A composite rod is made from stainless steel and iron and has a length of 0.521

Question

A composite rod is made from stainless steel and iron and has a length of 0.521 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 1.30 cm on a side. The temperature at one end of the rod is 79.1 °C, while it is 17.2 °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 four minutes.

NOTE: The answer is NOT 296.1 J. Please explain each step, Thank You!

Explanation / Answer

the total are of the composite corss section is:
A_total = (/4)D² = (/4)(21.30×10²m)²

= (/2)(1.30×10²m)²
= 2.65×104m²

The area of the steel cross section is:
A_steel = ( 1.30×10²m)² =1.69×104m²

So the are of the iron section is
A_iron = A_total - A_steel =2.65*104m² - 1.69*104m² = 0.69×104m²

he heat flow rate through each section can be found from integral form of fourier's law for stationary 1-D heat flow flow [1]:
H = Q/t = k*A*T/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 *1.30*10-4m² * (79.1 °C - 17.2°C) / 0.521m
=1.235 W

H_steel = 16 Wm²°C¹ * 1.69×104m² (79.1°C - 18.1°C) / 0.546m
=1.606 W

So heat conducted through the rod in 4minutes is :
Q = H_total*t = 1.606J/s *(4X60)s = 385.44 J

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