A hollow tube possesses a length l , inner radius R₁ , and outer radius R₂ . The material has a thermal conductivity K . Determine the heat flowing through the walls of the tube if the inside of the tube is maintained at temperature T₁ and the outside is maintained at T₂ .

Step-by-step solution

Consider an elemental cylindrical shell of radius r and thickness dr . The surface area of this shell is A = 2 r l . According to Fourier's law of heat conduction, the rate of heat flow H is given by: H = -K A dT dr = -K (2 r l) dT dr Rearranging the terms for integration: dr r = - 2 K l H dT Integrating both sides within the limits r = R₁ to R₂ and T = T₁ to T₂ : _ R₁ ^ R₂ dr r = - 2 K l H _ T₁ ^ T₂ dT ( R₂ R₁ ) = 2 K l H (T₁ - T₂) Taking the magnitude of heat flow, we get: H = 2 K l (T₂ - T₁) (R₂ / R₁)