Radon and Its Decay Products - American Chemical Society

Chapter 12

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Effects of Aerosol Polydispersity on Theoretical Calculations of Unattached Fractions of R a d o n Progeny Frank Bandi, Atika Khan, and Colin R. Phillips Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario M5S 1A4, Canada Theoretical calculations of unattached fractions of radon progeny require prediction of an attachment coefficient. Average attachment coefficients for aerosols of various count median diameters, CMD, and geometric standard deviations, σ , are calculated using four different theories. These theories are: (1) the kinetic theory, (2) the diffusion theory, (3) the hybrid theory and (4) the kinetic-diffusion theory. Comparisons of the various calculated attach ment coefficients are made and the implications of using either the kinetic or the diffusion theory to calculate unattached fractions for aerosols of various CMD and σ are discussed. Significant errors may arise in use of either the kinetic theory or the dif fusion theory. Large and unacceptable errors arise in calculating unattached fractions of a polydisperse aerosol by characterizing the aerosol as monodisperse. Unattached fractions of RaA are calculated for two mine aerosols and a room aerosol. g

g

Theoretical calculations of unattached fractions of radon or thoron progeny involve four important parameters, namely, 1) the count median diameter of the aerosol, 2) the geometric standard deviation of the particle size distribution, 3) the aerosol concentration, and 4) the age of the air. All of these parameters have a signifi cant effect on the theoretical calculation of the unattached frac tion and should be reported with theoretical or experimental values of the unattached fraction. The two fundamental theories for calculating the attachment coefficient, 3, are the diffusion theory for large particles and the kinetic theory for small particles. The diffusion theory pre dicts an attachment coefficient proportional to the diameter of the aerosol particle whereas the kinetic theory predicts an attachment coefficient proportional to the aerosol surface area. The theory 0097-6156/87/0331-0137$06.00/0 © 1987 American Chemical Society

Hopke; Radon and Its Decay Products ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

RADON AND ITS DECAY PRODUCTS

138

which i s usually adopted i s the k i n e t i c theory. However, when the diameter of the aerosol p a r t i c l e , d, i s of the order of the mean free path, £, of gas molecules (Knudsen number, i/d * 1), neither the d i f f u s i o n theory nor the k i n e t i c theory i s correct i n i t s assumptions. A hybrid theory has been used i n this region and may be applied over the whole p a r t i c l e - s i z e spectrum, since i t converges to the k i n e t i c theory for small p a r t i c l e sizes and to the d i f f u s i o n theory for large p a r t i c l e sizes. _ In this study, values of the average attachment c o e f f i c i e n t , 3, (averaged over the entire p a r t i c l e size spectrum) are calculated using a combination of the above theories. Unattached fractions are then calculated for uranium mine and indoor a i r p a r t i c l e - s i z e distributions and aerosol concentrations. C l a s s i c a l Diffusion Theory The o r i g i n a l theory of d i f f u s i o n a l coagulation of spherical aerosol p a r t i c l e s was developed by von Smoluchowski (1916,1917). The underlying hypothesis i n this theory i s that every aerosol p a r t i c l e acts as a sink for the d i f f u s i n g species. The concentration of the d i f f u s i n g species at the surface of the aerosol p a r t i c l e i s assumed to be zero. At some distance away, the concentration i s the bulk concentration. Solution of this diffusion problem, assuming quasi-steady state coagulation, leads to an attachment coefficient of the form: 3(d) d D

=

=

2*dD

(1)

diameter of aerosol p a r t i c l e d i f f u s i o n coefficient of the diffusing species.

In addition to the quasi-steady state assumption, the other assumptions required to arrive at equation (1) are: 1. the aerosol i t s e l f does not coagulate; 2. there i s a f u l l y developed concentration gradient around each aerosol p a r t i c l e ; and 3. the concentration of unattached radon progeny atoms i s much greater than the concentration of aerosol particles ( i n order that concentration gradients of radon progeny atoms may e x i s t ) . This last assumption i s usually not v a l i d since the radon progeny concentration i s usually much less than the aerosol concentration. From equation (1) i t can be seen that application of the d i f fusion theory leads to the conclusion that the rate of attachment of radon progeny atoms to aerosol particles i s d i r e c t l y proport i o n a l to the diameter of the aerosol p a r t i c l e s . Kinetic Theory The k i n e t i c theory of radon progeny attachment to aerosol particles assumes that unattached atoms and aerosol particles undergo random c o l l i s i o n s with the gas molecules and with each other. The attachment c o e f f i c i e n t , 3(d), i s proportional to the mean relative veloc i t i e s between progeny atoms and particles and to the c o l l i s i o n cross section (Raabe, 1968a):


12.

BANDI ET AL.

Theoretical Calculations of Unattached Fractions

3(d)

2

ir(d /2 + à/2)

=

r

/v

2

+ v

2

139

(2)

p

The attachment coefficient i s a function of the aerosol part i c l e diameter, d, and mean velocity, ν , as well as the unattached progeny diameter, d , and i t s mean velocity v. Since i n most situations d « d and ν » ν , equation (2) reduces to r Ρ 3(d)

=

(3)

ïïd^M

where the radon progeny mean velocity can be calculated from 8kT^

1/2

where Τ i s the absolute temperature, k i s Boltzmann's constant and m i s the mass of the unattached radon progeny. From equation (3) i s i s clear that the k i n e t i c theory predicts an attachment rate of radon daughters to aerosol particles propor t i o n a l to the square of the diameter of the aerosol p a r t i c l e . Hybrid Theory When the radius of an aerosol p a r t i c l e , r, i s of the order of the mean free path, £, of gas molecules, neither the d i f f u s i o n nor the k i n e t i c theory can be considered to be s t r i c t l y v a l i d . Arendt and Kallman (1926), Lassen and Rau (1960) and Fuchs (1964) have derived attachment theories for the t r a n s i t i o n region, r » i which, for very small p a r t i c l e s , reduce to the gas k i n e t i c theory, and, for large p a r t i c l e s , reduce to the c l a s s i c a l d i f f u s i o n theory. The underlying assumptions of the hybrid theories are summarized by Van Pelt (1971) as follows: 1. the d i f f u s i o n theory applies to the transport of unattached radon progeny across an imaginary sphere of radius r + I centred on the aerosol p a r t i c l e ; and 2. k i n e t i c theory predicts the attachment of radon progeny to the p a r t i c l e based on a uniform concentration of radon atoms corresponding to the con centration at a radius of r + λ. The attachment c o e f f i c i e n t , 3, corresponding to the hybrid theory can be shown to be (Fuchs, 1964) 9

ri + r \/D 1—2— 2

3(rx,r ) 2

=

16

1

+

D

2

C

(4)

where the subscripts 1,2 represent the radon progeny and the aero sol particles respectively and r D

X

= radius of species χ (χ = 1,2) = d i f f u s i o n coefficient of species χ (x = 1,2)

χ C =

r r + ôr/2

, +

a correction factor, where

4P G

r

r


140


r i + r2 s ,

_

Dl + D2

D -

ô

Ô

/-= >

G

-

r

'G? +

,

- /δ? + δ£

r

The thickness, δ , of the boundary layer corresponding to the spe cies x, the mean velocity of x, Gx, and the d i f f u s i o n c o e f f i c i e n t , D , can be calculated from the equations: χ

x

D χ where

= kTB χ

Β , the mobility, i s defined as: Β = x

( l + M. + 91 p [ - b r /Λ])/6πητ r r χ χ x x e X

r

A, Q and b are the constants 1.246, 0.42 and 0.87, respec t i v e l y (Fuchs, 1964) η = v i s c o s i t y of the medium i = mean free path of the gas molecules

/ V/2 8kT

G = χ

) \*mj

where m

i s the mass of the species χ

δ = - — L _ {(2r + I ) χ 6r A χ χ χ χ z

1

7

3

2

1

1

- ( 4 r + l )*' ) χ x 7

J

- 2r χ

where £ , the apparent free path of p a r t i c l e s , i s calculated from χ

I = G mΒ χ χ χ χ 1

Fuchs attachment various assumptions:

coefficient can be simplified by making

1) the velocity of radon progeny i s much greater than that of the aerosol p a r t i c l e s . (G\ » G2 so that * G χ = ν) 2) the radius of the aerosol p a r t i c l e s i s much greater than that of the radon progeny. (τ2 » *ι so that r « Γ2/2 = d/4) 3) the d i f f u s i o n c o e f f i c i e n t of radon progeny i s much greater than that of the aerosol p a r t i c l e s . (Di » D2 so that D « Di/2 = D/2) 4) δ , the thickness of the boundary layer i s approximately equal to the mean free path, £, of gas molecules i n a i r . χ

Using the above assumptions, equation (4) can be written as a func tion of aerosol diameter:


12.

BANDI ET AL.

141

Theoretical Calculations of Unattached Fractions 2

wd D(d/2 + I)

3(d) =

(5)

Zf - + 4D(d/2 + « Kinetic-Diffusion

Approximation

For small p a r t i c l e sizes the k i n e t i c theory i s applicable, whereas for large p a r t i c l e sizes the d i f f u s i o n theory applies. A useful approximation i s therefore to use the kinetic theory i n the small p a r t i c l e range and the d i f f u s i o n theory i n the large size region. The attachment coefficient then takes the form: 3(d) 3(d)

for 0 < d < d i for d i < d
a s Equation (20) i s v a l i d for t (the age of a i r ) > 20 min. Equation (20) can be s i m p l i f i e d further by assuming that \f « λ (Tj^ ,^ = 3.05 min, λ = 0.00379 s ) and therefore K * λ . This assumption i s reasonable since Pogorski and P h i l l i p s (1984) report \j for radon progeny i n a chamber as 0.00008 s and Porstendorfer (1984) estimates \j for radon progeny indoors to be between 0.000028 s and 0.000083 s depending on the v e n t i l a t i o n rate. Equation (20) can be rewritten as: 2

l

2

a

2

1

1

1

F

A

=

ΤΓΓΊΓ s

0.3 ym) which attach according to the d i f f u s i o n theory. In contrast, the k i n e t i c theory becomes more inaccurate as the aerosol becomes more polydisperse. The k i n e t i c - d i f f u s i o n approximation predicts an attachment c o e f f i c i e n t similar to the hybrid theory for a l l CMDs and f o r both Og = 2 and 3 (Figs. 3 and 4). The advantage of this theory i s that the average attachment coefficient can be calculated from an analyt i c a l solution; numerical techniques are not required. Figure 5 shows the variation of the hybrid theory with CMD for various Og. I t i s obvious that assuming an aerosol to be monodisperse when i t i s i n fact polydisperse leads to an underestimation of the attachment c o e f f i c i e n t , leading i n turn to large errors i n calculation of theoretical unattached f r a c t i o n . The variation of attachment c o e f f i c i e n t with a for CMD =0.2 ym and 0.3 ym i s shown i n Figures 6 and 7. Again i t i s apparent that the k i n e t i c theory or d i f f u s i o n theory are correct only at certain CMD and Og. Neither i s applicable under a l l circumstances. It i s also evident that the k i n e t i c - d i f f u s i o n theory i s a good approximation to the hybrid theory under a l l circumstances. Unattached fractions of RaA (at t = °°) for two mine aerosols and for a t y p i c a l room aerosol are shown i n Table I I I . It i s usually assumed that the attachment of radon progeny to aerosols of CMD < 0.1 ym follows the k i n e t i c theory. In Table III i t i s apparent that the hybrid and k i n e t i c theories predict similar unattached fractions for monodisperse aerosols. However, for more polydisperse aerosols, the k i n e t i c theory predicts lower unattached fractions than the d i f f u s i o n theory and thus the d i f f u s i o n theory i s the more appropriate theory to use. It i s also evident that the k i n e t i c - d i f f u s i o n approximation predicts unattached fractions simil a r to those predicted by the hybrid theory i n a l l cases. 1

g

g


156


Conclusions Calculation of the attachment coefficient is required for theoreti cal prediction of the unattached fraction of radon progeny. The hybrid theory, which is a form of Fuchs' theory with certain justi fiable assumptions, can be used to describe attachment to aerosols under all conditions of o and CMD. The kinetic theory and the diffusion theory may be used for certain aerosol distributions. However, these two theories begin to deviate from the hybrid theory as the aerosol polydispersity increases. Use of either the kinetic theory or the diffusion theory may therefore result in large errors. Although unattached fractions predicted using the kinetic and diffusion theory for high aerosol concentrations, such as mine atmospheres, are comparable, the same cannot be said for unattached fractions predicted at low aerosol concentrations, such as indoor air. For low aerosol concentrations, neither the kinetic nor the diffusion theory predicts unattached fractions close to those pre dicted by the hybrid theory. Exclusive use of either of these two theories results in large errors. Although the hybrid theory is the most correct theory to use in the prediction of unattached fractions, the error in using the kinetic-diffusion theory in place of the hybrid is small. The kinetic-diffusion theory has the advantage that the solution is in analytical form and thus is more convenient to use than the hybrid theory, which must be solved numerically. Finally, whenever theoretical or experimental unattached frac tions are reported, the aerosol characteristics o , CMD and Ν must also be reported to allow proper interpretation of the results. g

g

Acknowledgment This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Literature Cited Arendt, P. and H. Kallmann, Uber den Mechanismus der Aufladung von Nebelteilchen, Zeitschrift für Physik 35:421 (1926). Busigin, C., A. Busigin and C.R. Phillips, The Chemical Fate of Po in Air, in Proc. Int. Conf. on Radiation Hazards in Mining (M. Gomez, ed) pp. 1043, Soc. of Mining Engineers of Amer. Inst. of Mining, Metallurgical and Petroleum Engineers, Inc., Golden, CO (1982). 218

Fuchs, N.A., The Mechanics of Aerosols, Mcmillan Co., New York (1964) Khan, Α., F. Bandi, C.R. Phillips and P. Duport, Underground Measurements of Aerosol and Radon and Thoron Progeny Activity Distributions, to be published in Proc. 191st American Chemical Society National Meeting, New York, April 13-18 (1986).


12. BANDI ET AL.

157 Theoretical Calculations of Unattached Fractions

Lassen, L. and G. Rau, The Attachment of Radioactive Atoms to Aero sols, Zeitschrift für Physik 160:504 (1960). McLaughlin, J.P., The Attachment of Radon Daughters to Condensation Nuclei, Proc. Royal Irish Academy 72, Sect. Α., 51 (1972). Menon, V.B., P. Kotrappa and D.P. Bhanti, A Study of the Attachment of Thoron Decay Products to Aerosols using an Aerosol Centrifuge, J. Aerosol Sci. 11:87 (1980). Mohnen, V., Investigation of the Attachment of Neutral and Electri cally Charged Emanation Decay Products to Aerosols, AERE Trans. 1106 (Doctoral Thesis) (1967). Pogorski, S. and C.R. Phillips, The Transient Response of Radon and Thoron Chambers, Proc. Int. Conf. Occupational Radiation Safety in Mining, (H. Stocker, ed), vol. 2, 394, Can. Nucl. Assoc. (1985). Porstendörfer, J., Behaviour of Radon Daughter Products in Indoor Air, Radiation Protection Dosimetry 7:1 (1984). Raabe, O.G., The Adsorption of Radon Daughters to Some Polydisperse Submicron Polystyrene Aerosols, Health Phys. 14:397 (1968a). Raabe, O.G., Measurement of the Diffusion Coefficient of RaA, Nature 217:1143 (1968b). Raghunath, B. and P. Kotrappa, Diffusion Coefficients of Decay Pro ducts of Radon and Thoron, J. Aerosol Sci. 10:133 (1979). Smoluchowski, M. von, Versuch einer Matematischen Theorie der Koagulationshinetik Kolloider Losungen, Zeitschrift für Physik, Chemie XCII, 129 (1916). Smoluchowski, M. von, Drei Vortage über Diffusion, Brownsche Molekularbewegung and Koagulation von Kolloidteilchen, Physikalische Zeitschrift 17:557,585 (1917). Thomas, J.W. and LeClaire, P.C., A Study of the Two-Filter Method for Radon-222, Health Phys. 18:113 (1970). Van Pelt, W.R., Attachment of Rn-222 Decay Products to a Natural Aerosol as a Function of Particle Size, Ph.D. Thesis, New York Uni versity (1971). RECEIVED

September 24, 1986


Radon and Its Decay Products - American Chemical Society

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