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ST 0722800 PLAINTIFF'S EXHIBIT DOW-1685 i the Characteristics of Porous Beds and Structures R. E. MACMUIUN and G. A, MUCCINI 3t f >le t phase phase pineal tubin- 1. 1 Si ti t yj- 44, .a uu <d V 53>. ?, y.l*L ]] *. and Ckem. b. end 8. 334 4 J. D. .Vttre4 .nccriag rGraw* York .rnn(. I, . Ckem. o. 7. 73 32. 333 Kurd, . 42. 55 ige, and 32. 118 ird. and . SymjJr., and rogr., 44, rge, JA The structure of a porous body can be fold range in permeability), the correla characterised by three parameters, each tion applies to a wide range of porous of which can be directly measured by media, including packed granular beds simple experimental procedures. These and both rigid and flexible consolidated parameters are hydraulic radius m. vis media. The correlation is independent of cous-flow permeability P. and electrical- t.ha nnrrwity, which covers the range of resistivity ratio &/, which are shown to e. tut to 0.63. These are advantages be related by the simple equation >hicn appear to be lacking in previous m' = k-PR/Ro correlations by others. (1) Beyond the viscous-flow range of per or, ia dimensiooleu term*. meability the following modified forms of the Reynolds number Re and friction factor/ are suggested: (la) Re For beds of loose particles of known m'U,`p 1 fffy (28) size and shapes, the hydraulic radius m may be calculated from the relation n -- t/t, where t is the raid fraction &pd _ m-bp'Q____ /= u' L-p (R/Rt) (37) t the apeciaattriMfl^LtheDed. Since Oils -rffflB<Pi?i5Piuwally applicable to con solidated beds, an experimental procedure bated on surface-tension phenomena has been tested. This method, known as the The critical Reynolds number Re,,{,, beyond which the fluid flow cannot be considered as 100% viscous, is approxi mately " 6.5. break point for granular beds, or bubble point for consolidated beds, give* m di Tails A. Fields or Interest in Porous rectly by the relation m -- y/(Ap-f) Medu where 7 is the surface tension of the liquid employed and Ap the pressure required to break through the film of sealing liquid. The sealing liquid must easily wet the porous medium. The permeability coefficient P is a property of the porous medium only and is evaluated by measuring the resistance to viscous fluid flow by means of a fluid of Civil engineering, Permeability of earth and sand structures. such as dams, subssil drainage systems Flow of wells from water-bearing forma tions Intrusion of sea water in coastal areas Filter beds for water supply and sewage purification Aeration media for sewage known viscosity; that is. P m (u*L e)/ Geological and mining engineering (Ap-j). where u is the superficial fluid velocity, r is the viscosity, and Sp.'L the pressure gradient. The resistivity ratio R-R is measured by saturating the specimen with an elec Fluid flow ia gas- and petroleum-bearing formations Geophysical prospecting by electrical con ductivity method Leaching of ores and concentrates trically conducting fluid. R is the resist Chemical engineering ance in ohms, and is the resistance of the fluid occupying the same bounds of space as the specimen. Filtration of gases and liquids Fluid flow through packed beds Gas diffusion and dispersion The correlation was tested on a great variety of porous media, including beds of (last beads: sand: and such materials as bonded alumina, porcelain, carbon and graphite, fritted glass, sandstone, and polyvinyl chloride sheet. The mean value of Jfc was found to be 3.066 0.096. The probable error of a single set was found to be 0.65, or 17.6%. Covering a thousand fold range of hydraulic radius (a million- Electrochemical engineering Permeable and semipermeable dia phragms for electrolytic cells Medicine and biochemistry Biochemical filters Mechanical devices to simulate natural organs of the body with respect to fluid flow and electrical response Electro osmosis Miscellaneous JL *. MmMuIU l wHk JL 8, MmMuIIu .KmomikM, NJaean FaUa, Haw York, G- a. Mucetw. (opMr at Nbchra UniVffiitT. Nl*c**a FaO*. Saw Yaft, b to* Um Voivcnitr l Now Dana. Boat* Bead, ladiawa All industries engaged in manufacture of porous media, such as ceramic, metallic, plastic, rubber, leather, textile, etc. Agriculture, as in poroua character of soil This paper is concerned with the corre lation of viscous-flow permeability and electrical conductivity of porous media. It is believed that such a correlation would be of interest and importance in manv fields of technology, as listed in Table A. rMability <Unw The most generally accepted concepts of fluid flow through poroua beds are based on the Koseny-Cannaa (6 to 9) equations, which relate the pressure gra dient dp/dL of a fluid of density p, viscos ity 7, and superficial (approach) velocity u to the porosity and specific surface * of a bed of granular, nonconsoBdated par ticles. Various forms of these equations are m ii;l dL v (2) Re - = (3) In terms of sverage-particle diameter and sphericity 9, dp ^ 6/pw'fl dL ) (D Rp. ,, 0q(l - : In the viscous-flow region, with P ti-I-e) `(Ap-jl _fc, Re" Re m` = k, - The coefficient kr is variously reported s$ from 5 to 5.5. Useful as these equations may be. their limitations are often overlooked. For ex ample. the porosity t is an important factor. In loose beds of solid particles of known density, may be calculated read ily enough from the apparent density of the porous bed. In consolidated bodies of such materials as porous ceramics, porous carbon, or sandstone, there are complica tions. One must distinguish between pores that are open, semiclosed. and closed and between the microporosity contributed by the porous nature of the particles themselves and the macropo rosity of the interstitial space around the , 1956 v.t * AJ. ST0722801 particles. The porosity to be used in the point method should give the desired of the method to certain geometrically Kotenj'-Carmna equations, therefore, is value of m to be used in the Kozeov- predetermined beds is given in the ap frequently in doubt and sometimes im Carman equation (7). On the other hand, pendix.* measurable. the reciprocal average m calculated from likewise, the specific surface * is an the distribution curve is likely to give BoctrtoJ Inittivity AImo important factor. Obviously, the internal undue weight to the very small values of Electrical-resistivity, measurements surface of microporous particles contrib m, which results from forcing the sealing have been made on many porous noncon utes little nr nothing to the resistance to liquid (mercury) into semiclosed pores ducting bodies, the pores being filled with fluid Row. as compared with the external and microporea. a conducting medium (/, 10, It, IT, is, surface of the particles. Even in beds of Entirely aside from the question of SI, 22,29.32 to di). At first one might be irregular but nonporous granules the ex what values of m, <, and to use in the tempted to reason that if such a body i: ternal surfaces sometimes mate, and these Kozenv-Carman equations is that of the were, say, 40^ porous, then it would con- turfaeos are "blind" to the flow of fluid. organization of the network of pores, the d uct i0% as well as if the body were 100^ The total surface of a porous bed may be effective velocity u, of the fluid in the porous (i.e., no solids present at all). This :?. determined by the gas-absorption method, pores, and the effective length of path I, is far from the truth; the resistance always y'* but obviously this is not necessarily the of a parcel of fluid. These factors may be is much higher. The first record that could specific surface called for in the Kozeny- lumped under the genera) heading of (or- be found of this discovery was made br Carman equations. tuoiily, a term which conceals a great deal VeSisek and Vasicek (29), To explain this The hydraulic radius m of Equation (7) of ignorance. If the tortuosity is universal, result, they reasoned as follows: Since the is defined, for a porous body, as the ratio then the Kozen.v-Cereutn coefficient k, pore structure resembles a three-dimen /. It is therefore subject to the same should be a natural coustant. For an sional network, only one of the three di sources of error as and t separately. illuminating discussion of this point, the mensions is in a position to conduct, There is, fortunately, a method that can reader is referred to a recent article by namely, the one coinciding with the direc be used to measure hydraulic radius di Scheidegger (?{)> who concludes that the tion of the potential gradient. On this rectly. If a porous body is saturated with tortuosity can have values in an ex basis, < R/Rt = 3. For porous bodies con a liquid which easily wets the body, the tremely wide range and that it is not sur sisting chiefly of fissures, two dimensions liquid is locked into the pores by surface prising that the Kozeny-Carman relation conduct, and so R/Rt m 1.5. These tension forces. In the well-known bubble- is frequently in poor agreement with ac authors were qualitatively correct in their pressure method, and also the new break tual experimental results. This empha reasoning. In the present work < R/Rt point method described later, the force sises the danger of extending the very ranged from 1.5 minimum to 3.0 maxi required to break through this sealing useful Koseny-Carautn equations into mum. In some recent work by Cornell and liquid is measured. At the saturation level regions of application where they were Katz (11), <t R/Rt ranged from l & to 5.9 ia the bed, the following forces are in bal not intended. For treating permeability for sixteen samples of sandstone, and ance over a unit area of bed: in consolidated porous bodies, one must from 6.1 to 10.7 for eight samples of dolo Force supported = open area-dp- g Supporting force ** wetted perimeter of grains-ycos 9 The ratio, open area/wetted perimeter * bvdrauJic radius m therefore search for a key to tortuosity. It is recognized that much important work in this field has been published since the classical paper of Carman. For granu lar beds, Brownell aud Katz (.?, 3, 4) introduced arbitrary correction factors in mite and limestone. De LaRue and Tobias (12) have recently determined the influ ence of suspended nonconducting parti cles on the conductivity of a liquid. The equation that they evolved will be dis cussed later in relation to the present y cos 9 V'9 tbe form of exponents of the porosity. work. (8) Leva and eoworkeis f/T) used the Ko- When a porous body is itself an elec zeny-Carmaa relation to estimate surface trical conductor, the mechanism of con If this is true for any section of the bed, it is true for all sections of the bed, pro vided that (a) the structure of the bed is reasonably (or better, statistically) uni form and (b) the unit ares chosen is much larger than the areas of the open pores. By proper choice of sealing fluid, cos 6 may be taken as unity. Hence of tower packings. Ergun (14) considered the problem of mixed flow, partly viscous and partly turbulent, in packed columns, and Wagstaff and Xirmaier (30) further verified the Ergun equation. Brownell (5) deduced a modification of the KozenyCarman equation which applies to beds of spheres consolida ted with resin. Tiller (27, 2S) and Grace (J-5) have applied the duction is more complicated, as shown by Sauer et al. (?3). The conductivity of the pore structure, however, can be ndependlv measured by the use of direct current, as will be explained under Experi mental Method?. There are other dis turbing factors that can influence tbe measurement of resistivity ratio unless proper conditions are chosen to avoid open area X L wetted perimeter X L t s (9) Kozeny-Carman relation to tbe unit op eration of filtration, and Whitney (31) to the dewatering of wood pulp and fibrous them. The phenomena of superconduc tivity at low electrolyte concentration, doubly ionized layers, internal reactance, Equation {$) is also used as the basis for determining "pore-size" distribution in a porous body, sometimes by use of mercury or some nonwetting liquid. Opin ion varies as to whether m, given by Equation (S) by the bubble-point method, is identical with the reciprocal average value of m as calculated from the distribu tion curve or with the value of m calcu lated from independent measurement of < and i. In the experiments here reported, the equivalence ia quite good. The break point method is believed to selectively exclude closed and semiclosed pores from the numerator of Equation' (9), as well as to exclude the blinded surfaces from the denominator. If this is true, the break materials. Hatfield (id) studied the per meability of porous carbon and graphite. Much information is to be found in books by Dallavalle (13), Muskat (19), and Terzaghi (2$). It has become clear that no single general equation based only on the variables used by Kozeny-Carman has yet bees proposed that is adequate for such an extensive range of application. Before leaving this subject, the authors wish to point out aa approach that could give permeability solely in terms of po rosity and specific surface, provided that the fine structure of the bed is known. This can be done by evaluating the pointto-point variation of the hydraulic radius within the porous structure. Application and such are dealt with in the cited refer ences. hmwbiliir <m( Kkdriul Idhtiviir T(#il A very early attempt to correlate per meability with resistivity ratio was made by Wirth (32), who presented an evic tion that reduces to m* ** 2P- R 'fl*. The form of this equation is believed to be correct, but the coefficient 2 derives from the assumption of Poiseuille s law for capillary flow. This is known to give so erroneous result for porous beds. *Compleu dat* Buy b* obtaiMd u docuartt 49ZZ Iron lh Aswriru Doumauuoa laautui*. PtotodupUcatios Serne*. Library f Cobi**. WuhiOKfoa 2i, D. C.. hr t2JO toe pbaiopnou ar 11.71 for 13-atB. mietofila. Pag* 394 VI.Ch.E. J. September, 1934 15 notice*, si SHod*^ ' It, ' might b, h * Ixxjjr' . ould co- * >[ueh work has been done by Wyllie \ Gregory* (3.?, $4) and by Cornell and (ff) t whom introduce the --jgtivitv ratio as a correction factor for ^ous permeability equations. Cornell ipA KuU. for example, identify (./>)* ^th t'R/R* in the ideal form of tbe jj0Ieny-Carman equation 4z k,uy dl (10) ftnrh reduces to m* = kJ>R*R< (II) Two problems remain: (l) the theo retical justification of an equation of the foregoing type and (2) the determination of the coefficient over a wide range of con ations. The present authors have under taken to solve both problems. j%**ticai law* Ik Rvlatme p^*bitrty > RMitUvify !) Since fluid velocity in the 100^ viscous flow region is characterised by the rela tion Hi) ~ Im'x grad P (12) md electrical current density by the relation /. ~ <x grad E (13) Fig. 1. Assembly for simultaneous measurement of permeability, electrical resistivity, end hydraulic radius of pecked beds of unconsolidated particles. die ^ ng parti- * uid. Tim.-_ I be di^ -S|^ present " it can be seen that the phenomena are analogous. Equation (12) states that the velocity of a fluid of unit viscosity is proportional to the pressure gradient and three de termining structural dimensions of the network. These dimensions are porosity , hydraulic radius squared m* (which pro duces the drag in viscous flow), and tor tuosity x- which is the "fudge" factor introduced to make Darcy's law hold. Thu equation may be recast into the sore familiar form. uyL 9 kP *m\ (12a) Equation (13) states that the current flowing through unit area of a specimen the pores of which are filled with a con ductor of unit resistivity is proportional to the potential gradient and two de termining structural dimensions of the network. These dimensions are porosity and tortuosity *. which is precisely the same '`fudge" factor used io Equation (31). here introduced to make Ohm's law bold. This equation may be recast into tbe more familiar form. f(r)t -if.-<x = .xfl (13o) f&ttlrvM. CoOfTM. pnai* ar , 1954 So/ft - rc Combining (31a) and (32a) to u to Vl. 2, No. 3 Fig. 2. Details of cell shown in Figure 1 (for unconsolidated hods). A.I.CJ1.E. Journal Page 395 i A. f?;Qf trA C5 1 eliminate < and x. one obtains the simple relation m1 - kPH/R, (1) To prove this relation is the object of this paper, and therefore, independent measurements of hydraulic radius, per meability, and resistivity ratio have been made. EXMIIMCMTAL METXOOS UiKMMtidatarf faAod Mi It is of prime importance that the parked bed, once formed, must not in any way be disturbed during the course of the several measurements made upon it. The assembled equipment for accomplishing this is shown in Figure 1. Cell details are shows in Figure 2. The solution used for the tests was 0.4-V KjCOj, prepared from Baker's analysed K(COj HiO crystal and distilled water. This solution was analyzed frequently. This particular electrolyte was chosen for the fallowing reasons; (1) it it available in ea mst owm-romf kb still motiv uttars. highly pure form; (2) it is stable to atmos Fig. 3. Principle'/ break-point method for determining hydraulic radius of packed beds. pheric air; (3) it has no effect on cell parts, such as pyrex glass, rubber, Tygon, Monel, electrodes, stainless steel screens. (4) it has no effect on the materials tested, glare beads, silica sand: and 15} all the necessary physical properties, such as density, viscos ity, surface tension, and electrical resistivity, are accurately known as a function of con centration and temperature within the range of the work. These physical properties are listed in Table 13.* Material* Tested. Superbrite glass beads were obtained from Minnesota Mining & Minerals Co. in four graded sizes m follows: 60 to 80, were selected for testing. Since the plot of accumulative weight fraction against 6/D (Figure 12} gave practically t straight line, it follows that the reciprocal average diameter for the chosen cuts truly represents the diameter for which specific surface and sphericity can be calculated. The hydraulic radius and porosity of six different beds were measured by the break-point method and yielded specific surface by the relation The cell is then vibrated to pack the bed, and during that time the upper screen and electrode are placed and pressed down firmly. The cell ia then filled and connected to the constant-level feed tank. AU air is carefully vented from feed linea and celL Clamps c-1 sod c-3 are opened, and c-3 is closed. The flow of electrolyte is controlled by c-4. The depth of bed is measured by taking a Grade No. 108 109 no in i t*.$. sieve size Through On 40 45 50 60 60 80 100 120 Particle size* 23 Avg. s' -- ----* n` , sq. cm.^cc. of sand The specific surface of true spheres of diameter D,4 would be t 6/D,*, where D,t it the reciprocal average diameter for the cut. Sphericity is then defined m - series of ten readings around the cell. Since the outer edges of the electrodes are clearly visible, this distance is actually measured; from the average value, the calipered thickoe of the two electrodes and two aereens is subtracted. The inside diameter of the cell is. of course, known. Permeabthty. After steady state condi tions are reached at toy particular flow rate, the overflow is diverted to the liter-measur 302 300 410 397 271 290 ,325 295 228 200 214-- 214 136 150 157 148 11. Rtoieroetl sroras* at emn opc&ias. 2. Data fron roniiaar tpr*uabl> aueroai*- Ctrod). a Mitiiind by atitOan. ArtrafS at fifty bead* froa raadoo) aampl*. amutid uadsr bmto cap* with filar mierooMUr. Over BJ^ f th* bM> <r* pcrfMi tpherta. the bataK* rowwhat aUipaMdal. Five beds of sand gave the accompanying results. The specific gravity of the sand was meamired and found to be 2-640. Preparation of Btd. With the lower elec trode, screen, and wire seal ring in place, the cell was partly filled with potash solution, enough to cover the added beads or sand. An accurately weighed amount tsppro.vimntely 200 to 250 g.l of particles was added slowly through a funnel, the siem of which was swirled above the surface of the liquid. In this way a uniform and level bed ing cylinder, and the time to fill is measured with a stop watch. At regular intervals the loss in head i taken on the manometer, and the outlet temperature is recorded. Several flow rates i*re tried, and the proportionality of fiuw rate to head is checked immediately. Any deviation from proportionality would be cause for rejection of the data as not representing viscous-flow conditions. The correction for pressure drop through the screens has been shown to be negligible. Permeability is calculated from the equa The glass beads were found to be free of is formed, free from trapped air bubbles. tion extraneous material. A water extract was substantially as nonconducting as the water used. The specific gravity of the beads was measured and found to be 2.572. A bag of clean, dry, washed, sired quarts Mesh througb/on Dr.,* Rtsi'LTS ox Five Bros or Sand s' m,p cm."1 V cm."1 4 .sand of very* light color was obtained. A screen analysis on new U. S. standard screens gave the results shown in Figure 12.* This sand was then carefully and exhaus tively sized, and two ranges, <10 to 60 and 40/60 40/60 40/60 60/80 60/80 313 26.5 0 4423 300 190 0.634 316 26.8 0.4470 302 190 0 634 316 27.35 0.4325 281 190 0 677 207 19.25 0.4450 416.6 290 0.696 207 18.67 0.4570 450 290 0.645 Sro lootaoK h pas* 3W. Average 0.657 i.. . AJ.CH.E. Journal Soptombor, 1954 b*4a. >i ^rnut tu uk nci i*d ^ii ST 072 2804 v*l a ccfTar1 VtCIWI Fig. 4. Wster-peraeabUity test, falling-column method. may seal off some of the pores and give erroneous results; (3) specimens must be thoroughly dry before testing for gas per meability; <4} specimens must be freed of entrapped air (by evacuation^ in order in obtain 100*7 saturation with liquid before testing for liquid permeability or resistivity: (5) specimens must be carefully sealed st the edges to prevent bv-paasing of Quid: ami (6) when the bubble-point test is nude for hydraulic radius, the fluid chosen should easily wet (have on affinity far) the speci men. The specimens were prepared in the form of true cylindrical disks, approximately 1 in. in diameter and \i in. thick, with faces strictly parallel. All specimens were theu accurately calipered. MaUria.lt TttUd. Porous porcelain filter disks from the Seine Corporation are made, it is understood, by incorporating carbon in the ceramic mix. so that, on firing, the carbon burns out. leaving a porous struc ture. The following grades were tested: XF, XFF, 10, 01. 015. 02, and 03. Grades PC 40 and 60 and PG 40 and 60 of porous carbon and graphite from Nation'll Carbon Company were tested. For so esti mate of m, based on grain rise, distribution, and other data, the results shown in Table 4 should be consulted. Of fritted-glass filter tubes made by Corn ing Gloss Works and supplied by Will Corporation, three grades were tested, coarse, medium, and fine. IVaUr Permeability, Fallinp^oluma Meth od. In this method, also known as the method of Terxaghi (2), the specimen is 0.634 0 634 0.677 0.606 0 646 0.857 tiLq p &h g * (15) Feiaiirtiy. Simultaneously with the fore going permeability tests, the resistance R of the bed is taken on the impedance bridge. Sharp null points are obtained, by use of l.OOO-cycle current and telephone detection. R, is calculated from the dimensions of the bed and the specific resistivity of the solu tion. on the basis of Ohm's law. The re-, nstance of the electrodes, screens, and leads was measured and found to be negligible. Hydraulic Radius. At the conclusion of the ubove-meiitioned measurements of per meability and resistivity, and without any disturbance to the bed.Jbe liquid level is dropped to the top of the bed (with the vent open). Clamps c-1, c-3, and c-4 are tightly rlosed. and c-2 is opened. The leveling tube is then adjusted to the top of the bed. The cellgloid millimeter scale is then taped to the leveling tube, with the zero mark at the level of the water. The leveling tube is then lowered 2 cm. at a time, at 3-min. intervals, and two readings, are taken: the position of the sero mirk of the millimeter scale with reference to the stationary meter stick and the level of the water in the leveling tube with reference to the millimeter scale. The top of the bed. with reference to the meter stick, is, of course, noted and remains constant. The distance z and y are calculated and plotted immediately, in the manner shown in Figure 3 (D). As the breakpoint is ap proached, the increment Ay/Ar starts to increase. At this point the leveling tube is lowered 1 cm. at a time at 15-min. intervals. Past the breakpoint, ay/Ax increases rap. idly. Sufficient points are recorded to fix the slope of the new line accurately. The unaacuration line in the bed is usually clearly visible, because air has been drawo into the bed. It has been found impossible to reverse the procedure and to drive the air out of the bed by raising the leveling tube. It U. there fore. considered impractical to repeat the break-point tests, and it is preferable to start all over. The intersection of the two lines, as shown in Figure 3(D) is considered to be the break point, and J. gives the effective bead of liquid supported by the amd owing to sur face tension. Neglecting the angle of contact 9 gives m . cm. z.PF (16) Density of Particles. Density was de termined by comparing the weight in air and in water by use of a 200-g. sample and taking care to avoid errors due to entrapped air. ligid Perees Materials The testiog of rigid materials has the intrinsic advantage that the same specimens may be tested over and over again. It is necessary only to take the following precau tions: (1) only dear liquids or clean gases may be used, to prevent clogging of the pores with fines; (2) the faces of specimens, particularly the fine ones, must not be handled with the fingers, as grease or wax Fig, S. Apparatus for measuring gu per meability of rigid specimens (a- by use of rotameters, (b) by use of water displacement. Vol. 2, No. 3 A.I.Ch.S. Journal Pago 397 tionality between flow rate and pressure drop is proof that the flow is in the viscous range, otherwise the experiment is useless. If the pressure drop is only a matter of 10 to 40 cm. HiO head, then the correction for expansion of the gas in the specimen is negligible. Where the pressure drop through tl* specimen is appreciable, the following correction factor must be applied to each different flow rate, after which the true peruniahility may be calculated: AAOUT tcm. UOUiO OF KNOWN SURFACE TENSION. SPECIMEN WITH . LIGHT COAT OF RUBBER CEBENT. SATURATE SREC. AT START. >-TO aCRCURT UANOttETER correction factor C 2p> Pi +p> (18) 2p, + Ap whrre Pi is the upstream absolute pressure (cm. of H,0) P* in the downstream absolute pressure (cm. of H,0) D-TO DRY N, Ap - p, - p,, cm. of HjO WIRE ALL JOINTS CYLINDER PRESS FOR 40P.S.I. REG VALVE USE PRESSURE HOSE D CuU ,, . (19) Fig. 7. Resistivity cell for porous solid Fig. 0. Apparatua (or measuring babbit* Hydraulic Radius, BubbU-point Method- pour pressure of rigid specimens. Tltr* specimen is mounted as shown in Fig specimens. ) ure 0. When mounted, the specimen is cov ered with about 1 cm. of a liquid of known mounted below a column of water, and the height of the water column is measured against time. The inside area of the tube being known, the permeability can be cal culated. The apparatus used is shown in Figure 4. Cut A shows how the specimen is mounted, cut B shows how the specimen is evacuated sod saturated with water, cut C shows the aasembly during test, and cut D shows the method of plotting the data. If a m area water column, sq. cm. .4 area specimen, sq. cm. L thickness of specimeo. cm. H -- head of water, era., at time t, sec. tun <j, -- slope of line, .1 log H> At surface tension which is allowed to permeate. It is not necessary to evacuate the specimen and saturate, as this prooedure gives the same result as the simple permeation pro cedure. Tire cell is connected to a dry nitrogen cylinder through a sensitive pressure regu lator. The pressure is slowly increased until a alow but steady stream of bubbles is seen rising through the aealing liquid. Any bub bles from the edge do not count, only those arising from some part of the upper surface of the specimen. The bubble pressure is read on u mercury gauge or water gauge as Applicable. If the pressure is too high for the mercury gauge, for example 40 cm. of grmter, the Bourdon gauge on the lowpmwiire side of the regulator may be read. Tin- gas pressure is then reduced and the The hydraulic radiuses for fritted glass, with both water and CCU, are compared below: Med Coarse ium Fine m, (HfO), tt *, (CCU). a 16 3 2.76 1.52 18.1 2 08 1 28 This indicates that CCU tends to give a low reading oo glass, probably because of a small angle of contact which is ignored in the equation above. EUctncai-retitlictty Ratio. For noncon ducting porous bodies made from such ma terials as ceramics and plastics, the appara tus shown in Figure 7 was used. The speci men is mounted between copper electrodes nutisfactory for K-CO* solution i at the end J I Then P - --2.300 tan i (aLm'Ag), procedure repeated to give a series of read of a 1-in. 0.0. glass tube. The assembly is sq.cm. il7) ings, which sre averaged. Increasing the held together with Gooch rubber, and the \ ' For viscous-flow conditions the plot of log H vs. I must be straight; otherwise, the experiment must be rejected G<u PermtabdUy. In this test dry nitro* geo gas from a cylinder equipped with a sensitive gas-pressure regulator is passed through the specimeo. Pressure drop is measured by water manometer. Gas flow is measured either by a precalibrated Brooks laboratory flow meter (supplied by Ace Glass Company) or by the time required for water displacement from a calibrated pi pette. The apparatus u shown in Figure 5. In use of the Brooks flow meter only one float is used, not two, as suggested by the fluw ruts through the specimen until gas evolves U over the surface does not in- creuw t he bubble-point pressure appreciably. \ln < n certain point, however, the pressure inrrcjines owing to pressure drop through the Hfiecimen. With a little practice, the bubble point is reproducible. It is imjjortant to use a sealing fluid that wvl* the porous body. Water is excellent for gtn-w, porcelain, sandstone and, most ceremir (todies but is useless for porous carbon anil graphite. For these materials CCU or Cll| may be used. The hydraulic radius is calculated from tlic equation electrode? are pressed tightly aguinsr the specimeo. Each electrode is perfumed with two small holes to permit evacuation of the specimen and saturation with an electrolyte of known electrical resistivity. The rell is connected first to a Hyvar pump and the sur exhausted. Then the electrolyte is slowly introduced until the specimen is just immersed. The water will boil a little until all the air is displaced. Then the vacuum is partially released (until boil ing stops), aud additional electrolyte is sucked through to sweep out any partly con centrated electrolyte present in the speci men. The vacuum is finally released alto gether, the tightness of contact of electrode? manufacturer. The instruments were recali is checked, and the resistance is then meas brated before use. Moreover, the ball must be handled with care. If it is dropped on the m Ap g , cm. ured. The temperature is recorded, sod the electrolyte analysed. floor, the ealibntioD will change. The usual temperature and pressure corrections are where For this work the Heath-Kit Model BI-B1 was used with 1,000-cyele alternating cur applied to reduce readings to the tempera Y " surface tendon of liquid at tempera rent and telephone detection. Frequently a ture of the experiment. ture of experiment poor null point was encountered, which was For each specimen, several flow rates are Ap o bubble-point pressure, cm. H0 somewhat improved by use of a weaker used, covering a threefold range. Propor 9 080.6 cm.sec. electrolyte, i.e., about O.liV. The capacitance Pag* 391 A.I.Ch.5. Journal September, 1936 ' t -i CHCITa "MM** l orova aofig 1 itted |]y^ V(1 cotnpM^|^J Fla* CT C\*l 0 0 Qf\CL This method for RffU i* not so accurate O I U I (- lOU t%a the a.c. method previously described. The thicker the specimen to relation to the elec trode apecing, the greater the precision. \rOOOCH RUBBER ^SREOUCN $ When both methods have been used on ceramic bodies, a comparison indicates that in most cases the d.e. method gives a some what lower value for ??//?. The agreement is generally within 10%. ILc Tzm----------woto Vi ----- CONSTANT---- j PLATINIZED RLATINUtl ELECTRODES Fig. 8. Direct-current resis tivity ceU. NmkI|M Retain Sheets Porous sheets of polyvinyl chloride, rub ber, or woven fabrics are easily tested in Jh** |--M H >--l-i 8 apparatui shown in Figures 13* (permeabil ity and bubble-point testa) and 14* (resistiv ---------------------- ity test). Some physically weak materiiii* have to be supported by coarse screens to > prevent bending. The sheets are sealed a: CIRCUIT 0 too too too UtLLISMf a the edges where clamped between the facer of the glass-spool pieces. No new principles are involved, but :> <B) (C) word of caution is offered in dealing with hydrophobic materials, such as polyvinyl effect in the ceU was then balanced out, and ^though this sharpened the null point, it did not appreciably affect the value of the resistance. This means that the reactance of the circuit was small, certainly under 5%. The specific resistivity of the solution wa lead from the charts, the applicable concen tration and temperature being used. Inde pendent checks of the resistivity were made, isd these generally checked the estimated value to within 1 or 2%. If (r) - specific resistivity of the electrolyte, A* (r)L/A, gad from this R/$ is calculated. For electrically conducting porous bodice, such ae porous carbon and graphite or por ous metallic membranes, a different tech nique is required. The porous body itself would conduct alternating current, so much store than the electrolyte that the resistance of the latter could be estimated only by subtracting two large quantities of almost equal magnitude. Therefore direct current is used, and as long as the potential difference across the membrane is appreciably Jesa than the decomposition potential of the solution, the membrane itself provides no path for the direct current (except for a small leakage current at very low current density). The apparatus used is shown in Figure 8. The electrodes must not be placed in eontact with the specimen. Provision must be made to vent gases (Hi and 0) evolved at the cathode and anode. The specimen can be evacuated and saturated either before it i mounted or while it is in place- The authors favor the latter procedure. In principle, the total polarisation of the cell is measured with and without the specimen in the path. During the measurements the current (a matter of milliamperes) is kept constant. Voltages are read at various currents, and the slope of the volt-ampere curve is estab lished. A comparison of these slopes, with and without the specimen in the path, per mits calculation of the R/R* value. If ft, resistance of electrolyte with specimen in path, Rt - resistance of elec trolyte without specimen in path, and R* m (r)L/A, then R/R, Rx - R, + 1 R0 (20' chloride. In the resistivity measurement u welting agent had to be added to the electro lyte to achieve 100% saturation of the pore* during the evaluation and saturation pro cedures. Obviously the specific resistivity o: solution used had to be measured, rather than calculated. Is the bubble-point test do wetting agent should be used, as this would lower the surface tension of water to some indefinite value. The correct bubble-pom: pressure la obtained, because the same pres sure is required to force water into a hydrophobic specimen, as to force the water out of a hydrophilic specimen. EXPERIMENTAL RKfUllS The range of this experimental work is summarised in Table B. Also shown is the work of other investigators for which sufficient information is at hand to test the proposed reiatioo. Individual points trill be found plotted in Figure 10. Tables * foot&ot* os p*e sat. a Hyvae rhea the until the *ater will ed. Then tali! bodrolyte is .rtly coo lie (peri led allo> lectrodes so mi and the Srfeme* laW# EspariiwsMn Material tawtd Stnirtur* Clow boada Pak4 bod Number <4 oata ot data Table B. Soisiart or Resi lts Shows is Tables 1 to 0* tladindual poiata or* elettad is Figur* 10.1 2 Quant oaad Fackad bad 3 Porcalaia Rigid 4 ll> Carboo Rigid 5 (1) Glow Im Rigid 3 6 a> PVC abrrt Ftailbt* 7 (2l. (3` <4* Glaaa baada Boadad alumiaa kod bad Rigid 14 6 faadtvco* Rjgw lta|t o# preptrta Pamela riao. D*(p) 14fMt>7 Epbcricity,* Ponaitg (marra*. 1.00 0 36-0.43 BrSroube radiuaf. n(jti 20.7-41.6 Pemoafadtiv. /*(#*) 234-106.6 RrataUvri)' rmiia, A/R 3 ST-4 06 207-316 0 63-0.70 0.43-0 46 IS.7-27.4 40.6-57.7 3.73-4.57 0 22-0.57 0 35-33.7 0.007-30.4 4 10-12.SO 0.34-0.35 6.0-17. 4.0-23.6 4.10-460 1.5-16.3 0.05-5.62 6.23-14.49 0.65 1.65-4.52 0.13-1.04 2.76-3.47 156-C.100 1 oo 0.33-0 41 15.6-700 4.6-30.041 3.60-5.30 0.30-0 47 13.5-1002 81-226 4.S7-I440 0 11-0.25 050-444 0.002-0.42 12.10-36.20 Ocrivod fuaeuona * - (HiJto'i Boagt Arg. Ek Fa* 4** ft*. - /*- Ream A**. 3.43-4.5! 4,10 34 62 aos 3.206 U.Q5.7O-10.S3 7.20 1.61-3.47 1.14 16.00 3.12 3.615 S 92-6.64 6.66 33.SO 2.44-4.22 9.31 23.14 3.67 3.241 3 00-11.40 7.76 54.66 2.70-3.10 3.06 6.06 3.35 3.067 4.36-7.10 4.65 17.64 3.97-5.75 4.70 14.36 4.7T 6.775 7.20-16.10 10.64 31.90 3.66-6.06 4.61 13.63 2.53 9.714 6.00-19.SO 12.60 37.S0 2.42-4.67 6.44 48.19 7.47 6.664 6.43-7.86 5.41 76.64 2.76-6.36 3.60 21.63 6.71 0.135 0.00-16.00 12.40 >4.36 17.47 6.95 15.080 3-656 190.62 39 (H 55.479 5.100 421.23 EiWfiawtm: (1) MaeMetUa and Moedw; O) Plata aad MwbM; (6) UKua and Tofaiaa; (4) A. D. LilUa, lee.; (5) Conall tad Kata Urn (oetaota oa pace 394. tBydrauBa ndlm bj aurtm imdnq Mthed ttitmtylr-irl f-- n*---- 7 -ad 4 Vot % No. 3 A.I.Ch.9. Journal Page 399 Table C. CowfARiaov or R/Rt as Measured bt Present Authors, with R/Rt Calcu lated Fbou the Eouatio.v or Dc LaRie a.vd Tobias R/Rt &s measured, for packed beds. R/R% as calculated from Equation (21). which applies to random dispersions of floating particles. Glass beads (See Table 1) CoL 1 0 3835 2 0 4183 3 0.4266 4 0.4247 5 0.3975 6 0 4056 R/Rt calc. 4 20 3.70 3.56 3.60 3.98 3.88 R/Rt raeaa. 3 993 4 000 3.900 3.876 4.085 4.083 (7 Dev + 53 - 7.5 -61 -71 - 26 -50 s T 0722807 See Table 2^ 8 9 10 it 12 0.4470 0.4470 0 4325 0.4356 0 4570 3.32 3.32 3 50 3.47 3 21 Average 4.050 4.275 3 992 3 920 3.730 - 38 -18 0 -22 3 -12 3 -II 5 -13 9 Fig. 9. Resistivity ratios of random disper sions; plot of De LaRue-Tobias equation R/Rt - 1 to 9* contain the results of individual sets of data. ADDITIONAL SVIDCNCC ttothlivky tatia at fwwtiw # Void fraction Be LaRue and Tobias (IS) have meas ured the fractional decrease in conductiv ity of a fluid medium resulting from ran dom suspensions of various nonconduct ing particles in the fluid. The particles tested include glass spheres, polystyrene "cylinders. and sand, of varying sises. These were suspended in an aqueous solu tion of ZnBr, of approximately the same density as the particles, the slurry then being agitated to give random disper sions. The volume fractioa of dispersed phase ranged from zero to a value ap proaches that for loosely packed beds. The authors found the following relation to hold, to a high degree of precision: R 7f,, - "1/J (21) The relation is plotted in Figure 9. Now arises the question of whether this equation has validity when extrapolated to the region corresponding-to packed beds, where all the particles are more or less in intimate contact with surrounding particles. The answer is seen in the data presented in Table C. One concludes that the De LaRue-Tobias equation gives con sistently low values for R R* but is good to about 4(7 for packed beds of glass spheres. For packed beds of irregular grains, the equation can not be depended upon, even as an approximation. The equation cannot be used at all for consoli dated porous bodies. Ctcni Im4 Plain and Morrison (JO) measured the flow permeability of fourteen media com posed of packed spherieal glass beads, using the Terxaghi method. By using *Sw footoM* < 3M. Average -15 6 three fluids of widely different viscosity, they were able to determine the limits of viscous flow in terms of a critical Rey nolds number. In their Table I porosity < and sphere diameter D are given from which has been calculated hydraulic radius by the rela tion m - De$/6(l -- i). Several values for permeability P are given, and those values demonstrably within the viscousflow range were selected. Since Plain and Morrison did not measure the resistivity ratio R/R%, the present authors have esti mated it. using the De LaRue-Tobias relation. Equation (12), which has just been shown to be applicable to uncon solidated packed beds of spheres. This information is detailed in Table 7. sum marised in Table B, and plotted in Figure 10. The results support the proposed Equa tion (1) ven* well and give a mean value oft * 3.44. tondoe P*t*ea AJwia A series of measurements on porous matenalfr was corned out by A. D. Little. Inc. (IS). .KmoDg the properties measured were porosity, air permeability, aod elec trical-resistivity ratio. No attempt was made to measure hydraulic radius di rectly. However, among the materials tested were six samples of porous hooded 85(r alumina bodies, for which the manu facturer (Norton Company) supplies ac curate data as to grain site before bond ing. The grain size had been determined by microscope, with the average results given in Table S. The manufacturer also estimated sphericity of grains (before bonding) as 4 =* O.SO. During Bring, the bond spreads and tends to round off the grains and broaden the points of contact, thus causing cementation. This increases the sphericity to an estimated 4 " 0.90. The hydraulic radius m of the bonded specimen can then be estimated. Suffi cient data were available to calculate P and R/Rt. This information is detailed in Table 7, summarised in Table B, and plotted La Figure 10. The results support the pro posed Equation (1) very well and give a mean value of k * 3.59. Cornell and Katz (//) measured the porosity, gas permeability, aod resistivity ratio for twenty-four different samples of sandstone, dolomite, and limestone. The samples covered a wide range of these properties. Gas flow was in most eases in the viscous-turbulent region, and the pressure drop through the specimens was of considerable magnitude. Equations were developed to segregate the viscousflow contribution a from the turbulent flow contribution 6 to the total perme ability. Suitable corrections were made for gas expansion within the sample. Only a few specimens were tested for equivalent pore sise (diameter of circular capillary having same viscous-flow per meability). In their Figure 1 they show distribution of equivalent pore size vs. accumulative percentage of the total voids smaller than this size. No details of the method are given, other than that it was based on the water displaced from saturated specimens as air pressure was applied to the specimen. The authors did not make any direct measurements of hydraulic radius m or equivalent pore size. In handling their data, they used a modified form or the Koienv-Carman relation, given in their Equation (10). Noting that Oc ** Jot for a circular capillary of uniform crosssection and that the authors arbitrarily take , * 0.50, one finds that thi6 equation reduces to m' = --> = 1PR/R. (22) Ap g R* This is identical with Equation (1) save for the numerical coefficient. Thus Cor nell and KaU assumed a coefficient of 4 as being of the right order of magnitude (a very close approximation), whereas the Poge 400 AJ.Ch.G. Journal September, 19S6 Tabli? otted the pn>. ^ LEGEND GLASS SEAOS R.S.tt. GLASS BEADS GLASS FRIT SAND SANDSTONE PLAIN 4 MORRISON n.iu. R.BJt. CORNELLA KATZ 'SELAS" PORCELAIN R.S.& BONDED ALUMINA A.D.L. CARBON 6 GRAPHITE R.8.tl P.V.C. SHEET R.fctt. ST0722808 SOLID LINE------LOGtom - 0.26204 0.9000 LOG^f r/r, > *Mj.66G$aOG)* $>-R/ft LO0tO{?-A/ftJ Fig. 10. Correlation plot for MicMuliia equation m` - APA/A* direct J m or S their -ircuJv nd that * 0.50, to yl (22) 195* present Authors have experimentally de termined this coefficient as being 3.(ViG 0.09$. The results given hy Cornell and Katt in their Table HI ail fall exactly on the curve represented by Equation (22) since Og (or m) was calculated by means of this relation. MacMullin and Muecini have calcu lated m from the pore size distribution runes in Cornell and Katz's Figure I. for the four samples of sandstone for which this was measured. Since fine pores con tribute more to m than do coarse pores, the effective hydraulic radius must be the reciprocal average value of m. For equal increments of total void fraction, 1 ni(recip. avg.) = (23) 1(1 >m) The method of calculation is detailed in Table 10. The results are detailed in Table 9, summarized in Table B. and plotted in Figure 10. The data deviate widely (both plus and minus) front the expected values, although the average value of k -- 4.37 is not far from the expected value, k 3.67. This fact emphasises the desirability of direct experimental determination of hy draulic radius by reliable methods. MSCVIftJOM Of KHUU1 It was first necessary to establish the validity of the break-point and bubblepoint methods for the measurement of hydraulic radius. The data on glass beads, Table 1, indicate an approximate concor dance between m (breakpoint) and >n (D and ). Agreement is closest for uniformly sized beds, but m (breakpoint) gives high results for beds of mixed sizes. MacMullin and Muccini believe that their experimentally determined values of m have real meaning with respect to per meability. These values of m may not always be strictly the same a? those based on true values for total specific vuids and total specific surface, but they reflect the fact that some of the surface may be blanked off. contributing nothing to sup port of the sealing liquid in the break point method or to drag in permeability measurement. The justification for use of surfacetension method? of determining hydraulic radius rests on the demonstration that these methods can be applied to all man ner of porous media with reasonably con sistent results. The work on sand. Table 2. was pre ceded by many measurements on break point only. to arrive at the effective speci fic surface of each cut of sand and thus to arrive at the sphericity 6 of the sand in the permeability measurements. Inde pendent measurements of sphericity all checked closely, and the average value was usedjto check the m values found in the flow experiments. The close checks in all cases merely indicate the reproduci bility of the break-point method and are not aay inherent check os the "true" value of m. All the rest of the work was done on consolidated porous media, such as Sebs porcelain. Table 3; carbon and graphite, Table 4; fritted glass. Table 5; and poly vinyl chloride sheet. Table 6. To these experimental results are added the calculated results based on the work of others, os for example. beads. Table?: bonded alumina. Table 3; and sandstone. Table 9. Titus ooe is able to double the number of experimental data which can be used to test the validity of the proposed relation, m' TM kPR R.. Out of a total of fifty-two sets of data, twenty-eight are contributed by MacMullin and Muccini. The authors wish to point out that id their work each set of results represent? the average of several independent check: on m. P. and R 7t*. alternative fluids aad methods being frequently used for these testa. They are sure that in all cases the permeability measurements that they made were in the viscous-flow range, well below the critical Reynolds number. The entire fifty-two sew of results are plotted in Figure 10. log m rj. log PR. !?*. To be dimensionally consistent, the straight line through these points must have a slope of 0.30. The position of the line was determined by least square methods. In each column of each table is recorded the constant k of the equation, m- kPR/Rt, the deviation from the mean k -- 3.666, and the square of the devia tion. Using the relations VoL 2, No. 3 AJ.Ch.E. JoumeS Page 401 & HI 3ii a y.J ` & * u -M vn - probable error of the mean m - -7== VTTk V (n(n - 1) probable error of single observation, ftfr) EE VITk Vn -- 1 ber can be written in terms of the effective hydraulic radius m, Re . mu,? V (24) where u, is the "effective" velocity within the porous body. One recognizes that this velocity varies from point to point, just as m does. S T 0722809 Re" ffS IJ Therefore (31) Re - <*R/R*)'''-Ri" (32) For random-packed beds, where c approximates 0.40 and R/R* approximates 4.0, then one finds the following: Experimental work columns J-26 Whole array columns 1-52 k (mean) 3.690 (k) 0.122(3.3%) (k) 0.640 (17.6%) 3.666 0 098 (2 7%) 0.703(19.2%) The Authors are satisfied with the ac curacy of the mean value of k, although the probable error of any single set, when their technique is used, is os high as 0.646. or 17.6%. The expected error therefore in calculating any one of the three parameters m*. P, R/R from the other two is ] 7.0%, The expected error in calculating m from P and R-'R#, however, It has been pointed out by Carman () and others that if L, * the effective length of path of fluid through the bed and u -- the superficial velocity of the fluid then u L. " 7L Likewise it can be shown that (25) iH(rf TM Combining Equations (25) and (26) gives (R/R*\xn u. = u( ----j TM (27) Re = (0.4 X 4)1/'ft'' - 1.263 Re" Carman gives the critical Reynolds number as Rc".,,t -- 2; hence Re,, 1.265 X 2 - 2.53 Chilton (10) defines the Reynolds num ber as Re"' D,,yp --(33) Thus Jt____(R/SeY Re = Re"' (34) 6(1 - .) V is only half this, or 8.89c The expected error in estimating effective specific sur Substituting in (24) yields For random-packed spheres c will aver age 0.4, R/R* 4.0, and b " 1.0, and eo face ( = t/m) is about 10%. To test the alternative correlation of Koseny-Carman. they have listed kt * mup (R/R'V Re (28) 01 UY'!-Re"' Re (0.6) V0.4/ " an1?"1 in the tablet. For packed beds only, Experi mental work columns 1-12 Plain and Morrison columns 29--42 Whole array (26 points) 6.15 5.42 3.76 Plain and Morrison (20) defined their Revnolds number differentiv: Re' DjirP n (29) In their experiments on glass beads they arbitrarily take 0.325 Re" Chilton gives the critical Reynolds number as Re'"f,,, * 20. Hence. RetfU 0.325 X 20 - 6.50. Summing up. MacMuilin and Muccini prefer to use the Plain and Morrison data on critical Reynolds number, as they were Carman himself recommended the value kt = 5, although many other work L./L - V21 - 1.5S, aimed specifically at determining its value; that is. R(ir,, * 5.5. and below this ers seem to prefer a value of K m 5.5. so that Thus, the Koxeoy-Carman correlation value 100% viscous flow prevails. for packed beds is verified. However, in spection of the tables for consolidated porous media reveals a wide deviation, kt reaching values as high as IS on some samples of sandstone. The new correlation does not hold for sfraight-walJed capillaries, for which Also. 1 - "'1.5$ 1 (for beds of spheres) Pfirii** Pacts* In the viscous-flow region the produce of the friction factor and the Reynolds number must be a constant, in order that u be proportional to Jip\ that is. R/R. - 1, and m' =* UP. In such cases k -- 2.0 for round capiilariSS (PoiseuiUe's law) and may range from 1.2 to 3.0 for other than round cross sections. For por ous media the constant k * 3.666 appears to be outside this range. The new correlation covert a thousand fold range of hydraulic radius and a miUionfold range in permeability: it cov ers a wide range of types of porous media, including packed granular beds and both rigid and flexible consolidated medio, end the correlation is independent of the porosity, which covers a range of * 0.1 to a ** 0.85. This would seem to justify the use of this correlation, even though it and one can use the LaRue-Tobias realtion R/R0 = ~,/I (for beds of spheres) Combining yields ft-,-*-- rB.-u.'-rVM--) l-6 l.oS q \ / 1 *9.481 - < Re' (30) As Plain and Morriann reached a criti cal Reynolds number, Re' - 75 in tbeir bed 9, for which < was 0.33, f X Re ` 3.60*> (35 PR'R* Substituting the value of Rc [E'|uation (28)J, ' PuitR/R,)*-- (361 and breaking down P into its components yields m :lp g _ "u*U\R:Rt)`' (37) This form of the friction factor differs from that of Carman, wbo gives lacks some desired precision. Re,, 1 (0.33)*' 9.48' 0.63 73 - 5.34 r, ,, m Ap Q , 1 u*Lp (38) Since pore diameter has no meaning Carman (ff) defines the Reynolds num As a special cose, for packed beds of I within a porous body, the Reynolds num- ber I spheres, one may apply the LaRue-Tobiaa Pogt 403 A-I.Ch.E. Journal September, 1956 ft." (3 *> < appmdn^j J.265 Ktal ftej-aoVJ, 4 we T () ft." (30S s will an, -0. and so Rt'" I Heynotdr^s ce. Re,fit 1 ad Mucciai >rrboD data is they wtro *3 Mj 1 mining fa *-?ai I 1 below thh aiU. 1 -ip 1 he produet ^ I > Reynoldn order thsi ^ . -V) S66 (35) [Equatioa ; # ^(ion (RJt?) " *~tn, and Equation S1Q22ZBX&tuts crrio ^7) tale# the form >-^ (37a) specimen, ratio of spe cific voids to specific surface t '* i 1. Archie, G. ., Trane. ,4m. inti. Mining Met. Engrt., 144, 54 (1942'. 2. Brownell, L. E., and D. L. Katx, Chem. A * mean effective value of Bngr. Progr., 43, 537,(19471. ,rhich practically identical with Equa tion (38). The subject of this paper is limited to go*- in the viscous region. It is suggested, jgwever, that a new plot of / vs. Re, as defined bv Equations (28) and (37), rould yield rewarding results. In this -*per the proper location of the plot in ge viscous-flow region and the approxi mate point of departure where turbulent go* begins to take over have been est&bliihed. In order for the new equations to ^ valid, it is recognised that the hy draulic radius, as defined, is ao experi mentally determined property, based on surface-tension phenomena. Where it has frffp posable to cheek m from particle pit, sphericity, and porosity measuremeats, the agreement has been tolerably good. rotation Dimensions l " cross-sectional area of specimen cross-eectkmal area of liquid column in the Terzoghi method for liquid permeability c -* pressure-corTectioo f* l* m for permeability i N ~ normality, gram equi valents per liter P * permeability coefficient of specimen t p * pressure R * electrical resistance ohms R/Rt => electrical-resistivity ratio, a property of the specimen 0 Re Reynolds number 0 (r) specific electrical re sistance ohms i t specific surface (mac roscopic) of porous bed s' specific surface of (noa- porous) particles in bed /-* X m time t u * velocity; unless other wise noted, the super ficial fluid velocity through a unit cross section of specimen in direction of gradient U~l u, =* effective, or local, ve locity of fluid in pores of specimen lr* V " gross volume of porous specimen (* w " weight of specimen m 3. Ibid., 43. 601 (19*7). 4. Ibid., p. 703.' 5. Browned, L. E., D. C. O.imi. R. A. Miller, and W. T. Xekarvi*, A.I Ck.E. Journal. 2, No. 1, 79 0956>. 0. Carman, P. C., Trane. Inet. Chen. Eagre. (London), 15, 150 (1937 . 7. Ibid., 14, 168 (1938). 8. Ibid., 27. 237 (1949). 9. Carman. P. C-, J. Soc. C/um. Ind., 69. 134 (1950). 10. Chilton, T. H., and A. P. Colhum. Tran*. Am. In*t. Chem. Engn.. 26. 178 (1931). 11. Cornell. D., and D. L. Katz. Ind. Eng Chem., 45, 2145 <1933>. 12. De LaRue, R. E., and C. W. Tobi.i^ Paper 181. presented at Cincinnati meeting of Eleetrorbem. Soc. <Mjv 1 -5 1935). 13. Dallavalle, J. M., ``Micromemu-^' 2 ed., Pitman Publishing Corp., Chicago (1948). 14. Ergun, S. K., Chem. Eng. Freer.. 48. 89 (1952). 15. Grace, H. P., fee, ciL. 49. 303 <1953,. 16. Hatfield, M. R., Jnd. Eng. Chem.. 31. 1419 (1939). 17. Leva, Max et aL, US. Bur. Mint* Bull. 504 (1951). 18. Little, Arthur D., Inc., Rrpt. C-38493 (1953); private communication to R. B. MacMullin. factor in measurement x " distance in specimen, 19. Muakat. Irving, *'Tlow of Homogeneous of permeability to gas flow D * particle diameter 0.. -- reciprocal average par ticle diameter i density of solid part of 1 specimen -- voltage drop through m m specimen probable error of the mean value of k probable error of a sin gle determination of k 0 1 / mh> volts 0 0 measured in direction of gradient x, " head of liquid sup ported in break-point method y - liquid removed from bed. as measured in level tube, in break point method s specific viscosity rela tive to water y surface tension &k ** deviation from the 1 l l 0 mf~* Fluids Through Porous Media. 1 M'Graw-Hill Book Company, Inr.. Nw York <1937). 20. Plain, G. J.. and H. L. Morrison. .1 >/.. J. Phfie.. 22. 143 (1954i. 21. Rudolph, Hans, Koiloid Z.. 67. 93 <1934.. 22. Samartsev, A. G.. and V. Y. .truti* raov, Koiloid Zhur., 12. 130 '19.50 . 23. Sauer. M. C.. P. F. Southwii-k. K. ? Spiegler. and M. R. U'viiie. hoi. Chem.. 47, 2187 <1955 . * 24. Scheidegger. A. E., J. Appl. Ph .- . 23 994 <1954.. j ~ friction factor 0 mean value of k 0 25. Sbwinskv. A.. J. Chini PA-/*.. 23. 71* 1 acceleration of gravity ft-* i void fraction of speci <1920 . k m static heacTSf fluid ( men (total voids) 0 26. Tertigbi. Karl, ami R. B. P*--k. ` H -- hydraulic head in Terzaghi'S method for per- " macrovmd fraction (exterior to particles Mechanics." John Wiley k New York <1948 . In. meability 2 of bed) 0 27. Tiller. F. M., C4mi. Eng. P^r . 4* electric current amp. current density on superficial area of spec- men amp. I-1 * coefficient in MacMul- lin aquation for per meability 0 = coefficient in Koseny- c,, * microvoid fraction (in terior of particle of bed) 0 rf -- viscosity of fluid 6 * aogle of contact of fluid with solid part of specimen degrees * * specific electrical con ductance ohm"1!'1 467 <1953 >. 28. Ibid.. 51. 282 <1055 . 29. Velisek. von J.. and A. Varied. Kollon Z.. 71. Heft. 1. 36 11933 . 30. Wagsiaff. J. B-, and E. A. Nirmaier Ind. Eng. Chem., 47. 1129 il'J-So . 31. Whitney. R. P.. W. L. Ingnuasoo. an< S. T. Han, Trane. Am. Paper Puli Inti., 38. 157 (19551. Carman equation for m m microns. 10** cm. I 32. Wirth. 3. JC, Koiloid 2,% 116. 47 <1950 permeability 0 n =* permeability to a par 33. Wyllie, M. R., and A. R. Gregor. =* thickness of specimen ticular fluid, where Trans. .4m. Inti. Mining Mel. Engr. in direction of gradient effective length of path 1 II uLAp~l p TM density of fluid ml'1 196. 103 (1953). 34. --------- , Ind. Eng. Chem., 47, 137' of fluid in traversing 4 > sphericity of particle, (1955). l 6/(Ds') 0 35. Zhukov, L L, and D. A. Fridrikhsberg ' hydraulic radius of X * tortuosity function 0 Koiloid Zhur., 11, 163 (1949). Vf. 2, No. 9 A.KCH.B. Journal Page 402