Question 2.70: A geothermal pump is used to pump brine whose density is 105......
A geothermal pump is used to pump brine whose density is 1050 kg/m³ at a rate of 0.3 m³/s from a depth of 200 m. For a pump efficiency of 74 percent, determine the required power input to the pump. Disregard frictional losses in the pipes, and assume the geothermal water at 200 m depth to be exposed to the atmosphere.
Geothermal water is raised from a given depth by a pump at a specified rate. For a given pump efficiency, the required power input to the pump is to be determined.
Assumptions 1 The pump operates steadily. 2 Frictional losses in the pipes are negligible. 3 The changes in kinetic energy are negligible. 4 The geothermal water is exposed to the atmosphere and thus its free surface is at atmospheric pressure.
Properties The density of geothermal water is given to be \rho= 1050 \mathrm{~kg} / \mathrm{m}^3.
Analysis The elevation of geothermal water and thus its potential energy changes, but it experiences no changes in its velocity and pressure. Therefore, the change in the total mechanical energy of geothermal water is equal to the change in its potential energy, which is g z per unit mass, and \dot{m} g z for a given mass flow rate. That is,
Then the required power input to the pump becomes
\displaystyle \dot{W}_{\text {pump, elect }}=\frac{\Delta \dot{E}_{\text {mech }}}{\eta_{\text {pump-motor }}}=\frac{618 \mathrm{~kW}}{0.74}= \mathbf{835~kW}Discussion The frictional losses in piping systems are usually significant, and thus a larger pump will be needed to overcome these frictional losses.
