A composite rod is made from stainless steel and iron and ha

A composite rod is made from stainless steel and iron and has a length of 0.482 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 1.23 cm on a side. The temperature at one end of the rod is 77.4 °C, while it is 18.9 °C at the other end. Assuming that no heat exits through the cylindrical outer surface, find the total amount of heat conducted through the rod in three minutes.

My online homework says the CORRECT answer 1.95×10²J. No matter how I work the problem I cannot get that answer.

Thank you in advance!

Solution

Assuming that square has its maximum extension, i.e. its corners lie on the circular circumference of the cross section, the diameter of the circle equals the diagonal length of the square. The diagonal length of square is square-root od 2 times its side length.

So the total are of the composite corss section is:
A_total = (/4)D² = (/4)(2 1.23×10²m)²
= (/2)( 1.23 ×10²m)²
= 2.376457763 × 10^-4 m²

The area of the steel cross section is:
A_steel = ( 1.23×10²m)² = 1.5129 × 10^-4 m²

So the are of the iron section is
A_iron = A_total - A_steel = 2.376457763 × 10^-4 m² - 1.5129 × 10^-4 m² = 0.863557763 * 10^-4 m²

The heat flow rate through each section can be found from integral form of fourier\'s law for stationary 1-D heat flow [1]:
H = Q/t = kAT/x

The thermal conductivities are [2]
k_iron = 80 Wm²°C¹
k_steel = 16 Wm²°C¹

H_iron = k_iron A_iron T/x
= 80 Wm²°C¹ 0.863557763 * 10^-4 m² (77.4 °C - 18.9°C) / 0.482m
= 0.838475172 W

H_steel = 16 Wm²°C¹ 1.5129 × 10^-4 m² (77.4°C - 18.9°C) / 0.482m
= 0.293791369 W

The total heat flow rate is
H_total = H_iron + H_steel = 1.132266541 W

So heat conducted through the rod in 3 minutes is
Q = H_total t = 1.132266541 W 180 s = 203.8079774 J