Section 2.1.1 - Centrifugal Pump Theory - Scaling And Similtude

When a set of characteristic curves for a given pump stage is known, that machine can
be used as a model to satisfy similar conditions of service at higher speed and a different
size. Scaling a given geometry to a new size means multiplying every linear dimension
of the model by the scale factor, including all clearances and surface roughness
elements. The performance of the model is then scaled to correspond to the scaled-up
model by requiring similar velocity diagrams (often called “velocity triangles”) and
assuming that the influences of fluid viscosity and vaporization are negligible. The proportions
associated with Eqs. 27, 29, and 32 illustrate this. The blade velocity U (Eq. 30)
varies directly with rotative speed N or angular speed Ω—and directly with size, as
expressed by the radius r. For the velocity V (or W) to be in proportion to U, the flow rate
Q must therefore vary as Ωr³; hence, the “specific flow”Q_s must be constant (Eq. 28). Further,
as the head is the product of two velocities, it must vary as Ω²r²; hence, the head
coefficient ψ must be constant (Eq. 31). Finally, as power is the product of pressure-rise
and flow rate, shaft power P_s must vary as ρΩ³r⁵; hence, the power coefficient must be
constant.

and

(28)

Figure 7a is the result of following these similarity rules for a given pump that undergoes
a change in speed from full speed to half speed without a change in size. The similar
Q at half speed for a given Q at full speed is half that at full speed. At each such halfspeed
Q-value, the head ΔH is accordingly one-fourth of its full-speed value and the efficiency
η is unchanged. One can avoid replotting the characteristic curves in this manner
for every change in speed (and size) by expressing them nondimensionally in terms of Q_s,
ψ, η, and . They then all collapse on one another as illustrated in Figure 7b. Note that a
change in pump geometry or shape of the hydraulic passageways destroys this similitude
and necessarily produces a new set of curves—shaped differently but similar to each other.

Similitude enables the engineer to work from a single dimensionless set of performance
curves for a given pump model. This is a practical but special case of the more

general statement that pump performance as represented by efficiency, head, and power,
is more generally expressed in terms of the complete physical equation as follows:

(34)

where {li} is the infinite set of lengths that defines the pump stage geometry. A common
group of these lengths is illustrated in Figure 8. Dimensionlessly, Eq. 34 becomes

(35)

where the dimensionless quantities containing flow rate, viscosity and NPSH are respectively
defined as follows:

and

• {G_i} = {l_i/r} defines the dimensionless geometry or shape.
• {2-Φ} = the dimensionless quantities arising from the set of fluid, thermal, vaporization,
  and heat transfer properties {2-ph} that influence the flow of two-phase vapor and liquid.
  These quantities come into play when the NPSH is low enough for such flow to be
  extensive enough to influence pump performance.
• {Σ}= the dimensionless quantities arising from the set of properties {S} associated with
  entrained solids and emulsifying fluids that affect the performance of slurry pumps and
  emulsion pumps. Similarly, {Γ_p} arises from {g_p} for entrained gases.

When a set of characteristic curves for a given pump stage is known, that machine can
be used as a model to satisfy similar conditions of service at higher speed and a different
size. Scaling a given geometry to a new size means multiplying every linear dimension
of the model by the scale factor, including all clearances and surface roughness
elements. The performance of the model is then scaled to correspond to the scaled-up
model by requiring similar velocity diagrams (often called “velocity triangles”) and
assuming that the influences of fluid viscosity and vaporization are negligible. The proportions
associated with Eqs. 27, 29, and 32 illustrate this. The blade velocity U (Eq. 30)
varies directly with rotative speed N or angular speed Ω—and directly with size, as
expressed by the radius r. For the velocity V (or W) to be in proportion to U, the flow rate
Q must therefore vary as Ωr³; hence, the “specific flow”Q_s must be constant (Eq. 28). Further,
as the head is...