Question

An irrigation pump is used to draw water from a reservoir and deliver it to an irrigation ditch at a mass flow rate of 20 kg/s. The inflow from the reservoir and the outflow to the ditch are through pipes of diameters 10 cm and 7.5 cm respectively. If the water temperature is 15°C at the inlet and 20°C at the outlet, and the outlet is 100 m higher than the inlet, and one fifth of the input power to the pump is lost as heat transfer to the surroundings, what is the power that must be supplied to the pump? Neglect the kinetic energy of the water in your analysis.

Answer 1

The power that must be supplied to the pump can be calculated using fluid mechanics and thermodynamics principles. By using the formula P = 5/4 * m * g * h, substituting the mass flow rate, gravitational acceleration, and the height difference, we can determine the power needed, factoring in energy losses.

Step-by-step explanation:

To find the power that must be supplied to the pump, we can use the principles of fluid mechanics and thermodynamics. The power supplied to a fluid can be calculated using the equation P = ΔPQ, where ΔP is the increase in pressure provided by the pump and Q is the volumetric flow rate. In this case, since we are ignoring kinetic energy, the power is used to increase the water's gravitational potential energy due to elevation change and to overcome any losses. The potential energy change per unit mass is gh, where g is the acceleration due to gravity and h is the height difference. For a flow rate of m kg/s, the pump must provide mgh watts to lift the water. Accounting for temperature changes is not necessary as we're neglecting changes in density with temperature.

Since one-fifth of the power is lost to surroundings, the actual power required is 5/4 times the ideal power. So, the formula for finding the power needed by the pump becomes P = 5/4 * m * g * h. Plugging in the mass flow rate m = 20 kg/s, g = 9.81 m/s² (acceleration due to gravity), and h = 100 m, one can calculate the power required.