Dimensional analysis of centrifugal pumps?M. For centrifugal pumps of a given design (that is, those that are geometrically similar), there exists a functional relationship of the form: 1where P is the power required to drive the pump (with dimensions ML2/T3), 0 is the volumetric flow rate (L3/T), p is the density of the fluid being pumped (M/L3), N is the rotational speed of the impeller (T-1), and D is the impeller diameter (L).

By choosing p, N, and D as the primary quantities, we wish to establish two groups, one for Q, and the other for P, that can be used for representing data on all pumps of the given design. Verify that the group for Q is Ill = 0./NO3, and determine the group I12 involving P.

A one-third scale model pump (D1 = 0.5 ft) is to be tested when pumping 01 = 100 gpm of water (p1 = 62.4 Ibm/ft3) in order to predict the performance of a proposed full-size pump (D2 = 1.5 ft.) that is intended to operate at N2 = 750 rpm with a flow rate of Q2 = 1,000 gpm when pumping an oil of density p2= 50 Ibm/ft3. If dynamical similarity is to be preserved (equality of dimensionless groups):

(a) At what rotational speed N1 rpm should the scale model be driven?

(b) If under these conditions the scale model needs P1 = 1.20 HP to drive it, what power P2 will be needed for the full-size pump?

a) Dimensional analysis is useful in the design of centrifugal pumps. In this problem we will try to understand the relation between different operating conditions and the physical dimensions of the pump and obtain a relationship between the different parameters using dimensional analysis. The pressure rise across a pump Ap (this term is proportional to the "head" developed by the pump) may be considered to be affected by the fluid density p, the angular velocity w, the impeller diameter D, the volumetric rate of flow Q, and the fluid viscosity p. Find the pertinent dimensionless groups, choosing them so that Ap, Q, and p each appear in one group only.

b) Find 1 similar expressions replacing the pressure rise first by the power input to the pump Ws, and c) then by the efficiency of the pump g.

y (Q, P, p, N, D) = 0