Question

A Pelton wheel is supplied with water from a lake at an elevation H above the turbine. The penstock that supplies the water to the wheel is of length , diameter D, and friction factor f. Minor losses are negligible. Show that the power developed by the turbine is maximum when the velocity head at the nozzle exit is 2H/3. Note: The result of Problem 12.61 may be of use.

Answer 1

Following are the proving to this question:

Step-by-step explanation:

`(D_1)/(D) = \frac{1}{(2f((l)/(D)))^{(1)/(4)}}`

using the energy equation for entry and exit value :

`\to (p_o)/(y) +(V^(2)_(o))/(2g)+Z_0 = (p_1)/(y) +(V^(2)_(1))/(2g)+Z_1+ f (l)/(D)(V^(2))/(2g)`

where

`\to p_0=p_1=0\\\\\to Z_0=Z_1=H\\\\\to v_0=0\\\\AV =A_1V_1 \\\\\to V=((D_1)/(D))^2 V_1\\\\\to V^2=((D_1)/(D))^4 V^(2)_(1)`

`= (\frac{1}{(2f ((l)/(D) ))^{(1)/(4)}})^4\ V^(2)_(1)\\\\`

`= (1)/((2f ((l)/(D)) )) \ V^(2)_(1)\\`

`\to (p_o)/(y) +(V^(2)_(o))/(2g)+Z_0 =(p_1)/(y) +(V^(2)_(1))/(2g)+Z_1+ f (l)/(D)(V^(2))/(2g) \\\\`

`\to 0+0+Z_0 = 0 +(V^(2)_(1) )/(2g) +Z_1+ f (l)/(D) ((1)/((2f((l)/(D))))\ V^(2)_(1))/(2g) \\\\\to Z_0 -Z_1 = +(V^(2)_(1))/(2g) \ (1+f(l)/(D)(1)/((2f((l)/(D)) )) ) \\\\\to H= (V^(2)_(1))/(2g) ((3)/(2)) \\\\\to (V^(2)_(1))/(2g) = H((3)/(2))`

L.H.S = R.H.S