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Chapter 5: Problem 21

An appropriate turbulent pipe flow velocity profile is \\[ \mathbf{V}=u_{c}\left(\frac{R-r}{R}\right)^{1 / n} \\] where \(u_{c}=\) centerline véocity, \(r=\) local radius, \(R-\) pipe radias, and \(\hat{\mathbf{i}}=\) unit vector along pipe centerline. Determine the ratio of average velocity, \(\bar{u},\) to centerline velocity, \(u_{c},\) for \((\mathbf{a}) n=4\) (b) \(n=6\) (c) \(n=8\) (d) \(n=10 .\) Compare the different velocity profiles.

Short Answer

Expert verified
The ratios of average velocity to centerline velocity for different n values are 1.6 (for n=4), 1.71 (for n=6), 1.78 (for n=8), 1.82 (for n=10). As n increases, the average velocity gets closer to the centerline velocity, suggesting a more uniform flow profile.

Step by step solution

01

Understanding the given

The problem provides a velocity profile equation for a turbulent flow inside a pipe. Our task is to determine the ratio of the average velocity, \(\bar{u}\), to the centerline velocity, \(u_{c}\), for different values of \(n\). Here, \(u_{c}\) is the centerline velocity, \(r\) is the local radius, \(R\) is the pipe radius, \(n\) is the power in the velocity profile equation, and \(\hat{\mathbf{i}}\) is the unit vector along pipe centerline.
02

Determine the average velocity

The average velocity, \(\bar{u}\), is given by the integral of the velocity profile over the pipe's cross-sectional area divided by the area. This can be represented as:\[ \bar{u}= \frac{1}{\pi R^{2}} \int_{0}^{R} 2 \pi r u_{c}\left(\frac{R-r}{R}\right)^{1 / n} d r.\]
03

Simplify the average velocity expression

Substitute \(x = R - r\) into the average velocity expression and simplify it, you will get:\[ \bar{u} = \frac{2u_c}{R^n} \int_0^R x^{1/n} dx.\] Solve the Integral to get \(\bar{u} = \frac{2u_c}{R^n}[\frac{n}{n+1}x^{1+1/n}|_0^R\]\[= \frac{2n}{n+1}u_c.\]
04

Calculate ratios for given n values

Now we can calculate the ratio of average velocity to the centerline velocity for different n values as below:\n\n(a) For n=4, Ratio \(\bar{u}/u_c\) is \(\frac{2*4}{4+1} = 1.6\),\n(b) For n=6, Ratio \(\bar{u}/u_c\) is \(\frac{2*6}{6+1} \approx 1.71\),\n(c) For n=8, Ratio \(\bar{u}/u_c\) is \(\frac{2*8}{8+1} \approx 1.78\),\n(d) For n=10, Ratio \(\bar{u}/u_c\) is \(\frac{2*10}{10+1} \approx 1.82\).
05

Comparative analysis of velocity profiles

From these ratios, it can be observed that as the value of \(n\) increases, the ratio \(\bar{u}/u_c\) increases steadily. Therefore, it can be inferred that the average velocity gets closer to the centerline velocity as \(n\) increases. This indicates a less steep velocity gradient near the pipe centerline for larger \(n\) values, which suggests a more uniform flow profile.

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