Table II.
Experimental and Theoretical Ignition Energies Using Equation 9
Exptl.
Ez.Q
Exjlosice
Cal./ Sq. C m .
TNT Tetryl RDX PETN
>0.38 0 33 0 33 0 25
Computed E, and T ,
Et 0.46 0 30
T", OK. 934 689
0 26' 0 18h
631 531
E ~ , 4.9, T*
E, 0.36" 0 25d 0 30e 0 210
79 5
617 659 567
a From (I). Kinetics f r o m (5). ( 72). d ( 70). e (13). Autocatalytic, extrapolated to zero time. 0 ( 7 I ) . Autocatalytic.
Experimental data are not available to check the validity of Equation 10 quantitatively. Severtheless, the equation does predict that an increase in ambient temperature \vi11 decrease the ignition time. as is observed experimentally. T h e quantitative approach employed in this paper is that ignition is a pure thermal reaction. Ho\vever, it is knoivn that certain subsidiary factors such as chemical effects from certain gases, such as oxygen? or catalytic effects from the heating source may sometimes influence the ignition process. T h e inclusion of these factors is beyond the scope of this paper, but their general influence should decrease as the temperature of the source and its accompanying heat flux is increased. Acknowledgment
chemical reaction is pQL Z exp (-EIRTs), ivhere Q is the heat of chemical reaction a n d L is the length of the unit area column of explosive undergoing ignition. L may be approximated as the thermal thickness, kd:?J: and the thermal velocity, u, as L / t i . Thus, L = (kdt,)'I2. T h e heat flux through the surface of the explosive is obtained as before. Hence the ignition criterion is
K(T, - T o ) ,(Tkdt)"'
=
pQL Z exp(-EiRT,)
(9)
T h e ignition energies and surface temperatures calculated for the explosives of Table I using Equation 9 are given in ?'able 11. Q was taken as 500 cal. per gram in the calculations (76). T h e computed values in Table I 1 are in reasonable agreement with those in Table I. T h e uncertainties in the values used for the physical constants, and in the ignition temperatures obtained by extrapolating Figure 7 of ( 7 0 ' ) , are too large to conclude unambiguously which table contains the more valid temperatures. I t is kno\vn experimentally that the initial ambient temperature of a n explosive may under suitable conditions influence its ignition delay time. Equation 6 does not provide for this effect. However, Equation 9 does include the effect of initial temperature. Thus
?he Lvriter extends his appreciation to 0. R . Irkvin a n d P. K. Salzman of Aerojet-General Corp., where this \sork \vas initiated, for their stimulating discussions on this subject. literature Cited
(1) Bryan, G. J.. Noonan. E. C., Proc. Roy. Soc. 246, 167 (1358). (2) Carslaw, H. S.. Jaeger, J. C.. "Conduction of Heat in Solids," chap. 11. Oxford University Press, London, 1959. (3) Clarke. J . F.. J . E'luidMech. 13, 47 (1362). (4) Cook, G. B.: Proc. Roj. Soc. A246, 154 (1958). (5) Cook, M. A , . .%begg. M. T.. Ind. Eng. Chem. 48, 1090 (1956). (6) Frank-Kamenetski. D. A , , "Diffusion and Heat Exchange in Chemical Kinetics, chap. VI. Princeton University Press: Princeton, 1955. (7) Hicks. B. L., J . Chem. Phys. 2 2 , 414.(1954). (8) Jones, E., Proc. Roy. Sot. A198, 523 (1949). (9) Morgan, J . D.. Phil. M a g . 49, 323 (1925). (10) Rideal. E. K., Robertson. A. J. B., Proc. Roy. Sot. A195, 135 (1948). (11) Kobertson. A. J . B., J . Soc. Chem. Ind. 6 7 , 221 (1948). (12) Robertson, A. J . B.. Trans. Faraday Soc. 44, 977 (1948). (13) Ihid.. 45, 85 (1949). (14) Stout. H. P.. Jones. E.. "Third Syniposium (International) on Combustion." p. 329. LVilliatns and LVilkins Press. Baltimore. 1949. (15) Yang, C. H., Combust. Flame 6, 215 (1962). (16) Zinn. J., Mader, C. L.: J . Appl. Phyr. 31, 323 (1960). (17) Zinn, J., Rogers. K. N.; J . Phys. Chem. 6 6 , 2646 (1962). RECEIVED for review November 25. 1964 ACCEPTED February 16. 1965 Division of Fuel Chemistry, Symposium on Explosives. 145th Meeting, ACS,,New York. N. Y., September 1963.
AEROSOL FILTERS Pressure Drop across Single-Component Glass Fz'ber Fz'lters R
M
W E R N E R A N D
L. A. C L A R E N B U R G
Chemical Laboratory, .Vattonal Defence Research Organization T.VO. Rijsu ilk ( Z . H ). T h e .\-etherlands EVERAL
equations have been proposed to describe the
S pressure drop across fibrous filters for viscous
flohv.
From dimensional analysis of Darcy's law of flow through porous media. Davies (2) found an empirical expression relating pressure drop to filter porosity. T h e dimensionless group - W E 2 ~~
~
OLV is defined as the permeability coefficient! I;: in which p represents the pressure drop across the filter, '7 the viscosity of air, L the filter thickness. V the superficial velocity, and dE some 288
I&EC PROCESS DESIGN AND DEVELOPMENT
effective diameter. T h e effective fiber diameter is that fiber diameter \vith which the filter properties can be described as if the filter were composed of fibers of this diameter only. Davies' expression is given by
K = 64
x
(1 -
,)I5
[l
+ 56 (1 -
E ) ~ ]
(1 1
valid for fiber diameters ranging from 1.6 to 80 microns and filter porosities ranging from 0.700 to 0.994. Davies stated that for high values of the filter porosity. e . the effective fiber diameter. d E . should be taken as the arithmetic mean fiber diameter. dB%.,of the fibers in the filter.
The pressure drop w a s measured across glass f i b e r filters composed of Johns-Manville glass micro fibers. The fiber diameter distribution of all the codes used a p p e a r e d t o b e a fairly b r o a d log-normal distribution. T h e m e a n fiber diameters r a n g e d from a b o u t 0.1 to 3 microns. The filter porosities r a n g e d from 0.88 to about 0.96. A linear relation exists between the permeability coefficient a n d the porosity function, ( 1 e]'''. A resistance coefficient, defined a s the ratio of the permeability coefficient a n d the porosity function, a n d which is a function of t h e Cunningham slip factor, has been demonstrated to characterize a filter of given composition. As a result, a novel expression for the pressure d r o p across (1 fibrous filter has been proposed.
T h e porosity function, j ( 4 , here equal to (1 X [1 56 (1 - e)3], accounts for the variation of the pressure drop across the filter with the filter porosity only. I t approaches (1 - e)' 5 for high values of the porosity. Chen (7) found that the pressure drop depends not only on the filter porosity hut also on the structural arrangement of the fibers in the filter. He compared the drag force on an isolated fiber with the drag force on the same fiber in the filter. He concluded that the drag force on a fiber in the filter would not he equal to the drag force on an isolated fiber, because the neighboring fibers in the filter mutually influence the flow around each other. Therefore, he introduced a drag or resistance coefficient, k , which takes into account this structural interaction of the fibers in the filter. Chen derived the dependence of the pressure drop on the filter porosity from the drag force theory The equation of Chen for R is
+
R
= k
(-) 1
--6
The volume-surface mean diameter, d,,, is used as effective fiber diameter. Wheat (7) derived a relation for the pressure drop across a filter from an expression for the drag force on one individual fiber, situated perpendicular to the direction of flow. From the kinetic theory of gases and the experimental work of Stern ( 5 ) he deduced that filters containing fibers with diameters approaching the mean free path of the air molecules will show a higher flow rate at a given pressure drop than would be expected according to Darcy's law. This phenomenon is currently called slip flow. He concluded that the occurrence of slip flow should be introduced in the resistance coefficient, k , by making k a function of the Cunningham slip correction factor, C. The expression far C i s given in Equation 6. The expression far the porosity function was derived from the drag force theory, so that Wheat's expression for the permeability coefficient is given by Equation 2 in which he proposed for k the empirical relation
l/k
=
0.034 f 0.601 (C - 1)
various types of fibrous filters-i.e., those with fibers essentially oriented perpendicular to flow, with fibers parallel to flow, and probably with fibers randomized to flow. If Davies had grouped his data with respect to fiber orientation in the filter, he might have obtained different results. Grouping changes a t least the magnitude of the constants appearing in Equation 1. I t is the merit of Wheat (7) to have traced a relation between the resistance coefficient and the phenomenon of slip flaw. However, the scattered data used to derive this relation make the validity of Equation 3 rather doubtful. Although the studies concerned with the pressure drop across fibrous filters have given a better insight in the parameters related to it, calculated values of the pressure drop from pressure drop equations may occasionally differ appreciably from those actually measured. ?'his can be directly seen from the experimental results presented by Davies and Wheat. Since a reliable prediction of the pressure drop across a filter of given thickness, porosity, and fiber composition would be of great value, the relation of the pressure drop to filter porosity and fiber diameter was reinvestigated. This paper
(3)
As effective fiber diameter, dB, Wheat used the volumesurface diameter, d,,, for mixed-fiber filters and the some flowmean diameter for filters composed of fibers of almost uniform diameter. His experiments covered fiber diameters ranging from 0.18 to 14.4 microns and filter porosities from 0.940 co 0.750. Whereas the data of Davies ( ,2.)and Wheat (7). oartlv . . cover the same porosity and fiber diameter range, the dependence 01 the permeability coefficient, K , on the filter porosity should be essentially the same. This means that the porosity functions appearing in Equations 1 and 2 should he essentially equal. In fact, the porosity functions differ widely. Chen ( 7 ) has shown that the pressure drop across a fibrous filter depends not only on the porosity hut also a n the structural arrangement of the fibers in the filter. Davies derived an empirical expression far the porosity function from data on
Figure 1 .
Electron microscopical photograph of glass fiber
Johns-Manville code 102. Mognificotion -5000
VOL. 4
NO. 3
X
JULY 1965
289
Table I.
Codea
a
M a p . on Photo
No. of Fibers Counted
UQ
1196 964 490 361 271 41 3 444
2,45 2.35 2 59 2 70 2 80 3.21 3.40
100 10,500 102 10.500 104 10 500 106 10 500 108 5 500 110 5,500 112 5.500 Johns-Manuille code numbers.
Fiber Properties -
d , J! d,,
d',
(Eq. 4) 0.137 0.203 0 324 0 497 1 24 1.87 3.24
P
0.097 0.141 0 206 0 303 0 727 0.946 1.530
P
(Eq. 5) 0.0377 0.0855 0 260 0 665 4 42 17.3 47.0
8.60 14.1 23 4 38 7 69 5 8 4 , bb 117
3.46 2 77 2 30 1 92 1 39 1,35 1.19
125 140 15.0 170 2-0 420 1050
Calculated ualue.
is concerned with filters composed of glass fibers showing a lognormal diameter distribution. I n subsequent papers the relations found in this paper for single-component filters will be applied to multicomponent glass fiber filters of arbitrary composition. T h e Cunningham slip factor. C, is given (3, 7) by the equation
Experimental
Specification of Glass Fibers Used. All filters used in this study were prepared from Johns-Manville glass micro fibers. This factory grouped the glass fibers into codes according to the value of the mean fiber diameter. To establish the fiber diameter distribution u i t h i n each code, the fibers were electron microscopically photographed.
C = 1f
d
J
I
.&
z
(In d) 6 In d
3
L
S
C(d)$, (In d) 6 In d For small values of d-i.e.: d < 0.2 micron-Equation adequately approximated by
Figure 2. 290
C
z
3
~
=
1
+ dx
-
[2.46
+ 0.82 (1 - 0.44 d, A ) ]
sm
z
3
( 5 io 2 3 libre diam*t*r on pho1o.mm
Cumulative fiber diameter distribution of various codes
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
(6)
(7)
6 can be
(8)
For d > 0.2 micron a fair approximation is obtained by neglecting the exponential term; hence
(4)
IO
[2.46 f 0.82 exp(-0.44 d, A ) ]
In 0 05
With the values of d, and u, available, the values of d, and dlz, ivhere subscript z refers to the code under consideration, could be calculated from their statistical definitions m :
d
which ielates the Cunningham correction to the fiber diameter, d, and the mean free path of the air molecules, A. taken as 6.45 x 10-6 cm. a t 20' C . a n d atmospheric pressure. Since the smallest fiber diameter observed was about 0.05 micron. Cz was calculated according to the equation
A typical photograph is shown in Figure 1. From each code the diameters of a great number of fibers were measured. I n Figure 2 the cumulative fiber diameter distributions have been plotted on a log-probability scale. Evidently all codes exhibit a log-normal fiber diameter distribution. T h e geometric mean diameter, d,, and the geometric standard deviation, u g , of the distribution were determined with the aid of Figure 2. Their values have been put together in Table I. -
d, =
x
-
~
5 io
c' = 1
+ 2.46 X
d
(9)
LYith the aid of Equations 8 and 9, Equation 7 can be written as
the porosity of these filters. probably not too serious.
T h e errors thus introduced are
Results and Discussion
q 2 (In d ) 6 In d +
s" [ + a] 1
1.1
2.46
q f (In d ) 6 In d
Permeability Coefficient and Porosity. T h e permeability coefficient, K , was calculated according to its definition (10)
0.2
-
T h e average length of the fibers of one code. 1,: is estimated from the consideration that the number of fiber ends on a photograph, S. is proportional to the number of fibers on a photograph, n. and to the dimension of the photograph, but inversely proportional to the fiber length a n d to the photographic magnification. M . T h e square root of the surface of the photograph. s. is taken for its dimension. Hence, the number of ends per fiiber being 2.
in which
Ap = pressure drop, m m . of water column d2 = mean square diameter of fibers g = acceleration due to gravity, cm.t'sec.2 p
=
V in \vhich n i and LYlwt're determined by counting. T h e results obtained have been summarized in Table I
Preparation of Filters. Three grams of fibers were Lveighed and dispersed to a consistent!- of 0.37, in a laboratory mixer. I n a Rapid-Kothen paper-making apparatus ( 4 ) the fibers \\.ere deposited on a stainless steel screen. yielding circular sheets of 22-cm. diameter (80 grams per s q . meter). Lsually the top surface was smoothed by slightly pressing the sheet before drying. T o obtain filters of higher porosity the sheets \vere dried on the screen, removed there from. and left unpressed. T h e porosity \vas lo\vered by dry-pressing u p to 270 kg. per sq. c m . Filters prepared a t pH 2.6 to 3.0 shoxved relative1)- the best results \vith respect 10 mechanical strength and homogeneity. Fiber consistencies greater than 0.47,could not be handled Lvith the available apparatus. Filters composed of fibers of more than 2-micron diameter \vere difficult to prepare because of fiber clumping. Moreover, these filters had poor mechanical strength. From the sheets, samples of 6.0-cm. diameter Lvere selected and lveighed: and the average thickness was determined \vith a micrometer. 'To determine Ivhether the porosity of the filter changed as a result of the force exerted by the pressure d r o p across the filter area. samples of 10 different filters Lvere tested during 1 hour a t constant air flon.. Since the pressure d r o p across all samples as well as the filter thickness remained constant, the filter porosity is not time-dependent ( 6 ) . To measure the resistance to air f l o ~ ,each sample \vas placed in a brass holder with a flohv area of 19.63 sq. c m . T h e pressure drop across the filter was measured a t a number of different flow rates u p to superficial velocities of 35 c m . per second \sith a \vater-filled manometer. Occasionally, kvhen the pressure drop \vas very small. an alcohol-filled micromanometer \vas used. T h e observed pressure drop was corrected for the resistance to air flow of the empty holder. Determination of Porosity. If W represents the weight of a filter. A its surface area: L it sthickness, and po the specific density of the glass fi'bers. then the porosity may be calculated from the equation TI7 = ,4L p , ( l - e )
T h e thickness \vas determined with a micrometer; po xias determined to be 2.47. For porosities greater than 0.935 the top surface of the filters became markedly irregular, resulting in a less accurate determination of the thickness a n d hence of
Lvater density, g..'cc. taken as 1
7 = viscosity of air taken as 1.83 L = filter thickness, mm. =
X
g./cm. sec
superficial velocity, cm.,'sec.
L'alues of the permeability parameters are given in Table I1 for codes 102 and 108.
Table II.
Resistance Coefficient, k Ap, Mm.
Code
102
Filter Thickness, Mm.
0 47 0 46 0 49 0 82 0 62
Porosity
at
70 C m . /Sec. =
0 39 0 53
0 933 0 931 0 935 0 960 0 947 0 944 0 953 0 961 0 918 0 940
262 264 246 210 228 238 196 196 280 230
0 40 0 49 0 34 0 44 0.90 0 46 0 67 0.62
0 916 0 930 0 913 0 926 0 961 0 935 0 948 0.945
27 27 26 26 20 22 21 24
0 58 0 67 0 80
108
Hz0
v
K X 703
k = K / ( 7 - e)3
255 263 229 117 168 188 134 112 328 199 k,,
14 14 13 14 13 14 13 14 14 13 = 14 u = 0
5 2 0 5 6 0
5 0
1635 1310 1'65 1430 541 1065 765 920
8
5 8
6 9
2 0 5 0 5 1 6
6' 0 '0
2
68 8 71 2 70 2 68 1 64 9 '1
k,,
=
u =
2
69 5 2 2
About 50 different filters were prepared to establish the relation between the permeability coefficient and the porosity. I n Figure 3 the permeability coefficients have been plotted I n this study the against the porosity function, (1 porosities of the filters ranged from 0.960 to 0.880. T h e figure sho\vs linear relations for all codes investigated. Because of increasing inhomogeneity of the filters. deviations from the -
straight-line relationships increase with increasing value of d2. T h e correlation coefficient: however: remains highly significant. Resistance Coefficient. Since linear relations exist bet\veen K and (1 - c ) ~ for ' ~ the various codes: each code exhibiting a specific slope, it is obvious that this slope characterizes a filter of given composition. This slope is called the resistance coefficient and is denoted by the symbol k . Hence. with the aid of Equation 12. (1 3 )
VOL. 4
NO. 3
JULY
1965
291
Relation of permeability coefficient to porosity function, (1
Figure 3.
In Table I 1 the k values are given for individual filters made of code 102 and code 108 glass fibers, respectively; in Table I the k values for the various codes are given. As can be seen, the standard deviation of the individual k determinations is smaller than 570. It may well be that the characteristic k value depends on the structural arrangement of the fibers in the filter. In this study the structural arrangement does not vary, since in each filter the fibers are situated essentially radially at random and perpendicular to the direction of
14) for the relation between k and is not in agreement with Wheat's expression. Since the k values probably depend on the structural arrangement of the fibers in the filter. the constant and the power of ti appearing in Equation 14 will assume other values for different fiber orientations. Conclusion The results described above give rise to some interesting conclusions, valid for single-code glass fiber filters. The proportionality of the permeability coefficient and the porosity function (1 - c ) ~ ' *found in this study agrees well u i t h the porosity function found by Davies for the porosity range considered. For this porosity range the revealed proportionality seems to be firmly established now. It does not agree with the porosity functions of Chen and Wheat derived from the drag force theory. T h e resistance coefficient, defined as the ratio of the permeability coefficient and the porosity function. provides an accurate characterization of a filter of given composition. A strict relation between the resistance coefficient and the Cunningham slip factor is established. The relation found in is not in accordance with the work of this study, k = 180 ? . 5 , Wheat. Equations 13 and 14 give rise to the following over-all
fl0W.
Resistance Coefficient and Cunningham Slip Factor. I t seemed interesting to check Wheat's relation (Equation 3) between the resistance coefficient and the Cunningham slip factor. T o this end the values for the various codes, calculated by means of Equation 10, were plotted against their corresponding k values, calculated with the aid of Equation 13 on log-log paper. Figure 4 shoivs that log and log k exhibit a strict linear relationship represented by
c
c
where index i refers to the code.
-
The expression (Equation
k = 180C-2'5
51
c
'1 '
I
I
I
7
e
9 10'
I
I
1
1
I
I
I
2
3
4
5
6
7 8 9102
l
l
1
1 2
k Figure 4. 292
Relation of resistance coefficient to Cunningham slip factor
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
1
3
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 = \
