AN IMPROVED PARTICLE POPULATION BALANCE EQUATION IN THE CONTINUUM-SLIP REGIME

Pop u la tion bal ance equa tions (PBE) are gen eral math e mat i cal frame work for mod el ing of par tic u late sys tems [1]. The be hav ior of these sys tems is largely gov erned by the col li sion rate, i. e., the num ber of col li sions per unit time per unit vol ume of aero sol. The cal cu la tion of col li sion rate is sim pli fied by the fact that par ti cles are typ i cally found at low vol ume frac tions in the gas phase. Un der such di lute con di tions, col li sion can be mod eled as two-body in ter ac tions wherein ther mal/Brownian mo tion drives col li sions. In the frame work of mono-vari ants in ter nal co-or di nate and time for each par ti cle, the PBE char ac ter ized as Smoluchowski equa tion, which takes the form:


In tro duc tion
Pop u la tion bal ance equa tions (PBE) are gen eral math e mat i cal frame work for mod eling of par tic u late sys tems [1].The be hav ior of these sys tems is largely gov erned by the col li sion rate, i. e., the num ber of col li sions per unit time per unit vol ume of aero sol.The cal cu la tion of col li sion rate is sim pli fied by the fact that par ti cles are typ i cally found at low vol ume frac tions in the gas phase.Un der such di lute con di tions, col li sion can be mod eled as two-body in ter ac tions wherein ther mal/Brownian mo tion drives col li sions.In the frame work of mono-vari ants in ter nal co-or di nate and time for each par ti cle, the PBE char ac ter ized as Smoluchowski equa tion, which takes the form: ¶ ¶ in which n(u, t)du is the num ber of par ti cles per unit spa tial vol ume with par ti cle vol ume from u to u + du at time t; and b is the col li sion ker nel/col li sion rate co ef fi cient of co ag u la tion.
The PBE can be viewed as the Boltzmann's trans port equa tion in form.For its own non-lin ear integro-dif fer en tial struc ture, only a lim ited num ber of known an a lyt i cal so lu tions ex ist for sim ple co ag u la tion ker nel.The an a lyt i cal so lu tion of PBE, es pe cially in terms of a parti cle size de pend ent co ag u la tion ker nel, still re mains a chal leng ing is sue.Be cause of the rel a tive sim plic ity of im ple men ta tion and low com pu ta tional cost, the mo ment method has been ex tensively used to solve most par tic u late prob lems, and has be come a pow er ful tool for in ves ti gat ing aero sol microphysical pro cesses in most cases [2][3][4].In the con ver sion from PBE to the mo ment equa tion, the k-th or der mo ment M k is de fined: By mul ti ply ing both sides of the PBE, eq. ( 1), with u k and in te grat ing over all par ti cle sizes, a sys tem of trans port equa tions for M k are ob tained.In a spa tially ho mo ge neous sys tem, the par ti cle mo ments evolv ing with time due to the Brownian co ag u la tion can be ex pressed: The min i mum set of mo ments re quired to close the par ti cle mo ment equa tion is the first three, M 0 , M 1 , and M 2 .The zeroth mo ment rep re sents the par ti cle num ber con cen tra tion, the first mo ment rep re sents the par ti cle vol ume con cen tra tion, and the sec ond mo ment is a poly-dispersity vari able.It should be pointed out that M 1 re mains con stant due to the mass conser va tion re quire ment, and its ini tial con di tions for the par ti cle mo ment evo lu tion equa tion can be note das M 00 , M 1 , and M 20 , re spec tively.
Ac cu rate cal cu la tion of par ti cles mo ment evo lu tion re quires ac cu rate col li sion ker nel co ef fi cient.When the par ti cle ra dius of at least one ob ject is large rel a tive to the mean per sistence dis tance of the col lid ing en ti ties, the con tin uum ap prox i ma tion is sat is fied and Smoluchowski's b ap plies: where f i and f j are the fric tion fac tors of type i and j en ti ties, a i and a j -the ra dii of type i and type j en ti ties, re spec tively, k B -the Boltzmann's con stant, T -the tem per a ture of col lid ing en ti ties and back ground fluid.The fric tion co ef fi cient is a quan tity fun da men tal to most par ti cle trans port pro cesses.In the con tin uum re gime, the Stokes law form holds for a rigid sphere as f i = 6pma i , and m is the gas vis cos ity.
In fluid dy nam ics, the Cunningham cor rec tion fac tor or Cunningham slip cor rec tion fac tor is used to ac count for non-con tin uum ef fects when cal cu lat ing the drag on small par ti cles.The der i va tion of Stokes Law, which is used to cal cu late the drag force on small par ti cles, assumes a no-slip con di tion, which is no lon ger cor rect at high Knudsen num ber.The Cunningham [5] slip cor rec tion fac tor al lows pre dict ing the drag force on a par ti cle mov ing a fluid with Knudsen num ber be tween the con tin uum re gime and free mo lec u lar flow.The slip cor rec tion fac tor C c is given by: with A 1 = 1.165,A 2 = 0.483, and A 3 = 0.997 ob tained in ex per i ments [6].The Knudsen num ber is a dimensionless num ber de fined as the ra tio of the mo lec u lar mean free path length, l, to a rep resen ta tive phys i cal length scale (the par ti cle ra dius, a, in the pres ent study), i. e., Kn = l/a.Re cently, Yu et al. [7] pro posed a mo ment mode for par ti cle PBE in the con tinuum-slip re gime with a linearized slip cor rec tion fac tor as C c = 1 + AKn (A = 1.591), and have solved it an a lyt i cally.In the pres ent work, we will im prove their mo ment model in the con tinuum-slip re gime with out ne glect ing the non-lin ear terms in the slip cor rec tion fac tor, and an alyze its as ymp totic be hav ior.The re sults show that both the im proved mo ment model and its sim pli fied forms have the same as ymp totic so lu tion as that in the con tin uum re gime [8].

Math e mat i cal for mu la tions
The col li sion ker nel in the con tin uum-slip re gime can be re-writ ten based on par ti cle vol ume: The slip cor rec tion fac tor can be ex panded with Tay lor se ries at the point (the par ti cle mean vol ume u = M 1 /M 0 ): and the cor re spond ing func tions g 0 , g 1 , and g 2 are de fined: 2 9 1 18 where the con stant is de fined as B = Kn 0 (M 1 /M 00 ) 1/3 , with the ini tial Knudsen num ber based on par ti cle mo ment is Kn 0 = l/u 0 = l(4pM 00 /3M 1 ) 1/3 .Then the or di nary dif fer en tial equa tion for par ti cle mo ment evo lu tion can be ob tained: with the con stant B 2 = 2k B T/3m. Us ing the Tay lor se ries ex pan sion, the higher and fractal par ti cle mo ment can be cal cu lated by the first three-par ti cle mo ments M 0 , M 1 , and M 2 [9]: Their ac cu racy has been dis cussed in our pre vi ous work [10,11], and then the or di nary dif fer en tial equa tions for par ti cle mo ment can be ob tained: 11 10 with the dimensionless par ti cle mo ment The geo met ric stan dard de vi a tion, s, of par ti cle size dis tri bu tion is the func tion of dimensionless par ti cle mo ment can be noted as ln 2 s = ln(M C )/9 [2].The or di nary dif fer en tial eq. ( 11) can be solved nu mer i cally with fourth Runge-Kutta method with its ini tial con di tions [9,12].

Sim pli fied mo ment model
In the case of ne glect ing the non-lin ear term in the slip cor rec tion fac tor, the func tions g 0 , g 1 , and g 2 are sim pli fied: and the cor re spond ing mo ment model is re duced to: The sim pli fied mo ment model is that Yu et al. [9] have ob tained in their work, but with a lit tle dif fer ence in the co ef fi cient (i.e., A 1 = 1.257 or A 1 = 1.165, but A = 1.591).
In some cases, the col li sion ker nel may be writ ten [13]: which means It is only suit able for nar row par ti cle size dis tri bu tion or small Knudsen num ber.And the cor re spond ing or di nary dif fer en tial equa tion for par ti cle mo ment evo lu tion can be sim plified: The as ymp totic anal y sis of mo ment model It can be found that: lim lim ; lim ( ) ; lim ( ) ; then the par ti cle mo ment evo lu tion eq. ( 11) and its sim pli fied mod els, eqs.( 13) and ( 16), are also re duced to the same as the mo ment model in the con tin uum re gime, and its an a lyt i cal and asymp totic so lu tions have been ob tained by Xie and He [14] and Xie and Wang [8], re spec tively.

Dis cus sion and con clu sions
In the pres ent study, we have pro posed an im proved mo ment model for par ti cle PBE in the con tin uum-slip re gime.This model can be sim pli fied to the ex ist ing mod els in the lit er atures [7,9] us ing the linearized slip cor rec tion fac tor.It has the same as ymp totic be hav ior as that in the con tin uum re gime.The linearized mo ment model be comes sim pler so that its an a lyt i cal so lu tion can be ob tained.Due to the in tro duc tion of non-lin ear terms in the slip cor rec tion factor, the an a lyt i cal so lu tion of pres ent mo ment model is dif fi cult to ob tain.Even if the an a lyt i cal so lu tion is ob tained with some spe cial math e mat i cal tech nique, e. g., the ex po nen tial func tion sub sti tuted by a poly no mial or a Tay lor se ries [14], it can only proof that the pres ent model has some ad van tages in math e mat ics, but the struc ture of the an a lyt i cal so lu tion will be come too com pli cated to be used in prac tice.More over, the ex pan sion of the ex po nent func tion will bring more con straints in phys ics and math e mat ics, which makes the ef fec tive in ter val for par ti cle dimensionless mo ment, M C , of the an a lyt i cal so lu tion much smaller [7,11,14].In the ory, the geo met ric stan dard de vi a tion of aero sol par ti cle size dis tri bu tion can be an ar bi trary value; this con fined in ter val of M C re veals the in her ent draw back in the mo ment method.
An other rea son for the un nec es sary to ob tain the pres ent model's an a lyt i cal so lu tion is that the rel a tive er ror of the so lu tion be comes much larger at high Knudsen num ber.In es sence, the slip cor rec tion fac tor is a drag cor rec tion [15], rather than the cor rec tion of the col li sion kernel it self.Al though the cor rected for mula of par ti cle re sis tance in fluid can be well ap plied in the wide range from free mo lec u lar to con tin uum re gime, the col li sion ker nel based on the slip correc tion fac tor will be come in fi nite and un rea son able in phys ics at higher Knudsen num ber.Our pre vi ous works [16], have pro vides a use ful at tempt to deal with the non-phys i cal phe nom e non, but the anal y sis and cal cu la tion of the mo ment model have in tro duced some em pir i cal for mula.A better ap proach of math e mat i cal phys ics to cal cu late the mo ment model ac cu rately across the en tire par ti cle size re gime will be pre sented in the fu ture.

Ac knowl edg ment
This work is sup ported by the Na tional Nat u ral Sci ence Foun da tion of China with Grant No.11572138.