For the system outlined determine an expression for the critical radius of the insulation in terms of the thermal conductivity of the insulation and the surface coefficient between the exterior surface of the insulation and the surrounding fluid. Assume that the temperature difference, R1, R2, the heat transfer coefficient on the interior, and the thermal conductivity of the material of the sphere between R1 and R2 are constant.

Question

GIVEN

- An insulated hollow sphere

- Radii

- Inner surface of the sphere = R1

- Outer surface of the sphere = R2

- Outer surface of the insulation = R3

FIND

- An expression for the critical radius of the insulation

ASSUMPTIONS

- Constant temperature difference, radii, heat transfer coefficients, and thermal conductivities

- Steady state prevails

SKETCH

Answer 1

the rate of heat transfer through the sphere is



The rate of heat transfer is a maximum when the denominator of the above equation is a minimum. This occurs when the derivative of the denominator with respect to R3 is zero



The maximum heat transfer will occur when the outer insulation radius is equal to 2 k23/h3.