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

Physics & Space Science

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