A plane wall of thickness 2L has internal heat sources whose strength varies according to 0q is the heat generated per unit volume at the center of the wall (x = 0) and a is a constant. If both sides of the wall are maintained at a constant temperature of Tw, derive an expression for the total heat loss from the wall per unit surface area.

Question

GIVEN

A plane wall with internal heat sources Heat source strength: q

Gcos (ax) Wall surface temperatures = Tw Wall thickness = 2L

FIND

An expression for the total heat loss per unit area (q/A)

ASSUMPTIONS

Steady state conditions prevail

The thermal conductivity of the wall (k) is constant

One dimensional conduction within the wall

SKETCH

Answer 1

gives the equation for one dimensional conduction. For steady state, dT/dt = 0, therefore





With boundary conditions:



Integrating the conduction equation once



Applying the first boundary condition yields: C1 = 0

The rate of heat transfer from one side of the wall is



The total rate of heat transfer is twice the rate of heat transfer from one side of the wall



An alternative method of solution for this problem involves recognizing that at steady state the rate of heat generation within the entire wall must equal the rate of heat transfer from the wall surfaces