A BIMODAL TEMOM MODEL FOR PARTICLE BROWNIAN COAGULATION IN THE CONTINUUM-SLIP REGIME

In this pa per, a bi modal Tay lor-se ries ex pan sion mo ment of method is pro posed to deal with Brownian co ag u la tion in the con tin uum-slip re gime, where the non-lin ear terms in the Cunningham cor rec tion fac tor is ap prox i mated by Tay lor-se ries ex pan sion tech nol ogy. The re sults show that both the num ber con cen tra tion and vol ume frac tion de crease with time in the smaller mode due to the intra and inter co ag u la tion, and the as ymp totic be hav ior of the larger mode is as same as that in the con tin uum re gime.


In tro duc tion
Aero sol par ti cles are in creas ingly rec og nized as one of the most com mon un healthy com po nents of air pol lu tion, it has an aw ful im pact not only on the en vi ron ment but also on the health of hu man be ings.Mean while, there is also a strong re la tion ship be tween vis i bil ity and par ti cle size dis tri bu tion (PSD), es pe cially the par ti cle size from 0.1 to 100 mm [1].Par ti cle evolu tion in volves a wind range of phys i cal-chem i cal re ac tion pro cess, and the de tailed in for ma tion of PSD can be de rived from the par ti cle bal ance equa tion (PBE), which is char ac ter ized as the Smoluchowski equa tion of ir re vers ible co ag u la tion in terms of one-par ti cle den sity de pend ing on par ti cle size and time: ¶ ¶ where n(u, t) is the num ber den sity con cen tra tion of the par ti cles with vol ume from u to u + du at time t, and b(u, u 1 ) is the col li sion fre quency func tion be tween par ti cles with vol ume u and u 1 .
Theas equa tion is usu ally dis solved by nu mer i cal method due to the com pli cated non-lin ear par tial dif fer en tial-in te gral con struct, such as the mo ment method.It is to trans form the PBE into a se ries of mo ment equa tions and has be come a pow er ful tool for in ves ti gat ing aero sol microphysical pro cesses in most cases for its rel a tive sim plic ity of im ple men ta tion and low com pu ta tional cost, even though it is the par ti cle mo ment ob tained but not the real PSD.Recently, Yu et al. [2] pro posed a new mo ment method called as the Tay lor-se ries ex pan sion mo -ment of method (TEMOM), which could achieve the clo sure of mo ment equa tions with out any other prior as sump tion of PSD, and has been rec og nized as a prom is ing method [3][4][5].
Most stud ies fo cus on the evo lu tion of par ti cle sys tems with unimodal size dis tri bution, which lim its the ap pli ca tion of these tech nol o gies in some cases, for ex am ple, a smaller nucle ation mode with newly formed par ti cles and a larger ac cu mu la tion mode with par ti cles gen erated ear lier may co ex ist when the par ti cle for ma tion is not in stan ta neous [6,7].More over, re mov ing fine par ti cles through inter-co ag u la tion with coarse par ti cles also can't be rep re sented by a unimodal model be cause of the wide range of PSD [8].Modal aero sol dy nam ics (MAD) mod els rep re sent the PSD as a sum of two or more dis tinct modes, in which each mode is rep resented by a sep a rate size dis tri bu tion func tion and gov erned by the PBE [9].This method is success fully per formed for par ti cles un der go ing nu cle ation, co ag u la tion, and sur face growth si multa neously with the as sump tion that the PSD of each mode is monodisperse or log-nor mal dis tri bu tion [7,10].In the frame work of mo ment method, Lin and Gan [11] firstly in ves ti gated the Brownian co ag u la tion of par ti cles in the en tire size re gime with ini tial bi modal log-nor mal dis tri bu tion.Re cently, MAD mod els are ap plied to the TEMOM model in both the (near) contin uum re gime and free mol e cule re gime for Brownian co ag u la tion and ag glom er a tion [12][13][14].In this work, we will pro pose an im proved bi modal TEMOM model for Brownian co ag u la tion in the con tin uum-slip re gime with out ne glect ing the non-lin ear terms in the Cunningham correc tion fac tor.

Modal and the ory
For a bi modal model com posed of mode i (the smaller mode) and mode j (the larger mode), the num ber den sity con cen tra tion can be ex pressed as: n(u, t) = n i (u, t) + n j (u, t), and the cor re spond ing Smoluchowski equa tion takes the fol low ing form: ¶ ¶ The k th or der mo ment is de fined: ò n(u)du, and M 0 rep re sents the to tal par ti cle num ber con cen tra tion and M 1 the to tal par ti cle vol ume frac tion, while M 2 is a poly-dispersity vari able and u = M 1 /M 0 is the mean vol ume, not ing that the ini tial con di tion is M 00 , M 10 , M 20 , u 0 .Then eq. ( 2) can be trans formed into two sets of mo ment equa tions with the as sump tion that the new born par ti cles pro duced by inter-co ag u la tion be tween par ti cles in mode i and j be long to the larger mode j, Ob vi ously, the first term in the right side of eqs.3(a) or 3(b) rep re sents intra-co ag u lation in mode i or j and has no dif fer ence with the orig i nal unimodal mo ment equa tions, while the sec ond term in di cates the change by par ti cle mi gra tion from mode i to mode j due to inter-co agu la tion.
In the con tin uum-slip re gime, the col li sion ker nel func tion can be ex pressed: in which B 2 = 2K b T/3m, K b is the Boltzmann's con stant, T -the tem per a ture, m -the gas vis cosity, and C -the Cunningham cor rec tion fac tor: where Kn = l/r is Knudsen num ber de fined as the ra tio of the mo lec u lar mean free path length to the par ti cle ra dius; A 1 =1.165,A 2 =0.483, and A 3 = 0.997 [15].In some works, the non-lin ear term in C is ne glected, then C can be ap prox i mated as (1 + AKn) with A = 1.591 in case of Kn £ 5 [13,16,17].Knudsen num ber can also be writ ten: Kn 0 (u 0 /u) 1/3 if Kn 0 is set to be l/(3u 0 /4p) 1/3 , thus we can ap prox i mate the non-lin ear term g(u) = exp[-A 3 (u/u 0 ) 1/3 /Kn 0 ] at the point u us ing the Tay lor-se ries ex pan sion: in which g(u), g'(u), ¢¢ g (u) are the zeroth, first and sec ond de riv a tive of g(u), re spec tively.Truncate the right side of eq. ( 6) at the first three terms, and then C can be ar ranged: where f 0 (u), f 1 (u), and f 2 (u) are func tions of u: To gether with the eqs.(3-5), (7), the first three-or der bi modal TEMOM model in the con tin uum-slip re gime can be ex pressed as (k = 0, 1, 2): where  Fig ure 2 shows the par ti cle mo ment evo lu tion of mode j with dif fer ent Knj 0 and a, and the re sults of Mj 0 , Mj 1 , Mj 2 are nor mal ized as Mj k (t)/Mj k (0) for con ve nience.The de crease of Mj 0 is be cause of the intra-co ag u la tion in mode j and mainly de pends on the ini tial par ti cle num ber, while the inter-co ag u la tion has no di rect ef fect on Mj 0 due to the as sump tion that the new born par ti cles be long to mode j, but the con tri bu tion to mean par ti cle vol ume may have an in flu ence.When Knj 0 < 0.1, the char ac ter is tic of mode j can be treated as the same as that in con tin uum regime (usu ally de fines as Kn < 1 or 0.1), and the change caused by Knj 0 vary ing is neg li gi ble.The Mj 1 is in creas ing for the rea son of par ti cle mi gra tion from mode i to mode j, which will re sult in de creas ing Mi 0 and Mi 1 and then weaken the sig nif i cance of inter-co ag u la tion, thus Mj 1 will tend to a con stant [Mj 1 (0) + Mi 1 (0)] as time ad vances.In fig.2(d), the dimensionless mo ment M C tends to two at long time, which is as the same as that of a unimodal model in the con tin uum regime [18].

Con clu sion
In this work, we pro pose a bi modal TEMOM model for Brownian co ag u la tion in the con tin uum-slip re gime with out ne glect ing the non-lin ear terms in the Cunningham cor rec tion fac tor.The re sults show that num ber con cen tra tion is ex pected to have a ma jor con tri bu tion to inter-co ag u la tion, and the vol ume frac tion of mode i will de crease with time due to the inter-co - ag u la tion.For mode j, the de crease of num ber con cen tra tion is only caused by the intra-co ag u lation, while the inter-co ag u la tion will re sult in the in creas ing vol ume frac tion.As time ad vances, the Mi 0 and Mi 1 be come less and less, as well as the weak ened ef fect of mode i on inter-co ag u lation, thus the evo lu tion of mode j will tend to that of a unimodal model, and the as ymp totic behav ior be comes the same as that in the con tin uum re gime.

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.

Figure 1 .
Figure 1.The particle moment evolution of mode i with different mode j; (a) Mi 0 ; (b) Mi 1