Question

Air ( \(1 \mathrm{~atm}, 5^{\circ} \mathrm{C}\) ) with free stream velocity of \(2 \mathrm{~m} / \mathrm{s}\) is flowing in parallel to a stationary thin \(1-\mathrm{m} \times 1-\mathrm{m}\) flat plate over the top and bottom surfaces. The flat plate has a uniform surface temperature of $35^{\circ} \mathrm{C}\(. If the friction force asserted on the flat plate is \)0.1 \mathrm{~N}$, determine the rate of heat transfer from the plate. Evaluate the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

Short answer

Question: Determine the rate of heat transfer from the flat plate in the given convective heat transfer problem. Solution: Step 1: Calculate the Reynolds Number Using the given properties and the formula, we find: \(Re_{L} = \frac{4 \cdot 0.1 \cdot 1}{1.205 \cdot 2^2} = 83.4\) Step 2: Calculate the Prandtl Number Using the given properties and the formula, we find: \(Pr = \frac{1006 \cdot 1.81e-5}{0.0259} = 0.698\) Step 3: Calculate the Nusselt Number Using the Reynolds and Prandtl numbers, we find: \(Nu_{L} = 0.664 \cdot 83.4^{1/2} \cdot 0.698^{1/3} = 20.56\) Step 4: Calculate the Convective Heat Transfer Coefficient Using the Nusselt number and the formula, we find: \(h = \frac{20.56 \cdot 0.0259}{1} = 0.532 \mathrm{~W/(m^2 K)}\) Step 5: Calculate the Rate of Heat Transfer Using the obtained values and the formula, we find: \(\dot{Q} = 0.532 \cdot 2 \cdot 30 = 31.92 \mathrm{~W}\) The rate of heat transfer from the flat plate is approximately 31.92 Watts.

Step-by-step solution

Calculate the Reynolds Number

To calculate the Reynolds number, first, we need to evaluate the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\). We can refer to standard air property tables or use online sources to find the following properties: - Density (\(\rho\)) = 1.205 kg/m\(^3\) - Dynamic Viscosity (\(\mu\)) = 1.81e-5 kg/(m s) Then, we can use Laminar flow assumption to calculate the Reynolds number by utilizing the friction force (\(F_f\)) and the formula: \(Re_{L} = \frac{4F_f L}{\rho V^2}\) where: - \(F_f = 0.1 \mathrm{~N}\) - \(L = 1 \mathrm{~m}\) (width of the flat plate) - \(V = 2 \mathrm{~m} / \mathrm{s}\) (velocity of the air)

Calculate the Prandtl Number

The Prandtl number can be obtained from the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\): - Specific Heat (\(C_p\)) = 1006 J/(kg K) - Thermal Conductivity (\(k\)) = 0.0259 W/(m K) Then, we can calculate the Prandtl number (\(Pr\)) using the formula: \(Pr = \frac{C_p \mu}{k}\)

Calculate the Nusselt Number

As we have Reynolds number and Prandtl number, we can now calculate the Nusselt number (\(Nu\)) using the formula for laminar flow over a flat plate: \(Nu_{L} = 0.664 \cdot Re_{L}^{1/2} \cdot Pr^{1/3}\)

Calculate the Convective Heat Transfer Coefficient

Now that we have the Nusselt number, we can calculate the convective heat transfer coefficient (\(h\)) using the following formula: \(h = \frac{Nu \cdot k}{L}\)

Calculate the Rate of Heat Transfer

Finally, we can calculate the rate of heat transfer (\(\dot{Q}\)) from the flat plate using the formula: \(\dot{Q} = hA\Delta T\) where: - \(A = 2 \cdot 1^2 \mathrm{~m^2}\) (area of the top and bottom surfaces) - \(\Delta T = T_s - T_\infty = 35^{\circ} \mathrm{C} - 5^{\circ} \mathrm{C} = 30^{\circ} \mathrm{C}\) (temperature difference between the surface and the free stream) Now, plug in the values obtained from previous steps and calculate the rate of heat transfer from the plate.