Question

Air \(\left(1 \mathrm{~atm}, 5^{\circ} \mathrm{C}\right)\) with a free-stream velocity of \(2 \mathrm{~m} / \mathrm{s}\) flows in parallel with 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}\(. Determine \)(a)\( the average friction coefficient, \)(b)$ the average convection heat transfer coefficient, and (c) the average convection heat transfer coefficient using the modified Reynolds analogy, and compare with the result obtained in \((b)\).

Short answer

Answer: In this exercise, the following coefficients were determined for the flow of air over a flat plate: (a) The average friction coefficient was found to be 0.00182. (b) The average convection heat transfer coefficient was found to be 8.24 W/(m²K). (c) The average convection heat transfer coefficient using the modified Reynolds analogy was found to be 8.21 W/(m²K).

Step-by-step solution

Identify relevant parameters

First, list down the given information and define relevant parameters: - Air pressure: \(P = 1 atm\) - Air temperature: \(T_\infty = 5^\circ C\) - Free-stream velocity: \(V_\infty = 2 m/s\) - Flat plate dimensions: \(1m \times 1m\) - Surface temperature: \(T_s = 35^\circ C\)

Determine the air properties

Use the given values for temperature and pressure to determine the air properties like density (\(\rho\)), viscosity (\(\mu\)), and thermal conductivity (\(k\)) which will be needed based on the temperature of the flow. Here, we assume air to be an ideal gas. Then you can get: – Density: \(\rho = 1.217 kg/m^3\) – Viscosity: \(\mu = 1.85 \times 10^{-5} kg/(ms)\) – Thermal conductivity: \(k = 0.0262 W/(mK)\)

Calculate Reynolds number

Calculate the Reynolds number of the flow over the flat plate. The Reynolds number is defined as: \(Re = \frac{\rho V_\infty L}{\mu}\), where \(L = 1m\) \(Re = \frac{(1.217 kg/m^3)(2 m/s)(1 m)}{1.85 \times 10^{-5} kg/(ms)} = 131897.3\)

Calculate average friction coefficient

To calculate the average friction coefficient (\(C_{f}\)), use the Blasius equation for laminar flow over a flat plate: \(C_{f} = \frac{0.664}{\sqrt{Re}}\) \(C_{f} = \frac{0.664}{\sqrt{131897.3}} = 0.00182\) (a) The average friction coefficient is \(0.00182\).

Calculate average convection heat transfer coefficient

To calculate the average convection heat transfer coefficient (\(h_{avg}\)), use the Nusselt number relation for laminar flow over a flat plate: \(Nu = \frac{h_{avg}L}{k} = 0.664Re^{\frac{1}{2}} Pr^{\frac{1}{3}}\) First, determine the Prandtl number (\(Pr\)) for air at the given temperature: \(Pr = \frac{\mu C_p}{k} = 0.709\) (for air at 5°C) Then, calculate \(h_{avg}\): \(h_{avg} = \frac{Nu \cdot k}{L} = \frac{0.664Re^{\frac{1}{2}}Pr^{\frac{1}{3}} \cdot k}{L}\) \(h_{avg} = \frac{0.664(131897.3)^{\frac{1}{2}}(0.709)^{\frac{1}{3}}0.0262}{1} = 8.24 W/(m^2K)\) (b) The average convection heat transfer coefficient is \(8.24 W/(m^2K)\).

Calculate the average convection heat transfer coefficient using the modified Reynolds analogy

Using the modified Reynolds analogy (modified Colburn analogy), the average convection heat transfer coefficient can be determined using the friction coefficient: \(h_{avg} = \frac{C_{f} \rho V_\infty C_p}{2Pr^{\frac{2}{3}}}\) \(h_{avg_{mod}} = \frac{0.00182(1.217 kg/m^3)(2 m/s)(1005 J/(kgK))}{2(0.709)^{\frac{2}{3}}} = 8.21 W/(m^2K)\) (c) The average convection heat transfer coefficient using the modified Reynolds analogy is \(8.21 W/(m^2K)\), which is almost the same as the result obtained in (b). This indicates that the modified Reynolds analogy provides an accurate estimation of the convection heat transfer coefficient for this exercise.