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Chapter 6: Problem 42

Air flowing over a flat plate at \(5 \mathrm{~m} / \mathrm{s}\) has a friction coefficient given as \(C_{f}=0.664(V x / \nu)^{-0.5}\), where \(x\) is the location along the plate. Using EES (or other) software, determine the effect of the location along the plate \((x)\) on the wall shear stress \(\left(\tau_{w}\right)\). By varying \(x\) from \(0.01\) to \(1 \mathrm{~m}\), plot the wall shear stress as a function of \(x\). Evaluate the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

Short Answer

Expert verified
Answer: The wall shear stress (τw) is calculated using the formula τw = (1/2)ρV²Cf, where ρ is the air density, V is the velocity, and Cf is the friction coefficient given as a function of Vx/ν. By calculating τw for different locations along the plate (x), we can observe the effect of x on τw. The results can be plotted as a graph to visualize the relationship between the location on the plate and the wall shear stress.

Step by step solution

01

Formula for Wall Shear Stress

The formula for wall shear stress can be expressed as: \(\tau_w = \frac{1}{2} \rho V^2 C_f\), where \(\tau_w\) is wall shear stress, \(\rho\) is the air density, \(V\) is the velocity, and \(C_f\) is the friction coefficient.
02

Obtain the kinematic viscosity of air

The air properties are evaluated at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\). At these conditions, the kinematic viscosity of air is approximately \(\nu = 1.51 \times 10^{-5} \mathrm{~m^2/s}\).
03

Calculate the wall shear stress for different locations on the plate

To calculate the wall shear stress at different locations along the plate, first we need to find the air density at the given conditions. At \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\), the air density is approximately \(\rho = 1.205 \mathrm{~kg/m^3}\). For \(x\) varying from \(0.01\) to \(1 \mathrm{~m}\), use the given formula for friction coefficient \(C_{f}=0.664(V x / \nu)^{-0.5}\), and then use the formula for wall shear stress, \(\tau_w = \frac{1}{2} \rho V^2 C_f\).
04

Plot the wall shear stress as a function of location on the plate

After calculating the wall shear stress for different locations along the plate \((x)\), plot the results as a function of \(x\). This graph will show the effect of location on the wall shear stress. Note: In the specific case of this exercise with \(x\) varying from \(0.01\) to \(1 \mathrm{~m}\), it says to use EES (or other) software to solve the problem. However, the general approach to solving the problem is how it's described above.

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