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.
Understand the Problem
The question is asking for the determination of the power output of a pump given the inlet and outlet pressures, discharge rate, and height difference. It involves applying principles from fluid mechanics, particularly those relating to pumps.
Answer
The power output of the pump is approximately $13.97 \text{ kW}$.
Answer for screen readers
The power output of the pump is approximately $13.97 \text{ kW}$.
Steps to Solve
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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: $h = 2 \text{ m}$
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Convert Pressures to Pascals Convert the pressures from kPa to Pa for consistency in units:
- $P_A = -35 \times 10^3 \text{ Pa} = -35000 \text{ Pa}$
- $P_B = 120 \times 10^3 \text{ Pa} = 120000 \text{ Pa}$
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Calculate the Pressure Difference Find the pressure difference ($\Delta P$) using: $$ \Delta P = P_B - P_A $$ Substituting the values: $$ \Delta P = 120000 - (-35000) = 120000 + 35000 = 155000 \text{ Pa} $$
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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: $$ P_{\text{gravity}} \approx 1000 \cdot 9.81 \cdot 2 \cdot 0.08 = 1569.6 \text{ W} $$
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Final Power Calculation Substitute the pressure difference and the gravitational component: $$ P = 155000 \cdot 0.08 + 1569.6 $$ Calculate it: $$ P \approx 12400 + 1569.6 \approx 13969.6 \text{ W} $$
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Convert Power to kW To express power in kilowatts (kW): $$ P \approx \frac{13969.6}{1000} = 13.97 \text{ kW} $$
The power output of the pump is approximately $13.97 \text{ kW}$.
More Information
This calculation demonstrates the relationship between pressure differences, discharge rates, and height differences in determining the power output of pumps. The result shows how both the pressure energy and potential energy contributed to the total output power.
Tips
- Neglecting to convert units consistently (e.g., kPa to Pa).
- Forgetting to account for the gravitational potential energy when calculating power.
- Miscalculating the pressure difference by not considering negative values correctly.
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