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Chapter 7: Problem 38

The pressure rise, \(\Delta p=p_{2}-p_{1},\) across the abrupt expansion of Fig. \(\mathrm{P} 7.38\) through which a liquid is flowing can be expressed as $$\Delta p=f\left(A_{1}, A_{2}, \rho, V_{1}\right)$$ where \(A_{1}\) and \(A_{2}\) are the upstream and downstream cross-sectional areas, respectively, \(\rho\) is the fluid density, and \(V_{1}\) is the upstream velocity. Some experimental data obtained with \(A_{2}=1.25 \mathrm{ft}^{2}\) \(V_{1}=5.00 \mathrm{ft} / \mathrm{s},\) and using water with \(\rho=1.94\) slugs/ft \(^{3}\) are given in the following table: $$\begin{array}{l|l|l|l|l|r} A_{1}\left(\mathrm{ft}^{2}\right) & 0.10 & 0.25 & 0.37 & 0.52 & 0.61 \\ \hline \Delta p\left(\mathrm{lb} / \mathrm{ft}^{2}\right) & 3.25 & 7.85 & 10.3 & 11.6 & 12.3 \end{array}$$ Plot the results of these tests using suitable dimensionless parameters. With the aid of a standard curve fitting program determine a general equation for \(\Delta p\) and use this equation to predict \(\Delta p\) for water flowing through an abrupt expansion with an area ratio \(A_{1} / A_{2}=0.35\) at a velocity \(V_{1}=3.75 \mathrm{ft} / \mathrm{s}\).

Short Answer

Expert verified
The predicted value of \(\Delta p\) will depend on the equation obtained from the curve fitting, which cannot be determined without running the analysis. However, once the equation is known, it can be used to predict \(\Delta p\) for any given values of \(A_{1}/A_{2}\) and \(V_{1}\).

Step by step solution

01

Plot the Experimental Data

Collect the experimental data provided in the table into two arrays: one for \(A_{1}\), the other for \(\Delta p\). Additionally, create a third array for dimensionless parameters, where each entry is \((A_{2}/A_{1})V_{1}^{2}\rho\). Plot \(\Delta p\) against this dimensionless parameter.
02

Curve Fitting

Implement a curve fitting program (regression analysis) to this dataset to find an approximate function, which should yield a dimensionless form of the equation \(\Delta p=f\left(A_{1}, A_{2}, \rho, V_{1}\right)\). The equation should allow the calculation of \(\Delta p\) when the dimensionless parameter is given.
03

Predict \(\Delta p\)

Use the derived dimensionless form of the equation to calculate \(\Delta p\) for \(A_{1}/A_{2}=0.35\) and \(V_{1}=3.75 \mathrm{ft/s}\). Substitute these values into the equation and calculate \(\Delta p\).

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Most popular questions from this chapter

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