Water is drawn into the pump such that the pressure at the inlet A is -35 kPa and the pressure at B is 120 kPa. If the discharge at B is 0.08 m³/s, determine the power output of th... Water is drawn into the pump such that the pressure at the inlet A is -35 kPa and the pressure at B is 120 kPa. If the discharge at B is 0.08 m³/s, determine the power output of the pump. Neglect friction losses. The pipe has a constant diameter of 100 mm. Take h = 2 m. Read more

Steps to Solve

1. Identify Given Values The problem provides the following values:

   • Pressure at inlet A: $P_A = -35 \text{ kPa}$
   • Pressure at outlet B: $P_B = 120 \text{ kPa}$
   • Discharge rate: $Q = 0.08 \text{ m}^3/\text{s}$
   • Height difference: $h = 2 \text{ m}$

2. Convert Pressures to Pascals Convert the pressures from kPa to Pa for consistency in units:

   • $P_A = -35 \times 10^3 \text{ Pa} = -35000 \text{ Pa}$
   • $P_B = 120 \times 10^3 \text{ Pa} = 120000 \text{ Pa}$

3. Calculate the Pressure Difference Find the pressure difference ($\Delta P$) using: $$ \Delta P = P_B - P_A $$ Substituting the values: $$ \Delta P = 120000 - (-35000) = 120000 + 35000 = 155000 \text{ Pa} $$

4. Calculate the Power Output of the Pump The power output ($P$) can be calculated using the equation: $$ P = \Delta P \times Q + \rho \cdot g \cdot h \cdot Q $$ where $\rho$ (density of water) is approximately $1000 \text{ kg/m}^3$, and $g$ (acceleration due to gravity) is approximately $9.81 \text{ m/s}^2$. Thus:

   • Compute gravitational potential energy contribution: $$ P_{\text{gravity}} = \rho \cdot g \cdot h \cdot Q = 1000 \cdot 9.81 \cdot 2 \cdot 0.08 $$
   • Now calculate it: $$ P_{\text{gravity}} \approx 1000 \cdot 9.81 \cdot 2 \cdot 0.08 = 1569.6 \text{ W} $$

5. Final Power Calculation Substitute the pressure difference and the gravitational component: $$ P = 155000 \cdot 0.08 + 1569.6 $$ Calculate it: $$ P \approx 12400 + 1569.6 \approx 13969.6 \text{ W} $$

6. Convert Power to kW To express power in kilowatts (kW): $$ P \approx \frac{13969.6}{1000} = 13.97 \text{ kW} $$