Asymptotic Model of the Flow in Metallic and Elastomeric Progressive Cavity Pump

Transcription

1 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil Asymptotic Model of the Flow in Metallic and Elastomeic Pogessive Cavity Pump S. F. A. Andade, selmafaa@petobas.com.b J. V. Valéio, jujuvianna@gmail.com M. S. Cavalho, msc@puc-io.b Depatamento de Engenhaia Mecânica, PUC-Rio, R. Mq.S.Vicente 5, Rio de Janeio, , Basil Abstact. Pogessive Cavity Pump PCP is one of the most used technologies fo tanspoting diffeent fluid-like mateials, fom sandy cude oil to waste sludge. In its simplest fom, it consists of a single theaded scew oto tuning inside a double theaded nut stato, foming cavities sepaated by seal lines. The stato can be metallic o coveed with an elastomeic laye. This appaently simple mechanism poduces a moe-o-less pulsation-fee positive displacement flow, without the need fo valves, based on the movement of the cavities fom the suction to the dischage, as the oto tuns inside the stato. PCPs ae lagely used in the oil industy because of thei ability to pump liquids with high solid contents, high efficiency with high viscosity liquids, unique design, and simple installation and opeation. The pefomance of a PCP is a function of the oto diamete and eccenticity, stato diamete and pitch, the numbe of stages, and ubbe cove popeties, in the case of elastomeic PCPs. Oil poduction completions with PCPs ae geneally designed using chaacteistics pump cuves povided by manufactues. Howeve, seveal vaiables, such as viscosity, solid and gas contents - not taken into account in these cuves - can affect the volumetic efficiency of both metallic and elastomeic stato pumps. The analysis of pump pefomance at oil field conditions is the main goal of eseach on PCPs. The pessue dop - flow ate elationship of a PCP depends stongly on the pessue diven back flow between two consecutive cavities. Many models have been pesented to ty to descibe this back flow, commonly efeed to as slip flow. Only ecently, a complete thee-dimensional, tansient model of the flow has been pesented. The solution is extemely complex and computationally expensive mainly because of the 3-D chaacte of the flow, the complexity of the geomety and mesh motion to follow the oto movement. The esults agee well with expeimental data, howeve the use of the model fo testing diffeent opeating conditions and pump designs is limited, since the time to compute the flow at each conditions is extemely lage ove one week. The poblem becomes even moe complex in the case of elastomeic stato. Realizing that thee is a lage atio of length scales in the egion between consecutive cavities, we popose an asymptotic model to descibe the flow inside PCPs. The model educes the thee-dimensional Navie-Stokes equation to a two-dimensional Poisson s equation fo the pessue field inside the pump. At each time step, it was solved by a second-ode finite diffeence scheme. The esults obtained agee vey well discepancy less than 5% with the complete 3D model, howeve the computational time equied is appoximately 100 times smalle. The asymptotic model will make it simple to extend the model to PCPs with defomable stato, multiphase flow and diffeent geometies. Keywods: Pogessive Cavity Pump; Lubication Appoximation; Numeical Model 1. INTRODUCTION Almost evey fluid-like mateial can be pumped with Pogessive Cavity Pumps. Since the pumping pinciple conceived by the fench enginee René Moineau [1] in the 1930 s, a lage numbe of diffeent industial applications puts this positive displacement pump in the goup of the most used technologies fo moving fluids, fom sandy cude oil to waste sludge. Pogessive Cavity Pumps PCP in thei simplest fom consist of a single theaded scew oto, tuning inside a double theaded nut stato, foming consecutive cavities sepaated by seal lines. This simple configuation belongs to the singlelobe categoy of PCP. Moe complex geometies with a lage numbe of lobes ae also used, and any combination is possible as long as the stato has one moe lead than the oto. The fist geneation of PCP s had a metallic oto and stato, foming igid moving cavities in the space between the two sufaces. The following geneations pesented a ubbe coveed stato, which is common nowadays. The defomable stato ceates a compession fit with the oto, in contast with the metallic pump whee thee is a small cleaance leading to a lage leakage between consecutive cavities. Consideing that defomable statos have opeational limitations, elated with tempeatue and mechanical esistance of the elastomes, metallic PCPs have geat application oppotunity, and it has been ecognized by the oil industy that this pump could become an impotant technological altenative fo heavy oil poduction. This appaently simple mechanism poduces an almost pulsation-fee positive displacement flow, without the need fo valves, based on the movement of the cavities fom the suction to the dischage ends of the pump as the oto tuns inside the stato. The volumetic flow deliveed by a PCP at a constant oto speed and pessue diffeence depends on thee design featues: oto diamete, oto eccenticity and stato pitch. The pump pessue ating depends on the numbe of stages. The peculia chaacteistic of PCP is the occuence of back flow, also denominated slip flow, deived fom non pefectly sealed cavities. Due to thei unique design and pinciple of opeation, PCP systems povide many benefits in oilfield applications, such as high solid content toleance, best efficiency with high viscosity fluids, simple installation and opeation. Oil poduction

2 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil with PCP is geneally designed ove the knowledge of chaacteistic pump cuves povided by manufactue, but seveal vaiables can affect and change the volumetic efficiency of both metallic and elastomeic stato pumps. It is common knowledge that the chaacteistic cuves change significantly with liquid viscosity and gas content, theefoe pump cuves povided manufactues usually do not epesent the eal pump pefomance at down hole conditions. Moeove, in ode to design PCPs that can be opeated at exteme conditions, it is impotant to undestand the effect of each geometic design paamete on the pump pefomance. These ae the main easons behind eseach effots dedicated to study the flow inside PCPs. The pefomance of PCPs is a function of volumetic pump displacement and the slip flow ie, the backwad flow between consecutive cavities due to the advese pessue gadient along the pump. The limitation of simple models on pedicting pump pefomance is elated to the difficulties in calculating the intenal back flow. Fo any type of stato, igid o defomable, slippage is a function of the fluid chaacteistics, the diffeential pessue, the components dimensions and the kinematics. In the case of elastomeic statos the poblem become even moe complex, because the geomety of the flow channel becomes a function of the pessue field. Olivet et al., 00 [] pefomed an expeimental study and obtained chaacteistic cuves and instantaneous pessue pofiles along metal to metal pumps fo single- and two-phase flow conditions. The fist and simplest numeical model to descibe the flow inside the PCP was pesented by Moineau 1930 [1] and is based on calculating the back flow acoss the pump, consideing a Hagen-Poiseuille flow in the sealing lines, which is subtacted fom the volume displaced by the otating oto, giving the volumetic flow ate. As the diffeential pessue acoss the pump ises, so does the slippage, and the elation between diffeential pessue and net volumetic flow pumped, can be calculated. Since the slippage gap aea is not clealy defined, the model is able to descibe the qualitative behavio, but it is not accuate. The volumetic displacement associate with the oto movement can be easily calculated fom pump components geomety, but calculating a intenal slip is not a tivial poblem. In ode to impove Moineau s model, Gamboa and Olivet 003 [3] have modeled the slip as the supeposition of two diffeent mechanism: one due to the oto s movement and othe due to the diffeential pessue between two cavities. Howeve, limitations wee ecognized and the model was not able to fit expeimental data. Only ecently, a complete thee-dimensional, tansient model of the flow inside the cavities of a PCP has been pesented Paladino et al. 008 [4]. The numeical solution obtained with a commecial CFD softwae is extemely complex and computationally expensive, mainly because of the tansient and 3-D chaacte of the flow, the complexity of the geomety and the necessay mesh motion to follow the oto movement. The esults agee well with expeimental data, howeve the use of the model fo testing diffeent opeating conditions and pump designs is limited, since the time to compute the flow at a single opeating condition was ove one week. Theefoe, the time equied to poduce an entie pump pefomance cuve would be enomous. Realizing that the egion that defines the flow ate - pessue dop elationship is the small cleaance between the moving oto and the stato, whee the slip flow occus, and that the atio of length scale in this egion is vey hight, we popose an asymptotic model to descibe the flow inside PCPs. The poposed model educes the thee-dimensional tansient Navie-Stokes equation to a quasi-steady state two-dimensional Poisson s equation fo the pessue field inside the pump. This is the same idea behind simplified models of the flow in scew extude - see Li and Hsieh 1994 [5] and Shuesh et. al 008 [6]. The simplified models fo extude can pedict the qualitative behavio, but they ae still not accuate enough to use as a design tool, mainly because of the geometic and kinematic simplifications used. One of the hypothesis is to neglect the cuvatue effect of the coss section, and to descibe the geomety of the flow egion using catesian coodinates. Pina and Cavalho 006, [7] pesented an lubication appoximation model in cylindical coodinates to descibe the flow in annula space with vaying eccenticity and showed that neglecting he cuvatue effect can geatly compomise the accuacy of the model. The objective of this wok is to develop an asymptotic model fo the flow inside a metallic igid singlelobe pogessive cavity pump. The coss section is paameteized using cylindical coodinates and thee is no simplified assumption of the geomety and kinematics of the oto. The esulting D equation fo the pessue field at each time step was discetized with a second ode finite diffeence appoximation. The model was used to obtain pedictions of the pump pefomance cuve as a function of oto speed, liquid popeties and pump geomety. The esults obtained agee vey well discepancy less than 5% with the complete 3D model, but the computational time equied was appoximately 1000 times smalle.. MATHEMATICAL MODELLING The flow inside the pump cavitied is govened by the Navie-Stokes and continuity equations, which in cylindical coodinates ae:

3 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil u ρ t + u u z + v u v ρ ρ + w t + v v + u v z + w w t + v w + u w z + w u θ v θ w 1 v + 1 w θ + u z = 0, = ρg z p { 1 z + µ u + 1 } u θ + u z, = ρg p { 1 + µ v + 1 v θ } w θ + v z, = ρg θ 1 { p 1 θ + µ w + 1 w θ v θ + w z w θ vw whee u, v, and w ae the axial, adial, and tangential velocity components. To avoid solving the system of coupled thee-dimensional diffeential equations, dimensional analysis is used to eliminate some of the tems of the equations. The pocedue used hee is geneally known as lubication appoximation. The cleaance between the oto and stato is much smalle than the adius o length of the pump: F = R o R o L, as shown in the figue 1. Using continuity equation, it can be shown that the main flow is in the axial diection, the velocity component in the adial diection is much smalle than in the othe two, i.e., u, w v. Moeove, the vaiation of the velocity components in the axial and azimuthal diections ae much smalle than those in the adial diection. Thus, the deivatives with espect to the adial diection ae much lage than the othes u u z, u w θ and w z, w θ By applying dimensional analysis the appopiate tems of the Navie-Stokes tansient equations can be neglected and the system of diffeential equations becomes: 0 = p [ 1 z + µ u ], 0 = p [ ], 0 = 1 p 1 θ + µ w. 1 }. Figue 1. Components of PCP Since the pessue is not a function of, the velocities u and w can be analytically integated in the adius diection in tems of the pessue: p 1 u = z ρg 4µ + c 1 ln + c ; w = 1 p ln 1 + c 3 µ θ + c 4 ; whee c 1, c, c 3, c 4 ae constants calculated based on the bounday conditions..1 Pump Geomety and Integal Limits The movement of the fluid inside the pump is caused by the single-theaded helical oto olling eccentically inside a helical double-theaded stato. As the stato has a pitch length twice as long compaed to the oto, the fluid is tapped in consecutive quasi-sealed cavities. These cavities follow each othe and poduce a almost pulsation-fee, positive displacement flow, figue. The flow displacement inside of a PCP depends on thee design featues: to adius, eccenticity E and stato pitch P st. Note, in figue 3, that the cente of the coss section of the oto is not the cente of mass of the oto cente of the oto s helix due to its helical natue. And the distance fom the cente of the oto s helix and the cente of the stato defines the eccenticitye. Ou coodinate cente lies on the cente of the coss section of the oto C, theefoe in a coss section of the pump it tanslates as shown in the figue 4. Eventhough the velocity component in the adius diection, v, is much smalle compaed to the othes, in the bounday condition v is consideed, because of the pump kinematic. To descibe the

4 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil R s L b L Stato to Helicoide s coss section Figue. PCP longitudinal section Figue 3. PCP s coss section. geomety of the egion whee the fluid flows inside the pump, the position of the intenal adius of the stato has to be deived as a equation of θ and z, R o θ, z. The coss section of the flow egion vaies fom the oto adius, which in ou coodinate system is constant, to the intenal adius of the stato R o θ, z. Figue 4. tation and tanslation of the oto. Some impotant vaiables used to descibe this geomety, R o θ, z, ae shown in figues 5-6 and defined below: Θ S = πz = πz is the angle between an hoizontal axis fom the oto cente to a cental line of the stato; P st and P P st P ae the stato and oto piches; d CSR = C s C = E cosωt Θ S, so E d CSR E is the distance between the centes of the oto and stato; R s R s α 1 = actan and α = actan ae the angles between the cental oto line and the E d CSR E + d CSR segments: C A to C A and C B to C B, espectively. We divided the egion whee the fluid flows in a coss section in fou pats, as shown in figue 6. Each pat of the egion is epesented by a function. The functions that descibe R o θ, z ae witten in the table 1, which also shows the

5 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil Stato s coss section to s coss section Stato s cente to s cente Stato s axes of symmety Figue 5. Geometical elements of the PCP s coss section Figue 6. Each one of the fou egions: I, II, III and IV espectively. elated pat of the egion and the θ s limit of each pat, whee A = α1 Θ s ; B = π Θ s α; B = π + α Θ s ; A = π α1 Θ s. Table 1. The functions that descibes the geomety of a singlelobe metal PCP. Region θ limits Function R o θ, z I A θ < A E d cs cosθ Θ S + Rs E d cs sin θ Θ S II B θ < B E + d cs cosθ Θ S + Rs E + d cs sin θ Θ S [ ] III A θ < B R s + E d cs sin α 1 sinθ Θ S IV B θ < A [ ] Rs + E d cs sin α 1 sinθ Θ S

6 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil. Kinematic and Bounday Conditions Note that the coss section of the oto tanslates and otates with espected to the cente of the stato. The coss section of the PCP in fou diffeent times ae shown in figue 4. To deive the appopiate bounday conditions, we need to detemine the velocity of a point P in the oto and of a point Q in the stato, as shown in figue 7. Figue 7. Vaiables to descibe oto s kinematic The angula coodinate of P is θ = γ + Ωt, whee γ is the inicial angle and Ω is the clockwise otation. The position of the oto s cente ou cente of efeence elated to the stato s cente is: X cs = d cs cosθ s = +E cosωt Θ s cosθ s, Y cs = d cs sinθ s = E cosωt Θ s sinθ s. Thus the velocity of the oto s cente is: V cs = V Xcs + V Ycs : V Xcs = eω sinωt Θ s sinθ s, V Ycs = EΩ sinωt Θ s cosθ s. So, X P, Y P = X cs + cosγ + Ωt, Y cs + sinγ + Ωt is the position of the point P elated to the stato cente and its velocity is V P = + V YP : V XP = V Xcs Ω sin θ, V YP = V Ycs + Ω cos θ. V XP It is clea that the velocity of P elated to ou fame of efeence, in the oto cente, is: V Pcs = V XP V Xcs, V YP V Ycs = Ω sin θ, Ω cos θ. Finally, the velocity of the point Q located at the stato bounday, elated to ou coodinate system, is: V Qcs = V Q V cs V Qcs = V cs = EΩ sinωt Θ s cosθ s, EΩ sinωt Θ s sinθ s, note that V Q 0, because the stato is stationay. Applying the tansfomation matix below, any vecto can be witten in cylindical coodinates: [ ] [ ] [ ] V cos θ sin θ Vx = V θ sin θ cos θ V y Now, we ae able to pesent all velocities in = R o and = : vr o = V XQcs cos θ + V YQcs sin θ = +EΩ sinωt Θ s cosθ s wr o = V XQcs sin θ + V YQcs cos θ = EΩ sinωt Θ s sinθ s v = V XP cs cos θ + V YP cs sin θ = 0 3 w = V XP cs sin θ + V YP cs cos θ = Ω It is clea that P otates only and both, P and Q, do not have azimutal movement: ur o = u = 0. Back to the equations and applying the bounday conditions pesented in equations 3, we ae able to find

7 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil c 1, c, c 3, c 4 and wite the complete expessions of u and w: u = ρg p { R [ i /Ri ] } ln 4 z 4µ R i lnr o /R i R i W R o ΩR w = p { [ ln 1 θ µ + K ln 1 ]} + K + R o R R + Ω R 5 whee K = R o ln R o 1/ + R ln 1/ R o R. 6 Up to this point, the pessue field is still unknown. In ode to evaluate it, and since the expessions of u and w ae known, the continuity equation can be integated in the adius diection, fom to R o : { v + w θ + u } d = 0 z v R i d }{{} I whee I = R o vr o v, II = w d θ III = [ u d ur o R o z z u R ] z + w θ d }{{} II + u z d } {{ } III ] [ wr o R o θ w θ Applying the bounday conditions 3 in the equation 7, the continuity become: z u d + θ = 0 7 w d wr o R o θ R ovr o = 0. 8 Using the known equations of the velocity field eqs. 4 and 5 to integate w and u in the adius diection: w d = p { 1 [ R θ µ R o lnr o R ln R1 + K ] [ln 1 + K ln W + R [ R + o R R R ln with K in equation 6, u d = ] R Ω ln { [ p R z ρg R R o R 1 o R 8µ R and ]}, 9 1 R lnr o / R o ln R o R ]}. Finally, the equation can compactly be witten as: p C 1 + p C θ θ z z { 1 = θ C 0W + z ρgc + wr o R o θ + R ovr o ; 10 whee, C 1 = R [ µ R o lnr o R ln R1 + K ] [ lnr 1 + K ln { [ ]} R C = 8µ R 1 R o R 1 R + R lnr o/ ln R o R, and [ ] W R C 0W = o ΩR R o R R R ln RΩ ln. Note that K is the same used befoe eq. 6 and R o θ ]} can be diectly deived fom the functions pesented in table 1. The equation 10 is a Poisson equation with the pessue field as unknown. The bounday conditions ae: peiodicity: P θ = 0 = P θ = π; P P θ = 0 = θ θ θ = π and Diichlet condition: P z = 0 = P in, P z = L = P out, fo the suction and dischage of the pump. This equation was discetized using a cental finite diffeence and solved in the softwae matlab.

8 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil 3. NUMERICAL RESULTS In ode to compae and validate ou esults we use the same pump, fluid and physical conditions as used by Olivet et al, [] in thei expeimental study. The metal singlelobe PCP was chosen as the follow chaacteistics: pump length L b = 0.36 m, length of each oto pitch L = 0.06 m, numbe of the piches of the oto N = 6, adius of the oto s coss section = 0.0 m, adius of the stato s coss section R s = m, eccenticity E = m. In these numeical expeiments, the oto velocity 300 pm pumped a fluid of 4 cp against a diffeential of pessue of 345 kpa. 5 x pessue θ = 0 θ = π/ θ = π pump length Figue 8. Pessue distibution along pump length It is inteesting that the model povides detailed infomation of the flow inside the pump. The obtained esults wee as expected, the elevant physical effects occu when the flow goes thought the gap, i.e. pump pefomance is defined by the flow between consecutives cavities. It can be seen in figue 8, which shows the pessue distibution along pump length fo thee diffeent angles θ = 0, θ = π/ and θ = π. Inside the cavity the pessue is almost constant and a lage gadient occus when the fluid flows though the cleaance. This phenomena can be also confimed in figue 9. The left side of this figue shows the cleaance and the ight one the pessue distibution, both in a θ z plane and in the same moment. The fist plots show the cleaance and diffeential of pessue afte seconds and the second ones seconds afte, that is 3/4 of the pump otation x z z θ θ x z z θ θ Figue 9. Cleaance and pessue distibution in a θ z plane in two times: t 1 = 0, 013s and t = 0, 150s. The vaiation of the flow ate with time is pesented in figue 10. Compaing two diffeent imposed pessue s gadient, we can notice that the vaiation of the flow ate without diffeential of pessue P = 0 is smooth and it loses this

9 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil 4 x x Flow ate m³/s Flow ate m³/s Times Times Figue 10. Flow ate vesus time. P = 0, P = 345kP a. smoothness when the imposed pessue s gadient inceases P = 345kP a. Thee ae two peaks in the flow ate, one in the beginning and anothe in the middle of the pump cycle. The pump geomety suggest this vaiation in the flow ate. Notice in figue 4 that when the oto is in the beginning and in the middle of one otation it is located in the two extemes of the pump s coss section, touching the stato. Once the pessue field is evaluated we can use the analytical expessions eq. 4 and eq. 5 to descibe the velocity field. It is shown in a contou plot fig. 11 of velocity and also of pessue, both in the same time and position t = 0.15s and z 1 = 0.036m. The oto squeeze the fluid to flow against the pessue gadient, and it is clea, analyzing the velocity field, the slippage, which is the esponsible fo the loss of efficiency in the pump. And we can also notice that the fluid eciculates. To compae ou pump cuve with an expeimental data fom Olivet et al, [] and also with the numeical P U Figue 11. Contou gaphic of pessue and velocity fields in the same position z 1 = 0.036m and time t = 0.15s. solution obtained with a commecial CFD softwae by Paladino et al, 008 [4], a 300 pm of oto s velocity opeating in a fluid with the viscosity, µ = 4cp wee chosen. As shown in the compaative plot 1, ou esults agee well with the liteatue. The model can be used also to evaluate opeational paametes of the pump. Fo example, the influence of the gap in the flow ate can be analyzed in the figue 13. Defining a dimensionless flow ate as in equation 11 and vaying the diffeential of pessue fo thee gaps.. Q ad = Q m [ R s E + π R s R ] L/t FINAL REMARKS In this wok, an asymptotic model fo the flow inside a metallic igid singlelobe pogessive cavity pump was developed. The coss section is paameteized using cylindical coodinates and thee is no simplified assumption of the geomety and kinematics of the oto. The poposed model educes the thee-dimensional tansient Navie-Stokes equation to a quasi-steady state two-dimensional Poisson s equation fo the pessue field and povide detailed infomation about the flow inside the pump. The esulting D equation fo the pessue field at each time step was discetized with a second ode finite diffeence appoximation. The model was used to obtain pedictions of the pump pefomance cuve as a function of oto speed, liquid popeties and pump geomety. The esults obtained agee vey well with expeimental

10 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil datas and with the complete 3D model and the computational time equied was ode of magnitude smalle. Anothe inteesting chaacteistic of this asymptotic model is that it can be easily extended to PCPs with defomable stato, diffeent geometies and also multiphase flows. And also, because of the moe efficient computation, the model can be included into a PCP simulato to povide the pump cuve at any opeational conditions. 5. REFERENCES Moineau R., 1930, "A new capsulism", Doctoal Thesis. Olivet A., Gamboa J.A. and Kenvey F., 00, "Expeimental Study of Two-Phase Pumping in a Pogessing Cavity Pump Metal to Metal", Society of Petoleum Enginees SPE, Vol Olivet A. and Gamboa J.A., 003, "New Appoach fo Modelling Pogessive Cavity Pumps Pefomance", Society of Petoleum Enginees - SPE, Vol Paladino E., Lima J.A. and Almeida R.F.C.,008, "Computing Modeling of the Thee-Dimensional Flow in a Metallic Stato Pogessing Cavity Pump", Society of Petoleum Enginees - SPE, Vol Li Y., Hsieh F. 1996, "Modeling of flow in a single scew stude", Jounal of Food Engineeing, Vol. 74, pp Suesh A., Chakaboty S., Kagupta K., Ganguly S. 008, "Low-dimensional models fo descibing mixing effects in eactive extusion of polypopylene", Chemical Engineeing Science, Vol. 6314, pp Cavalho M.S. and Pina E.,006, "Thee-Dimensional Flow of a Newtonian Liquid Though an Annula Space with Axially Vaying Eccenticity", Jounal of Fluids Engineeing, Vol.18, pp Figue 1. µ = 4cP, oto velocity 300 pm

11 Poceedings of COBEM 009 Copyight c 009 by ABCM 0th Intenational Congess of Mechanical Engineeing Novembe 15-0, 009, Gamado, RS, Bazil dimensionless flow ate gap gap gap Figue 13. Q a P, thee diffeent gaps