Question

A radial flow hydraulic turbine whose design is based on a power specific speed, Ω_sp=1.707 is to produce 25 MW from a total head, H_E=25 m. The overall turbine efficiency η_o=0.92, the mechanical efficiency is 0.985, and the loss in head in the nozzles is 0.5 m. The ratio of the blade tip speed to jet speed is 0.90. Assuming the meridional velocity is constant and equal to 10 m/s and there is no swirl in the runner exit flow, determine

1. The volume flow rate through the turbine;
2. The rotational speed and diameter of the runner;
3. The absolute and relative flow angles at entry to the runner.

Answer 1

Determine the volume flow rate through the turbine

Given:

        P = 25 × 106 W
        ηo = 0.92, ρ = 1000 kg/m3
        g = 9.81 m/s2, HE = 25 m
    

Hence, the volume flow rate through the turbine is Q = 110.8 m³/s

Determine the rotational speed and diameter of the runner

Write expression for specific speed:

Calculate angular speed of turbine.

        Given:
        Ωsp = 1.707, g = 9.81 m/s2
        HE = 25 m, P = 25 × 106 W, ρ = 1000 kg/m3
    

Calculate rotational speed:

Therefore, rotational speed is N = 100.0567 rpm

Calculate blade tip speed.

        Given:
        K = 0.9, g = 9.81 m/s2, HE = 25 m, h = 0.5 m
    

Calculate diameter of runner.

        Given:
        U = 19.732 m/s, N = 100.0567 rpm
    

Therefore, the diameter is D = 3.766 m

Determine the absolute and relative flow angles at entry to the runner

Calculate hydraulic efficiency.

        Given:
        ηo = 0.92, ηm = 0.985
    

Calculate swirl velocity at runner exit.

        Given:
        ηh = 0.934, g = 9.81 m/s2, HE = 25 m, U = 19.732 m/s
    

Radial velocity at runner exit is the meridional velocity.

Calculate absolute flow angle at runner exit.

        Given:
        cθ2 = 11.6087 m/s, cr2 = 10 m/s
    

Calculate relative flow angle at runner exit.

        Given:
        cθ2 = 11.6087 m/s, U = 19.732 m/s, cr2 = 10 m/s
    

Hence, the absolute flow angle at runner exit is 49.26° and the relative flow angle at runner exit is -39.088°

There is the answer we calculate.