A water turbine delivering $$10$$ $$MW$$ power is to be tested with the help of a geometrically similar $$1:8$$ model, which runs at the same speed as the proto-type.

(a) Find the power developed by the model assuming the efficiencies of the model and the proto-type are equal.

(b) Find the ratio of the heads and the ratio of mass flow rates between the proto-type and the model.

In a hydroelectric station, water is available at the rate of $$175{m^3}/s$$ under a head of $$18m.$$ The turbines run at a speed of $$150$$ $$rpm$$ with overall efficiency of $$82\% $$. Find the number of turbines required if they have the maximum specific speed of $$460.$$

A Hydro turbine is required to give $$25$$ $$MW$$ at $$50$$ $$m$$ head and $$90$$ $$rpm$$ runner speed. The laboratory facilities available, permit testing of $$20$$ $$kW$$ models at $$5$$ $$m$$ head. What should be the model runner speed and model to prototype scale ratio?

A $$7MW$$ hydro turbine working under a head of $$10$$ $$m$$ at a rotational speed of $$125$$ $$rpm$$ is to be designed and developed by conducting model tests in a laboratory. Maximum possible discharge to the model is $$600$$ liters per second at a constant head of $$5m.$$ Determine the minimum scale of the model and its speed if the expected efficiency of the model is $$85\% .$$