arXiv:2104.06736v1 [physics.flu-dyn] 14 Apr 2021

Modeling pressure pulsation and backflow in progressing cavitypumps with deformable stator

Jens Muller, Yashar Kouhi, Sebastian Leonow, Martin Monnigmann*

Automatic Control and Systems Theory, Department of Mechanical Engineering,Ruhr-Universitat Bochum, 44801 Bochum, Germany.

E-mail: {jens.mueller-r55, yashar.kouhi,

sebastian.leonow, martin.moennigmann}@rub.de

April 15, 2021

AbstractThis contribution studies the impact of the rotor-statorinteraction in a single-stage progressing cavity pump onthe flow rate and pressure. Specifically, we investigatethe effect of the rotor movement on the sealings formedwith deformable stators for various speeds and pres-sures. Sealings are reconstructed with the help of a ge-ometric 3D model. We analyze the tangential and radialdeviation of the rotor from its reference path and showthat the radial deviation affects the flow rate, whereasthe tangential deviation affects the pressure dynamics.The conjectures are confirmed with a laboratory testsetup.

© 2021. This manuscript version is madeavailable under the CC-BY-NC-ND 4.0 licensehttp://creativecommons.org/licenses/by-nc-nd/4.0/

1 IntroductionProgressing cavity pumps (PCPs) belong to the groupof positive displacement pumps. They are used in in-dustrial applications for a wide variety of media. Thevolumetric flow rate of PCPs is known to be an almostlinear function of the rotational speed [12]. As the dif-ferential pressure increases, a growing amount of fluidflows against the conveying direction. The volumetricefficiency of the pump obviously decreases with grow-ing backflow. An accurate model for the backflow isrequired to predict the flow rate and the volumetric ef-ficiency of the pump. Furthermore, an understanding

*Corresponding author.

of the physical processes that cause backflow is advan-tageous, since results can be transferred to other pumpgeometries and can be used for the examination of thecorrelation between backflow and wear.

Methods for determining the backflow in a PCP are,for example, given in [2], [6], and [8]. However, theseapproaches are conservative in that they assume the ro-tor to move on an ideal path inside the stator. Con-sequently, the dimensions of the contact area (respec-tively channel geometry for clearance fit) between rotorand stator are calculated for the ideal case. This as-sumption seems to be sufficient for PCPs with metallicstators. Previous results in [4] and [5], however, revealthat neglecting any deviation of the ideal path can leadto incorrect results for PCPs with elastomer stators.

The investigations in [11] on PCPs with elastomerstators show there exists a correlation of the movementof the rotor to the shape of the sealings inside the pump.However, the authors of [11] claim that the shape of thesealings cannot be described quantitatively. In [1] and[5] tactile and inductive sensors, respectively, were usedto investigate the motion of the rotor. Since the move-ment of the rotor as a function of the rotational anglewas not investigated, the effect of the rotor movementon the sealings could not be analyzed.

It is the aim of this contribution to model the pressurepulsation and backflow of PCPs as a function of the ac-tual rotor movement. To this end, we study the relationbetween the actual movement of the rotor and the shapeof the sealings inside the pump for a single-stage PCPwith elastomer stator. We distinguish two componentsof the deviation of the rotor from its ideal path fromone another and show that one component governs the

1

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iv:2

104.

0673

6v1

[ph

ysic

s.fl

u-dy

n] 1

4 A

pr 2

021

rotor center axiscross section

rotor

'

P e

Figure 1: Superposed rotor movements with corresponding angles ψand ϕ for the ideal case.

amount of backflow, whereas the other component gov-erns the pressure dynamics. Both claims are verifiedexperimentally.

Section 2 introduces the pump geometry, the basicinteraction between rotor and stator, and preliminariesnecessary for the analysis of the load case. Section 3states the main results concerning the modeling of back-flow and pressure pulsation. Section 4 presents the lab-oratory test setup and verifies the given conjectures withmeasurement data. A conclusion is given in Section 5.

2 Progressing cavity pumpThe pump under consideration is a single-stage, sin-gle lobe progressing cavity pump without significantwear. It consists of a helical steel rotor and a deformableelastomer stator. The inner geometry of the stator is atwisted slot hole with the stator pitch PS . The length-wise orientation of the stator (set during the pump as-sembly) is specified by the stator orientation angle θ.The rotor is assumed to be rigid.

2.1 Rotor movement for the no-load caseThe rotor performs two superposed movements: the ro-tation around its center axis (angle ψ) and the move-ment of the center axis on a closed curve (angle ϕ). Fig-ure 1 illustrates the coordinates for both movements forthe ideal case. The closed curve in Figure 1 is referredto as the path P of the rotor. For ideal geometries andin the absence of disturbances, the path is a circle withradius e.

stator slot hole

moving rotor

contact

(a) (b)

Figure 2: Planar rotor-stator contact.

2.2 Rotor-stator interactionThe rotor-stator contact areas perform two tasks: theyact as a radial bearing for the rotor and thus force therotor to stay on, or close to, the ideal path P . At thesame time, the contact areas build up sealings insidethe pump that separate cavities. We analyze the inter-actions of the cross section of the rotor and the statorto investigate the shape of the sealings in the presentsection.

Two types of rotor-stator contact are sketched in Fig-ure 2. Any cross section of the rotor is either locatedbetween the straight walls of the slot hole (a) or locatedat one end of the slot hole (b).

These two contact types give rise to three types ofsealings [7]. The two areas highlighted in red color inFigure 2 (a) result in the spiral seal line (SPSL) and thewarping seal line (WSL), which are shown in Figure 3.Due to the helical geometry of the rotor, the WSL andthe SPSL do not have the same properties and must bedistinguished from one another. The area highlightedin Figure 2 (b) results in the semicircle seal line (SSL),which is also shown in Figure 3.

The three distinct types of sealings are connected toeach other and form a continuum in the case of an idealplacement of the rotor.

According to [2], [11], and [13] two flow compo-nents determine the amount of backflow through thesealings. The first component arises due to the relativevelocity between rotor and stator. The differential pres-sure across the sealings causes the second component.According to previous analyses in [8] and our analysisof the characteristic curve (Q/n-curve), the influence ofthe first component is small, so that we neglect it. Thepressure distribution inside the pump needs to be knownfor the investigation of the second component.

Let pp and ps refer to the pressure on the pressure and

2

A

B

C

A

AB

C

SSL 1

WSL 1

SPSL 1

SSL 2

SPSL 2

WSL 2 SSL 3

Figure 3: Ideal sealings inside a single-stage progressing cavity pumpfor a fixed ϕ.

1

23

3

Figure 4: Cavities inside a single-stage progressing cavity pump for afixed ϕ.

suction side, respectively. A cavity open to the pres-sure or the suction side attains the associated pressure.The cavity encapsulated inside the pump is assumed toattain the suction side pressure [4]. A sketch of a typ-ical situation is shown in Figure 4, where orange andblue colors indicate pp and ps, respectively. The cavi-ties labeled 3 are connected since they are both open tothe pressure side. Cavity 1 is open to the suction side,whereas cavity 2 is closed to both sides.

The differential pressure across the pump is definedas

∆p = pp − ps . (1)

A pressure difference ∆p 6= 0 across a sealing is a nec-essary condition for backflow. It is evident from Fig-ure 4 that ∆p 6= 0 holds for four of the sealings, specif-ically for SSL 2, WSL 2, SPSL 2, and SSL 3 shown inFigure 3. The differential pressure is zero (indicated bymatching colors) for the remaining sealings, leading toa vanishing backflow.

Backflow only occurs if one or more of the sealingdegenerate. This may happen because of wear, or be-cause the rotor deviates from its ideal position due todifferential pressure. We assume the pump is not sig-nificantly worn and manufactured in a way that no seal-

ϕ = 0 ϕ = 2π

p

s

Figure 5: Location of ideal sealings as a function of ϕ. Suction sideand pressure side indicated by s and p.

ing degenerates for ∆p = 0. We outline the effects of∆p 6= 0 on the rotor position in Sections 2.3 and 2.4.

All statements made so far apply to the locationsof the ideal sealings and cavities for an arbitrary butfixed value of ϕ. As the pump is operated, the angle ϕchanges and causes the sealings to move from the suc-tion side to the pressure side. Simultaneously, the seal-ings and cavities perform a rotational movement. Bothmovements are illustrated in Figure 5.

2.3 Geometric sealing model

We introduce a geometric 3D model to reconstruct thesealings inside the pump for the load case. Let R andS refer to sets of points (x, y, z) that represent the ro-tor and stator geometry, respectively. The rotor-statorinteraction can then be investigated by analyzing the in-tersection

C = R∩ S . (2)

The intersection C equals the shape of the sealingsformed by the rotor-stator contact. Figure 6 depicts Cand the intersection of the rotor center axis with the suc-tion side plane and the pressure side plane. The coordi-nates (xs(t), ys(t))

′ denote the intersection of the rotorcenter axis and the suction side plane and thus representthe position of the rotor end at the suction side. The co-ordinates (xp(t), yp(t))

′ refer to the intersection of therotor center axis with the pressure side plane. The angleψ and the coordinates (xs(t), ys(t))

′ and (xp(t), yp(t))′

3

suction side plane

pressure side plane

x

y

y

x

rotor center axis

(xs(t), ys(t))′

(xp(t), yp(t))′

Figure 6: Coordinates of the rotor position.

uniquely define the position of R, i.e., the position ofthe rigid rotor inside the stator.

The correct representation of the sealings C by (2)depends on the exact representation of rotor and statorin R and S, respectively. Due to manufacturing toler-ances, the real dimensions of rotor and stator typicallydiffer from the available data, e.g. the computer aidedmanufacturing data, in particular in the case of an elas-tomeric stator. We compensate for this mismatch withan effective rotor diameter

dR,eff = KdR , (3)

where dR is the rotor diameter according to the manu-facturing data and K is a correction factor. In this way,we ensure the geometric 3D model to resemble the realpump.

The evolution of the sealings over time with ro-tating rotor can be analyzed with time-series of ψ,(xs(t), ys(t))

′ and (xp(t), yp(t))′, and by determining

C by (2) for every sample of the time-series. If the rotormoves on its ideal path inside the stator, the reconstruc-tion of the sealings with the help of the geometric 3Dmodel results in the time series shown in Figure 5. Re-sults obtained under load conditions are shown in Fig-ure 13 and discussed in Section 4.

2.4 Load dependence of the rotor path

We refer to the rotor path that results for ∆p = 0 asthe reference path and only analyze deviations from thisreference path from here on. We distinguish a radial de-viation sR and a tangential deviation sT from the refer-ence path as depicted in Figure 7. It is convenient todefine a rotating coordinate system with one axis rep-resenting sR and the other axis representing sT at each

ϕ0

sT(ϕ0)

sR(ϕ0)

sT(ϕ1)

sR(ϕ1)

ϕ1

referencepath

Figure 7: RT-coordinate system for two rotational angles ϕ0 and ϕ1.

rotor end. For both rotor ends, the origin of the co-ordinate system is fixed to the center axis of the rotor.Figure 7 shows the rotating coordinate system and thetwo deviations sR and sT for two angles ϕ0 and ϕ1. Werefer to this coordinate system as RT-coordinate system.

The origin of the RT-coordinate system moves on thereference path of the rotor and rotates with the angularvelocity ϕ.

The following analyses are carried out in the RT-coordinate system unless stated otherwise. We explainhow to measure the rotor reference path and the deter-mination of K in section 4.1.

3 Modeling backflow and pulsa-tion in a PCP

The differential pressure across the PCP affects the tan-gential and the radial deviation of the rotor from its ref-erence path in distinct manners. The radial deviationresults in an almost constant tilt of the rotor with re-spect to the radial direction. The tangential deviationresults in a periodic tilting of the rotor. This was ob-served and modeled successfully by [4] and is valid forthe operating range of the pump. We claim the radialdeviation governs the backflow, while the tangential de-viation governs the pressure pulsation.

3.1 Backflow-conjecture

In Section 2.2 we showed that ∆p 6= 0 holds for twoSSLs. We will show in Section 4.2 that the shape of theSSL closest to the suction side degenerates most whendifferential pressure is applied. Thus, we analyze theshape of the SSL that is exposed to ∆p 6= 0 and closestto the suction side. We refer to this specific SSL asthe relevant SSL (see Figure 8). We assume that thebackflow of the PCP mainly depends on the shape of

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the relevant SSL. This is referred to as the backflow-conjecture.

In the present section, we model the shape of the rele-vant SSL as a function of the measured radial deviationand the angle ψ. A simple model for the flow throughthe resulting gap is used to calculate the amount ofbackflow. The approach will be verified in Section 4.3by comparing measured and calculated flow rates.

By definition, the R-axis of the RT-coordinate systempoints towards the SSL (see Figures 6 and 7). Thus, weexpect that the shape of the SSL is strongly influencedby the radial deviation of the rotor from its referencepath. The shape of the relevant SSL also depends on itsdistance to the suction side zSSL. Figure 8 illustrates thecorrelation of the radial deviations to the location of therelevant SSL. zSSL can be described as a function of ψby

zSSL = l − PS(

1− mod(ψ − θ, π)

2π

), (4)

with rotor length l, stator pitch PS and stator orientationθ (see [3] for additional information on why θ must betaken into account).

Rotor tilt leads to a vanishing contact between rotorand stator. The resulting gap height w of the relevantSSL can be calculated from

w =sR,p − sR,s

l· zSSL + sR,s , (5)

with radial deviation at the pressure and suction sidesR,p and sR,s, respectively. Figure 9 sketches the result-ing gap at the relevant SSL. Because the radii of therotor rR and the stator rS are known, w can be used tocalculate the gap area A and the perimeter U . The in-tersection of rotor and stator (resulting from rR > rS)is neglected in (5), so that any w > 0 leads to a break-ing sealing. Note that for the specific case shown inFigure 8, w is close to zero, since the height at zSSL isclose to the reference. Equation (5) reveals that w isa function of sR,p, sR,s and ψ. Consequently, w variesduring operation, even if the pump is operated at a fixedspeed and differential pressure. However, our investi-gations revealed that it is sufficient to use a mean gapheight wmean, calculated over several rotations for thecalculation of the backflow. This significantly reducesthe computational effort.

The flow through the gap is assumed to be turbu-lent since our calculations yield high Reynolds num-bers. We apply a model for the pressure loss in a pipe

+0

-

0

zSSL

sR,s

sR,p

relevant SSLl z

reference

Figure 8: Correlation of radial deviations sR,s and sR,p and the loca-tion of the relevant SSL.

rR

rS

wdynamics

Figure 9: Resulting rotor-stator gap (area A in blue, perimeter U inred).

flow and use the hydraulic diameter dh = 4AU to cal-

culate the pressure loss due to turbulences over the gap[10]. The pressure loss reads

∆p =ρ

2·(Qb

A

)2

· Ldh· λt , (6)

with differential pressure ∆p, fluid density ρ, back-flow Qb, gap length L, and pressure loss coefficientλt. λt can be described by the Blasius Equation λt =

0.3164(Redh )0.25 , with Redh = Qb·dh·ρ

η·A , where η denotes theviscosity of the fluid [9]. Inserting the Blasius Equationin (6) and solving it for Qb yields

Qb = A ·(

2 ·∆p · d1.25h

ρ0.75 · L · 0.3164 · η0.25) 1

1.75

. (7)

Equation (7) describes the backflow inside the PCPacross the relevant SSL taking into account the geome-try of the resulting gap due to the radial deviation of therotor from its reference path.

The calculated flow rate of the pump is determinedby

Qcal = Qi −Qb , (8)

where Qi is the ideal flow rate of the pump.

3.2 Pulsation-conjectureOur second conjecture states that the pulsation in differ-ential pressure, which can be observed during the oper-ation of a PCP, mainly depends on the periodic tilting

5

1

1

2a

2a2b

2b S

R

(a)

(b)

Figure 10: Increased cavity volume due to abrupt rotor tilt in tangen-tial direction evident from the comparison of the stator center axis (S)and the rotor center axis of the rotor (R).

deviation of therotor from areference path

radial deviation

strong influence onbackflow

tangential deviation

strong influence onpressure dynamics

backflow-conjecture pulsation-conjecture

Figure 11: Effects of the deviation of the rotor from its reference pathon the hydraulic conditions.

motion of the rotor in tangential direction. We use

τ(t) = sT,s(t)− sT,p(t) . (9)

as a measure for the periodic tilting motion in tangentialdirection. We will show in Section 4.2 that τ abruptlychanges twice per rotor rotation. Figure 10 sketches theposition of the rotor before (a) and after (b) this abrupttilting. The abrupt change in τ leads to a quick volumeincrease in the most recently opened cavity 2 (high-lighted by the green outline in Figure 10). The quickincrease in volume leads to an abrupt pressure drop onthe pressure side of the pump.

This conjecture will be verified in Section 4.4 bycomparing the measured periodic rotor tilt to the mea-sured pressure pulsation on the pressure side. Figure 11summarizes our two conjectures.

M

M

(a) (b) (c)

PTPT DT DT

AT

FT

Figure 12: Scheme of the laboratory test setup with the container(a), the control valve (b), the progressing cavity pump (c), the flowtransmitter FT, the angular transmitter AT, the pressure transmittersPT and the distance transmitters DT.

4 Experimental verification

This section presents the laboratory test setup and veri-fies the claims made so far experimentally.

4.1 Laboratory test setup

The experimental data used in this paper is measuredwith the laboratory test setup sketched in Figure 12.The progressing cavity pump1 (c) pumps water from thecontainer (a) through the control valve (b) and back intothe container. The pump is driven with a variable fre-quency drive. Various differential pressures can be setwith the control valve.

The angle ψ, the rotational speed n, the flow rate Q,and the pressure and suction side pressure are measuredas indicated in Figure 12.

Distance transmitters (8 at each side of the rotor)measure the distance between each end of the rotorand the pump housing during the operation (see [5] fordetails). This allows to determine (xs(t), ys(t))

′ and(xp(t), yp(t))

′ (see Section 2.3) as a function of time[4], thus measuring the rotor reference path as well asthe deviations due to differential pressure. We initializeS and R with manufacturing data and manually deter-mine K in (3) so that no sealing in C degenerates whenthe pump is operated with zero differential pressure (seeassumption in Section 2.2).

All signals are measured with a sample time of 1 ms.

110-6L, manufactured by Seepex GmbH

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4.2 Degeneration of the sealings due todifferential pressure

We analyze the shape of the sealings as a function ofdifferential pressure. Representative results obtainedwith the geometric 3D model2 and the correction fac-tor K ≈ 1.014 are shown in Figure 13.

Figure 13 reveals that the SPSL and the WSL are al-most insensitive to changes in the movement of the rotordue to increasing differential pressure. In contrast, theSSL significantly depends on the differential pressure.The contact area of the SSL 1 decreases with increasingdifferential pressure (see Figure 13 (a)). With increas-ing ϕ, SSL 1 moves towards the pressure side (see Fig-ure 13 (b)). At this location, the area of the consideredSSL only weakly depends on the differential pressure.The pressure dependence is reversed close to the pres-sure side, which leads to an increasing contact area ofthe considered SSL with increasing differential pressure(see Figure 13 (c)).

The results in Figure 13 suggest that the amount ofbackflow mainly depends on the shape of the SSL clos-est to the suction side, because any SSL closer to thepressure side exhibits a bigger contact area.

Additionally, both deviations of the rotor from its ref-erence path due to applied differential pressure werecalculated for various rotational speeds and differentialpressures ranging from 0 to 4 bar, varied by adjustingthe control valve (b) in Figure 12.

Figure 14 shows the radial deviation of the rotor forthe suction side and the pressure side for 200 rpm andvarious differential pressures. All deviations have beennormalized by dividing them by the maximum valuesmax that occurs in the tangential deviation of the pres-sure side.

The tilt of the rotor in radial direction, which dependson the applied differential pressure, is shown in Fig-ure 14.

The normalized tangential deviation of the rotor fromits reference path is depicted in Figure 15 for the samepoints of operation. Figure 15 shows the periodic tilt-ing motion of the rotor addressed in Section 3.2.

2We implement the geometric 3D model in the CAD SoftwareFreeCAD and use its python interface to place the geometric rep-resentation of the stator as well as the rotor according to measuredvalues.

0 bar 2 bar 4 bar 6 bar

ϕ

(a)

(b)

(c) p

p

p

s

s

s

(SSL 3)

SSL 2

SSL 1

SPSL2

SPSL1

WSL 2

WSL 1

Figure 13: Reconstructed sealings C for various differential pressuresand rotational angles ϕ. The pump was operated at 100 rpm. Suc-tion side indicated by s, pressure side indicated by p. Dashed boxeshighlight SSLs at various locations.

7

0 0.2 0.4 0.6 0.8 1

−0.4−0.2

00.20.4

time in snormalizeddeviation suction side

4 bar 3 bar 2 bar 1 bar

0 0.2 0.4 0.6 0.8 1

−0.4−0.2

00.20.4

time in snormalizeddeviation pressure side

Figure 14: Radial deviation of the rotor from its reference path for200 rpm and various differential pressures.

0 0.2 0.4 0.6 0.8 10

0.20.40.60.81

time in snormalized

deviation suction side

4 bar 2 bar 3 bar 1 bar

0 0.2 0.4 0.6 0.8 10

0.20.40.60.81

time in snormalized

deviation pressure side

Figure 15: Tangential deviation of the rotor from its reference pathfor 200 rpm and various differential pressures.

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

pressure in bar

flow

rate

inl/

s

100 rpm 200 rpm 300 rpm calc.

Figure 16: Comparison of measured and calculated flow rates for var-ious rotational speeds and differential pressures.

−1

−0.5

0

0.5

1

timead

j.norm

.tilt

−0.4

−0.2

0

0.2

0.4

adj.pressure

inbar 1 bar 2 bar 3 bar 4 bar

Figure 17: Time series of pressure pp and rotor tilt τ for 200 rpm andvarious differential pressures.

4.3 Verification of the backflow-conjecture

We compute the backflow according to the approachoutlined in Section 3.1. The calculation of Qb uses themean gap height wmean and is thus only performed oncefor each combination of speed and differential pressure.The specific value for the gap length L in (7) was foundby the comparison between the measured and calcu-lated flow rate for one arbitrary point of operation withnon-zero differential pressure and kept constant subse-quently. It was found to be L = 2.5mm, which is phys-ically reasonable and is in a similar magnitude as thevalue reported in [8].

Figure 16 shows the comparison of the calculated tothe measured flow rates for various speeds and differen-tial pressures. The error between the calculated and themeasured flow rate is below 3.5 %. The largest absoluteerror between measured and calculated backflow occursat 300 rpm and 4 bar differential pressure and amountsto 0.037 l/s.

The proposed approach reproduces the measuredflow rates with reasonable accuracy. This corroboratesthe proposed backflow-conjecture.

4.4 Verification of the pulsation-conjecture

Figure 17 shows time series of τ and pp for variouspoints of operation.

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The time series for each point of operation is depictedfor one revolution of the rotor. All time series in Fig-ure 17 are adjusted by subtracting their mean value foreasier comparison in a joint figure. The strong correla-tion of τ to pp, evident in Figure 17, corroborates thepulsation-conjecture.

5 ConclusionIn this contribution, a novel approach to modeling thebackflow of a PCP was proposed. The model takes theactual movement of the rotor due to its interaction withthe deformable stator into account. The deviation ofthe rotor from a reference path inside the elastomer sta-tor was divided into two components. We showed thatthe radial deviation can be used to model the backflow,whereas the tangential deviation mainly affects pressuredynamics. In contrast to existing approaches, the pro-posed model takes into account information regardingthe actual, non-ideal movement of the rotor.

AcknowledgmentsFunding by Ministerium fur Wirtschaft, Innovation,Digitalisierung und Energie des Landes Nordrhein-Westfalen is greatfully acknowledged.

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The formal publication of this preprint can be found viahttps://doi.org/10.1016/j.petrol.2021.108402

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