L

= representative linear dimension

g

= specific weight = pg

m

= dynamic viscosity

Au/Dy

u

= representative velocity

s

= surface tension

p

= density

g

= acceleration due to gravity

Noting that the kinematic viscosity, n, is given by m/p, substituting the above terms in Equation 1 and making it dimensionless by dividing by the inertial force, we obtain ai a2 a3 a4

a5 a5 a5 a5

gravitational force

2 m/p viscous force

s/p surface tension force

Because the proportionality factors a1( a2, and so on, are the same in model and prototype, complete dynamic similarity is achieved if the values of each dimensionless group, E, F, R, and W are equal in model and prototype. In practice, this is difficult to achieve. For example, to have Fmodel = F^iostÂ» and Rmodel = Rprototype requires either a 1:1 "model" or a fluid of very low kinematic viscosity in the reduced-scale model. Hence, the accepted approach is to select the predominant force and then design the model according to the appropriate dimensionless group. The influences of the other forces become secondary and are called scale effects.11,12

Froude Scaling Pump intake models are generally designed and operated using Froude similarity because the flow is controlled by gravitational and inertial forces. The Froude number is, therefore, made equal in model and prototype:

where m, p, and r denote model, prototype, and ratio between model and prototype.

In modeling a pump intake sump to study the formation of vortices, it is important to select a reasonably large geometric scale to achieve large Reynolds numbers. At a large Reynolds number, energy loss coefficients usually behave asymptotically with Reynolds number. Hence, with Fr = 1 and a sufficiently high Reynolds number, the Euler number E will be equal in model and prototype. This implies that flow patterns and loss coefficients may be considered similar in model and prototype. From Equation 3, the velocity, flow, and time scales are

For example, if a model-to-prototype scale of 1:10 is used, a prototype velocity of 1.0 ft/s (0.3 m/s) becomes a model velocity of (1 ^ 10)172 = 0.32 ft/s (0.09 m/s). In the model, all physical dimensions of the pump bays and approach channel are scaled to the ratio 1:10. Submergence, being a linear dimension, is also scaled to 1:10. A flow of 100 gpm (0.006 m3/s) in the model corresponds to a flow of 100 X 102 5 = 31,623 gpm (2 m3/s) in the prototype. Similitude parameters and laws are treated in detail in References 11 and 12.

Similarity of Vortices The fluid motions involving vortex formation in pump sumps have been studied by several investigators. It can be shown by principles of dimensional analysis that the dynamic similarity of fluid motion that could cause vortices at an intake is governed by the following dimensionless parameters:

d ud u2d

s n s/p where average axial velocity at the bell entrance circulation contributing to vortexing diameter of the bell entrance submergence at the bell entrance

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