Multi-criteria optimization of pitch curves for screw-type vacuum pumps

Transcription

1 Mult-crtera optzaton of ptch curves for screw-type vacuu pups Dpl.-Ing. D. Pfaller, Prof. Dr.-Ing. A. Brüer, Prof. Dr.-Ing. K. Kauder Fachgebet Fludtechnk, Technsche Unverstät Dortund, Gerany Abstract The present artcle on the energetc optzaton of dry-runnng screw-type vacuu pups deals wth the desgn of ptch curves for screw-type vacuu pups. An analyss s ade, n partcular, on the theoretcal power deand. For the developent of the coputaton odel, the hgh nuber of workng chabers on the rotors of the screw spndle s reproduced by eans of a seres connecton of ndvdual pup stages. The volues of the chabers are vared at constant overall chaber volue of the screw-type vacuu pup. The dsspatve gap ass flows arsng due to pressure dfferences between the workng chabers are represented by eans of a lossy gap flow. Wth regard to a nal specfc nternal work, a ptch curve turns out to be optal when t devates fro the current desgns of screw-type vacuu pups, n partcular n the central rotor secton. Introducton At the char of fludcs, the desgn possbltes of screw-type vacuu pups are surveyed wthn the fraework of a research project supported by the Geran Research Foundaton (DFG). The research project s aed at the developent of a ethod for the energetc desgn of dry-runnng screw-type vacuu pups. In practce, the screw-type vacuu pup s characterzed, above all, by ts hgh achevable pressure rato, the hgh sucton capacty and the possblty to be operated n a dry -runnng ode []. The large sucton-pressure range, reachng fro low to fne vacuu, represents a challenge for the therodynac descrpton of the copresson process. In addton to the hgh achevable pressure ratos of ore than one llon and the resultng gas teperatures, also the dfferent fors of flow, fro contnuu to olecular flow, coplcate the analytcal descrpton of flow processes nsde the pup. The frst spndle pups on the arket showed a constant ptch and were characterzed by an sochorc transport phase [2]. Fro an energetc pont of vew, however, a copresson of the workng gas s desrable, so that pups wth nternal copresson were developed [3]. A possblty to reduce the volue of the workng chaber pror to ts openng towards the pressure sde would be the desgn of a dscharge contour n the housng. Ths decelerates ) Ths work s kndly supported by the Deutsche Forschungsgeenschaft (DFG)

2 the openng of the workng chaber by a defned rotaton angle and thus generates the copresson of the enclosed workng gas. Another possblty would be the odfcaton of the workng chaber geoetry durng the conveyng phase. For ths purpose, ether the ptch along the rotor axs s changed or the rato of rotor head daeter and root daeter. Both ethods result n a varaton of the workng chaber volue along the rotor axs. Here, the queston arses of how should the ptch curve of the workng chaber volue for a gven pressure rato along the rotor axs be desgned to acheve an energetcally favourable operatng behavour. Apart fro the pressure rato of the adjonng workng chabers, also the gap ass flow s portant for the desgn of the ptch. A sple odel calculaton shall llustrate the nfluence of the gap ass flow on the dscharge ass flow of the stages. A screw-type vacuu pup wth a sucton pressure of 00 Pa and a sucton capacty of h - delvers a ass flow of approx kg s -. The gap ass flow for a gap n the area of the hgh-pressure sde (), known fro varous calculatons [4], s at about 0.02 kg s -. In ths exaple, the dscharge ass flow of the vacuu pup accounts for only 0.3% of the dscharge ass flow of the stage at the pressure sde. The odel calculaton shows that the ptch desgn of the workng chaber volue analogous to the ult-stage copressor structure, where a constant pressure rato s appled to the ndvdual stages, s not effectve n case of the screw-type vacuu pup. 2 Model concept and calculaton bases The rotor of a screw spndle for the applcaton as vacuu pup s characterzed by a hgh wrap angle, Fg.. Ths also results n a hgh nuber of workng chabers between sucton Fg. Par of rotors of a screw-type vacuu pup and pressure sde. The hgh nuber of workng chabers along the rotor axs leads to the assupton that the pressure dstrbuton wthn the screw-type vacuu pup can be calculated usng the conservaton of ass prncples and thus enables both nforaton about the attanable fnal pressure and the sucton capacty at deterned sucton pressures. The theral nfluences on the operatonal behavour of a screw-type vacuu pup wth the usual very hgh pressure ratos are not be neglected n steady-state operaton, but closely related to the calculaton of the gas teperatures arsng fro theral stress as well as to the heat transfer and to the coolng of the coponents.

3 Despte approaches whch nclude an ntegral calculaton of the heat balance of a screw-type vacuu pup [5], the heat transfer condtons n vacuu have not yet been fully researched. However, n order to enable the provson of verfable nforaton on the operatng characterstcs of the screw-type vacuu pup, a cold thus therally undefored achne s assued below. Ths dealzed state can be sulated n test operaton by short easurng cycles at the cold achne and can thus be verfed. For the calculaton of chaber pressures and ass flows, the closed workng chabers of the screw spndle are connected by pup stages (Fg 2). In ths case, the dscharge volue flow & of stage results fro the workng chaber volue V C, and the rotatonal speed n. V C, In cobnaton wth sucton pressure p C, and gas teperature T C, pror to stage, ths results n a dscharge ass flow The local gap ass flow the gven pressure rato & of stage. C, & of the -th stage results fro the respectve gap area A G, and G, p + Π = over the gap of stage for subcrtcal pressure ratos p Π < 0.528( Ar) to G p 2 κ, = α G, AG, C, + R T κ κ C, + ( Π ) ( Π ) κ R T κ C, + and for supercrtcal pressure ratos Π > 0.528( Ar) to G, 2 p 2 κ &. C, + = α G, AG, R TC, + κ + R TC, + κ + For a steady-state operatng pont wth defned sucton capacty and/or dscharge ass flow, the dscharged ass flow LP = s added to the local gap ass flow & G, of the ndvdual stage resultng n a dscharge ass flow & of the respectve stage C, & & + & C, = G,. For steady-state fnal pressure operaton, the local dscharge ass flow of each stage & C, ust correspond to the local gap ass flow &. G, The control factor n the calculaton of ass flows s the pressure n the workng chabers p C,. The calculaton of the nternal specfc work per stage w occurs on the assupton of an sentropc copresson process for the workng edu ar whch s regarded as deal gas.

4 w = c p T C, Π κ κ LP & G, V C, V C, V C, + V C, N+ = p C, = p LP p C, p C, + p C, N+ = p T C, = T LP T C, T C, + T C, N+ = T & C, Fg. 2: Abstracton of the spndle nto a seres connecton of workng chabers and stages wth gap and dscharge ass flow of the ndvdual stage [ N], Stage ndex & C, Conveyng ass flow over stage V C, Workng chaber volue p C,I Pressure pror to stage T C, Teperature pror to stage & Gap ass flow over stage G, Mass flow through -port LP Mass flow through LP-port The ultplcaton of the respectve dscharge ass flow & of stage results n the nternal C, power P of stage. The nternal power of the seres-connected pup stages deterned by the addton of the ndvdual stage powers P can be P = N P = N = = w. C, Snce nether the heat transfer condtons wthn the workng chabers nor the utual nfluence of the energy transported va gap ass flows are covered, the splfed approach of coplete re-coolng of the ar n the chabers between the stages s chosen. For the purpose of coparatve calculatons however, a re-coolng rate per chaber can be appled to the odel. The basc concept of the calculaton approach goes back to an artcle on screw achnes wrtten by K. Kauder and A. Rohe [6]. The adopted approach wll be expanded by the gap connectons of the blow hole, the profle gap and the radal gap. 3 Splfed odel calculaton for a two-stage screw-type vacuu pup In order to llustrate the nfluences of the odellng, a two-stage copresson s detaled n the frst step. Here, optzaton regardng the specfc nternal work w ges s carred out. For

5 calculaton purposes, the coplex gap stuaton n the spndle s splfed n order to ensure the traceablty of the calculaton steps. A lossy gap connecton, wth denson and connected chabers correspondng to the housng gap, s added to each pup stage. Gap connectons correspondng to a connecton between two workng chabers whch are not drectly seres-arranged n dscharge drecton are not taken nto account n ths odel calculaton. The gap areas A G, = A G = const. as well as the operatng paraeters are constant for all stages. Wthn the fraework of the splfed approach, the orfce coeffcents α G, = α G = const. of the gap are held constant. Thus, the nternal power of the screw spndle only depends on the stage pressure ratos and the dscharged ass flow The odellng s based on a constant al volue of the workng chabers V n = VC = =, const... Ths approach corresponds to the assupton of a constant rotor length and scoop area for the screw-type vacuu pup. Subsequently, the energetcally optal allocaton of the constant al volue to the avalable workng chabers.e. to the workng chabers V C, and V C,2 of the exaple - s searched for. Wth real rotor geoetry, the varaton of the workng chaber volue per stage corresponds Table : Paraeters for the calculaton of a two-stage vacuu pup Specfc heat capacty c p 005 J kg - K - Gas teperature T LP = T 293 K Gas constant R 287 J kg - K - Gap area A G Orfce coeffcent α G 0,8 Total volue of workng chabers V Dscharge pressure p 0 5 Pa Rotatonal speed n 5000 n - pup, pressure p C,2 and dscharge ass flow to a varaton n the ptch between the workng chabers at constant rotor length. For the purpose of the odel calculaton, all other geoetry and operatng paraeters are held constant, see Table. Thus, for the calculaton of the pressure dstrbuton and the dscharge ass flow of the screw-type vacuu pup an equaton syste of one equaton per stage results. Therefore, sucton and dscharge pressure (here n general 0 5 Pa) as well as the ntake teperature and rotatonal speed of the screwtype vacuu pup are predeterned. For the exaple of a two-stage screw-type vacuu for varably deterned chaber volues V C, and V C,2 at constant al chaber volue V are deterned by teraton. Fro the

6 subsequently gathered values for pressures and pressure ratos, the nternal powers per stage as well as the nternal power of the screw-type vacuu pup result. Apart fro the nternal power, the specfc nternal work w s of partcular nterest for the energetc desgn. Snce due to the varable allocaton of the al volue, the workng chaber volue on the sucton sde vares, the theoretcal dscharge ass flow s also varable even at constant sucton pressure. The calculated specfc works of the screw-type vacuu pup for dfferent ratos of the workng chaber volues are shown n Fg Specfc work w [J kg - ] A B C V C,2 / V C, Fg. 3 Curve of the specfc work w for two-stage copresson wth varaton of the volue rato between the frst and second workng chaber at a constant al volue of the workng chabers and dfferent sucton pressures A: Pa (200 bar), B: Pa (300 bar), C: Pa (400 bar) At frst, the calculated specfc work w shows a declne wth the ncreasng volue rato of the workng chabers V C,2 /V C,. At a rato of the workng chaber volues of approx. V C,2 /V C, = 0.54, a nu of specfc work s acheved at the studed sucton pressures, see Table 2. The rato ncreases to 0.58 only at a sucton pressure of Pa. The ajor nfluencng factor for achevng these nal values n the specfc al work w s, apart fro the dscharge ass flow & =, the al aount of the power per stage LP P, for the two stages consdered. The power C P, of a stage s calculated fro the appled C stage copresson Π and the dscharged ass flow of the stage & C,. In a frst approxaton, the stage pressure ratos vary analogously to the chaber volues.

7 Table 2: w The gap ass flow & relevant for the G, respectve dscharge ass flow per stage & depends both on the stage pressure C, alzed power P s attaned at a volue rato V C,2 /V C, of The dscharge ass flow LP rato and on the densty of the gas when flowng nto the gap. Wth ncreasng chaber sze, the stage pressure rato ncreases and, consequently, also the rato of the gap ass flows. Once the crtcal pressure rato over the gap s reached, the gap ass flow depends only on the densty of the workng gas at the gap nlet. Thus, the nternal work w C, of the frst stage ncreases wth the decreasng volue rato V C,2 /V C,, and the nternal work of the second stage w C, 2 decreases analogously to the ncreasng pressure rato of the stage. Under sad boundary condtons and for a sucton pressure of Pa the nal & = results fro the volue of the suckng workng chaber V C, and the returnng gap ass flow G,. At sall volue ratos V C,2 /V C,, the sze of the suckng workng chaber and thus the theoretcal dscharge ass flow are axal. Due to the sall workng chaber on the sucton sde, the stage pressure rato Π of the suckng workng chaber corresponds nearly to the al pressure rato Π ges or can even exceed t n case of extree values. For the volue rato, the returnng ass flow G, s axal and reduces the ass flow sucked fro the low-pressure range. Due to the reducton of the suckng stage, an ncrease n the volue rato of the workng chaber V C,2 /V C, results n a reducton of the theoretcal dscharge ass flow, but due to the lower stage pressure rato Πand the reduced workng gas densty between the two stages, also to the reducton of the gap ass flow G,. Because of the countervalng nfluences, the dscharge ass flow reaches ts axu value for a sucton pressure of Pa at a volue rato V C,2 /V C, of about 0.5. Paraeters for the poston of the specfc work na for dfferent sucton pressures p LP [Pa] [kg s - ] C, [kg s - ] C,2 [kg s - ] Π Π Π Π V V 2 C,2 C, P2 P P [W] P = J kg

8 The cobnaton of the dscharged ass flow and the al aount of nternal works of both stages results n the nu of specfc work for the descrbed sucton pressure of Pa at a volue rato V C,2 /V C, of about Table 2 shows a coparson of the poston of specfc work na pressures. w for dfferent sucton The pup varaton wth the hghest sucton pressure exaned does no longer show blockng of the gap flows. In the two varants wth low sucton pressure, a blockng of the gap ass flow occurs n the stage on the low-pressure sde. The rato of both stage pressure rato and absolute stage power s below. Accordngly, n both cases, a hgher value for pressure rato and nternal work s attaned n the stage on the low pressure sde. 4 Calculaton of optzed ptch curves for N-stage screw-type vacuu pups Upon copleton of the frst analyse of a two-stage screw-type vacuu pup, the next step shall be the transton to the realstcally desgned spndle geoetry. Ths requres the expanson of the nuber of stages and the odellng of further gap connectons. Furtherore, the odellng of the gap connectons coprses the housng, whch connects two drectly successve workng chabers on a rotor, the profle gap, whch connects two workng chabers of the studed two-teeth spndle that are not drectly successve n drecton of the pressure drop, and the radal gap whch reflects a connecton to the workng chabers on the counter rotor (Fg. 4) [4]. In order to cover the dfferent gap flows n the pressure range of the vacuu, the orfce coeffcent for the gap connectons s calculated fro an experentally deterned data base [6] both n dependence of pressure and geoetry. Due to the stage nubers of N = 5-2 stages coon on the arket, whch shall also be appled n the sulaton, t s no longer possble to calculate all pups upon a correspondng varaton of the chaber volues V C, at constant al chaber volue wthn a reasonable te frae, snce the nuber of pup varants ncreases exponentally wth the nuber of stages. However, n order to fnd a ptch curve wth the hghest possble resoluton whch ndcates the optu power, evolutonary algorths are appled. Here, the evolutonary approaches are based on the prncples of nhertance and utaton [7]. The pup synthess occurs n freely selectable group szes. Afterwards, the pressure profle and the specfc work and/or al power for all ebers of the group are deterned (ftness functon). In case of screw-type vacuu pups, a basc dstncton ust be ade between fnal pressure operaton and the operaton wth dscharge ass flow. For the evaluaton of the qualty of a pup varant n the feld of fnal pressure operaton, the fnal pressure attaned and the al power nput are used. Wthn the fraework of the artcle, however, only the operaton wth dscharge ass flow

9 Male rotor Feale rotor shall be analyzed. Here - analogously to the odel calculaton - a fxed ntake pressure and counter pressure are deterned for the optzaton. Another necessary paraeter n 2 2 V MC, V FC,j V FC,j+ addton to the gap geoetry s the rotor speed, whch are also predeterned. In contrast to the fnal pressure operaton, the al power related to the dscharged ass flow s used as evaluaton crteron, thus the specfc nternal work w of the whole screw-type vacuu pup. Fg. 4: Chaber odel wth a two-thread screw-type vacuu pup [4] : Housng gap 2: Radal gap 3: Profle gap Table 3: Paraeters for the calculaton of a 8-stage screw type vacuu pup Table 4 shows the coparson between the ntegral values for an 8-stage rotor wth a ptch curve optzed wth regard to the specfc nternal work (Fg. 5), a rotor wth a lnear ptch varaton and, as reference, a rotor wth constant ptch. For ths calculaton, the boundary condtons gven n Table 3 were predeterned. The coparson of the specfc work shows clearly that the evolutonary calculaton of the ptch causes a sgnfcant reducton of the specfc nternal work. In coparson, however the dscharge ass flow and thus the sucton capacty of the optzed rotor ncrease, whch causes a slght ncrease n the al power. Fg. 5 shows the related ptch curve of the optzed rotor after 00 evoluton steps. Gas constant R 287 J kg - K - Specfc heat capacty c p 005 J kg - K - Gas teperature T LP = T 293 K Total volue of workng chabers Dscharge pressure p Sucton pressure p LP 0 5 Pa ( bar) 00 Pa ( bar) Rotatonal speed n 5000 n - Table 4: Coparson of the ntegral values for dfferent ptch curves for boundary condtons accordng to Table 3. Spndle type P [W] [kg s - ] w [J/kg] V & [ 3 s - ] Constant ptch s = 30 (v, =) Lnear ptch (v, = 2.7) Optzed ptch (v, = 2.4)

10 As expected, the ptch on the LP-sde s at frst hgh. The declne n the ptch s followed by an ncrease n the ptch n the centre of the rotor. Then, at the transton to the workng chaber on the pressure sde, the ptch declnes agan Lead s [] LP-sde Stage [-] -sde Fg. 5 Ptch curve of the 8-stage rotor optzed wth regard to the specfc nternal work by eans of evolutonary algorths For the analyss of the ptch curve, the stage pressure rato Π and the stage ass flow & for the stages [,8 ] C, of the optzed rotor are shown n Fg. 6. The stage pressure rato between all stages s greater than one and reaches a axu value of four n the area of the LP-sde. A reversal of the pressure rato through the ncrease n the ptch n the central rotor secton does not exst. The curve of the stage pressure rato qualtatvely reflects the ptch curve (Fg. 5). The dscharge ass flow per stage ncreases analogously to the pressure fro the sucton to the pressure sde. The optzed rotor geoetry shown attans a dscharge ass flow of kg s -. A coparson wth the dscharge ass flows of the stages gven n Fg. 7 confrs that the workng chabers near the pressure sde are anly flled by returnng gap ass flows. The optzaton of the rotor results fro the countervalng curves of the stage pressure rato and the dscharge ass flow of the stages. Thus, at the sucton sde, the stage pressure rato s axal and a hgh specfc work s reached, however, due to the low densty of the workng gas n the sucton sde area, the conveyed ass flow ade up of gap flow and dscharge ass flow s low. The absolute power of the stage s therefore low, despte the hgh stage pressure rato. In the next stage, the stage pressure rato decreases, and the specfc work decreases accordngly.

11 Stage pressure rato Π [-] Π C, Dscharge flow of the stage C, [kg s-] LP-sde Stage [-] -sde Fg. 6 Stage pressure ratos and ass flows along the rotor stages for the rotor optzed wth regard to the specfc nternal work (Fg. 4) C 0 B 9 Pressure n the chabers p C, [Pa] A a b c Stage pressure rato Π [-] Fg LP-sde Stage [-] -sde Pressure curve and stage pressure rato over the rotor stages A: Pressure curve of the sochorc rotor B: Pressure curve of the rotor wth lnear ptch varaton C: Pressure curve of the optzed rotor a: Stage pressure rato of the sochorc rotor b: Stage pressure rato wth lnear ptch varaton c: Stage pressure rato of the optzed rotor The dscharge ass flow of the stage ncreases towards the pressure sde due to the ncreasng densty of the workng edu. Fg. 7 shows a coparson of the calculated 0

12 pressure n the stages and the stage pressure rato between a rotor wth constant workng chaber volue per stage, a rotor wth lnear ptch varaton and the rotor wth optzed ptch. Consderng the pressure curve of the sochorc rotor geoetry (a), the pressure rato n stage 3 takes the axu value. A coparson wth the pressure profle of the lnear ptch varaton (b) shows that the an porton of the pressure rato s further shfted towards the LP-sde. The optzed rotor (c) reaches the axal stage pressure rato n the area of the suckng stage, followed by a declne of the pressure ratos towards the hgh-pressure sde. Fg. 8 shows a coparson of the nternal power resultng fro the stage pressure ratos and the respectve dscharge ass flows, referrng to the dscharged ass flow. 4.E+07 4.E+07 A Specfc work w [J kg - ] 3.E+07 3.E+07 2.E+07 2.E+07.E+07 B C 5.E+06 Fg. 8 0.E+00 LP-sde Above all n the area of the workng chaber on the hgh-pressure sde, the optzed rotor (c) attans a sgnfcant reducton of the specfc stage power copared to the sochorc rotor (a) and the rotor wth lnear ptch varaton (b). The physcal reasons are to be found n the relatonshp between the stage pressure rato and the gap ass flow of the stages predonantly resultng fro the densty. 5 Suary and outlook Stage [-] -sde Curve of the specfc nternal work per stage w referrng to the sucked ass flow, plotted over the rotor stages A: Isochorc rotor B: Rotor wth lnear ptch varaton C: Optzed rotor In a splfed exaple, whch consders only the gap ass flow through the housng gap, the relatonshp between the workng chaber volues, the stage pressure ratos and the

13 sentropc copresson work for dfferent sucton pressures are llustrated. The analyss shows that t s possble to attan a nu of specfc work wth the varaton of the chaber volues at constant al volue. The expanson of the nuber of stages and the ncluson of further gap connectons as well as the applcaton of evolutonary algorths enable the subsequent calculaton of the ptch curve, optzed wth regard to the specfc nternal work, for a screw-type vacuu pup wth 8 stages. The coparson wth a rotor showng a lnear ptch varaton confrs a clear reducton of the specfc work. The experental verfcaton of the optzaton results s planned wthn the fraework of the upcong experental surveys n the char of fludcs. 6 Lst of sybols used Sybol Meanng Denson A Area 2 Cp Specfc heat capacty J kg - K - Mass flow kg s - n Rotary speed n - p Pressure Pa P Internal power W V V & Volue Volue flow 3 3 s - R Specfc gas constant J kg - K - s Ptch w Internal work J α Orfce coeffcent for gap ass flows - k Isentropc exponent - Π Pressure rato - C F G Workng chaber Feal rotor Gap 7 Lterature [] Wutz, ET AL Handbuch Vakuutechnk. Theore und Praxs, 7th extended verson Veweg, Braunschweg Wesbaden, 2000 [2] Patent DE [3] PATENT DE A, MATSUBARA KATSUMI, UCHIDA RIICHI, MATSUBARA MASATOSHI, SCREW VACUUM PUMP, OZAKI, MASAYUKI ; AKUTSU, ISAO, Schraubenkolbenaschne, [4] Wenderott, D Spaltströungen Vakuu, dssertaton, Unversty of Dortund, VDI Progress Reports, seral 7, Nr.423, VDI-Verlag, Düsseldorf, 200 [5] Rohe, A. ROHE, A.: Wärehaushalt von Schraubenspndel-Vakuupupen. dssertaton, Unversty of Dortund 2005, [6] Kauder, K. Rohe, A. Sulaton und Messung des Druckverlaufes a Bespel ener Schraubenspndel Vakuupupe, Schraubenaschnen No. 0, p , Unversty of Dortund, 2002 [7] Helpertz, M Methode zur stochastschen Opterung von Schraubenrotorproflen. Dssertaton, Unversty of Dortund 2003 LP M Hgh-pressure sde Stage ndex Low-pressure sde Male Rotor