We were discussing the pumps and basic pumping system, total head developed by the centrifugal pump, parts of centrifugal pump and their function in our previous post.

Now we will find out the work done by the centrifugal pump on water or we can say work done by the centrifugal pump impeller on water with the help of this post.

In our previous post, we have studied the basics concept behind the centrifugal pump. Now let us come to the point and let us see here how to determine the work done by the centrifugal pump on water.

### Work done by the centrifugal pump on water

In case of centrifugal pump, work will be done by the impeller on the water. Expression for the work done by the impeller on the water will be determined by drawing the velocity triangles at inlet and outlet of the impeller.

Let us see here a typical section of a centrifugal pump that indicates the impeller blade as displayed here in following figure.  We are interested here to find out the work done by the centrifugal pump on the water. Energy interaction will take place only in the rotor i.e. impeller of the centrifugal pump.
Velocity triangles will be drawn at the inlet and outlet tips of the vanes fixed to an impeller.

For best efficiency of the pump, water need to enter the impeller radially at inlet. Therefore, absolute velocity of water at inlet will make an angle of 90 degree with the direction of motion of the impeller at inlet.

As we have discussed above that absolute velocity of water at inlet will make an angle of 90 degree with the direction of motion of the impeller at inlet, therefore angle α = 0 and velocity of whirl at inlet Vw1 = 0.

Let us assume the following data as mentioned here.

N = Speed of the impeller in R.P.M
D1 = Diameter of the impeller at inlet
u1 =Tangential Velocity of impeller at inlet
V1 = Absolute velocity of water at inlet
Vr1 = Relative velocity of water at inlet
α = Angle made by absolute velocity of water at inlet with the direction of motion of vane
θ = Angle made by relative velocity of water at inlet with the direction of motion of vane

D2 = Diameter of the impeller at outlet
u2 =Tangential Velocity of impeller at outlet
V2 = Absolute velocity of water at outlet
Vr2 = Relative velocity of water at outlet
β = Angle made by absolute velocity of water at outlet with the direction of motion of vane
φ = Angle made by relative velocity of water at outlet with the direction of motion of vane

Tangential velocity of water at inlet, u1 = π D1 N/60
Tangential velocity of water at outlet, u2 =   π D2 N/60

As we know that in case of radially inward flow reaction turbine, the work done by the water on the runner per second per unit weight of the water striking per second will be given by following equation as mentioned here.

Work done by the water on the runner per second per unit weight of the water striking per second = (1/g) x [Vw1 u1 - Vw2 u2

As we know that a centrifugal pump is the reverse of a radially inward flow reaction turbine, therefore work done by the impeller on the water in case of a centrifugal pump will be given by following equation as mentioned here.

Work done by the impeller on the water per second per unit weight of the water striking per second = - (work done in case of turbine)

Work done by the impeller on the water per second per unit weight of the water striking per second = (1/g) x [Vw2 u2 - Vw1 u1

Work done by the impeller on the water per second per unit weight of the water striking per second = (1/g) x Vw2 u

Because, absolute velocity of water at inlet will make an angle of 90 degree with the direction of motion of the impeller at inlet, therefore angle α = 0 and Vw1 = 0.

Above equation also provides the head imparted to the water by the impeller or energy given by impeller to the water per unit weight per second.

### Work done by the impeller on water per second = (W/g) x Vw2 u2

Where, W = Weight of water
W = ρ x g x Q

Where, Q = Flow rate of water
Q = Area x Velocity of flow
Q =π x D1 x B1 x Vf1 = π x D2 x B2 x Vf2

Where,
B1 and B2 are the width of impeller and Vf1 and Vf2 are the velocities of flow at the inlet and outlet.

Therefore, we have seen here two very important terms i.e. work done by the centrifugal pump on water and head imparted to the water by the impeller or energy given by impeller to the water per unit weight per second.

### Head imparted to the water by the impeller or energy given by impeller to the water per unit weight per second = (1/g) x Vw2 u2

Do you have any suggestions? Please write in comment box.

Further we will find out, in our next post, Head and efficiency of centrifugal pump

### Reference:

Fluid mechanics, By R. K. Bansal