P3. 171 Consider a turbine extracting energy from a penstock in a dam, as in Fig. P3. 171. For turbulent pipe flow (Chap. 6),the friction head loss is approximately h,

= CQ², where the constant C depends on penstock dimensions and the properties of water. Show that, for a given penstock geometry and variable river flow Q, the maximum turbine power possible in this case is Pmax = 2pgHQ/3 and occurs when the flow rate is Q = (H/(3C))^1/2.