On the moments of the Boltzmann's collision operator

On the moments of the Boltzmann’s collision operator arising from chemical reactions Neeraj Sarna and Manuel Torrilhon Citation: AIP Conference Proceedings 1786, 140005 (2016); doi: 10.1063/1.4967636 View online: http://dx.doi.org/10.1063/1.4967636 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1786?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A discretization of Boltzmann’s collision operator with provable convergence AIP Conf. Proc. 1628, 1024 (2014); 10.1063/1.4902706 Automated Boltzmann collision integrals for moment equations AIP Conf. Proc. 1501, 67 (2012); 10.1063/1.4769474 A temperature and mass dependence of the linear Boltzmann collision operator from group theory point of view J. Math. Phys. 37, 6139 (1996); 10.1063/1.531768 Structure of the Boltzmann Collision Operator Phys. Fluids 8, 431 (1965); 10.1063/1.1761242 Eigenfunctions of the Boltzmann Collision Operator Phys. Fluids 7, 1388 (1964); 10.1063/1.1711389

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On the Moments of The Boltzmann’s Collision Operator arising from Chemical Reactions Neeraj Sarna, Manuel Torrilhon Center for Computational Engineering Science, Department of Mathematics, RWTH Aachen University, Schinkelstr. 2, D-52062 Aachen, Germany

Abstract. For any study of microflows it is crucial to understand the collision dynamics of the molecules involved. In the present work we will discuss the collision dynamics of chemically reacting hard spheres(CRHS). The inability of the classical smooth inelastic hard spheres, which have been extensively used in the past to study granular gases, to describe the collision dynamics of chemically reacting hard spheres has been discussed. Using the model of rough inelastic hard spheres as a motivation, a new model has been proposed for chemically reacting hard spheres which has been further used to derive certain useful velocity transformations. A methodology to compute the moments of the Boltzmann’s collision operator arising from chemical reactions, using Grad’s distribution function, has been discussed in detail. Finally explicit expressions for the rates of the reaction have been obtained which contain contributions from higher order moment and thus can be used for non-equilibrium chemically reacting flows.

INTRODUCTION In the present work we are concerned with bi molecular reactions of the type A1 + A2 ↔ A3 + A4 . The ChapmanEnksog expansion developed in [1] was first used in [2] to study the perturbation of Maxwell’s distribution function due to the presence of chemical reactions. The study of transport coefficients for a reactive quaternary gaseous mixture was made in [3] using the Chapman-Enksog expansion. A Grad’s type distribution function for chemically reacting systems was proposed in [4] where the moments of the chemical collision operator were computed by considering the molecules to be accommodated at the equilibrium distribution function. In addition to the work done in the context of understanding the continuum models for chemical reactions, the dynamics of binary collisions have also been studied extensively. The study of a simple reactive collision, which is simple in the sense that it does not involve any internal degrees of freedom, was conducted in [5] where the following two different conditions for a chemical reactions to occur- (i) the total kinetic energy is greater than the activation energy of the molecules and (ii) the translational kinetic energy along the line joining the centres of the two colliding molecules is greater than the activation energy of the reaction, have been discussed. Moreover, the line-of-centres model for the chemical collision cross-section has also been developed in [5]. A study involving binary collisions has been conducted in [6] where expressions relating the rates of a reaction with the phase density function have been developed. Despite of the exhaustive work done in understanding the chemical cross-sections, there still lacks a discussion concerning the contact mechanics of reactive hard spheres. Since chemical collisions by nature are inelastic so the work done in the field of granular gases can be used as a motivation to develop a better understanding of chemically reacting hard spheres. The contact behaviour of smooth granular gases has been discussed in [7] using Hertzian contact theory. A discussion on the collision dynamics of rough hard spheres and a study of transport coefficients can be found in [8]. In the present work will first discuss the inability of the smooth granular gas model in describing the inelastic nature of reacting hard spheres; further a different model, which endows several properties of the rough granular gas model, for reacting hard spheres will be proposed. We will then derive certain velocity transformations which will prove to be helpful during the computation of the moments of the chemical collision operator. In [9], the importance of the fourteenth moment in describing the flow properties of a granular gas has been discussed. Using the work in [9] as a motivation, we think that the fourteenth moment can also be helpful in describing the non-equilibrium chemical processes. Therefore to compute the moments of the collision operator we will use a

30th International Symposium on Rarefied Gas Dynamics AIP Conf. Proc. 1786, 140005-1–140005-8; doi: 10.1063/1.4967636 Published by AIP Publishing. 978-0-7354-1448-8/$30.00

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distribution function which includes the fourteenth moment. A study of the influence of the fourteenth moment upon the relaxation to equilibrium of a homogeneous chemically reacting mixture has been left as a part of the future work. The present article has been organised in the following fashion: In section-2 we discuss the collision dynamics of chemically reacting hard spheres and derive the velocity transformations, in section-3 we discuss a methodology to compute the moments of the Boltzmann’s chemical collision operator and finally in section-4 we will present the explicit expressions for the rates of the reaction which involve contributions from higher order moments.

DYNAMICS OF BINARY COLLISIONS Laws governing binary collisions In all the coming sections we will only discuss the molecular interaction corresponding to the forward reaction, all the relations for the backward reaction follow analogously and thus will not be studied for brevity. Consider an interaction between two particles with masses m1 and m2 moving with velocities c1 and c2 respectively. If the molecules have enough kinetic energy they will undergo a chemical reaction which will lead to the formation of two particles with masses m3 and m4 with velocities c3 and c4 respectively. This molecular interaction can be described with the help of the following laws of mass, momentum and energy conservation m1 + m2 = m3 + m4

Mass Conservation : Momentum Conservation :

m1 c1 + m2 c2 = m3 c3 + m4 c4 m12 g212

(1) (2)

m34 g234

= +Q (3) 2 2 where g12 = c1 − c2 , g34 = c3 − c4 and Q is heat of the reaction. Let f and r denote the activation energy of the forward and the back reaction respectively, then Q = f − r . Using (3) we also have Energy Conservation :

g234 = Qˆ 12 g212 , g212 = Qˆ 34 g234 (4) 2Q m12 ˆ 34 = m34 1 + 2Q2 and mαβ = mα mβ . Moreover, we will assume the following hard with Qˆ 12 = m 1 − , Q 2 m12 mα +mβ m12 g12 m34 g34 34 sphere interaction potential between the molecules.

Mechanics of Contact Let k represent a unit vector passing through the centres of two colliding molecules. Defining two unit vectors as en = −k and et = en⊥ ; we can decompose g12 and g34 as g12 = gn12 en + gt12 et and g34 = gn34 en + gt34 et . Let n and t represent the coefficient of restitution in the normal and the tangential direction then we have the following relations: gn34 = n gn12 and gt34 = t gt12 . Substituting these relations into (3) we have n2 cos2 (θ) + t2 sin2 (θ) = Qˆ 12

(5)

where θ is the angle between k and g12 . Now we can consider the following two ways in which n and t can be modelled • •

Unisotropic CRHS(model-1): n = n (gn12 ) and t = 1. Isotropic CRHS(model-2): n = t =

The first model of the above two is the smooth inelastic hard spheres model which has been extensively discussed in [7]. The second model is based upon the assumption of isotropic scattering. It is obvious that the second model will lead to a change in gt12 whereas the first one will not and so it appears as if the law of energy conservation is incomplete because it does not include an additional rotational degree of freedom. But it is crucial to keep in mind that though model-2 is inspired from the rough inelastic hard sphere model, the mechanism of energy exchange in a granular gas and in a chemically reacting system is completely different and therefore one can safely ignore the rotational energy. We can now look into the details of the above two models.


Unisotropic CRHS Without even going into the details of this model we can discuss about one of it’s eminent shortcoming in describing 2 chemical reactions. When we assume that t = 1, we always end up transferring kinetic energy equal to 12 m34 gt34 along the tangential direction without any consideration of the heat of the reaction (Q). As a result of this, it might happen in many physical situations that one ends up transferring so much energy along the tangential direction that 2 the following inequality is satisfied 21 m34 gt34 + Q > 21 m12 g212 If the above inequality holds then in order to satisfy energy conservation, the following inequality should also 2 hold true: 12 m34 gn34 < 0 which is physically absurd. This is one of the major limitations of the present model; we can now look into it’s details. By substituting t = 1 in (5), we have the following expression for n s n =

Qˆ 12 − 1 +1 cos2 (θ)

(6)

where only the positive root has been considered since n is positive by definition. Moreover, since n is a real number, we require the positivity of the following function κ = Qˆ 12 − sin2 (θ) Substituting the definition of Qˆ 12 in the above expression, we have the following relation for κ:   m12  2Q    − sin2 (θ) κ= 1 − m34 m12 g212

(7)

It is easy to see that κ ≥ 0 iff m12 r ≥1 m34 f

∀ (Q > 0) ,

m12 ≥ 1 ∀ (Q < 0) m34

(8)

Since (1) is the only restriction upon the selection of masses, one can easily choose all the masses such that r 12 → 0 or m m34 → ∞. Also, r and f can be chosen such that f ∈ [0, ∞). Clearly, (8) does not hold true for all the physical systems. Moreover, it can also be trivially concluded from (8) that the present model is not suitable for any system with m12 < m34 . It is crucial to note that for a granular gas, the heat exchange (Q) during collision depends upon the restitution coefficient n as a result of which (6) leads to a trivial relation. Thus this particular model works fine for smooth hard sphere inelastic granular gas. m12 m34

Isotropic CRHS Substituting n = t = into (3) we have =

q

Qˆ 12

(9)

Thus the two restitution coefficients are a function of the relative velocities, the heat of the reaction and the various masses. The positivity of Qˆ 12 in (9) is ensured since 21 m12 g212 ≥ Q for a chemical reaction to happen. Thus unlike the previous model, the present model does not have any limitation over the physical parameters of the system. It is interesting to note that we have deviated from the collision dynamics of rough hard spheres in the following sense. For a rough hard sphere granular gas, n ∈ [0, 1] and t ∈ [−1, 1] because of which there is a possibility for a change in both the magnitude and the direction of gt12 . Whereas for our present system ∈ (0, ∞), as a result of which there is only a possibility for a change in the magnitude of gt12 . Due to the physical limitations of model-1 we have chosen to use model-2 for further use. It is crucial to note that though the model-1 suffers from certain physical restrictions it has a crucial property of exhibiting un-isotropic scattering. Whether such a scattering is crucial or not depends upon the physical process begin studied therefore model-2 should only be used for processes where unisotropic scattering does not have a major role to play.


Velocity Transformations Using model-2, we can now derive relations between the pre and post collisional velocities. It is trivial to see that for model-2, k bisects g12 and g34 . Thus an explicit form of k can be given as p Qˆ 12 g12 − g34

k =

p (10)

Qˆ 12 g12 − g34

p Taking dot product of (10) with Qˆ 12 g12 − g34 and using k.g34 = −k.g12 we get

q

q

ˆ

Q g − g = 2 Qˆ 12 k.g12 (11)

12 12 34

Substituting the above expression in (10) we get q q g34 = Qˆ 12 g12 − 2 Qˆ 12 (k.g12 ) k Using (2), the center of mass velocity can be defined as: h = µ12 c1 + µ21 c2 = µ34 c3 + µ43 c4 where µαβ = Using h, c3 and c4 can be written as c3 = h + µ43 g34

c4 = h − µ34 g34

(12) mα mα +mβ .

(13)

Using the definition of h, the above expression can be written as c3 = µ12 c1 + µ21 c2 + µ43 g34 ,

c4 = µ12 c1 + µ21 c2 − µ34 g34

Finally substituting (12) in the above expression, we have the desired velocity transformation ! q q c3 = c1 + g12 µ43 Qˆ 12 − µ21 − 2µ43 Qˆ 12 (k.g12 ) k c4 = c2 − g12 µ34

q

! q ˆ Q12 − µ12 + 2µ34 Qˆ 12 (k.g12 ) k

The velocity transformations for c1 and c2 can be derived in a similar fashion and are given as ! q q ˆ c2 = c4 − g34 µ12 Q34 − µ34 − 2µ12 Qˆ 34 (k.g34 ) k c1 = c3 + g34 µ21

q

! q Qˆ 34 − µ43 + 2µ21 Qˆ 34 (k.g34 ) k

(14)

(15) (16)

(17) (18)

For completeness we can now study how the above relations change for a single gas and a binary mixture. For a single gas, the above mentioned physical parameters change to m1 = m2 = m3 = m4 = m,

Q=0

(19)

Using the above physical parameters, the velocity transformations change to: c3 = c1 − (k.g12 ) k and c4 = c2 + (k.g12 ) k. For a binary mixture, the physical parameters change to m1 = m3 ,

m2 = m4 ,

Q=0

(20)

Using the above physical parameters we find the following velocity transformations: c3 = c1 − 2µ21 (k.g12 ) k and c4 = c2 + 2µ12 (k.g12 ) k. The above results for a single gas and a binary mixture are similar to those obtained in [1]. Therefore if we have a computational framework for computing the production terms of a chemically reacting mixture then by proper selection of certain physical parameters, as stated above, we can compute the production terms corresponding to a binary mixture or a single gas.


THE BOLTZMANN’S EQUATION The evolution of a gas in phase space is described by the famous Boltzmann’s equation(BE). Let fα = fα (x, cα , t) denote the phase density function of component-α defined such that fα dxdcα denotes the number of particles in the phase space volume dxdcα ; then the time evolution of fα is given as ∂t fα + cα .∇x fα = I (α) + R(α)

(21)

where Iα is the elastic collision operator, Rα is the chemical collision operator and any external force on the molecules has been neglected. The explicit forms of Iα and Rα are given as(see [11] for full derivation) I (α) =

4 Z X

0 0 fα fβ − fα fβ gαβ σαβ sin(χ)dχddcβ

(22)

β=1 0

where fα denotes the phase density function corresponding to post collisional velocities, σαβ denotes the differential collisional cross-section, χ = π − 2θ is the scattering angle and is the angle made by the collisional plane. The explicit form of R(α) is given as Z Z R(α) dcα = να f1 f2 g12 σr12 sin(χ)dχddc1 dc2 − να f3 f4 g34 σr34 sin(χ)dχddc3 dc4 (23) f where να is the stoichiometric coefficient, σr12 and σ34 are the differential collisional cross sections corresponding to the backward and the forward reaction respectively. For the present bi-molecular chemical reaction we have the following values for να : −ν1 = −ν2 = ν3 = ν4 = 1. For σαβ we will use the elastic hard sphere differential cross-section given as

σαβ =

2 dαβ

(24)

4

and for the chemical differential cross-section we will use the line-of-centres model which is given as [5] σr12

  = U 1 −

 2 2 f  d f  m12 g212 4

  1 −

 2 f   , m12 g212

σr34

  = U 1 −

 2r  dr2  m34 g234 4

  1 −

 2r   m34 g234

The function U(x) appearing in the above expressions is the unit step function which is defined as ( 1, x > 0 U(x) = 0, x < 0

(25)

(26)

Moment Approximation The use of moment approximation to (21) is motivated from the fact that in most of the practical applications not fα but the moments of fα are of a greater importance. The R mass density ρα andRthe momentum density ρα vα are the zeroth and the first moments of fα respectively i.e. ρα = m fα dcα and ρα vα = m cα fα dcα . ρα and ρα vα can further be used P P to define the density ρ = 4α=1 ρα and the momentum density ρv = 4α=1 ρα vα of the mixture. The peculiar velocity of the molecules can now be defined as Cα = cα − v. Using Cα , one can define a general N-th order moment w(s)(α) i1 ...iN as w(s)(α) i1 ...iN =

Z R3

dCα Cα2sChi(α)1 . . . Ci(α) Ni

(27)

where the angular brackets denote the trace free part of the N-th order tensor [12]. In addition to the above defined (α) ρα and vα , we can also define the temperature θα (in energy units), the stress tensor σ(α) as: i j and the heat flux qi R R R (α) (α) (α) (α) (α) mα 3 1 2 2 (α) Cα fα dCα , σi j = mα Chi C ji fα dCα and qi = 2 mα CαCi f dCα . 2 ρα θα = 2


The moment equations for the component-α can be obtained by testing (21) against a test function say φα = φα (Cα ). It is well known that use of any moment method leads to the classical closure problem. In the present work we will assume a Grad’s type closure which is given as fα |G =

f0(α)

N X

(α) λ(α) i1 i2 ...i p C i1

. . . Ci(α) , p

f0(α)

= nα

p=0

1 2πθα

! 32

C2 exp − α 2θα

! (28)

where the coefficients λ(α) i1 ...i p are computed such that fα |G satisfies the basic definition of the moments. Since we are interested in a 14-moment theory, the test function φα will be φα |14 = mα {1, Ciα , 12 Cα2 , Chiα C αji , 12 Cα2 Ciα , Cα4 } with (α) (α) 2(α) 3 the following corresponding moments wα |14 = {ρα , ρα v(α) }. For convinience we can define a i , 2 ρα θα , σi j , qi , u 2(α) quantity ∆α which is the deviation of u from it’s equilibrium value i.e.

∆α = u2(α) − 15ρα θα2

(29)

It should be noted that in all the coming sections,the non-equilibrium moments have been linearised around zero. Moments of Chemical Collision Operator We can now discuss a methodology to compute the moments of R(1) (ν1 = −1) since the computation for all the otherR components follows analogously. Particularly we are interested in the moments of the gain part of R(1) i.e. ω = φ1 f3 f4 g34 σr34 sin(χ)dχddC3 dC4 since the moments of the loss part are trivial to compute. The computation of ω can be summed up in the following five steps but since the first four steps remain the same as the binary mixture case so we don’t discuss them in detail, see [13] for an elaborate discussion. A roadmap to compute ω now goes as follows • • •

Step-1: Express φ1 in terms of C3 and C4 using (18) Step-2: Integrate over the collision vector k Step-3: For convinience we can non-dimensionalize the resulting expression from Step-2 using the following scales gαβ f r Cα ˆα = √ C , ˆr = , gˆ αβ = p , , ˆ f = (30) m34 θ34 m12 θ12 θα θαβ σ ˆ r12

•

    2ˆ f  2ˆ f   = U 1 − 2  1 − 2  , gˆ 12 gˆ 12

σ ˆ r34

    2ˆr   2ˆr  = U 1 − 2  1 − 2  , gˆ 34 gˆ 34

(31)

2 where θ12 = θ1 +θ 2 . Step-4: Due to the structure of the Gaussian in (28), the following velocity transformation can be used to simplify the integral

r r   θ4 ˆ  ˆ3 + ˆh34 = 1  θ4 C C3  , 2 θ34 θ34

•

3

fα θα2 fˆα = nα

r gˆ 34 =

θ3 ˆ C3 − θ34

r

θ4 ˆ C4 θ34

(32)

The integration over hˆ 34 and the vectorial part of gˆ 34 can be performed in a similar fashion as explained in [13]. Only the integral over the scalar part of gˆ 34 needs special consideration. Step-5: Let Ig represent the generic form of the integral over the scalar part of gˆ 34 then Ig can be given as Ig =

Z gˆ 234 >2r /m34

(l−r) gˆ 34

 2  2r  gˆ  2 gˆ 34 + 2 ˆ f − ˆr × exp − 34  dgˆ 34 4

(33)

q p gˆ 234 + 2 ˆ f − ˆr arises from the presence of Qˆ 34 in the velocity where l, r ∈ Q. The term of the form transformation. An explicit expression for Ig is possible if r ∈ 2N and numerical integration is needed if r < 2N.


RESULTS We will now give the explicit expressions for the rates of the reactions and the mass production rates. Discussion regarding the higher moments of the collision operators is beyond the scope of the present article and thus has been left as a part of the future work.

Rates of the Reaction and Mass Production Let k f and kr represent the rate of the reactions for the forward and the backward reaction respectively then they could be related to the phase densities through the following relations [6] Z Z 1 1 kf = f1 f2 g12 σ12 dΩdc1 dc2 , kr = f3 f4 g34 σ34 dΩdc3 dc4 (34) n1 n2 n3 n4 One can easily obtain the Arrehnius Law from (34) by substituting the Maxwell-Boltzmann’s distribution function in place of fα but the resulting expression will only be valid for processes in chemical equilibrium. Expressions for the rate of the reactions which hold true in chemical non-equilibrium can be obtained by replacing fα in (34) by an appropriate non-equilibrium distribution function. Replacing fα by (28) in (34) we obtain the following relations for k f and kr " !# " !# f r kr = γr exp − + ∆˜ r , k f = γ f exp − + ∆˜ f (35) 2m34 θ34 2m12 θ12 √ √ where γr = 4 πθ12 d2f , γ f = 4 πθ34 dr2 , ∆˜ f and ∆˜ r are the contributions from the higher order moments which are given as   ! 4 2  β f f˜ θ12 m12 , f  X ˜ (θ34 m34 , r ) X β f r 2  ∆α θα , ∆˜ f = −  ∆˜ r = − ∆α θα2 (36) 120Mn3 n4 120Mn n 1 2 α=3 α=1 where M = m1 + m2 = m3 + m4 and f˜ (a, b) = (a2 − b2 + 2ab). The coefficients β f and βr are given as:   √ √ " " # #  πd2 Mn1 n2 s2   πd2 Mn4 n3 s2  r 12 f f r 34         β f =  , βr =   exp −  exp − 7 7 2θ12 m12 2θ34 m34 2 2 m212 m234 θ12 θ34 The mass production rates can be given in the following way in terms of the rates of the reaction P(α) = m ν k − k α α f r 0

(37)

(38)

CONCLUSION We have proposed a new model for studying the contact of two chemically reacting hard spheres which is based upon the assumption of isotropic scattering. Using this model we have derived certain velocity transformations which have been found to converge to the correct limit. Further, we have proposed a methodology to compute the moments of the chemical collision operator. Studying realistic problems using these production terms has been left as a part of the future work.

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