Question

Water is pumped from a lower reservoir to a higher reservoir by a pump that provides 20 kW of useful mechanical power to the water. The free surface of the upper reservoir is 45 m higher than the surface of the lower reservoir. If the flow rate of water is measured to be 0.03 m3/s, determine the irreversible head loss of the system and the lost mechanical power during this process. Take the density of water to be 1000 kg/m3.

Answer 1

According to the information presented, it is necessary to take into account the concepts related to mass flow, specific potential energy and the power that will determine the total work done in the system.

By definition we know that the change in mass flow is given by

$\dot{m} = \rho AV$

$\dot {m} = \rho Q$

Remember that the Discharge is defined as Q = AV, where A is the Area and V is the speed.

Substituting with the values we have we know that the mass flow is defined by

$\dot{m} = 1000*0.03$

$\dot{m} = 30kg/s$

To calculate the power we need to obtain the specific potential energy, which is given by

$\Delta pe = gh$

$\Delta pe = 9.8*45$

$\Delta pe = 441m^2/s^2$

So the power needed to deliver the water into the storage tank would be

$\dot {E} = \dot{m}\Delta pe$

$\dot {E} = 30*441$

$\dot{E} = 13230W = 13.23kW$

Finally the mechanical power that is converted to thermal energy due to friction effects is:

$\dot{W}_f = \dot{W}_s - \dot{E}$

$\dot{W}_f 20-13.23$

$\dot{W} = 6.77kW$

Therefore the mechanical power due to friction effect is 6.77kW