ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT in Equation 3 ; the over-all heat transfer coefficient, U , will be greater than that from Equation 3 by an amount- less than 4uTaay.. The extent of this additional contribution from radiant heat transmission through the solid will be controlled by the average distance the radiation must travel in penetrating the insulation and by the number of internal reflections made by the radiation. Thermal resistance i s independent of void shape
The value of insulation is usually expressed as an effective thermal conductivity, in the units B.t.u./(hr.)(sq. ft.)( F./inch). This measure is 12 times the reciprocal of our X / ( U A ) . Figures 3 and 4, predicated on opaque solids, predict effective thermal conductivities of about 0.2-0.3; these are common values for practical materials. Available data are not adequate for a more severe test of the theory. The idealization of cubical voids should be an adequate one for the analysis of highly porous materials. I n these, the thermal resistance of the gas-solid composite is essentially independent of the form of the individual voids. O
Acknowledgment
This problem was suggested for examination by R. K. N7itt of
The Johns Hopkins University. Valuable criticisms were given by F. E. Towsley of The Dow Chemical Co. Nomenclature
E
=
total fraction of voids thermal conductivity of gas and solid, respectively, B.t.u./(hr.)(Fq. ft.)(’ F./ft.) number of individual voids radius of spherical void, feet ”. = temperatures at opposite faces of a void; average tem erature, O R. over-ai heat flow per unit temperature difference, B.t.u./(hr.)( F.) length of cube of insulation feet equivalent heat transfer coefficient for radiation, B.t.u./(hr.)(sq. ft.)( F.) coordinates in spherical void, Figure 2 emissivity of solid at surface of void Stefan-Boltzmann constant, 0.173 X 10-8, B.t.u./ (hr.)(sq. ft.)(’ R.4) O
O
literature cited
(1) King, W. H., Mech. Eng., 54, 347 (1932). (2) McAdams, W. H., “Heat Transmission,” 3rd ed., McGraw-Hill. New York, 1954. (3) McIntire, 0. R., and Kennedy, R. N., Chem. Eng. Progr., 44,727 (1 948). RECEIVED for review May 18, 1954.
ACCEPTED February 25, 1955
Fluid Flow through Porous Aggregates E f f e c t of Porosity and Particle Shape on Kozeny-Carman Constants M. R. J. WYLLIE
AND
A. R. GREGORY
Gulf Research & Development Co., Pittsburgh, Pa.
ITERATURE on the permeability of porous media to fluid single alternative method, proposed by Rapoport and Leas (I9), flow is voluminous. For a general review, reference may utilizes the concept of stagnant fluid within consolidated porous be made to papers by Carman ( 5 ) , Sullivan and Hertel (W), media in a manner analogous to that discussed by Carman ( 4 ) and Hawksley ( I S ) . for clay aggregates. The constant of 5.0 is retained. During the last two decades, considerable interest has centered It was shown by Wyllie and Spangler (29) that if the constant on the Kozeny ( 1 4 ) equation as modified by Carman ( 5 ) . This of the Kozeny-Carman equation were increased to values conequation relates the permeability of a porous medium to its siderably greater than 5.0, the equation, when modified for the specific surface area and porosity. The equation is applicable effects of pore-size distribution, appeared to apply not only t o only under conditions of viscous flow. As derived, i t is also single-phase fluid flow in consolidated porous media but also descriptive only of flow in unconsolidated porous media. to multiphase flow. The Kozeny-Carman constants used were T o the oil industry, the problems of single- and multiphase based on electrical resistivity. measurements. In spite of the fluid flow in porous sedimentary rocks are basic to all oil recovery apparent agreement noted, doubt as to the general applicability processes. The sedimentary rocks that constitute petroleum of the equation remains. This doubt stems, in part, from the reservoirs are generally consolidated; unconsolidated sands are fact that the independent measurement of the surface areas of relatively rare. It is thus natural that a number of attempts consolidated porous media is extremely difficult. Thus, measshould have been made to apply the Kozeny-Carman equation urements of surface areas by gas adsorption methods, although t o consolidated porous media. A summary of the principal easily made, undoubtedly reflect in many cases a component of methods suggested has been given by Wyllie and Spangler (99). surface area which is not involved in flow processes (15). Many A more recent attempt is that of Cornel1 and Katz (6). measurements of permeability in systems involving two or more Attempts to render the Kozeny-Carman equation applicable fluid phases are also of dubious accuracy. t o consolidated porous media have centered on methods of deThe primary assumption made in extensions to the Kozenytermining an appropriate constant in the equation. The value Carman equation that involve altering the constant of the equaof 5.0 is generally employed for the constant when the equation tion is that the shape factor incorporated in the constant has a is applied to flow through unconsolidated porous media. For value of about 2.0 to 3.0 for all porous media. This assumption consolidated porous media, as suggested originally by Rose and appears to lack any serious experimental foundation. Bruce (Zl), much larger constants seem to be required, The I n this article the measurement of the Kozeng-Carman conJuly 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
1379
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT stants of unconsolidated aggregates composed of particles of regular geometrical shapes is described. The shapes of the particles employed differed widely. The shape differences and the rather wide range of porosities employed are both factors that should affect the average shapes of the pores in the aggregates. Electrical resistivity measurements suitable for the determination of the tortuosity ( 2 1 ) component of the Kozeny-Carman constants of the aggregates prepared are available in the
A.
Reservoir tank Fluid pump Constant head tank D. Overflow line P. Packed base F. Tube-packed section G. Test section H. Packed head 1. Beckman thermometer 1. Pressure lines K. Differential pressure gage I. Galvanometer M. Sampling tube N. Dye injector
6. C.
a D
nl
1 Figure 1.
Fluid flow system
literature ( 2 7 ) . If the electrical methods for determining the constants for consolidated porous media are valid, they should certainly also hold for simpler unconsolidated systems. Consequently, from experimentally determined Kozeny-Carman constants and tortuosities, shape factors may be calculated as a function of porosity. If the assumptions made in the application of the Kozeny-Carman equation to consolidated porous media and multiphase fluid flow are correct, it follows that the shape factors calculated should prove to be largely independent both of variations in the shapes of the particles constituting the aggregates examined and the porosities of the aggregates, and that the numerical value of the shape factors calculated should be in the range 2.0 to 3.0. Determination of shape factor, especially at low porosities, i s important to application of Kozeny-Carrnan equation
The basic Kozeny-Carman equation may be written in the simple form
K =
€3
kS%(1 -
E)2
(1)
In Equation 1, k is the Kozeny-Carman constant. For convenience and because of its familiarity, the term “Kozeny-Carman constant” is used throughout this work for the quantity, k. Actually, k is a parameter and the anomaly of a porosity-dependent “constant” will be encountered.
1380
According to Carman (6),the constant, IC, can be written in the form k = ko
L * (i)
where k o is B shape factor believed by Carman to lie within the range 2.0 to 3.0 with a probable average value of 2.5. The quantity, (Le/L)2,has been given the convenient appellation “tortuosity” by Rose and Bruce ($1). It is the square of the ratio of the actual average effective length of fluid flow in a porous medium, L,, to the geometrical length of the medium in the direction of macroscopic flow, L . Equation 1 reveals the importance of k; yet the value to be assigned to k is still controversial. Carman has suggested that (L,/L)a has a value of about 2.0 in all unconsolidated porous media. Then from Equation 2, if ko is about 2.5, the magnitude of the Kozeny-Carman constant, k , would always be about 5.0. Much evidence that is now available certainly suggests that for unoriented particle aggregates in the porosity range of 35 to 70%, k is 5.0 =k 10%. The recent work of Coulson (7), however, strongly indicates that even in the restricted porosity range of 30 to 45%, k is significantly dependent upon both porosity and particle shape. Sullivan and Hertel ( I S ) showed clearly that the orientation of particles in an aggregate affects k; for aggregates of glass fibers, k was found to be 3.0 when flow was parallel to the axes of the fibers and 6.0 when flow was perpendicular to the axes. For aggregates of fibers a t very large porosities ( E > 0.84), k was stated by Sullivan and Hertel to increase rapidly. Such an increase has been postulated theoretically by Emersleben (8). It is apparent that at very high porosities the constancy,of k at a value of about 5.0 for aggregates of nonoriented particles must also be suspect. Direct determinations of k for aggregates having porosities less than about 30% are almost entirely lacking, and for aggregates with porosities less than 26% none whatsoever appear to be available. Thus, k has never been determined for unconsolidated aggregates of particles a t the low porosities that are characteristic of many consolidated porous media. If Carman is correct and k o is substantially constant at a value of 2.0 t o 3.0, it is apparent that k may be determined with an acceptable degree of precision if independent means of measuring tortuosity can be found. It was first suggested by Thornton ( 2 4 ) that for multiphase fluid flow appropriate tortuosities were calculable from electrical resistivity measurements. Wyllie and Rose (28) extended this suggestion to all applications of the Kozeny-Carman equation. More recently, Cornell and Katz (6) modified the method of calculating tortuosity suggested by Wyllie and Rose while utilizing the same basic electrical measurements. The method used by Cornell and Katz to compute tortuosity essentially is similar to that proposed by Winsauer, Shearin, Masson, and Williams ( 2 5 ) . In the derivation of the Kozeny-Carman equation it appears that a porous medium is envisaged as a pipe of complex but nevertheless statistically constant cross-sectional area ( 4 ) . Wyllie and Rose have assumed that the equivalent Kozeny-Carman pipe is endowed with a constant cross-sectional area, EA,and an effective length, L,. Cornell and Katz, on the other hand, appear to have visualized a pipe with a constant cross-sectional area, eAL/L,, and an effective length, Le. Arguments in favor of both approaches may be adduced. Since the Kozeny-Carman equation is based on an analogy to flow in pipes and is not derived from fundamental principles, the ultimate justification for the use of either model with its concomitant tortuosity must be determined experimentally. Consequently, the usefulness of bot,h the Wyllie and Rose and the Cornell and Kate tortuosities is examined in this discussion. I n order to derive tortuosities, use is made by Wyllie and Rose and by Cornell and Katz of the parameter formation resistivity factor. The use of this parameter for characterizing natural
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 4’1, No. 7
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT constant cross-sectional area, 4, perpendicular to the direction of macroscopic current flow. Equation 4, however, defines the path length corresponding to the flow of current perpendicular to a constant cross-sectional area eAL/L.. Both equations assume that the average path length utilized in the flow of electrical current is the same as that utilized by a flowing fluid. The physical parameters that affect the magnitude of the electrical average path length, L,, have been discussed by Owen ( 1 7 ) . Design of apparatus provides uniform streamline flow over entire cross section of packed column
PRESSURE CONNECT I ON-
I
Figure 2.
'DOWEL
PIN
Lucite test sections of fluid flow apparatus
consolidated porous media first was suggested by Archie ( 1 ) . The formation resistivity factor reflects the pore structure of a porous medium. It is defined as the ratio of the resistivity of a porous medium saturated with an electrolyt,e to the resistivity of the saturating electrolyte. According to Wyllie and Rose,
F
=
(pWLe/eA)(pwL/A)-' = (L,/L)/e
(3)
and to Cornel1 and Kate,
F =
p w L e / ( d L / L e ) (P&/A
)-I
=
(Le/L)'/a
(4)
Thus, tortuosity, (LL/L)a,is F V according to Equation 3 and Fe according to Equation 4. Equation 3 defines a fictitious average path length for the flow of electrical current through a
Table I.
Particle
Nominal Dimensions, Mm.
Oil, water, and aqueous solutions of glycerol were used separately a t various times as the flowing fluid. Fluid passed through the test column from bottom to top by gravity flow from a constant-head tank. Pressures across five individual sections of the test column are measured by a differential pressure gage operating on the electrical strain gage principle. A simplified diagram of the flow system is shown in Figure 1. The apparatus used was based on one described by Coulson ( 7 ) . One important deviation from Coulson's design was an increase in the internal diameter of the test column from 2 to 5 inches. Large ratios of column diameter to particle diameter (Table I) were used in all experiments. Wall effects ( l 7 ) ,therefore, can be considered to be small or negligible. As shown in Figure 1, the column base, E , was packed with plastic particles about a/,,-inch in diameter. Section F was tightly packed with 4-mm. glass tubes, 3 inches in length and arranged in parallel. The discharge head of the column was also packed with plastic particles. The packed sections were designed to ensure that uniform streamline flow took place over the entire cross-sectional area of the test column. To test the efficacy of the packed sections in ensuring uniform flow, a blue dye was introduced a t point N through the tube indicated. The dye was alloFed to diffuse into a layer a t the bottom of the fluid column. When fluid was circulated, the dye layer was found to rise up the column with a pistonlike movement and was sampled at M . Samples were taken through a small movable steel tube which could be inserted into the fluid stream a t various points across the column. Dye concentrations of samples were checked colorimetrically. Any variation in velocity across the column could be examined by this sampling technique. In all experiments the flow rate was always kept below a Reynolds number of 1 defined by the following equation:
Properties of Particles Constituting Aggregates Specific Surface Area, Sq. Cm./Co., Determined by Optical comparPin Permeatore droppings ability6 Calipersf
Microscopeb
D/d"
Density
42
2.573
0 , 7 1 1 diam.
174
2.73
84.44
82.16
Grade 9
0.279 diam.
455
2.87
198.65
215.19
Grade 12 Grade 16 Grade 19
0 127 diani. 0 057 diam. 0 028 diam.
1016 2228 4536
2.77 2.79 2.39
466.82 961.0
478.48
Surface Area Used. Sq. Cm./Cc.
Glass Spheres 3 M Superbrite beads0 Grade 5
Cubes Cylinders Triangular prisms
3 . 0 diain.
3.22 X 3 . 2 1 X 3 . 2 1 3 , 3 0 diam., 3.21 height 6 . 3 8 length, 2 . 8 4 height A , 2 . 8 4 height E , 2.86 height C
40 39
58
,
..
19.23 83.56 203.12 214.20
19.81 , .
...
202 210
...
2nx _-
477
IS 8
83.5 208.0 474.0 9fil . o 3076.0
....
.... . . .
Lucite 1,176 .... 1,176 ....
....
....
..
18. 5 5 18.38
18.6
....
....
....
..
24.14
24.1
1.177
....
3045
....
3i07h
18.4
Plastic
....
Disks 3.18 diam., 1.58 height 54 1.434 .... .... .. 25.18 For nonsDherica1 oarticle. d = 6 Vn/An. b Average bf 200 &heres of each fiizemeasured by calibrated scale in eyepiece of microscope. 0 Average of 200 spheres of each siee measured by optical comparator. d Determined by statistical pin-dropping method using 600 in throws per specimen. a Determined by flow of oil through column packed with spleres; value of k taken from experimental k versus curve for 3.0 -mm. spheres. f Average of 200 particle measurements. diameters of 3-mm. spheres measured in three directions. g Obtained from Minnesota Mining & Manufacturing Co. and sorted for roundness b y inclined plane teohnique. h Determined by nitrogen absorption method b y P. H. E m m e t t , senior fellow a t Mellon Institute ( N z area assumed 16.2 sq. A.).
-
July 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
25.2
I
1381
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Re =
PUd -
9
where p is the density of fluid a t 25" C. The temperature of the flowing fluid was closely observed; for this purpose a Beckman thermometer located a t the top of the column was used. Measurements of fluid density and viscosity were made after each experiment, using a pycnometer and Ostwald viscometer. A range of temperatures encompassing the average test temperature was employed.
Figure 3.
Polished and etched section of plastic impregnated aggregate of disks
Test Section. The test section of the flow column was constructed of five sections of Lucite, each 1 ' / 2 inches in height, held together by bolts and sealed with O-rings. Pressure connections to each section were made through piezometer grooves cut into the face of each section. The construction is shown in Figure 2. Small rectangular slits provided pressure connection between the piezometer grooves and the fluid inside the section. The inside diameter of the sections was machined to very close tolerances and all sections were interchangeable. Wire gauze supporting screens were fitted across the top and bottom of the test section. The screens were further supported by rigid streamIined metal grids of 1/2-inch mesh imbedded in the Lucite column immediately above and below the test section. The electrodes shown in Figure 2 were for resistance measurements and are not pertinent to the present discussion. Measurement of Porosity. The porosities of aggregates were determined in all cases from measurements of the volume of the test section and the weight and density of the particles constituting the aggregates.
The determination of the specific surface areas of aggregates composed of nonspherical particles presents a more difficult problem. The surface areas of aggregates of smooth spheres are a function only of sphere size. The surface areas of aggregates of particles with plane sides are a function not only of the size of the particles but also of the porosity to which they are packed. The relationship between surface area and porosity results from the tendency, with diminishing porosity, for plane-sided particles to pack with planes in contact. This tendency is particularly evident in the case of aggregates of cubes. As the porosity of such an aggregate is decreased, more cube faces slide into contact. Eventually, definite indications of stacking or orientation may be observed. At a porosity of about 27%, orientation is quite evident. Since little if any fluid can pass between two flat surfaces which are in direct contact, it is necessary to determine as a function of porosity the effective area of aggregates of planesided particles. The surface area term in the Kozeny-Carman equation, aa conventionally derived, is the effective surface area of the particles exposed to the flowing fluid. It is not the total surface area prior to packing. The statistical pin-dropping method described by Rose and Wyllie may be utilized to determine effective surface area as a function of porosity. The method, however, is only applicable within the range of porosities in which the particles may be packed reasonably at random. Experimentally, the determination of effective surface area was carried out in the following manner. Samples of aggregates of plane-sided particles used in flow experiments were packed in a 5-inch-diameter Lucite cylinder. As far as possible, the particles were aggregated a t random and in such a manner that the porosity of each aggregate varied over the vertical height of the cylinder. Such a variation is easily obtained; indeed, it is difficult to avoid. A nonshrinking thermosetting liquid plastic was then introduced into the cylinder so as t o fill the voids between the particles. For this purpose, Selectron No. 5001 (Pittsburgh Plate Glass Go.) colored with suitable dyes was used. After the Selectron had
/
PLALTIC
Surface areas actually exposed to flowing fluid are determined
The determination of accurate surface to volume ratios is of primary importance if the Kozeny-Carman constant is to be determined from measurements of permeability and porosity by the application of Equation 1. For spheres of uniform size, the specific surface area is given by
so
=
z 6
For mixtures of spheres composed of two or more sizes of uniform spheres, the specific surface area of the mixture is
so
=
zlsea
+ zzsos + 23soc . . . . . f
ZnSon
The diameters of the spheres used were obtained in the following manner. Representative samples (200 spheres) were measured by different direct methods. The actual methods used depended on the diameters of the spheres. The results obtained by the conventional methods noted are recorded in Table I. I n addition, a statistical method described by Rose and Wyllie ( 2 2 ) was employed.
1382
Figure 4. Method of determining effective surface area Particles 1 and 2,4 and 5 are each considered as a single particle
set, the cylinder was sectioned at a number of points in a direction perpendicular to its axis. The surfaces of each section were then polished and etched lightly with acetone so as to bring the boundaries between particles and bonding plastic into clear relief (Figure 3). The polished and etched surfaces were then used for pin dropping in the usual manner. Pin lengths were chosen to be about ten times the longest dimension of a particle. The pins were cut from spring steel and were of square cross section of 1-mm. side. I n Figure 4, the method used to determine effective surface area is illustrated. As shown in Figure 4, the upper end of the pin lies wholly within a pore space while the lower end lies wholly within a single particle. Thus, the number
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 41, No. 7
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT of hits is unity. The number of cuts, if the total surface area were to be determined, would be 19, but the number of cuts for a determination of effective surface area is only 15. The latter total is arrived a t because the particles marked 1 and 2 and 4 and 5 are in such intimate contact that each pair can be considered to constitute a single particle of complex shape. Analogous conditions may be readily seen in Figure 3.
i:
=-$
uniform spheres whose surface area could be established with high accuracy by independent means. For this purpose, the 3-mm. glass spheres proved suitable (Table I). Approximately uniform spheres too small for convenient mensuration by other means were then packed to porosities in the range covered in the experiments with 3-mm. spheres, and the permeability of the packings was measured. The surface areas of the small spheres were then computed from Equation 1 by utilizing in the equation the Kozeny-Carman constants, IC, found experimentally for aggregates of 3-mm. spheres when packed to the same porosities as the aggregates of small spheres. For experiments with very tiny spheres it was found easier to use packed sections smaller in diameter than the 5-inch column shown in Figure 1. Since the spheres were very small, the ratio of container diameter to particle diameter remained very large. A 1-inch-diameter Lucite tube was used; a steel rod just fitting the tube was employed to compact the packings of tiny spheres to a porosity in the range 37 to 3901,. The length of the compacted aggregate was 5 cm. Such compaction is not only necessary when very fine particles are used but also appears to give more satisfactory data ( 1 5 ) . A nitrogen adsorption experiment using the Brunauer, Emmett, and Teller method (9) was utilized to measure the surface area of the smallest size of spheres. A result in good agreement with the area found by permeability experiments was obtained (Table I).
BAFFLE
PACKED SECTION
Figure 5. Apparatus for obtaining statistically random packing of aggregates
The determination of effective surface area by this method is partly subjective. Nevertheless, it has been found that excellent agreement is obtained if the same determinations are made by two workers. A minimum of 600 pin drops was found to be necessary in order to obtain reliable surface areas on each polished section. Since both surface area and porosity can be obtained for each section by applying the method of Rose and Wyllie (22),the examination of a number of sections from a particular cylinder gives rise directly to a porosity-effective surface area calibration for each aggregate of particles examined. While direct dicroscopic measurements of the diameters of spheres were satisfactory if the spheres were of reasonable size, these measurements became increasingly difficult as the diameters of the spheres decreased. These difficulties were enhanced by the unfortunate tendency noted for the size range of spheres of a particular nominal size to increase rapidly as the nominal size of the spheres decreased. This effect, which apparently is inherent in the production of very small spheres, was obviated as far as possible by close sieving and elutriation. I n spite of these precautions, the range of sizes remained wide by comparison with the size range of spheres of larger nominal diameter, In consequence, it was found necessary to measure larger and larger numbers of spheres in order to establish an average surface area of acceptable precision under conditions such that the measurements were inherently difficult to make. Use was made, therefore, of the permeability method of determining surface area-Le., by using Equation l. It is implicit in Equation 2, and it is the rationale of the present work, that k may be a function of the shapes of the particles composing an aggregate and the porosity of the aggregate. However, for any porosity and shape of particle, k must be a constant which is independent of the surface area, SO,of the particles constituting the aggregate. The method of utilizing Equation 1 was, therefore, to determine k as a function of porosity for a packing of July 1955
20 IO
I
1
L
4
CUBES
m _.
10
PRISMS *
O
F
10
Figure 6.
,
I
0.1
0.2
0.3 0.4 0.5 FRACTIONAL POROSITY, f
4
0.6
Effective surface areas as function of porosity of aggregates
Method of Packing. The characteristics of the particles composing the aggregates used are shown in Table I. Nonspherical particles were prepared by methods which have been described previously ($7). It was found impracticable to prepare nonspherical particles of more than one size. I n consequence, the range of porosity that could be encompassed when packing nonspherical particles was limited. Nevertheless, the widest possible range of porosity compatible with random orientation of the constituent particles was achieved for each nonspherical shape. Porosity was varied by the customary methods of altering the height from which particles were dropped into the test section, by vibrating the test section after packing, and by rodding. Gross uniformity of the test column was checked by confirming that the pressure drops across each of the five test
INDUSTRIAL AND ENGINEERING CHEMISTRY
1383
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table II.
Determination of Kozeny-Carman Constants Koeeny-car Constantman
Tortuosity,
T
Particles
=
