/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 80 A \(\frac{1}{10}\) -scale model ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wind Tunnel Testing
Wind tunnel testing is an essential method in fluid mechanics to analyze the aerodynamic properties of structures like airplanes. In a wind tunnel, air is blown past a scale model, allowing engineers to observe and measure critical factors such as drag and lift.

  • Wind tunnels enable precise control of wind speed, temperature, and pressure, which are difficult to replicate in real-world flight conditions.
  • Scale models are often used because full-scale testing would be expensive and complicated.
  • Engineers can test different theories and designs at a lower cost and with less risk.


By using a wind tunnel, an airspeed that matches expected airspeeds of an actual flight can be created, allowing for comparisons between the model and the real thing. This is crucial for determining how a full-sized aircraft will perform based on scaled test results.
Scale Model Analysis
Scale model analysis involves testing a smaller version of an object, like an airplane, to predict how the full-sized version will behave. This method is central when testing in wind tunnels.

  • Important scaling laws need to be applied to ensure that results are accurate and applicable to the full-size version.
  • These include maintaining similar Reynolds numbers, which account for velocity, length, and fluid viscosity.
  • The ratio of relevant forces and dimensions in a model must be preserved in the full-sized object to ensure predictive accuracy.


In order to make meaningful extrapolations from model to actual conditions, complex physical laws and mathematical equations are used to adjust the model results, making them valid for larger sizes and different environments.
Aerodynamic Drag
Aerodynamic drag is the force that opposes an object's motion through a fluid like air. It's a key parameter measured during wind tunnel and scale model testing.

  • Drag forces arise from pressure differences around the object and shear stress from the fluid's viscosity.
  • Lowering drag increases an airplane's performance and efficiency by reducing fuel consumption and improving speed.
  • The results from a scale model are adjusted using drag coefficients, allowing for accurate predictions of drag on the full-sized airplane.


Understanding and reducing aerodynamic drag are crucial for designing efficient aircraft and helps engineers optimize shapes and design features that minimize these adverse forces.
Compressibility Effects
Compressibility effects become significant when dealing with high-speed airflows, such as those around airplanes at high altitudes. These must be considered when extrapolating wind tunnel data to real flight conditions.

  • Air compressibility becomes relevant as an aircraft's speed approaches or exceeds the speed of sound.
  • Compressibility can affect things like lift, drag, and shock wave formations around the body of the aircraft.
  • Adjustments in drag and velocity due to compressibility ensure that scale model tests accurately reflect real-world dynamics at high altitudes.


By understanding and accounting for compressibility effects, engineers can better estimate and design for the pressures and forces an aircraft will experience during actual flight, ensuring safety and efficiency.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The pressure rise, \(\Delta p=p_{2}-p_{1},\) across the abrupt expansion of Fig. \(\mathrm{P} 7.38\) through which a liquid is flowing can be expressed as $$\Delta p=f\left(A_{1}, A_{2}, \rho, V_{1}\right)$$ where \(A_{1}\) and \(A_{2}\) are the upstream and downstream cross-sectional areas, respectively, \(\rho\) is the fluid density, and \(V_{1}\) is the upstream velocity. Some experimental data obtained with \(A_{2}=1.25 \mathrm{ft}^{2}\) \(V_{1}=5.00 \mathrm{ft} / \mathrm{s},\) and using water with \(\rho=1.94\) slugs/ft \(^{3}\) are given in the following table: $$\begin{array}{l|l|l|l|l|r} A_{1}\left(\mathrm{ft}^{2}\right) & 0.10 & 0.25 & 0.37 & 0.52 & 0.61 \\ \hline \Delta p\left(\mathrm{lb} / \mathrm{ft}^{2}\right) & 3.25 & 7.85 & 10.3 & 11.6 & 12.3 \end{array}$$ Plot the results of these tests using suitable dimensionless parameters. With the aid of a standard curve fitting program determine a general equation for \(\Delta p\) and use this equation to predict \(\Delta p\) for water flowing through an abrupt expansion with an area ratio \(A_{1} / A_{2}=0.35\) at a velocity \(V_{1}=3.75 \mathrm{ft} / \mathrm{s}\).

Assume that the flowrate, \(Q\), of a gas from a smokestack is a function of the density of the ambicnt air, \(\rho_{\alpha}\), the density of the gas, \(\rho_{x},\) within the stack, the acceleration of gravity, \(g,\) and the height and diameter of the stack, \(h\) and \(d\), respectively. Use \(\rho_{c}, d,\) and \(g\) as repeating variables to develop a set of pi terms that could be used to describe this problem.

At a large fish hatchery the fish are reared in open, water-filled tanks. Each tank is approximately square in shape with curved corners, and the walls are smooth. To create motion in the tanks, water is supplied through a pipe at the edge of the tank. The water is drained from the tank through an opening at the center. (See Video \(\vee 7.9 .)\) A model with a length scale of 1: 13 is to be used to determine the velocity, \(V\), at various locations within the tank. Assume that \(V=f\left(\ell, \ell_{i}, \rho, \mu, g, Q\right)\) where \(\ell\) is some characteristic length such as the tank width, \(\ell\), represents a series of other pertinent lengths, such as inlet pipe diameter, fluid depth, etc.. \(\rho\) is the fluid density, \(\mu\) is the fluid viscosity, \(g\) is the acceleration of gravity, and \(Q\) is the discharge through the tank. (a) Determine a suitable set of dimensionless parameters for this problem and the prediction equation for the velocity. If water is to be used for the model, can all of the similarity requirements be satisfied? Explain and support your answer with the necessary calculations. (b) If the flowrate into the full-sized tank is 250 gpm, determine the required value for the model discharge assuming Froude number similarity. What model depth will correspond to a depth of 32 in. in the full sized tank?

The fluid dynamic characteristics of an airplane flying \(240 \mathrm{mph}\) at \(10,000 \mathrm{ft}\) are to be investigated with the aid of a 1: 20 scale model. If the model tests are to be performed in a wind tunnel using standard air, what is the required air velocity in the wind tunnel? Is this a realistic velocity?

An equation used to evaluate vacuum filtration is $$Q=\frac{\Delta p A^{2}}{\alpha\left(V R w+A R_{f}\right)}$$ Where \(Q \doteq L^{3} / T\) is the filtrate volume flow rate, \(\Delta p \doteq F / L^{2}\) the vacuum pressure differential, \(A \doteq L^{2}\) the filter area, \(\alpha\) the filtrate "viscosity," \(V \doteq L^{3}\) the filtrate volume, \(R \doteq L / F\) the sludge specific resistance, \(w \doteq F / L^{3}\) the weight of dry sludge per unit volume of filtrate, and \(R_{f}\) the specific resistance of the filter medium. What are the dimensions of \(R_{f}\) and and \(\alpha ?\)
See all solutions

Recommended explanations on Physics Textbooks

Translational Dynamics

Read Explanation

Electricity

Read Explanation

Solid State Physics

Read Explanation

Scientific Method Physics

Read Explanation

Physics of Motion

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.