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The wall of a constant-diameter pipe of length 1 m is heated uniformly with heat flux $q''$ by wrapping a heater coil around it. The flow at the inlet to the pipe is hydrodynamically fully developed. The fluid is incompressible, and the flow is assumed to be laminar and steady throughout the pipe. The bulk temperature of the fluid is 0 $^\circ$C at the inlet and 50 $^\circ$C at the exit. The wall temperatures are measured at three locations P, Q, and R, as shown in the figure. The flow thermally develops after some distance from the inlet. The following measurements are made: Point: P, Q, R Wall temperature ($^\circ$C): 50, 80, 90 Among the locations P, Q, and R, the flow is thermally developed at

1. P, Q and R
2. P and Q only
3. Q and R only
4. R only

Correct answer: R only

Solution

In thermally developed flow with uniform heat flux, the difference between wall temperature and bulk temperature remains constant along the pipe. Since the bulk temperature increases from inlet to exit, the wall temperature should increase in a way that preserves the same local temperature difference only in the developed region. Among the given points, only R is consistent with the developed condition.

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