For a one-dimensional, steady-state, variable thermal conductivity heat conduction with uniform internal heat generation, develop a generalized finite difference formulation for the interior nodes, with the left surface boundary node exposed to constant heat flux and the right surface boundary node exposed to a convective environment. The variable conductivity is modeled such that the thermal conductivity varies linearly with the temperature as k(T) = ko (1?+??T) where T is the average temperature between the two nodes.

Question

What will be an ideal response?

Answer 1

Finite difference formulation for an interior node, boundary node subject to convection and constant heat flux in case of variable thermal conductivity is to be determined.

Analysis The one dimensional steady state heat conduction equation with variable thermal conductivity is expressed as



Using Eq. (5-6), the first derivative of the temperature at the midpoints surrounding the node with variable thermal conductivity can be expressed for x as



Simplifying above equation yields,



For left boundary node exposed to constant heat flux apply energy balance to the half volume around the boundary node with all the heat transfer entering the volume element.

Replacing k by k(T) in Eq. 5-22 we get,



For right boundary node exposed to convection environment, apply energy balance to the half volume around the boundary node with all the heat transfer entering the volume element.