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Chapter 7: Problem 58

For a certain model study involving a 1: 5 scale model it is known that Froude number similarity must be maintained. The possibility of cavitation is also to be investigated, and it is assumed that the cavitation number must be the same for model and prototype. The prototype fluid is water at \(30^{\circ} \mathrm{C}\), and the model fluid is water at \(70^{\circ} \mathrm{C}\). If the prototype operates at an ambient pressure of \(101 \mathrm{kPa}(\mathrm{abs}),\) what is the required ambient pressure for the model system?

Short Answer

Expert verified
The ambient pressure required for the model system is approximately 96.51 kPa. This value is computed based on Froude similarity, cavitation similarity and given fluid properties.

Step by step solution

01

Identify Relevant Formulas

First, the relevant formulas need to be identified. Under conditions of Froude similarity, the ratios of the speeds of the model and prototype scale as the square root of the reciprocal of the length scale ratio. This is given by: \(V_p/V_m=(L_m/L_p)^(1/2)\). The cavitation number (σ) for any system is given by: \(σ= (P_a−P_v)/0.5ρV^2\), Where P_a is the ambient pressure, P_v is the vapor pressure, ρ is the density of the fluid and V is the velocity of the flow.
02

Analyze the Froude Similarity

Froude similarity requires that the quantity \(L/V^2\) be the same for both model and prototype where L is the characteristic length and V is the velocity. However, in this case the prototype operates under different conditions from the model (temperature difference), so the Froude number will not be exactly the same.
03

Apply Cavitation Number Formula to the Prototype

By applying the cavitation number to the prototype and simplifying, we obtain: \(σ_p=(P_{a,p}-P_{v,p})/0.5ρ_pV_p^2\). One can refer to the standard water properties table for the other values: \(ρ_p=995 kg/m^3\) and \(P_{v,p}=4.24 kPa\) at 30 degrees Celcius.
04

Apply Cavitation Number Formula to the Model

Similarly, for the model scale, we have: \(σ_m=(P_{a,m}-P_{v,m})/0.5ρ_mV_m^2\). From the water properties table: \(ρ_m=977 kg/m^3\) and \(P_{v,m}=31.176kPa\) at 70 degrees C.
05

Equate Cavitation Numbers

Since the model and prototype must have the same cavitation number, equate \(σ_p\) and \(σ_m\) and solve for \(P_{a,m}\), the ambient pressure for the model system.
06

Solve for Required Ambient Pressure

By substituting \(V_p/V_m\) from Froude similarity into the equation obtained from equating cavitation numbers, and doing the necessary calculations, the final solution can be obtained.

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Most popular questions from this chapter

A very small needle valve is used to control the flow of air in a \(\frac{1}{8}-\) in. air line. The valve has a pressure drop of 4.0 psi at a flow rate of \(0.005 \mathrm{ft}^{3} / \mathrm{s}\) of \(60^{\circ} \mathrm{F}\) air. Tests are performed on a large, geometrically similar valve and the results are used to predict the performance of the smaller valve. How many times larger can the model valve be if \(60^{\circ} \mathrm{F}\) water is used in the test and the water flow rate is limited to 7.0 gal/min?

A 250 -m-long ship has a wetted area of \(8000 \mathrm{m}^{2} .\) A \(\frac{1}{100}\) -scale model is tested in a towing tank with the prototype fluid, and the results are: $$\begin{array}{|l|l|l|l|} \text { Model velocity }(\mathrm{m} / \mathrm{s}) & 0.57 & 1.02 & 1.40 \\ \hline \text { Model drag }(\mathrm{N}) & 0.50 & 1.02 & 1.65 \end{array}$$ Calculate the prototype drag at \(7.5 \mathrm{m} / \mathrm{s}\) and \(12.0 \mathrm{m} / \mathrm{s}\).

A breakwater is a wall built around a harbor so the incoming waves dissipate their energy against it. The significant dimensionless perameters are the Froude number Frand the Reynolds number Re. A particular breakwater measuring \(450 \mathrm{m}\) long and \(20 \mathrm{m}\) deep is hit by waves \(5 \mathrm{m}\) high and velocities up to \(30 \mathrm{m} / \mathrm{s}\). In a \(\frac{1}{100}\) -scale model of the breakwater, the wave height and velocity can be controlled. Can complete similarity be obtained using water for the model test?

The drag characteristics of an airplane are to be determined by model tests in a wind tunnel operated at an absolute pressure of \(1300 \mathrm{kPa}\). If the prototype is to cruise in standard air at 385 \(\mathrm{km} / \mathrm{hr},\) and the corresponding speed of the model is not o differ by more than \(20 \%\) from this (so that compressibility effects may be ignored), what range of length scales may be used if Reynolds number similarity is to be maintained? Assume the viscosity of air is unaffected by pressure, and the temperature of air in the tunnel is equal to the temperature of the air in which the airplane will fly.

The dimensional parameters used to describe the operation of a ship or airplane propeller (sometimes called a screw propeller) are rotational speed, \(\omega,\) diameter, \(D,\) fluid density, \(\rho\) speed of the propeller relative to the fluid, \(V\), and thrust developed, \(T .\) The common dimensionless groups are called the thrust coefficient and the advance ratio. Propose appropriate definitions for these groups.
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