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Aerosol filters-II theory of the pressure drop across multi-component glass fibre filters
Chemical Engineering Science, 1968, Vol. 23, pp. 773-781.
Pergamon Press.
Printed in Great Britain.
Aerosol filters -11 Theory of the pressure drop across multi-component glass fibre filters L. A. CLARENBURW and F. C. SCHIERECK Chemical Laboratory of the National Defence Research Organization, T.N.O., Rijswijk (Z.H.), The Netherlands (Received 4 January 1968) Abstract-Based on geometrical considerations a theoretical equation has been derived relating pressure drop to filter- and fibre properties of multi-component glass fibre filters. For a great number of filters theoretical predictions of the pressure drop are in excellent agreement with the experimental results for filter porosities not exceeding 0.94. These results constitute an indirect proof of the validity of the theory of tortuosity presented in an earlier paper. It was shown that by minor variations of the composition of a filter, almost without affecting the pressure drop, considerable gains in aerosolretention can be achieved. 1. INTRODUCTION
This paper is intended as a continuation of our previous article[l]: Theory of the pressure drop across single component glass fibre filters. A theoretical equation will be derived for the pressure drop across multi-component glass fibre filters of arbitrary composition. Again use will be made of the author’s theoretical work on pore size distributions [3] and tortuosity[4] in fibrous filters; this means that also the present theory is based on purely geometrical filter considerations. 2. THEORETICAL
Consider a n-component glass fibre filter. Let the weight composition of the filter be denoted as Xl,%,
- * -, “;F
xi
=
1.
If the mean fibre dia. of the i-th component is ai, and its mean fibre length is 6, then it will be supposed that
where the suffix 1 refers to the finest component and n to the coarsest. To derive a theoretical expression for the pressure drop across such a n-component filter the following assumptions have to be made: a. In a fibrous filter the fibres are orientated in planes perpendicular to the mean flow direction but in these planes at random. b. The properties of a layer of 2& thickness are representative for the properties of the whole filter; this means that the pressure drop across a filter of thickness L is L/2& times the pressure drop across one single layer. c. A filter is considered as a system of pores. If the filter properties are a mere superposition of the properties of the composing fibre layers, then the flow through such a system can be described starting from Poiseuilles’s law for the flow through circular capillaries. Based on these assumptions Clarenburg and Piekaar[ l] derived the following theoretical expression for the pressure drop across fibrous filters
ii~
i, < i2c . . ..c in
Ap= 11*4qL---=-
(1)
To apply this expression to calculate the pressure drop across a multi-component filter, an expres-
t Present address:Openbaar lichaam Rijnmond, Stationsplein 2, Schiedam.
773
and F. C. SCHIERECK
L. A. CLARENBURG
sion has to be found for the number of pores np on the surface area i,,“. Using the results of the rigorous mathematical theory of Miles[2], it was proved by Piekaar and Clarenburg[3] that filter properties can be derived from mathematical line models. Therefore it is advantageous to reduce the complexity of a n-component filter to a mathematical line model. To do so, first take the filter as a superposition of layers of 2& thickness of the finest component; a projection is made of alI the fibres of the finest component present in a layer of 2& thickness on a plane parallel to this layer, while the line thickness on the projection is being reduced to zero. In this stage of the operation all fibres having dia. & > Jr are considered to perturb the grid formed by the fibres of the first component and are left out of consideration. Next the filter is taken as a superposition of layers of 2& thickness of the second component and again a projection is made of all fibres of this second component present in a layer of 2d2 thickness on the same plane, superimposed on the projection of the first component; again the line thickness on the projection is being reduced to zero. All fibres having dia. (Tt> a2 are left out of consideration. This procedure is repeated until the fibres of all components, composing the filter, are being projected. Thus a mathematical line model is produced, of which the various lines may differ in length, according to the fibre properties. The number of lines N, of the i-th component present on the surface area 1,” of the line model, follows from the number of fibres of the i-th component present in a volume element of the filter of surface area 1,”and thickness 2& whence
(2) The total number of lines on the surface area i,,2of the line model is N=e
N,. f=l
It was shown by Piekaar and Clarenburg[3] that the number of polygons on the surface area in2is given by 3
in which n k = the total number of line intersections of type k on the surface area in2,and n& = number of polygons sharing one vertex of type k. There are 3 types of line intersections: the ‘normal-normal’ intersection, in which case one vertex is shared by 4 polygons; the ‘normalextreme’ intersection, in which case one vertex is shared by 3 polygons; the ‘extreme-extreme’ intersection, in which case one vertex is shared by only 2 polygons. Defining the total number of intersections on the square 1,”as n,= n,+n2+n3 it then follows that n,=4n,+4n2+3n3=
n,-n2-n3+9n2 -tfn, = n,-~n2--~n3.
If there are N lines, then there are 2N line ends; of these 2n3 are used for ‘extremeextreme’ intersections, whence n, = 2N - 2n,. Consequently np = nI -N/2. Now, let line of type and let the intersect on Pff, then
(4)
the number of intersections of a i with other lines of type i be ve, probability that 2 lines of type i the square of surface area 1,’ be
Consequently the total number of intersections of all lines with other lines of the same type on in2 is: nfi
=
Ni ci=l* -pf =; i
NOi--
1Pfi.
(5)
f=1
Similarly, let the number of intersections of a line of type i with lines of type j be vu, and let the probability that a line of type i is intersected
774
Aerosol filters-11
by a line of typej on the square of surface area 12bePU,then 1 vii= NjPU.
Subsequent substitution of Eq. (9) and (3) in (4) gives an expression for the total number of pores on the square i,2:
Recognizing that vu = vfi, the total number of intersections of all lines on 1,’ with other lines of different types, is
np--i& -
n-1
nU=x
)I
x
I=1 l=f+l
NfvU= ji; E
Iz
n-1
+2x
,$+W’M’u. (6)
Nj(Nf-
l)r;z
is1 n
-m
C NtNfifil
an
-;x
Nj
I
i=1 j=f+l
i=l
=
In a previous article (3) an expression was derived for the probability Pi that 2 lines of equal length Z,, whose centre points are situated within the boundaries of a square of surface area b2,intersect; it was found that
Njijy-i
=-&[(i f=1
Nji:]-fi f=l
Nf
(10)
14
in which the Nf’s are given by Eq. (2). The tortuosity factor L,/L of a n-component filter should preserve the mean between the extremes, i.e. the tortuosity (LE/L)( in single Pp=f. component filters of the composing fibres. The If lines of length 1; are randomly spread over a structure of the n-component filter can be taken surface area in*, & < i,,, then the probability as if the grids of the various components, each of finding a centre within the boundaries of a present in the weight fraction xi, are telescoped square 1;2is pIi ns. Consequently the probability into each other, while from each grid the fraction that 2 lines of length 4, whose centre points are (1 -xi) is being occupied by fibres of the other situated within the boundaries of square inz, components. Other, more direct methods lacking, intersect follows from the tortuosity factor for a n-component filter will be defined as (7) &IL = g1 x*W&)*. (11) Similarly, it can be easily _verified that the probability that a line of length 1, , with its centre In a previous article Piekaar and Clarenburg point situated within the boundaries of a square [4] derived the following theoretical expression &,2, (1, c i%) is intersected by a line of length for the tortuosity in single component filters 5 (& c I,),whose centre point is also within 1,2, is equal to the probability that under the - 1*024)0 (L,/L),2 = 1+ [ (ems J(F) same conditions a line of length & is intersected by a line of length ij : if2 -_
X 0.389 x nJi2
(8)
Substitution of Eq. (7) in (5) and (8) in (6) yields for the total number of intersections on the square i,,2: nl
= nfi+nu = $[;:
Nf(Ni-
R
f)i;p
i=1
+2n< f=l
i N,N,iJ*]. +*+I
(9)
1 ’
(12)
Here _npfis the number of pores on the surface area 1:. On calculating the various tortuosity factors (L,/L)j with the aid of Eq. (12), and on substituting these factors in Eq. (1 I), the tortuosity factor for the n-component filter is found. On substitution of Eq. (IO) and (1 I) in (1) a theoretical expression is obtained for the pressure drop across multi-component glass fibre filters. The resulting expression is a complicated
775
L. A. CLARENBURG
function of filter properties (L, E), of the fibre properties (6,s) and of the flow properties (77,v,) ; closer examination reveals that the pressure drop is almost independent of the fibre length, as should be expected as long as 1,” is sufficiently large to be a truly representative surface area of the filter. This limits the considerations presented in this paragraph to porosities not exceeding 094. To facilitate the experimental proof of the proposed pressure drop equation, again use will be made of the composition factor concept, as introduced by Clarenburg and Wemer[S]. The composition factor K, defined as: AP
K=~Lv8(l-e)~~~
and F. C. SCHIERECK
On substitution of Eq. (1) and (14) in (13) the final result becomes C(t) np LE 2 K=E(1-E)3/2’7-j’
r (
*
(1%
>
3. RESULTS
A great number of n-component glass fibre filters was manufactured from John’s Manville glass fibres, grouped by the factory into codes according to the value of the mean fibre diameter. The procedure followed in determining fibre and filter properties has been extensively described elsewhere[q The relevant fibre properties are summarized in Table 1. Table 1. Fibre properties
0.89 < E < OS% (13)
John’s Manville code number
was experimentally shown to be a filter characteristic, independent of the filter properties (L, E), dependent on the fibre composition only. This means, that the results obtained with a series of filters, having varying thicknesses and porosities, but of identical composition, can be directly compared. In our publication about the pressure drop across single component glass fibre filters[ 1I, a correction on Eq. (13) was introduced, owing to the fact that in an actual filter pinholes, exceeding in magnitude the normal size range of pores, are ever present. Based on our theoretical work[3] and on Corte’s[6] experimental work the correction factor C(t) for pinholes up to 25 p in dia. was derived to be
100 O-148 o-0501 105
102 0.207 o-0894 171
104 0.275 0.191 195
106 O-456 0.433 173
108 l-02 1.99 478
110 l-49 5.10 522
112 2.00 7.34 990
As an example the experimentally determined composition factors of 2 filters of entirely different composition are given in Table 2. As may be inferred from this Table the composition factor is a filter characteristic, at a given composition, independent of the filter properties. The composition factor thus permits a concise presentation of the experimental results, as rather than giving a list of hundreds of filters tested, only filters of different composition have to be listed, the experimental K-V~UeS being the mean of a number of experimental results (see Table C(t) = 11.4, for Mh = 6mho > 6.25 p 2). In this way the composition factors of to various filters were determined; results of the e-Z*/2& 2-component filters are listed in Table 3, of the & I 3, 4 and 5-component filters in Table 4. The C(t) = 6.75 ; , (14) theoretical values were calculated using Eq. (IS), in which Eq. (14, 12, 11, 10) had been &_* I e-x”2 dx substituted. For all calculations a porosity E = 0925 was maintained, being about the mean for M P= 6.25 p and M,, 6 6.25 p of our experimental porosity range. With the f4 = (ln 6.25 - hI mhs - 4h2 c&,)/h @ho aid of Eq. (12) the following tortuosity factors were then obtained. Co= (In 6.25 - ]n m&/In ohs code 100 102 104 106 108 110 112 LJL 2.81 2.59 3.08 2.63 2.43 2.88 2.34 rnt = O-0295i,,21np. 776
Aerosol filters - 11 Table 2. The composition factor as a filter characteristic
Code number
Filter composition Weight fractions
L (mm)
l
102 : 108
0.5 : 0.5
0.32 0.33 0.35 0.37 0.49 060
0.899 0906 0.910 0.914 0.936 0948
100:102:104: :108:112
0.25 : 0.225 : 0.15 : :0*125:0.25
0.37 0.38 0.39 0.39 040 0.49 0.50 0.69
0.888 0.890 0.893 0.895 0.895 0,913 0.914 0.938
x
AP at u, = lOcm/sec (in mm H90)
K (in p-*)
146 76.2 141.5 80.2 140 79.4 133 75.6 118 78.7 101 76.5 ~~~~= 77.82 1.8 248 95.3 236 91.2 237 92.8 246 99.1 246 96.2 221 94.0 204 86.6 179 90.3 K,~ = 93.2 k 4.4
x
x
200'
.200
100'
. 100
100
(10211
0
(101 0.2
Fig. 1. Composition
The agreement is striking.
between
M
06
0.g
1081
o-2
04
0.6
08
l(112)
factor K vs. the filter composition of two series of filters, i.e. code 102: 108 and code 100: 112.
theory and experiment
(L,/L)2 were derived theoretically, starting from the same purely geometrical filter model. Earlier an independent verification of the expression for nP/rn2 has already been given[3] ; from the results it may therefore be concluded that also the theory of tortuosity [4] holds. However, in the previous publication [ 11,
4. DISCUSSION
The data listed in Tables 3 and 4 constitute the experimental proof of the validity of Eq. (15). The two terms of this equation, i.e. n,/in2 and 777
L. A. CLARENBURG
and F. C. SCHIERECK
Table 3. Composition factors of two-component Codenr.
100: 102 loo:108 100: 110 100: 112
102 : 104 102 : 108
102:llO
102: 112
104:106 104: 108
104: 110 104: 112 106 : 108
106:llO 106: 112
Composition weight fractions
0*50:0~50 0.75 :0.25 0.75 :0.25 0~10:0*90 0~20:0-80 0~30:0~70 0.35:0.65 0*40:060 0.50 :o.so 060 : 040 0*70:0*30 0.75 : 0.25 0.80 : 0.20 0+0:0*10 o*so:o*so 0.05 :0.95 0.08 : O-92 0*10:0*90 0*17:0.83 0.33 : 0.67 o~so:o*so 0.67 : 0.33 0.75 : 0.25 0.80 : O-20 0*90:0*10 0.92 : o@K 0.06 : o-94 0*11:0.89 0.23 : 0.77 0.70 : 0.30 0.08 : 0.92 0.20:0*80 0.32 : 0.68 0.41 : 0.59 o-50 : 0.50 o-70 :0*30 0.50 : o-50 0~10:0*90 0.25 : 0.75 040 :040 0.25 : 0.75 0.33 :0*67 0.25 :o-7s 0~10:0+0 0*20:0.80 0*30:0.70 040:060 0.30 : 0.70 0.45 : 0-5s
040:040 0.45 : 0.55
Composition factor experimental 199 175 167 5.6 21.7 40.7 58.5 69.0 97.4 124 152 167 181 210 132 18.2 20.7 21.7 28.2 55.6 77.8 109 120 135 168 163 5.6 8.2 19.1 107 5.3 15.3 36.9 54.5 70.8 123 78.5 17.8 28.0 34.1 13.2 21-o 12.4 16.1 20.4 23.9 29.3 11.9 18.1 24*3? 19.3
filters
Composition factort calculated 235 204 21s 10.6 24.9 46.7 60.4 76.6 111 137 170 188 208 255 160 14.3 17.0 19.1 27.0 SO.6 85.1 117 130 139 158 163 8.9 13.9 29.8 130 6.9 17.5 33.5 49.2 68.3 113 82.8 15.0 24.7 37.7 18.8 25.7 13.3 12.7 15.5 18.6 22.0 13.7 20.1 13-s 15.5
tA1l theoretical K-values were calculated for a porosity e = 0925.
778
Aerosol filters - II Table 4. Composition factors of 3,4 and S-component filters
Code number
Composition weight fractions
Composition factor experimental
Composition factort calculated
100:102:108 100:104:108 100:108:110
0.375 :0530 : o*o!s 0595:0.256:0.149 0.050 : 0.500 : 0.450 0*050: 0.750 : 0,200 0*100:0*400:0*500 0~100:04aI:0~500 0*100:0600:0~300 0.200 : 0400 : o*m 0.330 : 0.330 :0.340 0~050:0*450:0*500 0~050:0600:0~350 0*100:06tM:0~300 0*100:0~750:0~150 0*300:0*300:0400 0.250 :0.250 : 0.250 :0.250 0.200 : 0.200 : 0.200 :0400 0.050:0.125:0.250 :0*200:0.375 0*250:0*225:0.150 :0*125:0*250 0~150:0~150:0~200 :0.250:0.250
183 150 12.1 16.5 16.0 7,8 7.6 37.6? 85.0 10.6 13.0 18.7 20.8 44.0
197 1% 13-o 15.1 19.4 12.9 12.9 77,o 97.3 11.2 12.2 17.3 18.9 43.6
52.0
59.1
22.2
26.2
36.5
45.3
100:110:112 102:104:108 102:108:110
104:106:108 102:104:108 :112 104:106:108 :llO 100:102:1@4 :108:112
104:1%:108 :110:112
117
93.2
17.7
17.9
tAll theoretical ~-values were calculated for a porosity e = O-925.
it was stressed already that the theory of tortuosity should lead to erroneous results for filter porosities exceeding 094, in fact composition factors of filters of these high porosities would be overestimated. In a subsequent article the theory of tortuosity will be completed as to include high filter porosities. Results presented in Table 3 give rise to some interesting observations. In Fig. 1 the composition factors of the various code 102 : 108 filters and of the code 100 : 112 filters are plotted vs. the filter composition, expressed as weightfractions of the various components. There appears to be a marked deviation from a linear increase of K with increasing weight fraction of the finest component. Up to a weight fraction of 20 per cent of the finest component the increase of K, and hence of the resistance to airflow of the filter, is “hardly’$?rceivable. In Table 5 the composition factors of filters
of compositions in this lower range are compared with aerosol penetration through these same filters. The data are taken from a study of Clarenburg and van der Wal[8]. The penetration is defined as the ratio of the number of penetrated ’ particles to the number of incident particles. All filters were tested under identical experimental conditions.
779
Table 5. Composition factor and aerosol penetration Code number
Composition weight fractions
K
Pen.
102 : 108
O:l*OO 0.05 :0*95 0~10:090 0% : 044 0.11:0*89 0.23 : 0.77 0.084 : 0.916 0.200 :o+IO
16.4 18.2 21.7 5.6 8.2 19.1 5.3 15.3
5.0 x 10-S 9.1 x10-4 1.0 x lo-’ 1.74 x 10-r 2.82 X 1O-2 5.24 X lo+ 1.41 x 10-l 4.17 x 10-4
102:llO
102: 112
L. A. CLARENBURG
and F. C. SCHIERECK
total number of line intersections of type k on the surface area indicated number of polygons sharing one vertex of type k total number of line intersections on the surface area indicated total number of line intersections of lines of the same type on the surface area indicated total number of line intersections of lines of different types on the surface area indicated number of lines of type i on the surface area indicated total number of lines on the surface area indicated pressure drop across a filter probability that 2 lines of type i intersects on the surface area indicated probability that a line of type i intersects a line of type j on the surface area indicated integration limit superficial velocity weight fraction of the i-th component filter porosity viscosity of the air composition factor number of intersections of a line of type i with other lines of type i number of intersections of a line of type i with lines of type j geometric standard deviation of the hydraulic radius distribution.
From the Table it may be observed, that by varying slightly the composition of a 102 : 108 filter, so that the composition factor hardly increases, a 50 fold gain in aerosol penetration is achieved. The same is even more conclusively so for filters of greater “inhomogeneity”, i.e. filters in which the mean diameter of the composing components diier more. Comparison of filters 102: 108= O-05:0*95 and 102: llO= 0.23 :0*77 reveals, that, although the composition factors are almost identical, the aerosol penetration through the latter is about 18 times as good as through the former. The results show that by a small variation of the composition of a filter, an appreciable gain in performance can be obtained. In a subsequent article it will be shown how by combining the theory of aerosol penetration[8] v&h the theory of the pressure drop presented in this paper, the composition of a filter can be computed with either maximal aerosol retention at fixed pressure drop, or minimal resistance to air flow at fixed aerosol penetration. We would like to introduce this as a first attempt to optimal filter design.
d
NOTATION
mean fibre dia. mean square fibre dia. i mean fibre length L filter thickness LEIL tortuosity factor mhg geometric mean hydraulic radius of the pores Mh maximum hydraulic radius of any pore in the filter, equal to 6mhg MP hydraulic radius of a pinhole np total number of pores (polygons) on the surface area indicated. d2
Acknowledgment- We gratefully acknowledge the permission of the Board of the National Defence Research Organization T.N.O. to publish the results of this study.
REFERENCES L. A. and PIEKAAR H. W., Chem. Engng Sci. 196823 765. [II CLARENBURG MILES R. E., Proc. nafl. Acud. Sci. 52 901 1157 (1964). t:,’ PIEKAAR H. W. and CLARENBURG L. A., Chem. Engng Sci. 1967 22 1399. L. A., Chem. Engng Sci. 1967 22 1017. [41 PIEKAAR H. W. CLARENBURG, L. A. and WERNER R. M., Ind. Eng Chem. Process Design and Development 1965 4 293 [51 CLARENBURG [61 CORTE H., Filtration and Separation 1966 3 396. L. A.,Ind. Engng Chem. Process Design and Development 1965 4 288. [71 WERNER R. M. and CLARENBURG L. A. and VAN DER WALJ. F., Ind. Engng Chem. Process Design andDevelopment 19665 110. ml CLARENBURG
780
Aerosol filters-II RCsumC-On a formule, sur la base ,de considerations geom&riques, une equation theorique qui Ctablit un rapport entre la chute de pression et les propi&% dun fihre en fibres de verre et des fibres composantes. Pour un grand nombre de filtres, les previsions theoriques s’accordent parfaitement aux r&hats exp&imentaux quand la porosit6 des filtres n’exc2de pas 0,94. Ces r&hats constituent une preuve indirecte de la validite de la th6orie de la tortuosite presentbe darts un article anterieur. On observe qu’en effectuant des variations mineures sur la composition dun filtre, presque sans affecter la chute de pression, on peut augmenter considerablement la retention aerosol. Zusammenfassung-Auf Grund geometrischer Uberlegungen wurde eine theoretische Gleichung abgeleitet, die den Druckabfail in Glasfaserfiltem, in eine Beziehung zu den Filter- und Fasereigenschaften bringt. Fur eine grosse Zahl von Filtem sind die theoretischen Vorhersagen beztiglich des Druckabfalles in hervorragender Ubereinstimmung mit den Versuchsergebnisse tiir Filterporositiiten bis zu 0,94. Diese Ergebnisse stellen eine indirekte Best’sitigung der Giiltigkeit der in einem friiheren Artikel dargelegten Kriimmungstheorie dar. Es wurde gezeigt, dass durch geringftigige Vertiderungen in der Zusammensetzung des Filters, sozusagen ohne Beeinflussing des Druckabfalles, eine betrachtliche Steigerung bei der Zuriickhaltung von Aerosolen erzielt werden kann.
781 C.E.S. Vol. 23 No 7-H
Pergamon Press.
Printed in Great Britain.
Aerosol filters -11 Theory of the pressure drop across multi-component glass fibre filters L. A. CLARENBURW and F. C. SCHIERECK Chemical Laboratory of the National Defence Research Organization, T.N.O., Rijswijk (Z.H.), The Netherlands (Received 4 January 1968) Abstract-Based on geometrical considerations a theoretical equation has been derived relating pressure drop to filter- and fibre properties of multi-component glass fibre filters. For a great number of filters theoretical predictions of the pressure drop are in excellent agreement with the experimental results for filter porosities not exceeding 0.94. These results constitute an indirect proof of the validity of the theory of tortuosity presented in an earlier paper. It was shown that by minor variations of the composition of a filter, almost without affecting the pressure drop, considerable gains in aerosolretention can be achieved. 1. INTRODUCTION
This paper is intended as a continuation of our previous article[l]: Theory of the pressure drop across single component glass fibre filters. A theoretical equation will be derived for the pressure drop across multi-component glass fibre filters of arbitrary composition. Again use will be made of the author’s theoretical work on pore size distributions [3] and tortuosity[4] in fibrous filters; this means that also the present theory is based on purely geometrical filter considerations. 2. THEORETICAL
Consider a n-component glass fibre filter. Let the weight composition of the filter be denoted as Xl,%,
- * -, “;F
xi
=
1.
If the mean fibre dia. of the i-th component is ai, and its mean fibre length is 6, then it will be supposed that
where the suffix 1 refers to the finest component and n to the coarsest. To derive a theoretical expression for the pressure drop across such a n-component filter the following assumptions have to be made: a. In a fibrous filter the fibres are orientated in planes perpendicular to the mean flow direction but in these planes at random. b. The properties of a layer of 2& thickness are representative for the properties of the whole filter; this means that the pressure drop across a filter of thickness L is L/2& times the pressure drop across one single layer. c. A filter is considered as a system of pores. If the filter properties are a mere superposition of the properties of the composing fibre layers, then the flow through such a system can be described starting from Poiseuilles’s law for the flow through circular capillaries. Based on these assumptions Clarenburg and Piekaar[ l] derived the following theoretical expression for the pressure drop across fibrous filters
ii~
i, < i2c . . ..c in
Ap= 11*4qL---=-
(1)
To apply this expression to calculate the pressure drop across a multi-component filter, an expres-
t Present address:Openbaar lichaam Rijnmond, Stationsplein 2, Schiedam.
773
and F. C. SCHIERECK
L. A. CLARENBURG
sion has to be found for the number of pores np on the surface area i,,“. Using the results of the rigorous mathematical theory of Miles[2], it was proved by Piekaar and Clarenburg[3] that filter properties can be derived from mathematical line models. Therefore it is advantageous to reduce the complexity of a n-component filter to a mathematical line model. To do so, first take the filter as a superposition of layers of 2& thickness of the finest component; a projection is made of alI the fibres of the finest component present in a layer of 2& thickness on a plane parallel to this layer, while the line thickness on the projection is being reduced to zero. In this stage of the operation all fibres having dia. & > Jr are considered to perturb the grid formed by the fibres of the first component and are left out of consideration. Next the filter is taken as a superposition of layers of 2& thickness of the second component and again a projection is made of all fibres of this second component present in a layer of 2d2 thickness on the same plane, superimposed on the projection of the first component; again the line thickness on the projection is being reduced to zero. All fibres having dia. (Tt> a2 are left out of consideration. This procedure is repeated until the fibres of all components, composing the filter, are being projected. Thus a mathematical line model is produced, of which the various lines may differ in length, according to the fibre properties. The number of lines N, of the i-th component present on the surface area 1,” of the line model, follows from the number of fibres of the i-th component present in a volume element of the filter of surface area 1,”and thickness 2& whence
(2) The total number of lines on the surface area i,,2of the line model is N=e
N,. f=l
It was shown by Piekaar and Clarenburg[3] that the number of polygons on the surface area in2is given by 3
in which n k = the total number of line intersections of type k on the surface area in2,and n& = number of polygons sharing one vertex of type k. There are 3 types of line intersections: the ‘normal-normal’ intersection, in which case one vertex is shared by 4 polygons; the ‘normalextreme’ intersection, in which case one vertex is shared by 3 polygons; the ‘extreme-extreme’ intersection, in which case one vertex is shared by only 2 polygons. Defining the total number of intersections on the square 1,”as n,= n,+n2+n3 it then follows that n,=4n,+4n2+3n3=
n,-n2-n3+9n2 -tfn, = n,-~n2--~n3.
If there are N lines, then there are 2N line ends; of these 2n3 are used for ‘extremeextreme’ intersections, whence n, = 2N - 2n,. Consequently np = nI -N/2. Now, let line of type and let the intersect on Pff, then
(4)
the number of intersections of a i with other lines of type i be ve, probability that 2 lines of type i the square of surface area 1,’ be
Consequently the total number of intersections of all lines with other lines of the same type on in2 is: nfi
=
Ni ci=l* -pf =; i
NOi--
1Pfi.
(5)
f=1
Similarly, let the number of intersections of a line of type i with lines of type j be vu, and let the probability that a line of type i is intersected
774
Aerosol filters-11
by a line of typej on the square of surface area 12bePU,then 1 vii= NjPU.
Subsequent substitution of Eq. (9) and (3) in (4) gives an expression for the total number of pores on the square i,2:
Recognizing that vu = vfi, the total number of intersections of all lines on 1,’ with other lines of different types, is
np--i& -
n-1
nU=x
)I
x
I=1 l=f+l
NfvU= ji; E
Iz
n-1
+2x
,$+W’M’u. (6)
Nj(Nf-
l)r;z
is1 n
-m
C NtNfifil
an
-;x
Nj
I
i=1 j=f+l
i=l
=
In a previous article (3) an expression was derived for the probability Pi that 2 lines of equal length Z,, whose centre points are situated within the boundaries of a square of surface area b2,intersect; it was found that
Njijy-i
=-&[(i f=1
Nji:]-fi f=l
Nf
(10)
14
in which the Nf’s are given by Eq. (2). The tortuosity factor L,/L of a n-component filter should preserve the mean between the extremes, i.e. the tortuosity (LE/L)( in single Pp=f. component filters of the composing fibres. The If lines of length 1; are randomly spread over a structure of the n-component filter can be taken surface area in*, & < i,,, then the probability as if the grids of the various components, each of finding a centre within the boundaries of a present in the weight fraction xi, are telescoped square 1;2is pIi ns. Consequently the probability into each other, while from each grid the fraction that 2 lines of length 4, whose centre points are (1 -xi) is being occupied by fibres of the other situated within the boundaries of square inz, components. Other, more direct methods lacking, intersect follows from the tortuosity factor for a n-component filter will be defined as (7) &IL = g1 x*W&)*. (11) Similarly, it can be easily _verified that the probability that a line of length 1, , with its centre In a previous article Piekaar and Clarenburg point situated within the boundaries of a square [4] derived the following theoretical expression &,2, (1, c i%) is intersected by a line of length for the tortuosity in single component filters 5 (& c I,),whose centre point is also within 1,2, is equal to the probability that under the - 1*024)0 (L,/L),2 = 1+ [ (ems J(F) same conditions a line of length & is intersected by a line of length ij : if2 -_
X 0.389 x nJi2
(8)
Substitution of Eq. (7) in (5) and (8) in (6) yields for the total number of intersections on the square i,,2: nl
= nfi+nu = $[;:
Nf(Ni-
R
f)i;p
i=1
+2n< f=l
i N,N,iJ*]. +*+I
(9)
1 ’
(12)
Here _npfis the number of pores on the surface area 1:. On calculating the various tortuosity factors (L,/L)j with the aid of Eq. (12), and on substituting these factors in Eq. (1 I), the tortuosity factor for the n-component filter is found. On substitution of Eq. (IO) and (1 I) in (1) a theoretical expression is obtained for the pressure drop across multi-component glass fibre filters. The resulting expression is a complicated
775
L. A. CLARENBURG
function of filter properties (L, E), of the fibre properties (6,s) and of the flow properties (77,v,) ; closer examination reveals that the pressure drop is almost independent of the fibre length, as should be expected as long as 1,” is sufficiently large to be a truly representative surface area of the filter. This limits the considerations presented in this paragraph to porosities not exceeding 094. To facilitate the experimental proof of the proposed pressure drop equation, again use will be made of the composition factor concept, as introduced by Clarenburg and Wemer[S]. The composition factor K, defined as: AP
K=~Lv8(l-e)~~~
and F. C. SCHIERECK
On substitution of Eq. (1) and (14) in (13) the final result becomes C(t) np LE 2 K=E(1-E)3/2’7-j’
r (
*
(1%
>
3. RESULTS
A great number of n-component glass fibre filters was manufactured from John’s Manville glass fibres, grouped by the factory into codes according to the value of the mean fibre diameter. The procedure followed in determining fibre and filter properties has been extensively described elsewhere[q The relevant fibre properties are summarized in Table 1. Table 1. Fibre properties
0.89 < E < OS% (13)
John’s Manville code number
was experimentally shown to be a filter characteristic, independent of the filter properties (L, E), dependent on the fibre composition only. This means, that the results obtained with a series of filters, having varying thicknesses and porosities, but of identical composition, can be directly compared. In our publication about the pressure drop across single component glass fibre filters[ 1I, a correction on Eq. (13) was introduced, owing to the fact that in an actual filter pinholes, exceeding in magnitude the normal size range of pores, are ever present. Based on our theoretical work[3] and on Corte’s[6] experimental work the correction factor C(t) for pinholes up to 25 p in dia. was derived to be
100 O-148 o-0501 105
102 0.207 o-0894 171
104 0.275 0.191 195
106 O-456 0.433 173
108 l-02 1.99 478
110 l-49 5.10 522
112 2.00 7.34 990
As an example the experimentally determined composition factors of 2 filters of entirely different composition are given in Table 2. As may be inferred from this Table the composition factor is a filter characteristic, at a given composition, independent of the filter properties. The composition factor thus permits a concise presentation of the experimental results, as rather than giving a list of hundreds of filters tested, only filters of different composition have to be listed, the experimental K-V~UeS being the mean of a number of experimental results (see Table C(t) = 11.4, for Mh = 6mho > 6.25 p 2). In this way the composition factors of to various filters were determined; results of the e-Z*/2& 2-component filters are listed in Table 3, of the & I 3, 4 and 5-component filters in Table 4. The C(t) = 6.75 ; , (14) theoretical values were calculated using Eq. (IS), in which Eq. (14, 12, 11, 10) had been &_* I e-x”2 dx substituted. For all calculations a porosity E = 0925 was maintained, being about the mean for M P= 6.25 p and M,, 6 6.25 p of our experimental porosity range. With the f4 = (ln 6.25 - hI mhs - 4h2 c&,)/h @ho aid of Eq. (12) the following tortuosity factors were then obtained. Co= (In 6.25 - ]n m&/In ohs code 100 102 104 106 108 110 112 LJL 2.81 2.59 3.08 2.63 2.43 2.88 2.34 rnt = O-0295i,,21np. 776
Aerosol filters - 11 Table 2. The composition factor as a filter characteristic
Code number
Filter composition Weight fractions
L (mm)
l
102 : 108
0.5 : 0.5
0.32 0.33 0.35 0.37 0.49 060
0.899 0906 0.910 0.914 0.936 0948
100:102:104: :108:112
0.25 : 0.225 : 0.15 : :0*125:0.25
0.37 0.38 0.39 0.39 040 0.49 0.50 0.69
0.888 0.890 0.893 0.895 0.895 0,913 0.914 0.938
x
AP at u, = lOcm/sec (in mm H90)
K (in p-*)
146 76.2 141.5 80.2 140 79.4 133 75.6 118 78.7 101 76.5 ~~~~= 77.82 1.8 248 95.3 236 91.2 237 92.8 246 99.1 246 96.2 221 94.0 204 86.6 179 90.3 K,~ = 93.2 k 4.4
x
x
200'
.200
100'
. 100
100
(10211
0
(101 0.2
Fig. 1. Composition
The agreement is striking.
between
M
06
0.g
1081
o-2
04
0.6
08
l(112)
factor K vs. the filter composition of two series of filters, i.e. code 102: 108 and code 100: 112.
theory and experiment
(L,/L)2 were derived theoretically, starting from the same purely geometrical filter model. Earlier an independent verification of the expression for nP/rn2 has already been given[3] ; from the results it may therefore be concluded that also the theory of tortuosity [4] holds. However, in the previous publication [ 11,
4. DISCUSSION
The data listed in Tables 3 and 4 constitute the experimental proof of the validity of Eq. (15). The two terms of this equation, i.e. n,/in2 and 777
L. A. CLARENBURG
and F. C. SCHIERECK
Table 3. Composition factors of two-component Codenr.
100: 102 loo:108 100: 110 100: 112
102 : 104 102 : 108
102:llO
102: 112
104:106 104: 108
104: 110 104: 112 106 : 108
106:llO 106: 112
Composition weight fractions
0*50:0~50 0.75 :0.25 0.75 :0.25 0~10:0*90 0~20:0-80 0~30:0~70 0.35:0.65 0*40:060 0.50 :o.so 060 : 040 0*70:0*30 0.75 : 0.25 0.80 : 0.20 0+0:0*10 o*so:o*so 0.05 :0.95 0.08 : O-92 0*10:0*90 0*17:0.83 0.33 : 0.67 o~so:o*so 0.67 : 0.33 0.75 : 0.25 0.80 : O-20 0*90:0*10 0.92 : o@K 0.06 : o-94 0*11:0.89 0.23 : 0.77 0.70 : 0.30 0.08 : 0.92 0.20:0*80 0.32 : 0.68 0.41 : 0.59 o-50 : 0.50 o-70 :0*30 0.50 : o-50 0~10:0*90 0.25 : 0.75 040 :040 0.25 : 0.75 0.33 :0*67 0.25 :o-7s 0~10:0+0 0*20:0.80 0*30:0.70 040:060 0.30 : 0.70 0.45 : 0-5s
040:040 0.45 : 0.55
Composition factor experimental 199 175 167 5.6 21.7 40.7 58.5 69.0 97.4 124 152 167 181 210 132 18.2 20.7 21.7 28.2 55.6 77.8 109 120 135 168 163 5.6 8.2 19.1 107 5.3 15.3 36.9 54.5 70.8 123 78.5 17.8 28.0 34.1 13.2 21-o 12.4 16.1 20.4 23.9 29.3 11.9 18.1 24*3? 19.3
filters
Composition factort calculated 235 204 21s 10.6 24.9 46.7 60.4 76.6 111 137 170 188 208 255 160 14.3 17.0 19.1 27.0 SO.6 85.1 117 130 139 158 163 8.9 13.9 29.8 130 6.9 17.5 33.5 49.2 68.3 113 82.8 15.0 24.7 37.7 18.8 25.7 13.3 12.7 15.5 18.6 22.0 13.7 20.1 13-s 15.5
tA1l theoretical K-values were calculated for a porosity e = 0925.
778
Aerosol filters - II Table 4. Composition factors of 3,4 and S-component filters
Code number
Composition weight fractions
Composition factor experimental
Composition factort calculated
100:102:108 100:104:108 100:108:110
0.375 :0530 : o*o!s 0595:0.256:0.149 0.050 : 0.500 : 0.450 0*050: 0.750 : 0,200 0*100:0*400:0*500 0~100:04aI:0~500 0*100:0600:0~300 0.200 : 0400 : o*m 0.330 : 0.330 :0.340 0~050:0*450:0*500 0~050:0600:0~350 0*100:06tM:0~300 0*100:0~750:0~150 0*300:0*300:0400 0.250 :0.250 : 0.250 :0.250 0.200 : 0.200 : 0.200 :0400 0.050:0.125:0.250 :0*200:0.375 0*250:0*225:0.150 :0*125:0*250 0~150:0~150:0~200 :0.250:0.250
183 150 12.1 16.5 16.0 7,8 7.6 37.6? 85.0 10.6 13.0 18.7 20.8 44.0
197 1% 13-o 15.1 19.4 12.9 12.9 77,o 97.3 11.2 12.2 17.3 18.9 43.6
52.0
59.1
22.2
26.2
36.5
45.3
100:110:112 102:104:108 102:108:110
104:106:108 102:104:108 :112 104:106:108 :llO 100:102:1@4 :108:112
104:1%:108 :110:112
117
93.2
17.7
17.9
tAll theoretical ~-values were calculated for a porosity e = O-925.
it was stressed already that the theory of tortuosity should lead to erroneous results for filter porosities exceeding 094, in fact composition factors of filters of these high porosities would be overestimated. In a subsequent article the theory of tortuosity will be completed as to include high filter porosities. Results presented in Table 3 give rise to some interesting observations. In Fig. 1 the composition factors of the various code 102 : 108 filters and of the code 100 : 112 filters are plotted vs. the filter composition, expressed as weightfractions of the various components. There appears to be a marked deviation from a linear increase of K with increasing weight fraction of the finest component. Up to a weight fraction of 20 per cent of the finest component the increase of K, and hence of the resistance to airflow of the filter, is “hardly’$?rceivable. In Table 5 the composition factors of filters
of compositions in this lower range are compared with aerosol penetration through these same filters. The data are taken from a study of Clarenburg and van der Wal[8]. The penetration is defined as the ratio of the number of penetrated ’ particles to the number of incident particles. All filters were tested under identical experimental conditions.
779
Table 5. Composition factor and aerosol penetration Code number
Composition weight fractions
K
Pen.
102 : 108
O:l*OO 0.05 :0*95 0~10:090 0% : 044 0.11:0*89 0.23 : 0.77 0.084 : 0.916 0.200 :o+IO
16.4 18.2 21.7 5.6 8.2 19.1 5.3 15.3
5.0 x 10-S 9.1 x10-4 1.0 x lo-’ 1.74 x 10-r 2.82 X 1O-2 5.24 X lo+ 1.41 x 10-l 4.17 x 10-4
102:llO
102: 112
L. A. CLARENBURG
and F. C. SCHIERECK
total number of line intersections of type k on the surface area indicated number of polygons sharing one vertex of type k total number of line intersections on the surface area indicated total number of line intersections of lines of the same type on the surface area indicated total number of line intersections of lines of different types on the surface area indicated number of lines of type i on the surface area indicated total number of lines on the surface area indicated pressure drop across a filter probability that 2 lines of type i intersects on the surface area indicated probability that a line of type i intersects a line of type j on the surface area indicated integration limit superficial velocity weight fraction of the i-th component filter porosity viscosity of the air composition factor number of intersections of a line of type i with other lines of type i number of intersections of a line of type i with lines of type j geometric standard deviation of the hydraulic radius distribution.
From the Table it may be observed, that by varying slightly the composition of a 102 : 108 filter, so that the composition factor hardly increases, a 50 fold gain in aerosol penetration is achieved. The same is even more conclusively so for filters of greater “inhomogeneity”, i.e. filters in which the mean diameter of the composing components diier more. Comparison of filters 102: 108= O-05:0*95 and 102: llO= 0.23 :0*77 reveals, that, although the composition factors are almost identical, the aerosol penetration through the latter is about 18 times as good as through the former. The results show that by a small variation of the composition of a filter, an appreciable gain in performance can be obtained. In a subsequent article it will be shown how by combining the theory of aerosol penetration[8] v&h the theory of the pressure drop presented in this paper, the composition of a filter can be computed with either maximal aerosol retention at fixed pressure drop, or minimal resistance to air flow at fixed aerosol penetration. We would like to introduce this as a first attempt to optimal filter design.
d
NOTATION
mean fibre dia. mean square fibre dia. i mean fibre length L filter thickness LEIL tortuosity factor mhg geometric mean hydraulic radius of the pores Mh maximum hydraulic radius of any pore in the filter, equal to 6mhg MP hydraulic radius of a pinhole np total number of pores (polygons) on the surface area indicated. d2
Acknowledgment- We gratefully acknowledge the permission of the Board of the National Defence Research Organization T.N.O. to publish the results of this study.
REFERENCES L. A. and PIEKAAR H. W., Chem. Engng Sci. 196823 765. [II CLARENBURG MILES R. E., Proc. nafl. Acud. Sci. 52 901 1157 (1964). t:,’ PIEKAAR H. W. and CLARENBURG L. A., Chem. Engng Sci. 1967 22 1399. L. A., Chem. Engng Sci. 1967 22 1017. [41 PIEKAAR H. W. CLARENBURG, L. A. and WERNER R. M., Ind. Eng Chem. Process Design and Development 1965 4 293 [51 CLARENBURG [61 CORTE H., Filtration and Separation 1966 3 396. L. A.,Ind. Engng Chem. Process Design and Development 1965 4 288. [71 WERNER R. M. and CLARENBURG L. A. and VAN DER WALJ. F., Ind. Engng Chem. Process Design andDevelopment 19665 110. ml CLARENBURG
780
Aerosol filters-II RCsumC-On a formule, sur la base ,de considerations geom&riques, une equation theorique qui Ctablit un rapport entre la chute de pression et les propi&% dun fihre en fibres de verre et des fibres composantes. Pour un grand nombre de filtres, les previsions theoriques s’accordent parfaitement aux r&hats exp&imentaux quand la porosit6 des filtres n’exc2de pas 0,94. Ces r&hats constituent une preuve indirecte de la validite de la th6orie de la tortuosite presentbe darts un article anterieur. On observe qu’en effectuant des variations mineures sur la composition dun filtre, presque sans affecter la chute de pression, on peut augmenter considerablement la retention aerosol. Zusammenfassung-Auf Grund geometrischer Uberlegungen wurde eine theoretische Gleichung abgeleitet, die den Druckabfail in Glasfaserfiltem, in eine Beziehung zu den Filter- und Fasereigenschaften bringt. Fur eine grosse Zahl von Filtem sind die theoretischen Vorhersagen beztiglich des Druckabfalles in hervorragender Ubereinstimmung mit den Versuchsergebnisse tiir Filterporositiiten bis zu 0,94. Diese Ergebnisse stellen eine indirekte Best’sitigung der Giiltigkeit der in einem friiheren Artikel dargelegten Kriimmungstheorie dar. Es wurde gezeigt, dass durch geringftigige Vertiderungen in der Zusammensetzung des Filters, sozusagen ohne Beeinflussing des Druckabfalles, eine betrachtliche Steigerung bei der Zuriickhaltung von Aerosolen erzielt werden kann.
781 C.E.S. Vol. 23 No 7-H