Simple Air-Permeability Method for Measuring Surface Areas of Fine

of the air- permeability method, the powder is so fine that the air flow rate is too low to measure conveniently. An experimental technique based on t...
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1623

V O L U M E 2 6 , NO. 10, O C T O B E R 1 9 5 4 Figure 3 shows that the turbidity of the 85 p.p.m. zirconium solution decreases 17 ith increasing concentration of fluoride ion. The limit of permitted concentration of other ions was found to be identical with that of Table I.

solution. KO critical conlrol is necessary of the length of heating, cooling, or hydrogen ion concentration. Thus the recommended turbidimetric procedure is a simple, direct, and rapid method for the microdetermination of zirconium. It can be modified to serve for the estimation of fluoride ion Concentration.

CONCLUSIONS

Zirconium in the prrsence of large amounts of various ions can be successfully determined by the suggested turbidimetric technique. This method makes use of readily available reagents and simple inexpensive equipment. illthough the Bausch and Lonib monochromatic colorimeter was selected for this investigation, any colorimeter or turbidimeter could be used. The suspension of zirconium phthalate is very stable and independent of the order of mixing the reagents, as the reaction takes place only above room temperature. Under these conditions the suspenqion is formed slowI?- from a homogeneous

ACKNOWLEDGMEYT

The authors wish to thank the Research Corp. for a grnnt in partial support of this work. LITERATURE CITED

(1) Purushottam, A , , and Rao, Bh. S. V. R., Analyst, 75, 654 (1950). RECEIVED for review January 18, 1954, hccepted June 29, 1954. Portion of a thesis submitted by Leroy E. Swim in partial fulfillment of the requirements for the degree of master of science, Kansas State College.

Simple Air-Permeability Method for Measuring Surface Areas of Fine Powders H.1. KAMACK Engineering Research Laboratory, Engineering Department,

Conventional techniques for measuring surface areas of powders cannot be applied to some materials either because, in the case of gas adsorption, some of the gas is taken up inside the material rather than on the surface of the particles or because, in the case of the airpermeability method, the powTder is so fine that the air flow rate is too low to measure conveniently. An experimental technique based on the air permeability method has been developed which is applicable to powders whose average size is below about 30 microns and which is especially useful for powders in the range from 5 to 0.2 microns. The method is simple and rapid. The method is useful for research work involving powders in the micron and submicron range, and also is useful as a rapid control test for such materials.

I

P; T H E

course of some grinding experiments in which ilmenite sand was used as a convenient test material, it became necessary to measure the surface area of the ilmenite during grinding. The two best-developed methods for measuring surface areas of powders are the nitrogen-adsorption method associated with the names of Brunauer, Emmett, and Teller, and the air-permeability method. The nitrogen-adsorption method depends on measuring the quantity of nitrogen needed to form a monomolecular layer on the surface of the porr-der. Besides being relatively expensive and time-consuming, this method was found to be inapplicable to ilmenite powder because nitrogen not only adsorbs on the surface of the particles of powder but also penetrates into the grains. Penetration was suspected when measurements by this method gave extremely high surface areas, and it was confirmed by testing the method on a sample of washed, unground ilmenite sand. These grains are in the range from 48 to 150 mesh and appear under the microscope as smooth, rounded, nonporous particles. The surface area calculated from the sieve analysis (which should be reliable, since the sand contained no dust) was 0.01 square meter per gram. Nevertheless, the apparent surface area obtained by the nitrogen-adsorption technique was 8 .square meters per gram. It is believed that the nitrogen may

E. 1.

du Pont d e Nemours

& Co., Wilmington,

Del.

have been held in tiny cracks between submicroscopic crystals of n hich the ilmenite is composed, and the width of these cracks is estimated to be less than 0.01 micron. Because of these limitations of the nitrogen-adsorption method. the author turned to the air-permeability method, which is based on measuring the flow rate of air through a compressed bed of the powder with a known pressure drop across the bed. This method measures only the external surface area of the particles, because air will not flow through the tiny cracks in the particles, even if they should happen to form continuous channels. However, the usual techniques of air-permeability measurement could not be applied to this special case, because some of the poffders to be measured were so fine that the air flow rate through the bed was too small to measure accurately. An attempt was made to increase the flow rate by using a relatively large area of sample bed and a large pressure drop across the bed, but this led to further difficulties in that the p o d e r bed tended to crack, which ruined the measurement. The method described beloFT is called, for convenience, the [‘manometermethod.” It involves no new theoretical principles, being based on the permeability theory which has been developed and confirmed by previous investigators, but it involves an experimental technique that avoids the difficulties of measurement described above; in fact, the only limitation on the fineness of ponders that can be measured is imposed by the limits of applicability of the theory rather than any difficulty of measurement. The main reason for this is that the surface area is made to depend on a measurement of the time for a known quantity of air to flow through the powder, and this time becomes longer and therefore easier to measure accurately as the powder becomes finer. The method has been found to be practical, reliable, rapid, and simple, and i t has been used on several hundred powder samples ranging in fineness from 30 to 0.2 microns average particle diameter in specific surface. METHOD AND APPARATUS

The apparatus shown in Figure 1 is used, consisting of a manometer of about 1-cm. bore with one arm about half the length of the other, a sample tube of about 0.5-inch diameter, and a stopper and adapter for connecting the sample tube to the short arm of

1624

ANALYTICAL CHEMISTRY

the manometer. The manometer is filled to a marked level which is as close as is practicable to the top of the short arm. When measuring powders for which D , is in the range from 30 to 3 microns, a xylene-filled manometer is used; from 3 to 0.2 microns, a mercury-filled manometer is used. The manometer scale is marked a t points which are 2.5 and 5.0 cm. on either aide of the zero point that corresponds to the equilibrium liquid level. Over this 10-cm. range, the manometer bore should be uniform and equal in both arms. The. sample tube used was of the type used in the Fisher Scientific Co.’s subsieve sizer, but any convenient sample tube is satisfactory. (The only requirement is that the volume Vo of free air space between the powder bed and the equilibrium liquid surface in the short arm of the manometer should not exceed about three times the volume enclosed in a 2.5-cm. length of the manometer tube. It is desirable also that V Obe the same in all determinations, because it enters into the calculation of an apparatus constant, as shown in Equation 30.) The sample tube referred to is a brass tube, 0.5 inch inside diameter and 5 inches long. Two perforated plugs slide in this tube freely. By placing a 0.75-inch filter paper disk over the face of each plug and crimping the paper around the edge of the plug, the plugs slide in the sample tube Kith sufficient friction to maintain their position against gravity and minor shocks. B R A S S SAMPLE TUBE 5 IN. LONG X 112-IN. I.D.

PERFORATED PLUG F I L T E R PAPER SAMPLE F I L T E R PAPER PERFORATED PLUG TO VACUUM

TO ATM,

RUBBER STOPPER GLASS ADAPTER GROUND GLASS JOINT

TIME IS MEASURED FOR L I Q U I D TO PASS BETWEEN THESE MARKS

EQUILIBRIUM LIQUID LEVEL

* FOR UNIFORM BORE, T H E SAME I N BOTH A R M S

w

-

I

2

MM. P Y R E X

(Equation 29). The modified Knudsen number, x, is then calculated from the equation:

where lip is a pressure correction factor (Equation 3 2 ) and Dk’ is the value of Dk converted to microns. The slip f l o correction ~ factor, f, is calculated from the equation:

( I t has been found convenient to construct a graph of this function for the purpose of obtaining f rapidly.) The specific surface average particle diameter, D,, of the powder is obtained from the formula: (5) The specific surface, S, is obtained from

S = 6/Ds~p (6) The basis for these formulas and the method of calculating the apparatus constant and correction factors that enter into some of them are discussed below. The apparatus requires no calibration, because the apparatus constant, a, can be calculated directly from the dimensions of the apparatus, as shown. The time required for a measurement increases with the specific surface of the sample and depends also on the manometer liquid used. Using mercury, a sample for which D, is 0.2 micron requires a measurement time of about 15 minutes. This may be considered as a maximum measuring time. To this must be added a sample weighing and preparation time of about 10 minutes and a calculating time of 5 to 10 minutes. THEORY

Kozeny-Carman Equation.

The air-permeability method for measuring surface areas is based on a relationship between the surface area of a powder and the rate of flow of fluid through a bed of the powder, which n-as developed originally by Kozeny (18) and independently by Fair and Hatch (14). This equation was modified by Carman (6, 7 , 1 0 ) and was used by him for the determination of surface areas; and it is now known as the KozenpCarman equation. It may be written as follows:

TUBING

Figure 1. Manometer for Surface Area Measurements on Powders

The powder sample is placed in the tube between these plugs, v,-ith a piece of filter paper between each plug and the powder sample. The sample is then compressed to a suitable porosity by pushing the plugs together with steel plungers inserted into the tube. One of these plungers serves as a spacer to locate the powder sample always a t the same point in the tube, in order to fix the volume V o . The length, L, of the powder bed is determined by measuring the positions of the faces of the plugs in the tube with a machinist’s dial depth gage. From this, the porosity can be calculated by the equation :

(7) This equation was derived from Poiseuille’s law for viscous (streamline) flow of a fluid through a tube by considering the powder bed as consisting of a large number of parallel flow tubes and making an appropriate choice of the hydraulic radius of the tubes in terms of the surface area and porosity of the powder. This derivation is available in the references to Carman (6, 7 , 1 0 )

where m is the mass of the powder sample. When the sample has been compacted to the desired porosity, the sample tube is connected to the short arm of the manometer, as shown in Figure 1, after the manometer liquid has first been displaced from equilibrium by applying vacuum to the long arm. When the sample tube is in place, the long arm is opened to the atmosphere and the liquid falls towards its equilibrium position, forcing air through the powder bed as it does so. The time At for the liquid surface in either of the arms to pass from the 5.0-cm. mark to the 2.5-cm. mark is measured with a stopwitch. The Kozeny-Carman particle diameter, Dk, is thrn calculated from the formuh:

0

100

200

300

400

500

600

Time, 1 , sec.

in which a is an apparatus constant (Equation 28), and k b is a correction factor for atmospheric pressure and temperature

Figure 2.

Variation of Pressure Drop across a Powder Sample with Time

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V O L U M E 26, NO. 10, O C T O B E R 1 9 5 4 and in many other places in the literature on this subject. The factor, k , in Equation 7 is a dimensionless number which in theory depends on the shape and tortuosity of the flow passages. Carman found experimentally a value of 5.0 for k , and this value does not seem to have been improved since. The Kozeny-Carman equation has been investigated extensively by many workers (see 19, for example) and may be considered as well established, within certain limits. The principal limitations are listed below. There are other limitations in the case of liquids, which need not be considered here. The flow through the powder should be viscous rather than turbulent. The criterion for this is that the modified Reynolds number, q / r S A (1 - e), should be less than about 2. If the mean-free-path length of the fluid molecules is appreciable in comparison with the size of the pores in the poTvder, a correction term must be applied, which is discussed below. This correction becomes significant when the Knudsen number, which is the ratio of the mean free path to the pore radius, exceeds about 0.1. The powder must be packed into a uniform cake free from cracks or other flam which would permit channeling of the fluid, for the theorv to apply. The variation of flow rate with porosity of the poader bed frequently does not follow the Kozeny-Carman equation a t all well. This is the msin difficulty with the equation, and is discussed further below. Porosity Function in the Kozeny-Carman Equation. Several investigators have criticized the porosity function in the KozenyCarman equation, and have suggested modifications of it. [The difficulty in accurately accounting for changes in porosity may be appreciated when i t is noticed that the porosity funct,ion e 3 / (1 - e ) P in the Kozeny-Carman equation varies as €6 when e = 50%, which is a reasonable value.] The factor $/(1 - . ) 2 in the equation arises from the definition of the hydraulic radius of the flow passages through the powder bed. The remaining factor of B arises in converting from the actual fluid velocity through the powder to the superficial velocity q / A , since the fractional free cross-sectional area in the bed is equal to the porosity. However, Furnas ( 1 6 ) and Chilton and Colburn ( I S ) have pointed out that some of the pore space in the bed does not contribute to the fluid flow, and that the effective fractional free cross-sectional area available for fluid flow is much smaller than the porositv. From their point of view, the velocity in the pores is related more nearly to the minimum cross section of each pore than to the average. Arnell ( 1 ) assumed that the porosity function in the IiozenyCarman equation should have the form e’../( 1 - e ) z , \There a is the effective fractional free cross-sectional area, and he obtained the empirical relationship: 01

= 25.7e/25.1

(8)

Iieyes ( 1 6 ) assumed that a layer of immobile fluid e\i-ted aiound each particle in the bed and obtained the relation: = 1.12

-

0.11)3/~2

($1)

Table I shons Fome values of 01 according to Furnaq. hrnell, and Keyes. There is not much agreement among the results, except a t low porosity, but they all indicate that the effective free area is considerably smaller than the porosity. I n using these modified porosity functions, it must be borne in mind that the value for k of 5.0 in the Iiozeny-Carman equation was determined empirically, assuming a = e, a t porosities ranging from 35 to 41% or, say, an average of 38%. If other values of a are used, another constant k’ must be introduced so that a / k ‘ has the same value as f / k when B = 38%. Table I1 shows some values of the ratio of the specific surface which would be calculated using Arnell’s or Keyes’ porosity function to the value using the Kozeny-Carman porosity function. These ratios are equal to ( k / k ’ ) ( . ~ / e ) . Because of the greater amount of evidence offered, the Arnell function is favored over Keyes’. It also deviates less sharply

Table I.

Fractional Free Cross-Sectional Area m,

5%

70

Furnas

Arne11

Keyes

63.8 55.1 41.9 38.0

10.8-1G 10.8-19

31.7 23.9 15,s 13.7

40.5 31.G 18.8 15.2

6%

15.4 , . .

Table 11.

Effect of Porosity Function on Surface Area

Porosity, % 38 40 45 50 60

Relative Specific Surface, k o / k ’ r Arnell : Carman Keyes : Carman 1 1.006 1.03 1.00 1.13

1

1.03

1,lO 1.15 1.24

from the Kozeny-Carman function, as Table I1 show. In fact, over the range from 40 to 50% porosity, in which most specific measurements are made, the maximum deviation is only 6%. Especially for fine powders, this correction is not of great importance, and Arnell himself dropped it in later work. I n some cases, considerably greater variation of the specific surface (calculat,ed from the Kozeny-Carman equation) with porosity than that indicated in Table I1 has been found. This variation is the main theoretical difficulty with the air-permeability method for measuring surface area. The difficulty may be minimized by taking measurements in a “normal” porosity range, as suggested by Carman and Malherbe ( 1 2 ) . Whenever possible, surface area measurements on similar materials should be compared a t the same porosity. Specific Surface Average Particle Diameter. In working with surface areas of powders, it is convenient to use the specific surface average particle diameter, D,, which is defined by the equation:

D, = 6 / S ,

=

6/Sp,

.

(10)

although Equation 10 is dimensionally consistent, it is also true that if S is in the usual units of square meters per gram and D , in the usual units of microns (and p p in grams per cubic centimeter) the equation is also correct. Some people object to the use of the specific surface average particle diameter on the legitimate grounds that, it gives a misleading idea of the average particle size of a nonuniform material. Hoxever, if it is thought of merely as another wag of expressing specific surface, in accordance with the definition above, there should be no confusion. -4part from particle shape factors, the specific surface average diameter lies closer to the weight median diameter of a powder than almost any other average diameter that might be used. Slip-Flow Correction to Kozeny-Carman Equation. \Then the permeability method of surface area measurement was first applied to powders with specific surface average particle diameters smaller than about 5 microns, it became evident that there were discrepancies between measurements with liquids and gasee, the latter giving lower values of specific surface (9, 20). The reason for this was not understood until 1946, when Rigden (22) explained it as being due to slip flow of gases The Poiseuille law for the viscous flow of a fluid through a tube is based on the assumption of a continuous fluid. When a gas flows through a tube under conditions such that the meanfree-path length of its molecules is much longer than the diameter of the tube, the assumptions on which Poiseuiile’s law is based are completely invalid, and viscosity has no meaning. Under such conditions, the molecules progress chiefly by bounding from wall to wall. Because of the extreme roughness of a wall on a molecular scale, the rebound of a molecule is random in direction (the molecules are said to be diffusely rather than specularly reflected) so the flow is controlled by the l a w of probabilityi e., by diffusion, and is therefore called molecular, or diffusive, flow. hlolecular flow occurs when h/rh is much greater than 2.

1626

ANALYTICAL CHEMISTRY

When X / ) . h is much smaller than 2 but still a significant quantity, the flow through a tube is intermediate between true molecular flow and true viscous flow. Under such conditions the flow resembles viscous flow in its behavior if it is considered that the velocity a t the tube wall, instead of being zero (as assumed in the laminar theory of viscous flow on which Poiseuille’s law is based) has a finite value. For this reason, this flow condition is known as slip flow. I n the theory of this tvpe of flow, the slip velocity a t the wall is related to the fraction, f d , of molecules striking the wall that are diffusely reflected. Between the region where the slip flow theory is valid and the region of pure molecular flow, the flow is more complicated. This is the region where X / r h is of the order of magnitude of 1. Knudsen (17’) was able to derive a partially empirical equation which expresses the flow rate over the whole range of X / T h from zero to infinity, which reduces to Poiseuille’s equation for viscous flow when X / T h = 0, to Knudsen’s equation for molecular flow when X / T h = a, and to the Kundt-Warburg equation for slip flow when X / T ~is small but not zero. Brown, DiNardo, Ching. and Sherwood (6) have also found an empirical correlation between the Knudsen number X / r h and the ratio of the actual flow rate to the flow rate predicted by Poiseuille’s law, which covers the entire range from viscous to molecular flow and which agrees quite well with Knudsen’s result. I n pipes, slip flow and molecular flow are phenomena associated with the flow of gases a t low pressures, but in the case of gases flowing through fine powders they can occur a t atmospheric pressure, since the criterion is the Knudsen ratio X / T h . For air a t atmospheric pressure, has a nominal value of 0.1 micron, and this ratio has a significant value for the pores of powders smaller than 5 microns in specific surface average particle diameter. Several investigators have derived equations to describe slip flow and molecular flow of a gas through a powder by drawing an analogy to flow through tubes in much the same way that the Kozeny-Carman equation was derived by analogy with viscous flow through tubes. I n the slip flow region, all of the equations that have been derived can be written in the form: I - e

1

q =

krS,2 (1 -

€)’

L

Under conditions where the last term of this equation can be dropped, i t reduces to the Kozeny-Carman equation, Equation 7 . Since Equation 11 is quadratic in S,, it is a little awkward to use. I t has been found convenient to solve it in the following way. Define the Kozeny-Carman particle diameter, Db, as the specific surface average particle diameter that would be obtained using the Kozeny-Carman equation without the slipflow correction:

DA

e

E

Now, replace S, in Equation 11 by D. in accordance with Equation 10, substitute Dk and z in Equation 11, and letf = D,/Dr. Equation 11 then becomes: f2

+ 2xf

-

1=0

(14)

So, solving for f Equation 4 is obtained, which permits the calculation of f , which is then applied as a correction factor to Dk to get D,. If Equation 14 is solved for x , we get

x

where ko is a tube cross-section shape factor that occurs in Poiseuille’s lay, which is close to 2 for most shapes. Rigden took 2 = 1.748, a value he did not explain. Carman ( 8 ) considered this value of Z was too low, since for granular beds k~ should be between 2.5 and 3. Using these values, and using f d = 0.84, which is the value found by Brown et al. ( 5 ) , the following values for 2 are obtained from Equation 16: 2.3 and 2.76.

Table 111. Comparison of \-slues of Slip Factor, Z

z Rigden (21, 82) Rigden, modified by Carman (8) See Equation 16 fd = 0 . 8 4 , ko = 2 . 5 ko = 3 fd = 0 . 7 9 , ko = 2 . 5 ko = 3 Arnell ( 2 ) using a from Equation 8 See Equation 17 0 = 0.9, k = 5, e = 0 . 4 e = 0.5 e = 0.6 Arnell (3) and Carman and Arne11 (11) See Equation 18 ku = 2 . 0 ko = 2 . 5 ko = 3 . 0 Carman and Arneli (1I), experimental (av 2 . jj, Lea and Xurse (BO). experimental (av 1 97)

1.745 2.3 2.76 2.55 3.06 1.86 2.06 2.38 2.04 2.55 3 06 1 .94-3.01

1 63-2.43

Lea and Nurse ( 2 0 ) measured Z euperimentally by measuring the flow rate as a function of mean free path, by varying the gas used and the pressure. Their results (corrected for their method of calculating mean free path, as discussed above) gave values of 2 ranging from 1.63 to 2.43, with an average of 1.97. Arnell (1) derived an equation for slip flow through a powder bed in the form p = pv #I pm, where pv is the flow rate according to the Kozeny-Carman equation, pm is the flow rate for pure molecular flow, and d is a numerical factor which Arnell concluded should have a value of 0.9. When Arnell’s equation is written in the form of Equation 11, it may be seen that

+

Kow, define a modified Ihudsen number, z,as follows:

z = 3ZX -1 -

Comparison of Values of the Slip Factor 2. Although the various investigators agree on the general form of Equation 11, there is considerable disagreement in the values that have been obtained for the numerical factor, 2. Consequently, their results have been reviewed in an attempt to determine the most reliable value to use. Different investigators have used different formulas to calculate the mean free path, A , or equivalent parameters, and this, of course, affects the value of Z they obtain. . Since only the product Z h occurs in Equation 11, it is a simple matter to adjust the value of Z to any chosen value of A. The value of X for air a t 1 atmosphere pressure has been arbitrarily considered to be 0.1 micron, and the value for Z found by different investigators has been adjusted to fit this value, whenever necessary. I t is also assumed that X varies inversely with pressure. I n Rigden’s original equation (WZ),

1-f’ = -2f

It is convenient to use Equation 15 to plot x against f, and then to use this plot for finding f.

in which 4 = 0.9 and k = 5.0. Values of cy corresponding to values of €aregiven by Equation 8 or Table I. Carman and Arne11 (11) compared Arnell’s slip factor as given by Equation 17 with Rigden and Carman’s slip factor, Equation 16, and concluded that the factor k a l e in Equation 17 is equivalent to the factor ko in Equation 16. They obtained, thereby, the modified slip factor: Z

=i

~

3% ko = 1.02 ko 9iT

(18)

1627

V O L U M E 26, NO. 10, O C T O B E R 1 9 5 4 for 4 = 0.9. They considered ko to lie between 2.5 and 3 wit,h a probable value of 2.5. I n order for Equation 18 and Equation 16 to yield the same value of the slip factor, 2 , it is necessary t’o havefd = 0.79 instead of 0.84, as found by Brown et al. Later, ilrnell ( 2 ) measured the flow rate through powders a t various pressures and Carman and Arne11 ( 1 1 ) used the results to obtain experimental values of m / e . I n view of Equation 17, this is equivalent to an experimental determination of Z by essentially the same method as that used by Lea and Nurse. The values of @ / e so determined ranged from 0.38 to 0.50, with an average of 0.50. The corresponding values of Z range from 1.94 to 3.01, with an average of 2.55. The values of Z, bot,h theoretical and experimental, obtained by these investigators, are summarized in Table 111. It may be seen that Carman and Arnell’s experimental values agree better with Amell’s modified slip factor (Equation 18) than with his original slip factor (Equation 17). Rigden’s slip factor gives lower values than Arnell’s unless a value o f f of 0.79 is taken to force agreement. Lea and Yurse’s experimental values of Z are also lower than Carman and .knell’s. Between the two sets of experimental values of Z, those of Lea and Kurse and those of Carman and A4riie11,the neight of evidence favors the latter for the following reasons: Arne11 (3) made measurements of flow rates through zinc oxide powders a t various pressures and found that the plot of mass flow rate against pressure is a straight line in each case. The specific surface can be calculated independently from either the slope or the intercept of such a line. iirnell found good agreement between these independent calculations, when Z was evaluated from Equation 18. The specific surface mean diameters ranged from 1.5 to 0.13 microns. The measurements of Lea and Nurse were made principally at porosities of 50 to 65’%’,,and, as shown by Carman and Malherbe ( 1 2 ) and discussed above, values of specific surface measured in this range will usually be low. Good agreement is obtained between the air permeability method and the nitrogen adsorption method of measuring surface area when the measurements are made in the 40 to 50% porosity range and ilrnell’s modified slip factor is used (3, 4,I d ) .

It is concluded that 2 = 2.5 is the most reliable single value for the slip factor. Application to Manometer Method. Equation 11, the KoeenyCarman equation modified for slip flow, was derived on the assumption that the flow rate, q) and the pressure drop, Ap, are constant in time. I n the manometer method, both q and A p vary continuously with time, and in order to apply Equation 11 to this method, it is necessary to express q / A p in terms of the measured time interval, A t . Although Equation 11 was derived for steady-state floy, it can also be considered to represent the instantaneous rate of flow in the present unsteady-state case, because, with t.he lo^ flow rates and low fluid density involved, it may be proved that a negligible error is involved in assuming that the air instantly adjusts itself to the equilibrium flow rate corresponding to the pressure drop. This situation is usually referred to as a quasistationary state. The mass Jf of air in the manometer upstream from the bed a t any time t is: .If where

p,

(Vo

+ AoAh/2)p

(po/ho)(ho

+ Ah)(To/T)

If the temperature change is adiabatic, T*/T1 = (p,/p1)O.2*8 = 0.984. I n this case, A q T = - 0.0176 VO. Now, this temperature change (at 300’ K. room temperature) is around 5’ C. Considering now temperature changes due to changes in ambient temperature, i t is obviously a simple matter to keep these well below 5” C. Consequently, a change of 5’ C. in the temperature of the air upstream of the powder bed may be regarded as a eafe upper limit under ordinary circumstances, and so I AqT 1 < 0.0175 Vo-i.e., the effect of temperature is normally less than one fourth the effect of pressure. To make the absolute value of the term APT smaller than, say, 5% of the sum Aqu Aqp, it is therefore sufficient that V Oshould not exceed 3 A V . If this precaution is taken in constructing the apparatus, the last term in Equation 21 may be dropped n i t h negligible error, giving:

+

q =

- hnl [‘&ha 2

+ V O+ AoAh]

ddt Ah

By plotting Ah against t, the slope of the curve, d Ahldt, a t any value of Ah could be used in Equation 22 to calculate q. But by making the change in the mean pressure, p,, relatively small, i t is possible to dispense with the plotting and to simplify the calculation. Combining Equation 11 and Equation 22, there results:

(19)

the density of t,he air in this region, may be espressed as P =

On the right-hand side of this equation, the first term represents the flow rate due to the volume of air displaced by the manometer liquid a t constant temperature and pressure, the sum of the second and third terms represents the flow rate due to isothermal expansion of the air as a result of the decreasing pressure on the upstream side of the bed, and the fourth term represents the flow rate due to an isobaric change in the temperature of the air. The temperature may change either from the work of expansion or by heat transfer to or from the surroundings. It will nolv be shown that, with a suitable choice of apparatus constant, the effect of temperature changes can be made negligible. Denote by Aqt, Aqp, and A q t the quantities of air flowing through the bed due to volume, pressure. and temperature changes, respectively, during the time At that the differential manometer reading falls from 10 to 5 em. The densest liquid likely to be used in the manometer is mercury, and this \vi11 give the greatest pressure and temperature changes. For this liquid, approximately: 1 Aq” = 2 AU(Ih1 - Ahz) = A V

(20)

The mass flow rate of air through the powder bed is -dJf/dt. I t has been shown by Carman ( 1 0 ) that, for a compressible fluid flowing through the powder bed, the Koeeny-Carman equation assumes the same form as for noncompressible fluids, provided that the flow rate is evaluated a t a pressure which is the arithmetic mean of the pressures above and below the bed. It is easily shown that the same rule holds true for Equation 11, provided that both X and q are evaluated a t the arithmetic mean pressure; therefore:

(Since isothermal flow is now assumed, air pressure is proportional to air density, so the factor h, A p which results from combining Equations 11 and 22 has been replaced in Equation 23 by pmAh.) From the kinetic theory of gases, the factor Amp, in the last term of this equation is independent of pressure. Therefore, the right side of Equation 23 is independent of t except for the factor p , in the first term. The relative change in the right side of Equation 23 will certainly be no greater than the relative change in p,, and will be less when the second term is important. Furthermore, since 1 p m = PO - AP (24) 2

+

ANALYTICAL CHEMISTRY

1628 the change in p , is half as great as the change in A p . Considering the worst case-Le., with mercurv as the manometer liquidthe relative change in p , for a change in A p from 10 em. of mercury to 5 cm. of mercury is 3%. So the change of the right-hand side of Equation 23 is less than 3%. The two terms in the right-hand side of Equation 23 are equal when D,‘ is about 1.5 microns. So for poviders finer than this, the change in the righthand side of Equation 23 is less than 1.5%. This was the basis for adopting the limits of 10 cm. and 5 cm. of manometer differential for measuring the time. With this limitation on Ah, the right-hand side of Equation 23 may be considered constant, and the equation is easily integrated. Writing the right side as p , y / l p , from Equation 11, the result is:

where Ahl and Ah2 are the values of Ah a t the beginning and a t the end of the measurement, respectively. Now in this equation p m should be assigned its average value during the test, which is: p, =

ha

+ 4 ( A h + Ah,)]

[ h ~

1

The equation may then be written:

and summing them up. For one particular apparatus, the measured dimensions were:

Vo A. A

= upstream free space at equilibrium = 7.1 cc. = cross-sectional area of manometer tube = 0.754 sq. em. = cross-sectional area of sample tube = 1.267 sq. cm.

Since Ahl = 10 em. and Ah2 = 5 cm., we have, from Equation 31: b

1 l --4 (10 + 5 )

= 3.75 cm.

and, from Equation 30: b,

=,

2 X 7.1 ___ 0.754

- 5) + 2(10 = 19 + 14.4 = 33.4 em. In (10/5) ~

Thus, the relative error in bl is only about half of the relative error in Va. The other quantities which enter into a are: fi’

= viscosity of air a t 25“ C. = 0.000183 poise

h o t = barometric height of mercury a t standard conditions =

76.00 cm. (For a xylene manometer, ho’ would equal 1203 em.) po‘ = standard atmospheric pressure = 1.013 X lo6 dynes per square cm. = empirical factor in Kozeny-Carman equation = 5.0 k

So, from Equation 28: a=

Substituting this equation into Equation 12 gives Equation 2 by giving a and ka the values:

+

76.0 76.0 33.4 18 X 5 X 0.000183 X 0.754 1.013 X lo6 76.0 3.76 1.267 10 X In- = 7.0 X lo-’ cm. sec. 5

+

Thus, the relative error in a due to the error in measuring VOis less than one fifth of the relative error in VO. In other xords, a 10% error in V Owill lead to less than 1% error in Dk. This degree of accuracy in measuring Vo is easy to obtain. So, for these specific apparatus dimensions, Equation 2 may be written, after substituting Dk‘ in microns for Dk in centimeters. (33)

where

in which L i s in centimeters and At is in seconds. It is often convenient in using the manometer method to use a powder sample whose mass in grams is numerically equal to the specific gravity of the particles, in grams per cubic centimeter. If this is done, Equation 1 becomes: Thus, a is an apparatus constant which is independent of the ambient conditions, while kb is a correction factor differing only slightly from 1, which corrects for deviations in the ambient conditions from normal atmospheric pressure and temperature. Since X in Equation 13 must be evaluated a t the pressure p , given by Equation 26, we can put X = X’ k , where

1-

B

=

1/AL, ( m =

pp)

(34)

and Equation 2 can be rewritten as follows:

Dk

=

dgt,

(m =

PP)

(35)

where a’ = a/A*. So, for the specific apparatus dimensions used above: In the last part cf the equality, the first factor in brackets represents a constant factor which enters into x (Equation 3) while the second factor in brackets is a correction for the atmospheric pressure. This second factor is ordinarily close to 1. Calculation of Apparatus Constant. The following example shows how the apparatus constant, a, is calculated for a mercury manometer. The three quantities that must be measured are the inside diameter of the manometer tube, the inside diameter of the sample tube, and the air volume V Obetween the equilibrium liquid surface in the manometer and the surface of the powder bed. The last quantity is the only one which presents any measurement difficulty. Fortunatelv, it need not be known very accurately (if it is small), as will appear below, Vowas obtained by dividing the volume into simple geometrical parts, measuring the appropriate dimensions of each part, calculating each volume,

Since the viscosity of air varies as the square root of its absolute temperature, the ambient temperature and correction factor kb may be written, from Equation 29: (37) The constants in this expression depend only on the initial and final pressure drops a h l and Ahn and the use of mercury in the manometer, and not on the dimensions of the apparatus. Under normal laboratory conditions of temperature and pressure, kb will be so close to 1 that it can be ignored. The modified Knudsen number (Equation 13) takes the form of Equation 3 by substituting 2 = 2.5, X = X‘k,, A ’ = 0.1 microns,

V O L U M E 26, NO. 10, O C T O B E R 1 9 5 4

1629

and replacing Dk by Dk'. The pressure correction factor k, (Equation 32) becomes (again, for a specific choice of Ah, and Ah2 and for mercury as the manometer liquid, but independently of apparatus dimensions):

This correction is somewhat more important than be included for accurate results.

kb

and should

VERIFICATIOV OF THEORY 4 Y D EXPERIMEhTAL RESULTS

The manometer method depends basically on the slip flow modification of the Kozenr-Carman theory which has been described above. Considerable experimental verification of this theory has been published by -4rnell ( 2 , S), Arne11 and Henneherry (d), and Carman and Malherbe (12), so that i t mas not considered necessary in this work to verify this theory further Two assumptions were made in applying the modified Kozeny-Carman equation to the manometer method. These m-ere: The temperature of the air upstream from the powder bed does not change significantly during a measurement; and the expression which appears on the right-hand side of Equation 23 is independent of pressure drop and time. Both these assumptions were checked experimentally as described below and found to be valid, provided the limitations imposed on the apparatus were complied with, The variation in the upstream air temperature was measured by sealing a thermometer bulb into this space. I n the course of over 100 experiments, the temperature variation from the start to the end of a measurement never exceeded 0 . 5 O C. and was usually from 0.0' to 0 3 " C. I n these tests, the manometer apparatus stood uninsulated on an open laboratory bench and only ordinaiy precautions n ere taken to protect it from changes in ambient temperature. I n the apparatus used, the upstream equilibrium volume, T'O, v a s about 7 cc. and the volume, AV, displaced by the manometer liquid a a s about 4 cc., so that Vol3 AV is less than 1. From the analysis of the temperature effect given above, it is seen that the observed temperature differences are not sufficient to cause significant error with this apparatus. A considerable error due to temperature changes could he introduced if T'o is not kept reasonably small with respect to A B . The second assumption mentioned above was tested in the following manner. For the present purpose Equation 23 can be a ritten in the form (7

+ 0 A h ) -dAh Ahdt

-

where -,, 8 , and K are constants. This equation may be rewritten:

K

data. It will be seen from this table that the slope in the 10 to 20 cm. of mercury pressure drop range is 2 3 % greater than the slope in the 5- to 10-em. range, while the slope in the 20- to 40-cm. range is i . 6 % greater. By'extrapolation, the slope over the 5to 10-cm. range is probably about 1% higher than the slope a t zero pressure drop, which is in agreement with the variation predicted by the theoretical analysis. I t is concluded that for a mercury manometer an initial pressure drop of 10 em. is satisfactory, 20 cm. would probably he satisfactory, but 40 cm. would not. In the case of coarser powders, where a lighter liquid such as xylene or water Iyould be used, a greater latitude in pressure drop could be tolerated. Reproducibility. The reproducibility of the surface areas measured by the manometer method \vas investigated briefly. -4t a specific surface level of 3 square meters per gram the standard deviation of the measurements was 4Yc; at a level of 2 square meters per gram it was 2%. Table IV.

Experimental Verification of Pressure-Drop Limitation of Manometer Method

Pressure Drop Range, cm. of hlercury 10 t o 5 20 t o 10 40 t o 20

Slope,

- K / (~x i 0 9 -3,68 -3.78 -3.96

Table V. Comparison of Manometer 3Iethod with Subsieve Sizer on Ilmenite Powder Samples Kozeny-Carman Diameter, Microns Manometer method 3.31 1.90 1.70 1.50

Subsieve sizer 3.20 2 03 1.77 1.54

Comparison with Other Methods. As discussed, the nitrogen adsorption method was not applicable to the material being studied. Consequently, no method of surface area measurement was available which was considered to have equal or greater reliability than the manometer method, which could be used to check the manometer method. Using coarse powders, comparisons ere made s i t h the Fisher subsieve sizer after carefully recalibrating the flowmeters and the porosity indicator on the latter instrument. I n making such a comparison, the Kozeny-Carman diameter, Dk', is used rather than the true specific surface average diameter, D8',because the Fisher apparatus does not take the slip-flow correction into account. Such a comparison is shown in Table V, for a few samples of ilmenite powder. The samples were measured on the two instruments as nearly as possible at the same porosity, the greatest difference being 0.06.

(39)

Y

Table VI. Comparison of Specific Surface -4verage Diameters by llanometer Method with Values Calculated from Size Distributions

where from Equation 23 v e see that

+

According to Equation 39, a plot of [In Ah (a/-,) Ah] against t should yield a straight line with a slope of - K / y , and conversely any variation in the slope of the line will indicate a variation in the assumed constant, K , since p and y depend only on the dimensions of the apparatus. Such a plot is shown in Figure 2. The test was made using a mercury-filled manometer, for which y / p was 45.65 em., and using a powder with a specific surface of about 2 square meters per gram for which the molecular flow term in Equation 11 has roughly tn-ice the magnitude of the viscous flow term. The slopes for three sections of the line in Figure 2 are given in Table IV,as calculated from the

Specific Surface Average Particle Diameter, D B hlicrons Calcd from Manometer method size distribution 1,23 1.28 0.03 0.5s 0.45 0.39 0.37 0.36 1.00 1.32 0.62 0.45 0.48 0.29 0.17 0.40 1.26 1.74 0.60 0.63 0.32 0.43 1.14 4.3 0.97 0.78 0.42 0.51

1630

ANALYTICAL CHEMISTRY

Measurements by the manometer method were also compared with specific surface average particle diameters calculated from particle size distributions of the powders, which were measured by gravity and centrifugal pipet sedimentation in water (Table VI). Each set of values in Table VI represents a sample of ilmenite ground progressively to different degrees of fineness. As is !yell known, if the particle size distribution curve is a straight line on logarithmic-probability paper, the specific surface average diameter can he calculated from the equation:

D, =

D,,a-(l/2) In a

(411

where u = D84.13/D60,and where 84.13% by Tveight of the powder is smaller than D8a.p. and 50% by weight is smaller than DbO. Surface average diameters calculat’ed in this way can be expected to agree with directly measured values only roughly, for several reasons, chief among which are the following: Equation 41 does not take into account the effect of particle shape on surface area; It is impossible to know whether the particle size distribution continues to follow a straight line below the limit (0.1 micron) of the measurement method; Individual particles in the permeability test may not correspond to individual particles in the sedimentation test, because of dispersion problems. In spite of such drawbacks, this comparison is very useful, because it can est’end into the submicron range. The powders that were analyzed practically all had size distribut,ions that became straight lines on logarithmic-probability paper, rvhich makes the comparison possible.

p,

second

rh

S S,

T

To t

At

Vo AV

z

Z 01

c

X

Am A’

A

= cross-sectional area of pou-der bed in a plane normal to

A,

=

direction of air flom-, sq. cm. cross-sectional area of manometer tube in a horizontal plane, sq. cm. a = apparatus constant in Equation 2, defined by Equation 28, em. sec. b,, b0 = apparatus constants used in defining a, kb, and k,. defined by Equations 30 and 31, cm. DS = partirle diameter determined by the Kozeny-Carman equation (Equation 2 or 12), em. DL‘= samp as Dk but in microns D, = specific surface mean particle diameter, defined by Equation 10, em. f = slip flow correction factor to the Koxeny-Carman particle diameter determined by Equation 4,dimensionless j d = fraction of molecules striking a wall that are diffusely reflected, dimensionless ho = prevailing barometric pressure in terms of height of manometer liquid, cm. ho’ = standard atmospheric pressure in terms of height of manometer liquid, cm. h , = mcan pressure in powder bed in terms of height of manometer liquid over course of a measurement, cm. Ah = pressure drop across powder bed in terms of height of mznometer liquid, cm. Subscripts 1 and 2 refer to the values a t the start and the end of a test, respectively k = empirical numerical factor in Kozeny-Carman equation, having value 5.0, dimensionless R, = channel shape factor in Poiseuille equation, dimensionless kb = atmospheric temperature and pressure correction factor in Equation 2, defined by Equation 29, dimensionless k , = pressure correction factor in Equation 3, defined by Equation 32, dimensionless L = length of po-ivder bed in direction of air flo-iv, cm. .If = mass of air in space between surface of manometer liquid and powder bed, grams m = mass of powder in powder bed, grams PO = prevailing barometric pressure, dynes per sq. em. PO’ = standard atmospheric pressure, 1.013 X 108 dynes per sq. cm.

mean pressure in povvdrr bed, dyncs per sq. em.

4% = volumetric flow rate for viscous flow, cc. per second pm = volumetric flow rate for molecular flow, cc. per second 4pL, AqpI Aqr = volumes of air flowing as a result of changes of

NOMEh-CLATURE

-411 of the equations are written to be dimensionally consistent except Equation 3, in which DI‘must be in microns, and Equations 33 and those following it. Examples of units in the c.g.s. system are given below.

=

A p = pressure drop across powder bed, dynes per sq. cm. p = volumetric air flow rate through powder bed, cc. per

fi p’ p PO pm pp

4

volume, pressure, and temperature changes, respectively, cc. = hydraulic radius of a flow tube, em. = specific surface of a po\vder, sq. em. per gram = surface area per unit volume of a powder, sq. em. per cc. = absolute temperature of a$ between manometer liquid surface and powder bed, K. Subscripts 1 and 2 refer to values a t the start and the end of a test, respectively = absolute temperature of air in room, K. = time during which air is flowing through powder bed, seconds = elapsed time for manometer liquid to fall from Ah1 to Ah*, seconds = volume of air in space between surface of manometer liquid and poxder bed n-hen manometer liquid is a t equilibrium, cc. = volume displaced by manometer liquid, cc. = modified Knudsen number (ratio of mean free path length of fluid molecules to diameter of tube through which fluid is flowing) defined by Equation 13, dimensionless = coefficient of the slip flow correction term in the modified Kozeny-Carman equation, value about 2.5, dimenpionless = effective fractional mean free crops-sectional area of a powder bed, dimensionless = fractional voidage of a bed of powder, dimensionless = length of mean free path of fluid molecules, cm. = length of mean free path of air molecules a t mean pressure of air in powder bed, em. = length of mean free path of air molecules at 1 atmosphere pressure, cm. = viscosity of air, poises = viscosity of air a t 23” C., poises = density of air in space betv-een surface of manometer liquid and powder bed, grams per cc. = density of air in room, grams per cc. = mean density of air in powder bed during course of a measurement, grams per cc. = density of particles of pouder, grams per cc. = factor in slip-flon equation, dimensionless LITERATURE CITED

(1) (2) (3) (4)

(5) (6)

(7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20)

Arnell, J. C., Can. J . Research, A24, 103-16 (1946). Ibid., A25, 191-209 (1947). Ibid., A27, 207-12 (1949). *4mell,J. C., and Henneberry, G. O., Ibid., A26, 29-38 (1948). Brown. G. P.. DiKardo, A . , Ching, G. K., and Sherwood, T. K., J . A p p l . Phys., 17, 802-13 (1946). Carman, P. C., J . Soc. Chern. I n d . ( L o n d o n ) ,57, 225-34 (1938). Ibid., 58, 1-7 (1939). Carman, P. C., ,Vature. 160, 301-2 (1947). Carman, P. C., “Symposium on S e w Methods for Particle Size -4nalysis in the Sub Sieve Range,” pp. 24-33, Am. Soc. Testing Materials, March 4, 1941. Carman, P. C., Trans. Inst. Chern. Engrs. (London), 15, 150-66 (1937). Carman, P. C., and .Imell, J. C., Can. J . Research, A26, 128-36 (1948). Carman’,P. C., and Nalherbe, P. le R., J . SOC.Chem. I n d . (London), 69, 134-43 (1950). Chilton, T. H., and Colburn, ;i.P., Trans. Am. Inst. Chem. Engrs., 26, 178-96 (1931). Fair, G. A I . , and Hatch, L. P., J . Am. Water Works Assoc., 25, 1561-65 (1933). Furnas, C. C., U. S.Bur. Mines, BUZZ. 307, (1929). Keyes, R. F., IND. ENG.CHEM.,ANAL.ED., 18, 33-4 (1946). Knudsen, >I,,Ann. P h y s i k , 28, 75-130, 999-1016 (1909). Kozeny, J., Sitzber. Akad. Wiss. Wien, Malh.-naturw. Klasse, 136 (Abt. IIa.), 271-306 (1927). Lea, F. hl., and Xurse, R. IT., J . Soc. Chem. I n d . (London), 58, 277-83 (1939). Lea, F. XI., and Nurse, R . W., “Symposium on Particle Size

Analysis,” Inst. Chem. Engrs. and SOC.Chem. Ind. (London). pp. 3 4 4 3 , Feb. 28, 1947. (21) Rigden, P. J., J . Soc. C h r m I n d . (London), 66, 130-6 (1947). (22) Rigden, P. J., Sature, 157, 288 (1946). RECEIVED for review October 24, 1057. .iccepted July 19, 19.54