relation between the pressure drop across a filter and the filter properties:
Ap
=
1f;O C-2.5 aLV (1 -
-~
e)3r2
e J P
T h e relation constitutes the basis of our study on mixed multicode glass fiber filters to be published in subsequent papers. Nomenclature
A
=
Acknowledgment
We are grateful to the board of the National Research Defense Organization TNO for permission to publish this study. \\’e thank L. C van Schie, who carried out the experimental work. We are very much indebted to J. Kramer of Profiltra N . V . for supplying the glass fibers We acknowledge the work performed a t the Technical Physics Department T N O and TH a t Delft, where all the electron microscopical photographs were taken with the aid of a Philips EM-200 electron microscope.
area of filter sample
c = Cunningham slip correction factor
= fiber diameter = arithmetic mean fiber diameter = mean square fiber diameter k = resistance coefficient E ; = permeability coefficient L = filter thickness Ap = pressure drop across filter superficial velocity-Le.: velocity across empty filter area T I 7 = \> (J1 + d,) Table I. -
d, Codea Microns
100
102 104 106 108 110 112 a
294
0,137 0,203 0,324 0,497 1.24 1.87 3.24
Fiber Properties
-
d’,
1,
Microns 125 0.0377 0,0855 140 150 0.260 0,665 170 270 4.42 17.3 420 1050 47.0 Microns
3.46 2.77 2.30 1.92 1.39 1.35 1.19
G,
-
k/d2 =
-
C
k 8.60 14.1 23.4 38.7 69.5 84.6 117
the approximation is made that all fibers considered are situated and of width Bo, the coarse in the same plane of length fiber being situated in the center of the plane. Consider a fine fiber situated in this plane. If it just does not cut the coarse fiber and if its center is situated at a distance ‘/p B from the coarse fiber, the angle a betlveen the t\vo fibers follows from the relation
K,
Microns 228 164 90 58
15.7 4.89 2.47
Johns-Manoille code numbers.
I&EC PROCESS DESIGN A N D DEVELOPMENT
sin a
=
BIZ
I n order not to cut the coarse fiber, all fine fibers situated in the plane of width B should have angles with respect to the coarse fiber
a
6 arcsin (B/'$)
(6)
T h e probability of finding a fine hber not cutting the coarse fiber P' is thus the product of the probability that the fine fiber makes an angle (Y 6 arcsin ( B / x ) with the coarse fiber and the probability of finding the whole fiber in the plane of width B .
In a two-component filter the corrected iveight fraction of the coarser component contributing to the resistance to air flow of the filter is 1 -
XI*
=
-
=
x1
11
-
arcsin
(7"
with Bo given by Equation 5. In Figure 1 His plotted against , t l (1 - xl) for all combinations of code 100 with the other codes. I t follo\vs from this figure, as can be demonstrated by numerical analysis, that the shadow effect within one code is negligibly small. I n the derivation of the expression for the shadow effect given above i t has been tacitly assumed that I, is infinitely long. As long as 2; this is a good approximation, becoming \\one as 1, approaches Consequently the shadow effect has been overestimated. However, the corrections required on the important shadox effects-e.g., between codes 100 still small, whereas the or 102 lvith codes 1113 or 112-are corrections required on the small shadow effects would not be of any practical value. Therefore Equation 8 has been used throughout this work for the calculation of the shadow effect.
1.>>
c.
-
xl)H
+ (1 - x1)H
The shado\v effect has a tendency to increase the resistance to air flow of a fibrous filter. as follows from the increase of xl: to XI*. As will be shown in a subsequent paper, this H effect plays a predominant role in aerosol penetration through fibrous filters. Multicomponent Filters. Consider an n-component filter where the ith component is present with the weight fraction x i . Suppose that the mean fiber diameters increase going from the first to the nth component. First the shadow effect on the nth component will be calculated. -4 two-component filter is considered as composed of components ( n - 1) and n: the weight fractions of the t\vo components having the same ratio as in the n-component filter, X ~ - ~ , ' X - ~ x , and x n , ; x n p 1 x,: respectively. Assuming that the porosity and the thickness of this filter have the same value as in the n-component filter, the shado\v effect of the (n - 1)th component on the nth component can be calculated according to Equation 8 and denoted as H , - , n , For calculation of the shadow effect of the ( n - 2)th component on the nth component i t should be recognized that the weight fraction of the ( n - 2)th component is divided over the (n - 1)th and the nth component in proportion to their relative weight fractions. Hence only the fraction
+
+
O n substituting Equations 2. 4, and 7 in Equation 3 the following expression is found for the shadow effect in a twocomponent glass fiber filter:
- (1
xi)
Considering that the sum of the weight fractions of the components should remain 1> the expression for the corrected weight fraction of the finer component is XI*
Since 01 is a function of B? the over-all probability, P? that a fine fiber \vi11 not cut the coarse fiber, is found by integrating P' over all values of B . HoLvever. there are a number of fibers partly in the plane of \vidth Bo and partly outside this plane. Therefore in order to include all fibers the integration limits should be extended from --B, to Bo. But in that case the surface area of the plane considered and with it the number of fine fibers become twice as large, Hence thr normalization factor of the integral becomes l , 4 . Thus with the aid of Equation 6 we have
(1
Xn xn-1
+
X
Xn-9
xn
of the (n - 2)th component is available for the screening of the nth component, the remaining fraction being involved in the shadow effect on the (n - 1 ) t h component. Again considering a t\vo-component filter, this time composed of components (n - 2) and n, the weight fractions of the two components can f x , ~ )and ( . x ~ - ~ be easily shown to be x , - z ; ( x n - 2 4- x , ~,),'(x,-~ x,-] x n ) , respectively. The shadow effect calculated with the aid of Equation 8 \vi11 be denoted as H n - z , n . However, this shadow effect applies only to the fraction of the nth component not yet screened by the ( n - 1 ) t h component; as a consequence H n - 2 . nacts on fraction x,(l - H n - l , n ) of
+
+
+
108
IU*
102
050
10
150
20
- x)
250
X
(1
Figure 1 .
Shadow effect of combinations of code 100 with other codes VOL. 4
NO. 3
JULY
1965
295
the nth component. Following a similar reasoning the shadow effects of all the other components on the nth component can be found. To find the expression for the total shadow effect on the nth component the following notations will be used. Let the corrected weight fraction of the nth component for the shadow effect of the (n - 1)th component be denoted as x ~ ( ~ - the ~ ) , weight fraction of the nth component corrected for the shadow effect of the (n - l ) t h , the (n - 2)th, the (n - 3)thcomponent as x ~ ( ~ -and ~ ) so , on, then
x n = x n (n) xn(n) x n ( n -1)
-
x n ( n ) ~ n - l , n= x , ( n - l !
(n-
U H , - ~ ,= ~
x n ( n -2)
2
n
(9) in which all H P , *are given by Equation 8 with
&,
=
1
=Pfl
As follows from Equation 9 and in agreement what has been mentioned earlier, the shadow effect has a tendency to increase the resistance to air flow of a filter. Pressure Drop and Structure Effect. From a great number of observations an empirical relation could be derived, describing accurately the pressure drop, 4p, across a homogeneously mixed multicomponent filter.
....., . log Ap = ~ ~ ( 2 )
xn(2)H1,,= xn(l) = x,*
Consequently the weight fraction of the nth component corrected for the shadow effects of all the other components may be expressed as xn* =
X,
-
-x ~ ( ~ ) H - x~ , -( ~~- ~, )~H , - ~ ,. ~. - X , ( ' ) H=~ , ~
(1 1)
k = 18OC-2.6 xn(t)ff-l,n
1
i n which the
xi* log Ap,
in Mhich Apt is the pressure drop across a single-component filter made of the zth component only having the same thickness and porosity as the mixed n-component filter. T h e weight fraction, x,*. has been corrected for the shadow effect. In a previous article (3) a relation was found between the resistance coefficient, k , and the Cunningham slip factor. C
2
xn -
i=1
=n
Substituting this relation into Equation 1 and substituting Equation 1 into 11, the following expression is obtained:
are given by Equation 8 with
log ( k m E =
c
xi* log (k/&
,=I
O n introducing the "composition factor:" Because of the shadow effect of the kth component on the nth component the fraction xn("+')Hk,,of the nth component is screened from the passing air flow, resulting in a corresponding decrease of the contribution of the nth component to the resistance to air flow of the filter. Since the condition
2
Xi* =
1
1-1
should be fulfilled, there is a corresponding increase of the contribution of the kth component to the resistance to air flow. Therefore the general expression for the corrected weight fraction of the kth component due to the shadow effect is
95
UI
0
k i ldZ Z K
the above expression reduces to
Since the composition factors of the various components are experimentally determined (see Table I), the effect:ve composition factor for the mixed-component filter can be'calculated according to Equation 12. With K~ known. the expression for the pressure drop across a mixed-component filter reads
Ap
25
30
=
K , ~ L V (-~ c ) ~ "
y
36
B
Figure 2. Experimental parameter, k / d 2 , vs. porosity function, ( 1 various combinations of codes 296
l & E C PROCESS D E S I G N A N D D E V E L O P M E N 1
K,
-
4 3 2 ,
for
(13)
~~
Table II.
Composition Factor,
K
Ap, Mm. HzO at
102:108
Wezght Fraction 0.5:0.5
100: 102: 104: 108: I’ 12
0.25:0.225:0.15:0.125:0:25
Filter Composition
We believe that the unusual way of calculating the effective composition factor for multicomponent filters is the consequence of an existing ”structure effect,” the physical significance of M hich is explained above Starting from a t\+o-component filter it is not difficult to give an expression for the pressure drop across such a filter containing the structure effect in an explicit form. From this empirical equation the structure effects of all possible combinations of codes can be calculated. As in the case of the shadow effect. the individual structure effects in a multicomponent filter have to be combined to find the total structure effect on a given component. We successfully predicted pressure drops across multicomponent filters using the combined structure effects. However. the physical background of the mathematical manipulations involved in the calculation of these combined structure effects is still lacking. Therefore we prefer to postpone publication of the results concerned.
Results and Discussion
Permeability Coefficient and Porosity.
The permeability
coefficient? K , is defined as
in which Ap is expressed in millimeters of water column, g is the acceleration due to gravity, p is the density of water, and L is expressed in millimeters. I n the above expression all parameters can be measured except the effective mean square fiber diameter of the composite filter. Therefore K / G 2 , being a purely experimental quantity, was plotted against the porosity function, ( 1 - e)3/2, to find the relation between pressure drop and porosity function for a mixed multicomponent filter. T h e porosities in this study ranged from 0.960 to 0.880. Over 200 different filters were prepared to establish the above relation. I n Figure 2 the results are given for five filters of different composition. There can be no doubt as to the linear proportionality of the pressure drop and (1 Values of the experim.enta1 parameter, K / @ , are given in Table 11. Composition Factor. Since linear relations exist between K i d 2 and (1 - e ) , ’ , for all combinations of codes investigated, each combination exhibiting a specific slope, it is obvious that
v
= 10
L, M m . 0.37 0.33 0,32 0.35 0,60 0.49
0.914 0,906 0.899 0.910 0.948 0.936
133.0 141.5 146.0 140.0 101 . o 117.0
0.40 0.37 0.50 0.69 0.39 0.39 0.38 0.49
0.895 0.888 0.914 0.938 0.893 0.895 0.890 0.913
246.0 248.0 204.0 179 .O 237,O 246.0 236.0 221 . o
E
Cm./Sec.
1.929 2.309 2.445 2.145 0 906 1.283
75.6 80.2 76.2 79.4 76 5 78.7 K ~ , = 77 8 u = 18 3.280 96.2 3.585 95.3 2.183 86.6 1.390 90.3 3.256 92.8 3.378 99.1 3.329 91 .2 2.415 94.0 K~~ = 93.2 u = 4.4
this slope characterizes a filter of given composition. This slope is called the composition factor, since it depends only on the composition of the filter. With the aid of Equation 14 the effective composition factor, KE, can be expressed as
I n Table I1 the K values are given for two sets of glass fiber filters of arbitrary composition. As can be seen, the composition factor can be determined with a fair degree of accuracy, the standard deviation being less than 5% for the individual determination. I t may well be that the characteristic composition factor depends on the structural arrangement of the
Table 111. Composition Factor of Two-Component Filters Filter H Composition XI* XI Kmlcd Kexptl
100:102 102:104 102:108 102:108 102:108 102:108 102:108 102:108 102:108 102:108 102:108 102:llO 102:llO 102:llO 102:112 102:112 104:106 104:108 104:108 104:108 104:llO 104:llO 104:110 104:112 106:108 106:108 106:108 106:108 106:llO 106:llO 106:tlO 106:112 106:112
0 50 0 50 0 05 0 083 0 10 0 167 0 50 0 667 0 75 0 80 0 917 0 06 0 11 0 228 0 083 0 20 0 50 0 10 0 25 0 40 0 167 0 25 0 336 0 25 0 10 0 20 0 30 0 40 0 297 0 452 0 50 0 40 0 45
0 010 0 013 0 011 0 019 0 023 0 042 0 209 0 418 0 627 0 836 1 000 0 05 0 09 0 205 0.11 0 30 0 016 0 015 0 035 0 065 0 075 0 12 0 18 0 25 0 006 0 014 0 024 0 037 0 085 0 16 0 195 0 29 0 35
VOL. 4
0.505 0,506 0.0605 0.101 0.121 0.201 0.605 0.806 0.907 0.967 1.000 0.107 0.190 0.386 0,189 0,440 0.508 0.114 0.276 0.439 0.230 0.340 0.456 0.438 0.106 0.211 0.317 0.422 0.357 0,540 0,600 0,574 0.643
NO. 3
194 122 18.2 20.0 20.9 25.4 65 0 104 130 151 164 7.1 9.5 19.0 5.5 15.7 72.4 19.1 25.4 33.6 96 13.1 18.5 11.9 18 . O 20.7 24.8 27,3 11 8 18.6 21.4 15.1 18.8
JULY
199 132 18.2 20.7 21.7 28.2 77.8 109 120 135 163 5.6 8.2 19.1 5.3 15.3 80.1 19.7 25.9 35.7 96 11.6 18.6 11 8 17.5 20.7 25 4 30.2 10.7 16.8 20.4 18.0 20.6
1965
297
~
Table IV. Filter Composition
Composition Factor of Multicomponent Filters Weight Fractions
100:102:108 100:104:108 100:108:110 100:108:110 100:108:110 100:110:112 100:110:112 102:104:108 102:104:108 102:108:110 102:108:110 102:108:110 102:108:110 104:106:108 102:104:108:112 104:106:108:110 100:102:104:108:112 100:102:104:108:112 104:106:108:110:112
0.15:0.15:0.20:0.25:0.25
fibers in the filter. I n this study the fibers are situated essentially radially a t random and perpendicular to the direction of flow. Pressure Drop across Multicomponent Filters. T o prrsent the results in a concise form it is of advantage to compare the calculated and experimental composition factors of the various filters, rather than to compare calculated and measured pressure drops. Since the composition factor, K ? depends only on the composition of the filter, the values given are the mean K ' S of a number of filters having the same composition but different porosities and thicknesses. T h e pressure drop across a great number of mixed multicomponent filters was measured. Csing Equation 15 the experimental composition factor was determined. T h e effective composition factor was calculated according to Equation 12, in which the corrected weight fractions were calculated with the aid of Equation 9. T h e results obtained with two component filters are summarized in Table 111; the magnitude of the shadow effect and the corrected weight fractions are given. I n Table I V the results obtained with multicomponent filters are given. From both tables it may be seen that the agreement betkveen calculated and measured composition factors is very close, the differences in most cases being less than 10%. Lt'ith the aid of Table 111 support can be given to the concept of the structure effect. I t was stated above that the pressure drop across a homogeneously mixed multicomponent filter is always lower than the pressure drop across a filter composed of layers of the same components in the same weight fractions. Since the pressure drop across two filters in series is additive, the pressure drop, ApL>across a filter composed of two layers may be expressed as
ApPL or
KL
=
+ (1 + (1 -
xi Ap1
= Xi ti1
XI) 4
Xi) K2
l&EC
Kexptl
181
183
150 12.1
150
12.1 16.5 16.0 7.8 7.6 37.6 85 10 6 13 0 18.' 20.8 44.0 52.0 22.2 36.5 93 17.9
16.1 15 2 9.2 9.8 29,4 76 10.9 13 0 16.0 18.1 40.5 52.0 22.0 39.7 101 15.4
it may be inferred that this is the case: giving support to our physical interpretation of the effect. The shape of the curves and the position of the minima will be discussed in a paper specially devoted to the structure effect. Conclusions
T h e pressure drop across any homogeneous glass fiber filter is proportional to the porosity function, (1 - e)3 z. in the range of porosities considered. Over 1200 individual filters were prepared to establish this proportionality. the largest deviation being 8Yc. This result agrees well with the tvork of Davies ( Z ) , but is in disagreement with the porosity function derived from the drag force theory ( 7 , 4 ) . T h e composition factor, defined as the ratio of the resistance coefficient and the effective mean square fiber diameter. is a filter characteristic depending only on the composition of the filter. T h e composition factor can be determined accurately. T h e pressure drop across any homogeneous glass fiber filter with fibers orientated essentially perpendicular to the air flo\v, can be predicted with the aid of the equation
1,o
/-
I
8
2
(1 6 )
Thus, the ratio K , , ~ ~ ~ / ' is K ~ a measure of the structure effect. I n Figure 3 this ratio is plotted against XI. T h e K,,,,~~ values Lvere taken from Table 111, the K~ values were calculated using Table I for the various K~ values. If the structure effect is to reduce the number of the smaller pores in the filter. replacing them by relatively wide pores. it has to be anticipated that the importance of the structure effect increases \vith increasing diameter ratio of the fibers mixed. From the minima of the various curves presented in Figure 3 298
Kcaled
0.375:0.530:0.095 0.595:0.256:0.149 0.05:0.50:0.45 0.05:0.:5:0.20 0.10:0.40:0.50 0.10:0.40:0.50 0.10 : 0.60:O.30 0.20:0.40:0.40 0.333:0.333:0.334 0.05:0.45:0.50 0.05:0.60:0.35 0.10:0.60:0.30 0 . 1 0 : o 7i:o 15 0 . 3 0 :0.30 :0. 40 0.25:0.25:0.25:0.25 0.20:0.20:0.20:0.40 0.05:0.125:0.25:0.20:0.375 0.25:0.225:0.15:0.125:0.25
PROCESS DESIGN A N D DEVELOPMENT
1 I
I
I
Figure 3. Ratio K,,,/KL vs. weight fraction, x i , of smallest component for filters of different compositions and various combinations of codes
L in which
T h e shado\v effect. introduced theoretically and contained in , proved to be essential in the corrected weight fractions. . x * ~ is the prediction of the pre'ssure drop across multicomponent glass fiber filters. I t has a tendency to increase the pressure drop. T h e experimental results give evidence of the existence of a structure effect. decreasing the pressure d r o p and overruling in magnitude the shado\v effect. Looking upon the effective composition factor as a statistical mean, the unusual way of averaging is striking. 'This must be the consequence of the phl-sical meaning of the effective composition factor, allowing for the structure effect. Nomenclature
Bo
= width of volume of air around fiber C = Cunningham slip correction factor (2 = arithmetic mean fiber diameter (12 = mean square fiber diameter E = subscript to a parameter. indicating that this parapeter stands for a property of a mixed multicomponent filter as a \\.hole ,q = acceleration d u e to gravity H = shadoiv effect k = resistance coefficient k-' = permeability coefficient 1 = mean fiber length
= thickness of glass fiber filter Ap = pressure drop across filter P = probability that two fibers within a given volume d o not "cut" each other S = surface fraction of a fiber screened from passing air flow by another fiber V = superficial velocity x = weight fraction x * = weight fraction corrected for shadow effect 01 = angle under Xvhich two fibers "cut" each other e = porosity of,filter 7 = viscosity of air K = composition factor Y = number fraction p = density of water
Acknowledgment
\Ye are grateful to the board of the National Research Defence Organization T N O for permission to publish this study. \\le thank L. C. van Schie, who carried out the experimental Lvork. \Ye are very much indebted to J. Kramei of Profiltra N.V for suppling the glass fibers. literature Cited
(1) Chen, C. Y.. Chem. Revs. 5 5 , 595 (1955). (2) Dabies, C. N., Proc. Inst. Mech. Engrs. (London) B 1 , 185 (1952). ( 3 ) LVerner, R. M . , Clarenburg, L. A,, IND.ENG.CHEM.PROCESS DESEXDEVELOP. 4, 288 (1965). (4) Lt'heat. J . A., Can. J . Chem. Eng. 41, 67 (1963).
RECEIVED for review May 14, 1964 ACCEPTED April 23, 1965 Second in a series of articles on aerosol filters.
COUNTERCURRENT EXTRACTION IN SALT-MEITAL SYSTEMS INVOLVING OX IDA T I O N R ED UCT ION R EACT IONS
-
PREMO C H l O T T l Institute for Atomic Research and Department of Metallurgy, Iowa State University, Ames, Iowa
The separation of solutes b y countercurrent techniques in a two-phase system wherein the transfer of solutes between the two phases depends on an oxidation-reduction reaction of fixed stoichiometry i s shown to b e analogous to the separation of components in a distillation column. A flow diogram for separating two solutes, such as uranium and thorium, in a KCI-LiCl/zinc system i s presented. Mathematical relations and graphical methods for determining the number of equilibrium stages required to effect a given separation are developed. The number of precipitate phases in the solvent streams must b e limited for the countercurrent system to b e operative.
XIDATION-reduction reactions in fused salt-liquid
metal
0 systems appear to hold promise as a method for separating fission products from reactor fuels (7-4). I n a KC1-LiC1 (44.4 weight 70LiC1)-zinc system some components such as uranium can be transferred from the zinc solvent to the salt by the addition of ZnCl?, which serves as a chlorinating or oxidizing agent. Similarly, uranium can be transferred from the salt to the zinc phase by the addition of a reducing agent such a s magnesium. Simple equilibration techniques followed by separation of the salt- and zinc-rich phases can be used to separate many metals from uranium. T h e zinc solvent can be removed by distillation a n d the
uranium recovered in the metallic state. Some metals, such as thorium, closely follow uranium in either the oxidation or reduction steps and cannot be effectively separated in a single batch process. T h e relative distribution of these components between the salt and metal phases will depend on competing reactions of the type U(in Zn soln.)
+ ThClS(in salt)
-+
+
UCla(in salt) T h ( i n Zn soln.) I n the absence of other oxidizing or reducing agents the transfer of uranium to the salt requires an equivalent transfer of thorium to the zinc-rich phase and similarly the transfer of thorium from the zinc to the salt requires an equivalent transfer VOL. 4
NO. 3
JULY
1965
299
