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A probabalistic theory of aerosol penetration through glass fibre filters
Aerosol Science, 1972, Vol. 3, pp. 461 to 490. Pergamon Press.
Printed in Great Britain.
A PROBABALISTIC THEORY OF AEROSOL PENETRATION
THROUGH GLASS FIBRE FILTERS L o u i s A . CLARENBURG Rijnmond Authority, Stationsplein 2, Schiedam/Tbe Netherlands
(Received 20 April 1972) Ahstraet--A theoretical model is developed capable of predicting aerosol penetration through glass fibre filters. The filter is described as an assembly of pores which conduct the flow. Some of the aerosol particles, the interception fraction, move along with the tortuous flow; the remainder, the inertial fraction, pursue their paths through the filter along straight lines until they are re=entrained in the flow. Oftbe interception fraction of particles, a part is captured as a result of pure interception, a part due to an increase of interception by both inertia and diffusion, and the rest by pure diffusion. From the inertial fraction of particles, part is captured due to pure inertia, part due to inertial interception and part due to inertial diffusion. The final expression for the penetration does not contain any adaption parameter, only directly assessable parameters related to flow-, filter-, fibre- and particle properties. The final relation has been tested experimentally, by varying particle diameter, face velocity, porosity fibre diameter, and filter composition. Theory and experim~t appear to be in good agreement. A suggestion is made for optimal filter design.
C
d~ d*E DaM
f,
gt hi H
J
k K*(i,k) K(i.k) K(m~,)
-i L mh~ mh~
Mh I'lp Hpm
N N* N**
Ap Pen P,
NOMENCLATURE* Cunningham factor mean fibre diameter mean square fibre diameter effective layer thickness in the m e a n flow direction effective layer thickness in thc direction pursued by the inertialfraction of the aerosol particles diameter of an aerosol particle diffusion parameter of an aerosol particle due to Brownian motion inertialfraction of the aerosol particles originating from a flow of hydraulic radius mht fraction of the aerosol particles outside the flow of hydraulic radius m~t due to diffusion fraction of the aerosol particlesoriginating from a flow of radius m~s arriving at a fibrelayer with velocities and directions different from the air flow shadow effect subscript, referring to the j-th fibre component in the filter Boltzmann constant; also index referring to the k-th fibre layer upstream or downstream from the fibre layer under consideration apparent probability that an aerosol particle originating from a flow of radius ms~ is captured in the k-th downstream layer due to inertia K*(i,k), corrected for the probability of re=entrainment probability that aerosol particles originating from a flow of radius mhi are captured due to inertia mean fibre length thickness of the falter hydraulic radius geometric mean hydraulic radius maximum observed hydraulic radius, taken equal to 6 rnhg number of the pores per unit surface area in a single component falter idem, in a multi-component filter number of fibres per unit volume effective number of fibres apparent number of fibres pressure drop across the filter penetration of aerosol particles through a filter probability of an aerosol particle of being captured in the unit layer
*Symbols not l'sted are explained in the text. 461
462
Lou~ A. CLARENBURG
probability of an aerosol particle originating from a flow of radius ma~ being captured due to interception P,,y (m,~) idem, due to diffusion P~n (mh,) idera, due to inertia overall probability of an aerosol particle being captured due to interception Pa int idem, due to diffusion Pa at! idem, due to inertia Pa In probability of re-entrainment of an aerosol particle originating from a flow of radius mh, in the Pr, (i,k) k-th downstream layer surface area S time available for an aerosol particle to traverse a stretch in a direction perpendicular to air flow ta due to its inertial momentum idem, due to diffusion lair absolute temperature T face velocity of the air flow Vs lia~tr velocity of a flow of radius m~l VI linear velocity of an aerosol particle originating from a flow of radius m~ at the k-th downstream vt (k) layer distance traversed by an aerosol particle due to inertia perpendicular to the flow direction during Sd the time td idem, due to diffusion during the time td,t Xdtf Xj weight fraction of the j-th fibre component in the filter effective weight fraction of the j-th fibre component X*~ apparent weight fraction of the j-th fibre component X**j a,{3 angle porosity of the filter ~s critical porosity specific density of the fibres; also of the aerosol particles p viscosity of the air i¢ filter composition factor geometric standard deviation of the pore size distribution og A(In m.3 normalized pore size distribution T tortuosity factor volume flow through a pore of radius 2mht mean volume flow through a pore reciprocal forth moment of the pore size distribution inertial momentum of an aerosol particle originating from a flow of radius m~l in the k-th up~I (k) stream layer mean free path of the air particles. THEORETICAL
Aerosol penetration through a fibrous filter. CONSIDrat a single c o m p o n e n t glass fibre filter. Let the m e a n fibre d i a m e t e r be d, t h e effective m e a n fibre d i a m e t e r dE. T o derive a theoretical expression for the a e r o s o l p e n e t r a t i o n t h r o u g h such a filter, the following a s s u m p t i o n s have to be m a d e : a. I n a fibrous filter the fibres are o r i e n t a t e d in planes p e r p e n d i c u l a r to the m e a n flow direction b u t in these planes at r a n d o m . b. T h e p r o p e r t i e s o f a layer o f 2d E thickness are representative for the p r o p e r t i e s o f the whole filter; this m e a n s t h a t e.g. the pressure d r o p across a filter o f thickness L is L/2de the pressure d r o p across one single layer. c. A filter is c o n s i d e r e d as a system o f pores. I f the filter p r o p e r t i e s are a mere superp o s i t i o n o f the p r o p e r t i e s o f the c o m p o s i n g fibre layers, then the flow t h r o u g h such a system c a n be described by Poiseuille's law for the flow t h r o u g h circular capillaries. Let the weight o f a glass fibre filter be W, the surface a r e a S a n d the thickness L ; let P be the specific weight o f the glass fibres a n d 8 the porosity, then
A probabalistic theory of aerosol penetration through glass fibre filters
463
W = S L p ( I - e). If a series of filters is manufactured, keeping W, S and p constant, then L (I - e) = constant. The more e increases, the more L increases, and more space is available for fibres in the unit layer 2dE. Since S is kept constant this space is found by an increase of 2dE; hence the effective layer thickness and the filter thickness increase in the same proportion, whence
L/2dr = constant. As a consequence 2dE(1 - 5) = 2a(1 - es) = constant. It will be assumed throughout this work that the thickness of the unit layer is 2dE = 2a(1 -- es)/(1 -- e).
(1)
The composition factor, as introduced by CLARENBURG and WERNER (1965, a, b) appears to be independent of porosity; therefore the value of ~s for any filter follows by equating the experimentally determined composition factor re, defined as
~p K
~I,
rlLV, (1 - e) 3/2
and the theoretical expression, derived by CLAP.ENBUROand PIEKAAR (1968) 11"4 re,h = ~s (1 --
[nps
(0"858 x/l-865/esa2- 1.024) 2 x 0.388]
"~,)3/2[72 +
with
Hps
N~ (N~ - 1) rr
and 8
N~ = ~ (1 - 53
N~ 2
/// d~
Let the probability that an aerosol particle is retained in a fibrous layer of unit thickness 2dE be Pa; suppose that L/2dE unit layers are contained in the filter thickness L. Then the aerosol penetration through such a filter is
Pen = (I - Pa)Z'/2'~E or
L L(1 - e) log Pen = ~ log (1 - Pa) = 2a(1 - 5s) log (1 - Pa)
(2)
In the following sections an attempt will be made to derive a theoretical expression for Pa. AIR
FLOW
THROUGH
A FIBROUS
MEDIUM
In a theoretical study it was shown by PIEKAAR and CLARENBURG(1967) that filter properties can be satisfactorily described using a geometrical filter model, in which a filter layer is considered as a system of "pores" of various polygon shapes. Moreover it was shown (CLARENBURG and PIEKAAR, 1968; PIEKAAR and CLARE/~BURG, 1967; CHEN, 1965)that the
464
Louis A. CLARENBURG
flow through these pores is conducted by the hydraulic surface area. This concept offers the possibility of converting the various polygon shapes of the pores to one uniform shape, the circular hydraulic surface. The hydraulic radii mh of these surface areas were shown to form a logarithmic normal distribution, A (In roD, with invariant geometrical standard deviation a~ = 1.9. In this paper only laminar flow will be considered. Although a parabolic velocity profile might be in order over the cross-section of the entire falter, the velocity profile over the crosssection of a pore is definitely not parabolic, the depth of a pore, 2dE, being too small for establishing such a profile. In a filter an air molecule is continuously accelerated and decelerated; at the "entrance" of each pore air molecules, originating from various pores of the upstream layer, hence with various initial velocities, come together to form the flow through the pore under consideration. Therefore a homogeneous velocity profile will be
assumed over the cross-section of the flow in a pore. It is generally accepted that the airflow through a fibrous medium follows a tortuous path. Starting from the above assumptions, a theoretical expression could be derived for the tortuosity z PmKAAR and CLARENBURG, i ~=
I +
(0-858 ~/1-865/~ - 1"024)2 x 7.262m hg 4d~
, t3J
with de given by equation (1), and m2hg = 0.0295 72/n~ np =
N(N - 1) rr
8 N=-(1
(4)
N 2
(5)
dJ -- ~) ~ •
(6)
It is also generally accepted that the mean linear flow velocity in a porous medium is v = -- z.
(7)
However, as follows from the probability of finding an air particle in a pore of hydraulic radius between mh~ and mh, + dmh,, the linear flow velocity through a pore of radius mh, is proportional to the ratio of the volume flow ~ through that particular pore and the mean volume flow ~bv, through a pore. The flow through a filter tries to find a path of minimal resistance. Therefore, one can visualize a filter as consisting of a number of interlocking pores which form channels to which Poiseuilles law could be applied. Doing so, the ratio q~i/q~v can be written as
C~"--=~tnMh
m~t A (In mhi)d In mhl
.... ~, m~ e st.~,g
I ~"
with Mh = 6mhg, the maximal observed hydraulic radius (4) and q, = (In Mh -- In mhg -- 41n2ag)/ln ~rg.
-~ =m~i~b~ -'T" d#
A probabalistic theory of aerosol penetration through glass fibre filters
465
Hence, the linear flow velocity in a pore of hydraulic radius mhi is Vs
;'i = - -
* m~*i e e .
(8)
It follows from equation (8) that the linear flow velocity is very unevenly distributed over the cross-section of a filter, the largest pores contributing most significantly to the conduction of flow. This will prove to be of fundamental importance in the description of the trapping efficiency of a fibrous layer. AEROSOL F L O W T H R O U G H
A FIBROUS MEDIUM
If it is complicated to give a description of the airflow through a fibrous medium, the result of inertial and diffusional forces upon the flow of aerosol particles through such a system is even more difficult to model. Consider a fibrous layer of thickness 2dE in the mean flow-direction and a pore of hydraulic radius rnh,. in the upstream layer. Assume that at time t = 0 the direction of flow changes from A.C. (Fig. 1) to AB on the average over an angle ~, related to the tortuosity by the equation cos ct = 1/z. (9) As a result of inertia, a fractionf~ of the particles conducted by the pore under consideration cannot follow this abrupt direction change and will pursue their course through the filter, in the first instance independent of the airflow.
Mean flow A
direction
FIG. 1. Tortuosity in a fibrous filter.
On going from A t o B the aerosol flow will lose a fraction g~ of the particles as a result of particles diffusing outside the airflow. Hence from any pore mhi in the first upstream layer a fraction 1 - f~ - gi of the aerosol particles will reach the layer under consideration in the airflow, i.e. with approximately the same direction and velocity. From a pore of hydraulic radius mhi in the first upstream layer a fraction h+ = fi + gi approaches the layer under consideration with direction and velocity different from the arriving airflow. However, going from A to C (Fig. 1) part of fraction h~ is re-entrained in the airflow, depending on inertial momentum of the particles and airflow velocity encountered. If the probability that a particle is re-entrained between the pore under consideration and the next down-stream layer is denoted as P,e (i,0,) then a fraction of theaerosol particles h* (1) = hl [1 - P,e (i,0)] (10) will arrive at the layer under consideration with velocity and direction different from the arriving airflow. Consider now a pore of hydraulic radius rnhl in the second upstream layer;
466
Lou~s A. CLARENBURG Fraction captured
Fraction
re-entrained in flow
h•l(2) [-~--~,-~ hi( I1 (I-K*(i,l)) P,e{i,I) h~'( I ) ( I - K ( i , I ) ) hT(
' I.}K * (t,l)
~
1I
PtuP Stream
h'~( I )
foyer
• h;P,,
i,O) 7'"~
upstream layer
Mean
lflow
] direction
FIo. 2. Inertial fraction of aerosol particles; losses due to capturing and re-entrainment. a fraction h* (0) arrives at the first upstream layer. Here a fraction K*(i,1) is trapped, so that a fraction h* (1) [1 - K(i,1)] approaches the layer under consideration. However, a fraction P,e (i,1) is re-entrained, so that finally the fraction h* (2) = h, (1 - P,e (i,O) (1 - P,, (i,1)) (1 - K* (i,1))
(11)
arrives at the layer under consideration from the second upstream layer (Fig. 2). Extending this reasoning to layers further upstream, it can be easily verified that the fraction of particles originating from a pore in the k-th upstream layer, arriving at the layer under consideration is k-1
h*(k) = h, I-I [1 - P,, (i,p)] [I - K*(i,p)],
K*(i,0) = 0
p=O
hi < 1.
(12)
It will be shown in a subsequent section that P,~ (i,p) and K*(i,p) are dependent on the size of the pore of origin only [equations (37) and (36)], so that the generalized equation (12) is an exact description of the fraction of aerosol particles, originating from a pore in the k-th upstream layer, arriving at the layer under consideration independent of the airflow. As can be seen from equation (I0) a fraction h~P,, (i,O) re-enters the airflow between the first upstream layer and the layer under consideration, l~tween these same layers a fraction [see equation ( I 1)] h~ (1 - e,e(i,O)) (1 - K*(i,1) P,~ (i,t) = h* (2)
P,e (i.1) 1 - - P r e (i,I)
re-enters the airflow originating from a pore in the second upstream layer. Originating from a pore in the k-th upstream layer [see equation (12)] a fraction
h* (k)
P,,(i, k - 1) 1 -
P,e
(i,k
-
1)
re-enters the airflow between the first upstream layer and the layer under consideration. In total, a fraction H~ re-enters the airflow between the first upstream layer and the layer under consideration, H~ given by
A probabalistic theory of aerosol penetration through glass fibre filters Hi = L h* (k)
k= 1
P~e (i,/~
1)
1 -- ere (i,k - 1) '
H* (i,0) = 0 .
467 (13)
Hence, the fraction of particles arriving at the layer under consideration in the airflow, i.e. with approximately the same velocity and direction, originating from a pore of hydraulic radius mhi in any of the upstream layers, is
Fi = 1 - hi + H i .
(14)
As a consequence, the fraction of particles arriving in the air flow at the layer under consideration is
f 'n~h Fi m ~iA (ln mh,) dln mhi F =
-
(15)
f.i~Mh m4hiA (ln m,i) dlnmh, This fraction of the particles will be referred to as the interceptional fraction; the sum of the fractions h* (k) averaged overall flows will be named the inertial fraction. THE
fi
FRACTIONS
AND
g~
Starting from Stoke's law it is easy to find an expression for fi. If at time t = o the airflow in a pore of hydraulic radius mhl changes direction over an angle ~, then the velocity of the particles at that moment in a direction perpendicular to the new flow direction vl sin ct. After a time t d the distance Xd travelled by the particles in a direction perpendicular to the new flow direction, can be immediately found by integrating Stoke's formula twice over the time, resulting in
Xdi (~) = ~ v sin l • (1 - e -~p'~) 2p with
(16) 18//
/.p = p dp2 The available time td can be found approximately, by recognizing that the distance covered by the airflow between two neighbouring layers is 2d E r and by assuming that the velocity remains constant hence tdl =
2dEz
(17)
Vi
The actual directions of the particles can assume any angle between ~ = 0 and ~ = ~max as defined by equation (9). Therefore, by integrating xdi (ct) over all possible values of ct an estimate is obtained for the mean distance Xdi, Vi[ '
Z--1
] [ 1 - - exp (2, tdi)].
(18)
According to the theory of tortuosity the flow originating from a pore of hydraulic radius mhi, splits up, on the average, in 4 smaller flows with a probability of over 99 per cent. If the hydraulic surface area of the pore is 4n m2ht, then, assuming circular flow (Fig. 3a), the cross-sectional area of the flow is n m2hi with radius mhl. As under the influence of
468
LOUIS A. C'~I~NeURC~
( 1
C
mh;
(a)
(b)
FIG. 3. Aerosol flow in a filter. (a) inertial shift of the aerosol particles; (b) expansion of the aerosol flow due to diffusion. inertia all the particles contained in this flow move a distance .rat in the same direction, a fraction of the particles represented by the surface area ABCD has come outside the airflow after a time t4,. According to elementary geometry the surface area ABCD is given by x,
S(ABCD) = x~
mR - -T + 2 m~, arc sin \2
ms,/
A=
~/
,
or
~m~
~-+-arcsin
>,~=~
F~hi
or in close approximation, as xd, ,~ mh~ most of the times, f~ = 2Y--2.
~'t9)
To find an expression for g~ use has been made of the statistical theory of Langmuir as reported by CrImq (1955). According to this theory, the average absolute value of the displacement of a particle in the time td~ along a certain direction is given by
xd~f,, = (4Dnu td,) I/z
(20)
with
CkT
Dnn = ~ 3~z r/dp and
C=
1 +~
°[
2.46+0.82exp
(21)
(
-0"44
.
(22)
Here C is the Cunningluml factor, with co the free path of the air molecules. As under the influence of diffmion the ~ r t i d e s have moved a distance xa,i in all directions (Fig. 3b), the fraction of the particles getting out of the airflow in the available time td, is represented by the surface area of the annulus with thickness xd,~,,i. Hence (m,, + xas,i) 2 - tr m~ gi =
7r rn~,
2 xd,I, i ~
~
mki
(123)
A probabalistic theory of aerosol penetration through glass fibre filters FILTER
469
MODEL
The mean number of vertices per polygon in a random line array is 4, as was theoretically shown by MILES (1964a, b); using a Monte Carlo technique this same number was experimentally found by PIEKAAR and CLARENBURG (1967). These last authors showed both theoretically and experimentally that the conversion factor from hydraulic pore area to real pore area is 1.28. This is almost exactly the ratio of the area of a square to the area of its inscribed circle. It is logical therefore to assume that the mean port shape is a square the flow being conducted by its hydraulic surface area (area of the inscribed circle).
INTERCEPTION
FRACTION
OF P A R T I C L E S
Consider the particles arriving in the flow at the " e n t r a n c e " of a pore of radius 2 mhi. According to the theory of tortuosity the total flow originates on the average from 4 different directions; hence in each pore 4 sub-flows will be distinguished (Fig. 4a), each of hydraulic area n m~i. Here the dominant influence of the larger pores on the description of the behaviour of a filter is clearly demonstrated, as was emphasized previously [equation (8)].
@ QJ MA=r AC=x S(ABCD}=S(r,x}
Flow in o pore
(o)
M~.=r MN=CF =n AC=z, S(BCDEFG)=S(r,x,o)
(b)
(c}
FIG. 4. Geometry of aerosol flow in a filter.
In the following, it will be assumed that an aerosol-particle of diameter dp is trapped if the trajectory of its centre would pass a fibre at a distance smaller than dp/2. The following notation will be used. In Fig. 4b the surface area of the segment ABCD will be denoted as S(r,x). According to elementary geometry S(r,x) = r2 arc cos ( 1 _ x ) _ (r _ x) ~/ x (2r _ x) '
x >~ O.
(24)
The probability that a particle is captured will be denoted as P(z), given by P(z)
=
S(r, x) 7~r 2
-
1
1 arccos (1 - z) - - (1 - z) -,/~ (2 - z).
x
7z
(25)
'
In Fig. 4c the surface area B ' C D ' E F G will be denoted as S(r, x, a); in close approximation S(r,x,a) = 2a x/x(2r-
x),
x i> 0.
(26)
470
Louxs A. CLARENBURG A
Infercepfion MA :r=m.~ CA=x=d /2 (o) F //////X/////'/~
Q) ..rncrease of infercepiion due t'o inertia MN = CF = xdt MN Ivf2 = C~Ft-" xd,l~
Decrease of interception clue t o iner-I'io MN = A B = x~ B~C~: '/2d o - x d / ~
MC = m . - d p / 2
BC = '/zd~- x,j
F ......
MC ~= m . - dp / 2 (b)
M A = N B : m., F /z/////y//z//A
F
N8 j ~ m , ~ (c) F
#III/I~I////A
7( ..................
Increase of intercep~rion clue to diffusion CF:x=fi i CF : x~,f,
2ncrease of interception due to diffusion
CF =x~,f,
CF = x ,~
NC=m.,-~/zdr~-xa/J~
NC -" mhi- ~'z d ~ - x~,
(e)
(dl F
MC = m. -I/z d~
(f) (g) FtG. 5. Mechanisms of aerosol capturing in a pore. The probability that a particle is captured P(w, z) is then
S(r, x, a) P (w, z) m
r2
2
= -w
x/-(2 - z).
(27)
T h e particles in the flow are subjected to various effects, all acting simultaneously. Each individual effect can only be described in a probabilistic sense, assuming the presence o f a great number o f pores and particles.
A probabalistic theory of aerosol penetration through glass fibre filters
471
The first effect to be considered is interception (Fig. 5a). Starting from a homogeneous distribution of particles over the cross-section of the flow the probability of interception P(z~) is given by the ratio of the area shaded to the cross-section of the flow. Superimposed on pure interception is the increase due to diffusion (Fig. 5t, shaded area); the probability of this effect is P(w2, z~). Under the influence of inertia the centre of the cloud can shift a distance M N = xa in either direction. As an approximation only 8 possible directions (Fig. 5g) are considered in this study; the resulting 8 overall effects will then be averaged as if representative for the behaviour of a filter. The increase o f interception due to inertia is in order if the centre shifts towards the fibres (Fig. 5b); the two probabilities considered are P ( % , z~) and P ( % , z~) respectively. Superimposed is the increase o f interception due to diffusion (Fig. 5d); the two probabilities considered are P(w2, z , ) and P(w2, z~) respectively. But also a decrease o f interception due to inertia has to be taken into account if the centre shifts from the fibres (Fig. 5c); the corresponding probablities are P ( z 3 ) a n d P(z2)respectively. The probabilities of the corresponding diffusional effects (Fig. 5e) are P ( w 2, z3) and P ( % , z2) respectively. The overall capturing probabilities in each of the 8 directions are given in Table 1, and averaged to yield the apparent probability of interception [equation (28)] in a pore of radius 2rnh~. TABLE I. COMPOSITION OF EFFECTS Direction
1 2 3
4 5 6 7
8
Capturing probability
Symbols.
2 P(zl) + P(wl.zl) + P(%,zi) + P(w2,z.O 2 P(zl) + 2 P (w3,zl) + 2 P (w2,zs) 2 P (zl) + P (wl,zl) + P (w2,zl) + P (w2,z~.) P (zl) + P (z2) + P (%,zl) + P (w2,z2) ~- P (w2,zs) P (zx) + P (za) -- P (wz,z3) 4- p (w2,zl) 2 P (z2) + 2 P (w2,z2)
zl = dp/2m~j z2 = (dp = xa~ "V2)/2mh~ za = (dr, = 2x,,)/2mht wl = xddmhl w2 = xals dmhi w3 = wlx/2. P (zl) + P (z3) + P (w2,z3) + P ( % , z t ) z . = (dp + 2xdt)/2m~l P (zl) + P (z2) + P (wz,z~) + P (%,z2) + P (w2,z~) z~ = (d~ + xa~X:'2-)/2m~
Pi*t (mhi) = ~g [10 P ( z l ) + 4 P(z2) + 2 P(z3) + 2 P ( w I, zl) + 4 P ( % , zl) + + 4 P ( % , zl) + 4 P ( % , z2) + 2 P ( w 2, z3) + 2 P(w 2, z4) + 4 P ( % , zs)]
(28)
However, as was discussed before, a fractionf~ of the particles escapes the airflow due to inertia, moreover a fraction g~ escapes due to diffusion, whereas a fraction H~, originating from a pore of same radius somewhere upstream, re-enters the flow. Furthermore, the probability of finding a particle in a flow of radius rnh~ should be proportional to the probability of finding a volume flow ~b~.As a result the probability of interception in a pore of radius between 2mh~ and 2mh~. + dmh~ is P i n , (mhi) =
(1 - f i
- gl + H,.) ~b--2/A(In trlhi) Pi*t (mh3dln mhi.
This probability averaged over all pores in a unit layer of the filter yields P~.i,~ = cbv
~ •/
In
In
Mh
rn'~i Pi,~ (mh,) A (in mhl) d In mhi m m
with Mh = 6 mhg and 2 rn m = dr~'2.
(29)
472
Louis A. CLAP.~r~SUP.G
Under the influence of diffusion the supposed circular aerosol flow expands from a radius mh~ to a radius (mh~ + xat, 3. According to Fig. 4a this flow is on two sides bordered by fibres, hence the apparent probability of an aerosol particle of being captured in a flow of radius mh~ is approximately Xdif, i
Pa~I (mh,)
mhi
Consequently
PcHf (mh3 = (1 - Ji - g~ + H3 ~2 A (In m,,) Pdi*f (mh~) d i n mhi and Pm d i f = ~ v
INERTIAL
f
in Ma
,din ma
m~i P d l f (mh/) A
FRACTION
(In m h i ) d In mhi
OF THE
(30)
PARTICLES
Consider now the fraction of particles emerging from theflow in a pore of radius 2mh. hence traversing the filter with velocities and directions different from the flow, in a way independent of flow. Let the inertial parameter of a particle originating from a pore of radius 2mhi in the k-th down-stream layer be defined in the usual way
C v~(k)
~, (~)
= ~
~31)
;.p d~
Here v~(k) is the velocity of a particle arriving at the surface of a fibre, after having traversed a stretch kd~. The initial velocity of this particle, when it emerged the flow, is v~ as defined by equation (8); as a result of friction with the air the velocity v~ (k) is approximately g i v e n by
v~(k)= vi -
,
*
Vs
k z p d E, v ~ ( k ) / > - - z ,
i32)
The directions, at which the particles emerge from the airflow, can assume any value between the angles ~ = 0 and ~ = ~ . . . . with ~m.x defined by equation (9). Hence x/r 2 d~ = 2delcos~ = 2 d e
1
r arc cos (t/z)
.
(33)
For the effectiveness of precipitation of a particle on a cylinder as a result of its inertial momentum, the expression proposed and experimentally verified by LANDAHL(1949) was adopted. Landahl's expression was shown to give an adequate description of the experimental data for Reynolds numbers in the vicinity of 10. As the major part of the flow is conducted by the largest pores, it can be verified using equation (8) that the Reynolds number in a porous medium can easily assume values not too far from 10. For this reason the effectiveness of precipitation is written as: [~i (k) 3
Pi.* (m~,i, k) = ~bi (k)3 + 0"77 Oi (k) 2 + 0"22"
(34)
A probabatistic theory of aerosol penetration through glass fibre filters
473
The reason that this expression was adopted for the effectiveness of inertial precipitation is that it has a reasonably good experimental foundation (LANDAHL and H~.gMAr~N, 1949; RANz and WONG, 1952). According to various authors (LANGMUm and BLODGETT, 1946; LEVIN, 1953; FUCHS, 1955) Pin* (mhi, k) = 0 for ~bi (k) ~< 1/16
(35)
Now, three effects should be considered. The probability that a particle is captured due to inertial precipitation is proportional to the ratio of the total surface area of the fibres perpendicular to the mean particle direction, N ' d / and the total surface area 72. The probability that a particle is captured due to inertial interception is proportional to the ratio of the surface area formed by the total length of the fibres and two times the particle radius N*dp7 and the total surface area 72. Finally, the probability that a particle is captured due to inertial diffusion is proportional to the ratio of the surface area formed by the total length of the fibres and a distance x*d~i, bringing the particles to within a distance d~/2 from the fibres, 2N* X*d~.e7 and the total surface area 12. Combining these three effects the apparent probability K* (i k) that the particle emerging from the flow in a pore of radius 2mh~ are captured as a result of inertia in the k-th downstream layer is N*
K* (i, k) = T
(3 + dp + 2 xd*I, i) Pi* (mhi, k) .
Here N* is the number of fibres in a layer of d E [equation (33)] thickness and is given by equation (6); x*diy, ~is given by equation (20), with t*, i given by
a~ ~'~(k)
t~i = ~
(36)
so that K*(i,k)
4(1
d~
~
5)~(a
+dp + 2
* ,)P* (rnh,,k)
Xdif,
•
(37)
However, as was shown before, only a fraction h* (k), as given by equation (12), arrives at the k-th downstream layer. The expression for h* (k) contains the probability of reentrainment. It will be assumed that a particle is re-entrained in the flow if it encounters on its path through the filter a flow of which the velocity is equal to, or greater than the actual velocity of the particle. The velocity of the particle, as given by equation (31), can be equated to the flow velocity in a pore of radius 2mhk as given by equation (8). Hence the probability of re-entrainment is equal to the probability of finding vk >/ v~ (k) or
v*m~ >>.v*m~i - k 2p d~ or mnk Thus
>>- (m~i --
k 2p d ~ / v * )
1/4 .
P,, (i, k) = e [Vk >>-V, (k)] = P [mhk ~ (m~i -- k 2p d~ /v*) x/*] Mh
fin =
A (In mhi) din rn,i
j In ,~.k
f ld~ ' "l A n( l nmr n hhi ) , _
(38)
474
Louts A. CLARENBURG
Now, the probability that particles originating from a pore of radius 2mh~ are trapped in the k-th downstream layer as a result of inertia, is
K(i, k) = h* (k) K* (i, k), K* (i, o) = O. Hence, the overall probability that particles originating from a pore of radius 2mh~ in some intermediate layer are captured as a result of inertia in the k-th layer, is given by k
K(mh~) = ~ K ( i ,
p), p = 1 , 2 , 3 . . . k .
p=l
The probability that a particle is captured in a unit layer of 2de thickness in the filter as a result of inertia is found by averaging K(mhi) over all pore sizes, whence 2
P.,~.
F?
K (mhi) A (In mh~) d In mhi
-
(39)
f'ThA(lnmh,)dlnmh,_ the factor 2 allowing for the fact that each fibre is part of two layers. THE
PROBABILITY
OF
CAPTURING
The probability that a particle is captured in a (mit layer of 2d~ thickness P~, is evidently given by Pa = Pa, i.t + P~, aly + P~, i. + P.l (40) where Po, ~., is given by equation (29), P., ail by equation (30), and P., i. by equation (39). Here P.t stands for the probability that a particle is captured due to electrical attraction. Since, in the present study, attempts were made to avoid this effect, Pet has been neglected. Although the final expression for P= is a very complicated one, it does not contain any adaption parameter; it contains the flow properties v,, q, k, T, and o~; the fibre properties d ~ a n d l ; the filter properties 8 and L; and the particle property dp. Hence equation (40) is a real predictor. AEROSOL
PENETRATION GLASS
THROUGH FIBRE
A MULTI-COMPONENT
FILTER
With the aid of the foregoing it is relatively easy to model the aerosol penetration through multi-component glass fibre filters. Consider a n-component glass fibre filter. Let the weight composition of the filter be denoted as Xj,X2 • • • X.; ~ Xj = 1 . (41) J
If the mean fibre diameter of the j-th component is aj and its mean fibre length is 7j, then it will be supposed that
~, <+72<... <,7, and 7~ <72 < . . . < 7 , where the suffix 1 refers to the finest component and n to the coarsest. The properties of a layer of 2dE thickness are no longer representative for the properties
A probabalistic theory of aerosol penetration through glass fibre filters
475
of the whole filter. Instead, the structure of the n-component filter can be taken as if the grids of the various components, each present in the weight fraction Xj are telescoped into each other, while from each grid the fraction 1 - X~ is being occupied by fibres of other components. Again the value of e, for any one filter follows from equating the experimental composition factor and the theoretical as defined by CLARENBURGand SCHmREeK(1968) 11 "4 =
glh
with
r/p~ .2
- - __
e, (1 - e~)3/2 1~ r = ~Xjr: j=l
and
1i(
] 1
n,~,, = 7z'1~ ~ Nj~1, - ~ Nj, 7~ - ~ ~ Njs j=l
,
j=l
7.2 a./
o gjs
=
(42)
j=l
-- X j (1 - e~)~-,j~-~.
(43)
7Z
The tortuosity factors ~j are given by equation (3), in which rn~g = 0-02#51~ /npm
(44)
with npm given by equation (42) if the subscript s is dropped. Keeping in mind the reasoning behind equations (42) and (43), i.e. that the properties of a filter are considered as a superposition of the properties of the unit layers 2 a j, leading to the filter property expressed by equation (44), it is evident that equation (29) expresses the probability of particle capture due to interception in a multi-component filter, equation (30) expresses the probability that it is captured due to diffusion. The description of capturing of particles due to inertia becomes somewhat more complicated, as the inertial parameter ff is a function of the fibre diameter C v i (j, k) ~k~(j, k) = - (45) 2 d*~.E with d*jEgiven by %IT 2
d*iE = 2 aj~
~o,coo,
--
1
(I/z)
(46)
and v~(j, k) by vi(j, k) = v i - k 2pd~:j.
(47)
As a consequence the probability of inertial precipitation P*i. on a cylinder becomes a function of the fibre diameter, P*~, (m~, j, k), given by equation (34) at the condition expressed by equation (35). The probability that a particle, pursuing its path through the filter, encounters a fibre with diameter a j, P(aj), within a stretch d*.~, is
3.
a.
P(aj) = ~ N'~.* l j/ j=l -~j N'j* 7j
(48)
with N ' j * the apparent number of fibres of diameter a: in a layer of d*j~ thickness as given by equation (46).
476
Loum A. CLARENBURG
Following an aerosol particle on its path through the falter, it is more than proportionally likely that it encounters a fine fibre, as part of the surface area of the coarse fibres is screened from the flow by the fine fibres and vice versa. This effect, introduced by CLARENBURGand WeRNER (1965b), was named the Shadow-effect; it was shown to play a predominant role in aerosol filtration. Consider a 2-component glass fibre filter. Let the surface area of a Coarse fibre screened by one single fine fibre be denoted as S ~ ) , with fl the angle between the coarse and the fine fibre; moreover, let the maximum number of possible intersections of fine fibres with one coarse fibre be denoted as NI, and the probability of intersecting as Pc, then the Shadow-effect H can be expressed as
s~)
Ht2 = ~
Ny Pc.
(49)
Consider a volume element (at + a2) It 12. The number of fine fibres in this element is 4 N 1 = - X I (1 - 8) (aa +
a32)72/d2-
7t
If they all intersect the coarse fibre then N1 = N¢. The probability of intersecting was calculated by I~eKXAR and CLAm~BtmG (1967) to be Pc = 2/n. If a fine fibre intersects a coarse fibre at an angle/L then the surface area screened is
S([3) = dl ~]sin ft. At very small angles 0 t> p t> /~g the area screened is S(p < p,) = a~ 1~ with fie defined as sin fig = a2fll" The probability of finding an angle between fl and fl + dfl is 2/n dfl, consequently
S
2 [',/z al ~2
2
Recognizing that for 7j >> d 2, flo approaches to a2/ll, and solving the above equation, ,~ is approximated by S = 2 ~1 a2(1 + ln271~ "~-2] as a result 16 a,( in 211~ H12--~-~X1 ( 1 - e ) ( a t +a2)d-~ 1 + ~2]" In the 2-component filter considered the fraction Hi2 of the coarse fibres is screened from the aerosol flow, whence the effective fraction of the coarse component is (1 - x 0 *
= (l - x 0 (l - / - / , 2 ) .
(50)
Similarly, it can be proved that the ser~ning of the fine fibres by the coarse, can be expressed as
A probabalistic theory of aerosol penetration through glass fibre filters
477
16 ~2 ( In 211~ H21 =~-3(1 -- X~)(1 - ~ ) ( a I +~2)d~22 1 + "~'2] " Hence the effective fraction of the fine component is X~ = X 1 (1
(51)
/-/21) .
-
From equations (50) and (51) it follows that the apparent weight fractions are
Z*p = X , / { X ~
+(I -X0*}
andX** = l - X * *
consequently N** = 8X** ( l n
- e)~--~"
(52)
For multi-component filters the considerations are more complicated, though qualitatively the same. The apparent probability K* (i, j, k) that particles emerging from the flow in a pore of radius 2mh~ are captured as a result of inertia in the k-th downstream layer by a fibre of diameter a: is
K'* (,,j,k) = ~
(~j +dp + 2 Xdi*f , i,j) Pi* (mhi, j, k) P (aj) ,
where X*dif, i, S is defined by equation (20), with
td,.j=
d]*r,
(53)
v~ (j, k)"
The apparent probability K* (i, k) that particles emerging from the flow in a pore of radius 2mh~ are captured as a result of inertia in the k-th downstream layer is evidently
1. a&
K*(i,k) = ~ K*(i,j,k) = ~ 4 Xj(1 - s) j=l j=l lj d~
(a: +dp + 2 Xd,*1.i.j)P,* (mh,,j,k) P (aj).
(54)
As the particle velocity v~ (j,k) is considered, according to equation (47) relative to a layer of d*jE thickness and if the probability of re-entrainment is considered relative to the same layer, equation (38) remains valid; also equation (12) still holds if expression (54) is substituted for K* (i,p). Therefore equation (39) expresses the probability of particle capturing due to inertia in a multi-component fibrous filter. The aerosol penetration through a multi-component glass fibre filter follows from equation (2), recognizing that the properties of a multi-component filter are derived from a superposition of the properties of the unit layers formed by the individual components. As only the effective fraction of each component takes part in the filtration process
logPen = 7
=1 kaj]J (1
log(1 - Po) ;
(55)
with, for a 2-component filter, X1 and X2 given by equations (51) and (50) respectively, and P, by equation (40).
478
Louis A. CLARENBURG EXPERIMENTAL
To test the theory presented in the foregoing section an aerosol generator should be available capable of producing monodisperse aerosols of various diameters. Moreover, the aerosol particles should be presented to the test filter at various face velocities. Finally the filter properties should be varied. Preparation of the filters has already been described in detail by WERNER and CLARENBURG (1955a), the experimental set up by CLARENBURG and VAN Dr.R WAL (1966). All relevant experimental conditions have been summarized in Table 2. TABLE 2. EXPERIMENTALCONDITIONS
Aerosol generation
Aerosol Droplet diameter Droplet size determination Aerosol concentration Concentration m c a s ~ e n t Penetration m ~ s ~ t Volume flow Face velocity Filter Fibre diameter in microns Square fibre diameter in sq. microns Fibre length in microns Determination fibre properties Filter weight Filter porosity Discharge of filter electricity Computer used
Modified La Mer generator (CLARENBURGand VAN DER WAL, 1966; LASSEN,1960; RAPPAPORTand WEaNS'rOCK.1955) Dibutyl phtalate (DBP) monodisporse, 0"3-1"6 micron higher order Tyndall spectrometer 40 t,g/liter Forward scattering optical concentration meter (CLARENBURG and PmNC~N, 1963) 450 1/hour 2-90 cm/sec. Prepared from John Manville's #ass fibres code 102 104 106 108 110 i12 0"207 0"275 0'456 1"02 1"49 2"00 0"089 0"191 0"433 1"99 5"10 7"34 171 195 377 478 522 990 Electron microscope (CLARENBURGand WERNER,1965a, b) 80 g/m2 0.88-0.96 Prometium source, 100 mC CD 3200
RESULTS
As was mentioned above, particle, flow, filter, and fibre properties are the constituents of the formula to be tested. For practical reasons, all of the single-component experiments were carried out with filters manufactured from code 108 fibres; filters prepared from code 106 fibres had a penetration too low to measure. Filters prepared from code 110 or 112 fibres appeared to be too weak, mechanically. In Fig. 6 to 10 results are shown when the aerosol particle diameter is varied (variation of particle properties) at constant face velocity; the experimental conditions are indicated on the figures. The points are experimentally determined, the drawn lines are calculated using equation (2). Figures 11 to 14 show the results when face velocity is varied (variation of flow properties) at constant particle diameter. In Fig. 15 results are depicted when porosity is varied (variation of filter properties) at constant particle diameter and face velocity. F o r the variation of fibre properties, two-component glass fibre filters were studied. O f the many available data, only a few were calculated using equation (55). They were selected to cover a wide range of fibre diameters (see Table 2). Figures 16 and 17 show the results of aerosol-penetration through filters manufactured of mixtures of code 102-110, 102.-112 and 106-112 respectively with varying particle diameter and at constant
A probabalistic theory of aerosol penetration through glass fibre filters x
] 0 -b
~, I:L
x
i0 -2
Vs = 2.I cm/sec
\
L = 510/.c { = 0.943
I 0 -~
I
f
I
\ \
\
I
i
i
I
I
I
I
0 0.2 0.4 0-6 0.8 1.0 F2 1.4 I-6 I-8 2-0 dp, ~.
FIG. 6. Aerosol penetration versus particle size at face velocity of 2-1 cm/scc.
Q X
~_I~~
×~×
Code [08 Vs= 6.4 cm/se¢
L =430/~ e =0"932
x
IO -3
I
1
l
I
I
I
l
o-z 0.4 o.6 o.e I-O 1.2 i-4 dp.
F
FIG. ?. A e r o s o l penetration v e n u s particle size at face velocity o f 6-4 cm/scc.
479
480
Louis A. CLARIENBURG
x\
0,,.
Code 108 V~=6-4 cm/sec L = 510~. =0-941
jO-~
L
00"Z
i,
]
x
I
f
0 . 4 0"6 0"8
~
[
I
I'0 1.2 1.4
f
1"6
dp,
Fro. 8. Aerosol pen~ration versus particle size at face velocity of 6.4 cm/sec.
t0"1
~
\
tc~2i x a.
Code108
'~ ' g2 o!. g., °'8 io ,'2.4~'6 d~, ~.
Fzo. 9. Aerosol penetration versus particle size at face velocity of 30 cm/sec.
A probabalistic theory of aerosol penetration through glass fibre filters
\ io -1
\ 8 {L Code 108 \ Vs = 5 4 crn/sec x
L
=
530 ~
= 0'943 16:
x
I~ 4
I
I
l
;
I
I
I
0 0.2 0.4 0.6 0.8 t-o 1.2 dp, p.
FIG. 10. Aerosol penetration versus particle size at face velocity of 54 cna/sec.
x
Code
lOB
dp = 0 . 3 6 #
4T
~L
/~-~-~"~--.~-
L (
• 530/~ = 0'943
~
r×
X
Id' 9
o..
,to
3
~o
31o 20
~o
&
;o
Jo
•x
2 /"
Code I 0 8
d~0,.0'38 5 3 0 /~ 0'943
1Of'
r×
x
8
0
x T"
I ro
I 20
I 30
I 40
1 50 Vs~
I 60 crn/sec
r 70
f so
x
F1o. 11. Aerosol penetration versus face velocity at panicle diameter of 0.36 ~ and 0'38 ~.
481
482
Loura
A. CLARENllURG
cod, ,08 x
dl = 0 , 6 6 ~ E = 5 1 0 p. • = 0-937.
x
~.
--<<.
I02
I~ ~
I I0
0
I 2CI
I 30
I 40
f 50 V~
T 60
t 70
80
FIG. 12. Aerosol penetration versus face velocity at particle diameter of 0.66 ~,.
x ~
× ~
Code 108 dp = 0 " 6 6 ~ L "- 5 6 0 ,~ = 0 - 9 4 5
10
I l0
l 20
[ 30
I 40
x
I SO
Ys)
T 60
; 70
i 80
: 90
c m / sec
~ O . | 3 . A e r o s o l l x ~ e ~ r a t J o n v e n u s face v e l o c i t y at p a r t i c l ¢ d i a m e t e r o f 0.66 ~.
A probabalistic theory of aerosol penetration through glass fibre filters
"x ×
Code
108
d.p = 1 . 4 0 F ×
L = 530/~ ~ = 0.945
-->..
i5'
IO 5
i 20
;
0
I0
' 30
! 40
I 50
I 60
[ 70
T BO
Vs FIG. 14. Aerosol penetration versus face velocity at particle diameter of 1.40/~.
5
~< × ×
4 13_
l Code
2
I08
x
dp=O.66/z Vs= 6 , 4 cm/se¢
I
x~ 'xx~ × ×
x
L = 500 F
I
0-88
~
x
0.89
0.90
I
0.91
I
0-92
I
0.93
I
0.94
FIG. 15. Aerosol penetration versus porosity.
]
0.95
483
484
Louis A. CLARENBURG
Code 102,112 X 1: 0.0834
\ 10-3
%
~ x ~
x
L
~
\
: 450
= 0"936 V5 = 6 ' 4 cm/sec ~t
eL B.
Code10210~ Eo"
f
}
io-z
X, • 0'23
L =.380/~ %
• "0"921%* V3 • 6;4¢~e//.
IG5
f l ~ I II I 0.2 0.4 04 08 1.0 t.2
I i, T i l l l l I [ 0"2 0.4 06 0.8 1.0 l.Z 1.4 1'6 1.8 2.0 dp,
F,o. 16. Aerosol penetration versus particle size at face velocity of 6.4 cm/sec for filters 102 : 112, respectively
i0
Code 106 x, = 0 4 0 16"
x
\
-'0"929 V3 = 6 . 4 c m / s e c
I
[
~
I
I
0.2 0.4 0.6 0 8 I-o i-2 d~,, M.m
it4
1.'6
FIG. 17. Aerosol penetration versus particle size at face velocity of 6.4 cm/sec for filter 106 : 112
A probabalistic theory of aerosol penetration through glass fibre filters
485
1
×
io-
xX x
X
Code 1 0 2 : 1 1 2 X b = 0.0834 L
= 440 F = 0.927
d~ = 0 . 6 6 F
~ 2Lo 35 4'o 5'o 6~o 7'o Io Vs
FIG. 18. Aerosol penetration versus face velocity at particle diameter of 0.66 ~ for filter 102 : 112.
face velocity. In Fig. 18 an example is given when face velocity is varied at constant particle diameter. It is surprising to see how well experiment and theory agree over a wide range of velocities, particle sizes, porosities and fibre diameters; and over a range of 4 orders of magnitude of aerosol penetration. DISCUSSION
It seems beyond doubt, that a pore model--as contrasted with an obstacle model--can adequately describe the complex process of aerosol capture in a fibrous filter. The characteristic features of each relation determined experimentally were also found by calculation. The actual agreement between observation and calculation is for most cases well within a factor of 2, being near to the experimental error; in one case (Fig. 17) the difference is a factor 2-5. Results of calculation depend very much on the accuracy of the input data. As appears from Figs. 6-10, 16 and 17, results are very sensitive to accurate particle size determination; using the OWL this precision seems not better than 0.01-0.02 p for the small particles, and 0.02-0.05/~ for the bigger. The porosity, also a very sensitive parameter (Fig. 15), was determined by measuring the thickness of the filter with the aid of a micrometer; the precision is probably better than 0.002. The sensitivity for face velocity increases as the performance of the filter improves (compare Fig. 11-14, 18); accurate determination of this parameter is no problem. By far the greatest problem is the determination of fibre properties. This is illustrated in Fig. 19 for a relatively insensitive relation of penetration versus face velocity at constant particle diameter. A small error of 5 per cent in the mean fibre diameter and a correspondingly small error of 10 per cent in the mean square diameter from their true mean values lead to'
486
Loum
A. CLARENBURG
h-i"
: I~
F
Code 1 0 8 d~ = 0 . 6 6 / z L = 5 6 0 ,u. = 094.5
0.969 0.969
--'--
1.791 2 ;89
-----
I •02
I
....
I 07I
I79t 2.189
1.071 ] V~
4C
,
~r
99
6?
cm/sec
FIG. 19. Sensitivity o f calculations t o c o r r e c t estimate of fibre properties.
variations in the computed results of a factor of more than 2; for more sensitive relations, miscalculation as a result of small errors in the determination of a and d-~ may be of the order of a factor of 5-10 (see Fig. 17). Unfortunately, determination of these fibre properties by electron microscopy is very tedious work, moreover, these fibre properties vary from batch to batch, and probably within a batch from sample to sample. Fortunately, results are insensitive to fibre length. With all this in mind, it is really surprising to find a reasonably good correspondence between experiment and calculation. In the theoretical part 8 mechanisms of particle capture were distinguished. The relative importance of some of these mechanisms is shown in Figs. 20 and 21 (those not shown did not contribute significantly). It becomes very clear that interception is the dominant mechanism of filtration under most conditions. Only at very small flow velocities, or for very small particle sizes does the mBchanism of diffusion becomes important and even dominant. Even at low velocities the increase of interception by inertia is a significant factor, while at high velocities the mechanisms of inertial precipitation and of inertial interception become important. When the mechanism of interception is dominant--according to Figs. 20 and 21, at v above 6 cm/sec and at dp above 0.6 micron, the probability of particle capture, Pc, should be [equations (28) and (29)] approximately inversely proportional to mho, given by equation (44). Then, from equation (55) log Pen should in first approximation be inversely proportional to mh,. In Fig. (22) log Pen is plotted against 1/mhgfor a number of filters, the data being given in Table 3. From Fig. 22 it follows that a simple calculation, which can readily be made by hand, offers a first orientation about the efficiency of a filter. The systematic deviations betwecn experiment and calculation at high velocities and large particle size (Figs. 9, 10 and 14) may be attributed to the departure from viscous flow within the largest pores. According to equation (8) the linear flow velocity within a large pore may easily attain a value as high as 50 m/see at face velocity of 30-50 ema/sec. This means that the Reynolds number of particles larger than 1 micron become in the order of 10
A probabalistic theory of aerosol penetration through glass fibre filters
1.. ~
.,~ *~-,.......,.
~ , ,~.~- " "
/. ~": -'-~._ j.~ ~
~.~,.'~
487
...__.. ~
. . . . . /-___ _,._<.......... /
/
/
f
/
//
/
Pmf mt
......
/ ,-" P0 ,.
I
/I #o@#
o o,.
/
,"
/
#s"
Code 108
##4, Id
V s =2I cmlsec = 0-943 L : 510,~
/
/
/
/
/
/
# #
t
t
0.2
01.4 :
i a 0'6
t
i
O.B
; I0
'
i i2
' I 4
', I C,
dp,
FIG. 20. Relative importance of various particle capturing mechanisms at low face velocity and varying particle diameter. P , = overall probability of capturing, all mechanisms combined. P° ~., -- overall probability of capturing by interception. P~., -- probability of capturing by pure interception. P~,,~,t = probability of capturing by interception increased by diffusion. P atl ~,, = probability of capturing by interception increased by diffusion. P , J~1 = overall probability of capturing by pure diffusion. P , ~, = overall probability of capturing by inertia.
TABLE 3. PENETRATION AS A FUNCTION OF HYDRAULIC PORE RADIUS
Filter composition
Xa
102 102 102 104 104 104 104 104 106 106 106 106 106 106 106
0.05 0.10 0.20 0.10 0"25 0-25 0"33 0"25 0"10 0-20 0"30 0.40 0"30 0-45 0.40
: : : : : : : : : : : : : : :
108 108 112 108 108 110 110 112 108 108 108 108 110 110 112
c
- log Pen
1/m~g
0-925 0-925 0-919 0"926 0-929 0"925 0"925 0"924 0"925 0"932 0"918 0"926 0"930 0-928 0"929
3.04 4-00 4.18 3-44 4"11 2"33 3"63 2"00 3-00 3"00 5"33 5"09 1"60 3-29 2"26
0.362 0.421 0.451 0.365 0-431 0"350 0"410 0-347 0.344 0"342 0"451 0"437 0"293 0"368 0"322
488
Lou~ A. CLARENBURG
~ -
/
/ J"
f
4_~ 10-3
P~ ,nt
f--~L..---~7-;'°'L o
.-."L...I-__.. -'-
.,I
I I
Pa
...,~
,,,"
~
/ P
.....
"-
...'/
" LT •4 '
/"~----
a .Q o . . . . . . . . . . . .
Code
t
108
V~ = 5 4
id 4
L
Pdff inr
cm/sec
= 530/~ = 0.943
,.. i 02
I, 0 4
i 0.6
; 08 do.
I I'0
', I 2
f 1.4
i 6'
/,~
Fro. 21. ReMtiv¢ importaac~ of various particle capturing mechaBisma at high face v¢|ocity and varying particle diamet¢r. Significance of symbols, ~ legend to Fig. 19. P,,,, s, = probability of capturing by inertial interception.
It should be pointed out that the present results were obtained with a dibutylphthalate aerosol. Some experiments have been repeated using paraffin and oleic acid aerosols; results were identical with those obtained with the DBP-aerosol. However, on using a di-octyl-phthalate aerosol penetration was significantly higher, while a benzyl-beazoate aerosol showed a remarkably different behaviour, penetration being more than an order of magnitude less. This may be due to evaporation of the more volatile aerosols. In the present study only fresh filters have been used. Duration experiments very clearly show clogging effects. Especially when solid aerosols are lacing filtered, clogging is the dominant factor. It is desirable that the present theory is extended as to include clogging. This the more so, as C L ~ U R O and SCHIERECK(1968) showed that small variations of the filter composition lead to orders of magnitude improvement of the filtration capacity of a filter, hardly affecting the resistance to air flow o f the filter. The present theory, combined with the theory of the pressure drop across a filter (PIEKAARand CLARENBURG,1967; CLARENBURGand ScmERECK, 1968), gives the tools to compute the optimal fibre composition of a filter. Either a composition can be computed giving minimal penetration at a stated pressure drop or, inversely, minimal resistance to air flow at a stated penetration.
A probalistic theory of aerosol penetration through glass fibre filters
V =6 25cm/sec d ; = 0 . 6 6 ~.rn
x
-5
x
489
×
-4
g
x
×
x
×
B I
X
-5
)o<
x
-2
-I 050
• 0.40
o 3o I /r%Q
FIG. 22. Aerosol penetration versus the inverse geometric mean hydraulic pore radius.
CONCLUSIONS
1. A pore model of a fibrous filter, as contrasted with an obstacle model, can adequately describe the complex process of aerosol filtration. The final expression for aerosol penetration does not contain any adaptation parameter; instead, directly measurable quantities related to particle, flow, fibre and filter properties only are employed. 2. To describe aerosol penetration through a fibrous filter, eight capturing mechanisms play a role, namely, pure interception, increase of interception due to inertia and diffusion, decrease of interception due to inertia, pure diffusion, pure inertia, inertial interception and inertial diffusion. The relative importance of these mechanisms of aerosol capture depends on the experimental conditions. 3. The present theory, combined with the theory of pressure drop, offers the possibility of computing the optimal composition of any filter with respect to some criteria and hence offers the opportunity for the design of optimal filters. 4. It would be desirable to expand the present theory to include electrical interaction between aerosol particles and fibres. For optimal filter design, the effect of clogging should also be considered. 5. There is no obvious reason why the present theory should not describe the filtration of particles dispersed in a liquid.
490
Lou~ A. CLARENBURG
Acknowledgements[ am very much indebted to the Director of the Chemical Laboratory of the Defense Research Organization TNO who offered me his facilities for computation. Without the invaluable help of the mathematicians G. J. Holland and later J. J. van Rijswijk, who wrote the computer programs, I would not have had any chance to perform this study.
REFERENCES
Cm~N, C. Y. (1955) Chem. gev. 55, 595. Ct.~ENBtmO, L. A. and Pmr,,~R, H. W. (1968) Chem. Engng Sci. 23, 765. C t ~ , m t m G , L. A. and PRtNCEN, L. H. (1963) Staub 23, 234. C t ~ N B t m o , L. A. and SCt-n~ECK, F. C. (1968) Chem. Engng ScL 23, 773. CLA~t,mtntG, L. A. and VAN DER WAL, J. F. (1966) Ind. Engng Chem. Process Design Develop 5, 1t0. Ct.M~NBURG, L. A. and WERNEIt, R. M. (1965b) Ind. Engng Chem. Process Design Develop 4, 293. DAWSON, S. V. (1969) Theory o f collection of airborne particles by fibrous filters. Harvard School of Public Health. Fucas, N. A. (1955) The Mechanics of Aerosols, Academy of Sciences USSR. L A r , a ~ , H. and H~rOO,NN, K. (1949) J. Collord Sci. 4, 103. LEVl~, L. (1953) Dokl. Akad. Nank. USSR 91, 1329. LANGMOIR,I. and BI..Ol~rr, K. (1946) Army Air Force Techn. Report no. 5418. L~d.~N, L. (1960) Z. angew. Phys. 12, 157. Mmm, R. E. (1964a) Proc. Natn. Acad. Sci. 52, 901. Mu.~, R. E. (1964b) Proc. Natn. Acad. ScL U.S.A. 52, 1157. ~ , H. W. and Ct.A~,lBtmG, L. A. (1967) Chem. Engng Sci. 22, 1399. ~R, H. W. and Ct.~NI)UI~G, L. A. (1967) Chem. Engng Sci. 22, 1817. R.ANZ, W. and WONG, J. (1952) Ind. Engng Chem. 44, 1371. R~d~PM'ORT, E. and WEINS'rOCK,S. E. (1955) Experientia 11, 363. VAN D~R W~d., J. F. (1968) Private communication. WERNER, R. M. and CtAI~EN6URG,L. A. (1965a) Ind. Engng Chem. Process Design Develop 4, 288.
Printed in Great Britain.
A PROBABALISTIC THEORY OF AEROSOL PENETRATION
THROUGH GLASS FIBRE FILTERS L o u i s A . CLARENBURG Rijnmond Authority, Stationsplein 2, Schiedam/Tbe Netherlands
(Received 20 April 1972) Ahstraet--A theoretical model is developed capable of predicting aerosol penetration through glass fibre filters. The filter is described as an assembly of pores which conduct the flow. Some of the aerosol particles, the interception fraction, move along with the tortuous flow; the remainder, the inertial fraction, pursue their paths through the filter along straight lines until they are re=entrained in the flow. Oftbe interception fraction of particles, a part is captured as a result of pure interception, a part due to an increase of interception by both inertia and diffusion, and the rest by pure diffusion. From the inertial fraction of particles, part is captured due to pure inertia, part due to inertial interception and part due to inertial diffusion. The final expression for the penetration does not contain any adaption parameter, only directly assessable parameters related to flow-, filter-, fibre- and particle properties. The final relation has been tested experimentally, by varying particle diameter, face velocity, porosity fibre diameter, and filter composition. Theory and experim~t appear to be in good agreement. A suggestion is made for optimal filter design.
C
d~ d*E DaM
f,
gt hi H
J
k K*(i,k) K(i.k) K(m~,)
-i L mh~ mh~
Mh I'lp Hpm
N N* N**
Ap Pen P,
NOMENCLATURE* Cunningham factor mean fibre diameter mean square fibre diameter effective layer thickness in the m e a n flow direction effective layer thickness in thc direction pursued by the inertialfraction of the aerosol particles diameter of an aerosol particle diffusion parameter of an aerosol particle due to Brownian motion inertialfraction of the aerosol particles originating from a flow of hydraulic radius mht fraction of the aerosol particles outside the flow of hydraulic radius m~t due to diffusion fraction of the aerosol particlesoriginating from a flow of radius m~s arriving at a fibrelayer with velocities and directions different from the air flow shadow effect subscript, referring to the j-th fibre component in the filter Boltzmann constant; also index referring to the k-th fibre layer upstream or downstream from the fibre layer under consideration apparent probability that an aerosol particle originating from a flow of radius ms~ is captured in the k-th downstream layer due to inertia K*(i,k), corrected for the probability of re=entrainment probability that aerosol particles originating from a flow of radius mhi are captured due to inertia mean fibre length thickness of the falter hydraulic radius geometric mean hydraulic radius maximum observed hydraulic radius, taken equal to 6 rnhg number of the pores per unit surface area in a single component falter idem, in a multi-component filter number of fibres per unit volume effective number of fibres apparent number of fibres pressure drop across the filter penetration of aerosol particles through a filter probability of an aerosol particle of being captured in the unit layer
*Symbols not l'sted are explained in the text. 461
462
Lou~ A. CLARENBURG
probability of an aerosol particle originating from a flow of radius ma~ being captured due to interception P,,y (m,~) idem, due to diffusion P~n (mh,) idera, due to inertia overall probability of an aerosol particle being captured due to interception Pa int idem, due to diffusion Pa at! idem, due to inertia Pa In probability of re-entrainment of an aerosol particle originating from a flow of radius mh, in the Pr, (i,k) k-th downstream layer surface area S time available for an aerosol particle to traverse a stretch in a direction perpendicular to air flow ta due to its inertial momentum idem, due to diffusion lair absolute temperature T face velocity of the air flow Vs lia~tr velocity of a flow of radius m~l VI linear velocity of an aerosol particle originating from a flow of radius m~ at the k-th downstream vt (k) layer distance traversed by an aerosol particle due to inertia perpendicular to the flow direction during Sd the time td idem, due to diffusion during the time td,t Xdtf Xj weight fraction of the j-th fibre component in the filter effective weight fraction of the j-th fibre component X*~ apparent weight fraction of the j-th fibre component X**j a,{3 angle porosity of the filter ~s critical porosity specific density of the fibres; also of the aerosol particles p viscosity of the air i¢ filter composition factor geometric standard deviation of the pore size distribution og A(In m.3 normalized pore size distribution T tortuosity factor volume flow through a pore of radius 2mht mean volume flow through a pore reciprocal forth moment of the pore size distribution inertial momentum of an aerosol particle originating from a flow of radius m~l in the k-th up~I (k) stream layer mean free path of the air particles. THEORETICAL
Aerosol penetration through a fibrous filter. CONSIDrat a single c o m p o n e n t glass fibre filter. Let the m e a n fibre d i a m e t e r be d, t h e effective m e a n fibre d i a m e t e r dE. T o derive a theoretical expression for the a e r o s o l p e n e t r a t i o n t h r o u g h such a filter, the following a s s u m p t i o n s have to be m a d e : a. I n a fibrous filter the fibres are o r i e n t a t e d in planes p e r p e n d i c u l a r to the m e a n flow direction b u t in these planes at r a n d o m . b. T h e p r o p e r t i e s o f a layer o f 2d E thickness are representative for the p r o p e r t i e s o f the whole filter; this m e a n s t h a t e.g. the pressure d r o p across a filter o f thickness L is L/2de the pressure d r o p across one single layer. c. A filter is c o n s i d e r e d as a system o f pores. I f the filter p r o p e r t i e s are a mere superp o s i t i o n o f the p r o p e r t i e s o f the c o m p o s i n g fibre layers, then the flow t h r o u g h such a system c a n be described by Poiseuille's law for the flow t h r o u g h circular capillaries. Let the weight o f a glass fibre filter be W, the surface a r e a S a n d the thickness L ; let P be the specific weight o f the glass fibres a n d 8 the porosity, then
A probabalistic theory of aerosol penetration through glass fibre filters
463
W = S L p ( I - e). If a series of filters is manufactured, keeping W, S and p constant, then L (I - e) = constant. The more e increases, the more L increases, and more space is available for fibres in the unit layer 2dE. Since S is kept constant this space is found by an increase of 2dE; hence the effective layer thickness and the filter thickness increase in the same proportion, whence
L/2dr = constant. As a consequence 2dE(1 - 5) = 2a(1 - es) = constant. It will be assumed throughout this work that the thickness of the unit layer is 2dE = 2a(1 -- es)/(1 -- e).
(1)
The composition factor, as introduced by CLARENBURG and WERNER (1965, a, b) appears to be independent of porosity; therefore the value of ~s for any filter follows by equating the experimentally determined composition factor re, defined as
~p K
~I,
rlLV, (1 - e) 3/2
and the theoretical expression, derived by CLAP.ENBUROand PIEKAAR (1968) 11"4 re,h = ~s (1 --
[nps
(0"858 x/l-865/esa2- 1.024) 2 x 0.388]
"~,)3/2[72 +
with
Hps
N~ (N~ - 1) rr
and 8
N~ = ~ (1 - 53
N~ 2
/// d~
Let the probability that an aerosol particle is retained in a fibrous layer of unit thickness 2dE be Pa; suppose that L/2dE unit layers are contained in the filter thickness L. Then the aerosol penetration through such a filter is
Pen = (I - Pa)Z'/2'~E or
L L(1 - e) log Pen = ~ log (1 - Pa) = 2a(1 - 5s) log (1 - Pa)
(2)
In the following sections an attempt will be made to derive a theoretical expression for Pa. AIR
FLOW
THROUGH
A FIBROUS
MEDIUM
In a theoretical study it was shown by PIEKAAR and CLARENBURG(1967) that filter properties can be satisfactorily described using a geometrical filter model, in which a filter layer is considered as a system of "pores" of various polygon shapes. Moreover it was shown (CLARENBURG and PIEKAAR, 1968; PIEKAAR and CLARE/~BURG, 1967; CHEN, 1965)that the
464
Louis A. CLARENBURG
flow through these pores is conducted by the hydraulic surface area. This concept offers the possibility of converting the various polygon shapes of the pores to one uniform shape, the circular hydraulic surface. The hydraulic radii mh of these surface areas were shown to form a logarithmic normal distribution, A (In roD, with invariant geometrical standard deviation a~ = 1.9. In this paper only laminar flow will be considered. Although a parabolic velocity profile might be in order over the cross-section of the entire falter, the velocity profile over the crosssection of a pore is definitely not parabolic, the depth of a pore, 2dE, being too small for establishing such a profile. In a filter an air molecule is continuously accelerated and decelerated; at the "entrance" of each pore air molecules, originating from various pores of the upstream layer, hence with various initial velocities, come together to form the flow through the pore under consideration. Therefore a homogeneous velocity profile will be
assumed over the cross-section of the flow in a pore. It is generally accepted that the airflow through a fibrous medium follows a tortuous path. Starting from the above assumptions, a theoretical expression could be derived for the tortuosity z PmKAAR and CLARENBURG, i ~=
I +
(0-858 ~/1-865/~ - 1"024)2 x 7.262m hg 4d~
, t3J
with de given by equation (1), and m2hg = 0.0295 72/n~ np =
N(N - 1) rr
8 N=-(1
(4)
N 2
(5)
dJ -- ~) ~ •
(6)
It is also generally accepted that the mean linear flow velocity in a porous medium is v = -- z.
(7)
However, as follows from the probability of finding an air particle in a pore of hydraulic radius between mh~ and mh, + dmh,, the linear flow velocity through a pore of radius mh, is proportional to the ratio of the volume flow ~ through that particular pore and the mean volume flow ~bv, through a pore. The flow through a filter tries to find a path of minimal resistance. Therefore, one can visualize a filter as consisting of a number of interlocking pores which form channels to which Poiseuilles law could be applied. Doing so, the ratio q~i/q~v can be written as
C~"--=~tnMh
m~t A (In mhi)d In mhl
.... ~, m~ e st.~,g
I ~"
with Mh = 6mhg, the maximal observed hydraulic radius (4) and q, = (In Mh -- In mhg -- 41n2ag)/ln ~rg.
-~ =m~i~b~ -'T" d#
A probabalistic theory of aerosol penetration through glass fibre filters
465
Hence, the linear flow velocity in a pore of hydraulic radius mhi is Vs
;'i = - -
* m~*i e e .
(8)
It follows from equation (8) that the linear flow velocity is very unevenly distributed over the cross-section of a filter, the largest pores contributing most significantly to the conduction of flow. This will prove to be of fundamental importance in the description of the trapping efficiency of a fibrous layer. AEROSOL F L O W T H R O U G H
A FIBROUS MEDIUM
If it is complicated to give a description of the airflow through a fibrous medium, the result of inertial and diffusional forces upon the flow of aerosol particles through such a system is even more difficult to model. Consider a fibrous layer of thickness 2dE in the mean flow-direction and a pore of hydraulic radius rnh,. in the upstream layer. Assume that at time t = 0 the direction of flow changes from A.C. (Fig. 1) to AB on the average over an angle ~, related to the tortuosity by the equation cos ct = 1/z. (9) As a result of inertia, a fractionf~ of the particles conducted by the pore under consideration cannot follow this abrupt direction change and will pursue their course through the filter, in the first instance independent of the airflow.
Mean flow A
direction
FIG. 1. Tortuosity in a fibrous filter.
On going from A t o B the aerosol flow will lose a fraction g~ of the particles as a result of particles diffusing outside the airflow. Hence from any pore mhi in the first upstream layer a fraction 1 - f~ - gi of the aerosol particles will reach the layer under consideration in the airflow, i.e. with approximately the same direction and velocity. From a pore of hydraulic radius mhi in the first upstream layer a fraction h+ = fi + gi approaches the layer under consideration with direction and velocity different from the arriving airflow. However, going from A to C (Fig. 1) part of fraction h~ is re-entrained in the airflow, depending on inertial momentum of the particles and airflow velocity encountered. If the probability that a particle is re-entrained between the pore under consideration and the next down-stream layer is denoted as P,e (i,0,) then a fraction of theaerosol particles h* (1) = hl [1 - P,e (i,0)] (10) will arrive at the layer under consideration with velocity and direction different from the arriving airflow. Consider now a pore of hydraulic radius rnhl in the second upstream layer;
466
Lou~s A. CLARENBURG Fraction captured
Fraction
re-entrained in flow
h•l(2) [-~--~,-~ hi( I1 (I-K*(i,l)) P,e{i,I) h~'( I ) ( I - K ( i , I ) ) hT(
' I.}K * (t,l)
~
1I
PtuP Stream
h'~( I )
foyer
• h;P,,
i,O) 7'"~
upstream layer
Mean
lflow
] direction
FIo. 2. Inertial fraction of aerosol particles; losses due to capturing and re-entrainment. a fraction h* (0) arrives at the first upstream layer. Here a fraction K*(i,1) is trapped, so that a fraction h* (1) [1 - K(i,1)] approaches the layer under consideration. However, a fraction P,e (i,1) is re-entrained, so that finally the fraction h* (2) = h, (1 - P,e (i,O) (1 - P,, (i,1)) (1 - K* (i,1))
(11)
arrives at the layer under consideration from the second upstream layer (Fig. 2). Extending this reasoning to layers further upstream, it can be easily verified that the fraction of particles originating from a pore in the k-th upstream layer, arriving at the layer under consideration is k-1
h*(k) = h, I-I [1 - P,, (i,p)] [I - K*(i,p)],
K*(i,0) = 0
p=O
hi < 1.
(12)
It will be shown in a subsequent section that P,~ (i,p) and K*(i,p) are dependent on the size of the pore of origin only [equations (37) and (36)], so that the generalized equation (12) is an exact description of the fraction of aerosol particles, originating from a pore in the k-th upstream layer, arriving at the layer under consideration independent of the airflow. As can be seen from equation (I0) a fraction h~P,, (i,O) re-enters the airflow between the first upstream layer and the layer under consideration, l~tween these same layers a fraction [see equation ( I 1)] h~ (1 - e,e(i,O)) (1 - K*(i,1) P,~ (i,t) = h* (2)
P,e (i.1) 1 - - P r e (i,I)
re-enters the airflow originating from a pore in the second upstream layer. Originating from a pore in the k-th upstream layer [see equation (12)] a fraction
h* (k)
P,,(i, k - 1) 1 -
P,e
(i,k
-
1)
re-enters the airflow between the first upstream layer and the layer under consideration. In total, a fraction H~ re-enters the airflow between the first upstream layer and the layer under consideration, H~ given by
A probabalistic theory of aerosol penetration through glass fibre filters Hi = L h* (k)
k= 1
P~e (i,/~
1)
1 -- ere (i,k - 1) '
H* (i,0) = 0 .
467 (13)
Hence, the fraction of particles arriving at the layer under consideration in the airflow, i.e. with approximately the same velocity and direction, originating from a pore of hydraulic radius mhi in any of the upstream layers, is
Fi = 1 - hi + H i .
(14)
As a consequence, the fraction of particles arriving in the air flow at the layer under consideration is
f 'n~h Fi m ~iA (ln mh,) dln mhi F =
-
(15)
f.i~Mh m4hiA (ln m,i) dlnmh, This fraction of the particles will be referred to as the interceptional fraction; the sum of the fractions h* (k) averaged overall flows will be named the inertial fraction. THE
fi
FRACTIONS
AND
g~
Starting from Stoke's law it is easy to find an expression for fi. If at time t = o the airflow in a pore of hydraulic radius mhl changes direction over an angle ~, then the velocity of the particles at that moment in a direction perpendicular to the new flow direction vl sin ct. After a time t d the distance Xd travelled by the particles in a direction perpendicular to the new flow direction, can be immediately found by integrating Stoke's formula twice over the time, resulting in
Xdi (~) = ~ v sin l • (1 - e -~p'~) 2p with
(16) 18//
/.p = p dp2 The available time td can be found approximately, by recognizing that the distance covered by the airflow between two neighbouring layers is 2d E r and by assuming that the velocity remains constant hence tdl =
2dEz
(17)
Vi
The actual directions of the particles can assume any angle between ~ = 0 and ~ = ~max as defined by equation (9). Therefore, by integrating xdi (ct) over all possible values of ct an estimate is obtained for the mean distance Xdi, Vi[ '
Z--1
] [ 1 - - exp (2, tdi)].
(18)
According to the theory of tortuosity the flow originating from a pore of hydraulic radius mhi, splits up, on the average, in 4 smaller flows with a probability of over 99 per cent. If the hydraulic surface area of the pore is 4n m2ht, then, assuming circular flow (Fig. 3a), the cross-sectional area of the flow is n m2hi with radius mhl. As under the influence of
468
LOUIS A. C'~I~NeURC~
( 1
C
mh;
(a)
(b)
FIG. 3. Aerosol flow in a filter. (a) inertial shift of the aerosol particles; (b) expansion of the aerosol flow due to diffusion. inertia all the particles contained in this flow move a distance .rat in the same direction, a fraction of the particles represented by the surface area ABCD has come outside the airflow after a time t4,. According to elementary geometry the surface area ABCD is given by x,
S(ABCD) = x~
mR - -T + 2 m~, arc sin \2
ms,/
A=
~/
,
or
~m~
~-+-arcsin
>,~=~
F~hi
or in close approximation, as xd, ,~ mh~ most of the times, f~ = 2Y--2.
~'t9)
To find an expression for g~ use has been made of the statistical theory of Langmuir as reported by CrImq (1955). According to this theory, the average absolute value of the displacement of a particle in the time td~ along a certain direction is given by
xd~f,, = (4Dnu td,) I/z
(20)
with
CkT
Dnn = ~ 3~z r/dp and
C=
1 +~
°[
2.46+0.82exp
(21)
(
-0"44
.
(22)
Here C is the Cunningluml factor, with co the free path of the air molecules. As under the influence of diffmion the ~ r t i d e s have moved a distance xa,i in all directions (Fig. 3b), the fraction of the particles getting out of the airflow in the available time td, is represented by the surface area of the annulus with thickness xd,~,,i. Hence (m,, + xas,i) 2 - tr m~ gi =
7r rn~,
2 xd,I, i ~
~
mki
(123)
A probabalistic theory of aerosol penetration through glass fibre filters FILTER
469
MODEL
The mean number of vertices per polygon in a random line array is 4, as was theoretically shown by MILES (1964a, b); using a Monte Carlo technique this same number was experimentally found by PIEKAAR and CLARENBURG (1967). These last authors showed both theoretically and experimentally that the conversion factor from hydraulic pore area to real pore area is 1.28. This is almost exactly the ratio of the area of a square to the area of its inscribed circle. It is logical therefore to assume that the mean port shape is a square the flow being conducted by its hydraulic surface area (area of the inscribed circle).
INTERCEPTION
FRACTION
OF P A R T I C L E S
Consider the particles arriving in the flow at the " e n t r a n c e " of a pore of radius 2 mhi. According to the theory of tortuosity the total flow originates on the average from 4 different directions; hence in each pore 4 sub-flows will be distinguished (Fig. 4a), each of hydraulic area n m~i. Here the dominant influence of the larger pores on the description of the behaviour of a filter is clearly demonstrated, as was emphasized previously [equation (8)].
@ QJ MA=r AC=x S(ABCD}=S(r,x}
Flow in o pore
(o)
M~.=r MN=CF =n AC=z, S(BCDEFG)=S(r,x,o)
(b)
(c}
FIG. 4. Geometry of aerosol flow in a filter.
In the following, it will be assumed that an aerosol-particle of diameter dp is trapped if the trajectory of its centre would pass a fibre at a distance smaller than dp/2. The following notation will be used. In Fig. 4b the surface area of the segment ABCD will be denoted as S(r,x). According to elementary geometry S(r,x) = r2 arc cos ( 1 _ x ) _ (r _ x) ~/ x (2r _ x) '
x >~ O.
(24)
The probability that a particle is captured will be denoted as P(z), given by P(z)
=
S(r, x) 7~r 2
-
1
1 arccos (1 - z) - - (1 - z) -,/~ (2 - z).
x
7z
(25)
'
In Fig. 4c the surface area B ' C D ' E F G will be denoted as S(r, x, a); in close approximation S(r,x,a) = 2a x/x(2r-
x),
x i> 0.
(26)
470
Louxs A. CLARENBURG A
Infercepfion MA :r=m.~ CA=x=d /2 (o) F //////X/////'/~
Q) ..rncrease of infercepiion due t'o inertia MN = CF = xdt MN Ivf2 = C~Ft-" xd,l~
Decrease of interception clue t o iner-I'io MN = A B = x~ B~C~: '/2d o - x d / ~
MC = m . - d p / 2
BC = '/zd~- x,j
F ......
MC ~= m . - dp / 2 (b)
M A = N B : m., F /z/////y//z//A
F
N8 j ~ m , ~ (c) F
#III/I~I////A
7( ..................
Increase of intercep~rion clue to diffusion CF:x=fi i CF : x~,f,
2ncrease of interception due to diffusion
CF =x~,f,
CF = x ,~
NC=m.,-~/zdr~-xa/J~
NC -" mhi- ~'z d ~ - x~,
(e)
(dl F
MC = m. -I/z d~
(f) (g) FtG. 5. Mechanisms of aerosol capturing in a pore. The probability that a particle is captured P(w, z) is then
S(r, x, a) P (w, z) m
r2
2
= -w
x/-(2 - z).
(27)
T h e particles in the flow are subjected to various effects, all acting simultaneously. Each individual effect can only be described in a probabilistic sense, assuming the presence o f a great number o f pores and particles.
A probabalistic theory of aerosol penetration through glass fibre filters
471
The first effect to be considered is interception (Fig. 5a). Starting from a homogeneous distribution of particles over the cross-section of the flow the probability of interception P(z~) is given by the ratio of the area shaded to the cross-section of the flow. Superimposed on pure interception is the increase due to diffusion (Fig. 5t, shaded area); the probability of this effect is P(w2, z~). Under the influence of inertia the centre of the cloud can shift a distance M N = xa in either direction. As an approximation only 8 possible directions (Fig. 5g) are considered in this study; the resulting 8 overall effects will then be averaged as if representative for the behaviour of a filter. The increase o f interception due to inertia is in order if the centre shifts towards the fibres (Fig. 5b); the two probabilities considered are P ( % , z~) and P ( % , z~) respectively. Superimposed is the increase o f interception due to diffusion (Fig. 5d); the two probabilities considered are P(w2, z , ) and P(w2, z~) respectively. But also a decrease o f interception due to inertia has to be taken into account if the centre shifts from the fibres (Fig. 5c); the corresponding probablities are P ( z 3 ) a n d P(z2)respectively. The probabilities of the corresponding diffusional effects (Fig. 5e) are P ( w 2, z3) and P ( % , z2) respectively. The overall capturing probabilities in each of the 8 directions are given in Table 1, and averaged to yield the apparent probability of interception [equation (28)] in a pore of radius 2rnh~. TABLE I. COMPOSITION OF EFFECTS Direction
1 2 3
4 5 6 7
8
Capturing probability
Symbols.
2 P(zl) + P(wl.zl) + P(%,zi) + P(w2,z.O 2 P(zl) + 2 P (w3,zl) + 2 P (w2,zs) 2 P (zl) + P (wl,zl) + P (w2,zl) + P (w2,z~.) P (zl) + P (z2) + P (%,zl) + P (w2,z2) ~- P (w2,zs) P (zx) + P (za) -- P (wz,z3) 4- p (w2,zl) 2 P (z2) + 2 P (w2,z2)
zl = dp/2m~j z2 = (dp = xa~ "V2)/2mh~ za = (dr, = 2x,,)/2mht wl = xddmhl w2 = xals dmhi w3 = wlx/2. P (zl) + P (z3) + P (w2,z3) + P ( % , z t ) z . = (dp + 2xdt)/2m~l P (zl) + P (z2) + P (wz,z~) + P (%,z2) + P (w2,z~) z~ = (d~ + xa~X:'2-)/2m~
Pi*t (mhi) = ~g [10 P ( z l ) + 4 P(z2) + 2 P(z3) + 2 P ( w I, zl) + 4 P ( % , zl) + + 4 P ( % , zl) + 4 P ( % , z2) + 2 P ( w 2, z3) + 2 P(w 2, z4) + 4 P ( % , zs)]
(28)
However, as was discussed before, a fractionf~ of the particles escapes the airflow due to inertia, moreover a fraction g~ escapes due to diffusion, whereas a fraction H~, originating from a pore of same radius somewhere upstream, re-enters the flow. Furthermore, the probability of finding a particle in a flow of radius rnh~ should be proportional to the probability of finding a volume flow ~b~.As a result the probability of interception in a pore of radius between 2mh~ and 2mh~. + dmh~ is P i n , (mhi) =
(1 - f i
- gl + H,.) ~b--2/A(In trlhi) Pi*t (mh3dln mhi.
This probability averaged over all pores in a unit layer of the filter yields P~.i,~ = cbv
~ •/
In
In
Mh
rn'~i Pi,~ (mh,) A (in mhl) d In mhi m m
with Mh = 6 mhg and 2 rn m = dr~'2.
(29)
472
Louis A. CLAP.~r~SUP.G
Under the influence of diffusion the supposed circular aerosol flow expands from a radius mh~ to a radius (mh~ + xat, 3. According to Fig. 4a this flow is on two sides bordered by fibres, hence the apparent probability of an aerosol particle of being captured in a flow of radius mh~ is approximately Xdif, i
Pa~I (mh,)
mhi
Consequently
PcHf (mh3 = (1 - Ji - g~ + H3 ~2 A (In m,,) Pdi*f (mh~) d i n mhi and Pm d i f = ~ v
INERTIAL
f
in Ma
,din ma
m~i P d l f (mh/) A
FRACTION
(In m h i ) d In mhi
OF THE
(30)
PARTICLES
Consider now the fraction of particles emerging from theflow in a pore of radius 2mh. hence traversing the filter with velocities and directions different from the flow, in a way independent of flow. Let the inertial parameter of a particle originating from a pore of radius 2mhi in the k-th down-stream layer be defined in the usual way
C v~(k)
~, (~)
= ~
~31)
;.p d~
Here v~(k) is the velocity of a particle arriving at the surface of a fibre, after having traversed a stretch kd~. The initial velocity of this particle, when it emerged the flow, is v~ as defined by equation (8); as a result of friction with the air the velocity v~ (k) is approximately g i v e n by
v~(k)= vi -
,
*
Vs
k z p d E, v ~ ( k ) / > - - z ,
i32)
The directions, at which the particles emerge from the airflow, can assume any value between the angles ~ = 0 and ~ = ~ . . . . with ~m.x defined by equation (9). Hence x/r 2 d~ = 2delcos~ = 2 d e
1
r arc cos (t/z)
.
(33)
For the effectiveness of precipitation of a particle on a cylinder as a result of its inertial momentum, the expression proposed and experimentally verified by LANDAHL(1949) was adopted. Landahl's expression was shown to give an adequate description of the experimental data for Reynolds numbers in the vicinity of 10. As the major part of the flow is conducted by the largest pores, it can be verified using equation (8) that the Reynolds number in a porous medium can easily assume values not too far from 10. For this reason the effectiveness of precipitation is written as: [~i (k) 3
Pi.* (m~,i, k) = ~bi (k)3 + 0"77 Oi (k) 2 + 0"22"
(34)
A probabatistic theory of aerosol penetration through glass fibre filters
473
The reason that this expression was adopted for the effectiveness of inertial precipitation is that it has a reasonably good experimental foundation (LANDAHL and H~.gMAr~N, 1949; RANz and WONG, 1952). According to various authors (LANGMUm and BLODGETT, 1946; LEVIN, 1953; FUCHS, 1955) Pin* (mhi, k) = 0 for ~bi (k) ~< 1/16
(35)
Now, three effects should be considered. The probability that a particle is captured due to inertial precipitation is proportional to the ratio of the total surface area of the fibres perpendicular to the mean particle direction, N ' d / and the total surface area 72. The probability that a particle is captured due to inertial interception is proportional to the ratio of the surface area formed by the total length of the fibres and two times the particle radius N*dp7 and the total surface area 72. Finally, the probability that a particle is captured due to inertial diffusion is proportional to the ratio of the surface area formed by the total length of the fibres and a distance x*d~i, bringing the particles to within a distance d~/2 from the fibres, 2N* X*d~.e7 and the total surface area 12. Combining these three effects the apparent probability K* (i k) that the particle emerging from the flow in a pore of radius 2mh~ are captured as a result of inertia in the k-th downstream layer is N*
K* (i, k) = T
(3 + dp + 2 xd*I, i) Pi* (mhi, k) .
Here N* is the number of fibres in a layer of d E [equation (33)] thickness and is given by equation (6); x*diy, ~is given by equation (20), with t*, i given by
a~ ~'~(k)
t~i = ~
(36)
so that K*(i,k)
4(1
d~
~
5)~(a
+dp + 2
* ,)P* (rnh,,k)
Xdif,
•
(37)
However, as was shown before, only a fraction h* (k), as given by equation (12), arrives at the k-th downstream layer. The expression for h* (k) contains the probability of reentrainment. It will be assumed that a particle is re-entrained in the flow if it encounters on its path through the filter a flow of which the velocity is equal to, or greater than the actual velocity of the particle. The velocity of the particle, as given by equation (31), can be equated to the flow velocity in a pore of radius 2mhk as given by equation (8). Hence the probability of re-entrainment is equal to the probability of finding vk >/ v~ (k) or
v*m~ >>.v*m~i - k 2p d~ or mnk Thus
>>- (m~i --
k 2p d ~ / v * )
1/4 .
P,, (i, k) = e [Vk >>-V, (k)] = P [mhk ~ (m~i -- k 2p d~ /v*) x/*] Mh
fin =
A (In mhi) din rn,i
j In ,~.k
f ld~ ' "l A n( l nmr n hhi ) , _
(38)
474
Louts A. CLARENBURG
Now, the probability that particles originating from a pore of radius 2mh~ are trapped in the k-th downstream layer as a result of inertia, is
K(i, k) = h* (k) K* (i, k), K* (i, o) = O. Hence, the overall probability that particles originating from a pore of radius 2mh~ in some intermediate layer are captured as a result of inertia in the k-th layer, is given by k
K(mh~) = ~ K ( i ,
p), p = 1 , 2 , 3 . . . k .
p=l
The probability that a particle is captured in a unit layer of 2de thickness in the filter as a result of inertia is found by averaging K(mhi) over all pore sizes, whence 2
P.,~.
F?
K (mhi) A (In mh~) d In mhi
-
(39)
f'ThA(lnmh,)dlnmh,_ the factor 2 allowing for the fact that each fibre is part of two layers. THE
PROBABILITY
OF
CAPTURING
The probability that a particle is captured in a (mit layer of 2d~ thickness P~, is evidently given by Pa = Pa, i.t + P~, aly + P~, i. + P.l (40) where Po, ~., is given by equation (29), P., ail by equation (30), and P., i. by equation (39). Here P.t stands for the probability that a particle is captured due to electrical attraction. Since, in the present study, attempts were made to avoid this effect, Pet has been neglected. Although the final expression for P= is a very complicated one, it does not contain any adaption parameter; it contains the flow properties v,, q, k, T, and o~; the fibre properties d ~ a n d l ; the filter properties 8 and L; and the particle property dp. Hence equation (40) is a real predictor. AEROSOL
PENETRATION GLASS
THROUGH FIBRE
A MULTI-COMPONENT
FILTER
With the aid of the foregoing it is relatively easy to model the aerosol penetration through multi-component glass fibre filters. Consider a n-component glass fibre filter. Let the weight composition of the filter be denoted as Xj,X2 • • • X.; ~ Xj = 1 . (41) J
If the mean fibre diameter of the j-th component is aj and its mean fibre length is 7j, then it will be supposed that
~, <+72<... <,7, and 7~ <72 < . . . < 7 , where the suffix 1 refers to the finest component and n to the coarsest. The properties of a layer of 2dE thickness are no longer representative for the properties
A probabalistic theory of aerosol penetration through glass fibre filters
475
of the whole filter. Instead, the structure of the n-component filter can be taken as if the grids of the various components, each present in the weight fraction Xj are telescoped into each other, while from each grid the fraction 1 - X~ is being occupied by fibres of other components. Again the value of e, for any one filter follows from equating the experimental composition factor and the theoretical as defined by CLARENBURGand SCHmREeK(1968) 11 "4 =
glh
with
r/p~ .2
- - __
e, (1 - e~)3/2 1~ r = ~Xjr: j=l
and
1i(
] 1
n,~,, = 7z'1~ ~ Nj~1, - ~ Nj, 7~ - ~ ~ Njs j=l
,
j=l
7.2 a./
o gjs
=
(42)
j=l
-- X j (1 - e~)~-,j~-~.
(43)
7Z
The tortuosity factors ~j are given by equation (3), in which rn~g = 0-02#51~ /npm
(44)
with npm given by equation (42) if the subscript s is dropped. Keeping in mind the reasoning behind equations (42) and (43), i.e. that the properties of a filter are considered as a superposition of the properties of the unit layers 2 a j, leading to the filter property expressed by equation (44), it is evident that equation (29) expresses the probability of particle capture due to interception in a multi-component filter, equation (30) expresses the probability that it is captured due to diffusion. The description of capturing of particles due to inertia becomes somewhat more complicated, as the inertial parameter ff is a function of the fibre diameter C v i (j, k) ~k~(j, k) = - (45) 2 d*~.E with d*jEgiven by %IT 2
d*iE = 2 aj~
~o,coo,
--
1
(I/z)
(46)
and v~(j, k) by vi(j, k) = v i - k 2pd~:j.
(47)
As a consequence the probability of inertial precipitation P*i. on a cylinder becomes a function of the fibre diameter, P*~, (m~, j, k), given by equation (34) at the condition expressed by equation (35). The probability that a particle, pursuing its path through the filter, encounters a fibre with diameter a j, P(aj), within a stretch d*.~, is
3.
a.
P(aj) = ~ N'~.* l j/ j=l -~j N'j* 7j
(48)
with N ' j * the apparent number of fibres of diameter a: in a layer of d*j~ thickness as given by equation (46).
476
Loum A. CLARENBURG
Following an aerosol particle on its path through the falter, it is more than proportionally likely that it encounters a fine fibre, as part of the surface area of the coarse fibres is screened from the flow by the fine fibres and vice versa. This effect, introduced by CLARENBURGand WeRNER (1965b), was named the Shadow-effect; it was shown to play a predominant role in aerosol filtration. Consider a 2-component glass fibre filter. Let the surface area of a Coarse fibre screened by one single fine fibre be denoted as S ~ ) , with fl the angle between the coarse and the fine fibre; moreover, let the maximum number of possible intersections of fine fibres with one coarse fibre be denoted as NI, and the probability of intersecting as Pc, then the Shadow-effect H can be expressed as
s~)
Ht2 = ~
Ny Pc.
(49)
Consider a volume element (at + a2) It 12. The number of fine fibres in this element is 4 N 1 = - X I (1 - 8) (aa +
a32)72/d2-
7t
If they all intersect the coarse fibre then N1 = N¢. The probability of intersecting was calculated by I~eKXAR and CLAm~BtmG (1967) to be Pc = 2/n. If a fine fibre intersects a coarse fibre at an angle/L then the surface area screened is
S([3) = dl ~]sin ft. At very small angles 0 t> p t> /~g the area screened is S(p < p,) = a~ 1~ with fie defined as sin fig = a2fll" The probability of finding an angle between fl and fl + dfl is 2/n dfl, consequently
S
2 [',/z al ~2
2
Recognizing that for 7j >> d 2, flo approaches to a2/ll, and solving the above equation, ,~ is approximated by S = 2 ~1 a2(1 + ln271~ "~-2] as a result 16 a,( in 211~ H12--~-~X1 ( 1 - e ) ( a t +a2)d-~ 1 + ~2]" In the 2-component filter considered the fraction Hi2 of the coarse fibres is screened from the aerosol flow, whence the effective fraction of the coarse component is (1 - x 0 *
= (l - x 0 (l - / - / , 2 ) .
(50)
Similarly, it can be proved that the ser~ning of the fine fibres by the coarse, can be expressed as
A probabalistic theory of aerosol penetration through glass fibre filters
477
16 ~2 ( In 211~ H21 =~-3(1 -- X~)(1 - ~ ) ( a I +~2)d~22 1 + "~'2] " Hence the effective fraction of the fine component is X~ = X 1 (1
(51)
/-/21) .
-
From equations (50) and (51) it follows that the apparent weight fractions are
Z*p = X , / { X ~
+(I -X0*}
andX** = l - X * *
consequently N** = 8X** ( l n
- e)~--~"
(52)
For multi-component filters the considerations are more complicated, though qualitatively the same. The apparent probability K* (i, j, k) that particles emerging from the flow in a pore of radius 2mh~ are captured as a result of inertia in the k-th downstream layer by a fibre of diameter a: is
K'* (,,j,k) = ~
(~j +dp + 2 Xdi*f , i,j) Pi* (mhi, j, k) P (aj) ,
where X*dif, i, S is defined by equation (20), with
td,.j=
d]*r,
(53)
v~ (j, k)"
The apparent probability K* (i, k) that particles emerging from the flow in a pore of radius 2mh~ are captured as a result of inertia in the k-th downstream layer is evidently
1. a&
K*(i,k) = ~ K*(i,j,k) = ~ 4 Xj(1 - s) j=l j=l lj d~
(a: +dp + 2 Xd,*1.i.j)P,* (mh,,j,k) P (aj).
(54)
As the particle velocity v~ (j,k) is considered, according to equation (47) relative to a layer of d*jE thickness and if the probability of re-entrainment is considered relative to the same layer, equation (38) remains valid; also equation (12) still holds if expression (54) is substituted for K* (i,p). Therefore equation (39) expresses the probability of particle capturing due to inertia in a multi-component fibrous filter. The aerosol penetration through a multi-component glass fibre filter follows from equation (2), recognizing that the properties of a multi-component filter are derived from a superposition of the properties of the unit layers formed by the individual components. As only the effective fraction of each component takes part in the filtration process
logPen = 7
=1 kaj]J (1
log(1 - Po) ;
(55)
with, for a 2-component filter, X1 and X2 given by equations (51) and (50) respectively, and P, by equation (40).
478
Louis A. CLARENBURG EXPERIMENTAL
To test the theory presented in the foregoing section an aerosol generator should be available capable of producing monodisperse aerosols of various diameters. Moreover, the aerosol particles should be presented to the test filter at various face velocities. Finally the filter properties should be varied. Preparation of the filters has already been described in detail by WERNER and CLARENBURG (1955a), the experimental set up by CLARENBURG and VAN Dr.R WAL (1966). All relevant experimental conditions have been summarized in Table 2. TABLE 2. EXPERIMENTALCONDITIONS
Aerosol generation
Aerosol Droplet diameter Droplet size determination Aerosol concentration Concentration m c a s ~ e n t Penetration m ~ s ~ t Volume flow Face velocity Filter Fibre diameter in microns Square fibre diameter in sq. microns Fibre length in microns Determination fibre properties Filter weight Filter porosity Discharge of filter electricity Computer used
Modified La Mer generator (CLARENBURGand VAN DER WAL, 1966; LASSEN,1960; RAPPAPORTand WEaNS'rOCK.1955) Dibutyl phtalate (DBP) monodisporse, 0"3-1"6 micron higher order Tyndall spectrometer 40 t,g/liter Forward scattering optical concentration meter (CLARENBURG and PmNC~N, 1963) 450 1/hour 2-90 cm/sec. Prepared from John Manville's #ass fibres code 102 104 106 108 110 i12 0"207 0"275 0'456 1"02 1"49 2"00 0"089 0"191 0"433 1"99 5"10 7"34 171 195 377 478 522 990 Electron microscope (CLARENBURGand WERNER,1965a, b) 80 g/m2 0.88-0.96 Prometium source, 100 mC CD 3200
RESULTS
As was mentioned above, particle, flow, filter, and fibre properties are the constituents of the formula to be tested. For practical reasons, all of the single-component experiments were carried out with filters manufactured from code 108 fibres; filters prepared from code 106 fibres had a penetration too low to measure. Filters prepared from code 110 or 112 fibres appeared to be too weak, mechanically. In Fig. 6 to 10 results are shown when the aerosol particle diameter is varied (variation of particle properties) at constant face velocity; the experimental conditions are indicated on the figures. The points are experimentally determined, the drawn lines are calculated using equation (2). Figures 11 to 14 show the results when face velocity is varied (variation of flow properties) at constant particle diameter. In Fig. 15 results are depicted when porosity is varied (variation of filter properties) at constant particle diameter and face velocity. F o r the variation of fibre properties, two-component glass fibre filters were studied. O f the many available data, only a few were calculated using equation (55). They were selected to cover a wide range of fibre diameters (see Table 2). Figures 16 and 17 show the results of aerosol-penetration through filters manufactured of mixtures of code 102-110, 102.-112 and 106-112 respectively with varying particle diameter and at constant
A probabalistic theory of aerosol penetration through glass fibre filters x
] 0 -b
~, I:L
x
i0 -2
Vs = 2.I cm/sec
\
L = 510/.c { = 0.943
I 0 -~
I
f
I
\ \
\
I
i
i
I
I
I
I
0 0.2 0.4 0-6 0.8 1.0 F2 1.4 I-6 I-8 2-0 dp, ~.
FIG. 6. Aerosol penetration versus particle size at face velocity of 2-1 cm/scc.
Q X
~_I~~
×~×
Code [08 Vs= 6.4 cm/se¢
L =430/~ e =0"932
x
IO -3
I
1
l
I
I
I
l
o-z 0.4 o.6 o.e I-O 1.2 i-4 dp.
F
FIG. ?. A e r o s o l penetration v e n u s particle size at face velocity o f 6-4 cm/scc.
479
480
Louis A. CLARIENBURG
x\
0,,.
Code 108 V~=6-4 cm/sec L = 510~. =0-941
jO-~
L
00"Z
i,
]
x
I
f
0 . 4 0"6 0"8
~
[
I
I'0 1.2 1.4
f
1"6
dp,
Fro. 8. Aerosol pen~ration versus particle size at face velocity of 6.4 cm/sec.
t0"1
~
\
tc~2i x a.
Code108
'~ ' g2 o!. g., °'8 io ,'2.4~'6 d~, ~.
Fzo. 9. Aerosol penetration versus particle size at face velocity of 30 cm/sec.
A probabalistic theory of aerosol penetration through glass fibre filters
\ io -1
\ 8 {L Code 108 \ Vs = 5 4 crn/sec x
L
=
530 ~
= 0'943 16:
x
I~ 4
I
I
l
;
I
I
I
0 0.2 0.4 0.6 0.8 t-o 1.2 dp, p.
FIG. 10. Aerosol penetration versus particle size at face velocity of 54 cna/sec.
x
Code
lOB
dp = 0 . 3 6 #
4T
~L
/~-~-~"~--.~-
L (
• 530/~ = 0'943
~
r×
X
Id' 9
o..
,to
3
~o
31o 20
~o
&
;o
Jo
•x
2 /"
Code I 0 8
d~0,.0'38 5 3 0 /~ 0'943
1Of'
r×
x
8
0
x T"
I ro
I 20
I 30
I 40
1 50 Vs~
I 60 crn/sec
r 70
f so
x
F1o. 11. Aerosol penetration versus face velocity at panicle diameter of 0.36 ~ and 0'38 ~.
481
482
Loura
A. CLARENllURG
cod, ,08 x
dl = 0 , 6 6 ~ E = 5 1 0 p. • = 0-937.
x
~.
--<<.
I02
I~ ~
I I0
0
I 2CI
I 30
I 40
f 50 V~
T 60
t 70
80
FIG. 12. Aerosol penetration versus face velocity at particle diameter of 0.66 ~,.
x ~
× ~
Code 108 dp = 0 " 6 6 ~ L "- 5 6 0 ,~ = 0 - 9 4 5
10
I l0
l 20
[ 30
I 40
x
I SO
Ys)
T 60
; 70
i 80
: 90
c m / sec
~ O . | 3 . A e r o s o l l x ~ e ~ r a t J o n v e n u s face v e l o c i t y at p a r t i c l ¢ d i a m e t e r o f 0.66 ~.
A probabalistic theory of aerosol penetration through glass fibre filters
"x ×
Code
108
d.p = 1 . 4 0 F ×
L = 530/~ ~ = 0.945
-->..
i5'
IO 5
i 20
;
0
I0
' 30
! 40
I 50
I 60
[ 70
T BO
Vs FIG. 14. Aerosol penetration versus face velocity at particle diameter of 1.40/~.
5
~< × ×
4 13_
l Code
2
I08
x
dp=O.66/z Vs= 6 , 4 cm/se¢
I
x~ 'xx~ × ×
x
L = 500 F
I
0-88
~
x
0.89
0.90
I
0.91
I
0-92
I
0.93
I
0.94
FIG. 15. Aerosol penetration versus porosity.
]
0.95
483
484
Louis A. CLARENBURG
Code 102,112 X 1: 0.0834
\ 10-3
%
~ x ~
x
L
~
\
: 450
= 0"936 V5 = 6 ' 4 cm/sec ~t
eL B.
Code10210~ Eo"
f
}
io-z
X, • 0'23
L =.380/~ %
• "0"921%* V3 • 6;4¢~e//.
IG5
f l ~ I II I 0.2 0.4 04 08 1.0 t.2
I i, T i l l l l I [ 0"2 0.4 06 0.8 1.0 l.Z 1.4 1'6 1.8 2.0 dp,
F,o. 16. Aerosol penetration versus particle size at face velocity of 6.4 cm/sec for filters 102 : 112, respectively
i0
Code 106 x, = 0 4 0 16"
x
\
-'0"929 V3 = 6 . 4 c m / s e c
I
[
~
I
I
0.2 0.4 0.6 0 8 I-o i-2 d~,, M.m
it4
1.'6
FIG. 17. Aerosol penetration versus particle size at face velocity of 6.4 cm/sec for filter 106 : 112
A probabalistic theory of aerosol penetration through glass fibre filters
485
1
×
io-
xX x
X
Code 1 0 2 : 1 1 2 X b = 0.0834 L
= 440 F = 0.927
d~ = 0 . 6 6 F
~ 2Lo 35 4'o 5'o 6~o 7'o Io Vs
FIG. 18. Aerosol penetration versus face velocity at particle diameter of 0.66 ~ for filter 102 : 112.
face velocity. In Fig. 18 an example is given when face velocity is varied at constant particle diameter. It is surprising to see how well experiment and theory agree over a wide range of velocities, particle sizes, porosities and fibre diameters; and over a range of 4 orders of magnitude of aerosol penetration. DISCUSSION
It seems beyond doubt, that a pore model--as contrasted with an obstacle model--can adequately describe the complex process of aerosol capture in a fibrous filter. The characteristic features of each relation determined experimentally were also found by calculation. The actual agreement between observation and calculation is for most cases well within a factor of 2, being near to the experimental error; in one case (Fig. 17) the difference is a factor 2-5. Results of calculation depend very much on the accuracy of the input data. As appears from Figs. 6-10, 16 and 17, results are very sensitive to accurate particle size determination; using the OWL this precision seems not better than 0.01-0.02 p for the small particles, and 0.02-0.05/~ for the bigger. The porosity, also a very sensitive parameter (Fig. 15), was determined by measuring the thickness of the filter with the aid of a micrometer; the precision is probably better than 0.002. The sensitivity for face velocity increases as the performance of the filter improves (compare Fig. 11-14, 18); accurate determination of this parameter is no problem. By far the greatest problem is the determination of fibre properties. This is illustrated in Fig. 19 for a relatively insensitive relation of penetration versus face velocity at constant particle diameter. A small error of 5 per cent in the mean fibre diameter and a correspondingly small error of 10 per cent in the mean square diameter from their true mean values lead to'
486
Loum
A. CLARENBURG
h-i"
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variations in the computed results of a factor of more than 2; for more sensitive relations, miscalculation as a result of small errors in the determination of a and d-~ may be of the order of a factor of 5-10 (see Fig. 17). Unfortunately, determination of these fibre properties by electron microscopy is very tedious work, moreover, these fibre properties vary from batch to batch, and probably within a batch from sample to sample. Fortunately, results are insensitive to fibre length. With all this in mind, it is really surprising to find a reasonably good correspondence between experiment and calculation. In the theoretical part 8 mechanisms of particle capture were distinguished. The relative importance of some of these mechanisms is shown in Figs. 20 and 21 (those not shown did not contribute significantly). It becomes very clear that interception is the dominant mechanism of filtration under most conditions. Only at very small flow velocities, or for very small particle sizes does the mBchanism of diffusion becomes important and even dominant. Even at low velocities the increase of interception by inertia is a significant factor, while at high velocities the mechanisms of inertial precipitation and of inertial interception become important. When the mechanism of interception is dominant--according to Figs. 20 and 21, at v above 6 cm/sec and at dp above 0.6 micron, the probability of particle capture, Pc, should be [equations (28) and (29)] approximately inversely proportional to mho, given by equation (44). Then, from equation (55) log Pen should in first approximation be inversely proportional to mh,. In Fig. (22) log Pen is plotted against 1/mhgfor a number of filters, the data being given in Table 3. From Fig. 22 it follows that a simple calculation, which can readily be made by hand, offers a first orientation about the efficiency of a filter. The systematic deviations betwecn experiment and calculation at high velocities and large particle size (Figs. 9, 10 and 14) may be attributed to the departure from viscous flow within the largest pores. According to equation (8) the linear flow velocity within a large pore may easily attain a value as high as 50 m/see at face velocity of 30-50 ema/sec. This means that the Reynolds number of particles larger than 1 micron become in the order of 10
A probabalistic theory of aerosol penetration through glass fibre filters
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FIG. 20. Relative importance of various particle capturing mechanisms at low face velocity and varying particle diameter. P , = overall probability of capturing, all mechanisms combined. P° ~., -- overall probability of capturing by interception. P~., -- probability of capturing by pure interception. P~,,~,t = probability of capturing by interception increased by diffusion. P atl ~,, = probability of capturing by interception increased by diffusion. P , J~1 = overall probability of capturing by pure diffusion. P , ~, = overall probability of capturing by inertia.
TABLE 3. PENETRATION AS A FUNCTION OF HYDRAULIC PORE RADIUS
Filter composition
Xa
102 102 102 104 104 104 104 104 106 106 106 106 106 106 106
0.05 0.10 0.20 0.10 0"25 0-25 0"33 0"25 0"10 0-20 0"30 0.40 0"30 0-45 0.40
: : : : : : : : : : : : : : :
108 108 112 108 108 110 110 112 108 108 108 108 110 110 112
c
- log Pen
1/m~g
0-925 0-925 0-919 0"926 0-929 0"925 0"925 0"924 0"925 0"932 0"918 0"926 0"930 0-928 0"929
3.04 4-00 4.18 3-44 4"11 2"33 3"63 2"00 3-00 3"00 5"33 5"09 1"60 3-29 2"26
0.362 0.421 0.451 0.365 0-431 0"350 0"410 0-347 0.344 0"342 0"451 0"437 0"293 0"368 0"322
488
Lou~ A. CLARENBURG
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Fro. 21. ReMtiv¢ importaac~ of various particle capturing mechaBisma at high face v¢|ocity and varying particle diamet¢r. Significance of symbols, ~ legend to Fig. 19. P,,,, s, = probability of capturing by inertial interception.
It should be pointed out that the present results were obtained with a dibutylphthalate aerosol. Some experiments have been repeated using paraffin and oleic acid aerosols; results were identical with those obtained with the DBP-aerosol. However, on using a di-octyl-phthalate aerosol penetration was significantly higher, while a benzyl-beazoate aerosol showed a remarkably different behaviour, penetration being more than an order of magnitude less. This may be due to evaporation of the more volatile aerosols. In the present study only fresh filters have been used. Duration experiments very clearly show clogging effects. Especially when solid aerosols are lacing filtered, clogging is the dominant factor. It is desirable that the present theory is extended as to include clogging. This the more so, as C L ~ U R O and SCHIERECK(1968) showed that small variations of the filter composition lead to orders of magnitude improvement of the filtration capacity of a filter, hardly affecting the resistance to air flow o f the filter. The present theory, combined with the theory of the pressure drop across a filter (PIEKAARand CLARENBURG,1967; CLARENBURGand ScmERECK, 1968), gives the tools to compute the optimal fibre composition of a filter. Either a composition can be computed giving minimal penetration at a stated pressure drop or, inversely, minimal resistance to air flow at a stated penetration.
A probalistic theory of aerosol penetration through glass fibre filters
V =6 25cm/sec d ; = 0 . 6 6 ~.rn
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-5
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CONCLUSIONS
1. A pore model of a fibrous filter, as contrasted with an obstacle model, can adequately describe the complex process of aerosol filtration. The final expression for aerosol penetration does not contain any adaptation parameter; instead, directly measurable quantities related to particle, flow, fibre and filter properties only are employed. 2. To describe aerosol penetration through a fibrous filter, eight capturing mechanisms play a role, namely, pure interception, increase of interception due to inertia and diffusion, decrease of interception due to inertia, pure diffusion, pure inertia, inertial interception and inertial diffusion. The relative importance of these mechanisms of aerosol capture depends on the experimental conditions. 3. The present theory, combined with the theory of pressure drop, offers the possibility of computing the optimal composition of any filter with respect to some criteria and hence offers the opportunity for the design of optimal filters. 4. It would be desirable to expand the present theory to include electrical interaction between aerosol particles and fibres. For optimal filter design, the effect of clogging should also be considered. 5. There is no obvious reason why the present theory should not describe the filtration of particles dispersed in a liquid.
490
Lou~ A. CLARENBURG
Acknowledgements[ am very much indebted to the Director of the Chemical Laboratory of the Defense Research Organization TNO who offered me his facilities for computation. Without the invaluable help of the mathematicians G. J. Holland and later J. J. van Rijswijk, who wrote the computer programs, I would not have had any chance to perform this study.
REFERENCES
Cm~N, C. Y. (1955) Chem. gev. 55, 595. Ct.~ENBtmO, L. A. and Pmr,,~R, H. W. (1968) Chem. Engng Sci. 23, 765. C t ~ , m t m G , L. A. and PRtNCEN, L. H. (1963) Staub 23, 234. C t ~ N B t m o , L. A. and SCt-n~ECK, F. C. (1968) Chem. Engng ScL 23, 773. CLA~t,mtntG, L. A. and VAN DER WAL, J. F. (1966) Ind. Engng Chem. Process Design Develop 5, 1t0. Ct.M~NBURG, L. A. and WERNEIt, R. M. (1965b) Ind. Engng Chem. Process Design Develop 4, 293. DAWSON, S. V. (1969) Theory o f collection of airborne particles by fibrous filters. Harvard School of Public Health. Fucas, N. A. (1955) The Mechanics of Aerosols, Academy of Sciences USSR. L A r , a ~ , H. and H~rOO,NN, K. (1949) J. Collord Sci. 4, 103. LEVl~, L. (1953) Dokl. Akad. Nank. USSR 91, 1329. LANGMOIR,I. and BI..Ol~rr, K. (1946) Army Air Force Techn. Report no. 5418. L~d.~N, L. (1960) Z. angew. Phys. 12, 157. Mmm, R. E. (1964a) Proc. Natn. Acad. Sci. 52, 901. Mu.~, R. E. (1964b) Proc. Natn. Acad. ScL U.S.A. 52, 1157. ~ , H. W. and Ct.A~,lBtmG, L. A. (1967) Chem. Engng Sci. 22, 1399. ~R, H. W. and Ct.~NI)UI~G, L. A. (1967) Chem. Engng Sci. 22, 1817. R.ANZ, W. and WONG, J. (1952) Ind. Engng Chem. 44, 1371. R~d~PM'ORT, E. and WEINS'rOCK,S. E. (1955) Experientia 11, 363. VAN D~R W~d., J. F. (1968) Private communication. WERNER, R. M. and CtAI~EN6URG,L. A. (1965a) Ind. Engng Chem. Process Design Develop 4, 288.