A centrifugal pump delivers water at the rate of 0.22 m3/s from a rese...
Given Data:
- Rate of water delivery, Q = 0.22 m3/s
- Diameter of the pipe, D = 0.2 m
- Power input to the pump, P = 90 kW
- Efficiency of the pump, η = 75%
- Fanning friction factor, f = 0.004
- Gravitational acceleration, g = 9.8 m/s2
- Density of water, ρ = 1000 kg/m3
Calculating the Water Head:
The water head is the total energy head required to deliver the water from one reservoir to the other. It can be calculated using the following formula:
Water Head = Pressure Head + Velocity Head + Elevation HeadSince both reservoirs are open to the atmosphere, the pressure head can be neglected. Therefore, the water head is given by:
Water Head = Velocity Head + Elevation HeadThe velocity head is given by:
Velocity Head = (v2)/(2g)where v is the velocity of water in the pipe.
The velocity of water can be calculated using the formula:
Velocity = Q / (π * (D/2)2)Substituting the given values, we can calculate the velocity of water.
The elevation head is given by:
Elevation Head = Hwhere H is the height to which the water is delivered.
Calculating the Power Output:
The power output of the pump can be calculated using the formula:
Power Output = Q * Water Head * ρ * gSince the efficiency of the pump is given by:
Efficiency = Power Output / Power Inputwe can rearrange the formula to calculate the power output:
Power Output = Efficiency * Power InputSubstituting the given values, we can calculate the power output.
Calculating the Water Head:
Substituting the calculated power output and the given values into the power output formula, we can solve for the water head:
Power Output = Q * Water Head * ρ * gSolving for Water Head, we get:
Water Head = Power Output / (Q * ρ * g)Substituting the calculated values, we can solve for the water head.
Calculating the Height H:
Since the water head is equal to the velocity head plus the elevation head, we can rearrange the formula to solve for H:
H = Water Head - Velocity HeadSubstituting the calculated values, we can solve for H.
Final Result:
After substituting the values and performing the calculations, we find that the height H, to which the water can be delivered, is approximately 36 meters.