2.7 Aerosols and coagulation

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Samuel Corey Collins
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1 1 Note on 1.63 Advanced Environmental Fluid Mechanic Intructor: C. C. Mei, 1 ccmei@mit.edu, December 1,.7 Aerool and coagulation [Ref]: Preent, Kinetic Theory of Gae Fuch, Mechanic of Aerool Friedlander, Smoke, Dut and Haze Seinfeld, Atmopheric Chemitry and Phyic of Air Pollution Levich: Phyio-Chemical Hydrodynamic.7.1 Brownian diffuion of particle Particle of ub-micrometer ize collide with air molecule randomly. and behave collectively a a ga. Let n(z) = number of particle per unit volume, i.e, the number denity of Brownian particle in air. By the perfect ga law, where k i Boltzmann contant and p = nkt, (.7.1) k = R/L, R = Univeral ga contant = erg/degree/mole L = Avogadro number = molecule /mole = calorie /degree /mole When the cloud of particle i in hydrotatic equilibrium, dp = ρg = nmg (.7.) dz where m i the ma per particle. Combining the preceding two equation, dp p = mg kt dz

2 hence, µ p(z) =p() exp mgz (.7.3) kt and µ n(z) =n() exp mgz (.7.4) kt which i Bolzmann law. Alternatively, random colliion on the microcale give rie to diffuiononthemacrocale. At the equilibrium tate, diffuion and gravitational convection mut balance each other o that, D dn nv = (.7.5) dz where V = fall velocity. Equating Stoke drag with the particle weight 6µaV = mg, weget V = mg 6µa (.7.6) Solving (.7.5) and uing (.7.6), n(z) =n() exp mgz! D6µa (.7.7) Upon comparion with (.7.4), the Brownian diffuivity can be identified D = kt 6µa (.7.8) Thi formula, due to Eintein (195) and Smoluchowki (196), i valid if a i maller than the mean free path ` of air molecule. Otherwie Cunningham empirical correction i needed,! kt D = C c (.7.9) 6µa where µ C c =1+` exp 1.1a (.7.1) a ` i the correction factor. For aerool particle in air under normal temperature, the diffuivity i D a = a (.7.11) A a rough order-etimate (Levich), we ue Eitein formula for water at room temperature, the Brownian diffuivity for colloidal particle i D a (.7.1)

In the following table taken from Seinfeld, p. 35, D and ν are compared. For gae, the mean free path i typically ` =1 5 1 6 cm. a (µ m) D (cm / (T = ) V (cm/ec) (ρ =1g/cm 3 ).1 5.14 1.1 5.5 1 4.1 6.75 1 6 8.

3 In the following table taken from Seinfeld, p. 35, D and ν are compared. For gae, the mean free path i typically ` = cm. a (µ m) D (cm / (T = ) V (cm/ec) (ρ =1g/cm 3 ) Coagulation due to Brownian diffuion When mall particle are bounced around randomly by urrounding fluid molecule, they may come o cloe to one another that Van der Waal force bind them together. Thi i coagulation. In a moving fluid additional factor uch a fluid hear and Columb force may intervene. A imple model (by Smoluchowki) for a tationary fluid with identical pherical particle of radiu a i a follow. 3 Figure.7.1: Left: A pherical hell. Right: Two pherical partical in colliion. Let u focu attention on a fixed particle. Conider a pherical hell from r to r + dr, Figure.7.-left. The rate of increae of particle inide the hell i n t 4r dr which mut be equal to the net influx through the two urface of the hell r 4r D n r! dr thu, n t = D r n! r r r (.7.13)

4 4 Aume that whenever two particle come into contact they tick to each other and become one. Therefore the pherical urface of radiu a concentric with the tationary particle act a a ink, on which n =, i.e., n = r =a (.7.14) See Figure.7.-right. The initial condition i n r (.7.15) n =, a <r< t = (.7.16) Th olution can be facilitated by introducing w = n µ r a (.7.17) and it i hown in Appendix A that where The boundary condition become, x = r a a (.7.18) w t = D w x (.7.19) D = D (a) (.7.) w =1, x =, (.7.1) while The initial condition i w =, x = (.7.) w =, t =,x> (.7.3) The olution, which can be obtained by the imilarity method (ee Appendix B), i: w =1 erf x Dt! =1 Z x Dt e z dz (.7.4) or, or n(r, t) =1 1 n n =1 a r 1 Z r a Dt Z r a Dt e z dz e z dz

5 5 Finally, the number concentration near a fixed particle i n(r, t) =1 a r + a r Z r a Dt o e z dz =1 a r + a µ r a r erf Dt (.7.5) We now ue thi information to find the rate of coagulation when all particle are moving, by calculating the number denity of particle in the proce of colliion. Starting from one particle, the rate of flux of particle acro the phere of radiu r =a i " # n J(t) =4r D =4D(a) 1+ a! (.7.6) r r=a Dt When we get the teady tate limit, t À (a) D J( ) =4D(a) =8Da (.7.7) Let u etimate D =1 4 cm /ec, anda =1 6 cm, then the time to teady tate i (a) D =1 8 ec and i very hort. Each tationary particle will be hit by, hence coagulate with, 8D a particle per econd. Since all particle are moving, the teady rate of colliion (coagulation) mut be doubled, i.e., 16Da. A the conequence, the number denity of particle. mut decreae. Each colliion reduce the number of particle by 1. Hence Thu or which may integrated to d dt d dt = 16aDn d n = 16aDn (.7.8) where D = kt 6µa = 16aDdt " 1 1 # = 16aDT () Note that Finally, (t) = 16aD = 16akT 6µa = 8 kt 3 µ () 1+[16aD] ()t = () 1+K o ()t (.7.9)

6 6 where i the coagulation contant while i the coagulation time. From Fuch, Table 8, p 91. K o =16aD = 8 kt 3 µ T coag = 1 K o () (.7.3) (.7.31) a(cm) K o 1 1 (cm 3 /ec) How long doe it take for (t) to drop to one-tenth of it initial value? t = () 1 K o () (.7.3) where It can be etimated that t 1 1 K = 4 kt 3 µ.7.3 Appendix A: Proof of (.7.19) 15 ec, if a =.1µm, andt =93 K. w t = r 1 n a t w x = w/r " 1 x/r =aw r =a n r a a = n r n r # " w x =a 1 n r 1 n r r n r = a r # 1 n r Subtituting thee reult in (.7.19), we get n t = D (a) n r +! n = D r n! r r r r r " # n r r + n r (.7.33) with D = D (a)

7 .7.4 Appendix B: Solution of (.7.19) by the method of imilarity Let u eek a tranformation x = λ a x t = λ b t w = λ c w uch that the initial-boundary-value problem retain the ame form. w t = w D x d t λ b t λ b+c x Forinvariancewerequire, a = b, a = b/. Clearly ξ = x λ a! w = D λ a+c w! t x x D t = λb/ x D λ b t = x D t atifie the requirement. From the boundary condition, which require that x λ a = λ c w =1 c = (.7.34) The initial condition a well a the boundary conditon at x λ a = are trivially atified. The imilarity olution i! x w = w(ξ) =w D t Some algebra: hence or Integrating w t ξ = w t = x w D µ 1 = w ξ t /3 t D w x = w D 4D t = w 4t w ξ = w 4 w w d log w dξ = ξ = ξ log(w )= ξ +Cont, 7

8 8 and Thu o that w( ) =. Since The integral i /. Hence w = w = ce ξ = dw dξ Z w = c e z dz z w =1, ξ = 1= c Z e z dz c = " Z Z e z dz = ξ =1 Z o x Dt e z dz Z # ξ e z dz

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