The hydraulic geometry or shape of a pump stage can in principle be chosen for given values
of the other independent variables in Eqs. 34 or 35 so as to optimize the resulting performance;
for example, to maximize the best efficiency η_BEP under certain conditions on the head
and power.Two such conditions that are common are a) no positive slope is allowed anywhere
along the ΔH-vs.-Q curve of Figure 7 (called the “no drooping nor dip” condition) and b) the
maximum power consumption must occur at the BEP (often called the “non-overloading”
condition). A fundamental and generally typical pumping situation involves a) negligible
influence of viscosity, (that is, high Reynolds number) b) the absence of two-phase fluid effects,
(that is, the existence of sufficient NPSH or τ) and c) the absence of entrained gases, solid particles
and emulsion-related substances in the fluid. In this situation, Eq. 35 has one remaining
significant independent variable; namely, the specific flow Q_s, which in the definitions of
Eq. 36 contains the volume flow rate Q, the pump speed Ω, and the characteristic radius r₂.
Most users don’t know the size of the pump stage a priori; so, r₂ is eliminated by replacing Q_s
in Eq. 35 with a new quantity that is the result of dividing the square root of Q_s by the -power
of the head coefficient ψ. Thus, from the definitions just given of Q_s and ψ, one arrives at 3/4 the
specific speed Ω_s as the independent variable in terms of which the geometry is optimized³:

(37)

For convenience, specific speed is usually expressed in terms of the conventional quantities
N, Q, and ΔH that correspond to the factors in Eq. 37, which quantities are expressed

in the units commonly used commercially. For example, forms found in the United States
and in Europe and the relationship of these to the truly unitless “universal specific speed”
Ω_s defined in Eq. 37 are as follows:

Thus,

(38c)

Rotor Shape as a Function of Specific Speed   Optimization of pump hydraulic geometry
in terms of the BEP specific speed has taken place empirically and analytically
throughout the history of pump development. An approximate illustration of the results
of this process for pump rotors or impellers is shown in Figure 9. Not only does the geometry
emerge from the optimization process but also the head, flow, and power coefficients
for each shape as well. Approximate values for the optimum BEP head coefficient ψ are
shown on the figure. The actual rotor diameter can then be deduced as noted—from the
ψ-definition of Eq. 31. The relationship among the various rotors is illustrated in the
figure—assuming that they all have the same speed and head; that is, as one moves along

the abscissa or specific speed axis of Figure 9, only the flow rate is changing as far as the
illustrations of the rotors are concerned. As would be expected, therefore, high-specific-
speed impellers need to have large passages relative to their overall diameter. Conversely,
the passages become smaller as the centrifugal range is traversed toward lower specific
speed, the low limit effectively being Ω_s = 0.2 (N_s ≈ 500), at which point an open impeller
of the “Barske” type is illustrated as being typical for this part of the range. Yet centrifugal
impellers are found at values of Ω_s as small as 0.1, at which point the impeller is sometimes
simply a disk with drilled holes for passages, as illustrated.

For simplicity, the common “double-suction” arrangement, wherein two centrifugal
impellers are effectively configured back-to-back, is not shown in Figure 9. Double-suction
impellers commonly discharge into a single, surrounding collector designed to accommodate
twice the flow rate of one side. Because this collector can be fed by an equivalent
single-suction impeller, the specific speed of the whole pump stage is computed using the
combined flow rate of the two sides of the double-suction impeller. While this is common
practice in the United States, the practice in some other parts of the world is to compute
the specific speed of such a pump stage using the flow rate of only one side of the impeller.
The former approach is convenient for associating the exit geometrical features and flow
behavior of the stage with the magnitude of the specific speed, while the latter favors the
suction or inlet characteristics.

At the lower end of the specific speed range shown in Figure 9, rotodynamic pumps (that
is, centrifugal pumps, in which category mixed- and axial-flow geometries are generally
included) would be too low in efficiency to be practical. Rather, rotary positive displacement
pumps take over, there being a transition through the drag pump domain. Sometimes
called a regenerative or periphery pump, the drag pump is actually a rotodynamic machine,
developing head peripherally around the impeller through successive passes radially
through the blades on both sides until a barrier is reached at some point on the periphery,
where the fluid is then discharged.⁶

The screw pump, on the other hand, is a truly positive displacement (rotary) machine.
It can have two, three, or more meshing screws and can move large quantities of fluid—
both single- and multiphase—against a large pressure difference Δp, giving it a specific
speed range that extends well into centrifugal pump territory. Screw pump rotors are
often configured back-to-back in a double-suction arrangement, as this balances the considerable
axial thrust of these rotors. Not shown is the progressive cavity pump, which
has a single screw surrounded by an elastomer sealing member.

Lower flow rates are readily accommodated by the vane pump, whereas gear pumps
handle a higher range of pressure differences at such flow rates. Finally, extremely high
pressures are produced by reciprocating pumps, the specific speed range of which extends
off the figure on the left.

Positive displacement pumps appear in Figure 9 in order to provide perspective. The
concept of specific speed is not generally applied to these machines, because a given positive
displacement pump can have such a wide range of pressure-rise capability at a chosen
flow rate and speed as to make it difficult to associate a given rotor geometry with a
particular value of specific speed. On the other hand, a unique rotodynamic pump geometry
is readily associated with the specific speed of the BEP of such a machine.

Performance of Optimum Geometries   Figure 9 enables one to easily identify the pump
stage types associated with required pumping tasks in terms of head, flow rate, and rotative
speed. Beyond this general picture is the related performance of a real pump geometry
in a real fluid. Although, for centrifugal pumps, the specific speed has the major effect on
performance, the available NPSH and the viscosity of the pumpage also have an influence.
These are evident in the following formal statement of the efficiency of an optimized pump
(cf Eq. 35)

(39)

where the radius r₂, representing the size, has been eliminated from the other variables in
Eq. 35 by introducing the suction specific speed Ω_ss and the head-flow Reynolds number
R_e,H,Q, which are defined in Eqs. 40 and 41:

where, the common form of the suction specific speed, called N_ss, is given in commercial
U.S. units by (Eq. 42)

(42)

Size Effect   For sufficiently high NPSH (or sufficiently low suction specific speed) and
low viscosity (or high Reynolds number), real pumps also possess a strong size effect on
efficiency. This is because, in normal manufacturing processes, the clearances d preventing
internal leakage Q_L (for example, past the impeller sealing rings in Figure 2) do not
scale up in proportion to the size (represented by r₂), nor do the surface roughness heights e.
Thus, a larger pump tends to be more efficient. Strictly speaking, however, the geometry
of the larger pump is not the same as that of the smaller pump, and this forces one to
modify Eq. 39 by reintroducing two of the length ratios G_i that were part of the set {G_i}
in Eq. 35 which characterize the hydraulic shape of the machine. Thus, Eq. 39, revised to
reflect these realities, becomes

(43)

A study of a large number of commercial centrifugal pumps by H. H. Anderson⁷ has
quantified Eq. 43 for such machines. These pumps were all operating in water and had

sufficient NPSH for performance not to be influenced by Ω_ss. The results are given by
Eq. 44, which is plotted in Figure 10 for the quantity X = 1:

Eq. 44 is a combination of separate relationships developed by Anderson for efficiency and
speed as functions of flow rate⁷. Included is a correction for specific speed that is too conservative
for N_s,(US) > 2286 or Ω_s greater than about unity.With this qualification, Figure 10 is
a useful representation for centrifugal pumps and is often as far as many users go in determining
the performance of these machines.

Viscosity Effects   Centrifugal pump geometries have not generally been optimized versus
Reynolds number—often because the effect on hydraulic shape is not very great except
for the highest viscosities of the pumpage, and a given application can sometimes experience
a substantial range of viscosity. Studies of conventional centrifugal pumps over a
range of Reynolds number have been combined in nomographic charts in the Hydraulic
Institute Standards, which yield correction factors to the head, efficiency, and flow rate of
the BEP of a water (low viscosity) pump in order to obtain the BEP of that pump when
operating at higher viscosity⁸. Figure 11 is a presentation of these correction factors in
terms of the head-flow Reynolds number. Values of the head correction factor C_H are also
given for flow rates other than that of the BEP, enabling one to produce a completely corrected
curve of head vs. flow rate. Strictly speaking, in view of Eq. 43, each pump geometry
has a unique set of such correction factors, yet the data presented in Figure 11 have
been widely utilized as reasonably representative of conventional centrifugal pumps.

NPSH Effects   In many cases, the available NPSH is low enough, or the suction-specific
speed Ω_ss at which the pump stage must operate is high enough for significant two-phase
activity to exist within the impeller. This is to be expected in centrifugal impellers of water

pumps if the available Ω_ss is greater than about 3 to 4 (or N_ss,(U.S.) = 8,000 to 11,000). In
such a case, Ω_ss and the vaporization quantities {2-Φ} in Eq. 35 dictate a profound change
in the impeller geometry into that of an inducer. The inducer has an entering or “eye”
diameter that is significantly enlarged—together with tightly wrapped helical blading.
Often the inducer is a separate stage that pressurizes the two-phase fluid as needed to
provide a sufficiently low value of Ω_ss at the entrance of the more typical impeller blading
that is immediately downstream of the inducer. If the two-phase fluid is near its thermodynamic
critical point, the {2-Φ} operate to greatly reduce the amount of two-phase
activity within the pump. (At the critical point, the liquid and gas phases are identical,
and therefore both have the same specific volume.) An example is the pumping of liquid
hydrogen, for which an inducer is unnecessary until much higher values of Ω_ss are
reached. Moreover, inducers—typically limited to Ω_ss-values of about 10 (N_ss,(U.S.) = 27,000)
in water—can, at sufficiently low tip speed, operate at zero NPSH, which corresponds to
an infinite value of Ω_ss⁹.

Pumping Entrained Gas   In addition to the liquid’s own vapor (which is the gas involved
in the NPSH-effects discussion), many pumping applications deal with a different gas; that
is, a different substance from the liquid being pumped. The effects of this gas on performance
arise from a) the volume flow rate of the gas at the inlet, b) the pressure ratio of
the pump, which determines how far into the impeller this gas volume persists; that is,
how much it gets compressed, and c) how much of the gas dissolves in the liquid as the
pressure increases within the pump, which depends on both the solubility and the degree
of agitation of the fluid produced by the pump. The set of fluid properties associated with
these gas-handling phenomena are represented by {g_p} in Eq. 34, the dimensionless form
(Eqs. 35 and 43) of this set being {Γ_p}. Generally, for typical commercial centrifugal pumps,
the performance under such conditions usually manifests itself as a loss of pressure rise,
which is reasonably stable up to an inlet volume flow rate fraction of gas to liquid of 0.04
to 0.07¹⁰. Inducers can handle larger inlet volume fractions of gas, and, under Dalton’s law
for partial pressures, the liquid’s own vapor also occupies the volume of the gas bubbles.
Single and multistage centrifugal pumps have been built that handle far greater gas volume
than these single-stage values^11,12; moreover, multiphase rotary positive displacement
screw pumps can handle gas volume fractions up to 1 (100% gas)¹².

Effects of Slurries and Emulsions   Finally there is the influence of the dimensionless
quantities {g} in Eq. 43. Impeller and casing design are altered so as to reduce wear-
producing velocities if the pumpage is a slurry of solids contained in a carrier liquid.
(Slurry pumps are usually single-stage machines with a collector or volute casing surrounding
the impeller.) This usually means a smaller impeller eye diameter (which, as can
be seen in Figure 6, reduces the inlet relative velocity W₁,) and a larger radial distance
from the impeller to the surrounding volute because the circumferential velocity component
V_θ of the fluid emerging from the impeller (also seen in Figure 6) slows down with
increasing radial position and is then lower in the volute passageway¹³. Performance also
is altered, depending on the composition and concentration of the slurry. These are complicated
non-Newtonian flows and are covered in detail elsewhere in this book in conjunction
with a thorough treatment on solids-handling pumps. Emulsions are another
example of such flows, many of which are destroyed by excessive local shear in the fluid.
For this reason, screw pumps are sometimes utilized for emulsions rather than oversized,
slow-running centrifugal pumps. Except for thin layers of the fluid at the clearances, most
of the flow in a screw pump experiences very little shear in comparison to the flow through
a centrifugal pump.

Electromagnetic Effects   Not appearing in Eq. 43 are quantities associated with electromagnetic
phenomena. For example, electric current flowing radially outward through
fluid contained in an axially directed magnetic field is capable of producing a rotating flow.
Called a hydromagnetic pump, this device is therefore “centrifugal,” yet it has no moving
parts. Such pumps have been used for liquid metals and could be made reasonably efficient
for any pumpage with high conductivity.

The hydraulic geometry or shape of a pump stage can in principle be chosen for given values
of the other independent variables in Eqs. 34 or 35 so as to optimize the resulting performance;
for example, to maximize the best efficiency η_BEP under certain conditions on the head
and power.Two such conditions that are common are a) no positive slope is allowed anywhere
along the ΔH-vs.-Q curve of Figure 7 (called the “no drooping nor dip” condition) and b) the
maximum power consumption must occur at the BEP (often called the “non-overloading”
condition). A fundamental and generally typical pumping situation involves a) negligible
influence of viscosity, (that is, high Reynolds number) b) the absence of two-phase fluid effects,
(that is, the existence of sufficient NPSH or τ) and c) the absence of entrained gases, solid particles
and emulsion-related substances in the fluid. In this situation, Eq. 35 has one remaining
significant independent variable; namely, the specific flow Q_s, which in the definitions of
Eq. 36 contains the volume flow rate