Chapter 6, Problem 26P

Using the analyses of Example 6.1 and Problem 6.25, plot the discrepancy (percent) between the flow rates obtained from assuming uniform flow and the free vortex (irrotational) profile as a function of r₂/r₁.

25. Repeat Example 6.1, but with the somewhat more realistic assumption that the flow is similar to a free vortex (irrotational) profile, V_θ = 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

$k=w\text{ln}\left(\frac{r_2}{r_1}\right)\sqrt{\frac{2r_2^2{r}_1^2}{\rho (r_2^2-r_1^2)}}$

and w is the depth of the bend.

P6.25