Sap = surface rate of deformation tensor, sec.-1 t = time, sec. T a p = surface stress tensor, dynes/cm. T’j = bulk phase stress tensor, dynes/sq. cm. ua = surface coordinates, cm. U = (p,/pgL2d)v, , dimensionless surface velocity ~
Y =v
va
=
vi
=
u s ?u y = w’ = x?y = y’ = X =
surface velocity vector, cm./sec. three-space velocity vector fluid velocities in x and y directions three-space surface vector, cm./sec. rectangular Cartesian coordinates, cm. spatial coordinates, cm. x / L , dimensionless distance
GREEKLETTERS cy = pL*/pc,d, dimensionless viscosity ratio p = y o (ba/by)/pgdL,dimensionless elasticity parameter y = surface mass density, grams/sq. cm. r = y / y o , dimensionless surface mass density A = volume fraction of fluid drained from film e = shear surface viscosity coefficient, dyne-sec./cm. q = half thickness of film, cm. K = dilational surface viscosity coefficient, dyne-sec./cm. I.( = bulk viscosity, dyne-sec./sq. cm. pa = K e, surface viscosity p = bulk fluid density, grams/cc. g = surface tension, dynes/cm. 6 = pgdzt/pL, dimensionless time = dimensionless time a t which A = ‘/Z $1,2 = kinematic viscosity, sq. cm./sec. Y = K / d , dimensionless equilibrium constant K
+
Literature Cited (1) Aris, R., “Vectors, Tensors, and the Basic Equations of Fluid Mechanics,” Chap. 10, Prentice-Hall, Englewood Cliffs, N. J., 1962.
( 2 ) Berkman, S., Egloff, G., “Emulsions and Foams,” Reinhold,
New York, 1941. (3) Bird, R. B., Stewart, W.E., Lightfoot, E. N., “Transport Phenomena,” p. 503, Wiley, New York, 1960. ( 4 ) Crank, J., “The Mathematics of Diffusion,” Oxford University Press, London, 1956. (5) Davies, J. T., Proceedings of 2nd International Congress on Surface Activity, Vol. I, p. 220, Butterworths, London, 1957. (6) Davies, J. T., Rideal, E. K., “Interfacial Phenomena,” Academic Press, London, 1961. ( 7 ) Hamming, R. W., “Numerical Methods for Scientists and Engineers,” McGraw-Hill, New York, 1962. (8) Hansen, R. S., J . Colloid Sci. 16, 549 (1961). ( 9 ) Johannes, W., Whitaker, S., J . Phys. Chem. 69, 1471 (1965). (10) Joly, M., “Surface Viscosity,” Chap. 1, “Recent Progress in Surface Science,” Vol. I, Panielli, Pankhurst, and Riddiford, eds., Academic Press, New York, 1964. (11) Kitchener, J. A,, “Foams and Free Liquid Films,” Zbid., Cham 2. (12) capidus, L., “Digital Computation for Chemical Engineers,” McGraw-Hill, New York, 1962. (13) McConnell, A. J., “Applications of Tensor Analysis,” p. 167, Dover Publications, New York, 1931. (14) Milne-Thomson, L. M., “Theoretical Hydrodynamics,” p. 390, Macmillan, New York, 1960. (15) Mysels, K. J., Shinoda, S., Frankel, S., “Soap Films,” Pergamon Press, London, 1959. (16) Peaceman, D. W., et al., Trans. A Z M E 198, 79 (1953). (17) Quinn, J. A., Dept. Chem. Eng., Univ. of Illinois, personal communication, 1964. (18) Scriven, L. E., Chem. Eng. Sci. 12, 98 (1960). (19) Slattery, J. C., Zbid., 19, 379 (1964). (20) Zbid., p. 453. (21) Sutherland, K. L., Australian J . Sci. Res. 5 , 683 (1952). (22) Whitaker, S., IND. ENG. CHEW FUNDAMENTALS 3, 132 (1964). (23) Whitaker, S., Jones, L. O., submitted to A. Z. Ch. E. J . (24) Wylie, C. R., Jr., “Advanced Engineering Mathematics,” p. 591, McGraw-Hill, New York, 1951. RECEIVED for review January 27, 1965 ACCEPTED March 11, 1966
TURBULENT PIPE FLOW OF MAGNESIA PARTICLES IN AIR S
. L . S0 0
A ND G
.J . T R E 2 E K ,
1
University of Illinois, Urbana, Ill.
Results on concentration and mass flow distributions in turbulent gas-solid pipe flow are presented. The system consists of nominal 30-micron magnesium oxide particles in air. Experimental range includes mass ratios up to 3 pounds of solids per pound of air a t Reynolds numbers from 1.3 X 1 O5 to 2.95 X 105. The density distribution of solid particles is strongly influenced by their average charge to mass ratio, while the velocity distribution of the particles remains in the range where the collision among the solid particles is negligible.
BASIC phenomena in gas-solid flow were recognized from a number of studies (3-7) on turbulent pipe flow of a suspension. They include differences between the velocities of the phases ( 3 ) ,slip motion of solid particles a t the wall (3, 5 ) , electric charges and drift of solid particles as opposed by turbulent diffusion of particles and the resultant concentration distribution (4,7), and the effect of solid particles on the state of turbulence (6). Since a semi-empirical method has to be used in dealing with turbulent pipe flow, a limited number of experiments afforded determination of empirical parameters for only a few cases. Extensive measurements in the present study make possible determination of these parameters of con1 Present address, Department of Mechanical Engineering and Astronautical Sciences, Northwestern University, Evanston,
Ill. 388
l&EC FUNDAMENTALS
centration and velocity distribution of the solid phase over wide ranges of loading (mass ratio of solid particles to gas) and flow velocities. Experimental systems reported earlier (7) were usetl in the present investigation. They include the 5-inch pipe flow system, the electrostatic probe for mass flow distribution, the fiber optical probe for concentration distribution (7), and the sampling tape (Kleen Stick) measure of collision rate a t the wall (6). The fluid used in the present study consists of a nominal 30-micron (7) magnesia dust in air. The actual distribution in particle size (longer side) is shown in Figure 1 on a logarithmic probability plot, giving a logarithmic mean particle size of 36 microns, The mass ratios used were up to 3 pounds of solids per pound of air a t Reynolds numbers of pipe flow from 1.3 X 105 to 2.95 X to5.
*ggb
.999r
r,, is given by the mass collision rate of solid particles, hPw, per unit area (6),or
0
.98
Tpw
(2/di)niPWUPW
(7)
and the friction factor is given by cpj
= ~PW/(PP0~PO2/2)
(8)
Further, we have mpw
= Dpw(dpp/dy)w
(9)
giving diffusivity by measuring density and collision rate. With the experimental data, pertinent correlating parameters (3, 7) are tested. T h e pipe flow parameter, K z ,
K2 = 1.56 X 10-6 ( P , / P ) ( ~ / R ) ( U ~ R( * / ~P) p ~ ’~~~ / k(10) ,) where v and p are the kinematic viscosity and density of the fluid, respectively, d is the mean size of the particles, jjp is the density of the solid material, u, is the mean velocity of pipe flow, and M, is the total flow rate of the solid particles. T h e particle slip parameter (7), K3,is modified as
.02t
-0I Figure 1.
Cumulative distribution curve of M g O tested Logariithmic probability chart
Following the consideration of collision among particles
( 5 ) , the friction due to particle impact, cp,, is correlated acSemi-empirical Relations and Parameters
T h e density, p , , distribution of particle cloud in pipe flow is presented by (3) Pp
= Ppo
-
(PPO
- Ppw)(Y/R)Q
(1)
where Y is the radius, R is the pipe radius, p p o and p p w are the densities of particles in the turbulent core and a t the wall, respectively, and CY is a n empirical constant. T h e velocity distribution is given by up = upw
+
- u p W ) [ ( R- r)/Rl1lm
(typo
(2)
where m is a n empirical constant, and upoand upware velocities of particles in the core and wall, respectively, a t fully developed flow, by analogy with the turbulent velocity law of the fluid phase ( 3 ) . For average charge to mass ratio, q/m,, of the particles, the electrostatic field, E, a t radius r for the above density distribution is given by (4):
E
=
(q/mpe)[bpor/2)
+ R ( ~ p w- ~ p o ) ( +a 2 ) - ’ ( ~ / R ) ~ + ’ l(3)
where e is the permittivity and the drift velocity is u7 = ( q / m p )( 2 / e ) 1 / 2 r [ ( ~ p+0( /p 4P w )-
p p 0 ) (a
+ 2)-*(r/R)a11/2
=
+ 2B(r/R)a - 2B2(r/R)2a+ . . .]
( q / m p )(pp0/2e)1’2~ [1
(4) with
B = 2bPW
- PPO)/PPO((Y
+ a2
Substitution of the above relations into the diffusion equation,
(d/dr) D,r(dp,/dr) = v,(dp,/dr)
(5)
where D, is the diffusivity of the particles, gives the diffusivity, D,,, a t the wall as:
Dpw = ( q / m p ) ( ~ p 0 / 2 € ) ‘ / ~f~ 2)-’ [(a
+ B(a f
2B2(3a
+ 2)
cording to cp~(upo/1~2) ; is the intensity of turbulent motion. T h e electrostatic effect is generalized according to So0 (7), the turbulent electroviscous number for space charge :
-1
+ ..,]
(6)
T h e shear stress due to collision of solid particles a t the wall,
Ne.D = VPpw/4.€(q/mp) RZ/D,w
(12)
Other relations are the friction factor, f,as influenced by the mass ratios of solid to gas, Mp*, and the pipe flow Reynolds numbers, 2Ru,/v. Some general behaviors of the pipe flow of a gas-solid suspension are seen from correlating experimental data with the above parameters. Experimental Results
Starting with clean 5-inch diameter brass pipe, the wall becomes deposited with a tenacious layer of the thickness of 1 to 2 magnesia particles after a prolonged run with the airmagnesia suspension. T h e potential of the wall increases to a terminal value of about 4000 volts from ground, and no further deposition is found. After shutdown, restarting does not lead to measurable rise in potential from ground, and no further deposition occurs. This suggests that after the layer is deposited, the wall becomes insulated and the magnesia particles in suspension do not give up their charge a t the wall and become deposited; rather, they become re-entrained by turbulent diffusion. T h e deposited layer constitutes a “roughness.” An increase of friction factor (based on air alone) by 10 to 12% was noted when compared to clean air flow. Flow rates were calibrated as previously (7). As was observed previously ( 7 , 6, 8 ) , the addition of solid particles led to a decrease in pipe flow friction factors up to a certain mass ratio of solid to gas, but increased again as mass ratio was increased (Figure 2). Over the range of the present experiments, the velocity profile of the gas phase is closely represented by the l/,th velocity law, but the velocity of the solid phase is represented by Equation 2, with m 2 1 (Figure 8). This, together with the density distribution of solid particles, led to a difference between mass ratio and mass flow ratio of solid to gas (7). T h e result of over-all measurements and computations from inVOL. 5
NO. 3 A U G U S T 1 9 6 6
389
3
Moas r a t i o
Figure 3.
M;
Relation between mass ratio and mass flow ratio
Figure 2. Friction factors a t various Reynolds numbers and mass ratios
tegrating measured density and mass flow profiles (7) is shown in Figure 3. For pipe flow, these would be identical if a suspension behaved as a gaseous mixture. T h e present results show that the mass flow ratio is consistently lower than the mass ratio. T h e comparison of the curve of 140 to 6 2 feet per second shows that with similar charge-mass ratio the electrostatic effect is felt more strongly a t low flow velocity, This confirms Thomas’ concept of mininium transport velocity (9)and explains the need for higher suspension velocity for smaller particles below a certain size range than larger particles as observed by Zenz (70). The results of Zenz apply to the particular combination of surfaces of particles and wall, and hence the charge of the particles. Stagnancy of shear layer should not be a factor; shear motion of fluid exerts a lift force ( 2 ) on the particles away from the wall. The collision rate of solid particles with the wall as measured by the sticky tape is shown in Figure 4. This rate is strongly affected by the turbulent intensity near the wall as well as the particle density. For 30-micron size particles, the time con18l/d2pp = stant for momentum transfer of the particle is F 1340 sec.-l; thus the radial stopping distance is, for a n intensity of 4 feet per second, only 0.036 inch, which is smaller than the boundary layer thickness of the gas. The collision rate with the wall is therefore nearly two orders of magnitude smaller than an estimate based on random motion according to the estimated intensity of turbulence. This gives a small contribution of collision of particles to the shear stress a t the wall or the pressure drop in pipe flow. The correlation on the over-all flow rate, h;ip, or the pipe flow parameter, Kz, and the particle slip parameter, KI, are shown in Figure 5 for various mass ratios, M p * , and flow velocity in the pipe. As M,* increases, the carrying capacity of the fluid, h,,, tends to a n asymptotic value, the limit being sedimentation and clogging. Figure 6 shows the values of particle diffusivity, D,,, a t the wall for various mass ratios and flow velocities. The calculated result is based on Equation 6 from the density profile and ratio of electric charge to mass. The measured data were obtained from Equation 9. Consistency of D,, obtained from these two sources shows that a similar degree of accuracy was maintained in the measurement of average charge-mass ratio and the collision rate a t the wall. The previous results (7)
-
390
I&EC FUNDAMENTALS
M;
Figure 4. mass flow
Rate of collision of particles at the wall vs.
are included for comparison. The particle diffusivity and the friction factor parameter obtained from the rate of collision in Equations 7 and 8 are plotted against the slip flow parameter, K I , in Figure 7. The data show that the density of solid particles is such that the whole range of experiment is in the “free particle” range by analogy to free molecule flow (5) due to large interparticle spacing and lack of collision among the particles. The velocity profile of the solid particle cloud is therefore analogous to that of free molecule flow with slip a t the wall and nearly linear velocity gradient. The fact that the density and velocity distribution of the particle phase are influenced by the ratio of charge to mass of particles and the particle diffusivity is demonstrated in Figure 8. With the density and velocity distribution parameter in Equations 1 and 2 plotted against the turbulent electroviscous number, Ne,, it is seen that the particle velocity profile is not strongly affected by the electrostatic effect. At large Ne,, (up,, upw)/upoand m tend to one or the case of “viscous” motion of solid particle phase (5). However, the density profile is strongly affected by N e , , At large Ne,, steady flow of
-
io-? NQ
i
140 ft./sec.
5
3
F r i c t l on Diffurivlty Factor
6 2 ft./sec.
0
140 ft./rec.
0
62 ft./sec.
Ref. ( 7 )
'
IO4
10-6
a
I , , , , ,
' ' a ( o a '
10-4
3
10-5
K 3 = 72 JT(tbp,/ppUpoXZR/d)
Figure 5 .
Variation of K z and Ks with loading ratio M,*
Figure 7. sivity
Particle friction factor and particle diffu-
I
Figure 6. Variation of wall diffusivity with particle loading ratio
and n-1/3
-
n-"S
-
@/2) [(4s/3)j5,lPMP*]'/~
'
1 ' 1
2
Discussion
d &
I
:./ i'
the suspension cannot Ibe maintained; deposition of particles will occur as in an electrostatic precipitator.
Figure 8 shows that pipe flow of a gas-solid suspension is basically a phenomenon of interaction between electrostatic and hydrodynamic effects, the pertinent parameter being the turbulent electroviscous number, Ne,, the ratio of electrostatic force to turbulent force. The measured average charge to mass ratio is, in general, in the 10- coulomb per kg. range. Unless the particle charge is truly negligible, steady fully developed laminar pipe flow of a suspension cannot be maintained. The corresponding N e , for laminar flow is (qlm,)P/v. For a given mass ratio, Mp*,the mean interparticle spacing is
'
I
Figure 8. Variation of a, m, Au*, and Ap* with electroviscous number Ft./
Sec.
See.
(7)
Z t C
cy
m
A,,*
62
140 Ft.1
'Po-
v
r
a
8d for M," = 3 for magnesia in air, the fraction VOL 5
NO. 3
AUGUST 1 9 6 6
391
solid is only 0.001. Hence, as expected, the suspension is entirely in the “free particle” range. The particle phase is in laminar slip motion in spite of the fact that the fluid is turbulent (5). literature Cited (1) Richardson, J. F., McLeman, M., Trans. Znst. Chem. Engrs. 38, 257 (1960). (2) Saffman, P. G., J . FluidMech. 22, 385 (1965). ( 3 ) Soo, S. L., IND.ENG.CHEM.FUNDAMENTALS 1 , 3 3 (1962). (4) Zbid., 3, 75 (1964). (5) Soo, S. L., Proceedings of Symposium on Interaction between
Fluids and Particles, Institution of Chemical Engineers, London, p. 50 (1962). (6) soo, s, L., Proceedingsof Symposium on Single- and Multicomponent Flow Processes, C. F. Chen and R. L. Peskin, eds., Rutgers University, New Brunswick, N. J., Rutgers Eng. Pub. 45,l (1965). (7) Soo, S. L., Trezek, G. J., Dimick, R. C., Hohnstreiter, G. E., IND.ENG.CHEM. FUNDAMENTALS 3,98 (1964). D. G.$A.z.Ch.E. J . 6, 631 (1960)* (*) (9) Zbid., 8, 373 (1962). (10) Zenz, F. A., IND. ENG.CHEM. FUNDAMENTALS 3, 65 (1964). RECEIVED for review June 21, 1965 ACCEPTED March 14, 1966
EFFECT OF DIFFERENTIAL PRESSURE ON FLOW OF GRANULAR SOLIDS THROUGH ORIFICES W I L L I A M
RESNICK
,
Y l T S H A K HELED, AHARON KLEIN, AND EPHRAIM P A L M
Department of Chemical Engineering, Israel Institute of Technology, Haifa, Israel
The effect of differential pressure on the flow rate of several free-flowing granular solids through horizontal orifices has been investigated experimentally. An equation developed from energy balance considerations describes the experimental results. TWO parameters appear in this equation, one that is a function only of the particle properties and one that is a function of particle properties and the orifice diameter. The two parameters behave as would b e predicted from the factors that define them,
N INCREASING number
of commercial processing operations Although data are available on the gravity flow of solids through orifices (7-4, 7-9, 77, 72), there are practically no data on the effect of differential pressure on efflux rate. Kuwai (70) investigated the effect of pressure on the efflux rate of limestone, glass beads, iron ore, and coal. H e used relatively low bed depths and most of his results are reported for experiments carried out in tubes less than 2 inches in diameter. Bulsara, Zenz, and Eckert ( 5 ) investigated the effects of pressure and suction on the efflux rate of non-free-flowing as well as free-flowing solids through a horizontal orifice situated a t the bottom of a n 8inch diameter pipe. They also developed a flow model based on the orifice equation for simple fluids. I n this work a formula is developed that correlates experimentally obtained efflux rates. T h e formula is developed for a circular aperture in a flat-bottomed hopper but could be generalized to include other aperture geometries. Twodimensional parameters appear that must, for the present, be determined experimentally.
A involve the pressurized flow of solids.
Development of Formula
The experimental facts available on gravity flow of solids through apertures were recently summarized by Harmens ( 9 ) . T h e moving bed of particles forms a bridge or dome over the aperture; this dome breaks as soon as it is formed, the material falls out, and a new dome is formed. I n the present analysis it is assumed that similar behavior characterizes particle flow under the influence of differential pressure. I t is assumed, after Harmens, that a continuous Aow of particles issues a t zero velocity from a conical surface and falls 392
l&EC FUNDAMENTALS
down past the aperture. This imaginary cone is supported by the edge of the orifice. T h e pressure a t the orifice level is PI and the pressure surrounding the cone is PZ (Figure 1). An energy balance can be written on any particle between its point of issue a t height h and the orifice level:
The drag force term can be readily calculated and, for low air rates, is negligible compared to the other terms in Equation 1. Neglecting drag force and solving for the velocity of the particle a t the orifice level gives: u = [2h(g
+ APApg,/m)]1’2
The velocity profile along the radius can be obtained by substituting the geometrical relationship: h = (R
- I) tan /3
(3)
into Equation 2 and this can, in turn, be converted to the number profile if it is assumed, as does Harmens (Y), that the number of particles per unit volume a t the orifice level is constant across the orifice and inversely proportional to the mean individual particle volume, V. const n(7) =
*-
V
d2(R
- r) tan /3
[g 4- APApg,/m]
(4)
The number of particles released per unit time, N , is obtained by integrating the number profile over the effective area of the orifice
