Gas distribution through injection manifolds in vacuum systems

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1 Gas distributio through ijectio maifolds i vacuum systems Jeremy A. Theil Cetral Research Laboratory, Johso Cotrols, Ic., Milwaukee, Wiscosi Received 24 May 1994; accepted 22 December 1994 Whe ijectig gas ito a vacuum system, quite ofte the gas is distributed through a gas ijectio maifold. However, desigs ormally rely upo practical experiece. By cosiderig the maifold arragemet as a etwork restrictios it is possible to optimize the distributio of gas throughout the maifold. The methodology for determiig the flow distributio through the two simplest topologies of gas maifold, sigle- ad double-opeig maifolds from a sigle-gas ijectio poit, is derived i this article. It is show that the double-opeig maifold topology teds to provide more uiform flow distributio tha the sigle-opeig maifold topology for similar coductace ratios. The results of this work iclude a summatio formula for the sigle-opeig maifold. I additio, guidelies for oe type of tailored flow maifold are give. Fially, three basic desig rules are preseted: 1 use as few holes i the maifold as possible, 2 use a double opeig maifold whe possible, ad 3 specify tube dimesios such that the tube/spray hole coductace ratio is maximized America Vacuum Society. I. INTRODUCTION A commo problem i vacuum reactors is the kowledge of the dispersio patter of ijected gases ito the reactor through gas maifolds. Normally a maifold is utilized for ijected gas compoets that eed to be distributed uiformly alog some sigificat axis i the reactor. Oe way to view a maifold is as a etwork of idividual flow restrictig elemets that passively cotrol the flow of gas. It is widely recogized that Kirchoff s laws of electrical circuits apply equally well for flow impedace i the molecular flow regime. 1 5 Some work has bee performed for large compoets such as electro microscopes ad particle accelerators, but little work has bee published o gas delivery systems. 3 5 This article cosiders two simple types of maifolds ad presets the results of the distributio of the gas flow alog the maifold. The first sectio explais how the fuctios describig the gas distributio are derived. It details the physical reasoig behid the use of the matrices that describe the system, as well as how the equatios are applied. The secod sectio discusses case studies for determiig the flow profile. Here the behavior of the differet maifold topologies is demostrated. I additio, a useful simplificatio for oe spray bar topology is preseted. The last sectio discusses how the uiformity ca be cotrolled, either producig uiform gas ijectio alog the legth of the maifold, or the more geeral case of producig a tailored flow distributio. II. DERIVATIONS The problem of steady-state gas flow through a series of tubes ad holes ca be mathematically idetical to resistive electrical etworks. Ohm s law, V1 V2 IR where V is voltage, R is resistace, ad I is curret, is aalogous to P1 P2 Q where P is pressure, Q is mass flow rate, ad is the iverse coductace or flow impedace, i.e., 1/C. 1 3,5,6 For ay resistive etwork, the flow impedaces add as series elemets, ad the coductaces add for parallel elemets. Aalogies of Kirchoff s laws apply, i that all flow rates goig ito ad out of a poit add to 0, ad the pressure drop aroud a loop adds to 0. With this kowledge, it is possible to create the system of equatios that describes gas distributio through ay imagiable etwork. Certai assumptios have bee made i order to simplify calculatios. The most importat oe is that the flow throughout the etwork is molecular, so coductace is heavily iflueced by the compoet geometry. It is also assumed that coductace through each segmet of the etwork is the same as a stad-aloe compoet. The first assumptio is used to decouple the pressure of coductace because the viscous i.e., lamiar or turbulet gas flow regimes have pressure-depedet coductace. 7 Sice molecular flow is assumed, the mea-free-path legth must be larger tha the characteristic dimesio for each compoet of the etwork. Fially, the system is assumed to be i steady state, so that wall loadig effects ca be eglected. 8 Strictly speakig, however, the secod assumptio is a approximatio. The resistive etwork aalogy to calculate a lumped-sum value for coductace of several objects eglects geometric combiatio ad molecular beamig effects. This ca lead to iaccuracies of as much as 16% i the estimatio of coductace. 9,10 For example, a stad-aloe compoet assumes that the icomig flux of gas molecules has a cosie distributio orieted o-axis with respect to the compoet opeig, but the output beam shape is a fuctio of compoet geometry that is geerally ot a cosie distributio. This leads to the deviatio for the icomig flux beam shape from a cosie for adjacet compoets. Eve though the purpose of this work is to uderstad how the coductaces of various elemets affect the throughput of the system, these poits should be kept i mid. The maifold simply cosists of a tube or maifold tube that distributes gas alog its legth to a series of holes distributio tubes. The two simplest types of maifold topologies are cosidered here: 1 a tube which has the gas ijected through oe ed ad the gas is distributed through 442 J. Vac. Sci. Techol. A 13(2), Mar/Apr /95/13(2)/442/6/$ America Vacuum Society 442

2 443 Jeremy A. Theil: Gas distributio through ijectio maifolds 443 FIG. 1. Schematic of a four-hole sigle-opeig maifold. The lies have ifiite coductace; the boxes have fiite coductace. The poits at the bottom are the chamber pressure. FIG. 2. Schematic of a four-hole double-opeig maifold. The lies have ifiite coductace; the boxes have fiite coductace. The poits at the bottom are the chamber pressure. the holes i the tube sigle opeig, ad 2 a tube with holes whose eds are both coected to the gas ijectio tube double opeig. Whe the coductaces of the holes are similar to those of the distributio tubes, the distributio of gas flow through the holes is ot close to uiform, ad the distributio of gases ca be affected by each additioal hole. A. Sigle-opeig maifold Figure 1 illustrates the case for the sigle-opeig maifold. Gas flows from the feed tube o the left, ad is distributed amog all of the holes i the tube accordig to the flow impedaces to flow of the various elemets. It is assumed that the chamber pressure at each hole is the same, i.e., P0 0. What will be determied is the depedece of the flow rate through various poits of the etwork based upo the coductaces. Accordig to Kirchoff s curret law, the flow through each ode is Q1 Q1 Q2, Q2 Q2 Q3, Q3 Q3 Q4, Q4 Q4, Q Q1. For odes 1 to 1, where i is the ith hole i the maifold, ad is the umber of holes i the maifold, the ode equatios are Qi Qi Q(i 1). The pressure drops across the spray holes are P1 P0 Q1 1, P2 P0 Q2 2, P3 P0 Q3 3, P4 P0 Q4 4, i other words, for holes 1 to, Pi P0 Qi i. The pressure drop through the tube creates the followig equatios: P1 P2 Q2 2, P2 P3 Q3 3, P3 P4 Q4 4, i other words, for segmets 2 to, P(i 1) Pi Qi i. These equatios ca be combied to form equatios to solve for the ukows of the flow through the maifold holes. Q Q1 Q2 Q3 Q4, 13 Q1 1 Q2 2 2 Q2 Q3 Q4, Q2 2 Q3 3 3 Q3 Q4, Q3 3 Q4 4 4 Q4. 16 The system of equatios traslates ito a matrix of Q1,Q2,...,Q that may be solved by Gaussia elimiatio Q Q Q1 Q The pivot array has 1 for the first elemet, ad 0 for all of the other elemets. Geeralizig the formulatio leads to the followig rules. The defiitio of total gas flow is from Eq. 13, where i is the hole umber, ad is the total umber of holes : Q Qi. i 1 Equatios are Q i 1 i 1 Qi i i Qj. j i B. Double-opeig maifold The aalysis of a double-opeig maifold etwork is similar to that of the sigle-opeig tube, except that the eds are coected to form a closed loop. The coectio adds a additioal circuit elemet to the sigle-opeig JVST A - Vacuum, Surfaces, ad Films

3 444 Jeremy A. Theil: Gas distributio through ijectio maifolds 444 maifold show i Fig. 2 as 5, hece a additioal flow path to the etwork. The easiest way to solve this matrix is to build upo the previous aalysis ad solve for the flow through this additioal elemet, thus requirig a additioal equatio. Accordig to Kirchoff s curret law, the flow through each ode is Q1 Q1 Q2, 20 Q2 Q2 Q3, Q3 Q3 Q4, Q4 Q4 Q5, Q Q5 Q1. 24 For odes 1 to, where i is the ith hole i the maifold, ad is the umber of holes i the maifold, the ode equatios are Qi Qi Q(i 1). However, for the ode at P5, where Q ad Q( 1) flow ito the ode ad Q1 flows out, the flow equatio is Q Q( 1) Q1. The pressure drops across the spray holes are the same as Eqs. 6 9 ; i other words, for holes 1 to, Pi P0 Qi i. The pressure drop alog the tube is the same as Eqs with the additio of Eqs. 25 ad 26 : P4 P5 Q5 4, 25 P5 P1 Q1 1 ; 26 i other words, for segmets 2 to, P(i 1) Pi Qi i. These equatios ca be combied to form equatios to solve for the ukows of the flow through the maifold holes. However, it has bee foud to be geerally easier to set up the matrix for 1 ukows by icludig the fractio of the gas stream that flows through tube segmet Q5. Q Q1 Q2 Q3 Q4, 27 Q1 1 Q2 2 2 Q2 Q3 Q4 Q5, Q2 2 Q3 3 3 Q3 Q4 Q5, Q3 3 Q4 4 4 Q4 Q5, Q4 4 1 Q1 Q2 Q3 Q4 Q5 Q Q5. 31 Geeralizig the formulatio leads to the followig rules: the defiitio of total gas flow is from Eq. 27 the rewritte form of Eq. 18, ad the other flow equatios for holes i 2 to ca be writte as Q i 1 i 1 Qi i i Q 1 Qj. j 1 32 Equatio 1 is Q Q1 1 1 Q 1 j 1 Qj 1 Q The pivot array has 1 for the first elemet, ad 0 for all of the other elemets Q Q Q1 Q4 Q III. CASE STUDIES Oe commo method of gas distributio is the use of a sigle-opeig maifold with a umber of evely spaced holes of idetical dimesios. I this case, all of the hole coductaces are the same as are all of the coductaces of the tube segmets betwee them, C C, ad C C, therefore ad. The matrix therefore simplifies to Q2 0 0 Q Q1 Q Because the system is a chai of may repeated uits each uit ca be thought of as formig a L composed of a grouded elemet dowstream from a series elemet, there is a high degree of symmetry. From a mathematical poit of view, the solutio is rather simple, but algebraically cumbersome. That the result ca be expressed as a summatio based upo biomial coefficiets, ad is summarized by Eq. 36. The formula has bee prove for systems with as may as 24 holes. The expressio determies what fractio of the total ijected flow will flow through the ith hole of a maifold with holes all of the same flow impedace, ad whose tube coductaces are also the same: f i, i i 1 j 0 j 1 j i! i j! 2 j! j i j j 1! j! 2 j 1! j j J. Vac. Sci. Techol. A, Vol. 13, No. 2, Mar/Apr 1995

4 445 Jeremy A. Theil: Gas distributio through ijectio maifolds 445 TABLE I. Gas distributio through a six-hole sigle-opeig maifold spray hole coductace l /s, tube coductace 0.44 l /s. Sigle-opeig maifold Double-opeig maifold Hole No. Fractio Normalized fractio Fractio Normalized fractio FIG. 3. a Normalized flow fractio through sigle-opeig maifold. b Normalized flow fractio through double-opeig maifold. The maifold dimesios are six holes spaced 2.3 i. apart, i. tube diameter, ad i. hole diameter Equatio 36 is quite useful i calculatig the gas flow distributio, by elimiatig the work required to solve large matrices. The followig umerical example is preseted to illustrate the differece betwee the two topologies; it assumes T 300 K, ad N 2 gas. Six holes of 0.76 mm diameter spaced 5.84 cm apart are drilled ito a 1/4 i. o.d. 4.7 mm i.d. stailess-steel tube to form a sigle-opeig maifold for gas ijectio ito a reactor. The coductace of the 4.7 mm i.d. is l /s, 10.5 s/l ; the spray hole coductace is l /s, s/l. Solvig for these coductaces produces the followig result see i Fig. 3 a, ad the values are listed i the first half of Table I. The plot shows that five times as much gas flows through the first hole as flows through the last hole. The other example is that of a double-opeig maifold with the same dimesios as the six-hole example previously discussed. I this case, with six holes of 0.76 mm diameter spaced 5.84 cm apart are drilled ito a 1/4 i. o.d. stailesssteel tube for gas admissio ito the reactor. I this case all hole coductaces are the same, as are all of the coductaces of the tube segmets betwee them C C, ad C C, ad. The matrix simplifies to Q2 0 Q3 0 0 Q Q5 0 Q Q Q1 37 As of ow, o summatio formula such as Eq. 36 has bee derived for the double-opeig maifold, so Gaussia elimiatio is employed. The symmetry is similar to the previous case, with the exceptio of the coectig elemet. The plot of the data is show i Fig. 3 b, ad the tabulated data are listed i the last two colums of Table I. Notice that the variatio i the distributio i the tube is oly 40%, as opposed to 80% for the sigle-opeig maifold. It is possible to compare the six-hole sigle-opeig ad the sixhole double-opeig maifold by imagiig that the sigleopeig tube maifold is shaped as a rig. The differece betwee the tube ad double-opeig maifolds would oly be that the sigle-opeig maifold has a wall blockig oe ed adjacet to the gas feed poit; it would be abset for the double-opeig maifold; see Fig. 4. There is a way to coceptualize the reasoig behid the relative uiformity of the two topologies. For the sigleopeig maifold, gas flows through the lie with a coductace give by the cross sectio of the tube. However, for the double-opeig maifold, gas flows i two directios at oce so that the cross sectio available for gas flow at the juctio is effectively doubled. There is o chage i tube cross sectio, but the additioal directio of gas flow provides a extra path for the gas to be trasported to each hole. It ca be thought of as addig a extra flow bar at the juctio ad shorteig the trasport legth of gas to ay exhaust hole. Oe commo miscoceptio that arises from this lie of reasoig, however, is that, sice the gas flow is evely divided ad if there is a hole exactly half way alog the tube from the ijectio poit, there will be o gas output from that hole sice there will be o pressure gradiet i the tube. It is correct that there is o tube pressure gradiet, but there still is a pressure gradiet betwee the tube ad the reactor that causes gas to flow ito the reactor at the halfway poit o the rig. JVST A - Vacuum, Surfaces, ad Films

5 446 Jeremy A. Theil: Gas distributio through ijectio maifolds 446 FIG. 4. Similarities i the shapes of the two maifold topologies. The arrows idicate directios of gas flow withi the maifold tube. Note that the tube maifold oly coects with the feed tube with oly oe feed path, while the double-opeig maifold coects with the feed tube with two feed paths. IV. MANIFOLD OPTIMIZATION A. Uiform output Quite ofte, it is desirable that the gas flow be evely distributed through the holes of the maifold, i.e., Qi Q/. This is a reasoable goal, for example, i cases i which the holes i the rig are o a plae that is ormal to the bulk gas stream. As show above it is easy to choose arbitrary dimesios that produce a maifold with ouiform output. Oe iferece of the data is that maifolds with uiform hole size ad spacig caot have uiform gas distributio if it is a sigle-opeig maifold with more tha oe hole, or a double-opeig maifold with more tha two holes. This sectio will discuss ways of producig maifolds that distribute the gas evely across the maifold. Figure 5 compares the maximum differece i flow betwee 12-hole sigle- ad double-opeig maifolds for a wide rage of coductace ratios. Though the geeral behavior of the plots is the same, the double-opeig maifold maitais a more uiform flow for each give coductace ratio. I additio, the plot shows that, to achieve better uiformity i both cases, the tube to hole coductace ratio should be maximized. By icreasig the coductace ratio, the pressure drop betwee segmets of the tube is miimized relative to the pressure drop across the hole. Miimizig the pressure drops miimizes the pressure drop variatio from hole to hole. The simplest ways to do this are to decrease the hole diameter, icrease the hole legth, ad icrease the ier diameter of the feed tube. This shows that doubleopeig maifolds produce more eve flow distributios tha sigle-opeig maifolds. The umber of holes employed i a maifold determies how uiform the gas flow is distributed. It ca be show that the greater umber holes that are used for a maifold, the larger the ratio of the flow betwee the first ad last hole i the maifold. For a give legth of tubig, the greater umber of holes that exist, the greater the hole desity alog the tube. Sice each tube is a opportuity for gas removal, the greater the umber of holes, the more depletio that occurs withi a give segmet of tube. Coversely, the fewer holes used, the more uiform the distributio of gas flow through each hole. Of course eve though the gas distributio through each hole may be similar, it says othig about the distributio of the gas i the reactor. The lower the diffusio coefficiet of the diluet, the more susceptible it will be to a ouiform cocetratio profile withi the reactor. Therefore, a set of toleraces of diluet cocetratio must be defied ad the diffusio dyamics must be kow i order to determie what the maximum desirable spacig of holes ca be. This balace is depedet upo the plug velocity of the reactor gas stream, the diffusio coefficiet of the ijected gas i the stream, ad the trasport distace of iterest. B. Nouiform output I theory it is possible to desig a maifold that produces ay desired flow distributio based upo the tube ad hole coductaces. I reality though, certai desigs would be impractical to fabricate. Oe approach to tailorig gas flow distributio would be to use a large coductace tube to miimize the pressure drop across the maifold, so that the flow through each hole would approach that of the ratio of the particular hole coductace over the sum of all the hole coductaces. C i f i, C C j 1 C i max. 38 j If C has to be withi the same rage as C i max, it becomes difficult to desig the maifold so that it has the desired flow distributio, because the tube coductaces must be icluded i the calculatios. I such a case, the oly solutio is to solve the appropriate matrix give the desired flows. FIG. 5. Compariso of the sigle- ad double-opeig maifolds miimum/ maximum flow distributio ratio for hole to tube coductace ratios of 1: to 1:0.1, for 12 holes. V. SUMMARY The methodology for determiig the flow distributio through a gas maifold has bee preseted. Two topologies have bee examied, sigle- ad double-opeig maifolds i which there is oly oe gas feed poit. It was show that the double-opeig maifold teded to provide more uiform flow distributio for similar coductace ratios ad umber of holes, compared to the sigle-opeig maifold. A formula was preseted that calculates the gas distributio values for a sigle-opeig maifold i which the holes coductaces are all the same, ad the tube coductaces be- J. Vac. Sci. Techol. A, Vol. 13, No. 2, Mar/Apr 1995

6 447 Jeremy A. Theil: Gas distributio through ijectio maifolds 447 twee the holes are all the same. There are three rules to cosider whe desigig a maifold that miimizes ouiformity i the gas flow distributio. 1 Use tube dimesios such that the tube/spray hole coductace ratio is maximized. 2 Use as few holes i the maifold as possible. 3 Use a double-opeig maifold whe possible. It is possible to tailor the gas flow distributio by varyig the coductaces of the various tube segmets ad spray holes by solvig the appropriate matrix. The most practical solutio, however, would be to use a tube with a large coductace relative to the hole coductaces ad make the hole coductaces proportioal to the fractio of gas flow through them. This article has demostrated how etwork theory ca be used to solve gas flow distributio problems through flow tubes, ad to determie geeralized criteria for desigig tubes with uiform gas flow distributio. This is a valuable first step i a geeralized theory of maifolds, ad it provides a basis from which to develop a geeralized etwork for aalyzig reactor flow modelig. 1 A. Roth, Vacuum Techology, 3rd ed. Elsevier, New York, 1990, pp S. Dushma ad J. M. Lafferty, Scietific Foudatios of Vacuum Techique, 2d ed. Wiley, New York, 1962, pp B. R. F. Kedall, J. Vac. Sci. Techol. A 1, S. Ohta, N. Yoshimura, ad H. Hirao, J. Vac. Sci. Techol. A 1, S. R. Wilso, J. Vac. Sci. Techol. A 5, Gas loadig/uloadig of the walls ca be treated as capacitors i the aalogy see Ref A. Roth, i Ref. 1, pp A. Berma, Vacuum Egieerig, Calculatios, Formulas, ad Solved Exercises Academic, New York, 1992, pp W. Steckelmacher, Vacuum 16, P. Clausig, A. Phys. 12, JVST A - Vacuum, Surfaces, ad Films

THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS

R775 Philips Res. Repts 26,414-423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated