A central hole of radius r₁ is created in a uniform circular disc having thickness d and radius r₂ . The inner surface (a cylinder of length d and radius r₁ ) is kept at a temperature ₁ , while the outer surface (a cylinder of length d and radius r₂ ) is kept at a temperature ₂ (with ₁ > ₂ ). Given that the thermal conductivity of the disc's material is K , determine the heat flowing per unit time through the disc.

Step-by-step solution

Consider a cylindrical shell of radius r and thickness dr within the disc. The surface area of this shell through which heat flows radially is A = 2 r d . The thermal resistance of this elemental shell is given by dR = dr KA = dr K(2 r d) . The total thermal resistance of the disc is obtained by integrating from r = r₁ to r = r₂ : R = _ r₁ ^ r₂ dr 2 K r d = 1 2 K d ( r₂ r₁ ) The rate of heat flow is given by H = R , where = ₁ - ₂ . H = ₁ - ₂ 1 2 K d ( r₂ r₁ ) = 2 K d ( ₁ - ₂) (r₂ / r₁)