Derive the control volume energy balance equation for three-dimensional transient conduction with heat generation in a rectangular coordinate system.
GIVEN
- Three-dimensional transient conduction with heat generation in a rectangular coordinate system
FIND
(a) The control volume energy balance equation
SKETCH
The control volume, in an x, y, z coordinate system, is shown in the sketch above. Define the nodal
indices as follows
and for simplicity define
Now, heat conducted into the control volume is
The heat generated in the control volume is
and the rate at which thermal energy is stored in the control volume is
Since the heat conducted into the control volume plus the rate at which heat is generated in the control
volume must equal the rate at which energy is stored in the control volume, the difference equation is
Dividing by k ?x ?y ?z we have
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