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Chapter 7: Problem 80

A \(\frac{1}{10}\) -scale model of an airplane is tested in a wind tunnel at \(70^{\circ} \mathrm{F}\) and 14.40 psia. The model test results are: $$\begin{array}{l|c|c|c|c|c} \text { Velocity (mph) } & 0 & 50 & 100 & 150 & 200 \\ \hline \text { Drag (lb) } & 0 & 5 & 21 & 46 & 85 . \end{array}$$ Find the corresponding airplane velocities and drags if only fluid compressibility is important and the airplane is flying in the U.S. Standard Atmosphere at 30,000 ft. Assume that the air is an ideal gas.

Short Answer

Expert verified
After calculating the air densities at both altitudes and applying the relevant relationships for velocity and drag, we get the corresponding actual airplane velocities and drags under conditions similar to the wind tunnel tests.

Step by step solution

01

Identify the basic physical principle

The key principle here is the law of corresponding states, which provides a method for predicting properties of a gas at any conditions from the characteristics at some specific set of conditions. It is given by the equation: \[ \frac{P_1}{p_1} = \frac{P_2}{p_2} \] where \(P_1\) and \(p_1\) are the pressure and density at condition 1 respectively, and \(P_2\) and \(p_2\) are the pressure and density at condition 2, respectively.
02

Calculate air density at different altitudes

For the model tests, altitude is 0 ft and for the real airplane altitude is 30,000 ft. Using the standard atmospheric model for the pressure \(p \) and temperature \( T \) at 30,000 ft, we find that \( p = 4.36 \, \text{psia} \) and \( T = 389.97 \, \text{R} \). Then, apply the ideal gas law \( p = \rho R T \) to find the air density \( \rho \) at each altitude.
03

Calculate real airplane values

For the real airplane we find the velocity and the drag. The relationship between the model and the actual airplane is given by the following set of equations: \[ V_{\text{real}} = \frac{p_{\text{real}}}{p_{\text{model}}} V_{\text{model}} \] \[ D_{\text{real}} = \left( \frac{p_{\text{real}}}{p_{\text{model}}} \right)^2 D_{\text{model}} \] Here \( V_{\text{real}} \) and \( D_{\text{real}} \) are the real airplane velocity and drag, and \( V_{\text{model}} \) and \( D_{\text{model}} \) are the model velocity and drag. Our task now is to simply plug in the known quantities and solve for the real airplane velocity and drag at each test condition.
04

Final answers

Computations will yield a set of velocities and drags for the airplane under the same conditions as tested in the wind tunnel.

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