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GATE 2024 ME – Question 19

Heat Transfer · Convection: boundary layer, dimensionless parameters and correlations · 1 mark · Multiple choice

Consider a hydrodynamically fully developed laminar flow through a circular pipe with the flow along the axis (i.e., $z$ direction). In the following statements, $T$ is the temperature of the fluid, $T_w$ is the wall temperature and $T_m$ is the bulk mean temperature of the fluid. Which one of the following statements is TRUE?

  1. For a thermally fully developed flow, $\frac{\partial T}{\partial z} = 0$, always.
  2. For constant wall temperature of the duct, $\frac{dT_m}{dz} = $ constant.
  3. Nusselt number varies linearly along the $z$ direction for a thermally fully developed flow.
  4. For constant wall temperature ($T_w > T_m$) of the duct, $\frac{dT_m}{dz}$ increases exponentially with distance along $z$ direction.

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Correct answer: (D) For constant wall temperature ($T_w > T_m$) of the duct, $\frac{dT_m}{dz}$ increases exponentially with distance along $z$ direction.

Explanation

For a thermally fully developed flow the temperature profile keeps its shape but the temperature itself still changes along the pipe, so A is false, and the Nusselt number is constant, so C is false. With a constant wall temperature, $T_w - T_m = (T_w - T_i)e^{-Pz\bar{h}/\dot{m}c_p}$, so $T_m$ varies exponentially with $z$ and $\frac{dT_m}{dz}$ is not constant, which rules out B. Of the four statements D is the only one that describes an exponential variation of the mean temperature with distance.