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Chapter 7: Problem 73

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?

Short Answer

Expert verified
The model should use the dimensionless parameters of Reynolds number, Froude number, and Discharge parameter. All similarity requirements can be met using water in the model, with lengths and discharge scaled down by the ratio 1:13. The model's discharge should be 0.557 \times (1/13)^{2.5} cubic feet per second, and the depth in the model equivalent to 32 inches divided by 13.

Step by step solution

01

Identify Dimensional Parameters

First, identify the dimensional parameters from the given function, including velocity \(V\), characteristic length \(\ell\), other lengths \(\ell_{i}\), density \(\rho\), viscosity \(μ\), gravity \(g\), and discharge \(Q\).
02

Formulate Dimensionless Parameters

Apply the Buckingham Pi theorem to derive dimensionless parameters using those identified in the initial function. In this case, we could arrive at dimensionless parameters like Reynolds number (Re = \(\frac{\rho V\ell}{\mu}\)), Froude number (Fr = \(\frac{V^{2}}{g\ell}\)), and Discharge parameter (\(Q' = \frac{Q}{\sqrt{g\ell^5}}\)).
03

Analyze Similarity Requirements

Next, analyze and validate the similarity requirements. In this case, using water for the model means that the similarity requirements regarding density and viscosity can be satisfied, since these remain constant. However, lengths must be scaled by the ratio 1:13 for the model to maintain similarity. Discharge needs to be modelled according to square root of this ratio because the dimension of discharge is volume per time.
04

Calculate Required Discharge and Depth for Model

In the full-sized tank, the discharge is given to be 250 gpm. This can be converted to cubic feet as 0.557 ft³/s. For the model, the discharge \(Q_m\) must satisfy the Froude similarity which implies \(Q_m = 0.557 \times (1/13)^{2.5}\). The depth should also be scaled down by the same ratio, giving a model depth as 32 inches divided by 13.

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