Question

Consider a uniformly tapered steel rod of circular cross-section of 1 m length. The diameter of the rod at one end is 5 cm, and that at the other end is 2.5 cm. If the heat flux at the end of the larger cross-section is 2500 kcal/m$^2\cdot$hr, the heat flux at the other end is equal to

• A. 2500 kcal/m$^2\cdot$hr
• B. 5000 kcal/m$^2\cdot$hr
• C. 7500 kcal/m$^2\cdot$hr
• D. 10,000 kcal/m$^2\cdot$hr

Hint:

In a tapered rod under steady conduction, if heat flow is constant, then heat flux is inversely proportional to the cross-sectional area — hence, smaller area means higher flux.

Answer 1

The Correct Option is D

Given that the rod is of circular cross-section and tapered uniformly, and assuming steady-state heat conduction with no internal heat generation, the total heat flow (Q) through any cross-section remains constant.
Heat flux ($q$) is defined as:
\[ q = \frac{Q}{A} \]
So if $Q$ is constant, then:
\[ q \propto \frac{1}{A} \Rightarrow q \propto \frac{1}{\pi d^2/4} \Rightarrow q \propto \frac{1}{d^2} \]
Let $q_1$ be the heat flux at the larger end ($d_1 = 5$ cm), and $q_2$ be the heat flux at the smaller end ($d_2 = 2.5$ cm):
\[ \frac{q_2}{q_1} = \left( \frac{d_1}{d_2} \right)^2 = \left( \frac{5}{2.5} \right)^2 = 2^2 = 4 \Rightarrow q_2 = 4 \cdot q_1 = 4 \cdot 2500 = 10,000 \ \text{kcal/m}^2\cdot\text{hr} \]
Thus, the heat flux at the smaller end is 10,000 kcal/m$^2\cdot$hr.