Section 2.1.1 - Centrifugal Pump Theory - Predicting The Performance Curves

The choices made in the foregoing design procedures can and should be verified analytically,
the objectives being first to generate the performance characteristic curves for head
and power at constant speed and second to ensure stable behavior of the various systems
in which the pump is to be applied. For the first objective, the solution involves analytical
or empirical approaches: a) at non-recirculating flow conditions; that is, from flow rates Q
somewhat below Q_BEP out to the maximum “runout” flow rate, b) at shut-off (Q = 0) and
low flow, or c) the complete set of curves for a given pump predicted by means of
computational fluid dynamics (CFD).

Generating Performance Curves   The fluid dynamical limitation on the deceleration
of the relative velocity W determines the shape of the head-versus-flow curves. This is
inherent in the choice made for the head coefficient ψ in Figure 12, which sizes the
impeller and is illustrated in Figure 22. The typical situation of zero (or nearly so) inlet
whirl V_θ,1 = 0 means that the ideal head coefficient ψ_i equals the most significant ratio of
the outlet velocity diagram because from Eqs. 15 and 31 (with V_θ,1 = 0):

(63)

(64)

Figure 22 illustrates how specific speed Ω_s affects the BEP value of ψ_i and therefore ψ.
Overall, only a small reduction of W occurs in most impellers. So, at low Ω_s, the low value
of W₁ associated with the small eye relative to the maximum diameter (Figure 9) enables
the outlet velocity diagram (Figure 22a) to have a high value of V_θ,2/U₂. On the other hand,
this ratio drops as Ω_s increases and the eye grows to be as large as the maximum diameter
of the wheel. Figure 22b is the result because the value of ψ at shut-off (about 1/2) is not
based on the one-dimensional concept of velocity diagrams but primarily on the pressure
generated by solid body rotation of stalled (though recirculating) fluid contained within
the impeller. The BEP values of ψ in Figure 22b are consistent with Figure 12 and illustrate
why a high-specific-speed impeller has such a substantial “rise to shut-off” of the
head curve. This is dramatically illustrated in Figures 8–10 of Section 2.1.3 in which the
head curves are normalized to that of the BEP³⁶.

a) Non-recirculating flows.   The BEP efficiency and head can be determined from correlations
for typical pumps or from computation of the losses. Fluid dynamic procedures
described in this section can be used to determine the shapes of the head and power curves
at all flow rates to runout, using the BEP as an anchor point for such computations. For
pumps designed conventionally, beginning with Figure 12, Anderson’s overall (BEP) efficiency
correlation (Eq. 44) as modified in Figure 10 is useful. Other similar charts, especially
Figure 6 in Section 2.1.3, are in widespread use. The breakdown of the losses
involved, as expressed by Eqs. 8–11, is quantified through the development of the three
component efficiencies η_HY, η_m, and η_v in Table 3. All three decrease with decreasing specific
speed—as might be expected from the charts just mentioned.

This can be seen in the η_HY-expression (a) of Table 3 because ψ_i is greater at low Ω_s as
discussed relative to Figure 22. Jekat’s η_HY-expression (b) of the table works surprisingly
well, largely because of the flow effect in Figure 10 (explained there as the “size effect” of

larger relative roughness and clearances in smaller pumps) and because low Ω_s tends to
go hand-in-hand with low flow rate Q.

To compute η_HY at Q ≠ Q_BEP (and, if required, at Q = Q_BEP as well), it is necessary to
go deeper into the prediction of η_HY by developing expressions for the losses noted in Eq.
21, which are basically expansions of the expression for the collector loss coefficient ζ_c²⁶
and for the impeller loss expression (c) of Table 3³⁷. In this expression, the incidence loss
coefficient k can be obtained from cascade data or developed as a combination of a turning
and a sudden expansion loss^5,9,27. The “pipe-type friction factor” f can be increased to
include secondary flow and diffusion losses due to blade loading (or turning³⁸ of the
absolute velocity vector V). The resulting f-value can thus be twice the usual pipe value
associated with the skin friction losses in the passage. (The pipe value of f is found
from the well-known pipe friction chart—Figure 31 in Section 11.1—by substituting a

representative average passage hydraulic diameter D_h = 4A_p/Ã for the pipe diameter d.)
A further increase in this f-value occurs if the impeller is missing one or both rotating
shrouds; that is, it is a semi- or fully-open impeller with blade tip leakage losses appearing
in the main flow stream³⁹. Multiphase flows in pumps often are accompanied by
greater than normal hydraulic losses; for example, increasing the concentration of solids
in the carrier liquid flowing through a slurry pump increases the f-value still further⁴⁰
(see Section 4.2).

Quasi three-dimensional (Q3D) analysis⁴¹ affords an assessment of the secondary flow
and diffusion losses and gives results similar to inviscid three-dimensional (3D) flow
analysis. Q3D analysis starts by solving the 2D meridional (hub-to-shroud) flow field. This
is followed by a series of 2D blade-to-blade inviscid solutions⁴² or approximations thereto²²,
each on a surface of revolution generated by one of the meridional streamlines of the hub-to-
shroud 2D solution and producing results like that of Figure 17. From this, one computes
the diffusion factors (Eqs. 57—59) and decides whether the diffusion losses are
significant—in which case a redesign is in order.

CFD Analysis finally enables one to make an accurate and complete hydraulic performance
evaluation of the impeller and accompanying stationary flow elements. This can be
done at all flow rates and NPSH-conditions for any fluid via a corresponding series of fully
viscous, single- or multi-phase solutions of the Reynolds-averaged Navier-Stokes equations
(as described in Section 2.1.2).

Mechanical efficiencyη_m, as stated earlier, is largely the result of impeller disk friction.
If the drag of bearings and seals is added, as in Eq. (d) of Table 3, the moment coefficient
C_m in the disk friction formula (e) can be increased over known disk friction
values^43,44 to include these effects. (On the other hand, the drag power loss of shaft seals,
though usually quite small, is generally directly proportional to speed. Such losses can
therefore be significant in small pumps running at lower-than-normal speeds.) The C_m
expression given in Formula (f) reflects this adjustment and includes the drag on both
sides of a smooth impeller at the usually high values of Reynolds number R_e for low viscosity
liquids and for a typical clearance ratio s/a = 0.05, where a is the disk radius.
This works well for most impellers: The drag at the ring fits roughly compensates for the
fact that the impeller eye has been cut out of the disk, and so on. (There is very little influence
on C_m of the gap width s between impeller shroud and casing wall, C_m being proportional
to (s/a)^0.1 in general.⁴³ For very small s/a, C_m instead grows as s/a decreases. See
Refs. 43 and 44 for formulas that apply to the entire range of Reynolds number R_e and
clearance ratio s/a.)

The value of C_m can be even larger for semi- or fully-open impellers, if the neighboring
fluid is rotating faster relative to the wall—as is the case with radial-bladed open
impellers. The fluid between a shrouded impeller and adjacent wall, on the other hand,
rotates at half speed⁴³. (In cases where the impeller surface and adjacent wall are both
rough, C_m is larger than just discussed⁴⁴.) Finally, notice in Eq. (h) that very low specific
speed Ω_s produces a dramatically low value of η_m. This drives ψ to the larger values of
Figure 12 at low Ω_s—also dictated by the W-deceleration considerations per Figure 22.
Overall there is a benefit, despite possibly lower η_HY [Eq. (a)] due to the consequently
greater ψ_i and collector loss.

Volumetric efficiencyη_v applies to leakage across impeller shroud rings or “neck rings”
and balancing drums. Eq. (j) in Table 3 is an approximation for the leakage across a typical
ring of a closed-impeller pump, assuming orifice-type flow at a discharge coefficient of 1/2.
Results can vary from this approximation of the more exact method reported by
Stepanoff.⁵ Referring to Figure 2, leakage Q_L occurs at r = r_R, (r_R being approximately 1.2
times r_e) under a pressure difference across the ring of about that of the pump stage. If
the shroud is removed and the open blades are fitted closely to the adjacent wall, as with
open impellers, the consequent leakage from one impeller passage to the next across the
blade tips does not affect η_v, and Eq. (j) should be modified accordingly. Rather, the tip leakage
causes a hydraulic efficiency loss as previously discussed. Finally, as with hm, Eq. (j)
indicates that low-Ω_s pumps have low η_v.

At flow rates Q other than Q_BEP, the analytical methods described previously for computing
the hydraulic efficiency are utilized, together with computation of the inlet and
outlet velocity diagrams, which yield the ideal head and power curves as illustrated in
Figure 6. In this procedure, the slip velocity V_s (Figure 15) applies to the BEP and, at other
flow rates, the exit relative flow angle β_f,2 can be assumed constant. This accords with the
fact that V_s for the narrower active jet at low flow rates must be smaller. A blockage model
for the thickening wakes and narrower active jets that develop as Q is decreased can be
introduced to compute the one-dimensional velocity diagrams, but ignoring this at non-
recirculating flow rates appears not to be serious in determining the shapes of the head
and power curves.

b) Shut-off and low flow. The foregoing analyses apply over that portion of the flow rate
range that does not involve recirculation, as illustrated in Figure 6. The complexity of
recirculation had not been readily handled analytically prior to the use of CFD, and this
has forced pump designers to estimate the low-flow end of the H-Q curve with the help of
empirical correlations. Insightful fluid dynamical reasoning about the physics of the flow
led to useful expressions for the head developed and the power consumed at shut-off. Shut-
off, then, in addition to the BEP, becomes the other anchor point of the head and power
curves; and this—together with the shapes established for these curves at the higher flow
rates—gives the analyst an idea of the intervening shapes.

Shut-off head H_s/o can be viewed as the sum of two effects occurring at Q = 0, each
being represented by a term in this equation:

(65)

or

(66)

where the first term is the centrifugal effect of essentially solid body rotation of the fluid
confined within the impeller; and the second term is the pitot effect of the recirculating
fluid from the impeller that impinges against the volute or diffuser throats which in turn
are connected through stagnant fluid to the exit port of the pump. While the factors k_imp
and k_ex associated with these effects vary with the hydraulic configuration, the values
involved can be estimated as follows: k_imp ≈ 1, as the radial equation of motion³ would indicate
for fluid rotating at Ωr everywhere within the blades, i.e., for r_h,1 < r < r_t,2 (Figure 8).
Thus, as indicated by Eqs. 65 and 66, increasing the minimum radius of the blades at inlet
r_h,1 tends to reduce the shut-off head. However, the presence or absence of fluid swirl in the
region upstream of the impeller blades at shut-off has been found experimentally to affect
the value of k_imp in surprising ways—sometimes increasing it above unity in such a way as
to minimize the effect on shut-off head of any non-zero value of r_h,1. The value of k_ex
depends on (r₃ - r₂)/r₂, or “Gap B” and other features of the impeller exit and collector
geometry. It is usually in the range 0.2 ± 0.1, any change in the geometry that increases
the shut-off power coefficient (see below) raising k_ex by driving more recirculating flow
from the impeller against the volute or diffuser throats. Thus the shut-off head coefficient
ψ_s/o (Eq. 66) for typical radial-flow pumps generally exceeds 1/2, the value of 0.585 being
advanced by Stepanoff⁵. Estimates for ψ_s/o are also indicated in Figure 22b.

Shut-off power consumption P_s/o includes disk, bearing, and seal drag power P_D and
that which drives the recirculation P_recirc. The latter is generally dominant by far. From
similarity arguments (Eq. 33), the shut-off power coefficient

(67)

is a constant for a given pump geometry. Mockridge, in a discussion attached to an ASME
paper by Stepanoff, reasoned that a wider impeller (larger b₂ at the same diameter D₂)
would recirculate more fluid at shut-off and therefore have a higher value of this coefficient.
His correlation is shown in Figure 23 and is probably the most significant quantitative
result available for predicting the performance of centrifugal pumps at shut-off
conditions short of implementing a full CFD solution of the impeller and accompanying
stationary flow elements³⁶.

The design task therefore resolves itself into an iteration between an efficient geometry-
generating scheme and a rapid CFD flow and performance analysis of the geometry
resulting from each iteration⁴⁵. This is especially useful if a nontraditional geometry is

involved, or if an efficient design is sought that will produce a desired performance curve
shape. Nevertheless, many turbomachinery designers can make more rapid and valid
judgments about their respective classes of machines through the time-honored iteration
between a proprietary direct or inverse design and a convenient inviscid performance prediction
scheme.^41,46 “Inverse” means that the blade shape emerges from an initial specification
of certain desired fluid velocity component distributions.¹⁷ This concept has been
expanded through more formal geometry optimization employing neural networks.⁴⁷ Thus,
designers have developed reliable diffusion criteria (computed, for example, from Eqs. 59a
and 59b) for interpreting the acceptability of the free-stream relative velocity distributions
W_s and W_p on the blade surfaces (Figure 17) produced by the Q3D blade-to-blade solutions⁴³.
Because CFD codes solve the actual viscous flow field, the boundary condition on the blade
surface is zero relative velocity. This can be at least partly overcome by displaying the
CFD-distributions of pressures on the blade surfaces, the interpretation of which would
require knowledge of the corresponding criteria for these pressures⁴⁸. Also, the velocities
at the edge of the boundary layer could be extracted from the CFD solution and displayed
in familiar terms.A useful design approach for the present may therefore be to a) produce
the final design by the more traditional methods and b) predict the performance curves
via CFD⁴⁹.

The choices made in the foregoing design procedures can and should be verified analytically,
the objectives being first to generate the performance characteristic curves for head
and power at constant speed and second to ensure stable behavior of the various systems
in which the pump is to be applied. For the first objective, the solution involves analytical
or empirical approaches: a) at non-recirculating flow conditions; that is, from flow rates Q
somewhat below Q_BEP out to the maximum “runout” flow rate, b) at shut-off (Q = 0) and
low flow, or c) the complete set of curves for a given pump predicted by means of
computational fluid dynamics (CFD).

Generating Performance Curves   The fluid dynamical limitation on the deceleration
of the relative velocity W determines the shape of the head-versus-flow curves. This is
inherent in the choice made for the head coefficient ψ in Figure 12, which sizes the
impeller and is illustrated in Figure 22. The typical situation of zero (or nearly so) inlet
whirl V_θ,1 = 0 means that...