Question

A thin, square, flat plate has \(1.2 \mathrm{~m}\) on each side. Air at \(10^{\circ} \mathrm{C}\) flows over the top and bottom surfaces of a very rough plate in a direction parallel to one edge, with a velocity of $48 \mathrm{~m} / \mathrm{s}$. The surface of the plate is maintained at a constant temperature of \(54^{\circ} \mathrm{C}\). The plate is mounted on a scale that measures a drag force of \(1.5 \mathrm{~N}\). Determine the total heat transfer rate from the plate to the air.

Short answer

Based on the given information about the square plate, the total heat transfer rate from the plate to the air is found to be approximately 5727 W using convective heat transfer equations and considering turbulent flow.

Step-by-step solution

Calculate the Reynolds number

First, we need to find the Reynolds number for the given conditions. To do this, we can use the Reynolds number equation. The density and viscosity of air at \(10^{\circ} \mathrm{C}\) are approximately \(\rho = 1.25 \mathrm{~kg/m}^{3}\) and \(\mu = 1.79 \times 10^{-5}\mathrm{~kg/m\cdot s}\), respectively. The characteristic length (D) for a square plate is the side length, which is \(1.2\mathrm{~m}\). \(Re = \frac{\rho VD}{\mu} = \frac{1.25 \times 48 \times 1.2}{1.79 \times 10^{-5}} = \approx 2.03 \times 10^6\)

Calculate the Prandtl number

Next, we need to find the Prandtl number for air at given temperature. The specific heat capacity of air at \(10^{\circ} \mathrm{C}\) is approximately \(c_p = 1005 \mathrm{~J/kg\cdot K}\). \(Pr = \frac{\mu c_p}{k} = \frac{1.79 \times 10^{-5} \times 1005}{0.0257} \approx 0.7\)

Calculate the Nusselt number

Now we can calculate the Nusselt number using the correlation for turbulent flow over a flat plate. \(Nu = 0.036 Re^{0.8} Pr^{1/3} = 0.036 \times (2.03 \times 10^6)^{0.8} \times 0.7^{1/3} \approx 4146\)

Calculate the convective heat transfer coefficient

The convective heat transfer coefficient can now be calculated using the Nusselt number. \(h = \frac{Nu \times k}{D} = \frac{4146 \times 0.0257}{1.2} \approx 88.84 \mathrm{~W/m^2\cdot K}\)

Calculate the total heat transfer rate

Now we can calculate the total heat transfer rate using the convective heat transfer equation. The area of the plate is \(A = D^2 = 1.2^2 = 1.44 \mathrm{~m}^2\), and the temperature difference is \((T_s - T_\infty) = 54 - 10 = 44\mathrm{~K}\). \(q = hA(T_s - T_\infty) = 88.84 \times 1.44 \times 44 \approx 5727 \mathrm{~W}\) The total heat transfer rate from the plate to the air is approximately \(5727\mathrm{~W}\).