FLOW THROUGH POROUS MEDIA correlation should aid in predicting the most economical combination of filt,ration variables,
S
= void saturation or fraction of cake voids filled with wet-
S,
= effective saturation = voids containing wetting fluid in
S, X
=
u
=
u’
= =
ting fluid
ACKNOWLEDGMENT
The authors wish to gratefully thank R. L. Sutherland of the Truax-Traer Coal Co. for his helpful suggestions and for supplying the coal used in this investigation. A special thanks is given to Marjorie L. Sutherland, statistical consultant, whose help and guidance made possible a thorough analysis of the data.
e
p p’
= =
=
active flow divided by voids containing both fluids in active flow residual saturation; limiting value of S cake porosity, fraction voids time, minutes surface tension, 1iq.-air interface, dynes/cm. surface tension, liq.-air interface, lb.//ft. liquid viscosity, cp. liquid viscosity, lb.,/(ft.)(min.) or lb.m/(ft.)(sec.)
NOMENCLATURE
B I BL IOG RAPHY
Cake moisture = wt. yo total moisture = CFM/sq. ft. .4ir rate CF/sq. It. = cubic ft. of air/sq. ft. of filter area drawn through cake and measured a t std. conditions of 1 atm. and 32” F. CFMlsa. ft. = cubic ft. of air Der min./sa. ft. of filter area drawn thrbugh cake and measured a t std. conditions ( C F M l s q . ft.),,. = cubic ft. of air/min./sq. ft. of filter area a t av. vake prpssure and 32” F. Ct = filtration constant d = cake thickness, ft. lb., ft. = conversion factor = 32.2 gc 1b.f sec.2 lb., cu. ft. K = permeability, lb.1 set.* L = cake thickness, inches AP = pressure drop across filter cake, inches Hg. AP‘ = pressure drop across filter cake, lb.j/sq. ft. A p = difference between final and initial pressures in the vacuum system after correcting for leakage, inches Hg. R = multiple coirelation coefficient
(1) Brown and Associates, “Unit Operations,” pp. 21C-55, New York,
John Wiley & Sons, 1950.
(2) Brownell, L. E., and Katz, D. L., Chem. Eng. Progr., 43, 601
(1947). (3) Brusenback, R. A., “Relationship of Filtration Variables to Filter
Cake Properties with Particular Reference to Cake Moisture Content,” Master’s thesis, Northwestern University, Evanston, Ill., 1952. (4) Gore, W. L., “Statistical Methods for Chemical Experimentation, New York, Interscience Publishers, 1952.‘ (5) Piros, R. J., Brusenback, R. A , , and Dahlstrom, D. 9.,Mining Engineering, 4, 1236-44, December 1952. ( 6 ) Silverblatt, C. E., Dahlstrom, D. A., “Economic Dewatering of Coal,” Annual Joint Fuels Conference, A.I.1W.E.-A.S.M.E. Chicago, Ill., October 29-30, 1953. (7) Snedecor, G. W., Statistical Methods, p. 220, Ames, Iowa, Iowa State College Press, 1946. (8) Sutherland, Dr. Marjorie L., statistical consultant, Chicago, Ill., private communication, 1953. (9) Sutherland, R. L., chief combustion engineer, Truax-Traer Coal Co., Chicago, Ill., private communications, 1952, 1953. RECEIVED for review November 23, 1953.
ACCEPTEDMarch 26, 1954.
Residual Equilibrium Saturation of Porous Media H.S . D O M B R O W S K l l
AND
L. E. B R O W N E L L
U N I V E R S I T Y OF M I C H I G A N , A N N A R B O R . M I C H .
A general correlation is presented t o predict t h e capillary retention of wetting fluids by porous media. T h e fraction of voids filled with t h e wetting fluid under equilibrium conditions is termed t h e residual equilibrium saturation. T h e correlation takes into account bed permeability and depth, liquid density, surface tension, and contact angle and t h e desaturating driving forces of gravity, centrifugal force, and pressure gradient of air as a displacing fluid. Static and dynamic end effects are incorporated i n t h e correlation.
1
N EARLIER studies ( 4 ) a general correlation was presented for the simultaneous flow of two fluid phases through porous media. Each fluid was treated as a single phase with modificatjlons for the effect of one fluid on the other. One fluid normally wets the solid (flows adjacent to the solid) and prevents contact of the other fluid with the solid. The second fluid flows through the remaining void space and is contacted by the first fluid rather than the solid. If the voids are completely filled with the first or “wetting” fluid, the porous medium is said to be “saturated” with the first fluid. If the voids are only partly filled with the wetting fluid, the fraction of voids filled with wetting fluid is termed the saturation. If, in the simultaneous flow of two fluid phases the flow of the 1
Present address, E. I. du P o n t de Semours & Co., Kewport, Del.
June 1954
wetting fluid to a porous medium is stopped, the flow of this fluid from this medium continues until an equilibrium value of saturation is reached; this is termed the residual equilibrium saturation. 811 the wetting fluid does not flow from the medium because capillary forces retain the wetting fluid in the smaller interstices of the porous medium. The determination of this residual equilibrium saturation is important as it can be used to predict the oil held in porous oil sands, the residual moisture in cakes from filters and centrifuges, and the holdup in packed towers. It can also be used to predict the relationships for the simultaneous flow of two homogeneous phases through porous media. This article describes the different variables and their influence on residual equilibrium saturation. Static and dynamic end effects are taken into consideration in addition to end effect-free porous beds.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1207
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
20/24 Mesh Crushed Quartz-IOX
32/35 Mesh Glass Spheres-IOX
Glass He1ices-5X
Figure 1 EXPERIMENTAL PROCEDURE
I n the experimental study a variety of particles including glass spheres, quartz sand, glass helices, aluminum cylinders, and nickel saddles, as shown in Figures 1 and 2, were used. These particles had a wide range of sphericity and produced porous media having a wide range of porosities as shown in Table I. Table I1 lists the properties of the various liquids used. -~
Properties of Porous Media
Table I.
Permeability hfaterials
Particle Diam. ( L I P ) ,Ft. 0 0 0 0 0 0 0 0 0 0
Glass spheres, mm. 3 38 4 42 5 30
0000316 0000445 0000630 0000746
000126 0001;10 000232 000300 000355 000703
0 00111 0 00143 0 00174
Porosity (X) 0 0 0 0 0 0 0 0 0 0
10i12 16/20
Aluminum mesh
(i )
379 367 376 356 367 370 362 384 435 433
1 0
0 398
1 0
0 408 0 130
(K),
Cu. Ft./ Sq. See. 1.276
2.88 7.20
1.350 2.64 4.35 1.388 1.72 2.04 1.81
x x x
10-9 10-9
x X x
10-8 10-7
10-9 X 10-8 X 10-8
10-7 X 10-7 X 10-6
3 98 X 10-6 7.88 X 10-0 1.275 X 1 0 - j
7.37
x
0 3iO
2.37
X 10-8
0 349
5.74
x
10-5
0 92
9.30 2.32
x
x
10-7 10-7
x
10-7
0 359
Crushed quartz, mesh
Sphericity
1 0
0 000591 0 000252
0 138
0 000300
0 372
0 88
2.57
0.00676
0 699
0 63
1.04
0.0300
0 930
0.14
5.30
0 441
10-1
cylinders,
14/16
able calibration was made of the dry porous medium and the liquidused. The density of the film was determined by the microphotometer which, when compared with the calibration curve, gave the point saturations. The x-ray film provided a permanent record of the saturation measurements. The description of the calibration and technique is rather lengthy, and the reader is rpferred to the thesis for further information. Figure 3 shows a print of a radiograph of a test bed and a refprence bed, each consisting of 35/42 mesh glass spheres. The drain height and difference in saturation between the upper and lov-er portions of the bed is indicated by the change in darkness of the print along the center line of the test bed. rigure 4 is a copy of n microphotometer recording of the density of film used for Figure 3 : it indicates the sensitivity of this method of determining point saturation. CORRELATION OF RESULTS
Residual equilibrium saturation is not a simple phenorncnon but is influenced by many variablep. Two general types of saturation distribution curves may be determined for porous media. I n the “static” type saturation distribution, the wetting fluid flows as a result of a desaturating force such as that of gravity or centrifugal force. The second nonwetting fluid is present hut does not f l o ~ . This condition is reached when an initially saturated porous medium is subjected t o drainage by gravity or centrifugal force. Figure 5 illustrates a typical saturation distribution curve for this case and shows a comparison n-ith the retention of liquid in a capillary tube. The second general type of curve i- associated ~ i t hporous media subjected to a pressure giadient of the nonnctting fluid. This pressure gradient can result in f l o of ~ the nonwetting fluid and also provides a desaturation driving force in addition to that of gravity and. perhaps, c.entrifuga1 force. Saturation dis-
Glass helices, inch ‘/lE 8/16
h’ickel saddles, mm. 0.912
x
10-6
Table I I . Liquid Cenco high vaciiuin oil !io. 93050-C
In the experimental measurements, pressure gradients, centrifugal force, the physical properties of the porous media, and the physical properties of the liquids mere measured by conventional methods. However, i t was necessary to develop a new method of measuring point liquid saturations within the porous media in order to determine saturation distribution curves. This new technique involved the use of x-rays, x-ray film, and a microphotometer. A heavy element was added to the liquid contained in the porous bed. Iodine waE used in the oils, and lead salts in the aqueous solutions. These liquids absorbed x-rays sufficientlv to permit determination of the amount of liquid present if it suit-
1208
Properties of Liquids
X 10-6
Mineral oil 3Iineral oil 10% iodobenzene hIinera1 oil 20% iodobenzene Kerosine Distilled water Distilled water 40% lead acetatenitrate Acetic acid 10% Acetic acid’ 30% Calcium c‘hloride solution, 7 .lf
+ +
+
Temp., Density O F. Lb./Cii.
fp)
bt.’
Surface Tension ( y ) Viscosity DynedCin. ( p ) , CP.
68 0 68 0
j4.S
68 0
60 4
35 5
55.0
88 0
65.5 50.0 62.1
35.3
22.8
28.6
i :009
91.5 63.0
2.124
82 0 68 0
30 J
36.4 36.6
72.7
68 0 76 5 78 0
64.4
69.6 56.7 45.6
82 0
92.5
93.6
INDUSTRIAL AND ENGINEERING CHEMISTRY
181.6 189.3
Vol. 46,No. 6
FLOW THROUGH POROUS MEDIA The desaturation driving forces are pa s P / L ) as given in the term ( p the sum of the forces of gravity, centrifugal force, and pressure gradient. If all these forces act simultaneously on a porous bed! the use of the sum of the forces would imply that all forces act colinearly and in the direction of desaturation or liquid flow from the bed. The term, therefore, should be interpreted as the vector sum of the gravity, centrifugal, and pressure gradient forces acting in the direction of desaturation. The solid line in Figure 7 best reprcsents the data for the glass spheres. Data for crushed quartz, aluminum cylinders, glass helices, nickel saddlcs, and various tower packing mat,crials are also included. Nickel Sadd les-5X 14-16 Mesh A l u m i n u m Cylinders-5X The sphcricit,ies of these particles ranged from 0.14 for the nickel saddles t o 0.925 for Figure 2 the crushed quartz. The porosities varied from 0.370 for the aluminum cylinders t o 0.930 for the nickel saddles. These particles exhibit a sonieivhat tribution curves of the form illustrated in Figure 6 result under wider spread in values of SOt'han tlic uniform glass spherrs, in part this dynamic condition. Referring t o Figure 5 , S O represents the minimum saturation to which a porous medium will drain under given conditions. Referring to Figure 6, a similar minimum saturation, SO, is observed. However, as a result of flow of a second fluid, the drain height, Ld, is replaced by a saturation distribution curve having a maximum between I .0 and So. The maximum sat,uration ie controlled by the dmaturation forces. I n practice, beds of a great variety of thiclinca.. -e$ are encountered, ranging from very thick beds of sand in oil reservoirs t o thin beds of filter cakes and centrifuges. For thick beds SO is of prime import,ance, and end effccte can be neglected. However, with thinner beds, such as filter cakes and eent,rifuge cakes, the end effect, may be of great importance. I n filter cakes, the cake thickness is usually less than the drain height,, Ld. Air flows through the cake, but, the sat,uration in no part of t,he cake is reduced to the minimum satural ion. Five syst,enis are considered in which different saturation distributions in porous media may be encountered.
+
+
1. Thick beds with no end rffects, rr-here S O is the only consideration 2 . Thick beds n-ith stat,ic ctnd effects, where SOand Ld are considered 3. Centrifuge beds x i t h static end effects, where 2 is modified for centrifuge cakes 4. Thick beds with static and dynamic end effects, where SO and Le are considered 5 . Thin beds with static and dynamic end effects, where bed thickness is less t'han drain height, and end effects control the saturation T H I C K BEDS W I T H NO END EFFECTS
The data for porous brds, drained by gravity and centrifugal force, are correlated in Figure 7 as the residual saturation, So, plotted against a dimensionless capillary number
which includes bed permeability, K ; liquid density, p ; surface tension, -/; and cont,act angle, 0; centrifugal force, a; and pressure gradient, A P / L . The limited data on liquid holdup in packed towers for drainage by gravity was useful in establishing the rurve at the extremity of high capillary numbers. A contact anglc of 180' between the liquid and the particles was assumed in this correlation. June 1954
Figure 3. Radiograph of Test Bed and Reference Bed of 35/42-Mesh Glass Spheres
because of the presence of crevices or irregularities in the surfaces of the particles which may t,rap liquid during drainage. The difference between values of SOfor these particles of va rious shapes and porosities and the values for uniform glass spher'es is
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
1209
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
!f I
.9
=
0 (gravity drainage)
Assume cos @ = 1
TEST BED
JP
Capillary No. =
9.0
(
x 10-6 32.2
) (30
52 X 6.85 X
10-5)
=
7.06
x
lo-*
From Figure 7 , So = 0.024 Sav. = 2.4% of the void volume Total void volume of tower = 20 X 0.7854 X 32 X 0.80 = 114 cu. ft. Liquid holdup = 114 X 0.024 X 52 = 142 pounds
FILM MARKER
T H I C K BEDS WITH STATICZEND EFFECTS
This special case presents beds whose total thickness is of such idative magnitude that the presence of the region of nearly 100% saturation is sufficient to affect the average saturation of the entire bed. Centrifuge beds of '/2-inch thickness, for example, AIR FLOW I
I1
*
----J3
I
'
DISTANCE
Figure 4. Microphotometer Recording of Density of F i l m Used i n Run on 100-115-Mesh Glass Spheres
h
not great. Therefore, it is recommended that the general correlation (Figure 10) be used for all unconsolidated porous media. Through proper interpretation, the correlation may be applied t o compressible beds. However the compressed-state bed permeability and liquid contained in internal pores (inside particle) must be known. Tables I11 and IV give the data used in the correlation.
Example Calculation. Calculate the liquid holdup in pounds of a 20 X 3 foot diameter tower packed with 1/2-inch Raschig rings after the tower has been shut down and allowed t o drain. The porosity of the packing is SO%, and the liquid is kerosine v i t h density of 52 pounds/cubie foot and a surface tension of 30 dynes/cm.
POROUS MEDIUM
CAPILLARY TUBE
Figure 5.
Saturation Distribution of Porous Bed with Gravity Drainage
Capillary tube, / I(D, p a , y cos 0) Porous medium, Ld = f(4, X , particle shape, pa, cos 8 )
A porous bed of this nature may be treated as a thick bed with no end effects as the length of the bed and the size of the particles are such t h a t average saturation is not significantly affected by the amount of liquid held in the bottom of thc bed by capillarity. Thus, the average saturation of thc bed is equal to SOand
Bed permeability, by method of Brown (I), will be
z
