density, grams/cc.
2.58 of aggregate network. dynes ’sq.
PIC =
P
=
U”
= compressive strength
Ty
= yield stress, dynes/sq. cm.
cm. For a plastic suspension, as shear rate-0 = aggregate volume concentration = floc volume concentration = kaolin volume concentration T-T~
Q.4
$‘F $‘K
literature Cited 11) Andreason. A. M. H.. Kolloid Z.48. 175 11929). \
I
(2j Brindley, G. W.,“Ceramic Fabrication Processes.” Chap. 1 , 11’. D. Kingery, ed., Wiley, New York, 1958. (3) Burgers, J. M., Second Report on Viscosity and Plasticity. pp. 113-84, VA 51. Academy of Sciences, Amsterdam, 1938. (4) Coe. H. S.. Clevenger, G. H., Trans. Am. Znst. Mzning Engrs. 55. 356-84 11916). (5) Fuerstinau.’ M. C., Department of Metallurgy, M.I.T., unpublished data, Mav 1960. (6) Gaudin, A. M., Fuerstenau. M. C., Eng. Mining J . 159, 110
(9) Georgia Kaolin Co., Elizabeth. K. J., “Georgia Kaolin Handbook,” Bull. TSBH-10 (1956). (10) Hawkesley, P. G. W., “Some Aspects of Fluid Flow,” Chap. 7, Arnold &.Co., London, 1951. (11) Kynch, G . J., Trans. Faraday S i c . 48, 166 (1952). 112) La Mer. V. K.. Smellie. R. H.. Jr.. Lee. P. K.. J . Colloid Sci. ’ 12, 230 (1957). ’ (13) Martin, R. T., Proc. 5th Conf. on Clays and Clay Material, Natl. Acad. Sci.-Natl. Research Council, Publ. 566, 23 (1958). (14) Mason, S.G., Pulp and Paper Mag. Canada 49, 13, 99 (1948). (15) Michaels, A. S.? “Ceramic Fabrication Processes,” Chap. 2, W. D. Kingery, ed.. Wiley, New York, 1958. (16) Reich, I., Vold, R. D., J . Phys. Chem. 63, 1497 (1957). (17) Richardson, 3. F., Zaki, \V. Tu’., Trans. Znst. Chem. Engrs. 32, 35 (1954). (18) Schofield, R. K., Samson, H. R., Discussions Faradqy Soc. 18, 135 (1954). (19) Smellie, R.H., LaMer, V. K., J . ColloidSci. 11, 704 (1956). (20) Steinour, H. A., IND.ENG.CHEM.36, 618, 840, 901 (1944). (21) Street, N., Buchanan, A. S., Australian J . Chem. 9, 4 (1956). (22) Wadsworth, M. E., Cutler, I. B., Mining Eng. 8, 830 (1956). ,
RECEIVED for review November 6, 1961 ACCEPTED December 15, 1961
/\ 1- ,0-5- A \ /.
(7) Gaudin, A. M., Fuerstenau, M. C., Preprint, Intern. Mining Proc. Congr. London, April 1960. (8) Gaudin. A. M., Fuerstenau, M. C., Mitchell, S. R., Mining Eng. 77, 613-6 (1959).
I
Division of Industrial and Engineering Chemistry, ACS, Symposium on Dynamics of Multiphase Systems, University of Delaware, Newark, Del., December 1961.
FULLY DEVELOPED TURBULENT PIPE FLOW OF A GAS-SOLID SUSPENSION S
Velocity and concentration distributions of solid particles were colculoted for the case o f low solid-to-gas mass ratios, small particles, and negligible gravity effect.
When
these conditions ore sctisfied, the solid particles slip a t the woll and lag behind the stream ot the center of the pipe.
Dependence of the particle velocity distribution on the concentration distribution is to be exoected.
from studies of motion of a single solid particle in the turbulent field of a fluid, studies of dynamics of a gas-solid suspension as a whole have either excluded turbulence (3, 72) or have been completely empirical (70, 77, 79). Studies on the gas dynamics (72) and compressible potential motion ( 3 ) have shown that rigorous forniulation of general turbulent compressible motion of a gas-solid suspension over a solid boundary will be very complicated indeed. A solution is not even available for the statistical .lormulation of simple turbulent pipe flow, where the turbulence is nonhomogeneous ( 7 ) . Even when one assumes a n isotropic turbulent field, solution of the problem of motion of a single particle is a t best academic (6). T h e present study proposes a n extension of the present semiempirical method of treating turbulent flow through circular
A
PART
. L.
S 0 0 , University of Illinois, Urbana, Ill.
pipes-namely, the 1/7th velocity law (7 7)-to the case of a gas-solid suspension. Data are utilized on distributed mass flow of solids in pipe flow a t air velocities from 50 to 100 feet per second, solid particles consisting of 100- to 200-micron diameter glass beads, and solid-to-air mass ratios of 0.05 to 0.15 (77). I n the range under consideration, compressibility of the fluid phase and the effect of gravity on the density distribution of solid particles are negligible. Concentrations of solid particles are small. such that the velocity distribution of the gas stream is not significantly affected by the solid particles. I t is sometimes tempting to suppose that the heterogeneous mixture of solid particles and fluid may behave as a homogeneous fluid, the dense particles playing the role of the heavy component of a homogeneous mixture. This assumption would greatly facilitate the development of a theory but is almost never true in practical situations, for it would require that the suspended particles be extremely small. I n fact, for particles as large as those of real interest, the solid-fluid mixture has to be treated as heterogeneous. Then, because the solid particles have greater density than the fluid, the concentration of particles is not uniform over a horizontal pipe, the average velocity of the particles near the wall is not zero even though the velocity of the fluid is zero a t the wall, the concentration of particles is velocity-dependent, and the excess of particle velocity over fluid velocity is positive near the wall and negative in the core of fluid, A realistic theory of the motion of solid-fluid suspensions must account for these facts. VOL. 1
NO. 1
FEBRUARY 1962
33
Density Distribution of Solid Particles Referring to Figure 1, we consider the one-dimensional random motion of solid particles up or down by a distance Ay in time T . The number going up is given by:
solids (75, 77) and not the concentration function, pv(,).] Equation 6 requires that
This conforms with the assumption that there is a continuous transport of small, energy-rich eddies from the wall toward the center and, simultaneously, a flow of energy-poor fluid lumps back toward the wall ( I ) . This nonuniform density distribution is therefore sustained by diffusion and drift of particles toward the center of the pipe.
The number going down is given by:
Mass Flow Distribution of Solids T h e net nurnber flowing upward in time
7
is given by: (3)
where (dn/b,l) corresponds to the number per unit thickness. The rate of flowing upward is given by
where D, denotes particle diffusivity. an elemental volume per unit time is
The number entering
where ud denotes the drift velocity in the y direction, which is not specified a t present. Further, for a steady state as in fully developed pipe flow, dn/dt = 0. Since we are dealing with a large number of solid particles, instead of dealing with the number, n, per unit volume, we shall multiply n by the mass of each particle (which is assumed to be uniform) to give the mass per unit volume occupied by the dispersed solid particles, p , (as distinguished from the density of the solid itself, &,). For fully developed pipe flow, all derivatives of time-average properties in the axial direction are zero, except that average pressure is a function of the axial coordinate only. Equation 5 becomes
A slowly moving solid particle tends to be accelerated when it is surrounded by a gas stream a t a higher velocity. T h e mean velocity of a solid particle near the center of the pipe will travel a t a velocity near but below that of the fluid. When this particle diffuses toward the wall, it encounters fluid of lower mean velocity and tends to be slowed down nearly to the fluid velocity. When a particle near the wall drifts toward the center of the pipe, it is accelerated. Constant exchange of solid particles between the regions near the wall and near the center of the pipe gives rise to a velocity of the solid particles which is higher than that of the fluid near the wall (''slip'' at the wall) and lower than that of the fluid near the center of the pipe. T h e mean velocity distribution of solid particles can be approximated by a n equation analogous to the th velocity law of the fluid. 1
(10) where Upoand UDmare mean velocities of the solid particles at the center and a t the wall, respectively; m is a constant to be determined. The mass flow distribution is therefore given by:
where x = y / R ,
(12)
As in previous treatments of pipe flow ( 7 7), we take (7)
y = R - r
from the wall of the pipe of radius R. We may postulate a density distribution of the form
c3 = ( ~ ~ ~ -1 ppoC6,)7i.R2/mp 6 , c4 =
Further, the mass flow a t the center, which can be measured directly, is given by c1
where subscript o denotes center of the pipe, w denotes the wall, and a is a constant to be determined; a > 1. This postulate remains to be justified from further development and comparison of the theoretical results with mass flow data. [Present available methods for the experimental determination of average distribution of solid particles give only the mass velocity of
+
(pp,C,, - p p y U p o- P ~ J Y , ~p p w U p W ) * R q / m p
+
c3
= PpoUpo/(mp/*R2)
(14)
= ppwUpW/(mp/aR2)
(15)
That at the wall is 61
- cz
Also
/ tn* Y -
Y
For total solid flow rate of m p the average mass velocity is given by
Figure 1. 34
Diffusion at y
I&EC FUNDAMENTALS
E% = *R
ppuPdr = Ji ppupdx
r
C2
-2
(k +
c1
r
1)
r(a + 2 )
The shear stress a t the wall, rW, is given by ( 7 7 )
(k
-+a+3)
where r is the gamma. function (2). Denoting the relative mass velocity at various distances from the wall as y ,
rw = 0.0225 pUo2(&)li4
for u =
Y = pPup.rrR2/mp
=
m)
-cl[c~fl(~,
(29)
U, being the velocity of the stream at the center of the pipe.
and substituting into Equations 11, 14, and 19, we get CP
U0Cy/R)''7
(20)
+ f d x , m ) l l [ c ? f ~ (m,x , a ) + f d x ,
T h e energy equation for the gas-solid suspension is given by
m, all ( 2 1 ) 0
where
= -u-
dP dz
+ p ( u + u )
Substitution 01 Equation 26 gives, at the wall ( y = 0, y e = 0)
which gives complete information on the density and the velocity distribution functions for the solid phase by substituting Up, into the relations in the previous section. T h e density and velocity of the solid particles at the wall and a t the center of the pipe are given by:
- c d 2 (mp/rrR2pU)( P / K ) - C P ) (m,/.rrRZpUo)( P / K ) Up,, = K U ~ / ( C-I C P ) U p , = KUdCl + C d / C l ( C l - c z ) (c1
(32)
cl(c1
(33)
pPw =
pp0
=
(34) (35)
where Thus by taking radial distributions of y from measurement, trial-and-error solution can be made to determine m, a , 61, and c2 to satisfy E q u a t i m 21 for a t least four selected points along a mass-flow profile. From trial values of m and CY, C I can be calculated by combining Equations 15 and 21 to give ci'(fi
+ fz) +
6
.(fa
+
f4
- ~ x f z )- 7tJ4
= 0
(23)
Calculations from measured quantities whose ranges of reliability are shown in Figure 3 gave approximate v a l u e of m = 1.25, LY = 2.30, and c 1 = 0.31.
Particle-Fluid Interacl'ion T h e solid particles are transported by the viscous drag of the fluid (4, 72), or p p &? dt = p p F ( u - u p )
0.0225
(24)
where u is the mean velocity of the stream and the drag coeficient, I;, is given by
and = m,/(ir/6 8p,) is the total number of particles passing through the pipe per unit time. The first expression for K in Equation 36 suggests that, within a limit to be discussed later. greater slip is expected a t the wall for larger density and diameter of the solid material; U,,,/U, is also larger, because of greater inertia. T h e second expression of K indicates that lag in the velocity a t the center of the pipe is greater for larger particles and greater numbers of particles per unit time. For the case calculated, the velocity, density, and mass-flow profiles are shown in Figure 2. I t is seen that choice of a smaller value of K leads to the prediction of smaller particle velocities and greater particle mass concentrations, starting with the same curve of particle mass velocity versus position.
Discussion where p is the viscosity or the fluid, d is the diameter of the solid particles, C, is the drag coefficient, and 6 , is the density of the solid material. T h e latter approximation is based on Stokes' IaW. For fully developed pipe flow, the momentum equation of the gas-solid suspension gives the pressure drop in the axial direction.
where v and p are the kinematic viscosity and the density of the fluid and Y, is the eddy viscosity. Integrating from y = 0 to Y = R gives
Another way of looking into the significance of parameter
K is to write
where 3 is the intensity of turbulence of the stream and d l / T u is the "particle Reynolds number" discussed previously ( 7 4 , a measure of relative turbulent motion between the solid particles and the stream. Greater slip as given by Equation 34 is owing to larger particle density, diameter, and greater intensity of turbulence, although this is opposed by the greater transport of turbulent energy (7rR2pU3,/mpU2)and larger values of the pipe-flow Reynolds number. VOL.
1
NO. 1
FEBRUARY 1962
35
t el 6 -
x 4 -
3
2
I
4
Figure 2. Velocity and density distributions at various values of K Since the above considerations do not include the details of particle-fluid interaction ( 8 ) and the gravity effect, the range of validity of the relations derived is necessarily limited. Also, since it is obvious that Upo/Lrocannot be greater than 1, Equation 35 gives
SPREPD OF MEASURED DPTA
10
~
-
8 -
as the upper limit of validity of the above relations. corresponds to UPOIL’”5 1
This 6 -
(39)
yR
as a limit with maximum slip velocity given by =
(L‘PlL/l.’O)*%X.
cx3 cd
(40)
-
4 -
The limiting values of the p p are pBID=
(CL
Ppo
- G?) (CI = (CI
+ cs)mplrR21Jocl
+ c3)mP/nR2Cr,
(41 1
2 -
(42)
The existence of an upper limit of K may be interpreted from Equation 37 to imply that not all the energy of turbulence serves to accelerate the solid particles. For
Figure 3. tributions
Velocity, density, and mass flow disCompared to experimental results (77)
because of very small d in the first expression of Equation 36, it is expected that the condition tends to that of a gaseous mixture and collisions between particles and fluid on a molecular level must be considered. For small values of K , however, because of either large d or large S in the second expression of Equation 36 or small ( U 0 R / v ) .it is expected that the assumptions of low particle concentration, negligible gravity effect, and the unaffected stream velocity profile will not be applicable; then clogging, sedimentation, pulsating motion, etc., will occur (79). The values of GI, m, and a! determined as above are, of course, not universal constants, but are expected to be valid over substantial ranges of velocity for a given gas-solid system. For p p near p, m should increase, with 7 as nearly the upper limit, a t which the system will behave like a gaseous mixture. T h e values of K derived from the data (77) range from less than 12.1 for 115-micron glass beads a t 0.15 mass ratio of solid to gas and 50 feet per second air velocity to less than 410 for 230-micron glass beads at 0.05 niass ratio and 100 feet per second air velocity. Therefore, the upper limiting values of U p , and U p ,iv U , should have been reached, as was observed 36
I&EC FUNDAMENTALS
kom intervals of successive exposure of particle position on a photographic plate [method presented in ( I S ) ] . The estimated profiles are shown in Figure 3. The fact that Up, rn U, may be attributed to such a small particle diffusivity that a particle near the center of the pipe travels far enough between successive contacts with the wall to be accelerated nearly to the fluid’s velocity (76). The existence of rather substantial values of U,, was noted in earlier successive-exposure photography (76), although the data were inconclusive. (Rather low particle concentration had to be maintained in order to identify the photographic records of individual particles.) Combining Equations 31 and 28, we get
Substitution of Equation 25 gives (45)
where n, is the number of solid particles per unit volume a t the wall. T h e quantity t / v p Fppwcorresponds to a “free path” of particle-stream interaction. Equation 44 is analogous to the boundary layer equations of slip flow ( 7 ) , which deal with the response to wall-particle interaction on a molecular level. Here the slip results from the slow response of solid particles to changes in the velocity of their fluid environment. The model under consideration is, by necessity, a simplified one. T h e idea of drift conforms to earlier considerations of wall interaction (78). T h e existence of slip near the wall was suggested by photographic observation (76). T h e present study suggests that a gas-solid suspension is a n interesting fluid-solid mixture with special velocity and concentration dependence. Such a dependence should be allowed for in studies of heat transfer, absorption. and ablation problems. 1he above suggests a method to obtain 5ome insight into these complicated problems. The intensity of turbulence of the stream in the experiments amounts to the order of terminal velocity of the particle. I n this range, the gravity effect of solid particles is sufficient to affect the intensity of turbulence of particles (16) but not to produce concentration distributions following the gravity model-that is, greatest concentration toward the bottom of the duct. To this end, concentration distribution as found by Longwell and Il’eiss (5) tends to give the distribution as shown in Figure 3. The particle diffusivity includes effects of turbulence and ‘.bounce” (9) from the wall. Photographic records (76), however, showed bounce to be small. T h e collision rate with the wall is small, and mutual collision between particles is negligible for the experimental range (73).
Acknowledgment
T. L. T u n g performed numerical computations. Nomenclature c3, c4 = constants defined by Equation 13 = drag coefficient = particle cliffusivity = fluid drag factor, 18 p / P p fl. 1 2 . f3. f 4 = functions defined by Equation 22 f; = parameter of fluid-particle interaction in pipe flow (Elquation 36) m = constant for assumed velocity profile of solid particles = total maGis flow rate of solid particles through mP tube x = total number of solid particles flowing through tube per unit time n = number of particles per unit volume P = preqsure R = radius of tube r = radial coordinate in tube t = time coordinate = fluid velocity at center of pipe uo U = time-average velocity of fluid a t a point = time-average velocity of solid particles a t a point UP U2 = turbulence intensity of fluid motion = radial drift velocity of particles (I cl,
cp,
CD
3
YIR
coordinate normal to wall of tube axial coordinate in tube constant in assumed solid concentration profile gamma function relative mass flow a t a point range of distance viscosity of fluid kinematic viscosity of fluid eddy viscosity of fluid density of fluid mass concentration of solid particles in fluid density of solid material time interval shear stress at wall of tube
a t center line of tube for solid particles at wall of tube
SYMBOLS
-
average
literature Cited (1) Hinze, J. O., “Turbulence,” p. 300, McGraw-Hill, New York, 1959. (2) Janke, E., Emde, E., “Tables of Functions,” p. 14, Dover Publications, New York, 1945. (3) Kliegel, J. R., “Flow of Gas-Particle Mixtures in Axially Symmetric Nozzles,” Inst. of Aerospace Science, paper 1913-61 (1 961 \ . \-. - - / .
(4) Lamb, H., “Hydrodynamics,” pp. 130-4, Dover Publications, New York, 1932. (5) Longwell, J. P., Weiss, M. A., IND.ENG.CHEM.45, 667-97 (1953). (6)’ Lumley, J. L., “Some Problems Connected with the Motion of Small Particles in Turbulent Field,” Johns Hopkins University, Baltimore, Md., Report on Contract NONR 248(38), 1951. (7) Patterson, G. N., “Molecular Flow of Gases,” p. 179, Wiley, New York, 1956. (8) Peskin, R. L., “Statistical Effects Resulting from the Presence of Many Particles in Motion in an Ideal Fluid,” Heat Transfer and Fluid Mechanics Institute, 1960. (9) Ranz, W. E., Talandis, G. R., Gutterman, B., A.Z.Ch.E. Journal 6, 124-7 (1960). (10) Rose, H. E., Barnacle, H. E., Engineer 203, 898-901, 939-41 (1957). (1 1) Schlichting, H., “Boundary Layer Theory,” Chap. 20, McGraw-Hill, New York, 1955. (12) Soo, S. L., A.Z.Ch.E. Journal 7,384 (1961). (13) Soo, S. L., “Boundary Layer Motion of a Gas-Solid Suspension,” Project SQUID Rept. 1LL-3P (1961). (14) Soo, S.L., J . Chem. Eng. Sei. 5 , 57-67 (1956). (15) Soo, S. L., Hohnstreiter, G. F., “Measurement of Concentration Distribution in Two-Phase Flow in Fiber Optical System,” Project SQUID Progress Rept., 1960. (16) Soo, S.L., Ihrig, H. K., Jr., El Kouh, A. F., Trans. A.S.M.E., J . Basic Eng. 82D,No. 3, 609-21 (1960). (17) Soo, S. L., Regalbuto, J. A., Can. J . Chem. 38,160-6 (1960). (18) Soo, S. L., Tien, C. L., J . Appl. Mechantcs 27E,5-15 (1960). (19) TVen, C. Y., Simons, H. P., A.I.Ch.E. Journal 5 , 263 (1959). RECEIVED for review November 6, 1961 ACCEPTED JANUARY 2, 1962 Division of Industrial and Engineering Chemistry, ACS, Christmas Symposium, Lniversity of Delaware, Newark, Del., December 1961. Work sponsored by Project SQUID, supported by the Office of Naval Research, Department of the Navy, under contract Nonr 1858 (25) NR-098-038. Reproduction in full or in part is permitted for any use of the United States Government.
VOL.
1
NO. 1
FEBRUARY 1962
37
