is
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