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1. A small hydropower plant had a number of Francis turbines installed in 1928. The plant is now under review for modernisation. Each of the existing turbines generates 300 kW at 250 rpm at an available head of 7.9 m for a flow rate of 5.8 m3 /s. The diameter of the draft tube is 1.34 m, and the outlet area of the turbine runner is the same as the cross-sectional area of thedraft tube.

Your task is to appraise the current performance and evaluate a number of different turbine replacements, all to operate under the same head, flow rate, and speed. Assume for all turbines a typical best efficiency of 95% and a combined efficiency of all other components of 94% for power conversion or transmission. The density of water is 1000 kg/m3

a) Evaluate the current performance of the installed turbines compared to that expected by a typical turbine under the given operating conditions. [71% or 67% overall]

b) Evaluate the choice of turbine type given the head it operates at and the power it provides.

c) Discuss whether a multi-jet Pelton Wheel with a specific speed per jet of NS= N (P/ρ) 1/2 (gH)5/4= 0.02 would have been a better choice. Calculate the number of jets required to generate the same output. [several hundred]

d) Would a Kaplan turbine with a specific speed of NS= 0.36 be an appropriate choice? [yes]

e) Would a Kaplan turbine with a specific speed of NS= 0.6 be an appropriate in the given operating conditions? [ No ]

f) By how much could the available head be increased by increasing the diameter of the draft tube by 10%? Would it be worth doing?

2. A tidal stream turbine is proposed, which looks like a wind turbine under water. The proposed site has a peak tidal stream velocity of 4 m/s. The density of water and air are 1000 kg/m3 and 1.2 kg/m3 , respectively.

a) Assuming a typical performance for such a turbine of 75% of the Betz limit, calculate the rotor diameter required to generate 500kW at peak velocity. Compare this to the diameter of a 500kW wind turbine with the same efficiency at a wind speed of 15m/s. [6.7m vs 27m]

b) If such turbines operate best at a ratio of the blade tip to the stream velocity of 6:1, calculate the tip speed velocity at peak velocity and the resulting rotation rate of the turbine. Compare this to a typical rotation rate of a wind turbine between 30 and 60 rpm. [68 rpm]

c) By modelling the rotor as a disk with a drag coefficient of CD= 1, estimate the drag force on the rotor at peak velocity and the resulting bending moment on the foundation if it is mounted 30 m above the sea bed. [8.5 MNm]

d) The fluid velocity at a time t in a tidal cycle is u= 4sin(2π t / T), where T is the tidal period. Estimate the energy generated during one tidal cycle. [5990 kWh] Assume a tidal cycle of duration 12.5 hours and use: ∫ sin3

(x) dx= – cos x + 1/3 cos3 x and x= (2π t / T)

Hint: integrate from 0 to π – What does that correspond to, and how do you get from there to a tidal cycle?

e) Compare the output found in (d) with the output from a turbine which generates 500kW

continuously, and with a 500kW wind turbine with a load factor of 30%.

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