Question

A centrifugal pump delivers water at rate of 0.22 m³/s from a reservoir at a ground level to another reservoir to height 'h' through a vertical pipe of 0.2 m diameter. Both are open to atmosphere. Power input to pump is 90 kW and it operates with 75% efficiency . f=0.004 use g=9.8 m/s² density=1000kg/m³. Find 'h'.

Answer 1

the problem is based on the concepts of fluid mechanics.

the pumps are used to provide the energy to fluid flowing which is lost because of the friction.

the general equation for the pump is derived by modifying the Bernoulli equation, as,

................................1

here,

P =pressure

v = velocity of flow

h = height of elevation

Ppump = pressure difference across pump

hf = frictional losses

= density of pump

and the efficiency of pump is given as,

..............2

here ,

Wpump = work done by pump

Win = power provided to pump.

so we will use these equation in this problem.

the data given to us:

1. Diameter of pipe = d = 0.2 m

2. efficiency of pump = = 0.75

3. density = = 1000 kg/m3

4. friction factor = f = 0.004

5. Win = 90 kW

6. flow rate = Q = 0.22 m3/s

now, the area of cross section of pipe = A = 3.14 * 0.2 m *0.2 m /4 = 0.0314 m2

and we know from equation of continuity that,

Q = area * velocity = A*V

so,

0.22 m3/s = 0.0314 m2 * V

so we get

V = 7.006 m/s..............3

let height be assume as 'h' meter.

the water is being pumped vertically from ground to 'h' meter. and the both the bottom tank inlet and discharge tank outlet is exposed to atmosphere. that means that P1 = P2 = atmospheric pressure.

and because the area of cross section is not changing the kinetic energy will also not change. so,

v1 =v2 =V

and ,

h1 = 0 m

and

h2 = h meter.

so, we substitute the values in the equation we get

we get

..............4

so, we have to find the frictional losses , which is given as,

here

f = friction factor

L = length of pipe

D = diameter of pipe

V = velocity

so here

L =h because the pipe is vertical.

so, we substitute the values we get

we get

................ 5

so, we have the hf value in terms of h, we substitute it in the above equation 4 we get

we get

.

so we have obtained the pressure difference across the pump in terms of h, now the work done by pump is given as,

here ,

Q = flow rate ,

so we substitute the values we get

......................... 6

we use the equation 2, to calculate the work done by pump as,

we substitute the values we have been provided in the question , we get

we get

.............7

so, we equate the eqiation 7 and 6, we get

we get

Pa.m3 = J

so,

we get

.........................[ans.]

so, the height of the pipe is 29.814 meter.