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Chapter 5: Problem 130

Calculate the kinetic energy correction factor for each of the following velocity profiles for a circular pipe: (a) \(u=u_{\max }\left(1-\frac{r}{R}\right)\) (b) \(u=u_{\max }\left(1-\frac{r^{2}}{R^{2}}\right)\) (c) \(u=u_{\max }\left(1-\frac{r}{R}\right)^{1 / 7}\)

Short Answer

Expert verified
The kinetic energy correction factors for the velocity profiles are computed by using the formula \(\beta = \frac{2}{^{2}}\). After the calculations, you should get one \(\beta\) value for each profile.

Step by step solution

01

Calculate for each profile

The average velocity for a circular pipe of radius R is given by the formula \(\int_{0}^{R} \frac{u(r)2\pi r dr}{\pi R^{2}}\), where \(u(r)\) is the velocity profile. Substitute \(u(r)\) with each function and perform the integration from 0 to R. Simplify the results.
02

Calculate for each profile

The average of velocity squared for a circular pipe is similarly calculated by the formula \(\int_{0}^{R} \frac{[u(r)]^{2} 2 \pi r dr}{\pi R^{2}}\). Substitute each function into the formula and integrate from 0 to R. Simplify the results.
03

Calculate the kinetic energy correction factor

Substitute and into the formula \(\beta = \frac{2 }{^{2}}\) for each velocity profile. Solve for \beta.

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