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

Physics & Space Science

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