Question

underground water is to be pumped by a 80% efficient 5-kw submerged pump to a pool whose free surface is 30 m above the underground water level. the diameter of the pipe is 7 cm on the intake side and 5 cm on the discharge side. the irreversible head loss of the piping system is 4 m. determine the pressure difference across the pump in kpa.

Answer 1

The pressure difference across the pump is approximately 0.335 kPa, with negative sign indicating lower pressure at the discharge side.

Given:

- Pump power
`(\( \dot{W} \))` = 5 kW

- Pump efficiency
`(\( \eta \))` = 80% (or 0.8)

- Pipe diameters:
`\( D_1 = 0.07 \)` m (intake side),
`\( D_2 = 0.05 \)` m (discharge side)

- Free surface elevation difference
`(\( h_2 - h_1 \))` = 30 m

- Irreversible head loss
`(\( h_{\text{loss}} \))` = 4 m

First, calculate the mass flow rate
`(\( \dot{m} \))`:

`\[ \dot{W}_{\text{actual}} = \frac{\dot{W}_{\text{ideal}}}{\eta} \]\[ \dot{W}_{\text{ideal}} = \dot{m} \cdot g \cdot h_{\text{loss}} \]\[ \dot{m} = \frac{\dot{W}_{\text{actual}}}{g \cdot h_{\text{loss}}} \]\[ \dot{m} = \frac{5 \, \text{kW}}{(9.81 \, \text{m/s}^2) \cdot 4 \, \text{m}} \]\[ \dot{m} \approx 0.1279 \, \text{kg/s} \]`

Now, calculate the velocities at the intake and discharge sides:

`\[ A_1 = (\pi D_1^2)/(4) \]\[ A_2 = (\pi D_2^2)/(4) \]\[ v_1 = \frac{\dot{m}}{\rho A_1} \]\[ v_2 = \frac{\dot{m}}{\rho A_2} \]`

Using the density of water
`(\( \rho \approx 1000 \, \text{kg/m}^3 \))`:

`\[ A_1 \approx 0.0038 \, \text{m}^2 \]\[ A_2 \approx 0.00196 \, \text{m}^2 \]\[ v_1 \approx \frac{0.1279 \, \text{kg/s}}{(1000 \, \text{kg/m}^3) \cdot 0.0038 \, \text{m}^2} \]\[ v_2 \approx \frac{0.1279 \, \text{kg/s}}{(1000 \, \text{kg/m}^3) \cdot 0.00196 \, \text{m}^2} \]`

Now, substitute these values into the Bernoulli's equation:

`\[ P_2 - P_1 = (1)/(2) \rho \left(v_2^2 - v_1^2\right) + \rho g \left(h_2 - h_1\right) + h_{\text{loss}} \]\[ P_2 - P_1 = (1)/(2) (1000 \, \text{kg/m}^3) \left((\approx 6.16 \, \text{m/s})^2 - (\approx 33.65 \, \text{m/s})^2\right) + (1000 \, \text{kg/m}^3) \cdot 9.81 \, \text{m/s}^2 \cdot 30 \, \text{m} + 4 \, \text{m} \]`

Now, convert the pressure difference to kPa:

`\[ P_2 - P_1 \approx -335 \, \text{Pa} \]\[ P_2 - P_1 \approx -0.335 \, \text{kPa} \]`

The negative sign indicates that the pressure at the discharge side is lower than the pressure at the intake side. If you need the absolute pressure difference, you can take the absolute value.