Derive the expression for critical radius of insulation for a cylindrical tube and explain its significance.

The critical radius of insulation for a cylindrical tube can be derived using the concept of heat transfer through insulation. Let's consider a cylindrical tube with inner radius \( r_i \) and outer radius \( r_o \), and let the thermal conductivity of the insulation material be \( k \). The tube carries a fluid at a certain temperature, and we want to minimize the heat loss ______ ______ ___ _________ ____ ______ ______ ________.
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