Thermo- and fluid dynamic model of a multiphase screw pump, operating at very high gas volume fractions

Transcription

1 Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions Dil.-Ing. (F) Klaus Räbiger Prof. Dr. T.M.A. Maksoud Prof. Dr. John Ward University of laorgan School of Technology Wales, UK Prof. Dr.. ausann eorg-sion-oh Fachhochschule Nürnberg Fachbereich Maschenbau und Versorgungstechnik Abstract To describe the erforance as ell as the therodynaic behaviour of a ultihase scre u, also oeratg at very high gas volue fractions u to 100 %, a ne thero- and fluid dynaic odel as develoed. The odel, hich ill be resented this aer, takes the tiedeendent gas-liquid heat transfer, the coressibility and acceleration effects of the ultihase ga flo as ell as the ossibility of critical flo conditions to account. For the siulation of the utual hase heat transfer, the solution of the ga flo conservation equations and the otiisation of the ga flo velocity, different nuerical techniques ere used. The odel as such gives iortant foration of the ressure and the teerature distribution, the ultihase leakage flos through the chaber connectg gas as ell as the effective flo rate or the voluetric efficiency of the u near the conveyance breakoff at very high gas volue fractions. Key ords : scre u - ultihase - gas volue fraction ISSN Sonderdruck Schriftenreihe der eorg-sion-oh-fachhochschule Nürnberg Nr. 35, Noveber 006

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3 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 3 1. Introduction A scre u is a articular tye of rotary dislaceent us, hich a nuber of scres rotate side a cyldrical housg. The geoetry and rotation of the scres generate a series of closed chabers, hich transort the fluid fro the lo ressure let to the high ressure let. The ressure distribution through the u and hence the flo characteristics and syste erforance is heavily fluenced by the leakage flo fro the discharge side to the suction side. This leakage flo occurs through three different gas side the scre u, naely the erieter ga beteen the scres and the housg, and the radial and flank ga beteen the atg surfaces of the scres, see Figure 1.1. Perieter a Radial a Flank a Figure 1.1 : Three different kd of gas side the scre u Previous vestigations of ultihase scre us have largely been concerned ith the general ug behaviour of these systes [1]. Moreover these studies have been restricted to relatively ediu sized us, hich the axiu oer consution and gas concentrations are relatively odest. In these situations the heat caacity and density of the gas-liquid ixture is doated by the liquid hase so that the ug rocess is essentially isotheral and therodynaic effects can be neglected [] and [3]. oever, this assution cannot be justified for larger, ore oerful scre us, hich are caable of conveyg to-hase fluids ith very high concentrations of the gaseous hase u to 100 %. There are existg to ne odels [4] and [5], hich clude the therodynaic effects by establishg ass and energy balance equations for each chaber. But no ultihase scre u odel can be found literature, hich deals ith the coressibility or the critical flo condition of the to-hase leakage flo through the gas. Due to the above-entioned lack of the correct odellg of coressibility effects, acceleration ressure dros and critical flo conditions, it is necessary to create a odel for coressible ga flos side of ultihase scre us, usg analytical and nuerical ethods, to redict the leakage flo and the theral behaviour ore accurately.. Chaber odel of the scre u The therodynaic scre u odel can be treated as a collection of several chabers, hich are connected by three different kds of gas. Due to the nature of a ositive dislaceent u - excet for the leakage aount, the theoretical volue flo is constant at all ties - only the leakage flo has to be vestigated, to detere the real volue flo or the voluetric efficiency factor. The drivg forces for leakage are the local ressure difference beteen to corresondg chabers and the rotatg scres. Because of the rotation of the scres, the chabers are transorted fro the suction to the discharge side of the u. In a real scre u, the leakage flo creases the ressure all the chabers creentally fro the let to the let. In the case of 100 % liquid hase, the ressure distribution is lear, because the fluid is coressible and a certa aount of leakage at the u let ust have the sae aount of leakage, hich flos at the u let. But if the gas volue fraction creases and the chabers are alloed to have different gas volue fractions, the ressure rofile becoes non-lear. The shae deends aly on the gas volue fraction and the rotational seed of the scres. To detere the correct ressure and teerature rofiles etc. along the scre axis, the chaber flo and flo rocess have to be vestigated for all chabers, begng at the u discharge side. The correct chaber connection beteen the suction (S) and the discharge (D) of a double-flight t scre u through the erieter ga (P) and the radial ga (R) is shon Figure.1. As a consequence of the or contribution of the flank ga to the hole leakage ass flo, the flank ga as not considered the current odel. Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

4 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 4 R 1S R 31 R D3 S P 1S 1 P 1 P 3 3 P D3 D R S R D Figure.1 : Chaber connection schee of a double-flight t scre u ( here ith 3 chabers ) The vestigation tie, hich is divided to sall tie stes, is equal to the life tie of the chabers. After this eriod, all chabers ove one chaber osition the let direction. This eans, that the forer last chaber No. 3 have oened to the high ressure side and there is no a colete ne chaber No. 1. The iteration of chaber changes, hich cludes all sub-iterations, has to be carried so long, until all therodynaic chaber variables reach a steady state. In the case of an odd nuber of chabers, the real ug behaviour has to be detered by a eightg rocedure beteen the next even uer and loer chaber nuber, deendg on their teoral occurrence. The calculation of the therodynaic conditions of a sgle chaber can be slitted to the follog orkg stages, see also Figure.: a) as flo ( constant secific total enthaly ) b) iquid flo ( constant secific total enthaly ) c) as coression by liquid flo ( gas teerature creasg ) d) iquid flo ( variable secific total enthaly ) e) as exansion by liquid flo ( gas teerature decreasg ) f) as flo ( variable secific total enthaly ) g) as-liquid heat transfer ( through the terfacial area ) The chaber conditions of both hases and also the volue fractions, after the flo (terediate stage) and after the flo rocess (fal stage) are calculated iteratively. The division of the total flo and flo rocess to sall tie stes assures a nearly contuous flo through all chabers. lo high d f a b Q & T c e g Direction of conveyance Sgle chaber as an oen and transient therodynaic syste Figure. : Oen and transient therodynaic syste Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

5 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 5 3. Fundaental therodynaic equations for oen systes Each chaber of the scre u as odelled as a therodynaic oen syste [6] and [7], hereas the flo and the escae of sgle or to-hase fluid fro one chaber to the other are considered as transient rocesses. To describe the therodynaic changes of the chaber condition, ass and energy conservation equations have to be alied, hich ere derived and troduced this and the next sections. Q 1 W1 = Etotal, Etotal, 1 total First la of therodyn. (3.1) E U E E = Total energy (3.) k ot U U cha / u / = Internal energy (3.3) E k / = Ek, cha / Ketic energy (3.4) E ot E ot, cha / g z / W = Potential energy (3.5) 1 = Wtech, 1 v v Syste ork (3.6) The energy conservation equation follos fro the first la of therodynaics for transient rocesses oen systes for a negligible ketic and otential energy of the chaber itself to h u 1 g z u 1 = Q 1 h W tech,1 g z (3.7) The heat transfer and the technical ork concerng the chabers ere not ileented the current adiabatic odel stage for the siulation of non-decreasg chaber volues. Nevertheless, both contributions ere troduced here for coleteness and to enable a future ileentation. The tie san for deterg the different ass flos and of the chaber has to be chosen sufficiently sall, the follog exale to 0. s, so that the fal ressure and teerature distribution becoe alost deendent of any further tie reduction of each sub-iteration and has to be adated certa circustances, deendg on the oeratg ot of the scre u. 4. Chaber flo rocess The chaber flo rocess consists of the flo of the gaseous hase, then the flo of the liquid hase - both at constant secific total enthaly - and fally the gas coression by the creased liquid volue. At any stage, both fluids are sharg the sae ressure value and durg the flo, the ultihase fluid is not able to flo of the chaber. Furtherore, the rocess can be assued as adiabatic, ith no alied ork energy fro side and a negligible difference otential energy. The chaber flo is atheatically described belo a sequential anner. Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

6 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 6 Forulas for the chaber flo u 1 u1 = h (4.1) = 1 conservation of ass (4.) h = e = constant total sec. enthaly (4.3) const. 1 e u 1 = chaber flo equation (4.4) e u as flo,, c c,, T T c c v, v, T T = resultg T, (4.5) T, = resultg 1 T, (4.6) iquid flo,,1 T T c c T T,1 = resultg T, (4.7), as coression by liquid flo V V α, = Vcha gas volue - before (4.8), = α Vcha gas volue - after (4.9) V V κ = 1 isentroic change of ressure (4.10), Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

7 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 7 V ( κ 1) T, = T,1 isentroic change of te. (4.11) V, 5. Chaber flo rocess The chaber flo rocess consists of the flo of the liquid hase at variable secific total enthaly, then the gas exansion by the decreasg liquid volue and fally the gas flo also at variable secific total enthaly. As the case for flo rocess, both fluids are sharg the sae ressure value and durg the flo, ultihase fluid is not able to flo to the chaber. Furtherore, the rocess can also be assued as adiabatic, ith no alied ork energy fro side and a negligible difference otential energy. The chaber flo is atheatically described belo also a sequential anner. Forulas for the chaber flo 1 u1 u = h (5.1) 1 = conservation of ass (5.) h = e const. not constant total sec. enthaly (5.3) d cha cha = h du cha u cha chaber flo equation (5.4) iquid flo =,, Tcha,,1 ln c v,, T cha (5.5),1 as exansion by liquid flo V V α V, = cha gas volue - before (5.6), = α Vcha gas volue - after (5.7) V V κ = 1 isentroic change of ressure (5.8), Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

8 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 8 V ( κ 1) T, = T,1 isentroic change of te. (5.9) V, as flo, = R T R T, 1 κ 1 resultg T, (5.10), T = resultg 1 T, (5.11) 6. as-liquid heat transfer side a sgle chaber After the chaber flo and flo rocess, a tie-deendent utual heat transfer [6] through the terfacial area beteen reag fractions of the liquid and gaseous hase lead to a artial teerature or energy balancg. enerally, the teerature of the gaseous hase ill decrease and the liquid hase ill be heated u. After defg a tie san - the heat transfer duration is equal to a sgle tie ste - and the itial conditions for the teeratures of the liquid and the gaseous hase, a syste of ordary differential equations has to be solved. The heat fluxes for both hases as ell as the heat transfer flux through the terfacial area can be defed as Q dt dt & = c eat flux of the liquid hase (6.1) Q dt dt & = c, eat flux of the gaseous hase (6.) Q T = h T A T ( T T ) & eat transfer flux through the terfacial area (6.3) The syste of ordary differential equations, hich results fro the equatg of Equations (6.1) and (6.) ith Equation (6.3), has the follog for: dt dt h A ( T T ) T T = (6.4) c dt dt = h T A c T, ( T T ) (6.5) The greatest robles deterg the utual heat transfer and thus the teerature of both hases as a function of tie is the correct defition of the heat transfer coefficient and the terfacial area. The latter deends aly on the kd of the current ultihase flo attern side the chaber, hich can develo Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

9 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 9 fro a sile stratified flo to highly disersed liquid dros the gaseous hase. Therefore, a correlation ust still be found, hich takes the different flo atterns and the corresondg heat transfer coefficients to consideration. Because of the resent lack of knoledge concerng the utual hase heat transfer coefficients and the terfacial area, the roduct of both araeters as chosen a ay, that the total heat balance as reached durg a sgle tie ste, so that both hase teeratures ill be identical at the end of the heat transfer. The Figure 6.1 shos the tie-deendent hase teeratures durg the utual heat transfer rocess side the last chaber near by the discharge ort. Figure 6.1 : Tie-deendent heat transfer beteen the liquid and gaseous hase 7. The ultihase ressure-driven ga flos The erieter and radial ga The erieter ga, shon the left hand side of Figure 7.1 and Figure 1.1, is the distance beteen the er scre diaeter and the ner diaeter of the housg and it connects to chabers back-to-back. Because of the large er scre diaeter coared ith the absolute ga height, the curvature of the ga can be neglected, so that it can be treated as a rectangular clearance. The radial ga, shon the right hand side of Figure 7.1 as ell as Figure 1.1, is the sace beteen the er and the ner radius of the scres. Deendent uon the nuber of flights, it connects to chabers back-to-back or one chaber ith the next but one. The basic theory of the erieter ga is also valid for the radial ga. Only the geoetric boundary conditions are different. y dis 0 R 1 high lo high s R, lo l R Figure 7.1 : The coressible ressure-driven flo side the erieter ga ( left ) and the radial ga ( right ) Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

10 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 10 Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions To change the forer coressible sgle-hase equations to a coressible ga flo forulation, the oogeneous Equilibriu Model [9], [10] and [11] for ultihase flos, here both hases have the sae ressure, velocity, and teerature at an arbitrary osition of the ga, as used. The steady state conservation equations - adiabatic flo ith a negligible change of otential energy - for a constant or also variable ga height along the flo ath follo to: ( ) = 0 l s Conservation of ass (7.1) ( ) = s l s s l λ Conservation of oentu (7.) 0 1, = dl d c dl dt Conservation of energy (7.3) The friction factor Equation (7.) as defed for laar and rough turbulent flos, deendg on the accordg Reynolds nuber side the erieter or radial ga. To defe the ass fraction or the hoogeneous density of the fluid at the ga let, a ne correlation as develoed, hich relates the gas volue fraction at the suction side (global VF) of the scre u to the VF at the corresondg ga let. The correlation resents a sooth transition fro a liquid sgle hase leakage flo, hich still exists at a global VF of arox. 85 %, to a urely gaseous backflo at a global VF of 100 %. Solution of the conservation equations as a syste of ordary differential equations For a convenient solution of the ODE syste, the conservation equations ere transfored to a atrix forulation, see also Equations (7.4) to (7.7), and afterards solved by a fourth-order Runge-Kutta schee. All riitive variables are solved siultaneously fro the ga let ste-ise to the ga let. b X A = l (7.4) = c R x T R x, A (7.5) = T X (7.6) ( ) = 0 s 4 s dl ds λ b (7.7)

11 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 11 The follog Figures 7. and 7.3 resent exelary distributions of the three riitive variables ressure, ean velocity and teerature as ell as the distributions of the coefficient atrix deterant firstly for the erieter ga and secondly for the radial ga. Figure 7. : Distributions of the riitive variables side the erieter ga ( left ) and the radial ga ( right ) Figure 7.3 : Distributions of the coefficient atrix deterants side the erieter ga ( left ) and the radial ga ( right ) The advantage of this atrix forulation is the fact that the coefficient or the left-hand side atrix A of the syste of conservation equations serves as an dicator for the critical flo condition. If the ga flo reaches anyhere the critical seed, the ressure gradient ill be fite at this location. Siultaneously, the deterant of atrix A ill becoe zero. Therefore, the let velocity ust be creased ste by ste until the deterant of the coefficient atrix ill be zero, then the critical ass flux density is detered. The erieter ga and the radial ga are both affected by this critical flo effects. For the scre u odel it ill be assued, that the critical seed or the critical ressure is reached at the erieter ga let or the radial ga ni region. Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

12 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 1 8. Otiisation or adjustent of the ga flo velocity In the case of the coressible erieter and radial ga flo, the alication of boundary conditions at the ga let and let is not ossible. In the one-diensional coressible case, the ean velocity can be detered by the ga ressure difference and the friction factor. In the coressible case hoever, only boundary conditions at the let could be alied. In contrast to the therodynaic variables let ressure and teerature, the let velocity is unknon. But there is an additional variable, hich is knon: the let ressure or the ressure of the next chaber. Therefore, it is ossible to estiate an let velocity and calculate the corresondg let ressure and iterate the rocess until the calculated let ressure is equal to the ressure of the next chaber. To ake this iteration rocess faster and ore efficiently, an otiisation rocess, see Figure 8.1, is used. Velocity Estiation Otiisation Process ODE Syste of Coressible a flo Function Evaluations no If (criterion < ε) yes Correct Velocity Distribution Figure 8.1 : Schee of flo velocity adjustent The otiisation algoriths, hich are used to otiise a syste or to iise a criterion, deend on the kd of syste equations (lear or non-lear) and on restrictions for the otiisation variables. Very oular and also efficient algoriths are the Quasi-Neton ethod (Fletcher, Poell) for non-restrictive otiisation robles and the Sequential Quadratic Prograg (Poell, Schittkoski) for ore colex and restrictive robles, hich are both available the atheatical rograe syste MATAB. An advantage of the Quasi-Neton ethod is the sensitivity toards itial conditions for the otiisation variables, hich are located far aay fro the iu. This is the reason, that this ethod is often refered for global otiisation robles. The otiisation variable is the flo velocity 0, for hich additional forations like itial conditions, uer and loer bounds have to be defed. The otiisation criterion, hich has to be iised, is the difference beteen the calculated let ressure and the actual ressure the next chaber: f crit let real = (8.1) But for critical coressible flos, it ust be considered, that the ressure difference criterion is not valid anyore, if the critical ass flux is reached. A further ressure reduction ill not cause an crease of the let velocity, so that the velocity has to be fixed, if the deterant of the coefficient atrix decreases to zero at the erieter ga let or the radial ga ni region. Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

13 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite Results of the scre u siulation The follog exelary ultihase scre u siulation vestigates an arbitrary oeratg ot ith an overall differential ressure of 10 bar, a global gas volue fraction of 96 % and a rotational seed of 900 r. This rotational eriod is characterised by three closed chabers beteen the discharge and the suction side, contrast to to closed chabers durg the rest of a sgle rotation. The flo teerature at the suction side as set to 300 K for both hases. The ost iortant geoetric araeters of the scre u can be taken fro Table 9.1. Descrition Sybol Value Unit Outer scre diaeter D 100 Inner scre diaeter d 70 Thread itch h 50 Scre length l S 10 Nuber of flights N --- Chaber volue V cha 44 c 3 Perieter ga height s P 188 µ M. radial ga height s R 105 µ Table 9.1: Iortant geoetric araeters The ressure distribution side the scre u is resented by Figure 9.1. The horizontal stes dicate the hoogeneous ressure field side each chaber. Above a VF of arox. 85 %, the ressure distribution is loosg ste by ste the forer arabolic shae, hich had a large ressure gradient near by discharge side. At a VF of 96 %, the leakage flo is already characterised by to hases and has a decreasg density and viscosity, so that the sealg behaviour is also decreasg. As the backflo crease, the ressure rises each chaber and the ressure rofile returns to learity. Figure 9.1 : Pressure distribution side the scre u Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

14 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 14 Figure 9. and 9.3 are shog the therodynaic behaviour of the liquid and the gaseous hase beteen the suction and the discharge ort. The largest gradient at this high VF is occurrg beteen the suction side and the first chaber, because of the backflo of already heated fluid fro the discharge to the different chabers. At the oent, here the last chaber oens to the discharge ort, the reag gas hase is coressed by the fast exandg fluid of the discharge, so that the gas teerature ill be creased aga, before the chaber content is released to the high ressure side. Figure 9. : iquid teerature distribution side the scre u Figure 9.3 : as teerature distribution side the scre u Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

15 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 15 The gas densities as ell as the gas volue fractions each chaber are resented by the to follog Figures 9.4 and 9.5. The gas densities are sily detered by the chaber ressure, the gas teerature and the use of the equation of state for air as an ideal gas. While the gas densities are creasg along the conveyance direction, the VF s are decreasg because of the backflo of the liquid hase and the resultg coression of the gaseous hase. Figure 9.4 : Chaber gas densities side the scre u Figure 9.5 : Chaber VF s side the scre u Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

16 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 16 Figure 9.6 shos the convergence of the voluetric efficiency as a function of chaber changes. After arox. 60 chaber iteration loos or 30 sdle rotations, a steady state is reached. Figure 9.7 resents the residual histories of the chaber ressure and both hase teeratures for the last chaber near by the discharge ort as an exale. The residuals are the differences of a certa variable beteen to consecutive chaber changes and have to decrease tie order to obta good convergence behaviour. The chaber ressure converges after only 7 iterations, hereas both teeratures converge after 45 chaber changes. Figure 9.6 : Convergence history of the voluetric efficiency Figure 9.7 : Residual histories of the ressure and hase teeratures of the last chaber Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

17 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 17 The voluetric efficiency durg a conveyance eriod of three closed chaber is equal to 88.7 %, hich can be also seen by the steady state Figure 9.6. Weightg both nubers of existg chabers to calculate the real leakage volue flo, ill result an tegral real volue flo of /h. Considerg the theoretical volue flo of /h, this ould corresond to a fal voluetric efficiency of 64.4 %. 10. Conclusion The resented ultihase scre u odel as develoed for the rediction of the u conveyance behaviour esecially at very high gas volue fractions above 85 %. To satisfy also the hysical requireents on a ultihase, an alost urely gaseous flo, and coressible leakage flo, a ne ultihase ga flo sub-odel as troduced. This odel is able to take to account the coressibility and acceleration effects of the leakage flo. The odel is also caable of dealg ith an extree condition of critical flo, hich can occur a ga beteen to chabers havg a large difference of static ressure. The consideration of these effects allos a ore recise siulation of the therodynaic behaviour and u characteristics at very high gas volue fractions u to the conveyance break-off. Acknoledgeents The authors ould like to thank the coany EISTRITZ for all the foration ab ultihase scre us and their additional technical suort and also the Deartent of Mechanical and Utility Engeerg of the University of Alied Sciences Nureberg for the coutational assistance and facilities. Noenclature eneral sybols reek etters A area (general) α gas volue fraction c secific heat caacity difference d ner scre diaeter ε sall criterion value D er scre diaeter, deterant η efficiency e secific total enthaly κ isentroic exonent E energy (general) λ friction factor f criterion function density g gravitational acceleration h secific enthaly, heat transfer coefficient, thread itch Subscrits enthaly, height l length cha chaber ass crit criterion N nuber of flights gaseous static ressure h hydraulic q secific heat energy hoogeneous Q heat energy T heat transfer R dividual gas constant, radius flo s ga height, secific entroy k ketic t tie liquid T teerature iu u secific ternal energy flo U ternal energy constant ressure v secific volue ot otential V volue (general) S scre ga flo velocity tech technical W ork energy v constant volue x ass fraction vol voluetric x,y,z coordate directions 0 itial condition Vectors / Matrices Abbreviations A coefficient atrix dis distance b right hand side vector D discharge X vector of riitive variables VF gas volue fraction ODE ordary differential equation P erieter ga R radial ga S suction Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions

18 Schriftenreihe eorg-sion-oh-fachhochschule Nürnberg Seite 18 References [1] WINCEK, M.: Zur Berechnung des Förderverhaltens von Schraubensdeluen bei der Förderung von Flüssigkeits / as-eischen, Dissertation, Universität Erlangen-Nürnberg, 199 ( The calculation of the conveyance behaviour of scre us at the conveyance of liquid/gasixtures, Ph.D. thesis, University of Erlangen-Nureberg, 199 ) [] KÖRNER,.: Zu Förderverhalten von Schraubensdeluen für Zeihasengeische hohen asgehalts, Dissertation, Universität Erlangen-Nürnberg, 1998 ( The conveyance behaviour of scre us for to-hase ixtures ith high gas-volue-fractions, Ph.D. thesis, University of Erlangen-Nureberg, 1998 ) [3] ETZOD, S.: Verlustanalyse von Schraubensdeluen bei Mehrhasenförderung, Dissertation, Universität annover, 1993 ( eakage analysis of scre us durg ultihase conveyance, Ph.D. thesis, University of anover, 1993 ) [4] NAKASIMA, C.Y. / OIVEIRA, S. / CAETANO, E.F.: Thero-hydraulic odel of a t-scre ultihase u, ASME IMECE 04, Anahei, USA, 004 [5] RAUSC, T. / VAUT, T. / BRANDT, J.U. / MEWES, D.: A odel for the deliverg characteristic of ultihase us, 4 th North Aerican Conference on Multihase Technology, Banff, Canada, 004 [6] STEPAN, K. / MAYINER, F.: Therodynaik - rundlagen und technische Anendungen - Band 1 Estoffsystee, Srger-Verlag, Berl / eidelberg, 1986 ( Therodynaics - fundaentals and technical alications - vol. 1 sgle aterial systes ) [7] ANE, E.: Technische Therodynaik - Eführung und Anendung, Addison-Wesley, Bonn, 1993 ( Technical therodynaics - troduction and alication ) [8] WANER, W.: Wäreübertragung, Vogel Verlag, Würzburg, 1998 ( eat transfer ) [9] WAIS,.B.: One-diensional To-hase Flo, Mcra ill Inc., 1969 [10] EVY, S.: To-hase flo colex systes, John Wiley & Sons, Ne York, 1999 [11] MAYINER, F.: Ströung und Wäreübergang as-flüssigkeits-eischen, Srger-Verlag, Wien / Ne York, 198 ( Flo and heat transfer gas-liquid ixtures ) [1] BESTE, D.: Analyse und Otiierung von Mehrkörersysteen, Srger-Verlag, Berl / eidelberg, 1994 ( Analysis and otiization of ulti-body systes ) [13] CORRADINI, M..: Multihase Flo : as/iquid (The andbook of Fluid Dynaics), ed. R. W. Johnson, CRC Press C / Srger-Verlag, Boca Raton / eidelberg, 1998 Thero- and fluid dynaic odel of a ultihase scre u, oeratg at very high gas volue fractions