Question

A composite rod is made from stainless steel and iron and has a length of 0.532 m. The cross section of this composite rod is shown in the drawing below and consists of a square within a circle The square cross section of the steel is 1.19 cm on a side. The temperature one end of the rod is 76.8 degree c, white it is 19.5 degree c at the other end. Assuming that no heat exits through the surface, find the total amount of heat conducted through the rod in two minutes.

Answer 1

Let L be the length of a side of the square. So, area of the steel, A_steel = L^2 Area of the iron, A_iron = pi R^2 - L^2 now, the radius of the circle is one half the length of the diagonal of the square. So, R = D/2 where, D = Diagonal length of square. R = squareroot 2 L/2 = L/squareroot 2 A_iron = pi (L/squareroot 2)^2 - L^2 = pi/2 L^2 - L^2 A_iron = (pi/2 - 1) L^2 now, Q_total = Q_steet + Q_iron = [K A delta T t/L_compo]_steel + [K A delta T t/L_comp]_iron = delta T t/L_comp [(k A)_steel + (K A)_iron] here, L_comp = Length of the composite. Q_totai = [(14 times L^2) + (79 times (pi/2 - 1) L^2] times delta T t/L_comp we have, L = 1.19 times 10^-2 m, delta T = (76.8 - 19.5) degree C = 57.3 degree C, T = 120 s and L_comp = 0.532 m, K_steel = 14 J/s m degree C K_iron

$\\Q_{total} = (14+40\pi-79)\times \frac{(1.19\times10^{-2})^2\times 57.3\times120}{0.532}J \\ \\Q_{total} = 111.0317277\:J \\ \\Q_{total} \approx 111.032\:J\:\:\:Ans$