Question

A Kaplan turbine designed with a shape factor (power specific speed) of 3.0 (rad), a runner tip diameter of 4.4 m, and a hub diameter of 2.0 m, operates with a net head of 20 m and a shaft speed of 150 rev/min. The absolute flow at runner exit is axial. Assuming that the hydraulic efficiency is 90% and the mechanical efficiency is 99%, determine:

1. The volume flow rate and shaft power output;
2. The relative flow angles at the runner inlet and outlet at the hub, at the mean radius and at the tip.

Answer 1

 

Determine The Volume Flow Rate and Shaft Power Output

Calculate the rotational speed of the shaft.

Given: N=150rpm

Ω=π30N=π30×150=15.70rad/s

Calculate the power delivered to the rotor.

Given: ρ=1000 kg/m3,Ωsp=3.0rad,Ω=15.70rad/s,g=9.18 m/s2,HE=20 m

P=ρ(ΩpΩ)2(gHE)52=1000(315.70)(9.81×20)52=1000×0.036×539197.05=19.67×106 W

Calculate the shaft power delivered to the external device.

Given: ηshaft =0.99,P=19.67×105 W

Pshaft =ηmP=0.99×19.67×105=19.47MW

Hence, the power delivered to the shaft is 19.47 MW.

Calculate the volume flow rate.

Given: P=19.67×105 W,ρ=1000 kg/m3,g=9.18 m/s2,HE=20 m

Q=PηHρgHE=19.67×1060.9×9.81×1000×20=111.4 m3/s

Hence, the volume flow rate Q=111.4 m3/s

Determine The Relative Flow Angles at the Runner Inlet and Outlet at the Hub, at the Mean Radius, and at the Tip

Calculate the specific work done.

Given: P=19.67×105 W,ρ=1000 kg/m3,Q=111.4 m3/s

ΔW=PρQ=19.67×1061000×111.4=176.6 m2/s2

Calculate the axial velocity.

Given: Q=111.4 m3/s,D1=4.4 m,Dh=2 m

cx=4Qπ(D12−Dh2)=4×111.4π(4.42−22)=9.234 m/s

Calculate the blade tip speed.

Given: Ω=15.07rad/s,D1=4.4 m

U1=ΩD12=15.07×4.42=34.56 m/s

Consider the expression for the relative flow angle at various radii at the runner inlet.

Given: U1=34.56 m/s,cx=9.234 m/s,ΔW=176.6 m2/s2

β2=tan−1⁡[(rr1)U1cx−(r1r)ΔWcxU1]=tan−1⁡[(rr1)34.569.234−(r1r)176.69.234×34.56]=tan−1⁡[3.743(rr1)−0.5534(r1r)](∗)

Consider the expression for the relative flow angle at various radii at the runner outlet.

Given: U1=34.56 m/s,cx=9.234 m/s

β3=tan−1⁡(rr1U1cx)=tan−1⁡(3.743×rr1)(∗∗)

Tabulate the value of equation (*) and (**) in the table below.

Position	Hub	Mean	Tip
Radius/Ratio	0.4545	0.7273	1
β2∘	25.81	62.99	72.59
β3∘	59.55	69.83	75.04

There is the answer.