An incompressible fluid oscillates harmonically \(\left(V=V_{0}\right.\) \(\sin \omega t, \text { where } V \text { is the velocity })\) with a frequency of 10 rad/s in a 4-in.- -diameter pipe. A \(\frac{1}{4}\) scale model is to be used to determine the pressure difference per unit length, \(\Delta p_{\ell}\) (at any instant) along the pipe. Assume that $$\Delta p_{\ell}=f\left(D, V_{0}, \omega, t, \mu, \rho\right)$$ where \(D\) is the pipe diameter, \(\omega\) the frequency, \(t\) the time, \(\mu\) the fluid viscosity, and \(p\) the fluid density. (a) Determine the similarity requirements for the model and the prediction equation for \(\Delta p_{\ell}\) (b) If the same fluid is used in the model and the prototype at what frequency should the model operate?

Step by step solution

01

Identify the variables

The variables in the problem are the pipe diameter D, the fluid velocity \(V_{0}\), the frequency of oscillation \(\omega\), kinematic viscosity \(\mu\), and the fluid density \(\rho\). These are represented in the function as \(\Delta p_{\ell}=f\left(D, V_{0}, \omega, t, \mu, \rho\right)\).

02

Create a dimensionless expression

You can do this by rearranging the equation so the dependent variable, \(\Delta p_{\ell}\) is on one side. We have: \(\Delta p_{\ell}=\frac{D^2 \omega \rho}{\mu}g(D V_0 \omega, t)\) where \(g(D V_0 \omega, t)\) is a dimensionless function that we do not know yet.

03

Analyse the similarity requirements

Based on the Reynolds number \(Re=\frac{D V_{0} \omega}{\mu}\) we see that we can obtain similarity if \(Re_{model} = Re_{prototype}\). This gives us the equation \(D_{model} V_{0-model} \omega_{model} = D_{prototype} V_{0-prototype} \omega_{prototype}\). Given that the magnitudes of the velocities are the same in the model and prototype, and \(D_{model} = \frac{D_{prototype}}{4}\), we find that \(\omega_{model} = 4 \omega_{prototype}\). This is the similarity requirement for the model.

04

Find the required frequency for the model

We will use the frequency in the prototype (\(\omega_{prototype}\) = 10 rad/s) and the similarity requirement we found in Step 3 to find the frequency for the model. So, we will multiply the frequency of the prototype by 4: \(\omega_{model} = 4 \omega_{prototype} = 4 * 10 rad/s = 40 rad/s\).

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