Question

The power P to drive an axial flow pump depends on the following variables: Density of the fluid, p; Angular speed of the rotor, omega; Diameter of the rotor, D; Head, H; Volumetric flow rate, Q. Use dimensional analysis to determine a set of dimensionless groups that can be used to correlate data. Show all your work. (11%) A model scaled to one-third the size of the prototype has the following characteristics: p_m = p_p; omega_m = 96 rads; D_m = 10 cm: H_m = 3 m; q_m = 1m^3/s; p_m = 21 kW. Given that full-size pump is to run at 32 rad/s. What is the power P_p required for this pump? (3%) What head H_P will the pump maintain? (3%) What will the volumetric flow rate Q_p be? (3%)

Answer 1

Power in fluid flow is calculated using the equation (P₁ + ½ pv² + pgh) Q = power. Dimensional analysis helps determine dimensionless groups. To find the power, head, and volumetric flow rate for a full-size pump, we use the model scaled values and given values.

Step-by-step explanation:

Power in fluid flow can be calculated using the equation (P₁ + ½ pv² + pgh) Q = power, where P₁ is the power associated with pressure, pv² is the power associated with velocity, and pgh is the power associated with height. To determine a set of dimensionless groups, we use dimensional analysis. From the given variables, we can form the following dimensionless groups: P/ρω³D², H/D, Q/D³.

To find the power P_p required for the full-size pump, we use the model scaled values and the given values as follows:
P_m/((p_m * ω_m³ * D_m²) = P_p/((p_p * ω_p³ * D_p²). Solving for P_p, we substitute the given values to obtain the power required for the full-size pump.

To find the head H_p that the pump will maintain, we use a similar approach:
H_m/D_m = H_p/D_p. Substitute the given values to find H_p.

Lastly, to find the volumetric flow rate Q_p, we can use the equation:
(Q_p * D_p³)/(Q_m * D_m³) = 1. Substitute the given values to find Q_p.