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paper to be presented at the International Symposium on Surface Contamination,
Pune 8-12, 1964, Gatlinburg, Tennessee.
117195720
ORAL-P- 4.
1
CONF-555-4 MASTER
-
DEPOSITION OF SUBMICRON-SIZE
PARTICLES IN VENT ILATION DUCTS*
.
st
L. P. Davis**
.
Oak Ridge National Laboratory
Oak Ridge, Tennessee
.
017
4

Focsimile Price 3
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* Research sponsored by the U.S. Atomic Energy Commission under contract
with the Union Carbide Corporation.
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**Currently attending University of Tennessee, Knoxville, Tennessee.
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CONTENTS
Page
Abstract. .............................
Introduction. ................
Theoretical Concepts. .......
Transport to Duct Walls in Tiroulent Flow
The Colburn Ana.
10 y . . . . . . . . . . . . . . . . . . .
Effects of Agglomeration. . . . . . . . . . . . . . . . . .
.
.
...
.
Method of Evaluation. ..........
.
Discussion of Results ..........
.
Conclusions ............
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Appendix. ............................ 16
. .
DEPOSITION OF SUBMICRON-SVE
PARTICLES IN VENTILATION DUCTS
L. P. Davis
ABSTRACT
A
Oi2nt;
The purpose of this study was to investigate mathematically the
concentration decrease due to particle deposition phenomena in highly
concentrated monociispersed aerosols (mean particle size less than 1.0
u ) flowing through ventilation ducts. It was found that, from the
standpoint of removal, the decrease in concentration due to deposition
on duct walls was insignificant; out, when considering contamination
on the duct walls, the amount deposited, even though small when compared
with the amount in the bulk stream, should not be overlooked.
The rate of de position, characterized by the deposition velocity,
was calculated by using the Colburn analogy between momentum and mass
transfer. Because of the significant coagulation encountered in the
cases investigated, the mean particle size was found to be a strong
function of the residence time in the duct. This necessitated that
the deposition velocity be expressed as a function of residence time
rather than as a function of particle diameter (as is the usual case).
The mean diameter was expressed as a function of time and concentration
combined with a mass balance over a differential volume element, and
the results were integrated.
The integrated equation gives a conservative estimation of the
decrease in concentration to be expected when highly concentrated,
unstable, submicron size aerosols are transported through ventilation
ducts.
INTRODUCTION
ta
much of the work done to date on deposition velocities of micron
and submicron sized particles has been in the area of stable aerosols
where the effects of agglomeration are not significant. Also there is
an almost complete lack or information in the literature concerning
deposition velocities in the size range of 0.1 to 1 micron. The purpose
of this report is to investigate the deposition behavior of particles of
size 0.001 to 1.0 micron with concentration ranging from 1 to 1000 mg/m3
with consideration given to the effects of agglomeration. Special atten.
tion is given to the deposiiion of particles in a typical ORNL cell venti-
lation duct. This is a 54-in.-square concrete duct, approximately 1000
feet long, that vents HRLEL, FPDL,. and several other plants to a fan below
the 3039 stack. Flow in the line varies from 10,000 to 50,000 cfm.
TICORETICAL CONCEPTS
Transport to Duct Walls in Turbulent Flow
The mechanisms governing the deposition of particles from a turbulent
ai: stream can be divided into two modes depending on the size of the
particles being deposited. For particles of mean diameter greater than
1.0 micron the mode of transport is one in which the particles, by virtue
of the transverse eddy velocities, are carried across the boundary layer
in a projectile fashion. Various theories have been proposed to predict
the deposition velocities in this range. Notable of these is the theory
proposed by Friedlander and Johnstone. - For particles of mean diameter
less than 0.1 micron, Brownian motion provides a possible mechanism and
use is made of the Chilton-Colburn analogy between mass and momentum
transfer.
Chamberlain' reports that for particles greater than 0.1 micron the
transfer rate calculated from the Colburn analogy becomes very small and
that the problem is solved by application of the theory by Friedlander
and Johnstone, but recommends that it not be applied to particles as small
as 1.0 micron. This places the deposition mechanism for the size range
0.1 to 1.0 micron in a somewhat nebulous situation. For smoke concentra-
tions or 10 to 1000 mg/m the effect of coagulation causes this diameter
range to be important even for particle diameters that are much smaller
(see Fig. 6). The lack of available experimental data necessitates that
some approximation be employed. The Colburn analogy, utilizing the theory
of Brownian diffusion, was extended into this region of uncertainty. This
approximation is thought to be conservative.
w
.
3
The Colburn Analogy
2.
The relationship between mass and momentum transfer can be stated in
tac concise form
8/2 = dpi
Where:
f = Fanning friction factor, dimensionless,
Jo = Colburn factor for mass transfer, dimensionless.
If the dimensionless quantity J, is rewritten, according to its
definition, as
Jo -
NSO
where:
k = mass flux rate, mass/(time) (unit area),
c = concentration, mass/unit volume,
V = free stream velocity, length/time,
Ns. = Schmidt number (ratio of kinematic viscosity to diffusivity),
dimensionless;
it is seen that upon equating Eq. (2) to f/2 that
V = (f/2) v N52 3.
Where
v. - deposition velocity (ratio of k to c), length/time.
Effects of Agglomeration
Agglomeration is characterized by a coefficient which satisfies the
following relationship:
h - = Kt,
where:
n = initial number of particles per unit volume, cm ,
n = number of particles per unit volume at any time t, cm ,
K = acglomeration coefficient, cmº/sec,
t = time, sec.
Expressing the number of particles in terms of average diameter, density,
and mass concentration yields an expression for the time required to reach
a given diameter, a, as follows:S
Told3-23)
t = 7 m/V, K
.
a
-
.
....
where:
t = time, sec,
0 = density of particles, G/cm,
= diameter of particles at time t, cm,
- diameter of particles initially, cm,
m/v, = mass concentration of particles, g/cm",
K - agglomeration coefficient, cm/sec.
In order to use the aforementioned equations to predict the behavior of
aerosols it is necessary to employ some theoretical expression for the
agglomeration coefficients such as:
.
.
.
-
.
..
..
.
K (2 + AŞ,
where:
k = Boltzmann constant,
T = absolute temperature,
n = viscosity of medium,
À = nean free path of the gas molecules,
r = radius of tne particles,
A = Ao + Be C'r(see Ref. 4),
A. = a constant (w1.25),
B = a constant (-0.44),
C' = a constant (al.09).
From the above equation it can be seen that the agglomeration
coefficient is not a constant, but is dependent on particle diameter.
METHOD OF EVALUATION
The attenuation to be expected due to the deposition phenomenon can
be determined from a material balance written over a differential length
of the ventilation duct. The details of this balance can be seen in
Fig. 3. The result is the equation given below:
4LV
c/Co = exp (- BV),
(7)
where:
C = concentration at a point Lom from the duct entrance, g/cm,
Co = initial concentration at duct entrance, g/cm,
L - distance from duct entrance, cm,
v. - deposition velocity, cm/sec,
V = i'rce strcam velocity, cmscc,
B = dimension of one side of duct, cm.
The above form of the equation should predict the ratio of final to initial
concentration (within the limits of accuracy of the value of the predicted
deposition velocity) for stable acroso.lii.
For instable aerosols the nican dianeter of the particles is continu-
ously increasing. This situation adds to the complexity of the problem.
It will be recalled that the deposition velocity assumed to be applicable
to the range of particle diameters being investigated here is a function
of the diffusivity. However, from Fig. 2 it can be seen that the diffusi-
vity is a rather strong function of the particle diameter; thus, making
the deposition velocity diarre ter dependent. Curves showing this depen-
dence are given in Fig. 4 where the deposition velocity as a function
of particle diameter with free stream velocity as a parameter are piotted.
In order to relate the particle size to downstream position in the duct
use will be made of the agelomeration coefficient which relates particle
size to tire. Siowever, upon examining Eqs. (5) and (6), it is seen that
the expression for the agglomeration coefficient is coupled with the
expression for the particie diameter. This being the case, Eq. (5) cannot
be solved explicitly for diameter for increments of time such that there
is a significant change in particle diameter.
To overcome this the following steps were taken:
(1) A plot of particle diameter vs time was constructed as follows:
(a) Equation (5) was rearranged to give
(5a)
-
--
•-
-
*.**--
* *
-
.
-
-
<..
-
-
-
.
-
-
-
-
-
----
."
.
.
.
.
.
.
..
.
d/d = 1 + 6tKc 1/3
door
(b) Using a time increment or 0.1 sec, a trial-and-error
particle diameter that would give a diameter increase of no
more than 10 per cent. This time increment can be considered
TO
nocligibly small as compared to the total residence time in the
duct; ncnce, it may be concluded that any part 'clc dianter 1063
than this is so short-lived as to be of negligible importance.
(c) Using the above dctcrmined diameter as the initial particle
size the diameter was increased by 10 per cent and the time
required to achieve this growth was determined.
(a) Next, this diameter was incremented by 10 per cent, the
acglomeration coefficient adjusted for the new diameter, and
the new time increment was calculated. Proceeding in this manner
will yield the mean diameter for any time for not only an initial
particle size as determincd from part (b), but also for all
omallor initial partiolo dlamo toru (sino. 0.d so « rooidonc.
time).
(e) The above procedure was repeated for various concentrations
and the results plotted as shown in Fig. 6.
(2) Equations were written for the curves shown in Figs. 4 and 6 as
follows:
(a) For V, as a function of particle diameter with velocity as
a parameter
V = [H + w v7a.
(b) For d as a function of time with concentration as a parameter
d = ECPta,
where:
E, H, W, Q B, and y are constants.
(3) Combining the above two equations yields an expression for the
deposition velocity as a function of concentration, free stream velocity
and position in the duct (since L/V = t) as given below,
va = [H + w 17e%c9() Za
(10)
This expression for V, was used in the differential form of Eq. (7) and
the results integrated as shown in the Appendix. The result is the
rather formidable equation
B 4 (H + WJE” 1,2a + 1
(11)
1-Yayl + ja
(8)
(9)
YA
(C/C
DISCUSSIOOI OF RESULTS
The final result or tnis investigation ir Eq. (11) given above. Wạen
the various constants are inserted into the cquation it becomes
(c/c 0.276 1.37.1 1.37 x 10 + 7.93 x 1004 y,0.735
c0.276 70.735
? (12)*
where:
C, C = final, initial concentration, g/cm>,
V = free stream velocity, cm/sec,
L = duct length, cm.
It should be noted that the constants 37.1 and 0.735 will vary slightly
since a varies from 0.25 to 0.26 for concentrations or 1 mg/mol to 1000
me/m”, respectively. See Fig. 6.
Perdus a few explanatory comncnts are in order concerning the third
restriction. The curves in Fig. 6 were determined using the assumption
that for a given concentration all particlc diameters below a certain value
were so short-lived that their ci'fcct on the final result would be insigni-
ficant. Since Eq. (5) is dependent on concentration the limiting initial
diameter is also dependent on concentration - the nigher the concentration
the higher the allowed initial diameter. Therefore an initial diameter
that satisfies the imposed conditions (all smaller diame ters exist for a
very brief time) for a concentration C will also satisfy these same con-
ditions for a concentration ca, provided that CzC7, Hence, the notation
à 52 x 1024 for c? 1 mg/m3 imzlies that the equation representing the
curves in Fig. 6 is also valid for, say, C = 100 mg/m and do = 2 x 100CM.
Also implied in the notation is that the equation would not give the
dependence of diameter on residence time for, say, C = 1 mg/m and d. =
7 x 10°C. This is clearly seen upon examination of Fig. 6 where it is
evident that for C = 1 mg/mad the particle diameter has not reached 7 x
100% u even at the end of 100 sec. The values of initial diameter given
by restriction 3 above are not the initial values determined by the
method described in the section on Theoretical Concepts. Instead, the
values given are approximately the diameter associated with t = l sec.
It is thought that this extension of the permissible range of initial dia-
meters does not introduce appreciable error if the fourth restriction is
applied with discretion.
---
-
For cascs arising that co not satisfy the third restriction a new
curve would have to be determined starting with the given value or do
and applying stops 1(c) and 1(1) or the section on Theoretical Concepts.
. For situations which do satisfy the above rour restraints, Eq. (12)
can be applicd directly. Consider the case of a monodispersed aerosol
of concentration 1 mg/m2 and initial particle size 0.02 u moving at an
average velocity of 10 ft/sec in a 54-inch square duct, 1000 ft long.
Upon using Eq. (12) to determine the decrease in concentration, 1t 18
found that c/c - 0.9954. From this it can be seen that the predicted
decrease in concentration is neclicible.
CONCLUSIONS
From the proceeding development the following can be concluded:
(1) The method described and the resulting equation provide a
plausible means or predicting the decrease in concentration due to
deposition.
(2) The procedure used involves numerous approximations (e.g.,
curve fitting) aná the results obtained should be viewed in light of
this fact.
(3) The predicted decrease in concentracion due to deposition in
the duct for the diame ters and concentrations investigated can be con-
sidered negligible; however, from the standpoint of contamination on
the duct walls the amount deposited should not be overlooked.
-9
UNCLASSIFIED
ORNL-DWG 63-7289

- -
-
.
.
.
."..
DIFFUSIVE TRANSPORT FORMULA (Colburn)
VELOCITY OF DE POSITION (cm/sec)
INERTIAL TRANSPORT FORMULA
(Friedlander and Johnstone)
0.001
0.001
1.0
10.0
0.01 0.1
PARTICLE DIAMETER W) il
Fig. 1. DE POSITION OF PARTICLES ON VERTICAL PLANE SURFACES
88%
LMT
.
****
**
*****
VMTRAWW
..
.
!
UNCLASSIFIED)
ORNL-DWG 63-1928

PARTICLE DIAMETER (w)
10-31
10->
10-6
10-5
10-4
10-3
10-2
DIFFUSIVITY (cm/sec)
Pin ¿
DIFFUSIVITIES OF PARTICULATES AS A FUNCTION OF PARTICLE DIAMETER
-11-
UNCLASSIFIED
ORNL-DWG 63-4929
C + DC

Fodb
INPUT RATE = B2vC
where:
B = duct dimension, cm
V = average free stream velocity, cm/sec
C : concentration, particles/cms or a/cm3
OUTPUT RATE = 82v(c + dC)
ACCUMULATION
where:
L = distance from duct entrance cm
Vg = deposition velocity, cm/sec !
MATERIAL BALANCE FOR THE ELEMENTAL VOLUME YIELDS
BPCV = 82V(C + dC) + 4B/C + ) Vodl
INTEGRATING FROM C = C. AT L = 0 AND C = CATLEL
GIVES:
: c/c. = exp-ngL
..
Fig. 3. Equation for Particle Deposition in a Duct (Stable Aerosol).
.......
..
:
.
a
.
,
UNCLASSIFIED
ORNL-DWG 64-3798

V = 40 ft/sec
ŏ
Å
o
Vg. DE POSITION VELOCITY (cni/sec)
-
---
...::
::.!
10-4 _
Vo (1.37 x 10-9 + 7.93 x 10-12v] 01.06%
Vg - cm/sec
V - cm/sec
d - cm
VALID FOR d > ~2 x 10-2 MICRONS
10-5
10-2
10-1
PARTICLE DIAMETER (microns).
rii .. DE POSITION VELOCITY IN 54 SQUARE-INCH DUCTS
..
Y23
-
/3
-
:-.
-
UNCLASSIFIED
ORNL-DWG 63-4931
.

-.--
--
--
--
--
-
108
AGGLOMERATIO:: COCITICIO:VT (cm, sic)
10-10
10-3
10-2
10-1
PARTICLE DIAMETER ()
L
AGGLOMERATION COEFFICIENTS AS A FUNCTION OF PARTICLE DIAMETER
9LOMERATION CO
UNCLASSIFIED
ORNL-DVIG 61.-3989



· -....
......
..
.
-. .-..-.....
1000 mg/m3
...
.........
......
.
...
.
100 mg/m3
10 mg/m3
1 mg/m3
PARTICLE DIAMETER (microris)
.
.
:............
.....
......
...
.
.
...
.
a mentos adicio .
honest
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thing.
10-2
---
........
|
d = 4.67 x 10-4 C.26 ja
0
d - cm
a = .25 aic=1
0
C - g/cm3
= .26 al = 103
APPLICABLE FOR THE FOLLOWING RANGE
do < 2 x 10-2 v ic> 1 mg/m3
da <3.8 x 10-2vic > 10 mg/m3
de < 7.0 x 10-2 pic > 100 mg/m3
do < 1.3 x 10-10;c > 1000 mg/m3
Too-
100
0
1- sec
--------0
-------- 7.0
TIME (seconds)
Fig.do Particle size as a function of Time cind concentration.
15
REPERUNCES
1. S. R. Friedlander and 1. F. Johnstone, "Deposition of Suspended Parti-
cles from Turbulent Gas Streams," Ind. and Eng. Chem. 1), 1151 (1957).
2. A. C. Chamberlain, "Transport or Particles Across Boundary Layers,"
NERE-M 1122 (1962).
3. W. E. Browning, Jr., and M. H. Fontana, unpublished report (1963).
4. J. W. Thomas, "The Dirrusion Battery Method for Aerosol Particle size
Determination," CRNL-1648 (Dec. 14, 1953).
ló
AP?DIDIX
:
5.
Development or Equation to predict (c/c.)
with Acglomeration considered
.
- .
- ...
From the differential mass bulance dctcrmined in Fig. 3, 1t 18 seca
-
.
that
.
.
.-
,:>
.
-
(B/4) v dc + a(C + dc/2) V = 0.
(7.1)
Using the expression for v, given in Eq. (10), Eq. (7.1) becomes
(B/4) v dc + C AL(H + W V)E%C?(L/V)20 0. (7.2)
Separation of variables yields
ac
+ WE' 770 al.
(7.3)
YB 3v1 + ja
Integrating the left hand side from c. to C and the right hand side from
O to L yields
c1 + yß - -
) 78 = 1 + 2 I
14(11 + WB 7
Ic-yBI
Ila + 1
O OV
(7.4)
Where:
B = duct dimension, cm,
C = concentration L cm from entrance, gm/cm,
C. - initial concentration,
E = constant (4.67 x 10-4),
= constant (1.37 x 109),
i - distance from duct entrance, cm,
V = free stream velocity, cm/sec,
= constant (7.93 x 10^12),
a = constant (0.25 to 0.26),
3 = constant (0.26),
= constant (-1.06).

END
.