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Chapter 7: Problem 10

Experiments show that the pressure drop for flow through an orifice plate of diameter \(d\) mounted in a length of pipe of diameter \(D\) may be expressed as \(\Delta p=p_{1}-p_{2}=\) \(f(\rho, \mu, \bar{V}, d, D),\) You are asked to organize some experimental data. Obtain the resulting dimensionless parameters.

Short Answer

Expert verified
The three dimensionless parameters are \(\pi_1 = \Delta p / \rho \bar{V}^2\), \(\pi_2 = \mu / \rho \bar{V}D\), and \(\pi_3 = d/D\).

Step by step solution

01

Identify the independent and dependent variables.

In the task, the dependent variable is the pressure drop \(\Delta p\) and the independent variables are \(\rho\) (density), \(\mu\) (dynamic viscosity), \(\bar{V}\) (mean velocity), \(d\) (diameter of orifice), and \(D\) (diameter of pipe). Thus, we have a total of six variables. And there are three fundamental dimensions in the problem: Mass [M], Length [L], and Time [T]. Thus, we have \(n\) = 6 variables and \(k\) = 3 fundamental dimensions. Hence, the number of dimensionless parameters is \(j = n - k = 3\). The dimensionless groups have to be formed using the Buckingham \(\pi\) theorem.
02

Express each variable in terms of fundamental dimensions

The dimensions of each variable are expressed as follows: \(\Delta p\) or pressure has the dimensions of force per unit area \([ML^{-1}T^{-2}]\), \(\rho\) has dimensions \([ML^{-3}]\), \(\mu\) has dimensions \([ML^{-1}T^{-1}]\), \(\bar{V}\) (velocity) has dimensions \([LT^{-1}]\), and \(d\) and \(D\) (diameters) has dimensions \([L]\).
03

Select repeating variables

We need to select three repeating variables, one from each dimension. The repeating variables should not form a dimensionless parameter together. A good choice would be \(\rho\), \(D\) and \(\bar{V}\). Now, following the Buckingham \(\pi\) theorem, we can construct three dimensionless groups.
04

Form dimensionless parameters

The three dimensionless \(\pi\) groups are: \(\pi_1 = \Delta p / \rho \bar{V}^2\), \(\pi_2 = \mu / \rho \bar{V}D\), and \(\pi_3 = d/D\). These replaces the six original dimensional variables.

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Most popular questions from this chapter

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