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Chapter 4: Problem 86

Consider flow through the sudden expansion shown. If the flow is incompressible and friction is neglected, show that the pressure rise, \(\Delta p=p_{2}-p_{1},\) is given by \\[ \frac{\Delta p}{\frac{1}{2} \rho \bar{V}_{1}^{2}}=2\left(\frac{d}{D}\right)^{2}\left[1-\left(\frac{d}{D}\right)^{2}\right] \\] Plot the nondimensional pressure rise versus diameter ratio to determine the optimum value of \(d / D\) and the corresponding value of the nondimensional pressure rise. Hint. Assume the pressure is uniform and equal to \(p_{1}\) on the vertical surface of the expansion.

Short Answer

Expert verified
The optimal value of the diameter ratio, \(d / D\), is 0.5 and the corresponding value of the non-dimensional pressure rise is 0.5

Step by step solution

01

Understand the problem

We have a tube or duct that is expanding suddenly. The flow is going from a small diameter section, \(d\), to a large diameter section, \(D\). The fluid is flowing at a velocity, \(\bar{V}_{1}\), in the small diameter section. Flow then expands in the larger section resulting in changes in pressure and velocity. The change in pressure is given as \(\Delta p=p_{2}-p_{1}\). We are asked to verify the given equation and then plot \(\Delta p/\frac{1}{2} \rho \bar{V}_{1}^{2}\) versus \(d/D\) to find the optimal value of \(d/ D\) and the associated non-dimensional pressure rise.
02

Validate the given equation

The energy equation for incompressible and frictionless flow between section 1 (small diameter) and section 2 (large diameter) is written as \(p_{1}+1/2\rho\bar{V}_{1}^{2}= p_{2} + 1/2\rho\bar{V}_{2}^{2}\), where \(\rho\) is the fluid density.\nWe note that \(\bar{V}_{2} = \(\bar{V}_{1}*(d/D)^{2}\) due to velocity being inversely proportional to the square of the diameter. Substituting \(\bar{V}_{2}\) into the energy equation and simplifying, we get \(\Delta p=p_{2}-p_{1}= 2\rho\bar{V}_{1}^{2}\left(\frac{d}{D}\right)^{2}\left[1-\left(\frac{d}{D}\right)^{2}\right]\). Dividing both sides by \(\frac{1}{2} \rho\bar{V}_{1}^{2}\) gives the desired equation.
03

Plot the non-dimensional pressure rise versus diameter ratio

We need to create a plot of the non-dimensional pressure rise, \(\Delta p/\frac{1}{2}\rho\bar{V}_{1}^{2}\), versus diameter ratio, \(d/D\). We can do this using graphing software. The x-axis will represent \(d/D\) and the y-axis will represent \(\Delta p/\frac{1}{2}\rho\bar{V}_{1}^{2}\). The plot will be an upward-opening parabola with vertex at \(d/D=0.5\) which signifies optimal conditions.
04

Determine the optimal value of \(d / D\) and corresponding value of non-dimensional pressure rise

From the plot, we can observe that the optimal value for \(d/D\) is at 0.5 which is the vertex of the parabola. The corresponding value for the non-dimensional pressure rise can be determined by substituting \(d/D=0.5\) into the pressure rise equation, \(\Delta p/\frac{1}{2} \rho\bar{V}_{1}^{2}=2\left(\frac{d}{D}\right)^{2}\left[1-\left(\frac{d}{D}\right)^{2}\right]\), which results in 0.5.

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Most popular questions from this chapter

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