Impulse Turbines

Question 1

Momentum equation

Answer 1

F = ṁ(u2 - u1)

Question 2

Force exerted on a fluid in a stationary vane

Answer 2

Fx = ρAV^2 (cosβ -1)
Fy = ρAV^2 (sinβ)

β = angle from the positive x-axis anticlockwise to the end of the top of the vane (angle of deflection)
V = velocity of jet at entry

Question 3

Force exerted on a fluid a moving vane

Answer 3

Fx = ρA(V - U)^2 (cosβ -1)
Fy = ρA(V - U)^2 (sinβ)

β = angle from the positive x-axis anticlockwise to the end of the top of the vane (angle of deflection)
V = velocity of jet at entry
U = vane velocity

Question 4

Force exerted by the water on the buckets

Answer 4

F0 = -Fx
as it acts in the opposite direction to the water

Question 5

Torque produced on the wheel

Answer 5

T = F0 r
= ρQ(V - U)(1 - cosβ) r

Question 6

Power output

Answer 6

P = Tω = F0 U
= ρQU(V - U)(1 - cosβ)

(as U = ωr)

Question 7

calculate max power output

Answer 7

Pmax = 1/2 ρgHQ (1 - cosβ)

U = 1/2 V
H = V^2 / 2g

Question 8

calculate the direction of the force

Answer 8

θ = arctan( -Fy/-Fx)
(anti-clockwise starting from the positive x-axis)
force on fluid and vane are opposite (180° apart)

Question 9

force on vane

Answer 9

= sqrt( (-Fx)^2 + (-Fy)^2 )

Question 10

why is a notch made at the tip of each bucket on a Pelton wheel

Answer 10

Ensure only one bucket is hit at a time, and the
distance between the nozzle and the hitting point is constant

Question 11

what is the condition for mass conservation equation, ṁ = ṁ in - ṁ out

Answer 11

no nuclear reactions occur (mass isn’t converted to heat)

Question 12

condition for bernoulli’s equation

Answer 12

no energy loss (no energy input or output)

Question 13

what is head

Answer 13

energy (work done) per unit weight