The island also considers using hydrokinetic energy with a m

The island also considers using hydrokinetic energy with a marine turbine as the power source. A river
with average current velocity of VC. = 1.0 m/sec flows on the island. Assuming an ideal Betz turbine with
mechanical/electrical efficiency of 0.85, availability of 100% during the 10 hours per day that the turbine
powers the village water pump, and that the current velocity follows a Rayleigh distribution. A funnel
upstream of the turbine is used to accelerate the current velocity before it reaches the turbine. The funnel
area ratio AC/AT = 2

a) Size the turbine, e.g. find the required turbine diameter AT to generate the required net power output = 500 Watts
_______________
b) If the funnel is removed, resize the turbine by finding a new required turbine diameter.
________________

Solution

The ideal mechanical power from a continuous flow can be written as:

P_ideal-m = ½. . A. V³

Where,

density of water

A Area of front of turbine in m²

V Velocity in m/s

Ideal power is always more than the real mechanical power due to hydrodynamic losses in marine turbines. The ratio of these two powers is known as mechanical efficiency and is given as:

_mech = P_m / P_ideal-m

                                                                        = P_m / (1/2 . . A_F. V³)

Where,

P_m = mechanical power

A_F = Area of funnel in m²

Assuming, density of water as 1000 Kg/m³

Given that P_m = 500 watts

V = 1 m/s

_mech = 0.85

Using the given values in equation 2, we get

                                                  0.85 = 500/ (1/2. 1000. A_F. 1)

                                                A_F = 1.1764 m

Also, the ratio of turbine and funnel diameter is given as 2. So,

                                                A_T = 1.1764 x 2 = 2.3528 m (Ans. of part a)

If funnel is removed,

                                                A_F = A­_T

Therefore, A_T = 1.1764 m (Ans. of part b)