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Chapter 7: Problem 54

When a valve is closed suddenly in a pipe with flowing water, a water hammer pressure wave is set up. The very high pressures generated by such waves can damage the pipe. The maximum pressure, \(p_{\max },\) generated by water hammer is a function of liquid density, \(\rho,\) initial flow speed, \(U_{0},\) and liquid bulk modulus, \(E_{v}\). How many dimensionless groups are needed to characterize water hammer? Determine the functional relationship among the variables in terms of the necessary II groups.

Short Answer

Expert verified
The number of dimensionless groups needed is 1. The dimensionless group is \(p_{\max}/(\rho \cdot U_{0}^2)\) and the functional relationship among the variables is \(p_{\max}/(\rho \cdot U_{0}^2) = f(E_{v}/(\rho \cdot U_{0}^2))\).

Step by step solution

01

Identify Variables and their Dimensions

First, note the variables and their dimensions. For this problem, the variables are: Maximum Pressure \(p_{\max}\) [M L^-1 T^-2], Liquid Density \(\rho\) [M L^-3], Initial Flow Speed \(U_{0}\) [L T^-1], and Liquid Bulk Modulus \(E_{v}\) [M L^-1 T^-2].
02

Number of Dimensions and Number of Dimensionless Groups

The dimensions involved are M(ass), L(ength), and T(ime). Therefore, there are 3 fundamental dimensions. The number of dimensionless groups is found by subtracting the number of fundamental dimensions from the number of variables in the problem. This yields four variables - three fundamental dimensions = one dimensionless group.
03

Determine the Dimensionless Groups

To construct the dimensionless group, select \(\rho, U_{0}, E_{v}\) as the repeating variables since \(p_{\max}\) is the variable of interest. Then, according to Buckingham's Pi theorem, find the necessary power for each repeating variable to make it dimensionless when multiplying with \(p_{\max}\). The dimensionless group (denoted as \(\Pi\)) is \(\Pi = p_{\max} / (\rho \cdot U_{0}^2)\).
04

Write the Functional Relationship

Finally, express the functional relationship among the variables in terms of these dimensionless groups. So, the original variables can be expressed as a dimensionless group: \(p_{\max}/(\rho \cdot U_{0}^2) = f(E_{v}/(\rho \cdot U_{0}^2))\)

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