GATE 2022 ME (ME2) – Question 65
Consider a hydrodynamically and thermally fully-developed, steady fluid flow of 1 kg/s in a uniformly heated pipe with diameter of 0.1 m and length of 40 m. A constant heat flux of magnitude 15000 W/m$^2$ is imposed on the outer surface of the pipe. The bulk-mean temperature of the fluid at the entrance to the pipe is 200 °C. The Reynolds number (Re) of the flow is 85000, and the Prandtl number (Pr) of the fluid is 5. The thermal conductivity and the specific heat of the fluid are 0.08 W m$^{-1}$ K$^{-1}$ and 2600 J kg$^{-1}$ K$^{-1}$, respectively. The correlation $Nu=0.023Re^{0.8}Pr^{0.4}$ is applicable, where the Nusselt Number (Nu) is defined on the basis of the pipe diameter. The pipe surface temperature at the exit is ______ °C (round off to the nearest integer).
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Correct answer: 321
Explanation
**Step 1: bulk temperature at the exit.** A constant heat flux over the surface adds heat at a steady rate.
$$Q=q^{\prime\prime}\,\pi DL=15000\times\pi(0.1)(40)=188\,496\text{ W}.$$
$$T_{b,out}=T_{b,in}+\frac{Q}{\dot m\,c_p}=200+\frac{188\,496}{1\times2600}=272.5\ ^\circ\text{C}.$$
**Step 2: heat transfer coefficient.**
$$Nu=0.023\,Re^{0.8}Pr^{0.4}=0.023(85\,000)^{0.8}(5)^{0.4}=384.5$$
$$h=\frac{Nu\,k}{D}=\frac{384.5\times0.08}{0.1}=307.6\text{ W/m}^2\text{K}.$$
**Step 3: wall temperature at the exit.** In fully developed flow with constant flux, $T_w-T_b=q^{\prime\prime}/h$:
$$T_w=272.5+\frac{15000}{307.6}=272.5+48.8=\mathbf{321\ ^\circ C}.$$