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Centrifugal Pumps

vonJohann Friedrich Gülich

1. Auflage

Springer-Verlag Berlin Heidelberg 2010

Verlag C.H. Beck im Internet:www.beck.de

ISBN 978 3 642 12823 3

Zu Inhaltsverzeichnis

schnell und portofrei erhältlich bei beck-shop.de DIE FACHBUCHHANDLUNG

536 9 Hydraulic forces

9.2.9 Unsteady axial thrust

During start-up of a pump, transient axial forces exceed the forces in normal op-eration because it takes a few seconds till fluid rotation in the impeller sidewall gaps is fully developed, while the pressure rise in the impeller is proportional to the square of the speed with virtually no inertia effects. When starting a vertical pump, the axial thrust can thus be acting upwards during a short period. This fact needs to be taken into account when selecting the axial bearing.

If a vertical pump is started with an open discharge valve, the axial thrust FI due to the momentum at very high flow can also be acting upwards – in particular with high specific speeds.

The spectrum of unsteady axial forces exhibits low- and high-frequency com-ponents similar to the spectrum of pressure pulsations. In practice such axial force fluctuations rarely cause problems, with the possible exceptions of high-energy pumps operating at partload if gap A was selected too large.

Measurements on multistage pumps in [B.20] gave kax = 0.005 at q* = 0 and kax = 0.0025 at q* = 1, where kax is the coefficient of the unsteady axial thrust per stage according to Eq. (T9.2.15). It is defined as RMS-value in the frequency range f = (0.2 to 1.25) n/60. About half of these values were measured in the range f < 0.2 n/60.

9.3 Radial forces

9.3.1 Definition and scope

The radial force acting on an impeller (the “radial thrust”) must be known in order to calculate bearing loads, shaft stresses and shaft deflection. Fundamentally, ra-dial forces are generated when the circumferential distribution of the static pres-sure p2 at the impeller outlet is non-uniform. Flow asymmetries in the collector as well as a rotationally non-symmetric impeller inflow can thus create radial forces.

Since the pressure distribution at the impeller outlet is unsteady, its integration over the circumference yields a time average, the static radial thrust, as well as a spectrum of unsteady radial force components, which are called “dynamic radial thrust”. Thus the dynamic radial thrust represents hydraulic excitation forces lead-ing to forced vibrations, which are discussed in Chap. 10.7.

The radial forces created by time-dependent pressure distributions at the outlet of a closed impeller are influenced by various physical effects: Excitation forces with static and dynamic components which are independent of rotor vibrations:

1. Non-uniform flow in the collector acting on the impeller outlet width and on the projection of the shrouds unless these are perpendicular to the rotor axis.

2. Non-uniformities of the flow in the impeller sidewall gaps, which can be cre-ated by the pressure distribution in the collector, as well as asymmetries in the

9.3 Radial forces 537

leakage flow if the impeller runs with some eccentricity in the annular seal (as is usually to be expected). Perturbations of rotational symmetry are also created if the impeller shrouds and casing walls are not machined.

3. If the impeller runs with some eccentricity in the annular seal due to shaft de-flection (for example, sag caused by the rotor weight) the pressure distribution in the seal is non-uniform; hence a static force in the annular seal is created as discussed in Chap. 10.6.2.

Unsteady reaction forces created by rotor vibration:

• Hydraulic impeller interaction, Chap. 10.6.3 • Forces in annular seals, Chap. 10.6 • Forces in open (axial and semi axial) impellers created by clearance variations

due to orbiting of the rotor (“Alford effect”, Chap. 10.7.3).

All of these effects cannot be easily separated when measuring the radial forces, nor are they amenable to an exact theoretical prediction which would need model-ing the three-dimensional flows in impeller and collector. Therefore, empirical coefficients are commonly used for estimating the radial thrust; these were de-rived from measurements and thus correspond to statistical data. Mostly the pub-lished radial thrust coefficients are valid for impellers with annular seals built with common clearances; they represent a combination of the effects (1) to (3) above. Two definitions of radial thrust coefficients (kR and kRu) are in use:

tot22

RR bdHg

Fk

ρ= (9.6)

tot2222

RRu

bduF2k

ρ= (9.7)

The actual figures of kR and kRu differ by the factor of the pressure coefficient; that is, kRu = ψ×kR applies. FR is the radial force and b2tot is the impeller outlet width including the wall thickness of rear and front shrouds. The coefficient kR can also be interpreted as kR = ΔpLa/(ρ×g×H) where ΔpLa represents the average pressure difference which acts on the projected surface d2×b2tot.

Unless qualified by an additional subscript (“dyn” for unsteady thrust compo-nents, or “tot” for the sum of static and unsteady components), all subsequent ref-erences to kR only imply the static (i.e. steady) radial thrust coefficient. As dem-onstrated by various tests, the radial thrust coefficients are virtually independent of rotor speed and Reynolds number in the range of practical interest; for geomet-rically similar pumps they are independent of the impeller size, e.g. [10.22]. For a given pump the radial thrust depends primarily on the flow rate ratio q*.


9.3.2 Measurement of radial forces

Several methods have been developed for measuring the radial forces acting on an impeller. Their principal characteristics are reviewed below since testing method and conditions should be appreciated in order to be able to correctly assess the significance of experimental radial thrust data. The tests are always done on sin-gle-stage pumps. Details on instrumentation and test data evaluation can be found in the quoted literature as well as in [9.7] and [9.8]. Integration of the pressure distribution: The radial force can be derived by in-tegrating (over the circumference) the static pressure distribution measured at the impeller outlet by means of probes and/or pressure tappings in the casing, e.g. [9.9] and [9.10]. If only pressure tappings in the casing walls are used, this method is quite simple since no special instrumentation is required. The accuracy is moderate depending on the number of pressure tappings installed. If the static pressure is recorded in a tight grid of measuring points covering the entire impel-ler circumference, the effective radial thrust can be determined without falsifica-tion through the forces generated in the annular seals. Measurement of bearing forces: In most of the investigations, the bearing forces were measured by means of strain gages or force transmitters. Strain gages are applied to brackets which hold the bearing. The brackets must be flexible enough to allow the radial forces to produce sufficient elastic strain in order to get accu-rate measurements. Such flexible elements reduce the eigen frequencies of the test rig; the test speeds must therefore be selected such that the measurements are not falsified by resonances. This can be verified by mounting a known mechanical unbalance in lieu of the impeller: in the range where the forces recorded increase proportional to the square of the rotor speed, there is no falsification due to reso-nances. The mechanical unbalance serves also for calibrating the test rig.

The measurement of the bearing forces yields the resultant of all forces acting on the rotor. It is not possible to separate the radial thrust from forces created by the seal – unless a radial seal is used instead of an annular seal. The radial thrust determined in this way depends on the characteristics of the annular seal, in par-ticular on the clearance and type of surface of the seal (plain or serrated). Measurement of the shaft deflection: If the shaft deflection is measured by proximity probes, the radial forces acting on the impeller can be determined after proper calibration of the test rig and taking into account the shaft deflection under the rotor weight. The calibration can be done by means of a mechanical unbalance or static forces (e.g. dead weights). As when measuring bearing forces, a dynamic calibration is important in order to avoid falsification of the test results by reso-nance effects. Due attention must be paid to the bearing clearances because these impair the accuracy of the measurements. This testing method is relatively simple but not very accurate (bearing clearances, run-out and dynamic characteristics of the test rig). Only the resultant of radial thrust and annular seal forces can be measured (as with bearing force measurements). Measurement of shaft stresses: All forces and moments resultant from impeller and seals can be measured by means of strain gages applied to the shaft close to


the overhung impeller, [9.11] and [10.22]. This complex and expensive method may be used when the interest is focused on unsteady forces. The calibration is performed by a mechanical unbalance. The dynamic characteristics of the test rig must be checked as described above to avoid resonances. Force measurements by means of magnetic bearings: Active magnetic bear-ings can be used to measure the forces acting on a rotor. The rotor of the test rig is centered by two magnetic fields which are generated by electromagnets. The elec-trical current in the magnets is controlled by proximity probes and electronics so that the rotor is held in its centric position. The bearing forces can be determined from the measured current and the air gap between rotor and magnet (i.e. from the signal of the proximity probes), [9.12]. An advantage of this method is that the test pump can be built with sufficient stiffness to avoid resonances.

9.3.3 Pumps with single volutes

The physical mechanisms which generate radial thrust shall be discussed in detail using the example of a single volute dimensioned according to the conservation of angular momentum as per Chaps. 3.7 and 4.2. In this context consider Fig. 9.20, where flow conditions and pressure distributions in a developed volute are sket-ched for three flow rates: (1) Near the best efficiency point, the flow angle at the impeller outlet matches the cutwater camber angle z. The flow deceleration largely follows the conserva-tion of angular momentum and the pressure distribution is nearly uniform except for a local perturbation induced by the flow around the cutwater. (2) At partload (q* << 1) the flow area in the volute is too large at every point on the circumference. The fluid exiting the impeller with the absolute velocity c2 is decelerated to an average velocity in the volute given by csp = Q/A (Q and A be-ing the local values of flow rate and area). The fluid approaches the cutwater with a positive incidence, causing a local depression in the static pressure. The static pressure rises from a minimum downstream of the cutwater to a maximum which is reached near the throat area according to the flow deceleration from c2 to csp. At low load – in particular at Q = 0 – there is a strong mismatch between c2 and csp and the fluid enters the volute similar to a jet entering a plenum. Consequently, an increasing fluid volume circulates in the casing since the fluid cannot be at rest even at Q = 0 due to the rotation of the impeller. The circulation is impeded by the cutwater where the flow separates reinforcing the pressure minimum, Fig. 9.20b.

Since the pressure distribution at the BEP varies little over the impeller circum-ference, its integration only yields a small resultant radial force (with an infinitely thin cutwater the force would be theoretically zero). In contrast, the non-uniform partload flow induces a strong variation of the static pressure over the circumfer-ence which results in a radial force usually reaching its maximum at Q = 0. Due to flow separation, the pressure recovery immediately downstream of the cutwater is lower than further on in the volute. As a consequence, the radial force is directed towards this pressure minimum, Fig. 9.21a.


270180 90 0 360°

c2

0.5

22u

H g2 Δ

-0.5

22u

H g2 Δ270180 90 0 360

c2

-0.2

0.2

22u

H g2 Δ

c2

c8

pH-

+

+ -

q* = 1.0c8 < c2 α2 ≈ αz

q* = 0 α2 → 0

q* = 1.7c8 > c2 α2 > αz

a)

b)

c)

pH

ε [°]

ε [°]

270180 90 0 360 ε [°]

αz

Fig. 9.20. Distribution of the static pressure and flow conditions in volutes

(3) At high flow rates (q* > 1) the volute cross sections are too small. The flow is thus accelerated downstream of the impeller. Correspondingly, the static pressure decreases in circumferential direction from a maximum (stagnation pressure) at the cutwater. The approach flow angle to the cutwater is too large (negative inci-dence) generating a flow separation in the discharge nozzle. Downstream of the cutwater there is a stagnation point and a pressure maximum. The vector of the re-sultant radial force is directed away from that maximum, Figs. 9.20c and 9.21b.


Fig. 9.21. Radial thrust in single volutes

As discussed in Chap. 1.4.1, the pressure distribution in a centrifugal pump is a unique function of the velocity distribution. In the absolute system, the flow fol-lows curved paths and the local pressure distribution establishes itself in a way that it balances the centrifugal forces induced by the flow path curvature, Eq. (1.26). If variable pressures are measured in the volute over the impeller cir-cumference, the flow around the impeller blades consequently depends on the ac-tual circumferential position of the blades. The blade forces (lift) thus change over the circumference; their resultant yields the radial thrust created by the blades. From the very existence of a radial force created by the impeller it is to be con-cluded that the impeller channels operate at every circumferential position in a different operation point. The experimental prove of this conclusion has been pro-vided by [9.21]: In a volute pump, the local values of flow rate and head were de-termined in 12 circumferential positions by integration of the velocity profiles measured by hot wire anemometry (in a test with air). The variations of measured local head and flow are shown in Table D9.1. For three operation points, Ta-ble D9.1 shows the average flow and head as measured at the discharge nozzle as well as the range of flow and head variations over the circumference.

Table D9.1 Variation of flow and pressure around the impeller circumfer-ence, [9.21] Operation point of pump (averages) Local variation around the impeller circumference

q* ψ q* ψ

0.25 1.14 -0.55 to 0.95 1.17 to 1.04

0,5 1,06 0,15 to 1,1 1,17 to 1,07

1.0 1.05 0.9 to 1.13 1.03 to 1.08

1.25 0.92 0.99 to 1.57 0.98 to 0.78


According to these measurements, at a nominal flow rate ratio of q* = 0.25, there is a flow of 55% of Qopt back to the impeller from a position upstream of the cutwater (highest pressure according to Fig. 9.20b). But at the circumferential po-sition of 30° (lowest pressure according to Fig. 9.20b) there is a forward flow of 95% of Qopt (almost 4-times the flow through the discharge nozzle).

The static pressure per se can create a force only if it acts on a solid structure; thus it produces a resultant radial force on front and rear shrouds which is added to the resultant of the blade forces discussed above.

From the flow behavior described above and from Figs. 9.20 and 9.21 result the following relationships which have been confirmed by numerous investiga-tions:

• The radial force attains its minimum near the design point of the volute. The ra-dial thrust in the design point is caused by: (1) asymmetries induced by the flow around the cutwater (in particular with relatively thick cutwaters), (2) geometrical tolerances, (3) friction losses are not constant over the circumfer-ence; they influence the pressure build-up in the volute. As discussed in Chap. 7.8.2, not all volutes are designed strictly according to the conservation of angular momentum. The effective position of the BEP results from the de-pendence of the various losses on the flow rate. Therefore the minimum of the radial force does not necessarily coincide with the BEP-flow or the design point.

• The flow rate at which the radial forces have their minimum is essentially de-termined by the design point of the volute, because the most uniform volute flow is expected at that flow rate. If the design point of the impeller differs from that of the volute, the effect on the radial forces is only of secondary in-fluence. If two different impellers are tested in a given volute casing, the mini-mum of the radial thrust is observed at about the same flow rate, i.e. it is found on the volute characteristic, [9.14]. In contrast, if a given impeller is tested in two different volute casings, the minimum of the radial thrust shifts to a higher flow rate with increasing volute throat area.

• The radial force rises at partload and at high flow; it attains a relative maximum at partload which is usually found at Q = 0 (see curve for single volute in Fig. 9.22).

• The larger the volute cross sections for a given cutwater diameter, the less uni-form is the flow in the volute at partload. Therefore, the radial thrust coeffi-cients of single volutes increase with growing specific speed until a maximum is reached in the range of nq = 50 to 60, refer to Fig. 9.27 in Table 9.4. Pre-sumably this maximum is given by a maximum blade loading: if the non-uniformity of the partload pressure distribution in the volute exceeds some spe-cific threshold, back-flow from the volute to the impeller inlet limits the local pressures which the impeller is able to sustain.

• At low partload the radial force acting on the impeller is directed to a point downstream of the cutwater (Fig. 9.21a), while at high flow rates the radial force is directed to a point shifted by roughly 180° as compared to partload


(Fig. 9.21b). Consequently, the radial force changes its direction by roughly 180° in the range around the best efficiency point. Near the BEP the force di-rection is thus uncertain and difficult to predict since it depends on tolerances and the effects mentioned above.

• With a given pump, the direction of the radial force vector varies with the flow rate q*. Considering different pumps, the force direction depends on the form of the casings and thus on the specific speeds. This dependence is depicted in Fig. 9.21a for Q = 0 based on measurements in [9.2].

• All data concerning the magnitude and the direction of the radial forces are subject to uncertainties, because the area distributions in the volute, the impel-ler sidewall gaps and the shapes of front and rear shrouds all have an impact on the pressure distribution, hence the radial thrust.

• As discussed above, it is not possible to separate the forces acting in annular seals from the radial thrust proper without special devices. With annular seals the radial forces measured depend on the seal clearances. That is why Table 9.4 provides radial thrust coefficients for “normal” and double clearances; normal clearances can be considered to be roughly corresponding to Eq. (3.12).

• If the distance between impeller and cutwater is increased strongly, the circula-tion of the fluid at Q = 0 is less impeded by the presence of the cutwater; the radial thrust drops by a few per cent as compared to a small cutwater distance. In an infinitely large casing there are no radial forces.

0.01 0.02 0.03 0.04 0.050

ϕ

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

kRu

Single volute

Double volute

Annular casing with vaned diffuser

Annular casing (without diffuser)

0.22

Fig. 9.22. Radial thrust in various types of collectors, nq = 19, [9.15]


9.3.4 Pumps with double volutes

Double volutes are employed in order to reduce the radial thrust. If a second cut-water is introduced at a spacing of 180° from the first cutwater, the rotational symmetry is improved as per Fig. 9.23. The pressure distribution in both of the partial volutes (over a wrap angle of 180°) is analogous to Fig. 9.20, as shown by measurements in [5.37]. As demonstrated by Fig. 9.23, even a rather short rib is able to drastically reduce the radial thrust at shut-off and below q* < 0.5, (test 4). Increasing the length of the rib to obtain a full double volute (test 2) yields an al-most constant radial force over the entire flow range. In general, a flat and unsys-tematic distribution of the radial thrust can be expected with double volutes in the flow range of q* = 0 to 1.1, e.g. [9.2]. The directions of the radial forces are un-certain too.

As demonstrated by Fig. 9.22, the radial thrust in double volutes can rise strongly at flow rates well above the best efficiency point. This increase is caused by differences in the flow resistances of the outer and inner channels. Conse-quently, the flow rates through both channels will not be the same and the impel-ler operates in both partial volutes at different points on the Q-H-curve (refer to the above discussion of Table D9.1). The difference in the static pressure at the impeller outlet prevailing in both of the partial volutes generates a radial force. Since the resistances rise with the square of the flow rate, the radial force rises steeply at q* > 1, while the impact at partload is small.

If both cutwaters are not spaced by 180° apart, the circumferential symmetry is destroyed and the radial thrust increases in comparison to a 180°-volute. Thus a cutwater spaced by less than 90° brings no radial thrust reduction at all. This is proven by the tests shown in Fig. 9.29 (in Table 9.4) where the radial thrust reduc-tion factor FDsp achieved by a double volute is plotted against the volute wrap an-gle, [9.13]. Double volutes with wrap angles of less than 180° are built for design reasons, e.g. in axial split pumps or for allowing complete draining of the casing (pumps with the discharge nozzle vertically upwards). If the wrap angle is near 120°, as may be the case in axial split pumps, a rib in the upper half of the casing can reduce the radial thrust.

With reference to Fig. 9.23 there is an optimum length of the rib in the dis-charge nozzle: while the rib in test 3 was too short (leading to a force maximum at q* = 0.75) the rib of test 1 appears to be too long generating higher flow resis-tance and higher radial thrust at high flow.

9.3.5 Pumps with annular casings

As discussed above, the high radial thrust created by single volutes at shut-off is generated by the cutwater impeding the free rotation of the fluid in the casing and leading thus to flow separation and a pressure minimum downstream of the cutwater. In contrast, the fluid rotates freely at Q = 0 in a vaneless annular casing. Consequently, the radial thrust of pumps with annular casings reaches its mini-


1.0 0.5 0 0.25 0.75 1.25 q*

1.0

0.5

0

Single volute

Test 4

Test 3 Test 1

Test 2

FR/FR,0,sv

Fig. 9.23. Radial thrust balancing by means of double volutes; all forces FR are referred to the radial thrust of the single volute at shut-off, FR,0,sv, [B.9]

mum at Q = 0. The radial force rises in a slightly upward curve with increasing flow rate until the best efficiency point, Fig. 9.22. At overload, when the annular cross section is much too small for the flow, a pronounced pressure minimum is generated in the domain upstream of the discharge nozzle. The radial thrust rises accordingly; the discharge nozzle acts like a strong depression.

In Table 9.4 a formula is given for estimating the radial thrust in annular cas-ings. The available data do not allow to discern an influence of the specific speed. The relation given for the function kR = f(q*) is quadratic, predicting a strong in-crease at q* > 1.

9.3.6 Diffuser pumps

The radial thrust in diffuser pumps is created by geometric tolerances of the dif-fuser and by asymmetries in the flow downstream of the diffuser, as caused by the discharge nozzle, for example. The more so, if the diffuser channels are short and have a small overlap. Therefore a ring of stay vanes with little or no flow deflec-tion is scarcely able to reduce the radial thrust.

From the available radial thrust measurements no clear dependence of the ra-dial thrust coefficient from the specific speed or geometric parameters could be found. Only the eccentricity of the impeller with respect to the diffuser generates defined radial forces which are approximately proportional to the eccentricity. These forces are de-centering, [9.16]. Since only very small eccentricities are en-countered in practice, these radial force components are usually insignificant; they are implicitly covered by the statistical test data in Table 9.4.


9.3.7 Radial forces created by non-uniform approach flows

Inlet chambers of between-bearing pumps (as used in multistage or double-entry single-stage pumps) generate an impeller approach flow with a circumferentially non-uniform velocity distribution, Chap. 7.13. One half of the impeller is ap-proached with a pre-rotation while the other half is subject to counter-rotation, Fig. 7.50. This variation of the circumferential velocity component c1u leads, ac-cording to the Euler-equation, to a variable work transfer in different sectors of the impeller. Thus a steady radial force is created whose magnitude and direction depend on the flow rate. Figure 9.24 shows measurements from [B.20] with an impeller of nq = 33: test 1 was made with an inlet chamber common to multistage pumps, while in test 2 an insert with radial ribs was mounted in order to reduce the circumferential velocity components and to smooth out the impeller approach flow. According to these tests, the radial forces created by non-uniform approach flow increase strongly at high flow (q*>> 1), because the perturbations of the flow induced by the inlet casing increase with growing inertia (i.e. high flow ve-locities).

125

100

50

25

125

100

75

75

50

25

Test 1

Test 2

0.04 kR,Y

0.02

-0.02

0.02-0.02 kR,X

FFy

Fx

Fig. 9.24. Influence of the approach flow on the static radial thrust coefficients. The figures give q* in per cent. Test 1: with inlet chamber of a multistage pump; Test 2: with ribs, which provide essentially a rotational symmetric approach flow, [B.20]. A line from the origin to any point on the curves represents the force vector (direction and size). Example: the vector F represents the radial force coefficient at a flow rate ratio of roughly q* = 1.1 with the dimensionless force components in y-direction Fy = 0.033 and in x-direction Fx = 0.01.


H H

Q

Q Q

cax

Fig. 9.25. Effect of asymmetric approach flow on the radial force created by an impeller of high specific speed

Note the strong dependence of the direction of the radial force on the flow rate. Such force variations have an influence on the bearing load and thus on the vibra-tion behavior of the pump, see Chap. 10. These tests confirm that different sectors of the impeller operate (even under steady operation) at different flow conditions, which implies their working on different points of the Q-H-curve. Such asymme-tries – and the resulting radial forces – can be caused either by the geometry up-stream or downstream of the impeller.

The forces created by a non-uniform approach flow to an axial or a semi-axial impeller of high specific speed can be estimated tentatively from Eq. (9.8).

FR = kR,D g H d22 with

2

2

schD,R d

LHHcossinfk Δββ= (9.8)

H is the head of the considered operation point and H is the head difference read from the Q-H-curve for the difference in flow rate Q caused by the non-uniform approach. For example: if the axial flow velocity is asymmetrical by ± 5% of the average as shown in Fig. 9.25, H is read for Q = ± 0.05×Q. Lsch is the blade length and )(5.0 B2B1 β+β=β is the average blade angle.

The factor “f” in Eq. (9.8) needs experimental verification. Setting f = 1.0 gives factors kR,D of similar magnitude as given in Table 9.4 for axial pumps.

CFD studies on unsteady hydraulic forces acting on a semi-axial impeller were reported in [9.28]. These excitation forces were created by a highly distorted ap-proach flow. Some results are shown in Fig. 9.30 (last page of Chap. 9).

9.3.8 Axial pumps

Radial forces in axial pumps are essentially caused by non-uniform approach flows induced for example by inlet bends or vortexes. Perturbations of rotational symmetry downstream of the impeller equally contribute to the generation of ra-dial forces. The mechanisms were discussed in Chap. 9.3.7.

The experimental radial force coefficients given in Table 9.4 are defined with the impeller outer diameter. According to the tests in [9.15] the coefficients for


steady radial forces are kR,D = 0.02 at q* < 1.2. At even larger flow rates, the ra-dial forces rise strongly because of the increasing non-uniformity of the flows at the impeller inlet and outlet. The coefficients for unsteady radial forces are ex-pected in the order of kR,D = 0.01.

If the rotor of an axial pump runs in an eccentric position, radial forces are cre-ated by the impact of the flow through the gaps between the blade tips and the casing on the work transferred by the individual blades, Chap. 10.7.3.

9.3.9 Radial forces in pumps with single-channel impellers

Single-channel impellers are used in sewage pumps and other applications where solid matter, typically as large as the nozzle size, is required to pass through the pump, Chap. 7.4. Since there is only one blade, the pressure distribution on the blade cannot be rotational symmetric. Consequently, a radial force is created on the impeller which rotates with the frequency fn given by the rotor speed. It is called the “hydraulic unbalance”. The pressure distribution on the blade depends on the flow rate – and so does the hydraulic unbalance. To some extent (but not completely for all flows) it can be compensated by mechanical balancing.

The mass distribution of the single blade is not rotationally symmetric either. As a consequence, there is a resultant mechanical unbalance. Thickening the blade near the leading edge helps the balancing.

The radial force coefficients defined by Eq. (9.6) are plotted in Fig. 9.26. Tests with a single volute pump with nq = 52 were reported in [9.27] and [9.29]. The static force coefficient follows the typical trend for single volutes, as shown in Fig. 9.22. However, the force coefficients are smaller than in Fig. 9.27 because front and rear shroud of the impeller are perpendicular to the pump axis; hence no radial forces act on the shrouds.

The coefficient of the hydraulic unbalance force (i.e. the radial force with the frequency f = fn) exhibits a minimum near q* = 0.6 when the impeller is mounted in a volute, tests of [9.27], but it rises linearly with q* when the impeller is mounted in an annular casing as tested in [7.56]. In this case, the behavior is simi-lar to the curve of the static radial forces in an annular casing (Fig. 9.22). The tests in [7.56] were done with an annular casing with nq = 43; the wrap angle of the blade was varied between 289° to 420° with only moderate effect on the hydraulic unbalance. The law for the blade angle B = f(r) was varied too, but inlet and out-let angles were kept constant at 1B = 9° and 2B = 35°. Hence it may be concluded that the hydraulic unbalance depends little on the blade design and that the curves in Fig. 9.26 can be used for an estimate in the absence of more pertinent informa-tion, see also Chap. 7.4 for results of parametric study in [9.29].

In the range dz/d2 = 1.067 to 1.24, the cutwater distance had little influence on the hydraulic unbalance, while the static radial force decreased at partload when increasing dz, [9.27]. It is noteworthy that the type of casing exerts a strong influ-ence on the hydraulic unbalance as shown by Fig. 9.26: even at high flow, there is an interaction between the flow in the casing and the flow in the impeller.


The system and the approach flow can have a high impact on the hydraulic un-balance as demonstrated by tests in [7.57], where a resonance with flow fluctua-tions in the suction bay doubled the hydraulic unbalance force coefficient kR when the speed was reduced from 1450 to 1000 rpm. Seemingly, the scaling laws were violated in this case.

0.00.10.20.30.40.50.60.70.8

0.0 0.5 1.0 1.5q*

kR

kR,dyn volute [9.27]kR,stat volute [9.27]kR,dyn annular casing [7.56]

Fig. 9.26. Static radial force and hydraulic unbalance coefficients of single-channel impel-lers; note that kR is defined by Eq. (9.6)

9.3.10 Radial thrust balancing

If the mechanical components of a pump are not dimensioned and designed cor-rectly for the true steady radial thrust, operational problems may be encountered:

• Excessive shaft deflection and wear in the annular seals • Radial forces generate alternating shaft stresses which can lead to shaft break-

age due to fatigue • Bearing overload and bearing damage • Damage to the shaft seals (in particular to mechanical seals) due to excessive

shaft deflection. For pumps built according to [N.6], the shaft deflection at the mechanical seal must be limited to 50 μm.

The steady radial thrust can be reduced by the following design measures:

1. Double volutes or a rib spaced at 180° from the cutwater, refer to Fig. 9.23. The double volute casings tested in [9.2] produced very flat curves FR = f(q*). The radial thrust coefficients (over the entire flow range) amounted to 10 to 20% of the thrust measured with a single volute at shut-off.

2. Multiple volutes can be considered in special cases, for example for multistage pumps (depending on manufacturing costs).

3. With reference to Fig. 9.23 multiple ribs similar to test 4 may be considered. For example two ribs each with a wrap of about 90° spaced at 120° and 240° from the cutwater.

4. A combination of diffuser and volute. The diffuser must have sufficient vane overlap in order to achieve the desired pressure equalization.


5. A combination of vaned diffuser and annular casing, see curve in Fig. 9.22. 6. At partload pumps with annular casings experience much smaller radial forces

than single-volute pumps, Fig. 9.22. Since the annular casing at low specific speeds implies only a small penalty in efficiency, this option may have advan-tages when the pump operates mainly below the best efficiency point (as is fre-quently the case with process pumps). However, note that the radial thrust in annular casings increases strongly at q* > 1.

7. As mentioned in Chap. 7.12, also a combination of an annular casing with a vo-lute can be beneficial to reduce the radial thrust. Measurements were published in [9.2]. The tested casings consisted of a concentric annulus wrapped over 270° of the circumference followed by a single volute. The specific speeds were nq = 23, 41 and 68, the cutwater distances were dz* = 1.33 to 1.84 and the casing width ratios were b3/b2 = 1.39 to 1.75. The radial forces at shut-off amounted to between 20 and 50% of the values measured with a single volute, but (probably) the reduction of the radial force is achieved only with wide im-peller sidewall gaps. However, at the BEP (and above) the radial thrust in the combination of annular casing and volute is higher than in a single volute.

8. If the flow in the impeller sidewall gap is effectively decoupled from the flow in the volute casing by minimizing gap A, the non-uniform pressure distribu-tion does not act on the projected area of the shroud. The radial thrust has been reduced that way in pumps with double volutes and double-entry impellers. If the decoupling is effective, any circumferential pressure variations in the im-peller sidewall gaps tend to diminish unless they are induced by uneven flow through the annular seals (e.g. due to eccentricity of the rotor).

9. Wide impeller sidewall gaps foster the pressure equalization over the circum-ference; the radial thrust − in particular at partload − is reduced; the effect is maximum at Q = 0. This effect is important for the combination of an annular casing with a volute

The effects of items 8 and 9 are to some extent opposite; which effect is dominat-ing in a specific case depends on the design but is difficult to predict.

9.3.11 Radial thrust prediction

For dimensioning the bearings of single-stage pumps and for calculating the shaft and its deflection, the radial forces acting on the impeller must be known for the entire flow range. These forces are commonly determined from experimental ra-dial force coefficients according to Eqs. 9.6 or 9.7. Table 9.4 provides all data necessary for that purpose for single and double volutes, annular casings and dif-fuser pumps, [9.13]. Explanations to Table 9.4:

1. In Table 9.4, q* always refers to the design point of the collector which is usu-ally identical to the best efficiency point of the pump. The difference only be-comes relevant if various impellers and collectors are combined.


2. Figure 9.27 in Table 9.4 (2) gives the radial thrust coefficients kR0 of single vo-lute pumps operating against closed discharge valve as a function of the spe-cific speed. Note that the specific speed must be calculated with the design flow of the volute. Attention is required when dealing with double-entry impel-lers. Under this provision, Fig. 9.27 is valid for single- as well as double-entry impellers. The abscissa in Fig. 9.27 is therefore labeled nq,tot = nq×fq

0.5. 3. A finite radial thrust is also to be expected in the best efficiency point. The de-

pendence on the specific speed is weak, Fig. 9.28. 4. The data provided in Fig. 9.27 originate from [9.13] and [9.2]; they compare

reasonably well with the published data. When the annular seal clearances in-crease, higher radial force coefficients must be applied as stipulated by curve 2. These must also be used for radial or diagonal seals as applied in pumps which are exposed to abrasive wear, Fig. 3.15. Note: any radial forces calculated by CFD must be compared to curve 2, unless the forces created in annular seals are taken into account in the CFD computations.

5. When calculating the radial thrust in double volutes, the coefficient kR0 read from Fig. 9.27 must be multiplied by the correction factor FDsp from Fig. 9.29 (in Table 9.4) which depends on the wrap angle of the inner volute. The value thus obtained may be applied in the range of q* = 0 to 1.1. As discussed above, the radial thrust can increase strongly at q* >> 1 due to the pressure difference caused by different flow resistances in the outer and inner channels.

6. If the two channels of a double volute are exactly symmetrical (“twin volutes”) according to Fig. 7.39, the radial forces amount to only 30 to 50% of the forces created in double volutes with a common discharge nozzle. Twin vo-lutes are found in the series stages of multistage pumps according to Fig. 2.8.

7. Data for annular casings are scarce in the literature; great uncertainties in the thrust prediction are therefore to be expected. All the more so, because there are no generally accepted rules for dimensioning annular casings. The function kR(q*) depends on the design of the annular casing. In the range where the ra-tio cR/c2 is small, the radial thrust is expected to increase moderately with the flow rate. Where the fluid is strongly accelerated near the discharge nozzle, radial forces presumably rise with the square of the flow.

8. The dependency (if any) of the radial force coefficients of diffuser pumps on geometric parameters or the specific speed is not documented.

9. The unsteady radial forces are quite similar for all types of collectors. The data given in Table 9.4 may be considered as broadband RMS values for the entire load spectrum of practical interest. Figure 10.35 provides broadband values for specific frequency ranges, refer also to Chap. 10.7.

10. Open and semi-open impellers can experience slightly higher radial thrust co-efficients than closed impellers, because they lack the effect of some pressure equalization in the impeller sidewall gaps. On the other hand, there are no ra-dial forces on the front shroud.

11. Cavitation has a significant impact on steady radial forces only when the pump runs with full cavitation. However, extensive cavitation can generate unsteady radial forces which increase the level of vibrations, [9.13].


Table 9.4 (1) Radial thrust calculation q* refers to the layout point of the casing (volute or diffuser)

Radial force tot22RR bdHgkF ρ=

1. Steady radial forces

q* = 0 kR0 from Fig. 9.27

q* = 0.5 from Fig. 9.28

0 < q* < 1 kR = (kR0 - kR,opt) (1 - q*2) + kR,opt

q* = 1 from Fig. 9.28

or kR,opt = 0.03 to 0.08

b2tot

Single volute

q* > 1 kR = 0.09 q*2 (or use kR,opt instead of factor 0.09)

kR,Dsp = FDsp kR0 with FDSp = (1.75 - 0.0083 εspo) from Fig. 9.29

Double volute

Below q* ≈ 1.1, FDSp and kR depend little on flow rate; above q*>> 1 the radial thrust can increase sharply.

If wrap angle εsp is smaller than 180°, the radial force coefficient for q* = 0.5 or q* = 1.0 can be estimated from the following interpola-tion formula:

9090

kk1

kkf sp

SV,0R

Rx

SV,0R

Rx −°ε−+= valid for 90 < <180°

kRx stands for the kR-values of single volute pumps for either q* = 0.5 or q* = 1.0.

fFkk dspSV,0RR =

kR50 and kR,opt are taken from Fig. 9.28

Twin volute kR,Zsp = (0.3 to 0.5) kR,Dsp

Annular cas-ings

kR0 = 0.03 to 0.1

kR,opt = 0.1 to 0.2

kR = kR0 (1 + q* + a q*2) “a” depends on the

geometry; tentatively a = 0.18

Diffuser kR0 = 0.02 to 0.09 q* = 1: kR,opt = 0.01 to 0.06

2. Unsteady radial forces q* < 0.5 q* = 1

All types of volutes and annular cas-ings, except single-channel impellers kR,dyn = 0.07 to 0.12 kR,dyn = 0.01 to 0.05

Diffusers kR,dyn = 0.04 to 0.16 kR,dyn = 0.01 to 0.09

3. Axial pumps

22D,RR dHgkF ρ=

steady force: kR,D = 0.02 for q* < 1.2

unsteady forces: kR,D = 0.01


Table 9.4 (2) Radial thrust calculation

0

kR0 0.4

10

1

2

20 30 40 50 60 70 80 90

0.3 0.2 0.1

0.5

qqtot,q f n n =

Fig. 9.27. Static radial force coefficients for single vo-lutes at shut-off (Q = 0)

Curve 1: Seal clearance accord-ing to Eq. (3.12) Curve 2: Double seal clearances single-entry impellers: fq = 1 double-entry impellers: fq = 2

Approximation formula for curve 1: kR0 = 3.730E-08x4 - 7.274E-06x3 + 3.610E-04x2 + 2.041E-03x + 3.944E-02 Approximation formula for curve 2 (average): kR0 = -5.000E-05x2 + 9.200E-03x + 1.910E-01

kR100 = 4.562E-05x2 - 1.894E-03x + 3.179E-02

kR50 = -2.515E-06x3 + 2.633E-04x2 - 2.197E-03x + 3.429E-02

0.00

0.050.10

0.15

0.200.250.30

0.35

0.40

0 10 20 30 40 50 60 70nq,tot = nqfq0.5

k R

q* = 0q* = 0.5q* = 1.0

Fig. 9.28. Static radial force coefficients for single volutes, data from [9.2]

0 εsp°180°150°120°90° 0

FDsp

1.0

0.5

Fig. 9.29. Factors for radial force coefficients of double volutes

εsp = wrap angle of inner volute (Table 0.2) Double and twin volutes: below q* ≈ 1.1 FDSp and kR depend lit-tle on flow rate; above q*>> 1 the radial thrust can increase sharply.


12. It is prudent to use the sum of static and dynamic radial thrust for the calcula-tion of the shaft and the bearings, i.e. kR,tot = kR + kR,dyn. For an in-depth analy-sis of the bearings the dynamic loading due to unsteady radial forces may be considered separately. In the absence of specific measurements the data given Fig. 10.35 can be used for that purpose.

13. In general it is recommended to apply at least a coefficient of kR,tot = 0.15 for the mechanical design.

14. The radial forces acting on the shrouds and in the annular seals represent a large uncertainty in radial thrust prediction because their effects are ignored in the published radial force coefficients.

Pump inlet vorticesSide walls vortices

Side walls vortices

Pump inlet vorticesSide walls vortices

Side walls vortices

-10000

-5000

0

5000

10000

-10000 -5000 0 5000 10000

Fx [N]

Fy [N]

Sugozu T101 (Fx,Fy) Shaft (One rev.)

Flow from Bassin

Fig. 9.30. Unsteady radial forces created by the highly distorted approach flow to a semi axial vertical pump, CFD analysis [9.28]

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