( ) c(d p ) = 0 c(d p ) < c(d p ) 0. H y(d p )

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1 8.7 Gavimeic Seling in a Room Conside a oom of volume V, heigh, and hoizonal coss-secional aea A as shown in Figue 8.18, which illusaes boh models. c(d ) = 0 c(d ) < c(d ) 0 y(d ) (a) c(d ) = c(d ) 0 (b) Figue 8.18 Gavimeic seling of a monodisese aeosol in quiescen oom ai: (a) iniial condiion fo boh cases, = 0, (b) lamina seling model fo > 0, (c) well-mixed model fo > 0. Le c(d ) 0 be he iniial mass concenaion of aicles of diamee D in he oom (Figue 8.18a). In he lamina model all aicles of he same size fall unifomly a eminal velociy v such ha hose nea he boom sele o he floo. A any subsequen ime hee ae no aicles of ha size above a ceain heigh y(d ), and he concenaion emains a c(d ) 0 below his heigh (Figue 8.18b). The same agumen alies o aicles of ohe sizes, exce he values of y(d ) ae diffeen because hey sele a diffeen velociies. Fo he well-mixed model, on he ohe hand, some aicles sele, bu hose ha emain ae mixed houghou he oom volume such ha c(d ) deceases wih ime (Figue 8.18c) Lamina Seling Model Some algeba yields y(d ) = v Le he aveage mass concenaion of aicles be denoed by c (D ). Thus, ( ) Ay(D )c(d ) 0 A v c(d ) 0 v c(d) = = = c(d) 0 1 V V The aveage concenaion of hese aicles deceases linealy wih ime unil a ime ( c ) called he ciical ime elases, whee c = v and all aicles of he size whose eminal velociy is v have seled o he floo. Noe also ha c vaies wih eminal velociy v, which deends on aicle size, densiy, ec. i.e, in a olydisese aeosol, aicles of diffeen diamees sele a diffeen seeds, and heefoe have diffeen ciical imes. (c)

2 8.7.2 Well-Mixed Seling Model On he ohe hand, suose he well-mixed seling model is valid. As aicles of a aicula size fall o he floo wih velociy v, an idealized mixe insananeously edisibues he emaining aicles houghou he oom. The ae of change of mass of aicles of his size susended in he oom ai is equal o he ae of deosiion ono he floo, which can be inegaed o yield ( ) d c(d )V dc(d ) = V = vac(d ) d d c(d) c(d) v = = ex c(d ) c(d ) 0 0 Since he well-mixed model esumes ha he concenaion is he same houghou he enclosue a any insan of ime, he em c(d ) is also equal o he aveage concenaion, c (D ). The well-mixed model edics ha he aveage mass concenaion deceases exonenially, while he lamina model edics ha i deceases linealy. In he well-mixed model c (D )/c(d ) 0 = a = c, while in he lamina model i is zeo. In he well-mixed model, he aveage mass concenaion does no decease o (0.1%) of is iniial value unil nealy seven of hese ime consans, i.e. unil 7 c. The lamina model leads one o believe ha he dus will be emoved oo quickly because i ignoes unavoidable hemal cuens, dafs, diffusion, ec. ha edisibue aicles. The lamina model also edics an infinie concenaion gadien a an ineface (see Figue 8.18b) ha canno exis in naue. The well-mixed model oveesimaes he ime o clean he ai because i exaggeaes he mixing mechanisms. Fo small, inhalable aicles, i is howeve he moe ealisic model o use. Ceainly, i is he moe consevaive model. In eihe case, i has been assumed in he analyses above ha any aicle ha his he floo says hee. In ealiy, some of he aicles can be e-enained ino he oom by ai cuens; models of his e-enainmen ocess ae beyond he scoe of his ex. An addiional souce of eo is he fac ha some aicles ae deosied o adsobed on ohe sufaces in he oom, including he side walls, as discussed in Chae 5.

3 8.8 Gavimeic Seling in Ducs Gavimeic seling in ducs can also be analyzed using he conces of a lamina seling model and a well-mixed seling model. Figue 8.19 illusaes he lamina and well-mixed models fo flow in a hoizonal duc of ecangula coss secion (A = W). y c(d ) = 0 c(d ) in U 0 c(d ) = c(d ) in x y (a) c(d ) < c(d ) in c(d ) in U 0 (b) Figue 8.19 Gavimeic seling of a monodisese aeosol in a hoizonal duc wih unifom ai flow: (a) lamina condiions, (b) well-mixed condiions. I is assumed ha he hoizonal velociy of he gas eveywhee in he duc is equal o he aveage duc velociy (U 0 ), i.e. hee is lug flow in he gas hase. A he duc inle, he mass concenaion of aicles of diamee D is c(d ) in Lamina Seling Model In he lamina seling model, all aicles of he same size fall a hei eminal velociy (v ) and move wih a hoizonal velociy equal o ha of he caie gas, v x = U 0. Thus a a downseam disance x, he uemos aicles have fallen a disance ( - y), x y = v = v U 0 The aveage mass concenaion of aicles of a ceain size c (D ) a a disance x fom he inle can be wien as Combining he above wo equaions yields c(d) = c(d) c(d ) in c(d ) y x v = 1 U in 0 Since aicles sele o he floo of he duc, he duc can be hough of as a simle aicle colleco. The gade efficiency of he duc, η(d ), fo aicles of size D is defined as x

4 c(d ) x v η = = (D ) 1 c(d ) in U 0 The above equaion also alies o gavimeic seling in fully esablished flow beween aallel laes. See Flagan and Seinfeld (1988) fo he deivaion. A a ciical disance (x = L c ) downseam in he duc, he collecion efficiency is 100% and he duc conains no aicles of he size defined by he eminal velociy. The ciical disance is defined by and L c U = v η (D ) = = = 0 c 0 x v x v x U Lv L c Noe ha we can define a gade efficiency fo he gavimeic seling ocess! Well-Mixed Seling Model Afe some algeba see ex.. whee c(d ) v x v x W v η (D ) = 1 = 1 ex = 1 ex 1 ex As = c(d ) in U0 U0 W Q - A s = aea of lowe collecing suface, A s = xw - Q = volumeic flow ae, Q = U 0 W The gade efficiency can also be wien in ems of he ciical lengh, as defined above, x η (D ) = 1 ex Lc Comaison of he collecion efficiencies fo lamina and well-mixed seling models shows diffeences simila o hose concluded fo seling in ooms: (a) The lamina seling model oveesimaes deosiion because i ignoes ubulence and diffusion ha mix and edisibue aicles. (b) The well-mixed model exaggeaes mixing bu neveheless ovides a moe accuae and consevaive design esimae. (c) A he ciical downseam disance, L c, he well-mixed seling model edics a collecion efficiency of 63.2%, while he lamina seling model edics 100%. (d) A a downseam disance of aoximaely 7L c, he well-mixed seling model edics a collecion efficiency of 99.9% Ineial Deosiion in Cuved Ducs

5 Paicles have ineia, and can coss ai seamlines as skeched: v θ = U θ v 3 2 vθ D π ρ 6 ai seamline 2 2 v c D D π ρ C 4 2 ai seamline aicle ajecoy aicle ajecoy Figue 8.22 (a) (b) Quasi-saic equilibium of a aicle of diamee D and densiy ρ in cuvilinea flow; (a) aicle velociy comonens, and (b) foces acing on he aicle. The aicle s adial velociy comonen hen simlifies o U v = C θ τ The above equaion is he same as ha fo gavimeic seling exce ha gaviaional acceleaion is elaced by cenifugal acceleaion. Deosiion of aicles on he oue wall of he bend can heefoe be modeled in a fashion simila o gavimeic deosiion in hoizonal ducs, hough use of eihe he lamina (no mixing) seling model o he ubulen (well-mixed) seling model. The aicles fo which ineial seaaion is imoan ae usually sufficienly lage ha he Cunningham sli faco (C) is close o uniy, bu fo comleeness C is included in he analysis which follows Lamina Se ling Model 2 v Q, c in duc v θ θ in c = c in 2 θ imac Q c = 0 Figue 8.24 Paicle ajecoy (dashed line) and mass concenaion in a cuved duc of ecangula coss secion fo he lamina flow model; aicle enes a = in, θ = 0, and imacs he oue wall a = 2, θ = θ imac ; aicle shown a abiay ime. 1

6 Well-Mixed Model c = c in duc c = c θ v dθ Q, c in v θ θ in c = c ou 2 θ imac 1 Figue 8.25 Q Paicle ajecoy (dashed line) and mass concenaion in a cuved duc of ecangula coss secion fo he well-mixed model; aicle enes a = in, θ = 0, and imacs he oue wall a = 2, θ = θ imac ; aicle shown a abiay ime a locaion (,θ). Fou conol volumes ae shown wih degee of shading indicaing how mass concenaion deceases wih θ.

7 Examle Paicle Classifie Given: A comany ocesses agiculual maeials, gains, con, ice, ec. One of he ocesses is a milling oeaion. Significan fugiive dus is oduced. Enclosues and exhaus ai (Q, in CFM) ae needed o caue he dus. A classifie is needed o seaae no less han 50% of he aicles lage han 100 µm (D > 100 µm) which ae euned fo eocessing. The smalle aicles ae emoved by files (a baghouse see Chae 9). You sueviso suggess consucing a simle device consising of a 180-degee elbow of ecangula coss secion conaining louves on he ouside suface, as in Figue E8.9a. The volumeic flow ae of ai in he elbow is Q. Cenifugal foce sends lage aicles in he adial diecion; he aicles ass hough he louves and ae dawn off by a sli seam and emoved by ohe means. To do: Comue he gade efficiency cuves (simila o Figue 8.7) ha will enable oeaos o selec he oe volumeic flow ae Q o achieve a ceain emoval efficiency (η). Assume ha he gas flow is ioaional and well mixed. Plo he esuls fo a classifie whose dimensions ae: 1 = 0.30 m 2 = 0.70 m W = 0.40 m s = cm and which seaaes uni densiy aicles (ρ = 1,000 kg/m 3 ) aveling in an ai seam a 300 K. Soluion: Fom Eq. Eo! Refeence souce no found., 0.70 m 0.30 m = = = 0.70 m 2 ( 0.70 m) ln 2 ln 0.30 m K Fom Eq. Eo! Refeence souce no found., Sk τ Q τ Q τ Q avg = = = 3 2W ( 2 1) ( 0.70 m)( 0.40 m)( 0.70 m 0.30 m) m and fom Eq. Eo! Refeence souce no found., fo θ = π (180-degees), he facional efficiency is Figue E8.9a Cenifugal aicle classifie (fom einsohn & Kabel, 1999).

8 Figue E8.9b Facional efficiency of a cenifugal aicle classifie fo hee volumeic flow aes; classifie dimensions: 1 = 0.3 m, 2 = 0.7 m, W = 0.4 m, s = 0.07 cm (fom einsohn & Kabel, 1999). τ Q η = θ = π = τ m ( D) 1 ex ( Skavg) K C 1 ex (1.00) 1 ex( 22.3 Q) whee he Cunningham sli faco (C) is assumed o equal uniy fo such lage aicles, as was shown in Table 8.4. Figue E8.9b shows he facional efficiency a hee volumeic flow aes. Discussion: Clealy he lowes volumeic flow ae classifies aicles ooly. A he hee highe flow aes, he device emoves 200 and 300 µm aicles efficienly. Use of he well-mixed model is a easonable selecion since Reynolds numbes in he elbow ae suely lage enough o esablish ubulen flow. The assumion of ioaional flow, howeve, needs o be jusified. The nex level of sohisicaion is o use comuaional fluid dynamics (CFD) comue ogams o edic he ajecoies of aicles in he hee-dimensional velociy field of he 180-degee elbow. CFD is discussed in Chae 10.

Lecture 17: Kinetics of Phase Growth in a Two-component System:

Lecture 17: Kinetics of Phase Growth in a Two-component System: Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien