Question

Air at \(1 \mathrm{~atm}\) is flowing over a flat plate with a free stream velocity of \(70 \mathrm{~m} / \mathrm{s}\). If the convection heat transfer coefficient can be correlated by $\mathrm{Nu}_{x}=0.03 \operatorname{Re}_{x}^{08} \operatorname{Pr}^{1 / 3}$, determine the friction coefficient and wall shear stress at a location \(2 \mathrm{~m}\) from the leading edge. Evaluate air properties at \(20^{\circ} \mathrm{C}\) and $1 \mathrm{~atm}$.

Short answer

Given the free-stream velocity \(U_{\infty} = 70\, \text{m/s}\), the Nusselt number correlation \(\mathrm{Nu}_{x}=0.03\,\operatorname{Re}_{x}^{0.8} \operatorname{Pr}^{1 / 3}\), and air properties at \(T = 20^{\circ} \mathrm{C}\) and \(P = 1 \mathrm{~atm}\), calculate the friction coefficient and wall shear stress at a distance of 2 meters from the leading edge of the flat plate.

Step-by-step solution

List the given information and find air properties at the given condition

We are given: - Free stream velocity: \(U_{\infty} = 70\,\text{m/s}\) - Nu correlation: \(\mathrm{Nu}_{x}=0.03\,\operatorname{Re}_{x}^{0.8} \operatorname{Pr}^{1 / 3}\) - Temperature: \(T = 20^{\circ} \mathrm{C}\) - Pressure: \(P = 1 \mathrm{~atm}\) First, let's find the air properties at the specified temperature and pressure: - Dynamic viscosity (\(\mu\)) - Density (\(\rho\)) - Specific heat capacity (\(C_p\)) - Thermal conductivity (\(k\)) Given temperature and pressure, the air properties are as follows: - \(\mu = 1.81 \times 10^{-5}\, \mathrm{Kg / m.s}\) - \(\rho = 1.20\, \mathrm{kg/m^3}\) - \(C_p = 1004 \,\mathrm{J/kg.K}\) - \( k = 0.026 \,\mathrm{W/m.K}\)

Calculate the Reynolds number at the specified location

The location specified is \(2\, \mathrm{m}\) from the leading edge, so \(x = 2\, \mathrm{m}\). The Reynolds number (\(\operatorname{Re}_x\)) can be calculated using the formula: $$\operatorname{Re}_x = \frac{\rho U_{\infty} x}{\mu}$$ Substitute the given values and calculate \(\operatorname{Re}_x\): $$\operatorname{Re}_x = \frac{1.20\, \mathrm{kg/m^3} \times 70\, \mathrm{m/s} \times 2\, \mathrm{m}}{1.81 \times 10^{-5} \,\mathrm{Kg / m.s}}$$

Calculate the friction coefficient

The friction coefficient (\(C_f\)) can be calculated using the Blasius relation: $$C_f = 0.664 \times \operatorname{Re}_x^{-1/2}$$ Plug in the value of \(\operatorname{Re}_x\) and calculate \(C_f\).

Calculate the wall shear stress

The wall shear stress (\(\tau_w\)) can be calculated using the formula: $$\tau_w = \frac{1}{2}\rho U_{\infty}^2 C_f$$ Substitute the given values and calculate \(\tau_w\): $$\tau_w = \frac{1}{2} \times 1.20\, \mathrm{kg/m^3} \times \left(70\, \mathrm{m/s}\right)^2 \times C_f$$ To summarize, we can find the friction coefficient and wall shear stress at the \(2\, \mathrm{m}\) location from the leading edge using the Reynolds number and air properties provided.