The head extracted by turbine CR in Fig. 7-18 is 200 ft, and the pressure at T is 72.7 psi. For losses of 2.0(V24^2/2g) between W and R and 3.0(V12^2/2g) between C and T9 determine (a) how much water is flowing and (b) the pressure head at R. Draw the energy line.

Question

The head extracted by turbine CR in Fig. 7-18 is 200 ft, and the pressure at T is 72.7 psi. For losses of [latex]2.0\left(V_{24}^{2} / 2 g ight) ext { between } W ext { and } R ext { and } 3.0\left(V_{12}^{2} / 2 g ight)[/latex] between C and T9 determine (a) how much water is flowing and (b) the pressure head at R. Draw the energy line.

Accepted Answer

[latex] ext { Because the energy line at } T ext { is at elevation }\left(250.0+\frac{72.7 imes 144}{62.4}+\frac{V_{12}^{2}}{2 g} ight)[/latex] and well above the elevation
at W, the water flows into reservoir W.

(a) Using T to W, datum zero,

at T T to C R to W [latex]H_{T}[/latex] at W.

[latex]\left(\frac{72.7 imes 144}{62.4}+\frac{V_{12}^{2}}{2 g}+250 ight)-\left[3.0 \frac{V_{12}^{2}}{2 g}+2.0 \frac{V_{24}^{2}}{2 g} ight]-200=(0+ negl +150)[/latex].

[latex] ext { Substituting } V_{24}^{2}=\frac{1}{16} V_{12}^{2} ext { and solving, } V_{12}^{2} / 2 g=32.0 ft ext { or } V_{12}=45.4 ft / sec ext {. Then }[/latex]

[latex]Q=\frac{1}{4} \pi(1)^{2} imes 45.4=35.7 cfs[/latex].

[latex] ext { (b) Using } R ext { to } W ext {, datum } R, \quad\left(p_{R} / \gamma+\frac{1}{16} imes 32.0+0 ight)-2\left(\frac{1}{16} imes 32.0 ight)=(0+ ext { negl }+50) ext { and } p_{R} / \gamma=52.0[/latex] ft. The reader may check this pressure head by applying the Bernoulli equation between T and R.

To plot the energy line in the figure, evaluate the energy at the four sections indicated.

elevation of energy line at T = 168.0 + 32.0 + 250.0 = 450.0
at C = 450.0 - 3 × 32.0 = 354.0
at R = 354.0 - 200.0 = 154.0

[latex] ext { at } W=154.0-2 imes \frac{1}{16} imes 32=150.0[/latex].

It will be shown in the following chapter that the energy line is a straight line for steady flow in a pipe of constant diameter. The hydraulic grade line will be parallel to the energy line and V²/2g below it (shown dotted).

Question 7.36: The head extracted by turbine CR in Fig. 7-18 is 200 ft, and...

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