A pump with a power of 5 kW (pump power, and not useful pump power) and an efficiency of 72 percent is used to pump water from a lake to a pool through a constant diameter. The free surface of the pool is 25 m above the free surface of the lake. If the irreversible head loss in the piping system is 4 m, determine (a) the mass flowrate of water and (b) the pressure difference across the pump.

Answer:

a) The mass flow rate of water is 14.683 kilograms per second.

b) The pressure difference across the pump is 245.175 kilopascals.

Explanation:

a) Let suppose that pump works at steady state. The mass flow rate of the water ([tex]\dot m[/tex]), in kilograms per second, is determined by following formula:

[tex]\dot m = \frac{\eta \cdot \dot W}{g\cdot H}[/tex] (1)

Where:

[tex]\dot W[/tex] - Pump power, in watts.

[tex]\eta[/tex] - Efficiency, no unit.

[tex]g[/tex] - Gravitational acceleration, in meters per square second.

[tex]H[/tex] - Hydrostatic column, in meters.

If we know that [tex]\eta = 0.72[/tex], [tex]\dot W = 5000\,W[/tex], [tex]g = 9.807\,\frac{m}{s^{2}}[/tex] and [tex]H = 25\,m[/tex], then the mass flow rate of water is:

[tex]\dot m = 14.683\,\frac{kg}{s}[/tex]

The mass flow rate of water is 14.683 kilograms per second.

b) The pressure difference across the pump ([tex]\Delta P[/tex]), in pascals, is determined by this equation:

[tex]\Delta P = \rho\cdot g\cdot H[/tex] (2)

Where [tex]\rho[/tex] is the density of water, in kilograms per cubic meter.

If we know that [tex]\rho = 1000\,\frac{kg}{m^{3}}[/tex], [tex]g = 9.807\,\frac{m}{s^{2}}[/tex] and [tex]H = 25\,m[/tex], then the pressure difference is:

[tex]\Delta P = 245175\,Pa[/tex]

The pressure difference across the pump is 245.175 kilopascals.

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