Determine the rate of heat flow across a cross section of the rod illustrated in the figure ( ₂ > ₁ ). The material of the rod has a thermal conductivity K .

Step-by-step solution

Consider an elemental disc of thickness dx at a distance x from the end with radius r₁ . The radius of this disc is given by r = r₁ + ( r₂ - r₁ L )x . The thermal resistance of this elemental disc is dR = dx K A = dx K r^2 . The total thermal resistance R of the rod is the integral of dR from x = 0 to x = L : R = ₀^ L dx K [ r₁ + ( r₂ - r₁ L )x ]^2 Let y = r₁ + ( r₂ - r₁ L )x . Differentiating with respect to x , we get dy = ( r₂ - r₁ L )dx dx = L r₂ - r₁ dy . The limits of integration change from x=0 L to y=r₁ r₂