Problem 1: Energy from Flow. Before the advent of the steam-powered engine, most mechanical processes were driven by extracting power from a nearby river using a water wheel. A miller wants to construct a water wheel to grind grain into an hour. He needs a total power output of 0.5 kW to meet the demand of his mill. If a nearby waterfall will flow at a rate of 400 liters/minute onto the top of the wheel, what is the required diameter of the water wheel to achieve the desired power output?

Diameter will be 27394.76 m

Explanation:

Power P = 0.5 kW = 500 W

Time t required for grinding = 1 hr = 3600 sec

Energy required E = P x t

E = 500 x 3600 = 1800000 J

Flow rate of water Q = 400 ltr/min

We convert to m3/sec

400 ltr/min = 400/(1000 x 60) m3/ses

Q = 0.0067 m3/sec

Energy provided by flow will be

E = pgQd

Where p = density of water = 1000 kg/m3

g = acceleration due to gravity 9.81 m/s2

d = diameter of wheel.

Equating both energy, we have,

1800000 = 1000 x 9.81 x 0.0067 x d

1800000 = 65.73d

d = 1800000/65.73

d = 27394.76 m