Conduction thermal resistance in different coordinate system

Thermal contact resistance The heat transfer through the solid contact spots and the gaps in the contact areas can be expressed as It can also be expresses as

The value of thermal contact resistance depends on the surface roughness, material properties, temperature, pressure, and the type of fluid trapped at the interface. Thermal contact resistance decreases with decreasing surface roughness and increasing interface pressure. Values of thermal contact resistance fall between 5x10-6 to 5x10-4 m2.oC/W

Problem

Adding more insulation to a wall always decreases the heat transfer since the heat transfer area is constant and adding insulation will always increase the thermal resistance of the wall without increasing the convection resistance.

However, adding insulation to a cylinder pipe or a spherical shell is a different matter. It increases the conduction resistance and decreases the convection resistance because of the increase in the surface area. So, the heat transfer may increase or decrease depending on which effect dominates. Due to these opposite effects, a critical radius of insulation is defined as the outer radius that provides maximum rate of heat transfer.

The rate of heat transfer from a cylinder increases with the addition of insulation when r2<rcr , reaches a maximum when r2=rcr , and decreases when r2>rcr.

The critical radius of insulation for a cylinder is calculated as: The critical radius of insulation for a sphere is calculated as:

radius