Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978 571
UG= superficial gas velocity, cm/s UL = superficial liquid velocity, cm/s VG= gas volume in aerated bed, cm3 We = Weber number, DpUG2PL/U, DpUL’pLla Greek Letters t~
= gas holdup, HG/H
liquid holdup, HL/H liquid holdup defined by eq 3 tsp = void fraction in a dry packed bed pL = viscosity of liquid, g/cm s PI, = density of liquid, g/cm3 pp = density of packing, g/cm3 a = surface tension, dyn/cm tL = tSL =
Subscripts
L = liquid obs = observed Literature Cited Bahbekov, 0. S..Romankov, P. G., Tarat, E. Ya., Mikhlev, M. F.,J. Appi. Chem. U.S.S.R., 42, 1454 (1969). Blyakher, L. G., Zhivaikin, Ya., Yurovskaya, N. A., Int. Chem. fng., 7 , 485 (1975). Barile, R. G., Meyer, D. W., Chem. f n g . Progr. Symp. Ser., 87, No. 119, 134 (1971). Chen, 8. H.. Douglas, W. J. M., Can. J . Chem. f n g . , 48. 245 (1968). Gelperin, N. L., latyshev, Yu. M., Blyakham, L. I., Int. Chem. E-., 8, 691 (1968). Kto, M., Sawada, M. Shimada, M., Takata, T., Sakai, T., Sugiyama, S., Kagaku Kogaku Ronbunshu, 2 , 12 (1976a). Kito, M., Kayama, Y., Sakai, T., Sugiyama, S., Kagaku Kogaku Ronbunshu, 2 ,
..- \.’.-“,.
A7R IIQ7Rhl
Krainev, N. I., Niyazov, M. I., Levsh, I. P., Umarov, S. U., J . Appl. Chem. U.S.S.R., 41, 1961 (1988). Tichy, J., Wong, A., Douglas, J. M., Can. J . Chem. Eng., 5 0 , 215 (1972).
cal = calculated G = gas
Received for review January 16, 1978 Accepted June 13, 1978
A Comparison of Correlations for Saltation Velocity in Horizontal Pneumatic Conveying Peter J. Jones” and L. S. Leung University of Queensland, St. Lucia, Queensland 4067, Australia
Eight well-known published correlations for horizontal saltation velocity are compared using the accumulated solids-air data of many workers in the field. The correlation of Thomas (1962) is recommended as being the most accurate available for a priori prediction of saltation velocity. Other correlations tested were due to Zenz (1964), Matsumoto et al. (1974, 1975), Rizk (1973, 1976), Mewing (1976), Rose and Duckworth (1969), and Doig and Roper (1963).
Figure 1 is a schematic diagram of the variation of pressure drop along a horizontal pipe with gas velocity at various solids mass fluxes. Line AB is for gas only in the pipe. As solids mass flux increases, the pressure drop per unit length of pipe increases. For any particular solids rate (i.e., a line of constant solids mass flux CDEF) the pressure gradient in the pipe decreases as superficial gas velocity is decreased until a minimum is reached (D). This point is defined as the “saltation point” and the corresponding gas velocity is the “saltation velocity”. If the gas velocity is further reduced, the pressure gradient increases dramatically and the pipe operates in the dense phase regime with or without slugging. The “saltation point” coincides approximately with the point a t which particles are observed to drop out of suspension and remain in a stationary layer on the bottom of the pipe, or at which particles stop rolling or sliding along the bottom of the pipe. For economic operation of pneumatic conveyors it is not desirable to have particles on the bottom of the pipe. By the same token, the gas velocity should be as low as possible to minimize pipe erosion and power consumption in the blower. To quote Scott (1977): “... the greatest difficulty facing the designer is the choice of an acceptable gas velocity. If the minimum transport saltation velocity for a given material in a particular system is known a reasonable prediction can be made for that material in other systems. However, the choice of an acceptable minimum velocity for a new material must at present be based on experiments or on the designer’s experience.” 0019-7882/78/1117-0571$01.00/0
We shall therefore evaluate each of the well-used correlations for saltation velocity to give the designer a “best buy” expression for a priori estimation of a minimum acceptable air rate in horizontal conveying.
Correlations for Saltation Velocity Several workers have published correlations for saltation velocity as defined above. Each expression presented below has been converted to S.I. units for consistency and clarity. The first and simplest of these is due to Dallavalle (1942). He proposed the relation (for both vertical and horizontal pneumatic transport lines)
Us = k p s d p “ ’ / ( p s+ 1000)
(1)
(dp in mm, p s in kg m-3, and Us in m s-l) where the parameters k and n’and their respective vertical and horizontal values are: h, 8.96, 8.35; n’, 0.60, 0.40. Clearly, his correlation takes no account of the solids loading in the pipe, which has been shown to be an important variable by every other worker in the field. For this reason, the Dallavalle equation will not be considered further here. Barth (1954, 1958) and Welschof (1962) recognized that a dimensionless expression might be more fruitful. It was observed that the Froude number at saltation (defined as Us/(gD)’I2)varied in a power law relationship with the solid to gas flow ratio. Barth proposed that
W s /W ,
M* = KFr:
0 1978 American Chemical Society
(2)
572
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
Table I OL
a
b
Matsumoto et al. (1974) 0.488 0.50 -1.75 Matsumoto et al. ( 1 9 7 5 ) 1.11 0.55 -2.3
c
error
3.0 + 5 0 % 3.0 *40%
In two papers, Matsumoto et al. (1974, 1975) attack the problem of saltation velocity correlation using a similar method. They propose that the expression should take the form
M* = 4p,/pg)“(Frt/ 10)b(Fr,/ 10)‘
0
SUPERFICIAL AIR VELOCITY, U
Figure 1. Schematicpressure gradient curves for horizontal pneumatic conveying. Curves: upper, Wai;middle, Wal;lower, W, = 0.
Welschof generalized this to M* = KFr,” where n varies with the type and size of particles. Rizk (1973, 1976) used his extensive experimental studies in an attempt to correlate K and n empirically. For Polystyrol and Styropor of various grades and sizes he was able to correlate saltation velocity by M* = (1/lO6)Fr,X (3) where 6 = 1.44d + 1.96, x = l.ld, + 2.5, and d, is in millimeters. 6unther (1957) recognized a similar fourth-power dependence in his saltation data (primarily for wheat). He extended Barth’s analysis using the boundary layer theory of Schlichting and Prandtl mixing length concept to arrive at his expression for the Froude number at saltation. Frs2 = B,2/[(D/dp)5/7 (1 + M*’/’)] + AS2M*’i2 (4)
A , and B, took the values 10.0 and 23.4 for wheat. The Barth fourth-power dependence comes out strongly in the second term of the expression. Without the benefit of the extensive experimental work of Siege1 (1973),Rizk (1973, 1976), Duckworth (1975,1976), or Matsumoto (1974, 1975,1976), which was to follow, Doig and Roper (1963) accepted the form of the Barth expression and presented their relation (i) for 3ms-’ < ut < 12ms-’ log (Fr,) = (ut - 0.61)/8.5 + 0.25 log (M*) (5a) where ut is the terminal velocity (meters per second) of a single particle of the mean size under consideration, and (ii) for ut < 3ms-‘ log (Fr,) = 0.098~1,+ 0.25 log (M*) (5b) Following along this line, Mewing (1976) used the Doig and Roper expression and the previously unavailable Matsumoto and Rizk data to arrive at new estimates of the parameters, viz. log (Fr,) = (ut + 5)/13 + 0.25 log (M*) u,:ms-’ (6) Rose and Duckworth (1969) developed an expression to correlate their own data and those of Segler (1951) using a dimensional analysis approach. They maintain that their equation applies equally well to inclined or horizontal pipes and to fluids other than air (viz., water). Their expression is
(8)
where Fr, is a particle Froude number equal to ut/ (gd,)1/2. No account is taken of the ratio of pipe to particle diameter. Matsumoto et al. fitted the parameters of the above using their own data on two occasions with slight differences in the method of measuring the saltation point. The results are listed in Table I. They explain deviations in parameters as being due to the fact that (i) the definitions of saltation velocity for the two sets of experiments are not “strictly” the same and (ii) it is only possible to keep the solids to gas ratio approximately constant when finding the pressure gradient minimum. It will be seen later that the scatter of published data is so great that the difference between these two expressions in the prediction of Fr, is insignificant. The only investigator to consider the problem of mixed particle size distributions (in more detail than merely taking an average particle diameter for calculation purposes) was Zenz (1957,1960,1961,1964). He proposed an empirical method by which the saltation velocity of a mixed particle distribution could be estimated. His correlation is primarily based on his own experimental data with additional reference to those of Culgan (1952) and Bagnold (1941), who also used an air-solid system. He correlates single particle saltation data for both spherical and angular particles by plotting Uso/wvs. d,/A where U, is the single particle saltation velocity, d, is the equivalent particle diameter, w = [4gpg(ps - ~,)/3pp2]’/~, and A = [3CL,2!4gP,(Ps - P g P 3 . It is of interest to note that this curve exhibits a minimum which Zenz explains in terms of the viscid boundary layer at the wall of the pipe. “A particle which is sufficiently small to sink into the viscid boundary layer ...will be out of reach of the turbulence which could pick it up. The smaller the particle, the deeper it is engulfed ...and therefore the higher the velocity required to reach the particle” (Zenz, 1961). In a later paper, Matsumoto et al. (1977) describe the effect of changing particle diameter on the “minimum transport” velocity. His findings support the existence of this minimum. Korn (1950, 1951) also acknowledges this phenomenon and attributes it to surface effects and electrostatic charging. A schematic diagram for the Zenz single particle saltation velocity correlation at a particular tube diameter is shown in Figure 2. For mixed distribution of particle size, the parameter SAis evaluated as shown, and the saltation velocity of a suspension is given by
W,
-= Ps
0 . 2 1 ~ -~uso)/u80 1 ~ ~
(9)
for SA> 0.05 and W, is in kg m-2 s-’. Us, here is scaled up with pipe diameter using the fact that Us, is propor= 3.2(M*)O.’ (D/d,)o.6 ( p , / ~ , ) - ” . ~( U,/(gD)’/2)0.5 (7) tional to D“ with 0.4 < n < 0.6. Ut Using a semi-theoretical, hydrodynamic approach, Thomas (1961,1962) proposed an expression based on the They recommend that the minimum operating air velocity friction velocity. The data upon which the correlation is be then taken as two to three times that predicted by the based are primarily for aqueous suspensions. However, above.
us
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978 573
Slope
-
S
_Spheric
A\
I
I
I
LOG dp/A
Figure 2. Zenz’s single particle saltation correlation and the definition of SAfor nonuniformly sized mixtures. The two intersections refer to the “largest” and the “smallest” particles. “Largest” and “smallest” are taken here at the 95% and 5% undersize by weight bases.
some air-solids data have been included to complete the picture. The friction velocity of a suspension flowing in a pipe is defined by
u*= ( T w / P m )
(10)
112
where T , = APD/4L (wall shear stress) and pm is the density of flowing mixture. For the purposes of calculation, pm may be assumed to be approximately equal to p g , the gas density. Since the voidage in the pipe just prior to saltation is high, this is a fair assumption. A transition particle diameter is also defined (11) = 5P,/PlnU* = 5Pg/PgU* Thomas defines two flow regimes: regime I, d , < A, and regime 11, d , 1 X. The friction velocity for saltation at infinite dilution is calculated for each of the regimes using the following expressions: for d , 1 X
for d, < X
-U-t - O.OlO(
us,*
d,Uso*
2.71
7) (12b) Pg
This friction velocity for saltation of a single particle (infinite dilution) is then used to calculate the velocity for the concentrated suspension using
where e, = voidage in the pipe at saltation. The voidage is directly related to the solid velocity and may be quickly approximated by assuming a value of the solid to gas velocity ratio of about 0.5. For air-solids systems, the voidage is usually high due to the large density difference between the solid and gas. Finally, by using the universal velocity profile one can relate the friction velocity at saltation, Us*,with the mean stream velocity, Us.
US
-=
us*
where Re = pgDUs/Fg.
5 log Re - 3.90
(14)
The Thomas correlation is the only one considered which uses a proposed saltation mechanism for its foundation. It is the most fundamentally based of all the published correlations. Comparison of the Correlations The experimental data of many workers were collected to compare the published correlations on a statistical basis. Only saltation data for air-solid systems were considered, first, since there appeared to be a sufficient number of points with which to make a comparison, and secondly, because it was considered more important to arrive at the most accurate correlation for pneumatic systems rather than the generalized situation. Table I1 shows the experimental detail of the 390 pieces of data used for the comparison. The measured values of solid loading ratio and Froude number at saltation for each of these data points are to be published elsewhere (Leung et ala). The root mean square relative deviations based on Fr, were evaluated for each correlation. The R.M.S. relative deviation is defined as
The cumulative sum of squares of deviations was also calculated in order to allow a statistical comparison to be made. Table I11 compares the correlations on a relative deviation basis as defined by eq 15. Some interesting results are apparent. First, the Thomas correlation (1962), although developed primarily for water-solid systems, predicts the saltation velocity more accurately than all other correlations tested. This assessment has been confirmed statistically at the 95 70 confidence level. Secondly, the Rizk, Matsumoto, Mewing, and Zenz correlations appear to have much the same accuracy on the basis of R.M.S. relative error. However, the well-used and widely accepted Zenz correlation seems to have a minor drawback. With very small particles (approximately 10% of the data points) the parameter SAassumes a value less than 0.05 and the correlation cannot be used with confidence (Leung et al.). An attempt is made here to obtain an improved correlation by introducing an additional factor (D/d,) to the equation of Matsumoto et al. A multilinear least-squares regression analysis was performed to obtain the exponents in the following equation. Fr, = cu’(M*)”’(ps/pg)b’(D/dp)c’(Fr,)d’ (16) Statistical comparison of experimental and predicted values of Fist however, show that eq 16 is not significantly superior to the correlations of Matsumoto et al. Thus it may be concluded that the addition of (Dld,) to the Matsumoto et al. type correlation does not improve the correlation. Of the published correlations, the most accurate is that due to Thomas (1962). This correlation is recommended here for a priori estimation of horizontal saltation velocity. Clearly, if one is designing a system for a solid whose saltation characteristics have already been investigated, the simplest method for scaling up is to use the expression of Matsumoto. Little sacrifice in accuracy is incurred by doing this. Conclusions A statistical comparison of published correlations for horizontal saltation velocity in pneumatic transport based on data collected from several sources was carried out. A total of 390 data points were used to evaluate R.M.S.
574
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
Table 11. Published Experimental Saltation Dataa p ss
author Culgan (as reported by Zenz Duckworth
Year
1975
Gunther
1957
Hours and Chen
1976
Matsumoto e t al.
1974,1975
Matsumoto e t al.
1974,1975
Rizk
1973
Siege1 (as reported by Rizk, 1976) Welschof Zenz
1973
1952
1962 1961
solid
d,, mm
distribution
kg m-3
soya beans Tenite glass glass glass glass glass glass polystyrene polystyrene polystyrene polystyrene polystyrene polystyrene mustard seed mustard seed polypropylene polyester fly ash I fly ash I1 sodium bicarbonate aluminum silicate sand wheat wheat polyethylene polyethylene sand sand glass glass glass g1ass copper copper copper polystyrene polystyrene glass glass copper copper copper polystyrene Styropor 2 Styropor 2 Styropor 2 Styropor 3 Styropor 3 Styropor 4 Polystyrol 168N Polystyrol 168N Polystyrol 168N Polystyrol 475K Polystyrol 475K Polystyrol 475K Polystyrol 475K Polystyrol 475K Polystyrol 475K Polystyrol Polystyrol Polystyrol wheat Rice Krispies rape seed glass sand sand sand sand salt cracking catalyst cracking catalyst cracking catalyst
6.35 3.05 1.27 0.671 0.336 0.105 0.0787 0.0394 1.30 0.927 0.699 0.648 0.457 0.356 2.03 2.03 3.00 3.00 0.024 0.045 0.040 0.070 0.069 2.7 2.7 3.5 3.5 0.70 0.70 0.41 1.00 1.30 1.51 0.1 2 0.29 0.55 0.96 2.10 0.48 1.02 0.30 0.52 0.76 1.07 0.731 0.731 0.731 2.385 2.385 5.65 3.15 3.15 3.15 1.776 1.776 1.776 2.52 2.52 2.52 1.75 1.75 1.75 2.59 6.35 1.676 0.587 0.559 0.930 0.559 0.483 0.167 0.107 0.059 0.0521
uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform 3.0-4.0 3.0-4.0 uniform uniform a = 0.030 a = 0.79 u = 0.070 a = 0.83 u = 0.30 (J = 0.023 a = 0.059 u = 0.063 a = 0.29 u = 0.049 u = 0.093 a = 0.055 a = 0.088 a = 0.117 a = 0.063 uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform 1.0-2.5 1.0-2.5 1.0-2.5 2.2-3.5 uniform uniform uniform 0.15-2.38 0.59-1.55 0.203-1.55 0.089-1.55 0.051-0.419 0.089-0.180 0.0101-0.180 0.0101-0.89
1170 1130 2916 2916 2916 2916 2916 2916 952 95 2 952 952 952 952 1140 1140 876 1360 2290 1950 21 36 1456 2570 1282 1282 958 958 2655 2655 2500 2500 2500 2500 8700 8700 8700 1000 1000 2500 2500 8700 8700 8700 1050 1050 1050 1050 1050 1050 1695 1695 1695 1050 1050 1050 1050 1050 1050 1050 1050 1050 1280 160 1090 2484 2644 2644 2644 2644 2099 1763 1763 1763
no. of data D,mm points 77.1 77.1 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 12.7 25.4 12.7 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40 125 31.9 55.0 31.9 55.0 26.0 26.0 26.0 26.0 26.0 26.0 26 .O 26.0 26.0 49 49 49 49 49 49 49.6 52.6 51.7 49.6 52.6 51.7 49.6 52.6 51.7 49.6 52.6 51.7 49.6 52.6 51.7 400 200 100 61 31,8 44.5 44.5 31.8 44.5 31.8 31.8 44.5 31.8 31.8 31.8
4 4 8 18 11 3 7 7 9 8 9 9 9 9 5 4 5 4 7 4 8 6 7 9 6 2 3 2 2 7 6 7 7 5 4 5 5 6 3 2 4 3 2 2 4 5 5 5 5 4 5 5 5 5 5 5 5 5 5 3 3 8 9 1 5 4
4 6 3 4 3 4 3 1
a Notes: (1)The Rice Krispies data of Zenz (1961) were not used in the statistical comparison. ( 2 ) In all cases, the conveying fluid is air, nominally at atmospheric pressure and room temperature. (3)Measured vaiues of Fr, and M* for each data point are to be published elsewhere (Leung e t al.).
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
Table 111. Results of Comparison of Saltation Correlations Correlation year Doig and Roper 1963 Matsumoto et al., I 1974 Matsumoto et al., I1 1975 Mewing 1976 Rizk 1973 Rose and Duckworth 1969 Thomas 1962 Zenzb 1964 a E = R.M.S. relative error
E, %"
90 60 53 50 60 78 44 54
Calculations for 350 of the 390 data points. relative error and cumulative sum of squares of deviations for each of eight published correlations. Froude number, defined as U,/(gD)'i2,was used as the independent variable for all correlation testing. 1. The results show the correlation due to Thomas (1962) is more accurate than all other correlations tested. The correlations due to Zenz (1964), Matsumoto et al. (1974,19751, Mewing (1976), and Rizk (1973,1976) all have much the same relative error on an R.M.S. basis. 2. The Thomas correlation is recommended here for design purposes. 3. The correlations of Matsumoto et al. are simplest to use when scaling-up data already a t hand such as that shown in Table 11. When scaling-up, little sacrifice in accuracy is made by using a correlation based on dimensional analysis. Nomenclature a , b, c , d = parameters in eq 8 A,, B, = parameters in eq 4 D = pipe internal diameter d = mean equivalent particle diameter If=R.M.S. relative error defined in eq 15 Fr, = Froude number at saltation (= U,/gD)1/2) Fr, = particle Froude number (= ~ ~ / ( g d , ) " ~ ) g = gravitational acceleration k = parameter in eq 1 K = parameter in eq 2 L = length of pipe M* = solid to gas mass flow ratio (= W,/W,) n' = parameter in eq 1 N = total number of data points Re = Reynolds number SA= slope of U,,/w vs. d / A plot of Zenz (see Figure 2) U,, Us,= gas velocity at saftation point of a suspension and a single particle, respectively ut = terminal velocity of a single particle of mean size W,, W , = mass velocities of solids and gas, respectively
575
a = parameter in eq 8 and 16 6, x = parameters in eq 3 A = [311,2/4gpg(ps - Pg)11'3 t, = pipe voidage at saltation X = transition particle size for Thomas correlation eq 11 gg = gas viscosity a = t4gP ( p , - p g ) / 3 p , 2 1 1 / 3
= cfensities of gas and solid, respectively density of flowing gas-solid mixture 7, = wall shear stress * = indicates friction velocity calc = calculated meas = measured Literature Cited
pg, ps
,om =
Bagnold, R. A., "The Physics of Blown Sand and Desert Dunes", Methuen, London, 1941. Barth, W., Cbem. Ing. Tech., 26, 29 (1954). Barth, W., Cbem. Ing. Tech., 30, 171-180 (1958). Bohnet, M., VDI forscbungsh., 507 (1965). Culgan, J. M., DSc. Thesis, Georgia Institute of Technology, Atlanta, Ga., 1952. Daliavalie, J. M., Heat. Vent., 39, 28-32 (1942). Doig, I. D., Roper, G. H., Aust. Chem. Eng., 9-19 (Jan 1963). Duckworth, R. A. "Plesswe Gradient and Velocty Correlation and their Application to Design'' Proc. Pneumotransport 1, org. by BHRA Fluki Eng., Cranfield, paper R2, 1971. Duckworth, R. A. "United Kingdom Research in Solid-Gaseous Flows" Joint Symp.-Pneum. Transport of Solids, 5th Afric. Inst. Mech. Eng., paper 1.2, April, 1975. Duckworth, R . A. "The Influence of the Particle and Fluid Properties and the Inclination of the Pipe on the Minimum Transport Velocity" Proc. Pneumotransport 3 org. by BHRA Fluid Eng., Cranfieid, paper S5, 1976. Gunther, W., Doktor-Ingenieurs Dissertation, Technische Hochschule Karlsruhe, 1957. Hours, R. M., Chen, C. P., personal communication, 1976. Korn, A. H., Chem. Eng., 57, 108 (1950). Korn, A. H., Cbem. Eng., 58, 178 (1951). Leung, L. S.,Wiles, R. J., Jones, P. J. "Pneumatic Conveying of Bulk Solids" monograph under preparation, 1978. Lippert, A., Chem. Ing. Tech., 38, 350 (1966). Matsumoto, S.,Hara, M., Saito, S.,Maeda, S.,J . Chem. Eng. Jpn., 7, 425 (1974). Matsumoto, S.,Harada, S.,Saito, S.,Maeda, S.,J . Chem. Eng. Jpn., 8, 331 (1975). Matsumoto, S.,personal communication, 1976. Matsumoto, S.,et ai., J . Chem. Eng. Jpn., 10, 273 (1977). Mewing, S. F., B. E. Thesis, University of Queensland, 1976. Muschelknautz, E., Krambrock, W., Chem. lng. Tech., 41, 1164 (1969). Rizk, F., Mor-Ingenleufs Dissertation, Technische Hochschule Karlsruhe, 1973. Rizk, F., "Pneumatic Conveying at Optimal Operation Conditlons and a Solution of Barth's Equation A = 4 (A,,P)'', Proc. Pneumotransport 3, organlzed by BHRA Fluid Eng., Cranfieid, paper D4, April, 1976. Rose, H. E., Duckworth, R. A., Engineer, 227, 476-483 (March 1969). Scott, A. M., Proceedings, 4th International Powder Technology and Bulk Solids Conference, Harrogate, U.K., p IO, 1977. Segler, G . , Pneumatic Graln Conveying", National Inst. of Agric. Eng., Silsoe, Bedfordshire, 1951. Siegei, W., VDI Forschungsh., 538, (1973). Thomas, D G., AlChE J., 7, 423 (1961). Thomas, D. G., AlChE J.. 8. 373 (1962). Welschof, G., VDI Forschungsh., 492,(1962). Zenz, F. A., Pet. Refiner, 36, 175 (1957). Zenz, F. A,, Othmer, D. F., "Fluidization and Fluid-Particle Systems", Reinhold, New YOrk, N.Y., 1960. Zenz, F. A., D. Chem. Eng. Dlssertatlon, Polytechnic Inst. of Brooklyn, 1961. Zenz, F. A., Ind. Eng. Cbem. fundam., 3, 65-75 (1964).
Received f o r review January 30, 1978 Accepted June 21, 1978
