40 1
Ind. Eng. Chem. Process Des. Dev. 1902,21,401-403
Estimation of the Drag Coefficient of Regularly Shaped Particles in Slow Flows from Morphological Descriptors Gregory R. Carmlchael Chemical and Meterlals Engineering Program, Unlversi& of Iowa, Iowa City, Iowa 52242
Morphological descriptors which determine particle shape uniquely and quantitatively are introduced and used to characterize the effects of particle shape and orientation on the drag of freely settling particles in the Stokes flow region. The analysis is based on data obtained from studies on the settling rates of isometric particles consisting of tetrahedrons, cubes, cube octahedrons, octahedrons, and spheres, and nonisometric particles of cylinders, square prisms, double cones, and spheroids of different aspect ratios and settling orientation positions.
Introduction The shape characteristics of a solid particle exert a profound influence on its ability to absorb momentum from a moving fluid stream and therefore can greatly affect the settling velocity. However, the details of the complex interactions between fluid dynamics and particle shape remain largely unquantified. Therefore, it is often necessary in equipment and process design to rely heavily on empirical correlations derived from experimental observations. Furthermore, although hundreds of experiments measuring settling velocities of different shaped particles have been conducted, a correlation which can accurately predict the settling velocity from knowledge of the fluid and shape parameters for both regularly and irregularly shaped particles has not been found. One of the major difficulties in obtaining empirical correlations for the drag coefficient has been the inability to systematically and quantitatively describe shape itself. However, recent advances in particle shape characterization now make it possible to describe shape uniquely and quantitatively (Beddows and Meloy, 1980). This ability to quantify shape can be used to develop empirical correlations. As an example, in this paper new shape descriptors are used to develop a correlation for the drag coefficient of regularly shaped particles in the Stokes flow region. The derived correlation is shown to predict accurately both particle shape and orientation effects. It should also be possible by using these techniques to develop accurate correlations for the drag coefficients of irregularly shaped particles. Furthermore, the new shape descriptors and techniques discussed in this paper can be used as an effective general tool in quantifying shape effects from experimental data. Data Data from three experimental studies (Pettyjohn and Christiansen, 1948; Malaika, 1949; Heiss and Coull, 1952) were selected for analysis using new morphological descriptors. In these experiments the settling velocities of isometric particles consisting of tetrahedrons, cubes, cuboctahedrons, octahedrons, and spheres, and nonisometric particles of cylinders, square prisms, double cones, and spheroids of different aspect ratios and settling orientation positions were measured and the drag coefficients and Reynolds numbers were calculated. (Analysis in this paper is restricted to data in the Stokes flow region Re 6 1.) Different particle material, fluids, and experimental techniques were utilized in these experiments. The experimental data were analyzed and reported according to the model CD = 24/Re.K (1)
Table I. Data Set Description particle no. 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
K, description a circular cylinder circular cylinder circular cylinder square prism square prism square prism double cone double cone double cone spheroid spheroid spheroid rectangular parallelpiped square projected area square projected area square projected area square projected area square projected area rectangular parallelpiped rectangular projected area rectangular projected area rectangular projected area rectangular projected area rectangular projected area circular cylinders roundwise fall roundwise fall roundwise fall roundwise fall roundwise fall circular cylinders edgewise fall edgewise fall edgewise fall edgewise fall edgewise fall spheroids edgewise fall cube octahedron octahedron tetrahedron
mead 0.78 0.96 0.76 0.76 0.92 0.71 0.75 0.94 0.67 0.79 1.00 0.73 0.72 0.84 0.93 0.96 0.97 0.92 0.87 0.93 0.93 0.86 0.82 0.76 0.91 0.96 0.94 0.88 0.81 0.76 0.76 0.86 0.96 0.97 0.96 0.93 0.91 0.96 0.97 1.10 1.22
aspect ratio 4 1 0.25 4 1 0.25 4 1 0.25 4 1 0.25 0.25 0.50
1 2 3 4 0.25 0.5 1 2 3 4 0.25 0.5 1 2 3 4 0.25 0.5 1 2 3 4 0.25 4 1 1 1
a Particles 1-12 and 37 and 38 from Malaika(1949); 13-36 from Heiss and Coull (1952); and 39-41 from Pettyjohn and Christiansen (1948). Description is given as shape and orientation. For Malaika’s data all orientations are with the maximum projected area perpendicular to the flow.
where K is the shape correction term and is equal to the ratio of the actual terminal velocity, ut,to the terminal velocity of a sphere with the same equivalent diameter, %* For spherical particles, K = 1 and eq 1 reduces to the Stokes equation. The data from these studies are pres0 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
402
drag increases as the characteristic length in the direction of motion increases. At least 13 different shape parameters have been used in analyzing the effect of shape on settling velocity (see Table 11). These quantities include sphericity, circularity, flatness factor, and roundness factor. These parameters have been useful in obtaining first-order correlations for the settling velocity for regularly shaped particles of the same shape class (e.g., sphereoidal or nonisometric nonsphereoidal, etc.) (see Table 111),but they are of limited applicability for particles of a different shape class or orientation, or irregularly shaped particles. However, new morphological descriptors developed by Leurkins (1980) can describe particle shape uniquely and quantitatively. These descriptors arise from a procedure in which two-dimensional profiles of a particle are expanded as truncated Fourier series of the form
Table 11. Shape Factors Used in Correlating the Drag Coefficient and the Reynolds Number ~~~
~~
A , / A , = (normal projected area of particle)/normal projected area of equivalent sphere) P = AT/A, ; AT = surface area tangential to the flow K = U/U,= (settling velocity of particle)/(settling velocity of sphere of equal volume) l / s = (length taken I to its max. circular or square section)/ (diameter of max. circular or square section) A , (circularity) = (circumference of circle with same projected area)/(actual circumference of projected area) h = n/6dP3/dn3= (volume of particle)/(volume of sphere) d, = diameter of circle having same A, d, = diameter o f sphere of equal volume i (sphericity) = (surface area of sphere of equal volume)/ (surface area of actual particle) m (mean hydraulic radius) = A , /( perimeter of projected area) area ratio = (projected area)/(area of circumscribing circle) prismodial ratio = (mean thickness)/(max. thickness) F (flatness factor) = ( b t c)/2a, a, b , c shortest longest axis R, (roundness factor) = (radius of sharpest corner)/ (longest diameter) S F (Corey’s shape factor)= a / ( b . ~ ) * . ~
N
R(8) = A0
+
+ nX= lA n COS (ne - an)
(2)
where R(B) is the radius of the profile from the centroid at the angle 8 and an is the phase angle. The shape terms are
ented in Table I. In all instances each point represents the average response of a large number of repeated experiments. The reported variations within a given experiment were quite small (on the order of a few percent) and all reported K values were corrected for wall effects. The drag coefficients were evaluated based on the particle projected area normal to the flow direction and the Reynolds number based on the equivalent sphere diameter.
(4)
m=l n=l
N
EL, = EL&)
(5)
n=l
where 1N
Ro = Ao2 + -
Analysis In the model represented by eq 1 the influence of shape is lumped into a single parameter K. Some general observations can be made from the data in Table I. From Malaika’s data it is seen that for particles with aspect ratio 4 the spheroid has the lowest drag (i.e., the highest K value) followed by the cylinder, prism, and double cone, respectively. For aspect ratio 1 the ranking followed the series sphere, cylinder, double cone, and prism, while for aspect ratio l/., the progression was cylinder, sphere, prism, and double cone. In addition, for all orientations, the drag increases as the aspect ratio deviates further from 1. For example, K values for cylinders settling roundside decrease as the length of the cylinder increases for a fixed crosssectional area. Furthermore, for all represented geometries the drag decreases when the particle is rotated to the edgewise position (e.g., a cylinder falling flat-end first). The importance of tangential area can be seen in the fact that for constant normal cross-section area objects, the
A2
(6)
L2(n)= A,2/2RO2
(7)
2n=1
and
Ls(m,n) = (3/4R03)(A,AnA,+n
COS
(a,+, - an - a m ) ) (8)
These shape parameters contain most of the particle profile information, and the particle image can be accurately regenerated from the individual coefficients (Lo, L2(n),and L3(m,n)).By use of the Particle Image Analyzing System (PIAS) at The University of Iowa, these parameters can be determined directly and automatically from particle images. These shape descriptors were used to develop a model for K from the data in Table I. The data in this table represent many different types of particles (isometric, nonisometric, blunt bodies, smooth bodies, etc.) as well as
Table 111. Empirical Equations for Use in Predicting the Drag Coefficient in the Stokes Region expression
CD = A R e - ‘ + BRe-a.s -c C A = 24.66SF-0.‘57 B = 4.07SF-0.0‘62 C = 0.49SF-L.399 CD = 24/K,Re K = 0.843 In { i / 0 . 0 6 5 } K = 0.946SF-0.378 K = 2 . 1 8 - 2.09SF K = [ C Y . P ] / [ { O . ~ ~ ~ / A ~+. ~0.0891 ~} log ( K )= 0.25(idc/d,)’”(d,/ds - 1) + log ( d , / d s ( i ) ” a )
comments R e < 10
Brezina ( 1 9 7 9 )
Re c 1
Heiss and Coull ( 1 9 5 2 )
0 . 4 < S F < 0.8 SF < 0.4 spheroids A varies with orientation
Komer ( 1 9 7 8 ) Komer ( 1 9 7 8 ) Malaika ( 1 9 4 9 ) Pettyjohn and Christiansen (1948) Pettyjohn and Christiansen (1948)
A
See Table I1 for definition of shape parameters.
reference
constant
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 403
Table IV. Morphological Descriptors of the Normal and Tangential Projected Areas for the Particles Described in Table I and the Predicted K Values particle
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
dcl d,
1.24 0.99 1.39 1.36 1.20 1.44 1.27 1.26 2.25 1.26 1.00 1.59 1.44 1.15 0.92 0.72 0.63 0.57 0.72 0.81 0.92 1.02 1.09 1.14 0.78 0.80 0.99 1.11 1.18 1.24 1.39 1.10 0.87 0.69 0.61 0.55 0.80 0.63 0.97 1.08 1.22
LOT Lo 0.86 0.99 1.17 0.92 1.03 1.16 0.90 1.01 1.10 0.88 1.00 1.13 1.15 1.04 1.00 1.04 1.09 1.15 0.87 0.96 1.00
0.20 0.91 0.87 0.86 0.95 0.99 0.95 0.91 0.86 1.15 1.05 1.00 1.05 1.10 1.15 0.88 1.13 1.00 0.98 1.00
zLZN 0.248 0.011 0.000 0.171 0.001 0.011 0.184 0.000 0.000 0.221 0.000 0.000 0.011 0.011 0.011 0.011 0.011 0.011 0.249 0.085 0.011 0.085 0.170 0.249 0.249 0.085 0.011 0.085 0.170 0.249 0.000 0.000 0.000 0.000 0.000 0.000 0.220 0.000 0.011 0.043 0.043
zLzT 0.000 0.000 0.270 0.011 0.057 0.260 0.000 0.011 0.176 0.000 0.000 0.220 0.249 0.085 0.011 0.085 0.170 0.249 0.011 0.011 0.011 0.011 0.011 0.011 0.000 0.000 0.000
0.000 0.000 0.000 0.249 0.085 0.011 0.085 0.170 0.249 0.000 0.221 0.011 0.016 0.043
z L , ~ K,pred. 0.79 0.0000 0.0000 0.92 0.77 0.1520 0.0009 0.76 0.0008 0.95 0.1340 0.73 0.0000 0.81 0.86 0.0009 0.64 0.1400 0.0000 0.80 0.0000 0.95 0.1240 0.73 0.73 0.1300 0.86 0.0058 0.90 0.0009 0.97 0.0058 0.93 0.0478 0.93 0.1300 0.88 0.0009 0.90 0.0009 0.0009 0.90 0.0009 0.85 0.0009 0.81 0.0009 0.77 0.0000 0.89 0.0000 0.94 0.0000 0.92 0.0000 0.87 0.0000 0.83 0.79 0.0000 0.1300 0.77 0.91 0.0058 0.94 0.0009 1.03 0.0058 0.0478 0.98 0.98 0.1300 0.90 0.0000 0.93 0.1240 0.88 0.0009 0.0003 0.94 0.0097 0.76
e,g. loo/- =
(r) * 100 , 10% / Pred
Obs
1.0
D W
t; 0.9
a K
a
0.8 Y
0.7
0.6 0.7
0.6
0.8
1.o
0.9
1.1
K MEASURED
Figure 1. K values predicted by eq 9 plotted against observed K values. Each point corresponds to a particle described in Table I. The lines represent relative error bands. 0
RS
W RE 0 CR
1.0
-
0.9
-
0 W
+ 0
-
w 0.8 0 LL
a
-
Y
0.7 -
4
t
0 6 01
025
05
10
2
3
4
10
PIS
Figure 2. K values predicted by eq 9 for spheroids (S), cylinders (C), and rectangular parallelepipeds (R)as a function of aspect ratio at different orientations (superscript S refers to square end normal to flow, R refers to round side normal to flow, and E refers to edgewise fall). Experimental data is shown by the geometric symbols.
orientation effects, and therefore they provide a meaningful test set. For each particle listed in Table I, the projected particle profile normal to the flow and that tangent to the flow were subjected to the above analysis. The resulting shape parameters are summarized in Table IV. The following chemical reaction rates, wear rates, and light scattering. model for K was determined through statistical anaylsis Acknowledgment K = 0.69(de / ~ s ) + ' ~ ~ ~ ~ CL2N)+'*002( ( L ~ ~ / CL2T)0'060( L ~ ~ ) ~C . ~ ~ ~The ( author wishes to express his gratitude to the members of the Fine Particle Research Group and R. Allen for L3T)4072 (9) their valuable assistance. where the superscripts N and T denote values for the Literature Cited normal and tangential areas, respectively. Values of K Beddows, J. K.; Meloy, T. P., Ed. "TesHng and Characterization of Powders predicted from this equation are plotted against the oband Fine Partlcles"; Heyden and Son: London. 1980. served values in Figure 1. The regression coefficient for Brezina. J. Presented at the 2nd European Symposlum on Particle Characterlzation, Sept 1979. this correlation is 0.94. Helss, J. K.; CouU, J. Chem. Eng. Rog. 1952, 48, 133. The predicted K values as a function of aspect ratio for Leurkins. D. Ph.D. Thesis, Unhrersity of Iowa, Iowa City, Iowa, 1980. different orientations is plotted in Figure 2. These trends Komar, P. D.; Relmers, C. E. J . oeol. 1978, 86. Malaka. J. W. M.S. Thesis, Unlvemity of Iowa, Iowa City, Iowa, 1949. are also present in the measured data. This figure demPettyjohn, F. S.; Christiansen, E. B. Chem. Eng. Rog. 1948, 4 4 , 157. onstrates that not only can this model predict K values accurately, but it can also model the influence of particle Received for review September 12, 1980 orientation. Revised manuscript received September 14,1981 Currently this study is being extended to higher ReyAccepted March 22, 1982 nolds number flows and to irregularly shaped particles. In addition, these shape descriptors and analysis techniques This research was supported in part through an Old Gold Summer Fellowship provided by The University of Iowa. are being used to model the effects of particle shape on
