literature Cited
Arutyunyan, L. A., Khurshudyan, E. KH., Geochem. Int. 3, 479 (1966). Bell. R. E.. Herfert. R. E.. J . Amer. Chem. Soc. 79. 3351 119573. B e r h d , J: C., Tridot, G.,’BuZl.Sci. Chim. Fr. SlO‘(1961). Buchwald, H., Richardson, E., Talanla 9, 631 (1962). Kunda, W., Ruiyk, B., Planseeber. Pulvermet. 13 (3), 167 (1965). Levenspiel, P., Chemical Reaction Engineering,’’ p 146, Wiley, New York, N. Y., 1962. McAndrew, R. T., Peters, E., Can. M e t . Quart. 3 (2), 153 (1964).
Saxena, R. S., Jain, M. C., Mittal, M. L., Aust. J . Chem. 21 ( I ) , 91 (1968). Sutolov, A., Mol bdenum Extractive Metallurgy,” University of Concepcion, dhile, 1965. RECEIVED for review August 10, 1972 ACCEPTED February 15, 1973 The data in this paper were originall presented at the Annual Meeting of A.I.M.E., San Francisco, Zalif., 1972.
Estimating the Solid Particle Velocity in Vertical Pneumatic Conveying Lines Wen-ching Yang Research & Development Center, Westinghouse Electric Corporation, Pittsburgh, Pa. 16255
Literature data on solid particle velocities in vertical pneumatic conveying lines were correlated with a modified terminal velocity equation which takes into account the friction losses and the void fractions in transporting lines. The data have mean particle diameter ranges from 109 to 2024 p , particle density from 53.7 to 169 Ib/ft3, and tube i.d. from 0.267 to 1.023 in. Both air and COZ were used as carrier gases. The correlation is good to +20% if experimentally observed friction factors are used.
Interest in the pneumatic conveying of solids has existed for a period of years. Despite the considerable progress achieved recently, the design of a pneumatic transport system remains essentially a n experimental a r t rather than a well-founded science. This is due to the complex mechanics involved in particle-particle, particle-gas, part’icle-pipe interactions mhich are difficult to isolate experimentally. One of the important parameters for calculating the pressure drop and power consumption of the transport lines is the solid particle velocit’y. Thus, a knowledge of the velocity is necessary for calculating acceleration losses and friction losses in a pneumatic conveying system. Gnfortunat’ely,no unified and accurate approach exists. It is the intention of this paper to present a method to estimat’e the solid particle velocity in vertical pneumatic transport lines. Horizontal pileumatic conveying lines are excluded in this discussion because the solid particles are t’ransportedthrough a different mechanism compared to that in vertical lines (Davidson and Harrison, 1 9 i l ; Wen, 1965). Review of Previous Approaches
The study of particle dynamics in a fluid is usually started with analysis of forces acting on a single particle. -1single solid particle, falling under the action of gravity in a n infinite and nonmoving fluid, will eventually attain a constant terminal velocity ( C t ) when the resisting drag force is equal to the net gravitational accelerating force. The resulting equation is
the fluid, p, = density of the solid particle, nz = mass of a single particle, and g = gravitational acceleration. The drag coefficient, CDS, depends on the particle Reynolds number, (Re),, defined as !J
where d, = particle diameter. The relation between CDSand (Re), is given in many sources (see most references in literature cited). For spherical particles, eq l becomes rt =
p;picDs P PP
- Pf)
(3)
For (Re), > 1000 but less than approximately 250,000, the ralue of CDSvaries between 0.4 and 0.5 with an average value of 0.44, but for (Re), > 2.5 X 105, the drag coefficient drops by more than 50y0 to a value of less than 0.2. Equation 3 is then reduced to
pp(p:f-
rt = 1.74
pf)
(1000 < (Re),
< 250,000)
(4)
I n the intermediate region, 2.0 < (Re), < 1000, the drag coefficient can be expressed as (Wen and Yu, 1966)
CDS = 24(Re),-l
+ 3.6(Re),-0,313
(2.0
< (Re), < 1000) (5)
or as (Schiller and Ijaumann, 1933)
CDS = (CDS)Stokes[l
+ O.l50(Re),O 6”7]
((Re),
< 800) (6)
where CDS = drag coefficient on a single particle, A = surface area perpendicular to the direction of fluid, pf = density of Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
349
-2Qh Limit
" /
M
/ +20kLimi
@' * A i r ascarrier &s C02 a1 Carrier Gas
A
/
Reference: Hariu and Molstad
Calculated 50110Vtlaitj. nhrc
Figure 1 . Comparison between calculated and experimental values of solid particle velocity using eq 12
Figure 2. Comparison between calculated and experimental values of solid particle velocity
Substituting eq 5 into eq 3, we have
stad (1949). A large discrepancy was observed as shown in Figure 1, especially a t high fluid velocities, for large particle sizes and for large particle densities. Equation 12 predicts that the slipvelocity, U t - Cp,is independent of fluid:velocity, which contradicts the actual experimental observations. This is due to the fact that friction losses due to particle-particle and particle-wall collisions are not taken into account in deriving eq 4,8, and 9. A new derivation of a terminal velocity equation for a particle in a relatively dense fluid-solids mixture will correct the difficulty.
L-t =
0.153d,1.'4g0~71(pp-
pf)"."
.43pf0 .29
(2.0
< (Re), < 1000) (8)
For (Re), _< 0.1, the drag force can be represented by Stokes' law. This leads to the following expression for the terminal velocity (9)
Present Mathematical Development
Equations 3-9 are applicable only for single-particle and non-moving infinite fluid systems; however, the correction for nonspherical particles is available. Pettyjohn and Christiansen (1948) suggested obtaining the terminal velocity for norispherical particles a t laminar and intermediate regions by multiplying the values obtained from eq 8 and 9 with a correction factor K1. K1
=
0.843 log
CD
4 -
0.065
where Q = sphericity (surface area of a sphere divided by the area of the nonspherical particle having the same volume as the sphere). I n the Xewton's law region, the drag coefficient can be corrected by the equation (CDs)non
= 5.31
- 4.884
(11)
where = drag coefficient for a single nonspherical particle with sphericity 4. The average sphericity can be estimated using the method discussed by Wadell (1934). Thus theoretically, the minimum fluid velocity necessary to support a particle is equal to the terminal velocity of the particle. The relative particle velocity can then be calculated by subtracting the terminal velocity from the fluid velocity.
c, =
Uf -
ct
Ind. Eng. Chem. Fundarn., Vol. 12, No. 3, 1973
=
(13)
CDsE-4'7
where CD = drag coefficient on a single particle in a n assemblage of particles of voidage E. Making this correction in eq 12, we have P --
-
x
E2.11
(14)
This is comparable to the experimental relation of Richardson and Zaki (1954)
c, = ci - ct x
tn-l
(15)
where n = a function of particle Reynolds number based on particle terminal velocity. Let WTs be the solid flow rate in lb/sec, then dWs is the effective weight of solid particles in a pipe of length dL. The total number of solid particles (assuming spherical particles), d N , in dL is
(12)
Iiowever, experimental data show that eq 12 is only an approximation, that the difference between the fluid velocity and the solid velocity, known as slip velocity, increases with increasing fluid velocity. Equation 12 was used to correlate the solid particle velocity measurements by Hariu and Mol350
Equations 3-9 are applicable only when the particle concentration is extremely small, Le., as the voidage E approaches 1. When the specific solid loading in pneumatic transport lines increases, the drag force acting on a particle is larger and can be related to the voidage of the mixture by (Davidson and Harrison, 1971; Wen and Yu, 1966)
or
I
Table 1. literature Values of Solid Friction Coefficient Substance
.4-/,' e Air as Carrier Gas
C02 as Carrier Gas Reference: Belden & Kassei A
Calculated Solid V e i a i t y , ftlsec
Figure 3. Comparison between calculated and experimental values of solid particle velocity (small catalyst)
(PP
4w3 - Pf)TD2LTp
where D is the pipe diameter. The total drag force on dN particles in dL is the sum of the drag force on each individual particle. dl:d
= '/qCDse-4.7
PdCf - U , ) 2 dW, (PP - Pf)d,gc
=
(19)
9 dW3 X gc
The solid friction losses can be defined following the familiar Fanning equation with a particle friction coefficient, f,.
Under steady-state condition
dFd
=
dFg
Ottawa sand, sea .and, microepheroidal cracking catalyst, ground cracking catalyst Particle diameter, ft 0 00036-0 00165 Particle density, lb/ft3 61-169 Carrier gas Air and COz Tube diameter, in. 0.267 and 0.532 i.d. ~~
~~~~
~~
~
~~
~
Comparison with Literature Data
The gravitational force can be expressed as
dF,
Table II. Solid Properties Covered by Hariu and Molrtad
Material
The voidage, E , in section dL is then given by e = l -
Lit. source
f,
Tenite 0 0044-0 008 Hinkle 0 008-0 019 Hinkle Polystyrene 0 003-0 008 Hinklr Catalin Alundum 0 009-0 018 Hinkle Coal 0 005 Barth 0 005 Barth Coke 0 003-0 013 Barth Wheat particles Ottawa sand 0 010-0 021 Hariu and AIolstadSea sand 0 008-0 019 Hariu and LIolstada Microspheriodal cracking catalyst 0 008-0 023 Hariu aiid 1Iolstada Ground cracking 0 008-0 018 Hariu arid Molstad. catalyst a f p reported in Hariu and Molstad is '/4 of f p defined in this paper because of different definition in Fanning equation. Also f p reported in Hariu and LIolstad includes pressure drop due to acceleration of particles and thus is generally larger than that reported elsewhere.
+ dFf
aiid we obtain
Comparing eq 23 and 12, we find that they are essentially identical if the terminal velocity is redefined by
Equation 24 is a modified equation of eq 3 with correct'ing for drag coefficient and fpL~p*/2Dg,correcting for particle friction losses in an assemblage of solid part'icles. Although eq 19 does not really account for all the forces on all the particles, eq 23 proves to be a better equation for estimating solid particle velocity in vertical pneumatic conveying lines. Thus, eq 23 should be of value in design.
The solid friction coefficient, fp, in eq 23 is hard to obtain, partly because of the complex mechanics involveti and partly because of the difficulty in isolating the friction losses from other losses during experimental pressure drop measurement. Hinkle (1953) reported f,, = 0.0044-0.008 for tenite, 0.0080.019 for polystyrene, 0.003-0.008 for catalin, and 0.009-0.018 for alundum in his horizontal pneumatic conveying studies. H e also observed that J, is independent of the particle Regnolds number. Barth (1960, 1962) a i d Hariu and Xolstad (1949) also obtainedf, for the other materials which are listed in Table I. The values reported by Hariu and Molstad as fs include pressure drop due to accelerat,ionof particles. Its effect is difficult to isolate. Thus their values of fs are generally larger than values reported elsewhere. (Because of different definition in Fanning equation, fp in t'his paper is equal to four times f .) Equation 23 was used to calculate the solid particle velocity for the materials and conditions shovl-n in the paper by IIariu and Molstad (1949). jp was taken to be four times the f 5 values reported in the paper. Comparison between the esperimental and calculated solid velocities is presented in Figure 2. The comparison covers 116 experimental data points. X o r e than 96% of t'he calculated values fall within +209& of the esperimental values. The solid properties covered in the experiments are summarized in Table 11. Equation 23 is also used for correlating the solid particle velocities reported by Belden and Kassel (1949). Sirice the value of fP is not reported in the paper, a constaiit value of 0.005 was assumed for the calculation. The results for the small catalyst are shown in Figure 3 which covers 59 data points with mean particle diameter a t 1005 p , particle density 53.7 lb/ft3, and with tube inside diameters of 0.473 and 1.023 in. About 90% of the calculated values are within = 2 0 7 ~of Ind. Eng. Chern. Fundam., Vol. 12, No. 3, 1973
351
Acknowledgment
Encouragement and helpful discussion from Dr. D. L. Keairns and Dr. D. H. Archer are gratefully acknowledged. Nomenclature
A
surface area perpendicular t o the direction of fluid motion, f t 2 drag coefficient on a single particle in an assemblage of particles of voidage drag coefficient on a single particle drag coefficient on a single nonspherical particle with sphericity $ drag coefficient from Stokes’ law mean particle diameter, ft inside diameter of vertical conveying lines, ft solid particle friction factor drag force on a single particle, ft-lb/sec2 net gravitational force on a single particle, f t-lb/sec gravitational acceleration, ft/sec2 terminal velocity correction factor for nonspherical particles mass of a single particle, lb an exponent in Richardson and Zaki equation particle Reynolds number actual fluid velocity, ft/sec actual particle velocity, ft/sec particle terminal velocity, ft/sec modified particle terminal velocity, ft/sec solid flow rate, lb/sec
CD
9 10
K1
M
20 Calculated Solid Velocity, ftlsec
Figure 4. Comparison between calculated and experimental values of solid particle velocity (large catalyst)
the experimental values. The results for the large catalyst, which has a mean particle diameter of 2024 p and particle density of 60.9 lb/ft3, are presented in Figure 4. Kotice that the line for -40y0 limit passes through the data points. The data presented by Belden and Kassel are not experimental measurements. They are the results of calculation from the equation
GREEKLETTERS =
I*
=
e
= =
Q Cf - Cp
= 1.32
d g d p ( p p
-
Pt)/Pf
= fluid density, lb/ft3
Pf Pn
(25)
said to be the result of their laboratory observations; however, no data on solid velocity and no description of measuring technique were presented. Without accurate measurement of f,, it is difficult to evaluate the solid velocity data of Belden and Kassel further. Nevertheless, the values calculated from eq 23 show consistent results.
particle density, lb/ft3 fluid viscosity, lb/sec-ft voidage in transporting lines sphericity (area of a sphere divided by the area of the nonspherical particle having the same volume as the sphere)
literature Cited
Barth, W., Chem. Ing. Tech. Z . 30, 171 (1960). Barth, W., Chem. Ing. Tech. 2. 32, 164 (1962). Belden, D. H., Kassel, L. S.,Ind. Eng. Chem. 41(6) 1174 (1949). Davidson, J. F., Harrison, D., Ed., “Fluidization,” Chapter 16, Academic Press, Kew York, N.Y., 1971. Hariu, 0. H., Molstad, AI. C., Ind. Ens. Chem. 41 (6). 1148 . ( 1949). Hinkle, B. L., Ph.D. Thesis, Georgia Institute of Technology, 1953. Pettyjohn, E. S., Christiansen, E. E., Chem. Eng. Progr. 44, 157 (1948). Richardson, T. F., Zaki, W. X., Trans. Inst. Chem. Eng. 32, 35 (1954). Schiller, L., Naiimann, A., 2. Ver. Deut. Ing., 77, 318 (1933). Wadell, H., J. Franklzn Inst., 217, 439 (1934). Wen, C. Y., U.S. Bur. Mines Inform. Circ. 8314 62 (1963). Wen, C. Y., Yu, Y. H., Chem. Eng. Progr. Symp. Ser. 62, KO. 62, 101 (1966). RECEIVED for review August 14, 1972 ACCEPTED February 9, 1973 I ,
Conclusions
Solid particle velocities in vertical pneumatic conveying lines can be predicted with a modified terminal velocity equation including a solid friction factor, f,. and the void fractions in transporting lines. The data correlated have mean particle diameter ranges from 109 to 2024 I*, particle density from 53.7 to 169 lb/ft3, and tube id. from 0.267 to 1.023 in. Both air and C o nwere used as carrier gases. The correlation is good to +20% if experimental values off, are used.
352
Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
