Development of a finite volume model for the compressible gap flow inside a screw pump

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1 Deeloment of a fnte olme model for the omressble ga flo nsde a sre m Dl.-Ing. (FH) Klas äbger rof. Dr..M.A. Masod rof. Dr. John Ward Unersty of Glamorgan Shool of ehnology Wales UK rof. Dr. G. Hasmann Georg-Smon-Ohm Fahhohshle Nürnberg Fahbereh Mashnenba nd ersorgngstehn Abstrat Sre ms are dely sed n ondtons of oeraton here a onstant flo rate and lo lsaton are desrable. he tye of sre m hose gas are theoretally nestgated n ths aer s a tn sre m hh s sed for mlthase oeratons. De to the la of noledge onernng the mng behaor at ery hgh gas olme fratons to % the omressble ga flo nsde a sre m has to be nestgated th tehnqes sh as omtatonal fld dynams more arately. he deeloment of a fnte olme model for the omressble flo n eah ga allos a redton of the thermodynam behaor and of the nner leaage flo rate and ths the effete flo rate of the m. he resltng ressre and temeratre dstrbtons ll enhane the nderstandng of the m oeraton. Key ords : sre m - fnte olme - gas olme fraton ISSN Sonderdr Shrftenrehe der Georg-Smon-Ohm-Fahhohshle Nürnberg Nr. 3 Jl 5

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3 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 3. Introdton and lteratre srey A sre m s a seal tye of rotary dslaement ms n hh a nmber of sres rotates nsde a ylndral hosng. he geometry and rotaton of the sres generate a seres of losed hambers hh transort the fld from the lo ressre nlet to the hgh ressre otlet see fgre.. Lo ressre area Hgh ressre area Dreton of oneyane Fgre. : Mlthase sre m he ressre dstrbton throgh the m and hene the flo haratersts and system erformane s healy nflened by leaage of the fld from the dsharge sde ba to the ston sde. hs leaage flo ors throgh three dfferent gas nsde the sre m namely the ermeter ga beteen the sres and the hosng and the radal and flan gas beteen the matng srfaes of the sres see fgre.. ermeter ga adal ga Flan ga Fgre. : hree dfferent nd of gas nsde the sre m reos nestgatons of mlthase sre ms hae largely been onerned th the general mng behaor of these systems []. Moreoer these stdes hae been restrted to relately medm szed ms n hh the maxmm oer onsmton and gas onentratons are relately modest. In these statons the heat aaty and densty of the gas-lqd mxtre s domnated by the lqd hase so that the mng roess s manly sothermal and thermodynam effets an be negleted [] and [3]. Hoeer ths assmton annot be stfed for larger more oerfl sre ms hh are aable of oneyng to-hase flds th ery hgh onentratons of the gaseos hase to %. On the other hand ast nestgatons sed a more smle theory for the flo throgh the gas and often neglet the omressblty of the gaseos hase. Another area of researh s the alaton of sre ms n blood oneyane. Wth the fos on lo hemolyss a hannel flo model as ntroded to nestgate the eloty feld the flo rate and the shear stress dstrbton [4]. he model as soled by a fnte analytal method th the assmtons that the fld s nomressble and the flo s flly deeloed n hannel dreton. Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

4 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete Deeloment of a fnte olme model for the omressble ga flo nsde a sre m 4 De to the aboe-mentoned la of arate models t as neessary to reate models for the ga flo sng omtatonal fld dynam to redt the leaage flo more arately. Bease of aldaton reasons and the fat that ths frst modellng stage refers only to sngle-hase flds the CFD model for the omressble ga flo as analysed for ar as an deal gas.. Conseraton eqatons for the ga flo he goernng eqatons for the flo feld nsde the three dentfed gas an be reresented by the omressble Naer-Stoes eqaton n a etor form as L K J I z y x t (.) he omonents of eah etor reresent seqentally the onseraton eqatons of mass momentm n three sae dretons and fnally energy. etor hh reresents the nsteady term of eqaton. ( ) I (.) etor hh reresents the steady omonent of eqaton. n the x - oordnate dreton ( ) x xz xy xx xz xy xx J (.3) etor hh reresents the steady omonent of eqaton. n the y - oordnate dreton ( ) y yz yy yx zy yy xy K (.4)

5 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 5 etor hh reresents the steady omonent of eqaton. n the z - oordnate dreton xz yz zz ( ) zx zy zz z L (.5) 3. Fnte olme aroah he fnte olme method s smlar to fnte dfferenes method (FDM) n that t s a ery effent allaton method n the feld of omtatonal fld dynams and s haratersed as hang a hgh aray omared to fnte element method (FEM) and by a hgher flexblty omared to the FDM. he fnte olme aroah [5] and [6] s to ntegrate the derate terms n the Naer-Stoes eqatons th reset to the three sae oordnates oer the hole flo doman and then transform the reslts th the dergene theorem by Gass. Integraton of the Naer-Stoes eqaton t I d J d d d x K y L (3.) z Dergene theorem by Gass d d F da ( F n)da F (3.) A A A n A (3.3) l l l esltng eqaton (formlaton for eah ell of the hole flo doman) d dt 3 6 ( Fml Aml ) m l I (3.4) th m... 3 for eah derate n the three oordnate dretons l... 6 for eah fae of the ell here th a bod s shae he resltng eqaton 3.4 s already formlated for a 3-dmensonal fnte olme ell hh ll be generated by a strtred dsretsaton of the hole flo doman. As an examle a sngle ell th all notatons an be seen belo n fgre 3.. Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

6 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 6 n 6 n SW_U NW_U n 4 NE_U z NW_L SE_U y SW_L x n 3 NE_L n SE_L n 5 Fgre 3. : 3-dmensonal fnte olme ell he follong form of eqaton 3.4 allos ther dret se for the nmeral allaton. d dt ( J A ) ( J A ) ( J 3 A3 ) ( ) ( ) ( ) J 4 A4 J 5 A5 J 6 A6 ( K ) ( ) ( ) A K A K 3 A3 ( K 4 A4 ) ( K 5 A5 ) ( K 6 A6 ) ( L A3 ) ( L A3 ) ( L3 A33 ) ( L A ) ( L A ) ( L A ) I (3.5) In the aboe exresson 3.5 the etor I s defned by the flo ondton n the entre of the ell (ellentred formlaton). he etors J K and L are allated aganst t n the mddle of the sx ell faes A to A 6 by aeraged state ales of to neghborng ells Flx-dfferene slttng sheme he flx etor J (see eqaton.3) an be sltted - as ell as the flx etors K and L - nto to etors J and J d hh ontan on the one sde all the onete and on the other sde all the dffse arts. ( ) J (4.) Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

7 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 7 xx xy xx xz xz x J xy d (4.) In ontrast to the dffse flxes at the ell faes hh an be determned by a entral dfferene sheme the onete flxes hae to be allated n a seal manner hh sles eseally the nge-ktta sheme th enogh dssaton to reent ossbly arsng nstabltes. An often sed algorthm to allate the onete flx s oe s aroxmate emann soler [7] hh belongs to the flx-dfferene slttng shemes. he onete flx throgh an arbtrary estern ell fae see fgre 4. follos to J [ J ( I ) J ( I ) ] A ( I ) I W L oe W L (4.3) Western ell fae I (-) I () Left state ght state Atal ell Fgre 4. : Western ell fae and ther neghborng ells he exresson 4.3 s smlar to a normal entral dfferene sheme hh nldes a dssaton term. he oe-matrx A oe at the rrent ell fae ll orresond to the gradent of the onete flx etor th reset to the etor of the onserate arables f the oe-aeraged arables are sed. he oe-aeragng s a seal nterolaton sheme of the ell-entre arables of the to neghborng ells. For the onserate arables follos ~ L (4.4) L ~ L (4.5) L L ~ L (4.6) L L ~ L (4.7) L Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

8 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete Deeloment of a fnte olme model for the omressble ga flo nsde a sre m 8 L L H L H H ~ (4.8) th ( ) E H (4.9) 5. reondtonng For omressble flos at lo Mah nmbers the fld s almost nomressble. In ths ase the system of tme-deendent densty-based Naer-Stoes eqatons ll beome ery stff. he rate of stffness an be exlaned as the rato of the largest to smallest egenale. he egenales of sh a system are a th : total flo seed a : seed of sond a he entral dea of reondtonng [8] s a reaton of a remltlaton matrx for the nsteady terms n the Naer-Stoes eqatons hh hanges the egenales n sh a manner that they get loser together. De to the fat that the egenales ll hae a smlar magntde the transent etor ll be transformed from onserate to rmte arables and remltled by the reondtonng matrx. he transformaton roess nldes the follong stes: he orgn s eqal to the etor I hh ontans all transent terms reresented by onserate arables - see term.. he Jaoban matrx of ths etor th reset to the rmte arables H H I (5.) th (5.) and (5.3) and the etor th all nsteady terms reresented by rmte arables (5.4) ll be embedded n the follong exresson

9 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete Deeloment of a fnte olme model for the omressble ga flo nsde a sre m 9 d z d y d x d t L K J I (5.5) After some addtonal mathematal oeratons for frther detals see referenes [7] and [8] the reondtonng matrx and t s mlementaton are: Θ Θ Θ Θ Θ H H Q (5.6) th Θ ref (5.7) d z d y d x d t L K J Q (5.8) 6. me steng method for the ga flo he tme steng method hh as hosen for ths mlementaton s a forth-order nge-ktta sheme [6] (forth-order aray n tme). ( ) ( ) n (6.) ( ) ( ) ( ) t α (6.) ( ) ( ) ( ) t α (6.3) ( ) ( ) ( ) 3 3 t α (6.4) ( ) ( ) ( ) ( ) ( ) ( ) 6 t α (6.5) ( ) ( ) 4 n (6.6) th α α 3 α and 4 α he etor of the HS ontans all the onete and dffse flxes and s exressed by the onserate arables. Bease of the reondtonng the transent etor nldes the rmte arables so that a re-transformaton of arables s neessary at the end of eah ell allaton.

10 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 7. Dssson of reslts 7. he ermeter ga he fgres 7. and 7. sho the dstrbtons of the elotes n x- and y-dreton as a fnton of the ga heght and length nsde the ermeter ga hh has a total heght of mrometers. In the ase of a arabol eloty rofle n x-dreton the sos fore hh s defned by the gradent of the eloty omonent n y-dreton mltled th the dynam sosty and the etted area mst be eqal to the sm of the ressre fores normal to the hannel nlet and otlet. he fgres 7.3 to 7.5 reresent the normal behaor of a omressble ga flo. he densty dereases as ell as the ressre. he negate ressre gradent n flo dreton s the drng fore for the fld moton. he dfferene beteen the nlet and the otlet ressre ll lead to the aboe mentoned seond fore. he temeratre shold normally also dereases bt the sos fores hh at on the fld an heat the flo throgh the ga. An ndator for ths resmton an be seen n fgre 7.5 here the regon lose to the all (hghest shear stresses) has hgher temeratre ales n ontrast to the mddle of the ga here the shear stress s beomng zero. he fgres 7.6 and 7.7 sho the tme hstory of the flo arables densty elotes n x- and y-dreton and ressre and also ther resdals. All the tme-deendent arables shold reah a steady-state at the end. For a better erfaton also the resdals ere determned. he resdals are the dfferenes of the flo arables of to onsete teraton stes and shold beome smaller th eah tme ste. In the deal ase they beome zero. Fgre 7. : Contor lot of the eloty omonent Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

11 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete Fgre 7. : Contor lot of the eloty omonent g/m 3 Fgre 7.3 : Contor lot of the densty Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

12 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete a Fgre 7.4 : Contor lot of the ressre K Fgre 7.5 : Contor lot of the temeratre Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

13 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 3 g/m 3 a Fgre 7.6 : Flo arables as a fnton of tme-stes g/m 3 a Fgre 7.7 : esdals (logarthm) as a fnton of tme-stes Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

14 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 4 7. he radal ga he radal ga - see the hte ells n fgre s defned by to onter-rotatng heels and has a mnmm heght of 5 mrometers n ths ase. he drng fores n ths ga as ell as n the general ase of the ermeter ga are the rotatng or mong alls and the ressre gradent along the ga axs here the ressre dfferene lays the maor role. Fgre 7.9 shos the ressre dstrbton nsde the radal ga. he reasons for ths dstrbton are the frstly onergent ga shae and therefore the resltng fld aeleraton and the frtonal ressre dro. De to the large frtonal ressre dro the aeleraton ressre dro belo the otlet ressre an be seen only to a lttle extent n the last thrd of the radal ga. he tme hstory and the logarthm resdals of the onserate flo arables as a fnton of tme-stes an be seen fnally n the fgres 7. and 7.. Fgre 7.8 : Ga geometry n the n regon th srrondng dmmy ells a a Fgre 7.9 : ressre dstrbton n the n regon Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

15 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 5 g/m 3 a Fgre 7. : Flo arables as a fnton of tme-stes g/m 3 a Fgre 7. : esdals (logarthm) as a fnton of tme-stes Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

16 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete he flan ga he nflene of the rotatng flans hh defne the flan ga and also resent to eloty bondary ondtons as mong alls an be seen n fgre 7. as a eloty etor lot. he reason for the hgh onentraton of eloty etors at both ends les n hgh nmber of ells hh are beomng smaller and smaller. Fgre 7.3 shos the eloty etor lot of the fld hh flos beteen the rotatng flans and s also dren by a negate ressre gradent. he ontor lot of the total eloty n the mddle of the flan ga (y) s shon n fgre 7.4. he ostons of both eloty maxmms are self-edent from the eloty bondary ondtons of fgre 7. or the fat that large dstanes from the entre of rotaton ll lead to hgh all elotes. he ostons orrelate th the ostons of a large rads of the rotatng flans and therefore hgh elotes. Fgre 7. : eloty bondary ondtons Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

17 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 7 Fgre 7.3 : eloty etor lot (mddle seton) Fgre 7.4 : Contor lot of the total eloty Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

18 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 8 8. Conlson he omressble flo throgh the ermeter ga radal ga and the flan ga s resonsble for the leaage of the hole sre m. he exat allaton of the leaage flo rate and ths the net flo rate of the m old be arred ot by omtatonal fld dynams. A fnte olme model hh s rogrammed and soled n MALAB ll be able to determne the one- to- and three-dmensonal dstrbtons of the fld arables densty elotes n all doman dretons ressre and temeratre nsde the gas. he rogram has a ery flexble strtre and arametr stdes onernng the bondary ondtons are ossble. he bondary ondtons are easly defned by the densty and elotes at the nlet the ressre at the otlet the mong alls and the geometry of the orresondng tye of ga. De to the omressblty of the fld hh as taen nto onsderaton th ths fnte olme model the thermodynam behaor old be estmated. hs fat ll be artlarly mortant n the feld of mlthase oeratons th ery hgh gas-olme-fratons here the oor lqd hase s not able to absorb the omresson heat of the gaseos hase. Nomenlatre General Symbols a seed of sond A area of a ell fae sef heat aaty e sef nternal energy E total nternal energy H total enthaly thermal ondtty stat ressre gas onstant t tme temeratre eloty omonent n x y and z - dreton total flo seed ell olme x y z oordnates etors / Matres A normal etor mltled by the ell fae area ; oe Matrx I arts of the Naer-Stoes eqaton hh are dfferentated th reset to t J K L arts of the Naer-Stoes eqaton hh are dfferentated th reset to x y and z n normal etor etor of rmte flo arables Q reondtonng matrx Gree Letters α nge-ktta fator µ dynam sosty densty shear stress Θ axlary term n the reondtonng matrx Sbsrts onete d dffse E N S W ardnal onts : east north soth and est ell ndators Deeloment of a fnte olme model for the omressble ga flo nsde a sre m

19 Shrftenrehe Georg-Smon-Ohm-Fahhohshle Nürnberg Sete 9 l ell fae area ndex L loer ; left m oordnate dreton ndex onstant ressre ; deraton th reset to ressre at onstant temeratre rght deraton th reset to temeratre at onstant ressre U er onstant olme olme x y z oordnates Sersrts n rrent tme ste n follong tme ste eferenes [] WINCEK M.: Zr Berehnng des Fördererhaltens on Shrabensndelmen be der Förderng on Flüssgets / Gas-Gemshen Dssertaton Unerstät Erlangen-Nürnberg 99 ( he allaton of the oneyane behaor of sre ms at the oneyane of lqd/gasmxtres h.d. thess Unersty of Erlangen-Nremberg 99 ) [] KÖNE H.: Zm Fördererhalten on Shrabensndelmen für Zehasengemshe hohen Gasgehalts Dssertaton Unerstät Erlangen-Nürnberg 998 ( he oneyane behaor of sre ms for to-hase mxtres th hgh gas-olme-fratons h.d. thess Unersty of Erlangen-Nremberg 998 ) [3] EZOLD S.: erlstanalyse on Shrabensndelmen be Mehrhasenförderng Dssertaton Unerstät Hannoer 993 ( Leaage analyss of sre ms drng mlthase oneyane h.d. thess Unersty of Hanoer 993 ) [4] KILANI M.I. / JAW S.Y. / HAIK Y. / CHEN C.J.: Nmeral smlaton of flo n a sre m Forteenth Engneerng Mehans Conferene EM Ameran Soety of Cl Engneers May -4 [5] OEEL J. H. / LAUIEN E.: Nmershe Strömngsmehan ( Nmeral fld mehans ) Srnger-erlag Berln Hedelberg 995 [6] WEND J.F.: Comtatonal Fld Dynams Srnger-erlag Berln Hedelberg 996 [7] BLAZEK J.: Comtatonal Fld Dynams: rnles and Alatons Elseer Oxford [8] WEISS J.M. / SMIH W.A.: reondtonng aled to arable and onstant densty flos AIAA Jornal olme 33 Nmber ages 5-57 Deeloment of a fnte olme model for the omressble ga flo nsde a sre m