Calculation of Pressure Difference Across a Pump in a Piping System

What is the pressure difference across the pump in a piping system?

Given:- Pump power [tex](\\( \\dot{W} \\))[/tex] = 5 kW- Pump efficiency [tex](\\( \\eta \\))[/tex] = 80% (or 0.8)- Pipe diameters: [tex]\\( D_1 = 0.07 \\)[/tex] m (intake side), [tex]\\( D_2 = 0.05 \\)[/tex] m (discharge side)- Free surface elevation difference [tex](\\( h_2 - h_1 \\))[/tex] = 30 m- Irreversible head loss [tex](\\( h_{\\text{loss}} \\))[/tex] = 4 m

Calculation of Pressure Difference

To calculate the pressure difference across the pump in the piping system, we first need to determine the mass flow rate and velocities at the intake and discharge sides. Given the pump power, efficiency, pipe diameters, elevation difference, and head loss, we can apply Bernoulli's equation to find the pressure difference.

First, we calculate the mass flow rate using the pump power and efficiency:

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

Next, we determine the velocities at the intake and discharge sides:

[tex]v_1 \\approx \\frac{0.1279 \\, \\text{kg/s}}{(1000 \\, \\text{kg/m}^3) \\cdot 0.0038 \\, \\text{m}^2}[/tex]

[tex]v_2 \\approx \\frac{0.1279 \\, \\text{kg/s}}{(1000 \\, \\text{kg/m}^3) \\cdot 0.00196 \\, \\text{m}^2}[/tex]

Then, we substitute these values into the Bernoulli's equation to find the pressure difference:

[tex]P_2 - P_1 \\approx -0.335 \\, \\text{kPa}[/tex]

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.