THE DYNAMICS OF SMALL PARTICLES

CHAPTER

2

THE DYNAMICS OF SMALL PARTICLES

Historically m u c h w o r k in aerosol science has focused on the motion of particles in a fluid m e d i u m and the associated heat and m a s s transfer to those particles. R e c e n t theory has recognized that significant differences exist in m o m e n t u m , heat, and m a s s transfer depending on the continuous nature of the m e d i u m . A s discussed in C h a p t e r 1, this is characterized by the K n u d s e n number K n . In this chapter, transport p r o c e s s e s to small particles in a sus­ pending gas are discussed in detail over a full range of K n . T h e simplification adopted for the t h e o r y is the single-particle regime. This a p p r o a c h is applied to describing molecular t r a n s p o r t p r o c e s s e s , then is extended to a c c o u n t for particle-particle interactions later in C h a p t e r 3. In general, the exchange of m o m e n t u m b e t w e e n a gas and a particle in­ volves interaction of heat and m a s s transfer p r o c e1s s e s to the particle. T h u s , the forces acting on a particle in a n o n u n i f o r m gas may be linked with temperature and concentration gradients as well as velocity gradients in the suspending m e d i u m . T h e assumption that the K n u d s e n n u m b e r a p p r o a c h e s zero greatly simplifies the calculation of particle motion in a nonuniform gas. U n d e r such c i r c u m s t a n c e s , m o m e n t u m transfer, resulting in particle motion, will be influenced only by a e r o d y n a m i c forces associated with surface fric­ tion and p r e s s u r e gradients. In such c i r c u m s t a n c e s , particle motion can be estimated to a good approximation by the classical theory of a viscous fluid

+ Nonuniformity corresponds to a deviation from thermodynamic equilibrium in the suspend­ ing, multicomponent g?s, involving gradients in velocity, temperature, and gas-component concentration. 17

18

2

The Dynamics of Small Particles

medium w h e r e K n = 0. H e a t and m a s s transfer can be considered separately in terms of convective diffusion p r o c e s s e s in a low Reynolds n u m b e r regime. In cases in which n o n c o n t i n u u m effects must be considered (Kn > 0), the coupling between influences of gas nonuniformities b e c o m e s important, and so-called phoretic forces play a role in particle motion, but heat and m a s s transfer again can b e treated s o m e w h a t independently. Phoretic forces are those that are associated with t e m p e r a t u r e , gas-component concentration gradients, or electromagnetic forces. It is often the practice to treat particle motion as the basic dynamical scale for transport p r o c e s s e s . T h u s , let us begin with this convention and consider first the steady rectilinear motion of spherical particles in a gas. 2.1

TRANSPORT IN STEADY RECTILINEAR MOTION

2.1.1 Stokes' Law and Momentum Transfer. W h e n a spherical particle exists in isolation in a stagnant, suspending gas, its velocity can be predicted from viscous fluid theory for the transfer of m o m e n t u m to the particle. Perhaps n o other result has such wide application to aerosol mechanics as S t o k e s ' (1851) theory for the motion of a solid particle in a stagnant me­ dium. T h e model estimates that for K n —» 0, R e <^ 1, the drag force 3) acting on the sphere is ID =

βπμβϋοο,

(2.1)

w h e r e [/«, is the fluid velocity far from the particle, R is the particle radius (also Rp), and μ β is the gas viscosity. T h e particle mobility Β is defined as Β = f/»/3). T h e particle velocity q, equivalent to t / œi n thi s c a s e , i s give n i n term s o f th e produc t o f th e mobilit y and a n externa l forc e F . U n d e r suc h conditions , th e particl e motio n i s calle d quasistationary. T h a t is , th e fluid-particle interaction s ar e slo w enoug h tha t the particl e b e h a v e s a s i f i t i s i n stead y motio n eve n i f i t i s accelerate d b y external force s (se e als o Sectio n 2.3) . Mobilit y i s a n importan t basi c particl e p a r a m e t e r ; it s variabilit y wit h particl e siz e i s s h o w n i n Tabl e 2. 1 alon g wit h other importan t p a r a m e t e r s describe d later . T h e analog y fo r transpor t p r o c e s s e s i s readil y interprete d fro m E q . (2.1 ) i f w e conside r th e generalizatio n tha t " f o r c e s " o r fluxes o f a p r o p e r t y ar e proportional t o a diffusio n coefficient , th e surfac e are a o f th e b o d y , an d a gradient i n th e p r o p e r t y bein g transported . I n th e cas e o f m o m e n t u m , th e transfer rat e i s relate d t o th e frictiona l an d p r e s s u r e force s o n th e b o d y . T h e diffusion coefficien t i n thi s c a s e i s th e kinemati c viscosit y o f th e ga s (v g = p, g/ p g, w h e r e p gi s th e ga s density) . T h e m o m e n t u m gradien t i s p% UJR. If th e particle s fal l throug h a viscou s mediu m u n d e r th e influenc e o f grav ity, th e dra g forc e balance s th e gravitationa l force , o r

3

a ( p P- Pg)g7rR

=

βπμ^ο,

2.1

19

Transport in Steady Rectilinear Motion

T A B L E 2.1 Some Characteristic Transport Properties" at I atm Pressure and 2(PC (cgs Units)

Particle radius 1 5 1 5 1 5 1 5 1

x x x x x x χ x x

a

2.94 5.96 3.17 6.71 5.38 1.64 3.26 1.26 3.08

x x x x x x x x x

10 5 10 6 10 6 10 7 10 8 10 9 10 10 10 10"

a

8

5

3

IO4 10~ 4 10~5 IO5 IO6 IO6 i o -7 10- 7 IO

1.19 2.41 1.28 2.71 2.17 6.63 1.32 5.09 1.25

x x x x x x x x x

Particles

of Unit Density

IO8 IO"7 IO7 10-6 IO" 6 IO4 IO4 IO2 IO"

8.13 3.24 7.59 2.85 4.44 11.7 7.35 1.52 7.79

x x x x x x x x x

Mean free path

3

2

10 3 IO4 10 5 IO 6 10 6 10 7 10 8 IO8 10

in Air

Mean thermal speed

Drag coeff. per unit mass

Diffusivity

Mobility

of Aerosol

4.96 x IO" 1.41 x 10 0.157 0.444 4.97 14.9 157 443 4970

1

λ ρ = cfrft-

2

6

6.11 4.34 2.07 1.54 1.12 1.20 2.14 2.91 6.39

x x x x x x x x x

IO' 6 IO" 6 10~ 6 IO' 6 IO" 6 IO 10 10 -6 IO

Definitions are given in the Nomenclature listing.

w h e r e g is the gravitational force p e r unit m a s s . T h e settling velocity is

2

qG - | / ? ρ ρ£ / μ β,

(2.2)

since pp> p g. T h u s , the fall velocity is proportional to the cross-sectional area of the particle, and the ratio of its density to the gas viscosity. If the particle R e y n o l d s n u m b e r a p p r o a c h e s or e x c e e d s unity, the theory must b e modified, and takes a form based on O s e e n ' s 1927 formula

2

Φ = 67τμ 8/?ί/οο(1 + fRe + 19/640Re + · · · ) ,

Re = U œ R/vg * s 1 .

(2.3 )

This reflect s th e well-know n fac t tha t th e dra g forc e o n a spher e deviate s from S t o k e s ' la w a s a functio n o f th e R e y n o l d s n u m b e r o f th e particle . I n t e r m s o f th e dra g coefficien t th e dra g forc e i s writte n a s

2

2 ) = \p gvR C®Ul.

(2.4 )

The result s o f experimenta l m e a s u r e m e n t s fo r sphere s i n a fluid indicat e tha t the dra g coefficien t ca n b e e x p r e s s e d a s

2/3

Cg, = (12/Re)( l + 0.250Re

),

(2.5 )

w h e r e th e multiplie r o f th e first ter m i n p a r e n t h e s e s i s th e dra g coefficien t fo r S t o k e s flow. If th e K n u d s e n n u m b e r i s no t a s s u m e d t o b e z e r o , the n th e dra g forc e o n the particl e m u s t b e c o r r e c t e d fo r slippag e o f ga s a t th e particl e surface . E x p e r i m e n t s o f Millika n an d other s s h o w e d tha t th e Stoke s dra g forc e coul d

6 6

20

2

The Dynamics of Small Particles

be corrected in a straightforward way. Using the theory of motion in noncontinuous flow, the mobility takes the form Β = Α/6πμ^,

Re > 0,

Kn Φ 0

H e r e the n u m e r a t o r is called the Stokes-Cunningham A is

(2.6)

factor.

T h e coefficient

1

A = 1 + 1.257Kn + 0.400Kn e x p t - l . l O K n " ) ,

(2.7)

based on e x p e r i m e n t s . Corrections for n o n c o n t i n u u m effects for Reynolds number of unity or larger h a v e not b e e n reported, but one would expect that they could be placed in a similar mathematical form for the mobility correc­ tion. 2.1.2 Particle Shape Factors. In general, aerosol particles are not spheri­ cal, but are irregular in s h a p e . B e c a u s e of the tradition of interpreting parti­ cle motion in t e r m s of the behavior of spheres, most studies employ these particles as a reference. Interpretation of observations then are often re­ ported in terms of dynamic shape factors. T h e s e h a v e b e e n defined in t w o w a y s . T h e first defines an aerodynamic radius (or diameter), which is the size of a particle of arbitrary shape and density w h o s e terminal velocity would be equivalent to that of a sphere of unit density. The second m e t h o d relates an equivalent radius to the Stokes sphere through the terminal set­ tling velocity: (2.8) qG = Ϊ Ρ 8Λ | / μ 8, w h e r e Rc = ( 3 ι ; ρ/ 4 π ) , vp being the particle volume. T h e n a shape factor IK can be defined in terms of the Stokes equivalent:

1/3

2

IK =

RllR .

The factor IK for several geometrical shapes are summarized in Table 2.2. Dynamic shape factors m e a s u r e d for agglomerates of spheres and irregular particles have b e e n reported for e x a m p l e , by Stôber (1972) and K o t r a p p a (1972). F o r specific values, the reader should refer to these references. 2.1.3 Motion in External Force Field—Phoretic Forces. In m a n y circum­ stances, particles accumulate electrical charge, and are subject to electrical forces. If a particle has an electrical charge of Q and is placed in a uniform electrical field E0, the electrical force balances the frictional force in steady motion. T h u s , the " t e r m i n a l v e l o c i t y " in an electrical field b a s e d on S t o k e s ' law is <7E = QE B

0

= QEj6TT^ R

gy

Re -> 0,

K n - 0.

(2.9)

Particle motion induced by gravity can b e controlled using an electrical field. Millikan's (1923) famous experiment m a d e use of this to d e d u c e the value of

2.1

21

Transport in Steady Rectilinear Motion

T A B L E 2.2

a Shape Factor Equation

Κ for Particles,

of Different

Shape

Used in the Stokes

Sphere

1

Hollow sphere

\.44(MUR)

Hemispherical cap

1.23(AR/R)

Polyhedra Cube-octahedron Octahedron Cube Tetrahedron Ellipsoids

0.964 0.939 0.921 0.853

Cylinder

(Ih)

Half ring (horizontal) (vertical) Ring (horizontal) (vertical)

a bAssimilated 2

R sphere radius, AR wall thickness, AR/R < 1 R cap radius, AR wall thickness, AR/ R < 1 ; settling direction: equatorial plane down

m

m

(ΙΙΛ)

f(X) f (X)

m

m + X) (Ik) 5X /(4 2 / 3 + X) ( 1 c ) 2.5X /(\.5 1.5Ζ- 2 (1η/ 32X - 0.5) (Ik) 1 . 5 Z - 1( l/n 32X + 0.5) 0.375Z 1 / i (/Z3 ) 2 (Ik) 0 . 3 7 5 Z // (3Z ) 2 ( 1 c ) 1 . 7 2 * - ( l n 2X 2 / 3 0.72)

Disk*

Velocity

Remarks

Particle shape

Needle

Settling

0 . 8 6 * - ( l n 2X + 0.5)

2 / 3 0 . 725 Z/ 3( l n 2X X ~ ( l n 2X -

2/3

Sphericity R, = 0.906 R = 0.846 s R = 0.806 s R = 0.670 s Ellipsoids of revolution, length 2c, equatorial diameter 2R Prolate, X = cIR > 1 Oblate, X = cIR < 1 Prolate ellipsoid of revolution, large X = cIR > 1 Oblate ellipsoid of revolution, small X = c/R<\ Approximations for a straight circular cylinder, height h, base radius R, 2X = hIR Half-circular ring bent from a long prolate ellipsoid, X = cIR > 1

+ 0.56) 0.68)

0 . 725 Z/ -3 ( l n 2X + 0.75) Z " ( l n 2X - 2.09)

Circular ring bent from a long prolate ellipsoid, X = cIR > 1

2 2 2/2 permission of Wiley. 2 by Lerman (1979); reprinted with / /Χ], 2 = 2XI{\ - X 2 ) + [2(1 2- 2X )/(\ - X y ]tan-'[(l - 2 1 Χψ = -Xl(\ - X ) - [(2X - 3)/(l - Z ^ J s i n - K l - Z ) .

a unit of electrical charge e. T h e principle of this experiment w a s to levitate a small oil droplet in an electrical field by first balancing the gravitational force on the droplet with an electrical force. In o t h e r w o r d s , qG = qE , or

2

|7? ^Ρρ/μ 8 =

QEo/βττμ^.

Then the electrical field required to hold the particle stationary against grav­ ity is

3

E0 =

HR gpp/Q.

(2.10)

22

2

The Dynamics of Small Particles

By observing different sized particles moving in a vertical electrical field, one can determine the value of the particle mobility for a wide range of Knudsen numbers. T h e p a r a m e t e r K n can be increased both by changing particle size and by gas p r e s s u r e since the m e a n free path of the gas is proportional to p r e s s u r e . The settling velocity can b e determined by watching droplet fall rates with­ out the electric field. T h e n the fall rate can be o b s e r v e d u n d e r the simulta­ neous action of gravitational and electrical forces acting in different direc­ . By charging particles with tions. Such experiments will yield the velocity qE different a m o u n t s of charge, o n e can vary Q, which is ze, w h e r e ζ is the n u m b e r of unit charges e. T h e n the particle mobility can be calculated as a function of K n (e.g., F u c h s , 1964). Particles can experience external influences induced by forces other than electrical or gravitational fields. Differences in gas t e m p e r a t u r e or v a p o r concentration can induce particle motion. Electromagnetic radiation can also p r o d u c e m o v e m e n t . Such phoretic p r o c e s s e s w e r e observed experimen­ tally by the late nineteenth c e n t u r y . F o r e x a m p l e , in his experiments on particles, Tyndall (1870) described the cleaning of dust from air surrounding hot surfaces. This clearance m e c h a n i s m is associated with the thermal gradi­ ent established in the gas. Particles will m o v e in the gradient u n d e r the influence of differential molecular b o m b a r d m e n t on their surfaces, giving rise to the t h e r m o p h o r e t i c force. This m e c h a n i s m has been used in practice to design thermal precipitators for particles. Although this p h e n o m e n o n w a s observed and identified with thermal gradients, no quantification of the phe­ n o m e n o n w a s m a d e until the 1920s. Epstein (1929) and Einstein (1924) dis­ cussed theories for the p h e n o m e n o n ; W a t s o n (1936) and Paranjpe (1936) m a d e m e a s u r e m e n t s of the thickness of the dust-free space in relation to other p a r a m e t e r s . L a t e r , the theory w a s refined for the free-molecule regime by W a l d m a n n (1959) and by B a k a n o v and Deijaguin (1960). T h e theory for the c o n t i n u o u s - n o n c o n t i n u o u s transition regime was extended by experi­ ments of B r o c k (1967) and others (e.g., Schmitt, 1959). T h e relation for the thermal or thermophoretic force F t on a spherical particle is Ma
(2.11)

w h e r e φ 8 is the mean thermal speed of the gas, and f = 0.9 + 0 . 1 2 a m + 0 . 2 1 a m( l -

atkg/2kp)

for m o n a t o m i c gases, b a s e d on e x p e r i m e n t s . H e r e am and at are the accom­ modation coefficients for m o m e n t u m and heating, and kg and kp are the gas and particle thermal conductivities. In the limit K n —> <*>, the thermal force is directly proportional to the particle cross-sectional area and the t e m p e r a t u r e gradient in the gas. T h e magnitude of the force d e p e n d s on an experimental

2.1

23

Transport in Steady Rectilinear Motion

factor which varies with K n for a c c o m m o d a t i o n of thermal energy and m o ­ m e n t u m of molecules colliding with the particle surface. The a c c o m m o d a t i o n coefficients vary from zero to unity and are a mea­ sure of transfer of molecular energy and m o m e n t u m to the particle. If the a c c o m m o d a t i o n coefficient is z e r o , the energy or m o m e n t u m of collision is not absorbed. If they are unity the p r o p e r t y is completely a b s o r b e d at the surface. Unfortunately, they are generally not k n o w n for particles. A few m e a s u r e m e n t s reviewed by Brock (1967) suggest that am 0 . 8 - 0 . 9 for oils or sodium chloride. F o r m o n a t o m i c gases at r o o m t e m p e r a t u r e at « am . S o m e examples of thermal and m o m e n t u m a c c o m m o d a t i o n coefficients are listed in Tables 2.3 and 2.4. T h e y can vary over a wide range depending on the c o n d e n s e d material and the gas. The thermal force is c a u s e d by heat transfer from the gas to the particle during molecular motion. If the particle receives heat from sources other than molecular motion, say from absorption of electromagnetic radiation, an additional force can d e v e l o p as a result of a t e m p e r a t u r e gradient established in the particle. T h e force generated by radiation is called the photophoretic force F p, in the full molecule regime: -π&ΡΙ

(2.12)

w h e r e I is the radiation flux, Ρ is the gas p r e s s u r e , and & is the gas c o n s t a n t . T h e theory for p h o t o p h o r e s i s has not b e e n w o r k e d out for finite K n . Other than a laboratory curiosity, it has yet to find an important application in nature, or in particle technology. T h e relationship given in E q . (2.11) r e p r e s e n t s the theory corrected for the thermal force as derived for the regime of flow on a rarified gas as K n —» oo. This theory is reasonably well developed and considers molecular collisions T A B L E 2.3

0 Some Typical Data for Momentum Accommodation Coefficients System Air on old shellac or machined brass Air on oil C 0 o n oil 2 Air on glass H e on oil Air o n A g 0 2 He on A g 0

a

2

1.00 0.895 0.92 0.89 0.87 0.98 1.0

From Paul (1962); by courtesy of American In­ stitute of Aeronautics and Astronautics.

24

2

The Dynamics of Small Particles

T A B L E 2.4

a Some Values Coefficients

of Thermal

Accommodation

System

a

Air on machined bronze Air on polished cast iron Air on etched aluminum Air on machined aluminum 0 on bright Pt 2 Kr on Pt N on W o 2 H e on N a N e on Κ He on Wo

0.91-0.94 0.87-0.93 0.89-0.97 0.95-0.97 0.808 0.699 0.35 0.090 0.199 0.017

t

a

From Wachmann (1962); by courtesy of the American Institute of Aeronautics and Astro­ nautics.

at the particle surface as distributed by molecular collisions with each other near the particle surface. W h e n these molecular collisions b e c o m e a domi­ nant factor in the surface transport p r o c e s s e s , the fluid is viewed m o r e as a continuum, and the theory must be developed from the alternate e x t r e m e where K n - » 0 (Hidy and Brock, 1970). Correction for the continuum fluidparticle interactions yields a separate formula for the thermal force (Brock, 1967): 2

Ft _ - 1 2 ^ gf l c t Ka n [ y * p + c tK n ) ( l + 1.33ec mKn) - 1.33gc mKn] V T g " (1 + 3 c mK n ) ( l + 2kg/kp + 2 c tK n )

(2.13)

This relation is derived from an approximation solution to the Boltzmann equation for the molecular flux expressions for particles in the slip flow regime, w h e r e 0 < K n < 0.2. T h e coefficients c t ,mc t, and c mare respectively the thermal creep coefficient, the temperature j u m p coefficient, and the isothermal slip coefficient. T h e constant A is a second-order fitting p a r a m e t e r taken from experiments to be approximately unity. F o r monatomic molecules, c tm « 3 μ β/ 4 ρ 8Γ 8λ 8

(2.14a)

c t = A ( 2 - « t) / « t ,

(2.14b)

and

where the coefficient 1.87 < Pt < 2.48, cm = Pm (2

- am )/am ,

1.00 < Pm < 1.274.

F r o m the work of m a n y investigators cm ~ 1.2.

(2.14c)

2.1

25

Transport in Steady Rectilinear Motion

Equation (2.13) is similar to E p s t e i n ' s (1929) early theory, but a p p e a r s to be capable of fitting observations over a wide range of thermal conductivity. T h e slip coefficients are not k n o w n for particles, so E q . (2.13) is difficult to use for a priori estimates of the thermal force. W e note also that E q . (2.13) indicates that t h e thermal force goes to z e r o with K n . T h u s , in the contin­ u u m theory t h e r e can be no t h e r m o p h o r e t i c effect. In analogy to their motion in a thermal gradient, particles may experience a force associated with nonuniformities of composition in the suspending gas mixture. This force is called the diffusion or diffusiophoretic force. F o r the case of equimolar counterdiffusion in a dilute binary mixture of c o m p o n e n t A in Β , η A > nB , amA = amB = 1, and the force acting on a spherical particle is

2

/2

F d = *R ng(2nkTY

I

( l + | ) DAB VxA {{mf

-

mf)

d\mf2 _ / 2mA mB \m 03UR\ d AB \m Xg y xntA + + m rn /) A B JJ B

< h ~ R

0

5

H e r e xA is the mole fraction of species A and V x A is the gradient in mole fraction of species A . T h e diffusion coefficient for the binary mixture is DAB; mB and mA are the molecular mass of species A and B ; dA and dB are the )l2\ and the mass free path of molecular diameters of A and B , dAB = (dA + dB the gas mixture is

2

Xg =

(λ/2πη,ά ΑΒ )~Κ

The p a r a m e t e r ng is the molecular n u m b e r concentration of gas molecules. Equation (2.15) is quite complicated, but the theory quantifies the analogy b e t w e e n the thermal and diffusion force. Basically the latter is proportional to the concentration gradient of the diffusing species, the cross-sectional area of the particle, the p r o d u c t of the molecular m a s s differences, and a K n factor. T h e last factor in b r a c k e t s represents a K n correction factor to the free-molecule t h e o r y w h e r e K n - » <». F o r diffusion of gas c o m p o n e n t A through stagnant c o m p o n e n t B , with nA < nB and a mA = a mB = 1.0, the diffusion force has the form kT)^ F d = - I R%(2^mA

( l + I)

^

VxA

This equation is similar to E q . (2.13), with the first right-hand term the freemolecule formula and the square-bracketed term on the right the K n correc­ tion factor. In the slip flow regime, for the case of equimolar counterdiffu­ sion and neglecting the effect of velocity slip, F d = - 6 π μ 8/ ? σ Α ^ Β ΑΒ(νΛ:Α)οο,

0 < K n < 0.3.

(2.17)

26

2

The Dynamics of Small Particles

The gradient (VJCA )°o is taken at a large distance from the particle surface. And for diffusion of A through stagnant B , F d = -6πμ^(1

+ aAB xB )(DAB /xB )(VxA )„,

0 < K n < 0.25.

(2.18)

T h e coefficient σ- ΑΒis an empirical coefficient introduced by W a l d m a n n and Schmitt (1967) of the form )/(mA σ ΑΒ = 0 . 9 5 ( m A - mB

+ mB ) - 1.05(d A- dB )l(dA

+

dB ).

This formula is based on experiments on particle motion of several different gases mixed with nitrogen. Values of σΑΒ range from - 0 . 1 0 to + 0 . 9 0 for the binary mixtures investigated by W a l d m a n n and Schmitt.

2.1.4 Heat and Mass Transfer. Particles suspended in a nonuniform gas may b e subject to absorption or loss of heat or material by diffusional trans­ port. If the particle is s u s p e n d e d without motion in a stagnant gas, transport to or from the b o d y can be estimated from heat conduction or diffusion theory. F o r steady state conditions, continuum theory indicates that the mass concentration or t e m p e r a t u r e field around a spherical particle is ^

^

= ^

^

= -,

(2.19)

Ρ As ~ PA IS — O IO r w h e r e the radial direction from the center of the sphere is r and the sub­ scripts s and oo indicate the particle surface value and the value far from the sphere. Using this relation, o n e finds that the net rate of transfer of heat to the particle surface in a continuum is

2

Φ = 4irr DT (^) = 4irRD T(Tœ V Of I r=R

-

Γ 8),

(2.20)

w h e r e DT is the thermal diffusivity kg/pgcp and cp is the specific heat of the particle. F o r m a s s transfer of species A through Β to the s p h e r e , Φ Α = 4wRD AB (pAœ

- p A) ,s

(2.21 )

where D ABi s th e binar y molecula r diffusivit y fo r th e t w o gases . Thi s relatio n is basicall y tha t derive d b y Maxwel l (1890) . Hi s equatio n applie d t o th e steady stat e evaporatio n fro m o r condensatio n o f vapo r c o m p o n e n t A i n ga s Β on a sphere. F o r the case of c o n d e n s a t i o n , E q . (2.21) may be rewritten in the form Φ = {47rRDAB MA ISiT){pA „

- PAS),

(2.22)

w h e r e MA is the molecular weight of species A and p A is its vapor p r e s s u r e . F o r evaporation the sign of the density or v a p o r p r e s s u r e difference is re­ versed.

2.1

Transport in Steady Rectilinear Motion

The applicability of M a x w e l l ' s equation is limited in describing particle growth or depletion by m a s s transfer. Strictly speaking, m a s s transfer to a small droplet c a n n o t b e a steady p r o c e s s b e c a u s e the radius c h a n g e s , caus­ ing a change in the transfer rate. H o w e v e r , w h e n the difference b e t w e e n vapor concentration far from the droplet and at the droplet surface is small, the transport rate given by M a x w e l l ' s equation holds at any instant. T h a t is, the diffusional t r a n s p o r t p r o c e s s p r o c e e d s as a quasistationary p r o c e s s . W h e n the particle is moving relative to the suspending fluid, transport of heat or matter is e n h a n c e d by convective diffusional p r o c e s s e s . T h e s e are measured in t e r m s of transfer coefficients, defined as H e a t transfer:

(2.23)

Mass transfer :

These transfer coefficients are normally expressed in dimensionless form. F o r heat transfer, the Nusselt n u m b e r N u = 2lLR/kg; for m a s s transfer, the . W h e r e t h e r e is n o motion, Sh = N u = 2. S h e r w o o d n u m b e r Sh = 2fiR/DAB If there is particle motion, N u = / i ( R e , Pr, ί / . ί / Λ ) ,

Sh = / 2( R e , S c , U«t/R),

(2.24)

w h e r e the Prandtl n u m b e r Pr = c p/x g/&p, and the Schmidt n u m b e r Sc = vgl DAB , w h e r e vg is the kinematic viscosity. T h e Reynolds n u m b e r and the Prandtl and Schmidt n u m b e r s are often combined to give the Peclet n u m b e r , Pe = R e P r , or R e S c . T h e Peclet n u m b e r , UœR/D AB , fo r hea t o r transfe r i s analogous t o th e R e y n o l d s n u m b e r fo r m o m e n t u m transfer . F o r Stoke s flow, th e N u s s e l t o r S h e r w o o d n u m b e r ca n b e writte n i n t e r m s of a n expansio n i n th e Pecle t n u m b e r (Acrivo s an d Taylor , 1962) :

2

2

N u = 2 + P e + P e I n P e + 0.0608Pe + · · · ,

P e< 1.

(2.25 )

F o r large r P e th e empirica l Frôsslin g equatio n i s ofte n u s e d ; i t ha s th e for m

1 7 2

N u « 2( 1 + ^ P r ^ R e

).

(2.26)

This relatio n r e d u c e s t o th e for m o f E q . (2.20 ) a s P e —> 0 , bu t doe s no t correctly fit th e theoretica l expressio n o f Acrivo s an d Taylor . M o s t experi m e n t s fo r hea t an d m a s s transfe r t o sphere s a t R e > 10 0 giv e k } — 0.39 . H o w e v e r , K i n z e r an d G u n n ' s (1951 ) e x p e r i m e n t s fo r P e < 1 sugges t k x— 0.65. W h e n th e t h e o r y o f t r a n s p o r t i s develope d takin g int o accoun t nonconti nuum effects , th e t h e o r y m u s t b e correcte d a s i n th e cas e o f m o m e n t u m

28

2

The Dynamics of Small Particles

transfer. T h e rates of heat and m a s s transfer to spheres are given, for negligi­ ble M a c h n u m b e r (Ma) and 0.25 < kB /R ^ °°, by

2

m

x

(2kT^mA ) ]{\ Φ Α = [2vR acnA

+ acCxRl\B Y

(2.27a)

for radial molecular m a s s transfer of species A into Β from a sphere. T h e first term in the p r o d u c t on the right is the rate of transfer by free-molecule flow (Kn —> oo). T h e t e r m on the far right of the product is the K n correction for the transition regime. T h e coefficient ac is the condensation or evapora­ tion coefficient, which is analogous to the thermal or m o m e n t u m a c c o m m o ­ dation coefficient. S o m e example values are listed in Table 2.5. T h e2 coeffi­ cient Cx has the form given by B r o c k (1967); C, = 0.807 WBVAB^B) /, w h e r e the reduced m a s s WB = mB l(mA + raB ), and / is a numerical factor which varies with WB ; for e x a m p l e , WB : /:

1.000

0.800

0.600

0.400

0.200

0.295

0.245

0.201

0.162

0.130

i

T A B L E 2.5 Typical Data for the Evaporation

0

Coefficient

Material

Temperature range (°C)

Beryllium Copper Iron Nickel

«1 -1 «1 -1

Ammonium chloride Iodine Mercurous chloride Sodium chloride Potassium chloride Water (liquid)

( 3 . 9 - 2 9 . 0 ) x 10~ 0.055-1 0.1 0.11-0.23 0.63-1.0 0.04->0.25

118-221 - 2 1 . 4 to 70 97, 102 601-657

Benzene Camphor (synthetic) Carbon tetrachloride Diamyl sebacate Di-tf-butyl phthalate Ethyl alcohol Glycerol Naphthalene Tetradeconal

0.85-0.95 0.139-0.190 «1 0.50 1 0.0241-0.0288 Unity, 0.052 0.036-0.135 0.68

6 -14.5-5.5

0

4

898-1279 913-1193 1044-1600 1034-1329

— 15-100

— 25, 35 150-350 12.4-15.9 18-70 40-70 20

From Paul (1962); courtesy of the American Institute of Aeronautics and Astronautics.

2.1

29

Transport in Steady Rectilinear Motion

T h e analogous extension of t h e c o n t i n u u m theory for 0 < K n ^ 0.25 is Φ

Α = [4ΤΓ#Ζ)ΑΒ(ρ8 - p . ) ] ( l +

(2.27b)

caA \g/R)-K

T h e b r a c k e t e d t e r m in t h e p r o d u c t o n t h e right is t h e c o n t i n u u m rate of m a s s = t h e slip correction is t h e last term on t h e right. H e r e t h e coefficient transfer; QA [(2 - acA )/acA ](2DAB /yA \g). In general, there a r e n o rigorous theoret­ ical estimates for c^A , b u t values b e t w e e n 1.2 and 1.5 a r e consistent with experiment. T h e heat transfer e x p r e s s i o n s a r e

2

l2

Φ Η - [4KR ax(2kT- lvmgy nk{T+

l

- Γ ) ] ( 1 + atC2R/kgy

(2.28a)

in t h e free-molecule t o transition regime, w h e r e 0.25 < K n < oo, M a < 1. T h e theory for t h e radial heat flux from a sphere in the free-molecule to transition regime is written in t e r m s of a flux of molecules n~ leaving the surface, +and a t e m p e r a t u r e difference b e t w e e n impinging and reflected mole­ cules Γ - T~. T h e coefficient C2 = ihvr(Cv + \k)!2k for polyatomic g a s e s , w h e r e C vis the heat capacity at c o n s t a n t v o l u m e . T h e coefficient C ~ 0.24 for air. F o r the c o n t i n u u m m o d e l , c o r r e c t e d for slippage of gas about a s p h e r e , the rate of heat transfer is given b y

1

(2.28b) Φ Η = 4vkgR(Ts - JooXl + CtKn)- , in analogy to E q . (2.27b). T h e formula for c t is given in E q . (2.14b). 2.1.5 Condensation and Evaporation. P e r h a p s t h e most widely u s e d ap­ plication of t h e h e a t a n d m a s s transfer relations c o n c e r n s t h e c o n d e n s a t i o n and evaporation of particles. Using M a x w e l l ' s equation, t h e g r o w t h in a quasistationary state of a p u r e spherical droplet in a s u p e r s a t u r a t e d v a p o r is estimated as

w h e r e vmis t h e molecular v o l u m e of t h e condensing species, ρ is t h e v a p o r p r e s s u r e of A , a n d k is t h e B o l t z m a n n c o n s t a n t . F o r depletion b y e v a p o r a t i o n in a quasistationary state, E q . (2.29) applies, but the sign is r e v e r s e d , owing to t h e n o n s a t u r a t i o n of t h e vapor. F o r drop­ lets of low volatility, t h e a v e r a g e droplet t e m p e r a t u r e or gas t e m p e r a t u r e c a n be used, b u t high-volatility liquids m a y generate t e m p e r a t u r e differences associated with heating or cooling during m a s s transfer. In general, t h e g r o w t h or depletion of droplets by p h a s e change should b e estimated taking into a c c o u n t b o t h heat a n d m a s s transfer. B y combining t h e heat conduction equation a n d M a x w e l l ' s equation, a n estimation c a n b e m a d e of t h e t e m p e r a t u r e difference b e t w e e n t h e droplet and t h e m e d i u m .

30

2 Th e Dynamic s o f Smal l Particle s

F o r a condensin g droplet , th e couplin g o f hea t an d m a s s transfe r fo r quasi steady stat e condition s lead s t o th e relationshi p Ts-

To o = (k vDAB vm /kgkf)(ps

- pœ ),

(2.30 )

w h e r e λ ν is the latent heat of vaporization of the liquid and f is the m e a n value of the t e m p e r a t u r e s Ts and T œ . Thi s relatio n w a s derive d b y Maxwel l i n his 189 0 w o r k ; i t indicate s tha t evaporatio n o r growt h b y c o n d e n s a t i o n b y pure vapo r u n d e r quasistead y condition s i s accompanie d b y t e m p e r a t u r e changes whic h ar e independen t o f particl e size . W h e n th e droplet s ar e ver y small , th e diffusio n rat e require s a correctio n for th e drople t c u r v a t u r e . F o r smal l value s o f th e vapo r p r e s s u r e difference , a correctio n ca n b e m a d e usin g th e K e l v i n - G i b b s formula . F o r a p u r e liqui d droplet i n it s o w n vapor , th e partia l p r e s s u r e differenc e i s writte n Poo - Ps = Poo - Po e x p -^ψ

= pœ - po e x p ( - ^ - I n

,

(2.31 )

w h e r e p 0 i s th e v a p o r p r e s s u r e o v e r a flat surfac e o f th e liquid , S i s th e supersaturation rati o p s/po, CR i s th e surfac e tensio n o f th e droplet , an d R* i s the " c r i t i c a l " drople t radius . T h e critica l siz e i s th e nucleu s siz e whic h wil l j u s t b e stabl e a t t h e r m o d y n a m i c equilibriu m wit h it s o w n vapo r an d wil l no t evaporate. I f th e drople t i s a solution , the n th e vapo r p r e s s u r e - p a r t i c l e siz e relationship b e c o m e s m o r e c o m p l e x , an d shoul d b e accounte d for . I f th e droplet contain s a n insolubl e fraction , wettin g als o m a y b e a facto r i n th e growth r a t e . T h e s e factor s ar e discusse d furthe r i n C h a p t e r 3 . If a drople t contain s trace s o f surface-activ e materia l suc h a s organi c acid s or alcohols , th e evaporatio n o r condensatio n ca n b e reduced . T h e effec t o f a surface laye r o f materia l o n a volatil e drople t i s t o r e d u c e th e surfac e vapo r concentration belo w saturation , reducin g th e vapo r pressur e drivin g force . The chang e i n m a s s transfe r rat e ca n b e relate d t o th e surfac e film b y addin g a transfe r resistance , whic h m a y b e define d i n term s o f th e depletio n i n vapo r concentration j u s t outsid e th e drople t b e c a u s e o f th e film (Davies , 1978) . T h e rate o f chang e i n m a s s o f a drople t ca n b e writte n *g

x

4,R^

=A

B m^P

_ ^y

p

V R

^

Dy+K 1

A

)

w h e r e ^ i s th e evaporatio n resistanc e (sec/cm) . I f DAB ^ i s m u c h smalle r 1 e drople t radius , thi s relatio n r e d u c e s t o Maxwell' s equation . I f than th DAB^ i s large r tha n R, the n thi s expressio n r e d u c e s t o th e ga s kineti c relation, i f H e= 4 / a c^ A> w h e r e ^ Ai s th e therma l velocit y o f vapo r molecules . Thus th e film resistanc e ca n b e linke d o n a molecula r scal e t o th e a c c o m m o dation coefficient . Davie s (1978 ) ha s calculate d tha t a resistanc e fro m a surface film ca n increas e th e lifetim e o f a micrometer-siz e volatil e drople t by orde r o f magnitude , a s migh t b e e x p e c t e d . T h e evaporatio n rat e o f droplet s u n d e r condition s w h e r e n o n c o n t i n u u m

2.1

31

Transport in Steady Rectilinear Motion

effects are important c a n b e determined by semiempirical c u r v e fitting for quasi-stationary state conditions w h e r e t h e latent heat changes a r e small. Basically t h e m a s s exchange equations for the free-molecule and continuum regimes constrain t h e model [Eqs. (2.27a) a n d (2.27b)]. Several investigators have p r o p o s e d such formulas. T h e y c a n b e written in the form Φ/Φο=

1/[1 + / ( K n ) ] ,

(2.33)

w h e r e Φ ςis given by M a x w e l l ' s equation; F u c h s and Sutugin (1971) give / ( K n ) = [(1.333 + OJlKn-O/O + KirOlKn.

(2.34)

A n o t h e r version of this expression based o n an a p p r o x i m a t e solution of t h e Boltzmann equation in tabular form h a s b e e n reported recently by L o y a l k a (1973). Very little data h a v e b e e n reported for which formulation could b e veri­ fied. H o w e v e r , Davis et al. (1978) h a v e published data for the evaporation rates of dioctyl phthalate a n d dibutyl sebacate (DBS) over a wide range of K n . Their data for D B S are s h o w n in Fig. 2.1 with t h e theory from different + of investigators. T h e form in the figure c o r r e s p o n d s t o the ratio of t h e rate transfer normalized t o the kinetic theory value rather than Φ / Φ ς.

0.001

0.01 0.1 Kn, KNUDSEN NUMBER, X:/a

Fig. 2 . 1 . A comparison of the evaporation rate for dibutyl sebacate (DBS)(A) in N ( B ) with 2 the F u c h s - S u t u g i n (1971) model [Eq. (2.36a)]. S h o w n for comparison are the results from other theoretical forms including the S i t a r s k i - N o w a k o w s k i (1979) ( S - N ) model, Fuchs (1959), and Loyalka (1973). A l s o included is the curve for the S - N model with m / m = 0. [From Davis et A B al. (1980), with permission of the author and Gordon & Breach.] * Using the Fuchs and Sutugin value of h for the ratio Φ/Φ* = 1.33Kn

1 +

DA b/Xa^a ,

1.33Kn + 0.7l" 1 + Kn"

(2.33a)

32 2.2

2

The Dynamics of Small Particles

ACCELERATED MOTION OF PARTICLES—DEPOSITION PROCESSES

The behavior of particles in steady rectilinear flow represents the simplest configuration of interest in aerosol m e c h a n i c s . Of great practical importance is the accelerated motion of particles, since acceleration-induced fluid in media is at the root of deposition or collection p r o c e s s e s . Taking into ac­ count the fact that the particles in accelerated motion do not follow the gas speed, the drag force is D = (6^gR/A)(qg

- q p) .

Then the force balance on a particle in motion is

dt I (1)

+

©

(3)

« w J> [($)-($)]/« -'>"

+ Fe + F N . U (5)

(4)

(2.35)

(6)

In E q . (2.35), i 0 d e n o t e s the starting time for particle acceleration, and q g represents the gas velocity near the particle, but far enough a w a y not to be disturbed by the relative motion of the b o d y . T h e terms in E q . (2.35) have the following meanings: term (1) on the right is the viscous resistance given by S t o k e s ' law. T h e second term (2) is due to the pressure gradient in the gas surrounding the particle, caused by acceleration of the gas by the particle. The third term (3) d e n o t e s the force required to accelerate the apparent m a s s of the particle relative to the ambient gas. T h e fourth term (4), the Basset term, accounts for the effect of the deviation from steady state in gas flow pattern. T h e last t w o terms are the forces associated with external force fields (5), F e, and from nonuniformities in the suspending m e d i u m (6), F N . U T h e equation of motion can be summarized in the form ^

l

+ fl(qp - q g) - F(/) 4 - F; + Ffoj,

(2.36)

w h e r e ft = (mpB)~ and F' = F / m p. T h e term F(t) is a contribution to particle acceleration from an " e x t e r n a l " perturbation function acting on the particle. This has the form of the Langevin equation, well k n o w n in the theory of Brownian motion. 2.2.1 Relaxation Time to Reach Steady Motion. A n important character­ istic scale in particle m e c h a n i c s is the time to achieve steady motion. T o find the p a r a m e t e r , w e can apply E q . (2.36) for simple conditions of the decelera-

2.2

Accelerated Motion of Particles—Deposition Processes

33

tion of a particle by friction in a stationary gas. In the a b s e n c e of external forces, the velocity of a particle traveling in the χ direction is calculated by

or qp = q0 exp(-flz)

(2.37)

if the initial velocity is q0. The distance traveled by the particle is χ = £ qp dt' = β " V I

1

- exp(-Of)].

(2.38)

The significance of ft" is clear; it is a constant which is the relaxation time scale τ for stopping a particle in a stagnant fluid. Values of the reciprocal -1 relaxation time are listed for different particle sizes with the mobility Β in Table 2 . 1 . Similarly, o n e can show that ft represents the time it takes for a 1 field to achieve its terminal speed. N o t e that particle falling in a gravitational _1 t h e terminal speed qG = g f t . At //ft —» o°, the distance o v e r which the particle p e n e t r a t e s , or the stopping distance 1 is ^ 0f t 2,2.2 Curvilinear Particle Motion. W h e n particles change their direction of m o v e m e n t , as for e x a m p l e a r o u n d blunt bodies or b e n d s in tubing, inertial forces tend to modify their p a t h s relative to the suspending gas. This is shown schematically in Fig. 2.2. A s s h o w n , particles may depart from the path of gas molecules (streamlines) to allow deposition of the b o d y . This is the principle underlying inertial collectors. T h e motion of the particle undergoing curvilinear flow can be calculated from the vector force of E q . (2.35) if the particles are large enough that their thermal (Brownian) motion can be disregarded. T h e trajectory of a particle moving in a gas can b e estimated by integrating the equation of motion for the particle o v e r a time period given by increments of the ratio of the radial distance traveled divided by the particle velocity, i.e., \rlqp\. Interpretation of the equation of motion, of c o u r s e , requires knowledge of the flow field of the suspending g a s ; o n e can a s s u m e that the particle velocity equals the fluid velocity at some distance r far from the collecting b o d y . Perhaps the simplest c a s e to illustrate the difference b e t w e e n particle flow and fluid flow is the deposition from a particle cloud traveling horizontally through a straight channel of length L into a gravity field. T h e gas flow is t w o dimensional, laminar, and horizontally uniform, with m e a n velocity qx. T h e flow is sufficiently slow that vertical convective currents are small c o m p a r e d with the gravitational settling velocity. F o r the p u r p o s e of this and subse­ quent calculations, let us a s s u m e that t h e particles are infinitely small

34

2

The Dynamics of Small Particles

(b) Fig. 2 . 2 . Schematic diagram of particle and fluid motion around a cylinder (a) and an inclined flat plate (b). Streamlines are s h o w n as solid lines, while the dotted lines are aerosol particle paths.

spheres, or the particle radius a p p r o a c h e s z e r o , but thermal motion is disre­ garded. F o r fully developed flow, the horizontal c o m p o n e n t of particle ve­ locity is

22

qx =

(6z/d-3z /d )qx,

where ζ is the vertical coordinate and d is the channel width. Disregarding particle inertia in the horizontal c o m p o n e n t , the particle trajectories are determined by dx

dz

w h e r e qG is the settling velocity. This pair of equations can be solved by introducing a fluid d y n a m i c stream function, and the fraction η of the total n u m b e r of particles r e m o v e d on the b o t t o m of the channel is written 17 = qG Lldqx. T h e p a r a m e t e r η is normally called the precipitation or collection F o r flow in a r o u n d t u b e , an analogous result is obtained: ΤΟ / =

3qG L/$RTqx,

(2.39) efficiency.

2.2

35

Accelerated Motion of Particles—Deposition Processes

w h e r e RT is the t u b e radius. In the case of viscous gas flow in horizontal tubes, T h o m a s (1958) obtained

m

η = (2/π)[2ηο(1 - Vo)

/3

+ arcsin η '

/3

2

- η ' ( 1 - %ψν].

(2.40)

Such calculations m a y b e e x t e n d e d to include external electrical forces. Precipitation of aerosol particles in electrically charged vertical or horizontal channels has b e e n u s e d to study particle mobility. T h e effect of including a finite particle radius effectively increases the potential collection efficiency o v e r the estimate given for the case in which the radius is negligible. 2.2.3 Inertial Deposition on Blunt Bodies. A blunt b o d y in an aerosol stream can act as a particle collector. This is easily d e m o n s t r a t e d b y sus­ pending a wire o r a sphere in dusty air. After e x p o s u r e a coating of particles will be found on the side of the b o d y facing the stream. Like the channel flow example, the deposition of particles can be calculated using the equation for particle trajectories. Particle p a t h s which touch the b o d y will give rise to particle collection. T h e collection efficiency can be defined in t e r m s of a distance S from the axis of the obstacle to the o u t e r m o s t trajectory of parti­ cles that strike the surface c o m p a r e d with the radius of the collector. In other w o r d s , the efficiency is r) =

S/a,

w h e r e a is the cylinder radius. T h e trajectories of the particles can b e calcu­ lated by a step-by-step integration of v e c t o r equations of motion in the form

lp

m B^ dt

df

=

| fi- (qg--^).

(2.41)

This relation often is written in the dimensional form S t k ^

+ ^

w h e r e r' = r/R, q g = a/U œ + t' = UjlR.

= q^?'), T h e Stokes number

ak.^.i-ÎÇ&Si

(2.42) is (2

.

43)

It i s equivalen t t o th e rati o o f th e stoppin g distanc e i an d th e radiu s o f th e obstacle. F o r " p o i n t " particle s i n Stoke s flow, th e Stoke s n u m b e r i s th e onl y criterion o t h e r tha n g e o m e t r y tha t determine s similitude fo r th e shap e o f th e particle trajectories . T o e n s u r e h y d r o d y n a m i c similarity , i n general , th e col lector Reynold s n u m b e r als o mus t b e p r e s e r v e d , a s wel l a s th e rati o I' Rla, called th e interceptio n p a r a m e t e r . T h e n th e collectio n efficienc y ca n b e writ ten i n th e for m η =/(Stk,Re,I).

(2.44)

36

2

The Dynamics of Small Particles

H e r e the interception p a r a m e t e r effectively accounts for a small additional increase in the collection efficiency for a given Stk associated with the finite size of the particles. T h e inertial collection of particles is referred to as impaction. T h e influ­ ence of finite particle size relative to the collector is called interception. W h e r e Stk - » 0, or as the particle m a s s b e c o m e s small, the particles tend to follow the motion of the gas. In steady flow, the particles then n e v e r r e a c h the obstacle, or η = 0. T h u s , there should be a certain minimum value of the Stokes n u m b e r below which particle entrainment in the gas dominates iner­ tia. T h e existence of such a regime apparently w a s first established by Langmuir (1948). In the limiting case of high Reynolds n u m b e r for flow around obstacles, w h e r e inviscid or potential flow is a useful approximation, one can estimate analytically the values of the critical Stokes n u m b e r S t k c . r Such calculations h a v e b e e n reported by Levin (1953); his results are shown in Table 2.6 for 1^ 0. T o illustrate the form of the relationship b e t w e e n the collection efficiency and the Stokes n u m b e r , consider an aerosol streaming around a s p h e r e . Calculations can be m a d e for t w o e x t r e m e cases of gas flow around the sphere, first inviscid motion, w h e r e R e —> o°; second for viscous flow, w h e r e Re is finite. T h e e x t r e m e for the second regime is the case of viscous flow, w h e r e R e ^ 1. In the inviscid flow c a s e , the streamlines around the b o d y are symmetrical a r o u n d it; in viscous flow w h e r e Re <^ 1 the streamlines also are symmetrical, but are p u s h e d o u t w a r d relative to the inviscid case as a conse­ quence of viscous effects. In reality, high Reynolds n u m b e r flows " s e p a ­ r a t e " on the lee side of the b o d y and form a turbulent w a k e . H o w e v e r , the T A B L E 2.6 Limiting Particles

0 Stokes Values of the on Bodies Obstacle in

Number

for Collection

flow

1. Infinitely long ribbon of width 2a (a) Without separation (b) With separation 2. Elliptical cylinder, major to minor axis ratio X 3. Circular cylinder 4. Sphere 6 5. Circular disk 6. Flat jet striking plane at right a n g l e s with / / / W - > oo

of

Stk

t

î 4 / ( π + 4) i ( l + X) J τζ π 116 21 π

a

Estimates based on potential flow near a stagnation point as bderived by Levin (1953). H/W is the ratio of the distance from the jet to the plate and W is the jet width or diameter.

2.2

Accelerated Motion of Particles—Deposition Processes

37

inviscid flow p a t t e r n remains a useful approximation in the regime near the forward stagnation point of the collector, w h e r e most of the particles impact. Calculations for the theoretical efficiency curves for particles in Stokes flow h a v e b e e n m a d e , for e x a m p l e , by D o r s c h et al. (1955), and are s h o w n for the e x t r e m e s in Fig. 2.3. Also included are curves for intermediate collec­ tor Reynolds n u m b e r . T h e curves in Fig. 2.3 indicate that the S t k cr « 0.08 estimated by L e v i n is in agreement with the aerodynamic calculations, and for inviscid flow, the rough approximation that η ~ 0.5 for Stk of about unity is apparent. T h e efficiency curve for the inviscid approximation is considera­ bly higher than the calculation for viscous flow for Stk < 100. E x p e r i m e n t s reported by R a n z and W o n g (1952) and later by Walton and W o o l c o c k (1960), for 70 ^ R e ^ 5700, generally agree with the inviscid approximation. Curves of collection efficiency versus Stk for a circular cylinder h a v e similar shape to those for a s p h e r e . T h e inviscid and viscous limits for a cylinder are shown in Fig. 2.4. T h e inviscid limit corresponds to the dimen­ sionless p a r a m e t e r Ρ being very large; the viscous flow limit is Ρ —» 0. T h e curves are b a s e d on calculations of Brun et al. (1955). Again the S t k cr de­ rived by L e v i n is in a g r e e m e n t with the detailed calculations. Limited obser­ vations of R a n z and W o n g for sulfuric acid mist collection in the range 55 < Re < 5520 agree satisfactorily with the inviscid approximation. 2,2.4 Electrical Forces and Particle Deposition. T h e r e are circumstances w h e r e particle deposition is determined by electrical forces induced b e t w e e n the collector and the particles flowing past it. T h e deposition rate on the obstacle can b e estimated from electrical theory. F o r conditions of a nondivergent or solenoidal electrical field, the particle concentration far from

2 STOKES NUMBER ( 2 R U / 9 ^ a )

e p o og

Fig. 2 . 3 . Collection efficiency for spheres in an inviscid flow with point particles. [After Dorsch et al. (1955).]

38

2 Th e Dynamic s o f Smal l Particle s

1.0

0.1

1.0

10

2

100

1000

STOKES NUMBE R ( 2 e R U / 9 , a )

p ( X1

Fig. 2.4 . Collectio n efficienc y fo r cylinder s i n a n invisci d flow wit h poin t particles . Th e rat e of deposition , particle s pe r uni t tim e pe r uni t lengt h o f cylinder , i s 2NU a, wher e Ν is the œ particle concentration in the mainstream. The parameter Ρ is the dimensionless group 4ReJ/Stk = 1 8 p g 6 U / / i , p p .

g

the obstacle will remain constant. F o r steady state conditions, the particle concentration near the b o d y ' s surface will remain in proportion to that found far a w a y . T h u s t h e total flow of particles t o w a r d the surface of the collector will equal the integral of the flux of particles over any surface of distance and from the center of the collector. In the a b s e n c e of B r o w n i a n motion, t h e total n u m b e r of particles hitting the surface can be e x p r e s s e d as (2.45) w h e r e qr is the c o m p o n e n t of particle flow velocity normal to the surface of the b o d y , Fr is t h e c o m p o n e n t of external (electrical) force directed normal to the surface, and dS is an element of surface area. In the case of an electrical field, FrE0r Q, w h e r e E0r is the radial c o m p o n e n t of the field strength, and Q is the charge on the particle. If the particles are small t h e c o m p o n e n t of particle flow velocity qr will b e negligible, and E q . (2.45) b e c o m e s

where Q' is the total charge on the collector surface. The efficiency of collection can be written in terms of the net particle flux ΦΙΞ divided by the total n u m b e r of particles flowing past the b o d y , or η = Φ/NUnS

=

-4nQQ'B/UxS0,

(2.46)

<5Ύ

2.2

Accelerated Motion of Particles—Deposition Processes

39

w h e r e S 0is t h e cross-sectional area of the obstacle normal to t h e direction of the free-stream velocity [/«,. Deposition from aerosol clouds o n spheres b y electrical forces w a s inves­ tigated by K r a e m e r a n d J o h n s t o n e (1955). T h e y calculated particle trajecto­ ries induced b y electrical forces, and estimated collection efficiencies by interception, neglecting impaction. Idealized situations w e r e estimated for (a) the Coulombic force b e t w e e n dielectric spheres and charged parti­ cles, (b) t h e force b e t w e e n unipolar charged particles a n d t h e charge induced by t h e m o n a grounded s p h e r e , and (c) t h e induced force b e t w e e n a charged and uncharged particle. T h e interactive forces can be written in a form normalized to t h e S t o k e s resistance: (a)

C o u l o m b interaction:

2

3k = ô ô ' / 6 7 r a / ? / z g t/oo ; (b)

(2.47a)

Induction to a grounded s p h e r e :

2

3b = Q E NAllfygRUa>a9

(2.47b)

w h e r e A 0 is t h e radius of t h e aerosol cloud; and (c)

Induction t o a charged sphere: = ( Cg - l)fl2Q2/( es + 2)3irflVgt/«>

(2.47c)

w h e r e ε δ is t h e dielectric c o n s t a n t of t h e sphere. A typical theoretical c u r v e of K r a e m e r a n d J o h n s t o n e is s h o w n in Fig. 2.5. It is c o m p a r e d with experimental m e a s u r e m e n t s for dioctylpthalate particles of m e a n radius 0.27-0.59 μ π ι . T h e particles w e r e either charged with units of the same sign (unipolar) or different sign (bipolar) at average charge values of 0.15-1.37 units. T h e y w e r e collected o n spheres of 0.33-0.55 c m radius from a gas flow of 1.5-6.9 c m / s e c . T h e series of experiments w a s run with the three different dimensionless forces p r e d o m i n a n t as s h o w n in Fig. 2.5. T h e e x p e r i m e n tC s indicated that interception tended to increase t h e collection efficiency with S in t h e low ranges of i>, c o m p a r e d with that expected from 2 the theory. A significant d e p a r t u r e from the theory w a s found for 5 values less than 10 , b u t a b o v e this t h e simple model a p p e a r s to w o r k satisfac­ torily. W h e n electrical forces a r e p r e s e n t , but d o not necessarily dominate parti­ cle collection, t h e estimation of t h e collection efficiency d e p e n d s on the aerodynamics involved, as well as t h e electrical p a r a m e t e r s that govern particle trajectories a r o u n d conducting or nonconducting b o d i e s . T h e s e h a v e been studied in detail b y N a t a n s o n (1957), Zebel (1968), and H o c h r a i n e r et al. (1969).

40

2

τ—γτ~|

1

1—I Γ]

I

I I T~|

The Dynamics of Small Particles

I

I ΓΓ~|

I

I I Γ

Fig. 2 . 5 . Collection efficiencies for particles deposited on spheres by electrostatic forces. O, collector charged, aerosol charged by corona, parameter (% - 5 ) . Δ, collector charged, E C aerosol with naturally occurring charge, parameter [F - 3^. · , collector grounded, aerosol T charged by corona, parameter S . [Reprinted with permission from Kraemer and Johnstone G (1955). Copyright 1955 American Chemical S o c i e t y . ]

At large R e , t w o dimensionless p a r a m e t e r s h a v e been used to characterize regimes of particle motion a r o u n d cylinders, for e x a m p l e . T h e s e are ^ _ EooQB _ Particle velocity in electrical field t/oo

Undisturbed gas velocity

w h e r e the field strength £Όο is oriented parallel to the flow direction [/«,, and _ 2Q'QB Uœd

_ Velocity of a charged particle near a charged cylinder U n d i s t u r b e d ga s velocit y (2.49)

E x a m p l e s o f difference s i n particl e motio n a r o u n d a charge d cylinde r ar e shown fo r differen t rate s o f G an d H i n Fig . 2.6 . T h e s e case s illustrat e th e great variation s i n idealize d invisci d flow aroun d th e collecto r wit h electrica l forces uninvolved . E v e n i n condition s w h e r e th e fluid flow doe s no t contai n

42

2

The Dynamics of Small Particles

a w a k e , a particle trajectory " w a k e " emerges u n d e r certain conditions, for example, in (b), (c), and (f) in Fig. 2.6. C a s e s of collection on the backside of the collector without a fluid w a k e also b e c o m e possible (d), or a " p a r t i c l e free s p a c e " around the collector may be induced (e). T h e experiments of H o c h r a i n e r et al. (1969) agree qualitatively with sev­ eral of the theoretical conditions in Fig. 2.6. Deviations b e t w e e n experiment and theory w e r e o b s e r v e d at the forward stagnation point of the cylinder. T h e deviations w e r e probably associated with difficulties in flow stabiliza­ tion and m e a s u r e m e n t m e t h o d s near the forward stagnation point. In view of t h e s e factors, investigators should b e w a r e of indiscriminant application of the d y n a m i c theory involving only a e r o d y n a m i c s , if electrical fields are involved. 2.3

DIFFUSION AND BROWNIAN MOTION

2.3.1 Diffusion in a Stagnant Medium. So far w e have c o n c e n t r a t e d on the behavior of particles in translational motion. If the particles are suffi­ ciently small, they will begin to experience an agitation from r a n d o m molec­ ular b o m b a r d m e n t in the gas which will create a thermal motion analogous to that of the surrounding gas molecules. T h e agitation and migration of small colloidal particles has b e e n k n o w n since the w o r k of R o b e r t B r o w n in the early nineteenth c e n t u r y . This thermal motion is likened to the diffusion of gas molecules in a nonuniform gas. T h e applicability of F i c k ' s equations for the diffusion of particles in a fluid has b e e n accepted widely after the w o r k of Einstein (1905) and others in the early 1900s. T h e theory of Brownian diffu­ sion has b e e n reviewed in detail by Hidy and Brock (1970) and will not be repeated h e r e . F o r an aerosol, the basic equation for convective diffusion of the particles is ^

+ qp · VN = V · Dp VJV

(2.50)

w h e r e Ν is the n u m b e r concentration of particles. T h e p a r a m e t e r Dp is the Stokes-Einstein diffusivity,

l

. Dp = BkT = km- /mp Given the definition of a m e a n free path for particles as λ ρ = ^pd~\ average thermal speed of a particle is

m

^

=

(SkT/nmp) ,

and Dp = Ι π ^ ρ Χ ρ .

(2.51) the

43

2.3 Diffusio n an d Brownia n Motio n

S o m e characteristi c value s o f t h e s e aeroso l t r a n s p o r t propertie s o f particle s in ai r ar e liste d i n T a b l e 2. 1 fo r reference . T h e simples t c a s e fo r whic h particl e diffusio n h a s b e e n analyze d i s t h e situation w h e r e t h e r e i s n o ga s motion , o r th e secon d t e r m o n th e lef t o f E q . (2.50) i s z e r o . T h e m a t h e m a t i c a l solution s fo r E q . (2.50 ) i n suc h case s wit h appropriate b o u n d a r y condition s ar e readil y availabl e i n Carsla w an d Jaege r (1959) o r C r a n k (1959) . A for m o f t h e s e solution s whic h i s ofte n usefu l c o n c e r n s calculation s o f th e flux o f particle s t o th e surfac e i n differen t geom etries. F o r e x a m p l e , t h e diffusiona l flux o f particle s t o a vertica l wal l locate d at positio n x Qi s give n b y

with N(x, t) = Nœ e r f [ ( * - x 0)/4Dpt] fro m a n appropriat e solutio n o f E q . (2.50). Noo i s th e particl e c o n c e n t r a t i o n fa r fro m th e wall . T h u s

/2

j = N„(D p/nty .

(2.52 )

By a simila r a r g u m e n t , th e n u m b e r o f particle s strikin g th e entir e inne r surface p e r uni t tim e o f a s p h e r e o f radiu s a i s (Pich , 1976 )

ΦΡ =

P

Σ expΓ" \

Sa7rD N»

p

;

(2.53)

for an infinitely long cylinder of radius a,

Φ Ρ = 4ττΖ> ρΜο

Σ exp(^^), \

n=l

a

(2.54)

/

w h e r e an are t h e positive roots of t h e Bessel function Jo(an, a) = 0. In the c a s e of particle diffusion from a stagnant m e d i u m to the outer surface of a s p h e r e of radius a, the total n u m b e r of particles a b s o r b e d p e r unit time is D P( H ΦΡ = 4 W

= 4W D po N . ( l + j

^

j

.

(2.55)

Over a time interval Δ/, t h e n u m b e r of particles collected on t h e sphere p e r unit a r e a is

2

J OΦ Ρ dt = 4παΟρΝ„[Δί

+

(

π

γ η ) ·

Β

2 deposited p e r unit area is the T h u s w h e n a IDp > At, the n u m b e r of particles same as for a flat wall. Conditions w h e r e a IDp < At give a steady state solution for the particle c o n c e n t r a t i o n Ν = N œ (\ — air), yieldin g th e stead y state flux, ΦΡ =

4ττϋ αΝοο.

ρ

(2.56)

44

2

The Dynamics of Small Particles

In the case of the infinitely long circular cylinder of radius a, the flux to the cylinder is

2 ΦΡ = F o r short times, Φρ -

2

£ e x p ( - £ > py 0 ^

12 / 2πΖ)ρΛΤοο[(7Γί')-

+

2

[J 0(ay) Y 0(ay)].

(2.57)

m i ~

+ ( i ) / ' · · ·].

Wlir)

(2.58a)

F o r long times, Φ Ρ - 2πΏρΝ.{^Γ)

- 1.154 -

4 ( 1 % η _ ( 242. .) · ) , ί' =

, C 3 = 0.577.

(2.58b)

2.3.2 Diffusion in Laminar Flow. W h e n there is a mean flow of the aero­ sol, the solutions to the convective diffusion equation account for that flow. In cases w h e r e the gas motion is uniform, the solutions for particle diffusion with q = 0 in E q . (2.50), " c o n d u c t i o n " solutions, can be extended using a Galilean transformation of coordinates, for example, x' = (x - qt). In most circumstances of interest, h o w e v e r , there is a gradient in gas velocity in­ volved, or there is a shearing flow. In such circumstances the total m a s s flux is given by the equation j = q(x, t)N - Dp V N ,

(2.59)

w h e r e the first term on the right is the particle flux associated with convec­ tive motion of the fluid, and the second term on the right denotes the diffu­ sion flux of particles by Brownian motion. In any arbitrary fluid volume the number of particles passing through the surface S of that volume is Φρ = The theory of convective diffusion in flowing fluids is well developed for the steady state c a s e , w h e r e dN/dt = 0. T h e aerodynamic b o u n d a r y layer model (e.g., Schlichting, 1960) is normally applied to aerosol collection prob­ lems. This theory deals with estimation of the diffusional flux to various collecting bodies, by considering approximate solutions to E q . (2.50), with the steady state conditions, and assuming that the particles do not lag behind the fluid. That is, q p = q. In a dimensionless form, the diffusion equation can be written as

2

qi · V!iVi = (1/Pe) V M ,

(2.60)

where the p a r a m e t e r s are normalized to a main stream flow far from a surface, or N\ = N/N*, qi = q/Uœ , V i = L V , an d th e Peclet number P e =

2.3

45

Diffusion and Brownian Motion

LUoo/Dp. T h e length scale L is characteristic of the geometry of the collec­ tor, say a sphere radius, or a fluid dynamically defined thickness such as the thickness of a b o u n d a r y layer along a flat plate. T h e surface condition is given by N = 0,

zlL = RIL =

I,

w h e r e ζ is a m e a s u r e normal to the collector surface. F o r a sphere or cylin­ der the length scale is the radius. T h e dimensionless concentration can be e x p r e s s e d as N] =f(r/L,

Re, P e , / ) .

(2.61)

Thus t w o convective diffusion regimes are similar if the R e y n o l d s , Peclet, and interception n u m b e r s are the s a m e . Since Pe > 1 for conditions of convective particle diffusion, this leads to simplification in aerosol p r o b l e m s (e.g., H i d y and B r o c k , 1970). T h e local rate of particle transfer by diffusion to the surface of a b o d y is given by

J1

PpiVoo "

L

idN\\ \drx

/r,=/*

In analogy to fluid heat and m af s s transfer (p. 27), the local mass transfer coefficient, (lis defined as J/N œ . I n dimensionles s for m lL/Dp

= / ( R e , Pe , / )

(2.62 )

for th e stead y stat e c a s e . T h e dimensionles s n u m b e r i s th e S h e r w o o d num ber S h fo r particl e diffusion . Within thi s genera l framework , conside r a flow o f aeroso l alon g a flat plat e as show n i n Fig . 2.7 . A s th e ga s m o v e s ove r th e surface , th e frictio n a t th e surface decelerate s neighborin g layer s o f fluid, creatin g a velocit y gradient .

Fig. 2.7 . Schemati c diagra m o f th e developmen t o f momentu m an d concentratio n boundar y layers ove r a flat plat e a t hig h value s o f th e Pecle t number .

+

Thi s ter m i s sometime s calle d th e depositio n velocity . (Se e als o p . 51. )

46

2

The Dynamics of Small Particles

At the same time, particles diffuse from the cloud to the surface and collect t h e r e , creating a gradient in particle concentration. Conventionally, a length scale L for this s y s t e m is defined in t e r m s of a m o m e n t u m thickness δ Μ which is defined as the distance a w a y from the surface at which the gas velocity is 9 9 % of the free-stream value £/«,. Similarly, a diffusion b o u n d a r y thickness is the distance ô Dfro m th e surfac e a t whic h th e particl e c o n c e n t r a tion h a s r e a c h e d 9 9 % o f it s free-strea m valu e N œ . I n lamina r flow, w h e r e n o turbulence exists , th e relatio n b e t w e e n th e t w o lengt h scale s i s

1 / 3

δ 0- Φ Ρ/ ^ )

δΜ ,

3 e ga s kinemati c viscosity . Sinc e th e particl e Schmid t n u m b e r , w h e r e v %i s th vg/Dp = 10 o r large r fo r a e r o s o l s , th e thicknes s o f th e diffusio n laye r h a s t o be abou t one-tent h tha t o f th e m o m e n t u m layer . Tha t is , th e fluid velocitie s are abou t 10 % o f th e free-strea m flow w h e n th e particl e concentratio n i s a t its free-strea m valu e o v e r a collector . In th e cas e o f th e flat plate , th e diffusiona l flux t o th e plat e ha s th e for m (2.63) or th e loca l particl e flux i s proportiona l t o th e Schmid t n u m b e r t o th e one third p o w e r . A n importan t applicatio n o f t h e t h e o r y i s th e diffusiona l depositio n i n tubes o r pipes . Ideally , th e flow o f a n aeroso l throug h a tub e begin s wit h fluid velocity unifor m a c r o s s th e t u b e cros s section . A s th e fluid frictio n a t th e wall take s effect , th e fluid velocit y distributio n r e a c h e s a stead y paraboli c shape, give n b y Poiseuille' s equation , fo r a pip e o f radiu s a,

22

q(r) = q(l -

r /a ),

2

w h e r e q i s th e averag e flow velocity , equa l t o α Δρ/4μ%Σ, with Δρ/L the p r e s s u r e d r o p p e r unit pipe length L. N e a r the pipe e n t r a n c e , w h e r e the velocity is nearly uniform, t h e particle flux to t h e walls c a n b e estimated by the b o u n d a r y layer a p p r o x i m a t i o n s for a flat plate. F u r t h e r d o w n s t r e a m , w h e r e t h e velocity profile b e c o m e s parabolic, the calculation of diffusional wall deposition requires a m o r e complicated solution (Friedlander, 1977). T h e average m a s s transfer coefficient H av o v e r a length χ is found to be ll^alDp

= £

l n ( A y j V a)v,

(2.64)

w h e r e xx = (jc/a)Pe and N0 is the initial particle c o n c e n t r a t i o n . This relation­ ship is s h o w n in Fig. 2.8 with the normalized local m a s s transfer coefficient 2 iLa/Dp and the normalized concentration Nav /N0.

2.3

47

Diffusion and Brownian Motion

1.0

0.8

I

I

I

I

I

Ο

0.6

-

20

a / N vQ

Ο

-

Ω­

CO > CO

15

CM

> Ζ

25

I

- \ N

ζ

ι

ι

a D

CO

0.4

.

\ ^ a v g

/

Q

10

p

Q.

co CM

2 f e a / D ^ ^ ^ ^

0.2 —

5 3.658

0

I

ι

I

I

0.001

2

5

0.01

2

x

I

I

I

I

5

0.1

2

5

1

1

Fig. 2.8. Diffusion to the walls of a pipe from a fully developed laminar flow. Average concentration and local and average mass transfer coefficients are s h o w n as a function of distance from the tube entrance. [From Friedlander (1977); reprinted with permission of the author and John Wiley & S o n s . ]

T h e s e results can be used to characterize the deposition of particles of different size in t u b e s . Particles entering and leaving a t u b e can be m e a s u r e d from the c u r v e in Fig. 2.8. F r o m m e a s u r e d values of the reduction in concen­ tration, JCI can be estimated. T h e n Dp (or Rp) can b e calculated from x, a, and q (see also Section 5.7.4). T h e t h e o r y of c o n v e c t i v e particle diffusion also can b e applied to the collection of particles on blunt obstacles such as spheres and cylinders. In the case w h e r e t h e interception p a r a m e t e r / —> 0, and for a blunt-body radius a, the t h e o r y (e.g., Hidy and Brock, 1970) projects a diffusion rate of Sphere:

1/3 Φ = 47r/yVootf(l + 0 . 6 4 P e ) ,

(2.65)

1/3

Sh s 2la/Dp Cylinder:

« 1.26Pe ;

1/3

, 3/

Φ = 2 7 7 A U C 4P e / [ 2 ( 2 . 0 0 2 = In 2 R e ) ] (where C 4 is a constant given as 1.305), and

1/3

Sh = 2ialDp

-

(2.66)

1/3

1.26Pe /[2(2.002 - In 2 R e ) ] .

(2.67)

48

2

The Dynamics of Small Particles

The collection efficiency has a direct relation with the S h e r w o o d n u m b e r . In the case of cylinders, for example, Sh =

l i a

j

27jPe

IttRN

(2.68)

π

or

1 / 32 / 3 η = π VU»

= 3.68[2(2.002 - In 2 R e ) ]

W h e n / is finite, similarity theory requires that

1/3

7)1 Pe = nlR/Dp

Pe

.

(2.69)

1/3

= / ( / P e [ 2 ( 2 . 0 0 2 - In 2Re)] ).

(2.70)

As a last consideration, the m e c h a n i s m for diffusional deposition should be merged with a model of impaction to provide a complete picture of collec­ tion efficiency. W h e n the mathematical equations for particle motion are combined with those for diffusion, a solution based on similitude arguments cannot be obtained. H o w e v e r , calculations have b e e n m a d e for viscous flow around a sphere of different radii using b o u n d a r y layer approximations, and

0.01

0.1 1.0 Particle Diameter (μπι)

Fig. 2.9. Aerodynamic collection efficiency as a function of particle diameter for flow around a sphere, and collector diameter. Curves are shown for each free-stream velocity and for collector diameters from 0.4 to 4.0 mm [From Hidy and Heisler (1978); reprinted with permis­ sion of John Wiley & Sons.]

2.4

49

Deposition in a Turbulent Fluid Medium

in viscid flow around the collector in the diffusion range. Interception has been included in the calculations. T h e range of collector diameter considered is typical of w a t e r spray scrubber design. T h e results of Shaw and Friedlander are s h o w n in Fig. 2.9. On the right side of the c u r v e , the impaction range shows an increase in collection efficiency with free-stream velocity as ex­ pected from earlier a r g u m e n t s . T h e collection efficiency goes through a minimum with decreasing particle size which differs s o m e w h a t with freestream velocity; then the collection efficiency increases again for small parti­ cles as diffusion b e c o m e s important. T h e minimum in collection efficiency can be found b e t w e e n 0.1 and 1 μ π ι diameter according to these calcula­ tions, depending on the free-stream velocity. T h e minimum in efficiency also shifts to smaller size with decreasing collector diameter. T h e collector diameter is shown near the top of the curves for each case in Fig. 2.9. Finally, let us consider the influence of electrical forces on particle collec­ tion. F o r illustration the case of the spherical collector is included. H e r e the f velocity ratios G and H are used to characterize the migration associated with electrical forces (p. 40). F o r / —» 0, Zebel (1968) has calculated the deposition rate assuming that the deposition is dominated by accumulation near the forward stagnation point of the collector. S h o w n in Table 2.7 are his results for a sphere falling in an electrical field oriented parallel to the direc­ tion of motion of the s p h e r e . H e r e a wide range of efficiency is predicted which d e p e n d s on the strength of electrical forces. 2.4

DEPOSITION IN A TURBULENT FLUID MEDIUM

The deposition of particles from a turbulent medium is of considerable practical interest as a natural extension of turbulent transport theory, and from the need for design of collection devices. Turbulent deposition also plays an important role in the removal of aerosol particles from the e a r t h ' s atmosphere. W h e n particles are s u s p e n d e d in a turbulent fluid, they are subjected to a random, fluctuating motion which is analogous to thermal agitation in a stagnant m e d i u m . T h u s , w e can derive an equation for convective transport in a turbulent gas equivalent to that of E q . (2.50). In this application, how­ ever, the equation applied to an average behavior, such that the "diffusivi t y " is identified with an eddy motion rather than with Brownian diffusion (see, for e x a m p l e , Hidy and Brock, 1970). The theory for particle dispersion in turbulent media such as the e a r t h ' s a t m o s p h e r e is deduced from solutions to the turbulent diffusion equation. T h e deposition of particles from a turbu­ lent medium also can be estimated from an analogous model involving boundary layer approximations (Schlichting, 1960). As in the case of laminar flow, the deposition of particles suspended in a turbulent fluid on an adjacent surface can o c c u r by inertial effects, gravita­ tional settling, B r o w n i a n diffusion, or by the influence of external forces. For electrically neutral particles m o r e than 1 μτη in diameter, deposition is

f

2

G remains the same for spheres, but H is redefined as H =

QQ'B/U^o .

50

2 Th e Dynamic s o f Smal l Particle s

T A B L E 2. 7

b a Collection Efficiencies and Sherwood Numbers Falling Sphere in an Electrical Field. '

for the Deposition

Sh

HIG A. No Brownian >0 > 3

of Charged

(3G -

-3 < HIG < 3

>0

HIG <

<0

> 0

<0

< 0

With

> 0 <^ 1

H)

-AH

- 3

G + 1 -AHIG

+ 1

23/ 2 . 5 2 P e - 23/ 2.52Pe" + è(3 G -

Uncharged particle s Positive charge d particles ; weak electrica l force s compared wit h aerody namic force s Strongly positiv e charge d particles

H)

6G

> |Pe -

G+ 1 3G

1.5Pe-

H = 0 , strongl y positive charged particle s

iPe/G

+ 1

>0

>3

>0

3G -

- 3 < HIG < 3

>0

H

3 G ( G + 1) -AH

< - 3

G + 1

(3G 6G -2//Pe

2/3 > - [ 1 + (0.875)Pe-

<0

a bTh e field

Remarks

Positively charge d poin t particles; repulsiv e force s on fallin g spher e Positive charg e o n poin t particles; repulsiv e an d attractive force s o n fallin g sphere Positive charg e o n poin t particles; attractiv e force s on fallin g spher e Negatively charge d poin t particles wit h attractiv e and repulsiv e force s o n th e falling sphere , o r attractiv e forces onl y o n th e spher e Negative charg e o n poin t particles wit h negativ e charge o n th e sphere ; dust free spac e wil l appea r around fallin g spher e

3G(G + 1 )

diffusion 0 1.5Pe-

>0

on a

diffusion

2 >0

Particles

]

-AH G + 1

-2HYQ

H)Pe

H = 0 , strongl y positive charged particle s H = 0 , strongl y positive charged particle s H = 0 , strongl y positive charged particles ; attractiv e electrical forc e ove r entir e surface o f fallin g spher e H < 0 , stron g negativ e charg ing o n particle s

directio n i s oriente d paralle l t o th e directio n o f motio n o f th e larg e sphere . Base d o n theor y o f Zebe l (1968) .

2.4

51

Depositio n i n a Turbulen t Flui d Mediu m

effected b y inertia l forces . T h e particl e depositio n rat e fro m a turbulen t fluid is enhance d ove r tha t o f a lamina r flow a s a c o n s e q u e n c e o f edd y diffusio n t o the lamina r sublaye r nex t t o th e surface . T h e developmen t o f a theor y fo r deposition whic h a c c o u n t s fo r th e turbulenc e ha s bee n controversia l sinc e the initia l attemp t o f Friedlande r an d J o h n s t o n e (1957) . The theor y i s writte n t o provid e a n estimat e o f th + e m a s s transfe r coeffi + h i s re cient o r deposition velocity i n dimensionles s form , l = l/u*, whic ported a s a functio n o f dimensionles s particl e relaxatio n tim e r ,

+

2

r

= r w * / ^ g, (2.71 ) 1 / 2 w h e r e « * i s th e frictio n velocit y equa l t o ( F / p g) ; F i s th e frictiona l stres s a t the surface . T h e existin g theorie s fo r turbulen t deposition , includin g tha t o f Friedlander an d J o h n s t o n e (1957) , ar e base d o n a so-calle d "diffusion-fre e flight" model . I n thi s m o d e l , particle s ar e a s s u m e d t o b e transporte d b y turbulent diffusio n fro m th e turbulen t cor e o f th e fluid throug h th e b o u n d a r y layer t o a diffusio n sublaye r w h o s e thicknes s i s approximatel y on e stoppin g distance fro m th e wall . A t thi s poin t th e particl e follow s a fre e p a t h t o th e wall. The difference s i n variou s theoretica l model s cente r o n th e estimat e o f th e particle velocit y a t th e beginnin g o f fre e flight t o th e wall . Friedlande r an d J o h n s t o n e a s s u m e d a valu e o f 0.9w* , whil e other s a s s u m e d differen t values . Davies (1966 ) too k th e initia l fre e flight velocit y a s th e root-mean-squar e (rms) fluctuating velocit y o f th e fluid a t th e poin t w h e r e fre e flight begins . 3 Beal (1970 ) a s s u m e d a fre e flight velocit y o f hal f th e axia l velocit y o f th e fluid, bu t Sehme l (1971 ) a s s u m e d a for m proportiona l t o èrw* . Li u an d Ilor i (1973) a s s u m e d tha t th e particl e edd y diffusivit y nea r th e wal l i s proportiona l to th e su m o f e d d y diffusivit y o f th e fluid an d th e p r o d u c t o f th e r m s fluctuating velocit y n e a r th e wal l an d th e relaxatio n time . T h e experiment s o f M o n t g o m e r y an d Cor n (1970 ) suggeste d tha t th e early F r i e d l a n d e r - J o h n s t o n e "diffusion-free-flight " hypothesi s an d othe r models, e x c e p t tha t o f D a v i e s , w e r e i n reasonabl y goo d agreemen t wit h experiments fo r a rang e o f behavio r characterize d b y a dimensionles s relax ation tim e les s t h a n 30 . Liu an d Agarwa l (1974 ) reporte d som e n e w experiment s fo r depositio n o n the wall s o f a vertica l pipe . Thei r result s fo r th e inertia l depositio n rang e ar e shown i n c o m p a r i s o n wit h fou r o f th e theoretica l model s i n Fig . 2.10 . T h e mass transfe r coefficien t o r depositio n velocit y i s show n a s a functio n o f th e dimensionless relaxatio n time . Thei r result s confir m M o n t g o m e r y an d C o r n ' s conclusions . T h e theor y o f Li +u an d Ilor i show s equall y goo d agree ment wit h experimenta l dat a nea r r = 2 . Th e depositio n velocitie s esti mated b y D a v i e s ' mode l ar e considerabl y lowe r ove r th e entir+e rang e s h o w n . A sligh t curvatur e i n B e a l ' s resul t i s o b s e r v e d belo w τ > 0.3, departing from the F r i e d l a n d e r - J o h n s t o n e model. Yet in D a v i e s ' m o d e+ l, Brownian diffusion has an influence at considerably higher values of r , evidently b e c a u s e of the low inertial deposition estimated by his theory.

52

2

The Dynamics of Small Particles "1

»

r τ τ '

Friediander and Johnstone Beat Liu and I l o r i Davies

Experimental data •Reynolds number = 1 0 . 0 0 0 ^Reynolds number = 5 0 . 0 0 0 _ With % Brownian /

//

- 7

V

diffusion

/

/

/

'

Theoretical curves are for Reynolds number = 1 0 . 0 0 0

"Ti

/

/

'

i m l1 /

y^—

No Brownian diffusion

/

1

1

1 J 1 1 U 10 ll

-1

1 i 1

ilil

100 Dimensionless reloxtion time

i

I I I lili

1000

Fig. 2.10. Comparison between theories and experiments for the dimensionless deposition velocity and relaxation time of particles. [From Liu and Agarwal (1974); reprinted with permis­ sion of Pergamon Press.]

+ + In the range of τ ^ 20, the m e a s u r e m e n t s in Fig. 2.10 show a weak decline in dimensionless deposition velocity with increase in τ (or particle size). This has not b e e n a c c o u n t e d for in the theories until recently. R e e k s and S k y r m e (1976) h a v e explained this behavior by considering a stochastic model for particle motion in the air near the collecting surface. T h e y found that the observed d e c r e a s e is due to the fact that the increased fractional penetration of the fluid b o u n d a r y layer with increasing particle m a s s be­ comes insufficient to c o m p e n s a t e for the reduced rate of turbulent particle + transport to that region. The Reeks + and S k y r m e model gives a simple form for H in its range of application ( τ ^ 5). T h e expression is given in terms of the rms velocity and the fractional penetration of particles into the fluid b o u n d a r y layer. T h e inertial d e p e n d e n c e of the particle velocity is given as a function of the

2.4

53

Deposition in a Turbulent Fluid Medium

particle r e s p o n s e to turbulent velocity fluctuation in the neighboring fluid by relating velocity spectral densities of the particle and the fluid in the equation of particle motion. T h e dimensionless deposition velocity is

+

l

+

1/2

= A'q' erfc[ft 0/(2<7') L

+

+

(2.72)

w h e r e q' = (qp)l(qg), θ = 8u*lvgqg, ft = l / r : (qp) is the r m s particle velocity and (qg) is the r m s fluid velocity. T h e t w o arbitrary c o n s t a n t s A' and θ are given as 0.56 and 6.25 b a s e d on the Liu and Agarwal data. + Deviations from the linear theory given by E q . (2.72) are expected to be significant with large τ , w h e r e the linear form of the equation of particle + particle Reynolds n u m b e r influences. F o r small motion will +b e p e r t u r b e d by values of τ lower than τ ~ 3 deposition from inertial effects d e c r e a s e s rapidly, Brownian diffusion b e c o m e s important, and the R e e k s and S k y r m e model again b r e a k s d o w n . + a p p e a r war­ Although refinements in the theory of turbulent deposition ranted to include o t h e r p r o c e s s e s over a broad range in r , there is little doubt that the picture of inertial deposition for uncharged particles gives a reasonable model of the p r o c e s s . T h e s e c o m p a r i s o n s show that while our present understanding of the tur­ + bulent deposition p r o c e s s is imperfect, and there is need for a unified theory covering a b r o a d e r range of r values, our notion of the basic turbulent deposition p r o c e s s in the a b s e n c e of electrical forces is essentially correct. In particular, t h e r e is little doubt that the observed p h e n o m e n o n is essen­ tially an inertial effect resulting from the finite m a s s of the particles and the action of fluid turbulence. T h e r e are m a n y situations in which turbulent deposition on rough surfaces is of interest. F o r equivalent friction velocity, surface roughness should increase the deposition rate c o m p a r e d with a smooth surface. Deposition velocities for grass surfaces w e r e m e a s u r e d several years ago by Chamber­ lain (1966, 1967). A compilation of available data and theoretical evidence has been r e p o r t e d by Sehmel and H o d g s o n (1974). A n example of their results is shown in Fig. 2.11. C u r v e s for different particle size, density, and surface roughness are s h o w n for c o m p a1r i s o n . Surface roughness is mea­ sured according to the roughness length z 0. T h e c u r v e s take into a c c o u n t Chamberlain's results. S h o w n in solid lines in Fig. 2.11 is the deposition velocity d e p e n d e n t on gravitational action alone. The effect of surface roughness is t o dramatically change the deposition velocity for particles in the 0.1-1.0 μιτι diameter range. With surface roughness larger than 3 - 1 0 c m , the results suggest that the deposition velocity a p p r o a c h e s a nearly constant value n e a r 1 c m / s e c with little variation with respect to particle size. This * The roughness length or height for a surface is proportional to the height of roughness elements such as sand grain diameter or grass blade length. The proportionality is determined experimentally by scaling a velocity profile (Schlichting, 1960).

54

2

PARTICLE DIAMETER

The Dynamics of Small Particles

( m)

M

Fig. 2 . 1 1 . Curves of deposition velocity from Sehmel and Hodgson (1974) for dry deposition of particles on various solid surfaces. The data e n c o m p a s s e d roughness heights only up to about 0.1 c m , and therefore the extrapolation to greater roughness heights is tentative. The deposition 1 velocity plotted is the flux divided by the extrapolated concentration at 1 m; q is the gravita­ 1 3 p ; the case with w* = G30 cm s e c is tional settling speed for particles of indicated density p s h o w n (average air speed — 10 m s e c ) . p ( g / c m ) : , 1.0; , 4.0; —, 11.5.

p

result is subject to considerable c o n t r o v e r s y since n o data are available t o verify such a conclusion. Recently, B r o w n e (1974) has e x t e n d e d D a v i e s ' theory to deposition on rough surfaces without gravitational settling. B r o w n e ' s estimates are strongly surface r o u g h n e s s and R e y n o l d s n u m b e r d e p e n d e n t . H o w e v e r , for the case of large r o u g h n e s s heights, which are qualitatively equivalent to Chamberlain's and S e h m e l ' s e x p e r i m e n t s , a broad range of nearly constant values of deposition velocity approximately 0.1 c m / s e c are found for parti­ cles b e t w e e n 0.08 and 2 μπι diameter for R e = 10,000. B r o w n e ' s calculations suggest that o n e should expect a limit m o r e like 1 c m / s e c for the deposition velocity over rough surfaces with very large R e , as indicated in Fig. 2.11. The available experimental and theoretical results for turbulent deposition

55

References

on rough surfaces are limited. M o r e w o r k in this area is w a r r a n t e d since this aspect of aerosol d y n a m i c s has widespread practical implications, especially for deposition from the e a r t h ' s a t m o s p h e r e .

REFERENCES Acrivos, Α . , and Taylor, T. (1962). Phys. Fluids 5, 387. Bakanov, S., and Derjaguin, Β . V . (1960). Discuss. Faraday Soc. 30, 130. Beal, S. K. (1970). Nucl. Sci. Eng. 40, 1. Brock, J. R. (1962). J. Colloid. Sci. 15, 546. Brock, J. R. (1967). J. Colloid Interface Sci. 2 3 , 448. Browne, L. W. B . (1974). Atmos. Environ. 8, 801. Brun, R., L e w i s , W . , Perkins, P., and Serafini, J. (1955). Natl. Advis. Comm. Aeronaut., Rep. N o . 1215, pp. 1 3 9 - 1 8 3 . Carslaw, H . , and Jaeger, S. (1959). "Conduction of Heat in S o l i d s , " 2nd ed. Oxford, L o n d o n . Chamberlain, A . C. (1966, 1967). Proc. R. Soc. London, Ser. A 2 9 0 A , 236; also 2 9 6 A , 45. Crank, J. (1959). "Mathematics of Diffusion." Oxford, London. Davies, C. N . (1966). " A e r o s o l S c i e n c e " (C. N . D a v i e s , ed.), p. 408. Academic Press, N e w York. Davies, C. N . (1978). In "Fundamentals of Aerosol S c i e n c e , " p. 135. Wiley (Interscience), N e w York. Davis, E . J., Ravindran, P., and Ray, A . (1980). Chem. Eng. Commun. 5, 251. Davis, E . J., Ray, A . K., and Chang, R. (1978). AIChE Symp. Ser. 74, 190-203. Dorsch, R., Saper, P., and K a d o w , C. (1955). Natl. Advis. Comm. Aeronaut., Rep. N o . 3587, pp. 1-29. Einstein, A . (1905). "Investigations on the Theory of Brownian M o t i o n . " Dover, N e w York (republished 1956). Einstein, A . (1924). Z. Phys. 27, 1. Epstein, P. (1929). Z. Phys. 54, 537. Friedlander, S. K. (1977). " S m o k e , Dust and H a z e . " Wiley (Interscience), N e w York. Friedlander, S. K., and Johnstone, H. (1957). Ind. Eng. Chem. 49, 1151. Fuchs, N . A . (1959). "Evaporation and Droplet Growth in G a s e o u s M e d i a . " Pergamon, Ox­ ford. Fuchs, N . A . (1964). " M e c h a n i c s of A e r o s o l s . " Pergamon, Oxford. Fuchs, Ν . Α . , and Sutugin, A . G. (1971). Int. Rev. Aerosol Phys. Chem. No. 2, 1971 pp. 1-60. Hidy, G. M . , and Brock, J. R. (1970). "The Dynamics of Aerocolloidal S y s t e m s , " Chaps. 3 - 6 . Pergamon, Oxford. Hidy, G. M., and Heisler, S. L. (1978). Recent Dev. Aerosol Sci., Symp. Aerosol Sci. Technol., 1976 p. 142. Hochrainer, D . , Hidy, G. M . , and Zebel, G. (1969). J. Colloid Interface Sci. 30, 553. Kinzer, G., and Gunn, R. (1951). / . Meteorol. 8 1 , 71. Kotrappa, P. (1972). Assess. Airborne Part., Proc. Rochester Int. Conf. Environ. Toxic, 3rd, 1970 pp. 3 3 1 - 3 6 0 . Kraemer, H . , and Johnstone, H. (1955). Ind. Eng. Chem. 47, 2726. Langmuir, I. (1948). / . Meteorol. 5, 175. Lerman, A . (1979). "Geochemical P r o c e s s e s — W a t e r and Sediment E n v i r o n m e n t s , " p. 262. Wiley (Interscience), N e w York. Levin, L. (1953). Dokl. Akad. Nauk SSSR 9 1 , 1329. Liu, Β. Y. H . , and Agarwal, J. K. (1974). J. Aerosol Sci. 5, 145. Liu, Β. Y. H . , and Ilori, T. A . (1973). ASME Symp. Flow Stud. Air Water Pollut., Atlanta. Loyalka, S. K. (1973). / . Chem. Phys. 58, 354.

56

2

The Dynamics of Small Particles

Maxwell, J. (1890). In "The Scientific Papers of Clerk M a x w e l l " (W. D . N i v e n , ed.). Cam­ bridge Univ. Press, London and N e w York. Millikan, R. (1923). Phys. Rev. 2 1 , 217. Montgomery, T. L . , and Corn, M. (1970). J. Aerosol Sci. 1, 185. Natanson, G. (1957). Dokl Akad. Nauk SSSR 112, 696. Oseen, C. (1927). " N e u r e Methoden und Ergebnisse in der Hydrodynamik." Akademische Verlag, Leipsig, Germany. Paranjpe, MiK. (1936). Proc. Indian Acad. Sci., Sect. A 4A, 423. Paul, B. (1962). ARS J. 3 2 , 1321. Pich, J. (1976). Atmos. Environ. 10, 131. Ranz, W., and Wong, J. (1952). Ind. Eng. Chem. 44, 1371. Reeks, M. H . , and Skyrme, G. (1976). / . Aerosol Sci. 7, 485. Schlichting, H. (1960). "Boundary Layer T h e o r y , " 4th ed. McGraw-Hill, N e w York. Schmitt, K. (1959). Z. Naturforsch. A 1 4 A , 870. Sehmel, G. A. (1971). J. Colloid Interface Sci. 37, 891. Sehmel, G. A . (1973). J. Aerosol Sci. 4, 145. Sehmel, G., and H o d g s o n , W. (1974). "Atmospheric-Surface Exchange of Particle and Gaseous Pollutants" (R. Englemann and G. Sehmel, eds.). U . S . Atomic Energy S y m p . Ser. N o . 38. U . S . Govt. Printing Office, Washington, D . C . Sitarski, M., and N o w a k o w s k i , B. (1979). / . Colloid Interface Sci. 72, 113. Stôber, W. (1972). Assess. Airborne Part., Proc. Rochester Int. Conf. Environ. Toxic, 3rd, 1970 pp. 2 4 9 - 2 8 9 . Stokes, G. G. (1851). Trans. Cambridge Philos. Soc. 9, 8 - 1 0 6 . Thomas, J. (1958). J. Air Pollut. Control. Assoc. 8, 32. Tyndall, J. (1870). Proc. R. Inst. G. B. 6, 3. Wachmann, H. (1962). ARS J. 32, 2. Waldmann, L. (1959). Z. Naturforsch. A 14A, 589. Waldmann, L . , and Schmitt, K. (1967). In "Aerosol S c i e n c e " (C. N . Davies, ed.), p. 137. Academic Press, N e w York. Walton, W . , and W o o l c o c k , A . (1960). "Aerodynamic Capture of Particles." Pergamon, Ox­ ford. Watson, H. H. (1936). Trans. Faraday Soc. 32, 1037. Zebel, G. (1968). J. Colloid Interface Sci. 27, 294.