Question

Consider a rod of uniform thermal conductivity whose one end \( (x = 0) \) is insulated and the other end \( (x = L) \) is exposed to the flow of air at temperature \( T_{\infty} \) with convective heat transfer coefficient \( h \). The cylindrical surface of the rod is insulated so that the heat transfer is strictly along the axis of the rod. The rate of internal heat generation per unit volume inside the rod is given as \[ \dot{q} = \cos \left( \frac{2 \pi x}{L} \right). \] The steady-state temperature at the mid-location of the rod is given as \( T_A \). What will be the temperature at the same location, if the convective heat transfer coefficient increases to \( 2h \)?

• A. \( T_A + \frac{\dot{q} L}{2h} \)
• B. \( 2T_A \)
• C. \( T_A \)
• D. \( T_A \left( 1 - \frac{\dot{q} L}{4 \pi h} \right) + \frac{\dot{q} L}{4 \pi h} T_{\infty} \)

Hint:

In a steady-state system, the temperature at a given location is primarily controlled by the balance between heat generation and heat loss. Changing the convective heat transfer coefficient affects the temperature at the exposed end but does not change the temperature at the mid-location.

Answer 1

The Correct Option is C

This is a steady-state heat transfer problem involving heat generation within the rod and convective heat loss at the exposed end. Step 1: Understanding the Problem
The system has an internal heat generation term \( \dot{q} \), and the temperature at the mid-location of the rod is \( T_A \). The temperature at any point in the rod will be influenced by both the internal heat generation and the convective heat transfer at the exposed end. Step 2: Effect of Convective Heat Transfer Coefficient
The convective heat transfer coefficient, \( h \), determines how efficiently heat is lost from the surface of the rod to the surrounding air. When the convective heat transfer coefficient increases to \( 2h \), the rate at which heat is lost to the surroundings increases, meaning the temperature distribution along the rod will change. However, the temperature at the mid-location of the rod, \( T_A \), is primarily controlled by the internal heat generation, which remains the same. Since the temperature at the mid-location depends on the balance between internal heat generation and heat loss, the effect of increasing the convective heat transfer coefficient will be negligible at this location. The temperature at the mid-location remains \( T_A \). Thus, the correct answer is (C). Final Answer:
\[ \boxed{(C)} \]