ME 437/ME 537 PARTICLE TRANSPORT, DEPOSITION AND REMOVAL. Goodarz Ahmadi

Transcription

1 ME 437/ME 537 PARTICLE TRANSPORT, DEPOSITION AND REMOVAL Goodarz Ahmadi Deartment o Mechanical and Aeronautical Engineering Clarkson University Potsdam, NY

2 Air ollution and smog. Particle trajectories in a hot gas iltration vessel.

3 Samle glass iber articles. 3

4 Dust storm over the red sea. 4

5 Birth lace o stars. 5

6 INTRODUCTION TO AEROSOLS Deinition: Aerosol is a susension o solid or liquid articles in a gas. Dust, smoke, mists, og, haze, and smog are various orms o common aerosols. Aerosol articles are ound in dierent shaes (isometrics, latelets, and ibers) and dierent sizes. or irregular shaed articles, dierent equivalent diameters are deined. Examles o equivalent diameters are: Equivalent area diameter, eret s diameter (maximum distance edge to edge); Stoke s diameter (diameter o a shere with the same density and the same velocity as the article); Aerodynamic diameter (diameter o a shere with the density o water and the same velocity as the article). The range o diameters o common aerosol articles is between 0.01 and 0 µm. The lower limit o nm roughly corresonds to the transition rom molecule to article. Particles larger than 0 µm normally do not remain susended in air or a suicient amount o time. Noting that the mean ree ath or air is about 0.07 µm and visible light has a wavelength band o µm, the mechanical and otical behaviors o articles are signiicantly aected by their size. Particles greater than 5 µm are usually removed by the uer resiratory system. But articles smaller than 5 µm can enetrate dee into the lung and become a health hazard. Tyical ranges o values or aerosol arameters or aerosols are listed in Table 1. The corresonding values or air (N ) are also shown in this table or comarison. Table 1 - Parameters o Aerosol in the Atmoshere Aerosols Air Number Density (Number/cm 3 ) Mean Temerature (K) Mean ree Path Greater than 1m 0.06 µm Particle Radius 0.01 µm 4 µm Particle Mass (g) Particle Charge (in Elementary Charge Units) 0 0 Weakly Ionized Single Charge The imortant relevant dimensionless grous relevant the motion o aerosols are listed in Table. 6

7 Knudsen Number Mach Number Schmidt Number Brown Number Reynolds Number Table Dimensionless Grous λ Kn = d v v M = c ν n λd Sc = = D 4 v Br = ( v,, ) 1/ v' = v' v v d Re = = ν 4M K n Here the ollowing symbols are deined: λ = Mean ree Path ν = Kinematic Viscosity d = Particle Diameter D = Diusivity v = Particle Velocity v = Thermal Velocity v = luid (Air) Velocity n = Number Density c = Seed o Sound Here suerscrit " " corresonds to luid and suerscrit " " denotes article. In these equations the root mean square luctuation velocity is given by and v' = (8kT / π m ) 1/ ν = 0.5c λ The mean ree ath o the gas is given as 1 kt λ = = πnd m πd mp Here n is the gas number density, d m is the gas molecule (collisional) diameter, -3 k = 1.38 J/K is the Boltzmann constant, P is ressure, and T is temerature. or air, d m = 0.361nm and 3.1T λ ( µ m) =, P is in Pa, and T is K. P 7

8 Table 3. Aerosol Characteristics Particle Diameter, µm Electromagnetic x-ray UV Vis Inrared Microwaves Wave Deinition Solid ume Dust Liquid Mist Sray Soil Clay Silt Sand Gravel Atmosheric Smog Cloud/og Mist Rain Tyical Viruses Bacteria Human Hair Particles Smoke Coal Dust Beach Sand Size Analysis Microscoe Method Electron Microscoe Sieving x-ray Diraction Ultra Centriuge Sedimentation Gas Cleaning Ultrasonics Settling Chamber Centriugal Liquid Scrubber Air ilter HE Air ilter Imact Searators Thermal Searators Electrostatic Searators Diusion Coe. cm / s Air Water Terminal (S=) Air Velocity cm / s Water Particle Diameter, µm 8

9 HYDRODYNAMIC ORCES Drag orce and Drag Coeicient A article susended in a luid is subjected to hydrodynamic orces. or low Reynolds number, the Stokes drag orce on a sherical article is given by D = 3πµUd, (1) where d the article diameter, µ is the coeicient o viscosity and U is the relative velocity o the luid with resect to the article. Equation (1) may be restated as C D 4 = () 1 ρu A Re D = In Equation (3), ρ is the luid (air) density, sherical article, and πd A = is cross sectional area o the 4 ρ Re = Ud (3) µ is the Reynolds number. The Stokes drag is alicable to the creeing low regime (Stokes regime) with small Reynolds numbers (Re < 0.5). At higher Reynolds numbers, the low the drag coeicient deviates rom Equation. igure 1 shows the variation o drag coeicient or a shere or a range o Reynolds numbers. C D Eq. () Eq. (4) Re igure 1. Variations o drag coeicient with Reynolds number or a sherical article. 9

10 Oseen included the inertial eect aroximately and develoed a correction to the Stokes drag given as 4[1 + 3Re/16] C D =, (4) Re which is shown in igure 1. or 1 < Re < 00, which is reerred to as the transition regime, the ollowing exressions may be used (Clit et al., 1978): C D [ Re ] =, (5) Re or 4 4 C D = + (6) 0.33 Re Re 00 CD 0 Exeriment Oseen 1 Stokes Eq. (5) Newton Re igure. Predictions o various models or drag coeicient or a sherical article.

11 3 5 or < Re<.5, the drag coeicient is roughly constant ( C D = 0. 4 ). This 5 regime is reerred to as the Newton regime. At Re.5, the drag coeicient decreases sharly due to the transient rom laminar to turbulent boundary layer around the shere. That causes the searation oint to shit downstream as shown in igure 3. Laminar Boundary Layer Turbulent Boundary Layer igure 3. Laminar and turbulent boundary layer searation. Wall Eects on Drag Coeicient or a article moving near a wall, the drag orce varies with distance o the article rom the surace. Brenner (1961) analyzed the drag acting on a article moving toward a wall under the creeing low condition as shown in igure 4a. To the irst order, the drag coeicient is given as 4 d C D = (1 + ) (7) Re h d d U U h h (a) Motion normal to the wall (b) Motion arallel to the wall igure 4. Particle motions near a wall. or a article moving arallel to the wall as shown in igure 4b, the Stokes drag orce need to be modiies. or large distances rom the wall, axon (193) ound C 4 9 d 1 d 45 d 1 d = (8) Re 16 h 8 h 56 h 16 h D [1 ( ) + ( ) ( ) ( ) ] 11

12 Cunningham Correction actor or very small articles, when the article size becomes comarable with the gas mean ree ath, sli occurs and the exression or drag must be modiied accordingly. Cunningham obtained the needed correction to the Stokes drag orce: D 3πµ Ud =, (9) C c where the Cunningham correction actor C c is given by λ 1.1d / λ Cc = 1+ [ e ] () d Here λ denotes the molecular mean ree ath in the gas. Note that C c > 1 or all values o d and λ. igure 5 shows the variation o Cunningham correction actor with Knudsen number. It is seen that Cc is about 1 or Kn <0.1 and increases sharly as Kn increases beyond 0.5. Table 4 illustrates the variation o Cunningham correction actor with article diameter in air under normal ressure and temerature conditions with λ = 0.07 λ µm. Equation () is alicable to a wide range o Kn = 00 that covers sli, d transition and art o ree molecular lows. The article Reynolds number and Mach number (bases on relative velocity), however should be small Cc Kn igure 5. Variation o Cunningham correction with Knudsen number. 1

13 Table 4 Variations o C c with d or λ = 0.07 µm Diameter, µm C c µm µm µm µm µm 3.54 Comressibility Eect or high-seed lows with high Mach number, the comressibility could aect the drag coeicient. Many exressions were suggested in the literature to account or the eect o gas Mach number on the drag orce. Henderson (1976) suggested two exressions or drag orce acting on sherical articles or subsonic and suersonic lows. Accordingly, or subsonic low 1 C D = Re 4 Re + S ex 0.47 S 0.5M Re + ex Re Re Re ( Re) 8 M 0.1M 0.M 1 ex 0.6S Re (11) where M is Mach number based on relative velocity, V = V V, and S= M γ is the molecular seed ratio, where γ is the seciic heat ratio. or the suersonic lows with Mach numbers equal to or exceeding 1.75, the drag orce is given by C D M M Re S S S = (1) 1 M Re or the low regimes with Mach between 1 and 1.75, a linear interolation is to be used. 13

14 Carlson and Hoglund (1964) roosed the ollowing exression: C D ex( } = M Re (13) Re M Re 1+ { ex( 1.5 )} Re M Drolets or drag orce or liquid drolets at small Reynolds numbers is given as D 1+ µ 1+ µ /3µ / µ = 3πµ Ud (14) where the suerscrits and reer to the continuous luid and discrete articles (drolets, bubbles), resectively. Non-sherical Particles i.e., or non-sherical (chains or ibers) articles, Stokes drag law must be modiied. D = 3πµ Ud ek, (15) where d e is the diameter o a shere having the same volume as the chain or iber. That is, d 6 = (16) π 1/ 3 e ( Volume) and K is a correction actor. 1/ 3 or a cluster o n sheres, d e = n d. or tightly acked clusters, k < 1.5. Some other values o K are listed in Table 5. 14

15 Table 5 Correction Coeicient Cluster Shae Correction Cluster Shae Correction Cluster Shae oo K = 1.1 oooo K = 1.3 oo oo ooo K = 1.7 ooooo K = 1.45 o o o o o o K = 1.16 oooooo K = 1.57 oo o o oo oooooo o o K = 1.64 ooooooo K = 1.73 oo Correction K = 1.17 K = 1.19 K = 1.17 Ellisoidal Particles or articles that are ellisoids o revolution, the drag orce is given by D = 6π µuak' (17) where a is the equatorial semi-axis o the ellisoids and K is a shae actor. or the motion o a rolate ellisoid along the olar axis as shown in igure 6a, b b (a) a (b) a igure 6. Motions o rolate ellisoids in a viscous luid. 15

16 K' = (β 1) 1/ ( β 1) 4 ( β 3 1) ln[ β + ( β 1) 1/ ] β b ( β = ) (18) a where β is the ratio o the major axis b to the minor axis a. or the motion o a rolate ellisoid o revolution transverse to the olar axis, as shown in igure 6b K' = (β 3) 1/ ( β 1) 8 ( β 3 1) ln[ β + ( β 1) 1/ ] + β b ( β = ) (19) a Similarly or the motion o an oblate ellisoid o revolution along the olar axis as shown in igure 7a, K' = 4 ( β 1) 3 β( β ) 1 tan ( β 1) 1/ ( β 1) 1/ ] + β a ( β = ) (0) b b a b a (a) (b) igure 7. Motions o oblate ellisoids in a viscous luid. or the motion o an oblate ellisoid transverse to the olar axis as shown in igure 7b, 16

17 8 ( β K' = 3 β(3β ) tan 1/ ( β 1) 1 1) ( β 1) 1/ ] β a ( β = ) (1) b By taking the limit as β in Equations (17)-(1), the drag orce on thin disks and needles may be obtained. These are: Thin Disks o Radius a or motions erendicular to the lane o the disk as shown in igure 8a D = 16µ au () or motions along the lane o the disk as shown in igure 8b D = 3µ au /3 (3) a a (a) (b) igure 8. Motions o a thin disk in a viscous luid. Ellisoidal Needle o Length b or motions along the needle as shown in igure 9a 4πµ Ub b D =, ( β = ) (4) ln β a or side way motions o the needle as shown in igure 9b 8πµ Ub D = (5) ln β 17

18 b b (a) (b) igure 9. Motions o a needle in a viscous luid. Cylindrical Needle or a cylindrical needle with a very large ratio o length to radius ratio, moving transverse to its axis as shown in igure, the drag er unit length is given as D 4πµ U = (6) (.00 ln R ) e where R = au e and a is the radius. It is understood that ν igure. low around a cylindrical needle. 18

19 Particle Shae actor The ratio o the resistance o a given article to that o a sherical article having the same volume is called the dynamic shae actor o the article, K. The radius o an equal volume shere is reerred to as the equivalent radius r e. Clearly re 1/ 3 = αβ or rolate sheroids, (7) r e 1/ 3 = αβ or oblate sheroids. (8) Hence, K 1/ 3 = K' β or rolate ellisoids, (9) K 1/ 3 = K' β or oblate ellisoids. (30) The Stokes (sedimentation radius) o a article is the radius o a shere with the same density, which is settling with the terminal velocity o the article in a quiescent luid. Values o shaed actors or a number o articles are available (Hidy, 1984; Lerman, 1979). 19

20 AEROSOL PARTICLE MOTION Equation o Motion Consider an aerosol article in luid low as shown in igure 1. The equation o motion o a sherical aerosol article o mass m and diameter d is given as du m dt 3πµ d = ( u u ) + mg (1) C c Here u is the article velocity, u is the luid velocity, g is the acceleration o gravity and the buoyancy eect in air is neglected. Here it is assume that the article is away rom walls and the Stokes drag is assumed. Gravity Drag igure 1. Schematics o an aerosol motion in a gas low. Dividing Equation (1) by 3πµ d and rearranging, we ind du τ = ( u u ) + τg dt where the article resonse (relaxation) time is deined as mc d ρ C τ = = 3πµ d 18µ C c Sd C = 18ν c c c, πd 3 ρ where m =, ν is the kinematic viscosity o the luid and 6 ratio. In ractice, or non-brownian articles, C 1 and c S () (3) = ρ / ρ is the density 0

21 d ρ τ (4) 18µ Terminal Velocity or a article starting rom rest, the solution to () is given as t / τ u = ( u + τg)(1 e ) (5) where u is assumed to be a constant vector. or u = 0 and large t, the terminal velocity o article u t is given by u t ρ d gcc = τg = 18µ (6) Table 7 Relaxation time τ or a unit density article in air ( = 1 atm, T = 0 o C). Diameter, µm u t = τ g τ sec Sto Distance u o = 1 m/s Sto Distance u o = m/s µm/s µm 4 mm µm/s µm 9.15 mm µm/s µm 0.03 mm 1 35 µm/s µm mm mm/s µm mm 3.03 mm/s µm 3.09 mm cm/s mm 76. mm Stoing Distance In the absence o gravity and luid low, or a article with an initial velocity o, the solution to () is given by u o / x = u (1 e t τ o τ ) (7) u = u (8) e t / τ o where x is the osition o the article. As t, u 0 and x = u oτ (9) 1

22 is known as the stoing distance o the article. or an initial velocity o 00 cm/s, the sto distance or various articles are listed in table 7. Particle Path or constant luid velocity, integrating Equation (5), the osition o the article is given by x t / τ = x + u τ(1 e ) + ( u + τg)[t τ(1 e o o t / τ )] (11) Here x o is the initial osition o the article. or a article starting rom rest, when the luid velocity is in x-direction and gravity is in the negative y-direction, Equation () reduces to x y / u / u t / τ τ = [t / τ (1 e )] (1) t / τ τ = α[t / τ (1 e )] (13) where the ratio o the terminal velocity to the luid velocity α is given by α = τg τ u (14) igure shows the variation o vertical osition o the article with time. y/utau α =0.1 α =1 α = t/tau igure. Variations o the article vertical osition with time.

23 rom Equations (1) and (13), it ollows that y αx = (15) That is the article aths are straight lines. igure 3 shows samle article trajectories. y/utau α =0.1 α =1 α = x/utau igure 3. Samle article trajectories. Buoyancy Eects or small articles in liquids, the buoyancy eect must be included. Thus, Equation (1) is relaced by where a du 3πµ d (m + m ) = ( u u ) + (m m ) g (16) dt C c m is the mass o the equivalent volume luid given as m 3 πd ρ = (17) 6 and m a = a m is the aarent mass with ρ 1 m. being the luid density. or sherical articles, 3

24 Keeing the same deinition or article relaxation time as given by (3), Equation () may be restated as 1 du ( 1+ ) τ = ( u u ) + τg(1 S dt 1 ) S (18) The exression or the terminal velocity then becomes u t 1 ρ d gcc ρ = τg (1 ) = (1 ) (18) S 18µ ρ Note that the Basset orce and the memory eects are neglected in this analysis. 4

25 Lit orce Small articles in a shear ield as shown in igure exerience a lit orce erendicular to the direction o low. The shear lit originates rom the inertia eects in the viscous low around the article and is undamentally dierent rom aerodynamic lit orce. The exression or the inertia shear lit was irst obtained by Saman (1965, 1968). That is L (Sa ) 1/ du du = 1.615ρν d (u u ) 1/ sgn( ) (1) dy dy Here u is the luid velocity at the location o mass center o the article, u is the du article velocity, γ& = is the shear rate, d is the article diameter and ρ and ν are the dt luid density and viscosity. Note that L is in the ositive y-direction i u >u. Lit u u igure 1. Schematics o a article in a shear low. Equation (1) is subjected to the ollowing constraints: R R u u d Ωd = 1 R e Ω = << 1 ν ν γd = & 1/ R eg 1 ε = >> 1 ν R es << eg << Here Ω is the rotational seed o the shere. Dandy & Dwyer (1990) ound that the Saman lit orce is aroximately valid at larger R es and small ε. McLaughlin (1991) showed that the lit orce decreases as ε decreases. Based on these studies Mei (199) suggested the ollowing emirical it to the results o Dandy and Dwyer and McLaughlin. or large ε and R es, es 5

26 L L(Sa ) ( α = 1/ )ex( R 0.054( αr /) α es 1/ es ) 1/ or R or R es es 40 > 40 () where α = γ& d u u R esε = R = R eg es (3) or 0.1 ε 0 L L(Sa ) = 0.3{1 + tanh[.5log ( ε )]}{ tan[6( ε 0.3)]} (4) or large and small ε McLaughlin obtained the ollowing exressions ε or ε >> 1 L = (5) 5 L(Sa ) 140ε ln( ε ) or ε << 1 Note the change in sign o the lit orce or small values o ε. McLaughlin (1993) included the eects o resence o the wall in his analysis o the lit orce. The results or articles in a shear ield but not too close to the wall were given in tabulated orms. Cherukat and McLaughlin (1994) analyzed the lit orce acting on sherical articles near a wall as shown in igure. Accordingly where L(C = ρv d I L / 4 (6) L) V = u u = u γ& l and or non-rotating sheres, 3 IL = ( K 0.79K K ) ( / K K K ) (7) + G 3 ( K K ) G 6

27 Lit d l igure. Schematics o a article near a wall in a shear low. or rotating (reely) sheres, Here + IL = (1.7631s K K K ) ( / K K K ) (8) G 3 ( K K K ) G d K =, l G γ& d =. (9) v Lit orce on a Particle Touching a Plane Leighton and Acrivos (1985) obtain the exression or the lit on the sherical articles resting on a lane substrate as shown in igure 4. They ound 4 L(L A) = 0.576ρd γ& () which is always oint away rom the wall. Note that the Saman exression given by (1) may be restated as L (Sa ) 1/ 3 3/ = 0.807ρν d γ& (11) 7

28 Lit igure 3. Schematics o a shere resting on a wall in a shear low. Equation (6) with I L given by (7) reduces to () or K = 1, ٨ G = -1. or small articles in turbulent lows, using u u + = y +, u + = * u, * + yu y =, ν u γ = ν * (1) where u * is the shear velocity, equations () and (11) become where L(L A) = 0.576d (13) L(Sa ) = 0.807d (14) + L = ρν L, * + du d = (15) ν Exerimental studies o lit orce were erormed or generally larger articles in the range o 0 to several hundred µm. Hall (1988) ound + L(Hall) +.31 = 4.1d or d + > 1.5 (16) Mollinger and Nieuwstadt (1996) ound + L(MN) = 15.57d or 0.15 < d + < 1 (17) igure 4 comares the model redictions with the exerimental data o Hall. It is seen that the exerimental data is generally much higher than the theoretical models. 8

29 l+ 1.00E E+0 Exeriment 1.00E+01 Saman 1.00E+00 Mollinger 1.00E E-0 Hall 1.00E E-04 Leighton 1.00E d+ igure 4. Comarison o model redictions with the exerimental data. 9