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Chapter 7: Problem 53

(See The Wide World of Fluids article "Modeling Parachutes in a Water Tunnel," Section \(7.8 .1 .\) ) Flow characteristics for a \(30-f t\) diameter prototype parachute are to be determined by tests of a 1-fit-diameter model parachute in a water tunnel. Some data collected with the model parachute indicate a drag of 17 lb when the water velocity is \(4 \mathrm{f}\) Us. Lse the model data to predict the drag on the prototype parachute falling through air at \(10 \mathrm{ft} / \mathrm{s}\). Assume the drag to be a function of the velocity, \(V\), the fluid density, \(\rho\), and the parachute diameter, \(D\).

Short Answer

Expert verified
The above steps derived the relationship of the drag force, the fluid density, the velocity, and the diameter of the parachute. Using the given model data, we are able to calculate the constant of proportionality. With this, we can predict the drag on the prototype by substituting the known values into the equation.

Step by step solution

01

Identify the proportionalities

The drag force is a function of the velocity (\(V\)), fluid density (\(\rho\)), and parachute diameter (\(D\)). Therefore, we can write the force (\(F\)) proportionality as: \[ F \propto V^n \cdot \rho^m \cdot D^p \]where the powers \(n\), \(m\), and \(p\) need to be determined.
02

Use dimensional analysis

We use dimensional analysis to work out the powers. The basic dimensions for these quantities are - Force (\(F\)): (mass length/time²) or (density×length³ length/time²) = \[\rho^1 \cdot V^0 \cdot D^4\]- Velocity (\(V\)): length/time = \[\rho^0 \cdot V^1 \cdot D^0\]- Density (\(\rho\)): mass/length³ = \[\rho^1 \cdot V^0 \cdot D^{-3}\]- Diameter (\(D\)): length = \[\rho^0 \cdot V^0 \cdot D^1\]Replacing these in our proportionality gives \[\rho^1 \cdot V^0 \cdot D^4 \propto (\rho^0 \cdot V^1 \cdot D^0)^n \cdot (\rho^1 \cdot V^0 \cdot D^{-3})^m \cdot (\rho^0 \cdot V^0 \cdot D^1)^p\], equating coefficients we get \(n=2\), \(m=-2\), \(p=1\). Our proportionality then becomes \[ F \propto \rho V^2D\].
03

Use model data to find constants

Substitute the proportionality relationship into an equality by introducing a constant of proportionality (\(C\)):\[ F = C \rho V^2D \]We can use the given model data (that is, the force of 17 lb, the water density, the water velocity of 4 ft/s, and the model diameter of 1 ft) to calculate the constant of proportionality.
04

Predict drag on prototype

We can then use the determined constant of proportionality, the air density, the velocity of the prototype parachute (10 ft/s), and the diameter of the prototype (30 ft) to calculate the resultant force (drag) on the prototype.

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Most popular questions from this chapter

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