GATE 2023 CH – Question 54

Heat Transfer · Convection, boundary layers and heat transfer coefficients · 2 marks · Numerical answer

A fluid is flowing steadily under laminar conditions over a thin rectangular plate at temperature $T_s$ as shown in the figure below. The velocity and temperature of the free stream are $u_\infty$ and $T_\infty$, respectively. When the fluid flow is only in the $x$-direction, $h_x$ is the local heat transfer coefficient. Similarly, when the fluid flow is only in the $y$-direction, $h_y$ is the corresponding local heat transfer coefficient. Use the correlation Nu = 0.332 (Re)$^{1/2}$ (Pr)$^{1/3}$ for the local heat transfer coefficient, where, Nu, Re, and Pr, respectively are the appropriate Nusselt, Reynolds and Prandtl numbers. The average heat transfer coefficients are defined as $\bar{h}_l = \frac{1}{l}\int_0^lh_xdx$ and $\bar{h}_w = \frac{1}{w}\int_0^wh_ydy$. If $w = 1$ m and $l = 4$ m, the value of the ratio of $\bar{h}_w$ to $\bar{h}_l$ is _______ (in integer).

a thin plate with the length $l$ along $x$ and the width $w$ along $y$.

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Correct answer: 2

Explanation

From the correlation $h_x \propto x^{-1/2}$. The average over a length $L$ is $\bar{h} = \frac{1}{L}\int_0^LCx^{-1/2}dx = 2CL^{-1/2}$. So $\bar{h} \propto L^{-1/2}$ and $\frac{\bar{h}_w}{\bar{h}_l} = \sqrt{\frac{l}{w}} = \sqrt{4} = 2$.