Gas Flow, Conductance, Pressure Profile: Vacuum Technology ...

JUAS 2014, Archamps, France

Gas Flow, Conductance, Pressure Profile:

Vacuum Technology for Accelerators with Exercises

R. Kersevan, TE/VSC-IVM – CERN, Geneva

Content:

• Concepts of gas flow, conductance, pressure profile as relevant to the design of the vacuum system of modern accelerators;

• A quick definition of the terms involved;

• Some computational models and algorithms: analytical vs numerical;

• Simple exercises

• Conclusions;

• References to documents are given during presentation.

(*) Not endorsing products of any kind

or brand

Gas Flow, Conductance, Pressure Profile: Fundamentals of Vacuum Technology for Accelerators

Sources: “[1] Fundamentals of Vacuum Technology”, Oerlikon-Leybold (*); [2] “Vacuum Technology, A. Roth, Elsevier;

[3] “A User’s Guide to Vacuum Technology”, J.F. O’Hanlon, Wiley-Interscience; [4] “Vacuum Engineering Calculations, Formulas, and Solved Exercises”, A. Berman,

Academic Press;

2


Sources: [5] “Handbook of Vacuum Technology”, K. Jousten ed., Wiley-Vch, 1002 p.

3


Units and Definitions [1]

4



•Without bothering Democritus, Aristotles, Pascal, Torricelli et al… a modern definitionof “vacuum” is the following (American Vacuum Society, 1958):

“Given space or volume filled with gas at pressures below atmospheric, i.e. less than 2.5E+19 molecules/cm3 ”

• Keeping this in mind, the following curve [2] defines the molecular density vs pressureand the mean free path (MFP), a very important quantity:

Mean-free path: average distancetravelled by a molecule before hittinganother one (ternary, and higher-order,collisions are negligible)

• The importance of obtaining a lowpressure, in accelerators, is evident:

- reduce collisions between the particlebeams and the residual gas;

- increase beam lifetime;- reduce losses;- reduce activation of components;- reduce doses to experimenters;- decrease number of injection cycles;- improve beam up-time statistics;- more..

5


•A different view can be found here…http://hyperphysics.phy-

astr.gsu.edu/hbase/kinetic/menfre.html

Units and Definitions

More molecular dimensions, dm, can be found here:http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_4.html

Gas p (mPa) Gas p (mPa)

H2 11.5x10-3 CO2 4.0x10-3

N2 5.9x10-3 Ar 6.4x10-3

He 17.5x10-3 Ne 12.7x10-3

CO 6.0x10-3 Kr 4.9x10-3

6

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/menfre.html

http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_4.html


Definition of “flow regime”

The so-called “Knudsen number” isdefined as this:

And the different flow (pressure) regimesare identified as follows:

FREE MOLECULAR FLOW : Kn >1TRANSITIONAL FLOW : 0.01< Kn< 1CONTINUUM (VISCOUS) FLOW: Kn< 0.01

Most accelerators work in the free-molecular regime i.e. in a conditionwhere the MFP is bigger than the“typical” dimension of the vacuumchamber, and therefore molecularcollisions can be neglected.


DKn

7


Definition of “vacuum ranges”• Linked to the Knudsen number and the flow regimes, historically defined as in tablebelow [1];

•With the advent of very low-outgassing materials and treatments (e.g. NEG-coating), “Ultrahigh vacuum” (UHV) is sometimes split up in “UHV” and “XHV”(eXtreme High Vacuum) regimes

• (Ref. N. Marquardt,CERN CAS)

Units and Definitions

8


• Impingement rate and collision rates• The ideal gas law states that thepressure P of a diluted gas is given by

… where:V = volume, m3; m = mass of gas, kgM = molecular mass, kg/moleT = absolute temperature, °KR = gas constant = 8.31451 J/mol/KnM = number of molesNA = Avogadro’s number = 6.022E+23

molecules/molekB = Boltzmann’s constant = 1.381E-23

J/K

•Deviation from this law are taken careof by introducing higher-order terms…

… which are not discussed here.

Impingement and Collision rates, and Ideal Gas Law

TkNnRTnRTM

mPV BAMM

...)1( 2 CPBPRTPV

9


How does all this translates into “accelerator vacuum”?

• Let’s imagine the simplest vacuumsistem, a straight tube with constantcross-section connecting two largevolumes, P1>P2

Q= P·dV/dt

Q, which has the units of

[Pa·m3/s] = [N·m /s] = [J/s] = [W]

… is called the throughput. Thereforethe throughput is the power carried bya gas flowing out (or in) of the volume Vat a rate of dV/dt.

• dV/dt is also called “volumetricflow rate”, and when applied to the

inlet of a pump, it is called “pumpingspeed”.

• Therefore, we can also write thefirst basic equation of vacuumtechnology

Q= P·S

Volumetric Flow Rate – Throughput – Basic Equations - Conductance

•Having defined the throughput, we movenow to the concept of conductance, C:

Suppose we have two volumes V1 andV2, at pressures P1>P2 respectively,connected via a tube

…we can define a second basicequation of vacuum technology

Q= C·(P1-P2)= C·DP

… which, making an electrical analogy…

I= V / R

… gives an obvious interpretation of Cas the reciprocal of a resistance toflow.The higher the conductance the more “current” (throughput) runs

through the system.

P1 P2Q

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How can conductances be calculated? How does the dimension, shape, length, etc… of a vacuum component define its

conductance?

•We need to recall some concepts ofkinetic theory of gases:

• The Maxwell-Boltzmann velocitydistribution defines an ensemble of Nmolecules of given mass m andtemperature T as

… with n the molecular density, and kb

as before.• The shape of this distribution for air atdifferent temperatures, and fordifferent gases at 25°C are shown onthe right:


)2/(2

2/3

2/1

2

2

2 Tkvm BevkT

mN

v

dn

11


• The kinetic theory of gases determinesthe mean velocity, most probablevelocity, and rms velocity as

… with R and T as before.Therefore, cmp < cmean < crms .

For air at 25°C these values are:cmp = 413 m/scmean= 467 m/scrms = 506 m/s

For a gas with different m and T, thesevalues scale as


M

RTc

M

RTc

M

RTc

rms

mp

mean

3

2

8

gas

gas

air

air

M

T

T

M

The energy distribution of the gas is

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•Within the kinetic theory of gases, itcan be shown that the volumetric flowrate passing through an infinitely thinhole of surface area A between twovolumes is given by

... and by the analogy with the secondbasic equation we get that theconductance of this thin hole is

For holes which are not of zero-thickness, a “reduction” factor k,0<k<1, can be defined. K is calledtransmission probability, and can bevisualized as the effect of the “sidewall” generated by the thickness.It depends in a complicated way fromthe shape of the hole.So, in general, for a hole of area Aacross a wall of thickness L

Zero-thickness vs finite-thickness aperture in a wall…

… the transmission probability decreases as L increases…

Transmission probability

4

rmscAq

4

rmscAc

),(4

),( LAkc

ALAc rms

L

13


•Only for few simple cross-sections ofthe hole, an analytic expression ofk(A,L) exists.

• For arbitrary shapes, numericalintegration of an integro-differentialequation must be carried out

(ref. W. Steckelmacher, Vacuum 16 (1966) p561-584)

Transmission probability

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• Keeping in mind the interpretation ofthe conductance as the reciprocal of aresistance in an electric circuit, we maybe tempted to use “summation rules”similar to those used for series andparallel connection of two resistors.

• It turns out that these rules are not sofar off, they give meaningful resultsprovided some “correction factors” areintroduced

parallel

series

and they can be extended to moreelements by adding them up.

• The correction factor takes intoaccount also the fact that the flow ofthe gas as it enters the tube “develops”a varying angular distribution as itmoves along it, even for a constantsection.

•At the entrance, the gas crosses theaperture with a “cosine distribution”_

…while at the exit it has a so called“beamed” distribution, determined bythe collisions of the molecules with theside walls of the tube, as shown above(solid black line)

•As the length of the tube increases…

Sum of Conductances

21 CCC

21

111

CCC

q

F(q)≈cos(q)

15


… so does the beaming, and theforward- and backward-emittedmolecules become more and moreskewed, as shown here…

• The transmission probability of anyshape can be calculated with arbitraryprecision by using the Test-ParticleMontecarlo method (TPMC).

• The TPMC generates “random” moleculesaccording to the cosine distribution…

… at the entrance of the tube, andthen follows their traces until theyreach the exit of the tube.

• Time is not a factor, and residencetime on the walls is therefore not anissue.

• Each collision with the walls isfollowed by a random emissionfollowing, again, the cosinedistribution…

• … this is repeated a very large numberof times, in order to reduce thestatistical scattering and apply thelarge number theorem.

• The same method can be applied notonly to tubes but also to three-dimensional, arbitrary components,i.e. “models” of any vacuum system.

• In this case, pumps are simulated byassigning “sticking coefficients” tothe surfaces representing their inletflange.

• The sticking coefficient is nothing elsethan the probability that a molecule…

Molecular Beaming Effect

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… entering the inlet flange gets pumped,i.e. removed from the system.

• The equivalent sticking coefficient sof a pump of S [l/s] represented by anopening of A [cm2] is given by

… i.e. it is the ratio between the givenpumping speed and the conductance ofthe zero-thickness hole having the samesurface area of the opening A.

• The “interchangeability” of the conceptof conductance and pumping speed, bothcustomarily defined by the units of [l/s](or [m3/s], or [m3/h]), suggests that ifa pump of nominal speed Snom [l/s] isconnected to a volume V via a tube ofconductance C, the effective pumpingspeed of the pump will be given by therelationship

• From this simple equation it is clearthat it doesn’t pay to increase theinstalled pumping speed much morethan the conductance C, whichtherefore sets a limit to theachievable effective pumping speed.

• This has severe implications foraccelerators, as they typically havevacuum chambers with a tubularshape: they are “conductance-limited systems”, and as such need aspecific strategy to deal with them

Effective Pumping Speed

4rmsc

A

Ss

CSS nomeff

111

17


• The transmission probability of tubeshas been calculated many times. Thispaper (J.Vac.Sci.Technol. 3(3) 1965 p92-95)

… gives us a way to calculate theconductance of a cylindrical tube ofany length to radius ratio L/R >0.001:

… where Ptransm is the transmissionprobability of the tube, as read onthe graph and 11.77 is the “usual”kinetic factor of a mass 28 gas at20°C.

•Other authors have given approximateequations for the calculation of Ctransm,namely Dushman (1922), prior to theadvent of modern computers

… with D and L in m.•We can derive this equation byconsidering a tube as twoconductances in series: CA, theaperture of the tube followed by thetube itself, CB.

• By using the summation rule for….

Transmission Probability and Analytical Formulas

transminlettransm PcmslcmAslC )//(77.11)()/( 22

L

D

LDslCtransm

341

/4.12)/(

3

18


… 2 conductances in series…

… we obtain:

and

… with D and L in cm. Substitutingabove…

Beware: the error can be large!

• Exercise: 1) estimate the conductanceof a tube of D=10 [cm] and L=50 [cm]by using the transmission probabilityconcept and compare it to the oneobtained using Dushman’s formula.

2) Repeat for a tube with L=500 [cm].

3) Calculate the relative error.

BA CCC

111

33.9 DCA LDCB /4.12 3

)3/(41

/4.12 3

LD

LD

CC

CCC

BA

BA

• This fundamental conductance limitationhas profound effects on the design ofthe pumping system: the location,number and size of the pumps must bedecided on the merit of minimizing theaverage pressure seen by the beam(s).

• The process is carried out in severalsteps: first a “back of the envelope”calculation with evenly spaced pumps,followed by a number of iterationswhere the position of the pumps andeventually their individual size (speed)are customized.

• Step one resembles to this: a cross-section common to all magnetic elementsis chosen, i.e. one which fits inside allmagnets (dipoles, quadrupoles,sextupoles, etc…): this determines aspecific conductance for the vacuumchamber cspec (l·m/s), by means of, forinstance, the transmission probabilitymethod.

Dushman’s Formula for Tubes (see P.Chiggiato’s lecture for more refined models/formula, like his (21))

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Pressure Profiles

•We then consider a chamber of uniformcross-section, of specific surface A[cm2/m], specific outgassing rate of q[l/s/cm2], with equal pumps (pumpingspeed S [l/s] each) evenly spaced at adistance L. The following equations canbe written:

…which can be combined into

… with boundary conditions

… to obtain the final result

Aqdx

Pdc

2

2

Aqdx

xdQ

dx

xdPcxQ

)(

)()(

{

SAqLxP

Lxdx

dP

/)0(

0)2/(

{

S

AqLxLx

c

AqxP 2

2)(

L

20


• From this equation for the pressureprofile, we derive three interestingquantities: the average pressure, the

peak pressure, and the effectivepumping speed as:

• From the 1st and 3rd ones we see thatonce the specific conductance is chosen(determined by the size of themagnets, and the optics of themachine), how low the average pressureseen by the beam can be is limited bythe effective pumping speed, which inturn depends strongly on c.

• The following graph shows an exampleof this: for c=20 [l·m/s] and differentnominal pumping speeds for the pumps,the graphs show how Seff would change.

• This, in turn, determines theaverage… pump spacing, and

ultimately the number of pumps.

•Summarizing: in one simple step, witha simple model, one can get anestimate of the size of the vacuumchamber, the number and type ofpumps, and from this, roughly, a firstestimate of the capital costs for thevacuum system of the machine. Notbad!

Pressure Profiles and More…

1

0

)1

12( )

1

8

1(

)/1()1

12()(

1

Sc

LS

ScAqLP

SAqLSc

LAqLdxxP

LP

EFFMAX

EFF

L

AVERAGE

21


• From the previous analysis it is clearthat there may be cases when eitherbecause of the size of the machine orthe dimensions of (some of its) vacuumchambers, the number of pumps whichwould be necessary in order to obtain asufficiently low pressure could be toolarge, i.e. impose technical and costissues. One example of this was theLEP accelerator, which was 27 km-long, and would have needed thousandsof pumps, based on the analysis we’vecarried out so far.

• So, what to do in this case? Changemany pumps into one continuous pump,i.e. implement distributed pumping.

• In this case if Sdist is the distributedpumping speed, its units are [l/s/m],the equations above become:

•We obtain a flat, constant, pressureprofile.

• The distributed pressure profile in LEPhad been obtained by inserting aNEG-strip along an ante-chamber,running parallel to the beam chamber,and connected by small elliptical slots:


LSSPP

SAqP

distEFFAVGMAX

EFFAVG

/

22


Effect on pressure profile of Sdist

• Exercise: knowing that one metre ofNEG-strip and the pumping slotsprovide approximately 294 [l/s/m] atthe beam chamber, derive theequivalent number of lumped pumps of500 [l/s] which would have beennecessary in order to get the sameaverage pressure.

Input data:- Cspec(LEP) = 100 [l·m/s]- Sdist(LEP) = 294 [l/s/m]- ALEP = 3,200[cm2/m];- q = 3.0E-11 [mbar*l/s/cm2]

• Exercise: the SPS transfer line has a vacuum pipe of 60 [mm] diameter, and the distance L between pumps is ~ 60 [m]. The pumping speed of the ion-pumps installed on it is ~ 15 [l/s] at the pipe. Assuming a thermal outgassing rate q=3E-11 [mbar·l/s/cm2]

calculate:1) Pmax, Pmin, Pavg, in [mbar]2) Seff, in [l/s]


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•During this short tutorial we have discovered some important concepts andequations related to the field of vacuum for particle accelerators.

•We have seen that one limiting factor of accelerators is the fact that theyalways have long tubular chambers, which are inherently conductance limited.

•We have also seen some basic equations of vacuum, namely the P=Q/S whichallows a very first glimpse at the level of pumping speed S which will benecessary to implement on the accelerator in order to get rid of the outgassingQ, which will depend qualitatively and quantitatively on the type of accelerator(see P.Chiggiato’s lessons on outgassing and synchrotron radiation, this school).

• Links between the thermodynamic properties of gases and the technicalspecification of pumps (their pumping speed) as been given: the link is via theequivalent sticking coefficient which can be attributed to the inlet of the pump.

•One simple model of accelerator vacuum system, having uniform desorption, evenlyspaced pumps of equal speed has allowed us to derive some preliminary butpowerful equations relating the pressure to the conductance to the effectivepumping speed, and ultimately giving us a ballpark estimate about the number ofpumps which will be needed.

• Finally, an example of a real, now dismantled, accelerator has been discussed(LEP), and the advantages of distributed pumping vs lumped pumping detailed.

Conclusions:

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• P.Chiggiato, this school, and previous editions

• R. Kersevan, M. Ady: http://test-molflow.web.cern.ch/

Link to sample files discussed during the talk:

https://dl.dropboxusercontent.com/u/104842596/zipped.7z

• Y. Li et al.: “Vacuum Science and Technology for Accelerator Vacuum Systems”,U.S. Particle Accelerator School, Course Materials - Duke University - January2013; http://uspas.fnal.gov/materials/13Duke/Duke_VacuumScience.shtml

Thank you for your attention

References (other than those given on the slides):

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http://test-molflow.web.cern.ch/

https://dl.dropboxusercontent.com/u/104842596/zipped.7z

http://uspas.fnal.gov/materials/13Duke/Duke_VacuumScience.shtml

Gas Flow, Conductance, Pressure Profile: Vacuum Technology ...

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