Pressure field in a tube with a general and arbitrary time- and position-dependent gas source F. T. Degasperi, M. N. Martins, J. Takahashi, and S. L. L. Verardi Citation: Journal of Vacuum Science & Technology A 22, 2022 (2004); doi: 10.1116/1.1778408 View online: http://dx.doi.org/10.1116/1.1778408 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvsta/22/5?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 186.217.234.225 On: Tue, 14 Jan 2014 12:13:24 http://scitation.aip.org/content/avs/journal/jvsta?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/test.int.aip.org/adtest/L23/1291604448/x01/AIP/Hiden_JVACovAd_1640x440Banner_12_10and12_17_2013/1640x440_-_23874-BANNER-AD-1640-x-440px_-_USA.jpg/7744715775302b784f4d774142526b39?x http://scitation.aip.org/search?value1=F.+T.+Degasperi&option1=author http://scitation.aip.org/search?value1=M.+N.+Martins&option1=author http://scitation.aip.org/search?value1=J.+Takahashi&option1=author http://scitation.aip.org/search?value1=S.+L.+L.+Verardi&option1=author http://scitation.aip.org/content/avs/journal/jvsta?ver=pdfcov http://dx.doi.org/10.1116/1.1778408 http://scitation.aip.org/content/avs/journal/jvsta/22/5?ver=pdfcov http://scitation.aip.org/content/avs?ver=pdfcov Redistrib Pressure field in a tube with a general and arbitrary time- and position-dependent gas source F. T. Degasperi Faculdade de Tecnologia de São Paulo, Centro Estadual de Educação Technológica Paula Souza, São Paulo, Brazil M. N. Martinsa) and J. Takahashi Laboratório do Acelerador Linear, Instituto de Fisica da Universidade de, São Paulo, São Paulo, Brazil S. L. L. Verardi Instituto de Biologia, Letras e Ciências Exatas-Universidade Estadual Paulista, São José do Rio Preto São Paulo, Brazil (Received 6 February 2004; accepted 7 June 2004; published 23 September 2004) In this article we present analytical and numerical results for a pressure profile along the axis of a tube with a general and arbitrary time- and position-dependent gas source. The model is able to determine the pressure values along the tube, once the pumping speed at each extremity and the gas sources are specified. The time evolution of the pressure along a tube is presented for situations commonly found in high-vacuum applications, such as particle accelerators, colliders, storage rings, and synchrotron light sources.© 2004 American Vacuum Society.[DOI: 10.1116/1.1778408] s in suc ele wel rms ctio rate an ype with l- rticle sing eling sure and ts f first ich tens f the sive sen oth ean ob- ra im urce, ed in ular me, y- d r d by olu- ul- sible t part ce de- ns nd of the at I. INTRODUCTION Several areas of applied physics deal with problem high-vacuum technology that present tubular geometry, as particle accelerators, colliders, and storage rings, and tron devices, such as klystrons and photomultipliers, as as electron microscopes and mass spectrometers. In te vacuum modeling, one usually assumes a straight se with pumping at the extremes, and constant degassing This kind of model is adequate for steady-state situations where the materials used do not present very different t of general gas sources(e.g., constant degassing rates). Real- istic situations in vacuum tubes require the use of parts different degassing rates(e.g., insulating ceramics or be lows). The same phenomenon happens when beam pa or beam radiation strike the walls. In addition, the degas rate is time dependent, and in this case, realistic mod must take this into account. In this article we present analytical results for a pres profile in a tube with a general and arbitrary time- position-dependent gas source. We also present resul two typical situations found in vacuum applications. The situation deals with localized impulsive degassing, wh may represent a situation where there is sudden in radiation-induced degassing in a small area in the wall o tube. The second situation deals with a tube with exten axially dependent degassing, impulsive in time. We pre results for the pressure profile and its time evolution for b cases. The results are very powerful because they present m by which to deal analytically with generic degassing pr lems, for any extensive and time-dependent degassing can be represented by a linear combination of localized pulsive degassing rates. a) Electronic mail: martins@if.usp.br 2022 J. Vac. Sci. Technol. A 22 (5), Sep/Oct 2004 0734-2101/2004/ ution subject to AVS license or copyright; see http://scitation.aip.org/terms h c- l of n s. d s s or e t s te - II. ANALYTICAL SOLUTIONS We treat the case of a tube with lengthL and diameterD with a constant degassing rate, plus an arbitrary gas so which depends both on the time and position, as illustrat Fig. 1. The differential equation for the pressure in the molec gas flow regime as a function of the position and ti psx,td, can be written as c ]2psx,td ]x2 = − qsx,td + v ]psx,td ]t , s1d wherec is the specific conductance of the tube(c=CL, C is the total tube conductance); qsx,td includes both the stead state degassing from the walls,qs, and the time- an position-dependent source,qTsx,td; andv is the volume pe unit length of the tube.1 The gas source can be represente the following function: qsx,td = qS+ qT sx,td. s2d Since the differential equation is linear, the general s tion will be the sum of the particular solutions for the imp sive sources and can be written as pGsx,td = pSsxd + pTsx,td. s3d The steady-state part of the degassing will be respon for the usual parabolic pressure profile, and the transien can represent any gas source or process that can indu sorption from the walls. The following boundary conditio were assumed:2–4 (1) For the steady-state solution, the pressure at the e the tube,x=L /2, is the total throughput,qSL, divided by the total pumping speed, 2S. (2) All the gas reaching the pumps is pumped, both for transient and the steady-state solutions, so thatx =L /2, 202222 (5)/2022/5/$19.00 ©2004 American Vacuum Society conditions. Download to IP: 186.217.234.225 On: Tue, 14 Jan 2014 12:13:24 can w: state sien gas the cuum in a d to pen- n the ase, wing s ing the h an ge- 2023 Degasperi et al. : Pressure field in tube with a dependent gas source 2023 Redistrib − cU ]pTsx,td ]x U x=L 2 = SpTsL/2,td, ∀ t ù 0. (3) The geometry of the problem is symmetric, so we treat it in the interval 0øxø+L /2 and atx=0, cU ]pTsx,td ]x U x=0 = 0, ∀ t ù 0. (4) The transient gas source occurs att=0, so pTsx,0d=0, 0øxøL /2. The explicit form of the general solution is shown belo pGsx,td = − qS 2c x2 + qSL 2 S1 S + L 4C D + o m=1 ` exps− abmtdKsbm,xd 3E0 t exps− abmt8d a c q̄sbm,t8ddt8. s4d The first two terms correspond to solution of the steady- case; the last term corresponds to solution of the tran case, where Ksbm,xd = Î2F bm 2 + H2 2 L/2sbm 2 + H2 2d + H2 G1/2 cossbm,xd and q̄sbm, t8d = E 0 L/2 Ksbm, x8dqsx8, t8ddx8, wherebm are solutions of the transcendental equation, bm tanSbm L 2 D = S c = H2, and qsx,td is the function that represents the transient source for 0øxø+L /2 andtù0. The throughput at each point of the tube is given by FIG. 1. Schematic drawing of the tubular geometry studied. HVP—A neric high-vacuum pump. following expression: JVST A - Vacuum, Surfaces, and Films ution subject to AVS license or copyright; see http://scitation.aip.org/terms t Qsx,td = − c ]psx,td ]x . s5d We assume that any amount of gas that reaches a va pump is pumped, so that the integral of the throughput given time interval is given by QTsL/2,td =E 0 t − cU ]psx,t8d ]x U x=L/2 dt8, s6d whereQT is given in mbar l. This expression can be use evaluate the lifetime of ionic, nonevaporable getter(NEG), or cryogenic pumps commonly used in accelerators. A. Impulsive gas source, localized in position and time We consider a straight tube section with an axially de dent degassing rate, impulsive in time, superimposed o uniform degassing background of the tube walls. In this c we can represent the total degassing rate by the follo function: qsx,td = qS+ q8dsx − x8ddst − t8d, s7d whereq8 represents the amount of gas liberated atx=x8 in t= t8. Considering this degassing rate as occurring atx8=0 and t8=0, Eq.(4) yields the following solution: pGsx,td = − qS 2c x2 + qSL 2 S1 S + L 4c D + q8 2v o m=1 ` Bm 2 cossbmxdexps− abm 2 td, s8d with a=c/v and Bm 2 = 2 bm 2 + H2 2 fL/2sbm 2 + H2 2d + H2g . For the case described by Eq.(8), the total amount of ga reaching the pump is given by the solution of Eq.(6): QTSL 2 ,tD = qS L 2 t + q8 2 o m=1 ` Bm 2 bm sinSbm L 2 D 3f1 − exps− abm 2 tdg. s9d The part of Eq.(9) that represents the amount of gas com from the transient source is lim t→` Hq8 2 o m=1 ` Bm 2 bm sinSbm L 2 Df1 − exps− abm 2 tdgJ = q8 2 , showing that one half of the amount of gas produced by source reaches the pump atx=L /2. B. Extensive gas source, position dependent and impulsive in time In this case, we consider a straight tube section wit extensive gas source in space and impulsive in time super- conditions. Download to IP: 186.217.234.225 On: Tue, 14 Jan 2014 12:13:24 tube in- pa e a sien , a hing om- the rs, ples ch- de- en s th not f ga first pen ugh d, th de- hould ping stant lized ; ts lf of e pa- f gas ee gas is , the , go- file ssure notice s five d the 2024 Degasperi et al. : Pressure field in tube with a dependent gas source 2024 Redistrib imposed on the uniform background degassing of the walls. The throughput per unit length is given by qsx,td = qS+ q8sxddst − t8d. s10d In some situations in high-vacuum applications, for stance, in particle accelerators or storage rings, beam ticles or beam radiation may hit the walls and produc burst of gas. We can represent this situation with a tran gas source, extensive in space and impulsive in time shown below: q8sxd = 5 0, for − L/2 ø x , − a, q8, for − a ø x ø a, 0, for a , x ø + L/2. 6 s11d The general solution can be written as pGsx,td = − qS 2c x2 + qSL 2 S1 S + L 4c D + q8/vo m=1 ` Bm 2 bm sinsbmadcossbmxdexps− abm 2 td. s12d Like in the previous case, the total amount of gas reac the pump atx=L /2 can be calculated as QTsL/2,td = qSL/2t + q8o m=1 ` Bm 2 bm 2 sinsbmadsinsbmL/2d 3f1 − expsabm 2 tdg. s13d The part of Eq.(13) that represents the amount of gas c ing from the transient source is lim t→` Hq8o m=1 ` Bm 2 bm 2 sinSbm L 2 Dsinsbmadf1 − exps− abm 2 tdgJ= q8a, showing that one half of the amount of gas produced by source reaches the pump atx=L /2. III. RESULTS AND DISCUSSION We present two case studies, using realistic paramete5,6 to exemplify the results obtained in Sec. II. These exam illustrate situations commonly found in high-vacuum te nology, mainly in electron tube and particle accelerator vices. Other possible applications in high-vacuum equipm may be treated as well. The reader should be aware that the model assume all the gas arriving at the extremities is pumped, which is a realistic hypothesis and depends on the amount o qsx,td and the specific conductance of the tube. The parameter is important because it will define what hap with the pressure at the entrance of the pump. If the thro put is larger than the pressure times the pumping spee pressure will rise. The dynamics of this process will be fined by how the pumping speed depends on the pressur J. Vac. Sci. Technol. A, Vol. 22, No. 5, Sep/Oct 2004 ution subject to AVS license or copyright; see http://scitation.aip.org/terms r- t s t at s s - e which is a characteristic of the pump. So, each case s be treated according to specific conditions of the pum system. A. Impulsive gas source, localized in position and time In this case, we deal with a tube that has a con degassing rate along the whole length, plus loca impulsive degassing occurring atx=0 cm and t=0 s. The parameters are the following: L=400 cm D=3 cm; qS=4.7310−9 mbar l s−1 cm−1; q8=1.0 310−6 mbar l; c=324 l s−1 cm; v=7.1310−3 l cm−1; qSL =1.9310−5 mbar l s−1; a=4.563104 cm2 s−1; and S =100 l s−1. Taking those values into Eq.(8), we obtain the resul shown in Fig. 2 which shows the pressure only in one ha the tube, because the geometry is symmetrical. One can see from Fig. 2 that for times just abovet=0 s, the pressure profile is represented by the steady-stat rabola plus the rise in pressure due to localized burst o (solid line,t=3310−5 s). A very short time later, one can s the rise in pressure becoming smoother, because the already diffusing along the tube(dashed line,t=10−4 s). Af- ter this, we can see the pressure decreasing aroundx=0 cm, because the gas is moving toward the pump(dot-dashed line t=10−3 s). At later times, we can see the gas reaching pump, with a subsequent rise in pressure(long dashed line t=10−1 s), and the gas being pumped, with the pressure ing down toward the parabolic background profile(dot-dot dashed line,t=1 s), then reaching the background pro after t=3 s. The oscillations observed on thet=3310−5 s curve(solid line) are due to summation of Eq.(8), which was limited to 120 terms. To represent the steep peak in pre from the burst of gas at times close tot=0 s, it would be necessary to use a larger number of terms. One should that for t=10−4 s (and the same 120 terms), this artifact doe not occur. Figure 3 shows the time evolution of the pressure at different positions along the tube(x=0, 50, 100, 150, an 200 cm). The plot shows the pressure only in one half of FIG. 2. Pressure field along the tube forxù0 at different times. e,tube, since the geometry is symmetrical. conditions. Download to IP: 186.217.234.225 On: Tue, 14 Jan 2014 12:13:24 me er back , the bac due -sta val- con ffec m in n be rs b ent The the - ows etr e p rst o ee gas - de- ard the d . 2 should res- nt of , the five d the is here back- se to en than ssure erent erent 2025 Degasperi et al. : Pressure field in tube with a dependent gas source 2025 Redistrib We can notice two different kinds of behavior in the ti evolution of the pressure along the tube. For the centsx =0 cmd, the pressure rises suddenly and decays to the ground value in about 1 s. In the other parts of the tube pressure presents more general behavior: it stays at the ground value while the gas diffuses to that position, rises to the arrival of the gas, and decays back to the steady value. After aboutt=1 s, the pressure tends toward the ues observed before the impulsive degassing att=0 s, and presents a parabolic profile along the tube. We have not sidered absorption of the gas by the tube walls, but this e can be included in the model by changing the source ter Eq. (2). A negative term in the gas source equation ca physically interpreted as adsorption of gases and vapo the walls(pumping effect of the walls). B. Extensive gas source, position dependent and impulsive in time Results for a tube with extensive axially depend degassing, impulsive in time, are presented below. parameters, typical of particle accelerators, are following:5 L=400 cm; D=3 cm; a=30 cm; qS=4.7 310−9 mbar l s−1 cm−1; q8=1.0310−6 mbar l cm; v=7.1 310−3 l cm−1; c=324 l s−1 cm; qSL=1.9310−5 mbar l s−1; a=4.563104 cm2 s−1; andS=100 l s−1. Substituting these values in Eq.(12), we obtain the pres sure field along the tube, shown in Fig. 4. The plot sh only the pressure in one half of the tube, since the geom is symmetrical. One can see from Fig. 4 that, for times just abovet=0 s, the pressure profile is represented by the steady-stat rabola plus the rise in pressure due to the extensive bu gas(solid line, t=3310−5 s). A short time later, one can s the rise in pressure becoming smoother, because the diffusing along the tube(long dashed line,t=10−4 s and dot ted line, t=10−3 s). After this, we can see the pressure creasing aroundx=0 cm, because the gas is moving tow the pump, and subsequent rise in pressure atx=200 cm(dot −1 FIG. 3. Time evolution of the pressure in the tube observed at diff positions. dot-dashed line,t=10 s). At later times, we can see the gas JVST A - Vacuum, Surfaces, and Films ution subject to AVS license or copyright; see http://scitation.aip.org/terms - k- te - t y y a- f is being pumped, with the pressure going down toward parabolic background profile(dashed line,t=1 s and dotte line t=5 s). The same oscillatory behavior noticed in Fig appears here, for the same reasons. Nevertheless, one notice that the artifact is sensitive to both the maximum p sure and the extent of the burst. So, for a given amou gas liberated by the transient gas source, if it is extensive number of terms needed to smooth the curve is larger. Figure 5 shows the time evolution of the pressure at different positions along the tube(x=0, 50, 100, 150, an 200 cm). The overall behavior in this situation is analogous to localized (in space) burst that was shown in Fig. 3, but more intense due to the larger amount of gas liberated. T is also a small numerical artifact that appears as higher ground pressure at times less than 0.1 s for positions clo the extreme of the tubesx.180 cmd. This can also be se in Fig. 4, where the background parabolas for times less 0.1 s show a slight rise compared to the background pre profile st→`d for positions abovex=180 cm. FIG. 4. Pressure field along the tube forxù0 at different times. FIG. 5. Time evolution of the pressure in the tube observed at diff positions. conditions. Download to IP: 186.217.234.225 On: Tue, 14 Jan 2014 12:13:24 situa ithin m o ions ssin res- func ne ntia pro tura ctin sse ack pped ro à o ico Ex- Pro- 2001; F. pinas, . f 2026 Degasperi et al. : Pressure field in tube with a dependent gas source 2026 Redistrib IV. CONCLUSIONS This model can be used to represent transient-state tions where a localized impulsive burst of gas occurs w the system, as in the case, for instance, of a particle bea radiation beam striking the wall of the tube.3 It should be noted that this model presents a wide range of applicat because it can be generalized to conform to any dega profile the burst may present. Any given time or spatial p sure profile of a burst can be represented by a general tion like Eq. (4). Depending on details for the model, o might have to use numerical methods to solve the differe equation of Eq.(4). To improve the model used to calculate the pressure file along the tube, we can also consider the effect of na adsorption of gas, important in the case of supercondu devices, in the wall of the tube. In this case, the dega part of the wall serves as a vacuum pump, and the b ground pressure with a parabolic profile is reached late J. Vac. Sci. Technol. A, Vol. 22, No. 5, Sep/Oct 2004 ution subject to AVS license or copyright; see http://scitation.aip.org/terms - r , g - l - l g d - when the process of slow uniform desorption of gas tra in the walls is established(steady state). ACKNOWLEDGMENTS Work was supported in part by Fundação de Ampa Pesquisa do Estado de São Paulo(FAPESP) and Conselh Nacional de Desenvolvimento Científico e Tecnológ (CNPq). 1A. Berman,Vacuum Engineering Calculations, Formulas and Solved ercises(Academic, New York, 1997). 2M. N. Ozisik, Boundary Value Problems of Heat Conduction(Dover, New York, 1989). 3M. N. Martins, F. T. Degasperi, J. Takahashi, and S. L. L. Verardi, ceedings of the 2001 Particle Accelerator Conference, Chicago, http://accelconf.web.cern.ch/AccelConf/p01/PAPERS/WPAH051.PD 4F. T. Degasperi, M.Sc. thesis, Universidade Estadual de Cam Campinas SP, Brazil, 2002. 5J. Gómez-Goñi and A. G. Mathewson, J. Vac. Sci. Technol. A15, 3093 (1997). 6Surface Conditioning of Vaccum Systems, edited by R. A. Langley, D. L Flamm, M. C. Hseuh, W. L. Hsu, T. W. Rusch(American Institute o r Physics, New York, 1990). conditions. Download to IP: 186.217.234.225 On: Tue, 14 Jan 2014 12:13:24