Application of the Energy Equation

The energy equation relates the pump's shaft power input to the flow rate by accounting for gravitational head and head loss. The pump head is calculated by dividing net shaft power by the product of flow rate and water's specific weight. Substituting this into the simplified energy equation allows engineers to solve for the pump's flow rate directly using conservation of energy in control volume principles.

Both reservoirs are open to the atmosphere, so inlet and outlet pressures equal atmospheric pressure and cancel in the energy equation. Similarly, water surfaces in both reservoirs have negligible velocity. These simplifications reduce the energy equation to depend only on gravitational head and pump head, making calculations more straightforward.

Head loss, caused by friction and other resistances, represents energy dissipated in the system. In this example, the 5-meter head loss reduces the effective energy available for lifting water. The energy equation accounts for this loss, ensuring the calculated flow rate reflects the actual energy balance between pump input, gravitational lift, and system resistance.

Pump head equals the net shaft power input divided by the product of flow rate and water's specific weight. In this case, with 7500 joules per second of power input, the head represents the height equivalent of energy the pump can deliver. This relationship is central to solving for flow rate using the energy equation.

The 10-meter elevation difference between reservoirs represents the gravitational head the pump must overcome. This gravitational potential energy requirement is a key term in the energy equation. Combined with the 5-meter head loss, the total head needed is 15 meters, which the pump must supply through its shaft power input.

Specific weight, the weight per unit volume of water, directly relates shaft power to pump head through the energy equation. Dividing power by the product of specific weight and flow rate yields the pump head. This parameter ensures dimensional consistency and accounts for water's physical properties in converting mechanical power into hydraulic head.

After substituting pump head into the simplified energy equation and rearranging algebraically, flow rate becomes the unknown variable. Substituting known values—7500 joules per second power, 10-meter elevation, 5-meter head loss, and water's specific weight—yields the numerical flow rate. This final calculation determines the rate at which water is pumped between reservoirs.