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1 Quail, Franis J. and Sanlon, T.J. and Baumgartner, A. (2010) Design study of a regenerative pump using one-dimensional and three-dimensional numerial tehniques. Numerial Hear Transfer Part A: Appliations. ISSN This is an author produed version of a paper published in Numerial Hear Transfer Part A: Appliations. ISSN This version has been peer-reviewed but does not inlude the final publisher proof orretions, published layout or pagination. Strathprints is designed to allow users to aess the researh output of the University of Strathlyde. Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other opyright owners. You may not engage in further distribution of the material for any profitmaking ativities or any ommerial gain. You may freely distribute both the url ( and the ontent of this paper for researh or study, eduational, or not-for-profit purposes without prior permission or harge. You may freely distribute the url ( of the Strathprints website. Any orrespondene onerning this servie should be sent to The Strathprints Administrator: eprints@is.strath.a.uk

2 Design study of a regenerative pump using one-dimensional and three-dimensional numerial tehniques Dr. Franis J Quail¹, Dr. Thomas Sanlon 2 and Armin Baumgartner 2 Abstrat Regenerative pumps are low ost, ompat, able to deliver high heads at low flowrates. Furthermore with stable performane harateristis they an operate with very small NPSH. The omplexity of the flowfield is a serious hallenge for any kind of mathematial modelling. This paper ompares an analytial and numerial tehnique of resolving the performane for a new regenerative pump design. The performane harateristis omputed by a CFD approah and a new one-dimensional model are ompared and mathed to experimental test results. The approahes of both modelling tehniques are assessed as potential design tools. The approahes are shown to not only suessfully resolve the omplex flowfield within the pump; the CFD is also apable of resolving loal flow properties to ondut further refinements. The flow field is represented by the CFD as it has never been before. A new design proess is suggested. The new regenerative pump design is onsidered with a omparable duty entrifugal pump, proving that for many high head low flowrate appliations the regenerative pump is a better hoie. 1 Institute for Energy and the Environment, University of Strathlyde, Glasgow, G11XW, Sotland. 2 Department of Mehanial Engineering, University of Strathlyde, Glasgow, G11XJ, Sotland. Keywords: Regenerative pump, omputational fluid dynamis (CFD), helial flow Nomenlature A Cross setional area (m²) Supersripts P Power (kw) Q Volume flow rate (m³/s) * Intermediate variable U Mean fluid veloity (m/s) Mean value ρ Density (kg/ m 3 ) r radius (m) Subsripts H Head (m) p Pressure (kn/m 2 ) o open hannel N Rotational speed (rev/min) irulatory g Gravitation aeleration (m/s 2 ) sp Sution port η Effiieny dp Disharge port ω Angular veloity (rad/s) t tangential α Inidene fator e exit ζ Ellipse angle (rad) i inlet D Diameter (m) s Solid body θ Angle measured from inlet 0 Inner hannel σ Slip Fator 2 Impeller Z Number of impeller blades 3 Outer tip hannel Flow Coeffiient L Linear region Effetiveness of irulatory flow u upper K ψ Head Coeffiient Loss oeffiient b Impeller width (m) Abbreviations Impeller tip learane (m) d Channel width (m) CFD Computational fluid dynamis U Peripheral veloity (m/s) CAD Computer Aided Design NPSH Net Positive Sution Head (m) - 1 -

3 Introdution Regenerative pumps are rotor-dynami mahines [1], apable of developing high heads at low flowrates with a single impeller [2]. At low speifi speeds the pump also shares harateristis with positive displaement pumps and entrifugal pumps. Other benefits with this type of pump inlude its ompat design [3] and the ability to operate at low NPSH. Regenerative pumps, like entrifugal pumps, are kineti pumps. The head rise is ahieved through an exhange in momentum between the impeller and the fluid. There are a limited number of publiations for regenerative pumps in omparison to other kineti devies [4]. In spite of low effiieny, usually less than 50% [3,4], regenerative pumps an offer a more effiient alternative to other (entrifugal) pumps in many appliations [5]. Until reently only analytial, one-dimensional models were used to predit the performane harateristis and were limited in representing the omplex flow-field within the pump requiring signifiant experimental orretion [6]. CFD analysis and a newly developed one-dimensional model offer a muh more preise way of alulating the performane of a regenerative pump. This paper presents these methods whih are not dependent on experimental orretion, are valid for a range of pump geometries and are onsidered as design tools to optimise pump performane. Figure 1: The regenerative pump A regenerative pump onsists of a asing with an annular hannel and an impeller, Fig 1. The impeller is a dis with several, usually 20 to 50, radial vanes at the edge of the dis periphery. These vanes rotate in the annular hannel. The fluid enters the hannel through the sution port and repeatedly irulates through the impeller vanes due to the ation of a entrifugal fore field. The flow path approximates a toroidal helix, similar to the shape of a orksrew [3]. This effet of repeated irulation is also desribed as internal multi-staging [2], for eah passage through the vanes may be regarded as a onventional stage [4]. This multi-staging is where the pump derives the name regenerative pump. The fluid exits through the disharge port. The disharge port is loated immediately before the stripper, a short setion where the annular hannel is redued to a lose learane preventing the fluid from flowing from the disharge to sution port, Fig 2. Figure 2: The regenerative pump The pump detailed in this paper inorporates a new design for the sution port and the disharge port. Previous designs had straight ports leading into the hannel in a radial diretion. This arrangement has subsequent large losses for the flow into the pump [7]. The new design inorporates the advantages of the onventional geometries (multistaging) but realigns the fluid flow. The flow enters and exits the flow hannel in an axial diretion, Fig 2. The ports are aligned in axial diretion so that an arrangement with multiple stages is possible by assembling the asings in row using one single shaft. The flow in the ports of this arrangement is lead into a 90 turn and enters the annular hannel in a tangential diretion entri above the impeller. This permits flow guidane at the sution and disharge ports and minimises axial thrust while the irulatory flow develops[8]. One-dimensional analysis One-dimensional models, until reently have aounted for the majority of the publiations as the preferred method of alulating performane. Engels [9] in suession of Shmiedhen [10] and Ritter [11] suggested that the helial flow was responsible for the head rise. He developed an early one-dimensional model and predited the pump s harateristis. His equations expressed the performane however they relied on experimentally defined oeffiients. This fat limits the appliability of Engels model

4 Wilson et al.[12] developed a one-dimensional model that is the basis for most publiations to date. He suggested an equation for the momentum exhange and identified losses ourring in the pump. However these losses were defined experimentally and do not rely on geometrial or physially derived data. The simplifiations used in this model also proved to have a signifiant effet on the performane predition apability of the suggested model. Song et. al. [1] developed a one-dimensional model with a fundamental hange. Most publishers sueeding Wilson negleted the hange of irulatory veloity along the hannel and orreted this by introduing a linear region fator. This lead to an analytial model, but the linear region fator was a priori unknown and a soure of error. Song et. al. did not simplify the differential equations for the momentum exhange, but used a numerial tehnique to alulate the irulatory veloity at disrete points to orret the defiieny. Yoo et. al. [2] proposed equations to alulate the flow geometry namely, the entre of the irulation, and the inlet and outlet radius. Sine Wilson et. al., a primary assumption of previous analysis has been that the flow exits the blade only at the impeller tip. Yoo et. al. applied the ontinuity equation and alulated the irulation and derived inlet and outlet points that are independent of the impeller analysis. Apart from the more aurate desription of the flow geometry, the model by Yoo et al., still relies on the linear region fator to orret the error due to the simplifiation of the differential equation. The linear region fator presented in most published one-dimensional models does not represent the fluid behaviour satisfatory. In the urrent study a numerial modelling approah is presented that demonstrates the ability to suessfully resolve the flow field in a manner that has not been ahieved previously. The one-dimensional model used in this paper inorporates a new model whih does not rely on experimental data from onduted test, but an predit the performane only from the geometry and fluid properties. Six assumptions are made in the development of the mathematial model. 1. Steady flow 2. Inompressible fluid 3. All proesses are adiabati 4. There are no effets due to sution and disharge, exept for losses 5. The pump flow is haraterised by a tangential and a irulatory veloity 6. Tangential pressure gradient is independant of radius The head rise for the pump is alulated from the angular momentum exhange or irulatory head rise losses ourring in the pump. These are sution port head loss head loss ΔgH t. The head rise of the pump is expressed by gh = gh sp dp gh and the ΔgH sp, disharge port head loss ΔgH dp and tangential ΔgH ΔgH ΔgH Eq. (1) t where A V 2 2 gh = σu e αu i Eq. (2) Qs A is the irulatory flow area and Q s the solid body rotation of the fluid between the impeller vanes. V is the mean irulatory veloity evaluated by numerially averaging over the whole pumping region using the omposite Simpson s rule [13]. The irulatory veloity is alulated by disretising and linearising of the fist-order non-linear differential equation expressed by X X V +GX V + HX V = F 2 g, o g, o g, o g, o Eq. (3) using a preditor-orretor tehnique. The fators σ and α are the slip fator and the shok fator as detailed by [2]. CFD analysis Regenerative pumps, as with other kinds of turbomahinery, an be analysed using omputational fluid dynamis (CFD). This tehnique to predit performane and to examine flow harateristis has been suessful in standard regenerative pumps reently[7]. Turbomahinery appliations for CFD models are proving more robust and have found greater utilisation in the aademi and ommerial fields [8]. CFD models an offer benefits over experiments in lower osts and in auray as the flow features are not disturbed by measuring equipment [8]. This paper presents some of the benefits of the CFD analysis and also onsiders some of the limitations in this respet. In this study, a Multiple Referene Frame-modelling (MRF) approah is hosen. Only a stationary performane predition is desired and the MRF- model is suffiient in this ase. For turbomahinery the approah using MRF modelling is also referred to as the frozen rotor tehnique. The inner region lose to the impeller is a moving referene frame while the hannel region and inlet and outlet ports are treated as a stationary referene frame. The alulation is onduted as steady state and therefore fairly effiient in omputational effort. FLUENT appliation briefs [14-16] report a mathing of about 7% between CFD results and experimental data. Unsteady interation effets like wakes are negleted by this model but this an only be addressed by using the muh more expensive sliding mesh modelling. This is onsidered the remit of future work

5 A grid was hosen after onduting a grid independene analysis. Four separate meshes with 0.4 million, 1.3 million, 1.7 million and 2.3 million ells were ompared. All of the grids had been reated using the same tehnique but applying different ell sizes. Eventually, the grid with approximately 1.7 million ells was hosen for the omparative analysis. Further refinement did lead to less than one perent hange in parameter auray in the pump model. Fig. 3 shows the basi shape and refinements of the CFD mesh. The inner region is onstruted of mainly hexahedrons with refinements towards wall boundaries and the interfae region. The outer frame is meshed with tetrahedrons to model the geometry while obtaining an aeptable mesh quality [17]. The skewness riterion did not exeed 92 per ent at any point. The geometry for the grid is the same used for manufaturing the pump exported diretly from the CAD appliation. Figure 3: CFD Grid arrangement Varying volumetri flowrates of the regenerative pump was assessed for multiple differential pressures using the ommerial CFD ode FLUENT Version The Reynolds Averaged Navier Stokes Equations were solved on the domain using a pressure based solver and the SIMPLE algorithm [18] for pressure orretion. Turbulene was modelled using the k-ε-realizable model, a highly stable and well developed two-equation approah [19]. This turbulene model is suitable for omplex shear flows involving rapid strain, swirl, vorties and loally transitional flows. The applied disretisation sheme obtained seond order auray. The boundary layer was not resolved, but modelled using a + ontinuous wall funtion (Enhaned Wall Treatment). Therefore the y -value was kept below 300. Experimental proedure Until now regenerative pump port geometry have retained a fairly basi geometri onfiguration with simple radial inlet and outlet ports mahined into the asings [6]. For the pump used in this paper with a rapid prototyping tehnique robust enough to produe, for the first time, more omplex regenerative pump axial inlet and outlet ports were manufatured. The axial port arrangement differs to the lassial radial ports. The inflow into the pump and the outflow are parallel to the pump s axis (fig. 2) in ontrast to the 90 angle of the radial pump standard. Rapid Prototyping using `Objet Polyjet,[20], (photopolymer jetting tehnology) tehniques, builds omponents in thin layers (0.016mm) enabling smooth, aurate and highly detailed parts to be produed in a range of materials diretly from CAD models. Quail et. al. first demonstrated the ability to produe regenerative pump omponents in an aurate and robust enough manner to permit experimental testing for a radial standard pump [21]. The experimental rig arrangement, Fig 4, inorporated a reservoir tank whih stored and ultimately reeived the working fluid, in this ase water. The fluid was drawn to the pump from the tank and the fluid flow was adjusted via a flow ontrol valve in the return line to the tank. The fluid flowrate was measured using a Hall Effet turbine flow meter, situated upstream of the flow ontrol valve and upstream of the pump. The pump itself was driven by an indution motor operating at a onstant speed of 3000rpm (50 Hz). The motor housing was oupled to a dynamometer ontaining a load ell to measure strain and hene indiate input torque to be used in the pump effiieny alulations. The test impeller had 30 blades of width 11mm and diameter 74.5mm. The pump was of double sution shape designed with alignment of the blades to balane axial thrust. In the urrent study a radial impeller was used with teeth or vanes mahined into eah side at its periphery. The measurements were olleted via a data aquisition unit

6 Figure 4: Shemati of the experimental rig arrangement Results The results of the one-dimensional model and the CFD are ompared to the experimental results for two pumps. Pump 1 is the pump developed by Quail et. al., [7,8,21] inorporating a radial sution and disharge port arrangement. Here the flow enters the pump radially and joins with the flow in the pump with a 90 angle to the peripheral veloity. Pump 2 is the pump designed by the authors (fig 1). The sution and disharge ports are axially aligned to support a multistage arrangement. In addition the flow enters aligned to the peripheral veloity in the hannel. The harateristis are displayed in a dimensionless manner using a head oeffiient and a flow oeffiient. They are expressed by gh ψ = Eq. (4) 2 ω r g,o Q = Eq. (5) ωa o r g,o Fig. 5 shows the omparison between both pumps and the tools for performane predition. Figure 5: CFD / Experimental Results Comparison The CFD is apable of muh more than just predit the overall performane of the pump. As previously shown by Quail et al. [7] a powerful advantage of CFD is the full folwfield resolved. This approah of using generally valid equations also inludes all flow harateristis and is only restrited by the numerial tehnique. The one dimensional models an be defined in two main ategories, the ones following the irulation or liquid filament theory and the ones following the turbulene and mixing theory [22]. Neither model is able to fully represent the flow in a regenerative pump, but both have their strengths as it was shown by Senoo et al. [23]. Nevertheless no existing one- dimensional model inludes both theories of operation. The CFD on the other hand does not rely on one restrited model with simplifying assumptions, but aptures properties of both theories in a more abstrat way. As Figure 5 demonstrates, both the CFD and the one dimensional analysis math the experimental results quite well

7 It an be seen that the representation of the CFD is slightly better than the one-dimensional modelling. The new 1D model does not rely on any data from experiments onduted on regenerative pumps and is therefore a heap and easy-to-use way to alulate the performane. Figure 6 8 show aptured images of the CFD results. The streamlines plotted in figure 6 learly shows the helial flow path of the fluid in the pump. The pathlines are oloured by veloity, where a darker olour represents higher veloity. Here it an be seen that the veloity rises from pivot to perimeter as it is assumed in the one-dimensional models. This again shows the advantage of the CFD simulations over onedimensional modelling. Figure 6: Regenerative Pump heliity pathlines plot The helial flow an be quantified by the heliity. Heliity is the vortiity or irulation of the flow times the flow normal to the irulation. Formally it is expressed by H = u 3 ud r Fig 7. indiates a ross-setion though the hannel where the flow is fully developed. Blak indiates positive and white negative heliity. The flow is direted into the shown plane and the right hannel side shows lokwise, the left side ounter-lokwise irulation. Figure 7: Regenerative Pump heliity ontours plot

8 The helial flowpath or for that matter streamlines, until now, ould only be visualised by omplex flow visualisation experiments as onduted by Bartels [24], Lazo et al. [25] and Lutz [26]. CFD is able to apture this ruial fat of the regenerative pump in a way that was not possible before. This advantage an be used, not only to determine the overall performane, but to assess and refine ertain aspets of the flow. The key to a more effiient regenerative pump lies in the understanding of the omplex flow and in the knowledge of how to effet the flow. Figure 8 indiates the pressure distribution in the regenerative pump from blue (light) to red (dark). Unlike in entrifugal pumps, where the pressure rise ours in radial diretion, the head is developed in a peripheral diretion in regenerative pumps. The aim of designing regenerative pumps is to avoid loal pressure drops or peaks and to assure onstant pressure rises with a high gradient in peripheral diretion. The CFD analysis allows examination of the pressure very aurately, without the influene of pressure probes et. It is also possible to view the pressure distribution at any point. Figure 8 additionally shows the pressure distribution at the improved inlet and outlet ports. The aligned inflow and outflow of the pump lead to muh smaller pressure drops than in the version with radial inlet and outlet ports [27]. Figure 8: Tangential pressure rise through axial regenerative pump Comparison with a entrifugal pump Pumps are the single largest user of eletriity in industry in the European Union, and of those pumps, entrifugal pumps represent some 73% of all pump types [28]. The regenerative pump like the entrifugal pump is a kineti pump however the regenerative pump an in many appliations offer a more effiient alternative [5]. If high pressures are needed, the pump an be assembled to multiple stages. They are the best hoie for many appliations. Small entrifugal pumps have a low ratio between the ontained volume and the interior surfae resulting in a high level of frition between the moving fluid and the impeller and the pump asing walls. Large entrifugal pumps with impellers of greater diameter and width impart high veloity to the fluids they transport and have a higher ratio of between the ontained volume and interior surfae thereby reduing frition and improving effiieny [8]. Fig. 9 ompares the harateristis of a entrifugal pump to a regenerative pump. Considering a the use of both pumps at a flowrate of 16.6 l/min (2.77 x 10-3 m 3 /s) the regenerative pump is apable of developing 450 kpa while the entrifugal pump only delivers 167 kpa

9 Figure 9: Pump omparison plot. Figure10: Pump omparison plot. The effiieny of the entrifugal pump at 16.6 l/min is about 5%. The regenerative pump at this flowrate is able to perform with an effiieny of 13%. Figure 10 shows the effiieny over the flowrate range of the pumps. The delivered head of the regenerative pump is 2.7 times higher than the apabilities of the entrifugal pump while the effiieny is 2.6 times better. If three stages of the entrifugal pump are used, to gain a similar head rise, the effiieny of the arrangement is less than two per ent. Proposed Design Proedure Considering the fat that there is little published data about design riteria a design proedure, inorporating the findings of this work, is proposed. As a first step, after defining the performane speifiations, the one-dimensional model is used to math the performane. An optimisation algorithm an be used to automatially fit required head and flowrate while optimising effiieny or pump size. The pump geometry is quik to implement and a solution an be reahed in very little time. The seond step is to load the results of the one-dimensional optimisation into a parameterised CAD model to reate a fully three-dimensional geometry. With this geometry a CFD simulations are onduted. The pump s features an be refined loally and so adjusted to yield best flow through the pump further refining the design. True performane an however only be suffiiently proved by experiments the reation of a prototype is therefore inevitably. In the third step of the design proedure, this an be done by rapid prototyping. For regenerative pumps are usually rather small mahines, this an be done with little finanial effort. The speimens produed by today s rapid prototyping mahines proved to be suffiiently durable to ondut tests without resorting to slow and expensive traditional manufaturing tehniques. Mehanial data an easily be obtained from similar entrifugal pumps. In these three steps of the design proess, the advantages of all three proedures, one-dimensional analysis, CFD simulation and onduting experiments are used. The one-dimensional approah is fast and able to proess a great number of different geometries in short time. The flow features are introdued into a CFD analysis whih needs more time, but is also quite heap. Expensive experiments are minimised, beause only optimised pump geometries will be tested. Using rapid prototyping to reate speimen, ost and time an be redued here as well [21]. The use of rapid prototyping tehniques beomes even more important onsidering omplex port geometry arrangement defined above whih an only be ast or mahined via five-axis-milling

10 Conlusions The CFD analysis and the one-dimensional tool are alternatives to experimental testing. They offer potential as advaned design tools without the need for expensive tests. The one-dimensional approah is fast and an be used to evaluate different geometry parameters very quikly. The CFD method takes more time to develop and to alulate the performane, but is still faster and less ostly than onduting experimental test. The advantages offered by CFD analysis over all previous methods of alulation and flow visualisation are signifiant. The advanes in rapid manufaturing offer an opportunity to quikly orroborate the analytial models. The benefit of a orroborated three-dimensional model offers further apabilities for resolving the flow s properties (flow visualisation) and making it possible to fous on ourring loal effets within the pump. To design a regenerative pump a proedure is proposed. After evaluating the needs and given restritions, a three-step design proedure leads to good results in very little time at low ost. Future work The Multiple Referene Frame approah for the CFD analysis gives a good approximation, but neglets unsteady effets. A sliding mesh method (SMM) approah is urrently being undertaken and the results will be published in the oming months. The one-dimensional alulation algorithm will be expanded to inlude a Simplex Optimisation Algorithm [29]. It will be possible to fit the pumps geometry to the intended purpose automatially by the variation of speified variables. The Simplex algorithm proved to be very stable for other appliations and with the fast one-dimensional model a very preise examination of multiple variants of pump geometries will be possible. The regenerative pump design presented in this paper will be extended to onsider multiple stages. This will make the regenerative pump an even more attrative alternative to entrifugal pumps, onsidering fewer stages and therefore smaller dimensions and less weight and better effiieny in the regenerative pumps performane range. The new inlet and outlet port arrangement allows a multistage arrangement. The arryover effets from stage to stage will be examined by CDF analysis and finally a true multistage version of the regenerative pump will be built to ondut experiments

11 Referenes 1. J. W. Song, Abraham Engeda, and M. K. Chung. A modified theory for the flow mehanism in a regenerative flow pump. Pro. IMehE, Part A: J. Power and Energy, 217: , I. S. Yoo, M. R. Park, and M. K. Chung. Improved momentum exhange theory for inompressible regenerative turbomahines. Pro. IMehE, Part A: J. Power and Energy, 219: , M. Raheel, A. Engeda Systemati design approah for radial blade regenerative turbomahines Journal for Propulsion and Power Vol. 21 (2005) 4. T. Meakhail and S. O. Park. An improved theory for regenerative pump performane. Pro. IMehE, Part A: J. Power and Energy, 219: , S, Mueller, Consider regenerative pumps for low flow/low NPSH appliations. Hydroarbon proessing pg (2004) 6. A. Engeda, Flow analysis and design suggestions for regenerative flow pumps, ASME FEDSM (2003) 7. Quail, F. J., Stikland M. T., Sanlon T. J., Design optimisation of a regenerative pump using numerial and experimental tehniques, International Journal of of Numerial Methods for Heat and Fluid flow. In Press. 8. Franis Joseph Quail. Design Study Of A Regenerative Pump Using Numerial And Experimental Tehniques. PhD thesis, University of Strathlyde, H. Engels, Investigations of Ring pumps, Teh Hoh Hannover. (1940) 10. C. Ritter. Über selbstansaugende Kreiselpumpen und Versuhe an einer neuen Pumpe dieser Art. PhD thesis, Dresden, W. Shmiedhen. Untersuhungen über Kreiselpumpen mit seitlihem Ringkanal. PhD thesis, Dresden, W.A.Wilson, M.A. Santalo, J.A. Oelrih, A Theory of the fluid dynami mehanism of regenerative pumps, Trans. ASME Vol 77 PP (1955) 13. Atkinson, Kendall A. (1989). An Introdution to Numerial Analysis (2nd ed.). John Wiley & Sons. 14. FLUENT. Bakward inlined Centrifugal Fan - Ex143 Appliation briefs from FLUENT (2001) 15. FLUENT. Automotive Water Pump - Ex164 Appliation briefs from FLUENT Courtesy of TESMA Engine Tehnologies (2001) 16. FLUENT. Mixed Flow Pump - Ex232 Appliation briefs from FLUENT (2005) 17. FLUENT In., editor. Mesh Generation for Turbomahinery. FLUENT v6.2, Ferziger, J. H.; Peri, M. (2001). Computational Methods for Fluid Dynamis. Springer-Verlag. 19. T.-H. Shih, W. W. Liou, A. Shabbir, Z. Yang, and J. Zhu. A New k-ε-eddy-visosity Model for High Reynolds Number Turbulent Flows - Model Development and Validation. Computers Fluids, 24(3): , Objet, Quadra, PolyJet, FullCure, Eden350, Eden350V are trademarks of Objet Geometries Ltd. 21. F.J.Quail, T.J.Sanlon, M. Stikland, Rapid Manufaturing Tehnique used in the Development of a Regenerative Pump Impeller, Proeedings of World Congress On Engineering - London (2009) 22. Grabow, G. Comparison of several theories of the mehanisms of flow in regenerative pumps. Proeedings 2nd Conferene on flow mahines (1966) 23. Yasutoshi Senoo. A omparison of regenerative-pump theories supported by new 24. performane data. Transations of the ASME, 78: , 1956.J. Bartels, Performane of a peripheral pump -Assoiate Professor, Polytehni Institute of Brooklyn (1947) 25. L. Lazo, T. Hopkins, Theoretial and experimental analysis of a regenerative turbine pump, Massahusetts Institute of Tehnology (1953). 26. G.F. Lutz, Experimental Investigation of the pressure distribution in a regenerative turbine pump, Massahusetts Institute of Tehnology (1953). 27. F. J. Quail, M. Stikland, and A. Baumgartner. Design study of a novel regenerative pump using experimental and numerial tehniques. In 11th European Fluid Mahinery Congress Sept 2010, In Press. 28. IMehE, Energy Saving in pumps and Pumping, Fluid Mahinery Group Symposium (2007) 29. George B. Dantzig: Linear Programming and Extensions, Prineton University Press, 1966 Appendix H F X g,o G X A = 2Q θr 0 A g,o K + A tanβ A tanβ 2 r 0 r ri g, o = αu i Q θr g,o r r r i A 2 2 X o = 2 1 φσu e αu i 1 α 2 2 g, Ui 2Q0θrg, o