Repeat Example 6.1, but with the somewhat more realistic assumption that the flow is similar to a free vortex (irrotational) profile, \(V_{\theta}=c / r\) (where \(c\) is a constant), as shown in Fig. P6.25. In doing so, prove that the flow rate is given by \(Q=k \sqrt{\Delta p}\), where \(k\) is

\[k=w \ln \left(\frac{r_{2}}{r_{1}}\right) \sqrt{\frac{2 r_{2}^{2} r_{1}^{2}}{ho\left(r_{2}^{2}-r_{1}^{2}\right)}}\]

and \(w\) is the depth of the bend.

Data From Figure P6.25

Data From Example 6.1

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12 P6.25