The heat conduction equation in cylindrical coordinates is
(a) Simplify this equation by eliminating terms equal to zero for the case of steady-state heat flow
without sources or sinks around a right-angle corner such as the one in the accompanying
sketch. It may be assumed that the corner extends to infinity in the direction perpendicular to
the page. (b) Solve the resulting equation for the temperature distribution by substituting the
boundary condition. (c) Determine the rate of heat flow from T1 to T2. Assume k = 1 W/(m K)
and unit depth .
GIVEN
• Steady state conditions
• Right-angle corner as shown below
• No sources or sinks
• Thermal conductivity (k) = 1 W/(m K)
FIND
(a) Simplified heat conduction equation (b) Solution for the temperature distribution (c) Rate of heat flow from T1 to T2 ASSUMPTIONS
• Corner extends to infinity perpendicular to the paper
• No heat transfer in the z direction
• Heat transfer through the insulation is negligible
SKETCH
The boundaries of the region are given by
Assuming there is no heat transfer through the insulation, the boundary condition is
(a) The conduction equation is simplified by the following
Steady state
No sources or sinks
No heat transfer in the z direction
Substituting these simplifications into the conduction equation
(b) Integrating twice
The boundary condition can be used to evaluate the constants
Therefore, the temperature distribution is
(c) Consider a slice of the corner as follow
The heat transfer flux through the shaded element in the ? direction is
Multiplying by the surface area drdz and integrating along the radius
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