ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Subscripts W indicates n-alls alone R indicates rods alone S indicates spheres (beads) alone W R indicates walls rods WRS indicates walls rods spheres
++
+
REFERENCES
(1) Blake, I’. C., Trans. Am. Inst. Chem. Engrs., 14, 415 (1921-22). (2) Brinkman, H. C., A p p l . (London), Scz. Research, A l , 27 (1947); Al, 81 (1948); Research (London),2, 190 (1949). (3) Brownell, L. E., Gami, D. C., Miller, K. A, and Nekarvis, W. F., presented at a meeting of the .Am. Inst. Chem. Engrs., San Francisco. Calif.. SeDtember 1953. (4) Burgers, J. M., Proc. Kon;nkl. Akad. Wefenschap. Amsterdam, 44, 1045, 1177 (1941); 45, 9, 126 (1942) (5) Burke, S . P., and Plummer, TV. B., IND.ENG.C H E ~ I .20, , 1196 (1928) (6) Carman, P. C., Trans. Inst. Chem. Engrs. (London), 15, 150 (1937); J . Soc. Chem. Ind., 57, 226 (1938); 58, l(1939). (7) D’Arcy, H. P. G., “Les Pontaines Puhlicpes de l a Ville de Dijon.” Paris, Victor Dalmont, 1856. (8) Emersleben, O., Physii. Z., 26, 601 (19253. (9) Ergun, S.,Anal. Chern., 23, 151 (1981) ; 24, 388 (1952) ; Chern. Eng. Progr., 48, 89 (1952). (IO) Ergun, S.,and Orning. A. 8., IND.ENG.CHEM,,41, 1179 (1949). i l l ) Fowler. J. L.. and Hertel. K. L.. J . A.naZ. Phvs., 11,496 (1940). ( l 2 j Hancock, R. T., Mining Mag. (London),55, 90 (1936);’67, 1 i 9 (1942); Trans. Inst. ,!dining Engrs. (London); 94, 114 (1937); Proc. Inst. M e c h . Engrs. (London),153, 163 (1945), discussion; Trans. Inst. Chem. Engrs. (London),24, 36 (1946), discussion. (13) Happel, J., D. Ch.E. thesis, Polytechnic Institute of Brooklyn, 1948: IND.EKG.CHEX.,41, 1161 (1949). (14) Happel, J., and Ryrne, B.,J,, Ibid.. 46, 1181 (1954).
Happel, J., and Epstein, N., Chem. Eng., 57, 137 (1950). Hatch, L. P., J . A p p l . Mechanics, 5, A86 (1938), discussion: 7, A109 (1940); Trans. A m . Geophys. Union, 24, 537 (1943). Hawksley, P. G. W., “Some Aspects of Fluid Flow,” Paper 7 , London, Edward h l O l d , 1951. Hirst, -4.A , , Trans. Inst. Mining Engrs. ( L o n d o n ) , 85, (1932-33); 94,93 (1937-38). Kozeny, J., Sitzber. Akad. Wt’iss. Wien, Abt. IIa, 136, 271 (1927). L e r a , >I., Weintraub, M., Grummer, M., Pollchik, VI.. and Storch, H. H., U. S. Bur. Mines, Bull. 504 (1951). Lewis, E. W.,and Bowerinan. E. W., Chem. Eng. Progr., 48, 603 (1952). Lewis, W. K., Gilliland, E. R., and Bauer, W.C., IND.E x . C H E M .41, 1104 (1949). Martin, J. J., McCabe, W.I., and Monrad, C . C., Chem. Eng Pmgr., 47, 91 (1951). hlertes, T . S., and Rhodes, H . B., presented a t a meeting of the -4m. Inst. Chem. Engrs., Toronto, Canada, April 1953. Morse, R. D., ISD.EXG.CHEY.,41, 1117 (1949). Oman, 9.O., and Watson, K. M., Refinery Management arad Petroleum Chem. Techno/.,36 R795 (Nov. 1 1944). Perry, J. H., ed., “Chemical Engineers’ Handbook,’’ 3rd ed., p. 1018, New York, 1lcGraw-Hill, 1950. Steinour, H. H., ISD.EKG.CHEM.,36, 618, 840, 901 (1944). Sullivan, R. R., J . Appl. Phys., 12, 503 (1941) ; 13, 725 (1942) Sullivan, R. R., and Hertel, K . L., Ibid., 11, 761 (1940); Teatile Research, 11, 30 (1940): Advances in Colloid Sei., 1, 37-80 (1942). Uchida, S.,Rept. Inst. Sei. and Tecltnol., Univ. Tokyo, 3, 97 (1949); abstract, I N D . ESG. CHEX., 46, 1194 (1954) T’erschoor, H., Appl. Sei. Research, A2, 155 (1951). Wilhelin, R. H., and Kwauk. >I., C h m . Eng. Progr , 44, 201 (1948). RECEIVED for review December 30, 1953.
ACCEPTEOA i ) t i l I ? L Y i 4
(Viscous Flow in Multiparticle Systems)
Slow Viscous Flow throug Mass of Particles T h e following i s a n outline, written by John Happel f r o m a translation f r o m t h e Japanese by Tetsuji M o t a i , of a paper by Shigeo Uchida, University of Tokyo. T h i s paper was published i n Report of the Institute of Science ond Technology, University of Tokyo, 3, 97 (1949), and abstracted f r o m t h e Japanese by Chemical Abstracts. T h i s outline i s presented because t h e method used in t h i s paper is employed, in a modified form, in theaccompanying papers (pp. 1181 and 1187) as a basis for a theoretical t r e a t m e n t of flow through cubic assemblages. Uchida presents a rigorous solution, using spherical harmonics, of t h e Stokes-Navier and continuity equations for t h e case of creeping flow i n a n infinite cubic assemblage.
s
LOW motion of a mass of particles relative t o a viscous fluid
has been studied extensively because of its importance in practical applications. Previous studies of the behavior of particles falling through a viscous fluid a n d viscous flow through beds of compacted particles are reviewed. A correlation of data obtained from the literature on flow through packed beds is presented, employing the following formula for correlation:
where $c.p = resistance index L = length of bed measured in direction of flow Ap = pressure drop across bed p = density U = average velocity (superficial) d = mean diameter of particle
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An attempt is made t o derive analytically a relationship be$p> the resistance index, and CY, the void fraction, by the use of the Xarier-Stokes equations for viscous flow. For simplicity, an orderly array of spheres is assumed, though it is realized that the beds of particles found in practice consist of various sizes and shapes and also have varying degrees of compactness. For the present model it is assumed that space is divided into numerous cubes and a particle is placed in the center of each of them.
tween
SOLUTION OF BOUNDARY PROBLEM
Fundamental Equation and Its General Solution. In order to obtain the solution for sufficiently small Reynolds numbers -Le., R = 1 or less-consider the Navier-Stokes equation expressed as
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46,No. 6
FLOW THROUGH POROUS MEDIA Equations 7' and 8' become identical
Ab6/30p where u, v, and w are the velocity components in the x, y, z directions, respectively. The following continuity equation must also be considered simultaneously with Equation 1:
Ab2/5,u
-du+ - +dv- = odw bx
az
A general solution expressed in terms of spherical harmonics for Equations 1 and 2 is given by Lamb (1) and reproduced in the paper as Equations 3 to 5. Boundary Conditions. The center of the sphere of radius a is taken as the origin of the x, y, and z coordinates. The direction of flow is assumed to be parallel to the x-axis, and the cube formed by the x, y, and z axes, the side of which is 21 in length, contains the sphere of radius a. The boundary conditions are simplified as follows: On the surface of a sphere r = a
u = v =
w
=
0
v =w =0 v=O
On top and bottomendsof cubex = i l y=*l
On sides of cube ( z =
(6)
(7)
= 0
Approximate Solution by Finite Series (Tuboi's Method). The method employed is adapted from a paper by Tuboi ( 3 ) . This method is to express the general solution, which satisfies the differential equations involved, in the form of a finite series and to determine the coefficients for each term in such a way as to satisfy the given boundary conditions on a limited number of points arbitrarily chosen on the boundary. For the present case, the author considers only the spherical harmonics of zero and first order. By using Lamb's development ( I ) , the following expressions are obtained:
u v
+ E/2pr + A' + E'/r3 3,(Ar6/301- Er2/6p + E')xZ/r5 = -3(Ar6/30u - Er2/6rr + E )xu/@ (10)
=
Ar2/5p
w = 3(Ai6/30p - Er2/6p
+ E')x~/;~
where A, E, A', and E' are constants. The approximate solutions thus obtained in the form of finite series will not satisfy the given boundary conditions of Equations 7 and 8 a t all points on the sides of the cube. I t is necessary, therefore, to modify the boundary conditions given in Equations 7 and 8 in such a way that those conditions are assumed to be satisfied only along the intersection of the sides of a cube and the surface of a sphere of radius b. Hence, instead of Equations 7 and 8, the following boundary conditions are to be satisfied: x=&l
y=&l
z = f Z
r = b
v = w = O
r = b r = b
(7')
dxdz = 0 (11) =
0
By substituting the boundary conditions from Equation 6 into Equation 10
+ E/2pa + A' + E'/a3 = 0 Aa2/30p - E a2/6p + E' = 0
June 1954
+ E'
.
= 0
+ E / 2 P b + A' + E'/b3 =
Ub
(11)
= c =
0.
(15)
From Equations 12 to 15, A , E , A', and E' are computed as functions of a, b, Ub, and 1. By the use of Equation 1 1 , b can be expressed in terms of a / [ . The mean velocity of flow through the particles is defined RS follows:
With this expression, it is possible to derive a relationship between U and LTa in terms of a, 6 , and l. Resistance is then computed by using methods described by Lamb (1)and deriving the pressure components on the surface of the sphere from the expressions for u, v, and w. The force, D , acting on the sphere is obtained by integration of the pressure component on the surface of the sphere in the x direction, as follows:
D
=
f fp,,dS
= 6?r1aC[b(b5-
(b5 -
a5)
a5)/((9/4) (b
- ( 5 / 4 )( b 3 - u
- a) X ~ ) ~Ub/C' ) ] (21)
The resistance index, $s, is defined as $s = D / ( 1 / p p C 2 ~ a 2 ) . For a single sphere, $sibecomes $LS, = 24/R, where R is the Reynolds number. From these expressions $'S/$S,
=F
[(I
- h5)/((9/4) ( 1 - A) ( 1 - A6) ( 5 / 4 ) (1 - A 3 ) 2 ] ] U b / U (23)
Where A = a/b. The resistance index, $8, of a mass of p a r t d e s is always greater than that for a single particle, $sl. As the distance between particles increases infinitely, this reduces to the simple expression
It is shown that the ratio of the velocity of fall of a mass of particles to that of a single particle becomes 1/F. [This corresponds to V / V Oin the accompanying paper by Happel and Epstein (29.1 It is further demonstrated that, if wall effects can be neglected, the resistance to flow through a bed, $P, is given by the following expression:
w = O
J-i~lw,=ldxdy
Aa2/5p
62/61
v = O
The conditions of Equations 7' and 8', together with Equation 6 , can be satisfied by EI, proper choice of three of the above four arbitrary constants. Therefore, one additional restriction is necessary. I n addition t o these modified boundary conditions, it is also assumed that the quantity of flow through the sides of the cube is equal to zero, which is expressed as follows: l l l " = i
-E
This corresponds to the case where u is constant and u Therefore, if ur = b = ub
(12) (13)
The fractional void volume, a//l: CY
= 1
CY,
can be expressed in terms of
- i"
Table I of Uchida's article tabulates corresponding values of a / b , all, CY, F , and G. The author notes that the values obtained for G are larger than correspond to actual data on pressure drop through particle beds. [Figure 7 in the paper by Happel and Epstein ( 2 ) is a plot of l/F versus CY and indicates that the resistance predicted for sedimenting particles is also too great, as would be expected.] LITERATURE CITED
(1) Lamb, H., "Hydrodynamics," 6th ed., p. 594, Cambridge, England, University Press, 1932. (2) Happel, J., and Epstein, N., IND. ENG.CHEM., 46, 1187 (1954). (3) Tuboi, Ihati, Umi to Sora (Ocean and Sku), 15, 189 (1935).
INDUSTRIAL AND ENGINEERING CHEMISTRY
1195
