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Chapter 7: Problem 75

(See The Wide World of Fluids article titled "Ice Engineering." Section \(7.9 .3 .)\) A model study is to be developed to determine the force exerted on bridge piers due to floating chuaks of ice in a river. The piers of interest have square cross sections. Assume that the force, \(R\), is a function of the pier width, \(b\), the thickness of the ice, \(d\), the velocity of the ice, \(V\), the acceleration of gravity, \(g,\) the density of the ice, \(\rho_{i},\) and a measure of the strength of the ice, \(E_{i},\) where \(E_{i}\) has the dimensions \(F L^{-2}\) (a) Based on these variables determine a suitable set of dimensionless variables for this problem. (b) The prototype conditions of interest include an ice thickness of 12 in. and an ice velocity of \(6 \mathrm{ft} / \mathrm{s}\). What model ice thickness and velocity would be required if the length scale is to be \(1 / 10 ?(\mathrm{c})\) If the model and prototype ice have the same density, can the model ice have the same strength properties as that of the prototype ice? Explain.

Short Answer

Expert verified
The problem requires determining dimensionless parameters, calculating model ice thickness and velocity considering a 1/10 length scale, and analyzing whether the model ice can have the same strength as the prototype even if they have the same density. A) 4 dimensionless parameters can be determined through the Buckingham \(\Pi\) theorem. B) The model ice thickness is 1.2 inches and the model ice velocity is approximately 1.9 ft/s. C) The model ice cannot have the same strength as the prototype ice even if they have the same density.

Step by step solution

01

Identify the Dimensionless Parameters

To form the dimensionless parameters, we need to apply the Buckingham Pi theorem. This problem includes 7 variables (\(R, b, d, V, g, \rho_{i}, E_{i}\)) and 3 fundamental dimensions (mass \(M\), length \(L\), time \(T\)), so we can obtain 4 dimensionless parameters. To simplify the process, we can choose \(b, g,\) and \(\rho_{i}\) as recurring variables.
02

Calculate Model Ice Thickness and Velocity

If the scale of the model is 1/10th, the length ratio \(L_{r}\) is \(1/10\). Therefore, the model ice thickness \(d_{m} = d_{p} / L_{r} = 12 \, \text{inches} / 10 = 1.2 \, \text{inches}\). The velocity ratio is equal to \(\sqrt{L_{r}}\) , so the model ice velocity \(V_{m} = V_{p} / \sqrt{L_{r}} = 6 \, \text{ft}/s / \sqrt{10} = 1.9 \, \text{ft}/s\) approximately.
03

Analyze the Strength Properties of the Model Ice

If the model ice has the same density as the prototype ice, the model ice cannot have the same strength properties as that of the prototype. This is because the strength of the ice, \(E_{i}\) is a property that also depends on ice's microstructure, its impurities, and other factors. Therefore, it wouldn't be correct to assume that two ice samples with the same density have the same strength as well.

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Most popular questions from this chapter

A thin elastic wire is placed between rigid supports. A fluid flows past the wire, and it is desired to study the static deflection. \(\delta,\) at the center of the wire due to the fluid drag. Assume that $$\delta=f(\ell, d, \rho, \mu, V, E)$$ where \(\ell\) is the wire length, \(d\) the wire diameter, \(\rho\) the fluid density, \(\mu\) the fluid viscosity, \(V\) the fluid velocity, ind \(E\) the modulus of elasticity of the wire material. Develop a suitable set of pi terms for this problem.

Air bubbles discharge from the end of a submerged tube as shown in Fig. P7.57. The bubble diameter, \(D\), is assumed to be a function of the air flowrate, \(Q\), the tube diameter, \(d\), the acceleration of gravity, \(g\), the density of the liquid, \(\rho\), and the surface tension of the liquid, \(\sigma\). (a) Determine a suitable set of dimensionless variables for this problem. (b) Model tests are to be run on the Earth for a prototype that is to be operated on a planet where the acceleration of gravity is 10 times greater than that on Earth. The model and prototype are to use the same fluid, and the prototype tube diameter is 0.25 in. Determine the tube diameter for the model and the required model flowrate if the prototype flowrate is to be \(0.001 \mathrm{ft}^{3} / \mathrm{s}\).

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The excess pressure inside a bubble (discussed in Chapter 1 ) is known to be dependent on bubble radius and surface tension. After finding the pi terms, determine the variation in excess pressure if we (a) double the radius and (b) double the surface tension.

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