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Vacuum 103 (2014) 57e67

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DSMC simulation of rarefied gas mixtures flows driven by arrays ofabsorbing plates

Aldo Frezzotti a,*, Gian Pietro Ghiroldi a, Livio Gibelli a, Antonio Bonucci b

a Politecnico di Milano, Dipartimento di Scienze & Tecnologie Aerospaziali, Via La Masa, 34, 20156 Milano, Italyb SAES Group, Viale Italia, 77, 20020 Lainate, Italy

a r t i c l e i n f o

Article history:Received 16 September 2013Received in revised form11 December 2013Accepted 12 December 2013

Keywords:Rarefied gasesMixturesSurface absorptionCondensation

* Corresponding author. Tel.: þ39 (02) 23998358.E-mail addresses: aldo.frezzotti@polimi.it,

(A. Frezzotti).

0042-207X/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.vacuum.2013.12.004

a b s t r a c t

Gas flows induced by arrays of absorbing surfaces of various shapes are met in several vacuum tech-nology devices, like NEG or cryogenic pumps. In order to obtain a simplified model of low pressure gasdynamics driven by surface absorption, flows of rarefied gas binary mixtures past arrays of absorbingplates are studied by numerical solution of a system of coupled and spatially two-dimensional Boltz-mann equations. The overall absorption rate is obtained as a function of problem parameters whichinclude the characteristic flow Knudsen number and wall absorption probabilities. Particular attention isdevoted to mixtures flows in which one of the components is present in small amounts and it is weaklyabsorbed.

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1. Introduction

In a previous paper [1], the authors have studied one-dimensional steady flows of binary gas mixtures in contact withan infinite planar wall which absorbs the two gas components atdifferent rates. Flow properties have been investigated by solving asystem of two coupled Boltzmann equations [2,3], since the properformulation of boundary conditions and the presence of a Knudsenlayer above the absorbing plates require kinetic theory conceptsand methods. It has been observed that the one-dimensionalsteady absorption problem is virtually equivalent to a classicalcondensation problem [4e6] and that the presence of a smallamount of a weakly absorbed component can significantly affectthe absorption rate of the most abundant species, in completeanalogy with similar results about condensation flows [7].

Although many flow properties can be understood by the studyof the one-dimensional problem, the geometry considered inRef. [1] is far from the complex shapes present in vacuum tech-nology devices, like NEG or cryogenic pumps [8], in whichabsorbing surfaces are organized in more or less densely packedarrays. Gas flow properties in general and absorption rates inparticular are then expected to be affected not only by gas-walland mixture components interactions but also by devices

aldo.frezzotti@gmail.com

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geometry. In order to obtain a more precise description of thedynamics of low pressure gas mixtures flows in vacuum devices,in this paper we address the numerical computation of the two-dimensional steady solution of a system of two-coupled Boltz-mann equations for a binary mixture of hard sphere gases [3]. Gasflows are generated by the presence of absorbing surfaces ar-ranged in an infinite array of identical and parallel plates. Asshown below, the solutions of the problem are determined by arelatively large number of physical and geometrical parameters.Hence attention has been mainly focused on two aspects of theproblem. The first one is the assessment of the effects of the de-gree of gas rarefaction (defined through a characteristic flowKnudsen number [2]) on the overall absorption rate. The secondone is the effect of the interaction of the two components of abinary mixture on the absorption rate. In particular, the situationhas been investigated in which one of the components is presentin small amounts and is weakly absorbed by the array plates. Asmentioned above, in a one-dimensional model the presence of aspecies with small absorption probability can considerablydecrease the overall mixture absorption rate. However, the flowgeometry is expected to play a role and its influence needs to beinvestigated. Accordingly, the paper is organized as follows: Sec-tions 2 and 3 present the mathematical formulation and brieflydescribe the adopted numerical method; Section 4 presents anddiscusses collisionless solutions of the Boltzmann equation whichdescribe high vacuum operation of the absorbing surfaces array.Moreover, they provide reference conditions to judge about thedevice performances in the transition regime [2] discussed in

A. Frezzotti et al. / Vacuum 103 (2014) 57e6758

Section 5. Here, attention is restricted to steady solutions origi-nating from upstream subsonic unperturbed flow conditions. Re-sults are presented and discussed for single component gas flowsat different degrees of rarefaction as well as selected properties ofbinary mixtures flows. Results are finally summarized in theconcluding Section 6.

Fig. 1. Sketch of array geometry and coordinates system.

2. Problem formulation

In the problem considered here, a binary mixture ofmonatomic hard sphere gases occupies an infinite spatial region.The atoms of species i (i ¼ 1, 2) have mass mi and hard spherediameter si. It is assumed that the mixture stationary motion isgoverned by a system of two coupled, steady Boltzmann equations[3] in the form:

v+vfivr

¼X2j¼1

Qij

�fj; fi

�; (1)

where the collision integral [3] Qij, which describes the mechanicalinteraction between species i and j, has the following expression:

Qij

�fj; fi

�¼ 1

2

�si þ sj

2

�2 ZS2

d2bk ZR3

dwhfj�r; w*

���t�fi�r; v*���t�� fjðr; wjtÞfiðr; vjtÞ

i���vr+bk���:(2)

In Eq. (1), fi(r,vjt) denotes the distribution function of atomicvelocities v of species i, at spatial location r and time t. The unitvector bk specifies the relative position of two atoms at the time oftheir impact with relative velocity vr ¼ w � v. The velocities of v*

andw*, which are turned into v andw in a restituting collision, aredefined as [3]:

v* ¼ v þ 2mj

mi þmj

�vr+bk�bk; (3)

w* ¼ w � 2mi

mi þmj

�vr+bk�bk: (4)

It is worth observing that the behavior of rarefied gas mixturesis affected by intermolecular potentials. Hard sphere dynamicsprovides a description which might differ from more accuratecollision models [9]. However, the present choice, dictated bycomputational simplicity of the hard sphere model, does not affectthe general conclusions of the study. Macroscopic quantities asso-ciated with the mixture flow are obtained as moments of the dis-tribution functions. Partial and total number andmass densities arerespectively defined as

ni r; tð Þ ¼Z

fi r; vjtð Þdv; n r; tð Þ ¼ P 2i¼1ni r; tð Þ; (5)

riðr; tÞ ¼ miniðr; tÞ rðr; tÞ ¼ P2i¼1

riðr; tÞ: (6)

The molar fraction, ci(r, t), of the i-th component and its massfraction or concentration, ~ciðr; tÞ, are defined as

ciðr; tÞ ¼ niðr; tÞnðr; tÞ ; (7)

~ciðr; tÞ ¼ riðr; tÞrðr; tÞ : (8)

Average velocities of each component are then obtained as

uiðr; tÞ ¼ 1niðr; tÞ

Zvfiðr; vjtÞdv: (9)

It is then convenient to define the hydrodynamic velocity,associated with the overall local mixture momentum, as:

uhydðr; tÞ ¼P2

i¼1 riðr; tÞuiðr; tÞrðr; tÞ : (10)

Single component temperature Ti and overall mixture temper-ature T can be associated with velocity dispersions with respect tothe hydrodynamic velocity, as follows:

32niðr; tÞkBTiðr; tÞ ¼ 1

2mi

Z �v � uhyd

�2fiðr; vjtÞdv; (11)

32nðr; tÞkBTðr; tÞ ¼

X2i¼1

12mi

Z �v � uhyd

�2fiðr; vjtÞdv: (12)

The mixture is set into motion by gas absorption taking place atthe surfaces of an infinite array of identical and equally spacedrectangular plates, as sketched in Fig. 1. Each plate has negligiblethickness and sides of lengths Ly and Lz, respectively. The plates areparallel and their centers lie on a straight line (array axis) perpen-dicular to each plate, being D the distance between two nearestneighbor plate centers. A Cartesian reference frame is constructedby positioning its origin in the center of one of the plates. The x axiscoincides with the array axis, whereas the y and z axes are parallelto the plate sides of length Ly and Lz, respectively. The boundarycondition, describing the gas mixture behavior at a point rp on anabsorbing plate surface, is obtained by assuming that atoms ofspecies i impinging on the wall are absorbed with a velocity inde-pendent probability ai or reflected with probability 1 � ai. For eachspecies, the distribution functions of reflected atoms is a Maxwel-lian fwi, defined as:

fwi�rp; v

��t� ¼ nwi�rp; t

�ð2pRiTwÞ3=2

exp�� v2

2RiTw

�; v+n > 0 (13)

where Tw is the uniform temperature of the plates surface. The gasconstants are defined as Ri ¼ kB/mi, being kB the Boltzmann con-stant. The unit vector n is the normal to the plate surface at rp,pointing to the gas region. Each amplitude nwi is determined byrequiring that the reflected flux equals the impinging flux �J�imultiplied by the reflection probability 1 � ai:

A. Frezzotti et al. / Vacuum 103 (2014) 57e67 59

nwi

ffiffiffiffiffiffiffiffiffiffiRiTw

r¼ �ð1� aiÞJ�i ; J�i h

Zðv+nÞfi

�rp; v

��t�dv;

Fig. 2. Sketch of the two-dimensional, rectangular computational domain D. Arrowsnear top and bottom domain sides indicate the unperturbed inflow velocities uð�Þ

N atopen ends. Periodic boundary condition are applied at left and right domain verticalsides which separate D from its nearest images (dashed line rectangles).

2pv+n<0

(14)

In order to simplify an otherwise three-dimensional flow, it hasbeen assumed that the length Lz is much greater than the plateswidth Ly. In this case, it is reasonable to neglect flowgradients alongz and to assume fi to depend on the space variables x and y only. Theperiodic structure of the array allows limiting flowfield calculationto the rectangular strip S ¼ fr : �D=2 � x < D=2; y � 0g byimposing periodic boundary conditions on the sides at x ¼ �D=2and specular reflection for the velocity of molecules impinging onthe side at y ¼ 0. Furthermore, it is assumed that far from the arraythe mixture is in a state of equilibrium characterized by thefollowing Maxwellian form of the components distributionfunctions:

fNiðvÞ ¼ nNi

ð2pRiTNÞ3=2exp

"� ðv � uNÞ2

2RiTN

#: (15)

In the expression above, nNi is the upstream value of the i-thspecies number density, whereas TN and uN are the overall mixtureupstream temperature and bulk velocity, respectively. The bulkvelocity uN is written as

uN ¼ uNxbx þ uNyby; (16)

being bx and by unit vectors respectively parallel and normal to thearray axis.

The number of flow parameters and the relationships amongthem are more easily obtained by the non-dimensional form of thegoverning Eq. (1) and the associated boundary conditions (13), (14),(15). Reference values for mass, length and velocity can be obtainedfrom the equilibrium state at infinity as:

mN ¼X

2i¼1cNimi ; (17)

lN ¼X

2i¼1cNi

1P2j¼1 nNjps

2ij

ffiffiffiffiffiffiffiffiffiffiffimiþmj

mj

q0B@

1CA ; (18)

vN ¼ffiffiffiffiffiffiffiffiffiffiffiffikBTNmN

s: (19)

In the above equations, the quantities cNi are the upstreammixture molar fractions. Eq. (18) defines the mixture mean freepath, lN, being sij ¼ (si þ sj)/2 [10]. Adopting the above quantitiesas new units, readily shows that the solutions of the non-dimensional form of Eq. (1) depend on the following set of pa-rameters: Kn ¼ lN/Ly (the Knudsen number based on plate width),b ¼ D/Ly,m2/m1, s2/s1, a1, a2, TN/Tw, cN1,MNx andMNy. The paralleland normal upstream Mach numbers, MNx and MNy, are obtainedas the ratio of the corresponding upstream bulk velocity compo-nent to the mixture sound speed cN ¼ gkBTN/mN, being g¼ 5/3 thespecific heat ratio of amonatomic gas. The analogy between surfaceabsorption induced flows [1] and vapor condensation flows [4]suggests that the number of parameters which can be indepen-dently assigned in stationary conditions depends on the value ofupstream normal Mach number MNy. When the upstream normalvelocity component, uNy, is subsonic, MNy is a function of theremaining parameters, whose values can be assigned arbitrarily.When uNy, is supersonic, it is to be expected thatMNy can be freelyassigned within an interval determined by the remaining

parameters [4]. The large number of problem parameters preventsfrom obtaining a complete investigation of flow properties. How-ever, physical intuition and a sensible choice of exploratory simu-lations indicate that some important flow features are not verymuch dependent on the particular choice of the problem setting.Therefore, the necessarily limited sample of numerical calculationsshown below aims at highlighting a few specific aspects of theproblem, not at offering a thorough exploration of the flow pa-rameters space.

3. Sketch of the numerical method

Steady numerical solutions of Eq. (1) have been obtained by theDirect Simulation Monte Carlo (DSMC) method [10]. In DSMC, thedistribution function is represented by a collection of numericalparticles whose positions and velocities evolve in time by asequence of time steps, each of them consisting of a free flight and alocal collision sub-step. The former corresponds to the streamingoperator at l.h.s. of Eq. (1), whereas the latter is performed ac-cording to stochastic rules, consistent with the structure of thecollision terms given by Eq. (2). Macroscopic fluid properties areobtained by averaging particles microscopic states. The method isparticularly efficient in steady flows modeling where macroscopicquantities are obtained by time averaging microscopic states afterthe onset of steady flow conditions.

In the particular DSMC implementation used in the presentwork, the assumed specular symmetry of flow fields with respect tothe x axis has not been directly exploited to reduce the computa-tional effort. In order to increase the number of microscopic sam-ples and obtain flow properties with greater accuracy, thecomputational domain has been doubled. Flow properties havebeen computed in the finite rectangular domain D ¼ fðx; yÞ˛R2 :

�D=2� x < D=2; j:yj: < ymaxg, sketched in Fig. 2.D is constructed asthe union of the strip S, truncated at y ¼ ymax, with its specularimage in the half plane y < 0. Then, the boundary conditions atinfinity, as expressed by Eq. (15), and the specular reflection con-dition at y ¼ 0 are replaced by the following conditions aty ¼ �ymax:

A. Frezzotti et al. / Vacuum 103 (2014) 57e6760

fNiðx;�ymax; vÞ ¼ nNi

ð2pRiTNÞ3=2exp

264��v � uð�Þ

N

�22RiTN

375; (20)

uð�ÞN ¼ uNxbxHuNyby: (21)

The value of ymax has to be set sufficiently large to ensure theunperturbed flow conditions specified by Eq. (15). The domain iscentered on a single absorbing plate which is assigned infinitelength Lz and occupies the points of the segment P ¼ fðx; yÞ˛R2 :

x ¼ 0;�Ly=2 � y � Ly=2g. According to the boundary conditionexpressed by Eqs. (13) and (14), whenever a particle of species i hitsthe plate surface it is absorbed with assigned probability ai orinstantaneously re-emitted with probability 1 � ai at the samespatial position but with new velocity sampled from a probabilitydensity proportional to (v+n)fwi. Periodic boundary conditions areassumed to hold on the sides at x ¼ �D/2. Hence, when a compu-tational particle crosses any of the two sides parallel to the y, it is re-injected into the domain from the opposite side with the samevelocity.

The treatment of open ends boundary conditions at y ¼ �ymax isdifferent for upstream supersonic or subsonic values of MNy. Thelatter case requires special care, since uNy cannot be prescribed butit is an outcome of the calculation. The problem can be solved bydifferent methods [6]. In the present work the open boundary endshave been replaced by evaporating surfaces which emit atoms ofboth species into the domain. Thus, the flow is turned into anevaporation/condensation one [4], the array playing the role ofcondenser. The distribution functions associated with evaporatingatoms at y ¼ �ymax are the Maxwellians:

~fNi vð Þ ¼ ~nNi

2pRi~TN� �3=2 exp � v � ~uN

� �22Ri~TN

" #: (22)

The above expression holds for v+by < 0 at y ¼ ymax and forv+by > 0 at y ¼ �ymax. All atoms leaving D from open ends areeliminated, thus setting the evaporation coefficients of the ficti-tious evaporating surfaces equal to one. In Eq. (22) ~nNi, ~TN and~uN ¼ ~uNxbx are prescribed values of species number densities,temperature and velocity parallel to the array axis. It should benoted that, in general, ~nNi and ~TN do not coincide with the cor-responding quantities in Eq. (15). However, ~uNx and uNx do coin-cide as shown in Ref. [4] for steady evaporation flows with unitevaporation coefficient. If ymax is much larger than Ly then the flowfield can be divided into three regions. The first one is a one-dimensional Knudsen layer in the vicinity of each evaporatingwall. The Knudsen layer is followed by a more or less large plateauregion where the gas is in equilibrium, flowing with uniformdensity, composition, temperature and velocity towards the array.Finally, the mixture approaches the third region where the arraycause macroscopic quantities to have strong variations close to theplates. According to the flow picture outlined above, boundarycondition at infinity are set by tuning ~nNi and ~TN to obtain thedesired reference values of nNi and TN in the equilibrium plateauregion, uN being a result of the computations. It should be observedthat the treatment of the upstream boundary conditions describedabove and the analogy with evaporation/condensation problemsallow obtaining steady subsonic flow conditions in the plateau re-gion. Moreover, the extent of the Knudsen layers in the vicinity ofthe evaporating plates is an increasing function of MNy [11,12],ranging from a fewmean free paths for small values ofMNy to a fewhundreds mean free paths when MNy is close to unity.

Local flow properties are computed by dividing D into a grid ofnon-uniform rectangular cells. Smaller cells (typical size of lN/20)are used close to the plate surface, whereas cell size is allowed togrow far from the absorbing plate where gradients become negli-gible. It is to be noted that an accurate description of the Knudsenlayers close to evaporating surfaces is not required to build up thealgorithm for the control of upstream flow conditions.

The ymax value has to be chosen large enough to accommodatethe evaporation Knudsen layer, and awide plateau region in front ofthe array. For the flow settings considered here, a value of the orderof 100e200lN is sufficient for simulations with MNy below 0.6.Larger values (ymax ¼ 200e400lN) have been used for higher MNy

values to compensate the growth of the Knudsen layer thickness.Although larger values of simulation particles number have beenoccasionally used, a typical simulation has used about 2 � 106

unweighted particles. Sampling of microscopic particles propertiesis started after stabilization of the particles number in the domain.This is usually observed to occur in a time of the order of 5e10ymax/cN. After particles number stabilization, a sampling time of theorder of 10e20 ymax/cN is usually set, the number of samples beingaround 105. At the end of each simulation, average values of nN, cN1

and TN are computed within the plateau regions. Simulations arerestarted with corrected values of ~nNi and ~TN whenever the largestpercentage difference between the computed value of any amongthe quantities mentioned above and its corresponding presetnominal value exceeds 1.0� 10�3. In the following descriptions anddiscussions numerical results are labeled by nominal values ofprescribed flow parameters.

4. Collisionless flow properties

An important limit case of the general class of flows describedby Eq. (1) is the collisionless or free molecular flow regime, whenthe gas density is so low that atomic collisions in the gas phase canbe neglected. In this case, each species interacts only with theplates of the array and the problem reduces to solving two separatesingle component steady equations in the form:

v+vfivr

¼ 0 ; (23)

with boundary conditions at plate surfaces given by Eqs. (13) and(14). It will be further assumed that atoms coming from theambient gas surrounding the array do have the following Max-wellian distribution function prior to their first collision with aplate surface:

f 0ð ÞNi vð Þ ¼ nNi

2pRiTNð Þ3=2exp � v2

2RiTN

�: (24)

Since the governing equation and boundary conditions have thesame form for both species, the index imay be dropped. The problemdescribed by Eq. (23) is amenable to analytical treatment by a simpleextension of the classical solution methods for free molecular flows[13]. In particular, the determination of both local and global gasabsorption rates can be reduced, by straightforward manipulations,to the solution of the following linear integral equation:

~J ~yð Þ ¼ 1þ b2

2

Z12

�12

1� að Þ~J ~y0� �� 1

b2 þ ~y0 � ~y� �2h i3=2 d~y0 : (25)

Eq. (25), which applies to both gas components, has a dimen-sionless form obtained by setting: ~y ¼ y=Ly and ~Jð~yÞ ¼ J�=JN. Thedimensionless incident gaseous flux at position ~y on one of the

Fig. 4. Profiles of normalized incident flux along a plate surface for different values ofa. The plates spacing parameter b is set equal to 0.4. Curves are labeled by the cor-responding value of a.

A. Frezzotti et al. / Vacuum 103 (2014) 57e67 61

plate faces is ~Jð~yÞ, whereas JN ¼ nNRTN/2p is the reference fluxassociated with the ambient gas equilibrium state. The dimen-

sionless total absorbed flux is ~Jabs ¼ a

Z 12

�12

~Jð~yÞ d~y. It is worth

observing that the solutions of Eq. (25) depend on the parameters aand b, not on the temperature ratio Tw/TN. Accurate numericalapproximations of~JðzÞ can be obtained by elementarymethods. Theabsorbed flow rate can be converted into an effective flow velocityuabs (or Mach number) by equating Jabs to the net flux entering thedomain from a rectangular surface element of area D, far from thearray and normal to the y axis:

MNy ¼ uabsffiffiffiffiffiffiffiffiffiffiffiffigRTN

p ¼ 1b

ffiffiffiffiffiffiffiffiffi2gp

p ~Jabs : (26)

Fig. 3 shows MNy as a function of a, for different values of theplate spacing parameter b. For any fixed b, MNy is a monotonicallyincreasing function of the absorption probability. However its rateof change is a decreasing function of a. Actually, the linear growth,dictated by the factor a in the expression of~Jabs, is contrasted by thereduction of the incident flux~Jabs, as shown in Fig. 4. Such reductionis caused by the mutual shielding of nearest neighbor plates and bythe reduction of re-emitted molecular flux.

Fig. 3 also shows that the increase ofMNy caused by larger platespacking tends to saturate, particularly at high absorption proba-bilities. Again, the larger nominal absorption area, obtained byreducing b, is compensated by the interference of nearby plateswhich allow absorption to effectively occur only on the outermostsurface of each plate, as shown in Fig. 5.

5. Absorption driven flows in the transition regime

5.1. Single component gas flows

The absorption driven flows of a rarefied gas composed by asingle species is the second limit case to be examined before dis-cussing the more complex mixture flows. The effect of collisions inthe gas phase is now taken into account by solving the full Boltz-mann equation for a single species. The results of the simulationspresented and discussed in this section aim at providing a fewreference cases which allow a better understanding of mixturesflows. In particular, the influence of the Knudsen number Kn ¼ lN/

Fig. 3. Upstream Mach number, defined by Eq. (26), as a function of absorptionprobability a and array plates spacing to width ratio, b. B: b ¼ 0.4; ,: b ¼ 0.2; >:b ¼ 0.1;O: b ¼ 0.01; �: b ¼ 0.4 DSMC test collisionless flow simulations.

Ly, plates spacing ratio b and absorption probability a on MNy hasbeen explored for upstream subsonic solutions (MNy < 1). In thiscase, solutions are characterized by the following set of free flowparameters: Kn, b, a, Tw/TN and MNx. Once the above set of quan-tities is assigned a value, the upstream Mach number MNy (i.e. thenet absorption flow rate) is determined by solving the governingequation.

The temperature ratio Tw/TN and the Mach number MNx, asso-ciated with the upstream velocity component parallel to the arrayaxis, have been set equal to one and zero, respectively. Actually, it isreasonable to assume that MNy would respond to a variation of Tw/TN as shown in Ref. [1] for the one-dimensional geometry. Simi-larly, the analogy between the present flow setup and one-dimensional condensation flows [6,1] suggests that the presenceof a velocity component parallel to the array axis might have alimited effect on MNy. Fig. 6 shows the effect of flow parameters aand b on MNy, when Kn is set equal to 0.1. Corresponding freemolecular flow data are also displayed for easier comparison. For allof three plates spacing considered, the onset of gas advectioncaused by collisions has a strong effect on absorption rates whichare generally higher than companion values in the collisionless

Fig. 5. Profiles of normalized incident flux along a plate surface for different values ofb. The absorption probability ai is set equal to 0.8. B: b ¼ 0.4; ,: b ¼ 0.2; >: b ¼ 0.1;O: b ¼ 0.01.

Fig. 6. Single component gas. Upstream normal Mach number, MNy, as a function of a.Solid lines: Kn ¼ 0.1, b ¼ 0.1, 0.2, 0.4, Tw/TN ¼ 1.0 and MNx ¼ 0.0. Dashed lines, freemolecular flow regime. B: b ¼ 0.4; ,: b ¼ 0.2; >: b ¼ 0.1.

A. Frezzotti et al. / Vacuum 103 (2014) 57e6762

regime. Actually, gas phase collisions are expected to increase ab-sorption rates by reflecting towards the array plates molecules thatwould have escaped absorption in the free molecular flow regime.Moreover, gas convection favors the penetration of molecules in thegap between two nearby plates thus producing flatter absorptionprofiles, as shown in Fig. 7. The MNy profiles in Fig. 6 seem toindicate that free molecular flow and transition regime curvesapproach a ¼ 0 with identical slopes, for any fixed b. The expla-nation can be formulated by the following argument, which strictlyholds when Tw/TN ¼ 1.0. Let f and f be solutions of the full andcollisionless Boltzmann equations, respectively. Both f and f can beformally expanded in power of a around the value a ¼ 0. For thislimit value, no flow occurs and the Maxwellian f ð0ÞN , defined by Eq.(24), solves both the collisionless and full Boltzmann equations, ifTw/TN ¼ 1.0. Hence, one may write:

f ðr; vjaÞ ¼ f ð0ÞN þ avfva

����a¼0

þ O�a2

�; (27)

f r; vjað Þ ¼ f 0ð ÞN þ a

vfva

�����a¼0

þ O a2� �

: (28)

Fig. 7. Comparison of local absorption rate profiles along plate surface. Dashed line:free molecular flow, b ¼ 0.1, a ¼ 0.5. Solid line: collisional flow, Kn ¼ 0.1, b ¼ 0.1,a ¼ 0.5, Tw/TN ¼ 1.0 and MNx ¼ 0.0.

The absorbed fluxes associated with f and f can be respectivelywritten, at any point on an absorbing plate, as:

JabsðrjaÞ ¼ a

Zv+n<0

jv+njf ðr; vjaÞdv; (29)

Jabs rjað Þ ¼ a

Zv+n<0

��v+n��f r; vjað Þdv: (30)

Substituting the expansions (27), (28) into the above expres-sions, it is seen that the first order terms in the resulting expansionsof Jabs(rja) and Jabsðrj:aÞ are identical, being both determined by f ð0ÞN .

The effect of the flow parameters Kn and a on MNy is shown inFig. 8 for a fixed plates spacing ratio b ¼ 0.1. The increase ofabsorbing area, caused by the reduction of Kn, generally produces ahigher upstream flux. The results also show thatMNy curves can bedivided into two groups. Above a threshold value of Kn, steadysubsonic solutions are found for any value of a. Below thethreshold, steady subsonic solutions are only possible for a smallerthan a critical value of the absorption probability, ac(Kn, b) < 1,corresponding to exactly sonic upstream condition. For theconsidered geometry (b ¼ 0.1), the Knudsen number threshold isaround 0.05. The critical a decreases when Kn is reduced, beingaround 0.9 for Kn ¼ 0.05 and close to 0.6 for Kn ¼ 0.005. The valuesof ac mentioned above have been estimated as the highest a givingrise to nearly sonic uniform flow conditions in well defined up-stream plateau regions adjacent to the array.

Improving the estimate by going closer to the sonic limit leads toflowfields where the evaporation Knudsen layers near the twoopposite domain open ends have an extension of a few hundredsmean free paths and merge with the array flow region. In this case,no well defined upstream equilibrium flow region exists in thedomain and an improved approximation to ac can be obtained onlyby a considerable extension of the spatial domain and increase ofthe computational effort. A more detailed description of the flow-field structures is given in Figs. 9e13 for the smallest value of theKnudsen number, Kn ¼ 0.005. The plate spacing parameter is setequal to 0.1. The presented data refer to the two cases a ¼ 0.1 anda ¼ 0.6. The former corresponds to a relatively slow flow, whereasthe latter value produces nearly sonic upstream conditions. In theconsidered geometric setup, the plates array can be also viewed as aperiodic assembly of planar channels with absorbing walls. The gasenters the space between two nearby array plates from the two

Fig. 8. Single component gas. Upstream Mach number as a function of absorptionprobability and Knudsen number for fixed b ¼ 0.1. Tw/TN ¼ 1.0 and MNx ¼ 0.0. B:Kn ¼ 0.2; ,: Kn ¼ 0.1; >: Kn ¼ 0.05;O: Kn ¼ 0.005.

Fig. 9. Absorption driven flow of a single component gas, a ¼ 0.1, Kn ¼ 0.005, b ¼ 0.1, Tw/TN ¼ 1.0, MNx ¼ 0.0. (a) e Normalized number density field. (b) e Normalized temperaturefield.

A. Frezzotti et al. / Vacuum 103 (2014) 57e67 63

open ends with equal upstream density and temperature butopposite values of the upstream mean velocity. In absence of anupstream mean velocity component along the array axis, the flowsymmetry allows restricting the flowfield representation in therectangular domain fðx; yÞ˛R2Þ : 0 < x < D=2; 0 < y < ymaxg. Thedensity and temperature fields for a¼ 0.1 are shown in Fig. 9(a) and(b). The density keeps its upstream value until the gas enters theregion delimited by array walls whose absorption causes the den-sity to drop approximately to 70% of its upstream value. It is to benoted that the density gradient along x (normal to array plates) issmall. The temperature suffers very small variations from its up-stream value, TN. It is possible to observe a slight temperature in-crease near the plate surface and a slight temperature decrease inthe center of channel, near the inlet. The velocity field, separatelyrepresented by the x and y components of the gas mean velocity, isdepicted in Fig. 10(a) and (b). The velocity component normal toplates surface, ux, is very small. It suffers a sudden jump at thechannel inlet where the streamlines slightly turn towards thechannel center, in response to the small density bump shown inFig. 9(a). However, after a sudden drop, absorption forces ux tobehave roughly as a linear function of x, with little variations alongy. The velocity component uy is spatially uniform before the gasenters the channel region. Wall absorption causes a pressure dropin the space between plates. Accordingly, uy increases in the

Fig. 10. Absorption driven flow of a single component gas, a ¼ 0.1, Kn ¼ 0.005, b ¼ 0.1, Tw/TNsurface, ux. (b) e Map of the gas mean velocity component parallel to plates surface, uy. Ve

channel center and close to the inlet. In the vicinity of the platessurface, the opposite behavior is observed because gas-wall inter-action decreases the velocity component parallel to plates surface,as expected. On approaching the symmetry plane y ¼ 0, whereuy(x,0) ¼ 0, the flow decelerates almost linearly. The flow Machnumber map is given in Fig. 14(a). The flow is everywhere subsonicand the behavior of M(x, y) closely patterns the map of juy(x, y)j,since T(x, y) z TN and jux(x, y)j � 1. When a is increased up to 0.6,the density field, shown in Fig. 11(a), is qualitatively similar to theprevious one. However, the density drop is much stronger and aslight re-compression is observed near the symmetry plane atx ¼ 0. The temperature field, in the companion Fig. 11(b), exhibitsstronger variations. The gas temperature increases near the platewhere the gas velocity decreases but it decreases rapidly in thecentral region of the channel inlet following the gas expansion. Thesubsequent gas re-compression caused by the symmetric coun-terflow in the second half of the domain, causes the temperature toreach a minimum and increase monotonically till the symmetryplane at y ¼ 0. The behavior of the gas velocity components (seeFig. 12) is also very similar to that observed in the case of weakerabsorption rate. The magnitude of ux and uy is obviously muchgreater. The Mach number map, shown in Fig. 13(b) shows that theunperturbed upstream flow is nearly sonic and that becomes su-personic in the region between two plates.

¼ 1.0, MNx ¼ 0.0. (a) e Map of the gas mean velocity component orthogonal to plateslocity components are normalized to

ffiffiffiffiffiffiffiffiffiffiRTN

p.

Fig. 11. Absorption driven flow of a single component gas, a ¼ 0.6, Kn ¼ 0.005, b ¼ 0.1, Tw/TN ¼ 1.0,MNx ¼ 0.0. (a) e Normalized number density map. (b) e Normalized temperaturemap.

A. Frezzotti et al. / Vacuum 103 (2014) 57e6764

5.2. Binary gas mixture flows

The mixture flow simulations results discussed below aim atshowing the effects of molecular mass ratio, mixture compositionand absorption probabilities on the net absorption rate. To limit thesimulations number, a single value of the mass ratio has beenconsidered, m2/m1 ¼ 5. However, mixture flow results are con-trasted to the corresponding findings for a single species and for anartificial mixtures of mechanically identical species (m2/m1 ¼ 1),having different absorption probabilities. The array geometry hasbeen kept fixed, by setting Kn ¼ 0.1 and b ¼ 0.1. Furthermore, thetemperature ratio Tw/TN and the Mach number MNx have been setequal to 1.0 and 0.0, respectively. The effect of molecular mass ratioand mixture composition have different effects, depending on thevalues of the absorption probabilities ai. Fig. 14 presents profiles ofMNy computed by assigning the same value to the absorptionprobabilities to the species of a mixture whose mass ratio m2/m1has been set equal to 5. The upstream molar fraction cN1 has beengiven the values 0.25, 0.5, 0.75, 0.9783, corresponding to the con-centrations ~cN1 ¼ 0:0625; 0:1667; 0:375; 0:90. In the casesconsidered, the mixture absorption rates remain rather close to thesingle species absorption profiles. The differences between themixture and single species absorption curves shown in Fig. 14 arelarger than those found in the one-dimensional geometry [1] for

Fig. 12. Absorption driven flow of a single component gas, a ¼ 0.6, Kn ¼ 0.005, b ¼ 0.1, Tw/TNsurface, ux. (b) e Map of the gas mean velocity component parallel to plates surface, uy. Ve

similar flow settings and, when considered functions of mixturecomposition, they exhibit a maximum which seems to be locatedbetween cN1 ¼ 0.5 and cN1 ¼ 0.75.

Although the overall mixture absorption rates do not deviatemuch from the single species case, the different atomic mass valuesof the two components cause species separation in the vicinity ofthe array and between the array plates. As a matter of fact, theupstream velocities and temperatures are the same for both spe-cies. Hence, the absorption rate per particle is the same for bothspecies. However, the gas-wall collision rates are proportional to 1/mi therefore equality of absorption rates per particle is obtained byan increase of the concentration of the heavy component in thearray region. In principle, the concentrations could depend on bothx and y spatial coordinates but in the considered flow conditionsthe gradient in the direction x, normal to array plates, is negligible.Fig. 15 shows the x-averaged molar concentrations profiles of thelight component in the case a ¼ 0.5, for three different values ofupstreammixture composition. Species separation begins about 10reference mean free paths from the gap between the plates andreaches its maximum on the flow symmetry plane at y ¼ 0, asexpected.

In the more interesting situation when the wall absorptionprobabilities of the two mixture components are different, themixture separation in the vicinity of array plates is determined not

¼ 1.0, MNx ¼ 0.0. (a) e Map of the gas mean velocity component orthogonal to plateslocity components are normalized to

ffiffiffiffiffiffiffiffiffiffiRTN

p.

Fig. 13. Absorption driven flow of a single component gas, Kn ¼ 0.005, b ¼ 0.1, Tw/TN ¼ 1.0, MNx ¼ 0.0. (a) e Mach number field, a ¼ 0.1. (b) e Mach number field, a ¼ 0.6.

A. Frezzotti et al. / Vacuum 103 (2014) 57e67 65

only by the mass ratio m2/m1 but also by the different values takenby the coefficients ai. On the ground of previous investigations [7,1]based on one-dimensional flow geometries, it is expected that thepresence of a species with small absorption probability decreasesthe absorption rate of the most condensible component. Themechanism is very simple: The less condensible species accumu-lates in front of the absorbing wall thus slowing the absorptionprocess of the other. On the other hand, the absorption rate of theless condensible species is enhanced since the more condensiblecomponent behaves as a shepherd species which promotes itsaccumulation near the absorbing wall. In one-dimensional geom-etry, the effect is quite strong since a small amount of the compo-nent with small absorption probability can considerably inhibit theoverall mixture absorption rate. However, it is reasonable to expectthat the intensity of the effect is influenced by the flow geometrywhich may affect the accumulation of the less condensible speciesnear the absorbing plates.

In order to investigate this aspect of the problem, a series ofDSMC simulations have been performed on a fixed array geometry(Kn ¼ 0.1, b ¼ 0.1) and setting Tw/TN ¼ 1.0,MNx ¼ 0.0,m2/m1 ¼ 5, asabove. The simulations series consists of three groups. In the firstone, species 2 is assigned the role of less condensible species by

Fig. 14. Absorption driven flow of a binary mixture, Kn ¼ 0.1, b ¼ 0.1, Tw/TN ¼ 1.0,MNx ¼ 0.0, m2/m1 ¼ 5, a1 ¼ a2 ¼ a. Upstream normal Mach number MNy as a functionof the absorption probability, a and mixture composition. Solid line: reference singlecomponent case; +: cN1 ¼ 0.25 ð~cN1 ¼ 0:0625Þ; �: cN1 ¼ 0.5 ð~cN1 ¼ 0:1667Þ; ,:cN1 ¼ 0.75 ð~cN1 ¼ 0:375Þ; *: cN1 ¼ 0.9783 ð~cN1 ¼ 0:90Þ.

setting a2 ¼ 0.05; the upstream mixture composition has beenvaried by setting cN2 ¼ 0.1 ð~cN2z0:357Þ and ~cN2 ¼ 0:1(cN2 z 0.0217). The second group contains simulations in whichthe role of less condensible species is played by species 1. Accord-ingly, a1 is set equal to 0.05 and the upstreammixture compositionvaried by setting cN1 ¼ 0.1 ð~cN1z0:0217Þ and ~cN1 ¼ 0:1(cN1 z0.357). The third group contains simulations of a mixture ofmechanically identical atoms (m2 ¼ m1) in which the lesscondensible component is assigned an absorption probability equalto 0.05 and fixed upstream concentration equal to 0.1. The simu-lations of the third group have been performed to help separatingthe contributions of mass ratio and absorption probabilities dif-ference to overall mixture absorption rate. In all simulationsgroups, the absorption probability of the most condensible (andabundant) species, am, has been varied between 0.1 and 1.0.

The upstream Mach number curves obtained from the simula-tions described above are shown in Fig. 16 along with the referencecurve corresponding to the single component case. The reduction ofupstream flow Mach number is evident not only in the two caseswhere the less condensible species has the higher upstream con-centration, but also in the cases where its concentration is quitesmall. The comparison with the behavior of the mixture of

Fig. 15. Absorption driven flow of a binary mixture, Kn ¼ 0.1, b ¼ 0.1, Tw/TN ¼ 1.0,MNx ¼ 0.0, m2/m1 ¼ 5, a1 ¼ a2 ¼ 0.5, cN1 ¼ 0.5 ð~cN1 ¼ 0:1667Þ. x-averaged molarconcentration profiles of the light component. B: cN1 ¼ 0.25; ,: cN1 ¼ 0.5; >:cN1 ¼ 0.75. The vertical dashed line marks the right edge of region occupied by arrayplates.

Fig. 16. Absorption driven flow of a binary mixture, Kn ¼ 0.1, b ¼ 0.1, Tw/TN ¼ 1.0,MNx ¼ 0.0, m2/m1 ¼ 5, Overall upstream mixture Mach number MNy as function of theabsorption probability am of the most abundant component. The absorption proba-bility of the less abundant component, al, has been kept fixed and equal to 0.05. B:a2 ¼ al, cN2 ¼ 0:1ð~cN2 ¼ 0:357Þ; ,: a1 ¼ al, cN1 ¼ 0:1ð~cN1 ¼ 0:0217Þ; �: a2 ¼ al,~cN2 ¼ 0:1ðcN2 ¼ 0:0217Þ; -: a1 ¼ al, ~cN1 ¼ 0:1ðcN1 ¼ 0:357Þ; dashed line:m2 ¼ m1, a2 ¼ al, cN2 ¼ 0.1; dashed-dotted line m2 ¼ m1, a2 ¼ al, cN2 ¼ 0.357.

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mechanically identical atoms shows that the mass ratio value alsoplays a role in determining MNy. The dashed-dotted line and thesolid line marked with filled squares respectively refer to two gasmixtures with the same upstream molar composition but differentmass ratio. In the first case, the two species have equal atomicmasses and the presence of the less condensible species is quiteeffective in decreasing the overall absorption rate, by accumulatingin front of the array and within the plates, as shown by the com-panion concentration curve in Fig. 17. In the second case, the massratio is 5 and the heavy species is the most abundant one. Theoverall absorption rate is now higher because the heavy species canmore easily penetrate within the gap between the array plates, asseen from the corresponding concentration profile in Fig. 17. It isalso worth observing that, when the concentration of the lesscondensible species is small, the Mach number curves are not farfrom the single component curve. Such results are rather differentfrom those obtained from the one-dimensional problem consid-ered in Ref. [1] where the addition of a small amount of a species

Fig. 17. Absorption driven flow of a binary mixture, Kn ¼ 0.1, b ¼ 0.1, Tw/TN ¼ 1.0,MNx ¼ 0.0, m2/m1 ¼ 5, al ¼ 0.05, am ¼ 0.5. x-averaged molar concentration profiles ofthe less abundant component. Curve markers are the same of Fig. 16.

with low absorption probability causes large deviations from theabsorption rate of a single component gas. The two different be-haviors are easily explained by the respective flow geometries. Inthe one-dimensional absorption driven flow, the mixture is incontact with an infinite planar plate. Flow properties variations areassumed to occur only along the coordinate normal to the plate andthe velocity is normal to the plate surface. When one of the mixturecomponents is weakly absorbed, it accumulates and forms a uni-form and almost stationary gaseous sheet above the absorbingsurface. The absorption of the second component is reduced justbecause its atoms have to penetrate the sheet before reaching thewall at the position of minimum concentration.

In the two dimensional case, accumulation of the lesscondensible component is also present, as shown in Fig. 17, but nowthere exist a velocity component parallel to the plates whose lengthis finite in y direction. As a result, the concentration of the lesscondensible species along the plates surface is no longer uniformsince it reaches its maximum in on the symmetry plane y ¼ 0 but itis closer to its upstream value near the plates edges. Therefore theaccumulation of the less condensible species is less effective inreducing the overall mixture absorption.

6. Final remarks and conclusions

Flows of single component and binary gas mixtures generatedby absorption at the surface of an infinite array of planar parallelplates have been investigated in the early transition regime bynumerical solution of a system of coupled, spatially two-dimensional Boltzmann equations. Single component gas simula-tions have been performed to investigate the transition from thefree molecular to the collisional flow regime and the influence ofgas phase collisions on the overall absorption rate. Gas componentsseparation and interaction have been investigated, in particularwhen one of the gas mixture component is present in smallamounts and has a very low absorption probability. The resultsshow that the overall absorption rate in the two-dimensional ge-ometry shows a considerable deviation from the results of a simplerone-dimensional model of the same phenomenon. On the groundof the results obtained so far, it seems reasonable to conclude thataccurate modeling of the device geometry would be necessarybefore planning comparisons with experimental data. Simulationson simplified flowgeometries, although useful, are likely to provideonly a qualitative picture of the flow properties.

Acknowledgment

The authors gratefully acknowledge the support received fromFondazione Cariplo and SAES Getters within the framework of theprojects Surface interactions in micro/nano devices andMathematicalmodels for active barriers.

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