712
Ind. Eng. Chem. Process Des. Dev. 1980, 19, 712-715
COMMUNICATIONS Fluldization Characteristics of Cuboids The fluidization characteristics of cuboid plastic particles have been studied. The minimum fluidization conditions can be correlated by the equation 400(1
- t,)~,p de
+2
.
4 = ~( p p ~ - p)gtm3de ~ ~
At the terminal velocity for fluidization the coefficient of resistance has been correlated to the sphericity factor by the equation: C, = 7.754, - 4.31.
Introduction The pressure loss, the minimum fluidizing velocity, and the terminal velocity resulting from the flow of fluids through columns packed with granular materials has been the subject of a large number of experimental investigations and theoretical analyses. To analyze the minimum fluidizing velocity for fine particles, the pressure dropvelocity relationship according to Carman (1937) and Kozeny et al. (1938) may be used. For larger particles a more general equation such as the Blake (1922), Carman (1937) or Ergun (1952) equation must be used. In the case of nonspherical particles the previous equations have been used with a modified particle diameter, but there is a considerable lack of experimental data for large nonspherical particles. In the region of fluidization, a number of workers including Hancock (1937, 1938) and Richardson and Zaki (1954) have shown that a log-log plot of velocity against voidage is linear with a slope n. This index n is a function of the ratio d / D and of the Reynolds number. Correlations for n have been proposed by Davidson and Harrison (1971) in the case of spherical particles. Furthermore, Richardson and Zaki (1954) have shown that for cubes and cylinders n can still be correlated by the use of a shape factor. However, the fluidization of large particles of low sphericity, &, has not been extensively studied. The purpose of the present paper is to report the results of fluidization studies on large cuboids whose minimum dimension is 6 mm. The range of the variables investigated is given in Table I. The study shows that existing correlations must be modified for the prediction of fluidization parameters for such particles. Experimental Apparatus and Technique Apparatus. The apparatus, shown in Figure 1, consisted of a 150-mm glass working section A vertically mounted through which water was circulated by a centrifugal pump C. The flow rate was varied using control valve E and bypass valve F. A flow-measuringsection B, containing a sharp edged orifice plate conforming to BS 1042, was used to measure the water flow rate. The working section was lo00 mm long. The distributor plate H was made from 1.5 mm thick PVC sheet with rows of 1.5 mm diameter holes at 5 mm pitch. The distance between the rows of holes was 2.5 mm. A perforated screen N prevented carry-over of particles into the water collec-
Table I. Range of the Variables Investigated specific gravity of cuboids cross-sectional dimensions of cuboids aspect ratio of cuboids (length:thickness) bed mass of cuboids diameter of test section fluid medium water with temperature range
1.14-1.49 6 x 6 mm to 18.5 X 18.5mm 1:1-9:1 1.0-2.0 kg 150 mm 15-20 'C
Table 11. Range of Materials and Sizes Tested
material
cross section spy (width X cific thickgrav- ness), ity mm
nylon
1.14
Perspex
1.19
Tufnol
1.36
PVC
1.49
12.5 X 12.5 6X 6 9.5 X 9.5 12.5 X 12.5 18.5 X 18.5 12.5 X 12.5 12.5 X 12.5
length, mm
80, 60,39, 29.5, 19, 12.5 54, 26, 12.5 63,31,15
80,60,40, 29, 19, 12.5 114,. 56., 28
80, 51,39, 25, 19, 12.5 52, 25.5,12.5
tion tank G. Static pressure probes L and M were used to measure the pressure drop across the bed. These probes pressurized a pair of flasks J from which connections to a Furness micromanometer were made pneumatically. Experimental Technique. The particles were loaded and unloaded by separating the flanges at N and H in Figure 1, respectively. For a given size and density of particles, the pressure drop across the bed and the bed height were measured for a range of water flow rates. The flow rate was increased by small increments beginning with a fixed bed and finishing when entrainment occurred; Le., particles began to collect on screen N. Data were also taken for a range of decreasing flow rates. The pressure drop characteristics of the apparatus were obtained without the particles. Hence the pressure drop across the fluidized bed for a given flow rate was obtained 0 1980 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19,No. 4, 1980 713
Figure 1. Schematic diagram of apparatus: A, 150-mm diameter glass section; B, flow-measuring section; C, centrifugal pump; D, water reservoir tank; E, flow control valve; F, bypass valve; G, collection tank; H, perforated stainless steel screen; I, mercury U-tube manometer; J, pressurized flasks; K, Furness micromanometer; L and M, static pressure probes; N, retention plate; 0, sharp-edged orifice plate.
by subtracting the pressure drop of the empty column from the pressure drop measured with the particles inserted. Plastic particles of uniform size were used for each test. Particle dimensions (and the properties of the materials used are given in Table 11. Discussion General. For a fixed bed of particles, the pressure drop due to fluid solid friction was correlated mathematically by Ergun (1952) and shown to be related to voidage, sphericity, particle diameter, liquid viscosity, and bed height. The general equation takes the form
In the case of small spherical particles the constants $ and y in the Ergun equation are well established as 150 and 1.75, respectively, an.d eq 1 is usually presented as
-=[7--,1.[ Ap
L
150(1 -
PU
1.75(1 - 6) pu2 €3 d
The pressure drop due to fluid solid friction comprises two factors, the viscous isnd the kinetic energy losses. For nonspherical particles equivalent diameters have been proposed for substitution into eq 1 and 2. An effective diameter, de, may be used and correlations have been derived by Ergun (1952) and Davidson and Harrison (1971). The effective diameter is defined as the diameter of a sphere with the same surface area to volume ratio as the particle. A true nominal diameter, dp, defined by Waddell (1934) and Kunii and Levenspiel (1969) as the diameter of a sphere of the same volume as the solid has also been used in eq 1 and 2. Minimum Fluidization. At the onset of fluidization the drag force caused by the upward flow of fluid equals the weight of the particles. By substituting the relevant mathematical terms into eq 2 as shown by Michell (1970), the pressure drop at minimum fluidizing condistions can be expressed in the form of eq 3. (3) The Ergun equation is useful since it can be applied over all values of Reynolds number. Hence substitution of the
2 x 132
I 132
4 x 102
A
4
lo3
1 4
Figure 2. Plot of friction factor, fv, vs. Re/(l Figure 4.
c).
For key, see
minimum fluidizing conditions of eq 3 into eq 2 yields eq 3, which can be used to determine u, over all values of Reynolds number.
The results of fluidization tests on particles of various lengths and a square cross section of dimensions 12.5 mm X 12.5 mm gave values of u, which do not agree well with those predicted by eq 4. A wide range of particle sizes were then tested to determine the fluidization characteristics of cuboids. Plastic particles of four different densities and varying dimensions, as listed in Table 11, were studied. From the results of the investigations the friction factor, f,, proposed by Ergun, was plotted against Re/(l - 6 ) . For particles of length to thickness ratio of 4 or less this gave a linear characteristic. However, the slope and intercept differed from those obtained by Ergun; the plot is shown in Figure 2. Hence it is proposed that for large particles of square cross section the following equation may be used to calculate urn.
For length to thickness ratios greater than 4, divergence from this equation occurs. The divergence increases with particle length and a correlation for this variation has yet to be determined. Figure 3 compares the results for cuboids with a general correlation proposed by Wen and Yu (1966). The authors considered that their equation was a convenient means of obtaining a rapid estimation of the minimum fluidization velocity. Figure 3 indicates that the equation of Wen and Yu can be used to obtain an estimate of the minimum fluidization velocities of large cuboids. Terminal Velocity. The fluid flow rate through a fluidized bed is limited between u, and the entrainment velocity of solids in the fluid. At the terminal velocity, ut, carry-over commences and this represents the minimum transportation velocity of the system. The terminal velocity of the particles may be obtained from eq 6 given by Kunii and Levenspiel.
714
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 1 08
I
2.4
7
3
lo7
lo7
2.0
.
1.6
.
cr 1.2
.
0.3
.
0.4
.
Ga lo7
f
7 x 106
A 0
3 x 106
1 3
2.2
0.4
0.6
5.8
1.0
a; Figure 4. Plot of coefficient of resistance, C,, vs. sphericity factor, #s: 0, PVC, 12.5 X 12.5 mm cross section; 0,Tufnol, 12.5 X 12.5 mm cross section; Q, nylon, 12.5 X 12.5 mm cross section; 0,Perspex, 18.5 X 18.5 mm cross section; ,. Perspex, 12.5 X 12.5 mm cross section; A, Perspex, 9.5 X 9.5 mm cross section; A, Perspex, 6 X 6 mm cross section.
106
I
2
5
lo2
102
103
t
2
10:
Figure 3. Plot of Galileo number, Ga, vs. Reynolds number, Re,. For key, see Figure 4. C d is an experimentally determined drag coefficient, and Coulson and Richardson (1966) proposed that for spherical particles, the drag coefficient may be obtained from the following expressions.
< 0.2 (ii) Re > 500
(i) Re
(iii) Re = 0.2-500
= 24Re-'
(7)
Cd = 0.44
(8)
Cd
Cd
= 18.5Re4'.6
(9)
These expressions have been obtained from the free fall velocity of single particles. It has generally been the practice to use values of Cd obtained in this manner in fluidization calculations. For nonspherical particles other values of drag coefficients must be used. It has been shown by Waddell that the drag coefficient depends on the sphericity of the particle as well as its Reynolds number. Waddell used the sedimentation data of a number of investigators to show the dependence of the coefficient of resistance, C,, on Reynolds number for a range of sphericity factors from 0.125 to 1.00.
c, =
2V(P, - P)g
the particle. Using the experimentally derived values of ut to determine the coefficient of resistance, C,, a graph of C, vs. 4, was obtained for values of +8 from 0.60 to 0.81 as shown in Figure 4. A straight line was obtained according to eq 11. c, = 7.754, - 4.31 (11) This result is completely different from the results of Waddell, in which the coefficient of resistance decreased with increasing sphericity factor. The reason for this difference is due to the turbulence which is present during fluidization. A similar effect has been obtained in studies of the settling rates of mineral particles in water by Needham and Hill (1947). These data illustrated the effect on the ratio BVI TV of both the tube and particle Reynolds numbers. When the tube Reynolds number was turbulent BVITV was shown to be greater than unity. When both the tube and particle Reynolds numbers were laminar, the ratio BV/TV was approximately unity. In all the tests carried out in this investigation both the tube and particle Reynolds numbers were turbulent, namely, Re(partic1e) > 500 and Re(tube) > 4000. Conclusion The fluidization characteristics of large cuboid shaped particles have been studied. A minimum fluidization conditions the data correlate to the following form of the Ergun equation.
(10)
APU?
It was found in the present investigation that a drag coefficient used by Waddell and called the resistance coefficient, C,, gave a better correlation of the results than the more conventially used coefficient, cd. In the calculation of C, the actual volume of the particle and the projected area, A , of the particle in its most stable orientation are used. For cd,the volume is taken as ad216 and the area perpendicular to flow as ad,2/4. For a spherical particle C, and Cd Will be equal, but the values will diverge as the sphericity factor decreases. The most stable orientation of the particles was with the largest side perpendicular to the direction of flow so that the projected area, A , was the product of the length and the breadth of
At the terminal velocity the coefficient of resistance has been correlated linearly with the sphericity factor to give the equation: C, = 7.754, - 4.31. Nomenclature A = cross-sectional area of particle in its stable orientation, m2 BV = balancing velocity, m s-' C d = drag coefficient, dimensionless C, = coefficient of resistance, dimensionless d = particle diameter, m de = particle equivalent diameter, m d, = particle true nominal dimeter, m
Ind. Eng. Chem. Process Des. Dev. I W Q , 19, 715-717
D = bed diameter, m f,= frictional resistancle g = gravitational constant 9.81, m s-' G a = Galileo number, d: t(p, - p ) g / p 2 L = bed depth, m p = pressure, N m-2 Ap = pressure drop, N m-2 R e = Reynolds number R e = Reynolds number based on diameter d, and u, T S = terminal velocity in free fall u = velocity m s-l u, = minimum fluidizing velocity, m ut = terminal velocity in s-l V = volume of particle, m3
715
L i t e r a t u r e Cited Blake, F. C. Trans. Am. Chem. Eng. 1922, 74, 415. Carman, P. C. Trans. Inst. Chem. Eng. 1937, 15, 150. Coulson, J. M.;Richardson, J. F. "Chemical Engineering", Voi. 2, 1966. Davidson, J. F.; Harrison, D. "Fluidization", Academic Press: New York, 1971. Ergun, S. Chem. Eng. Prog. 1952, 48, 89. Hancock, R. T. Trans. Inst. Mining Eng. 1937, 1938, 94, 114. Kozeny, J. Sitzber. Akad. Wiss. Wien, Math-Naturw. Kl., Abt. I I a 1927, 736, 271 [Discussed in Carman, P. C. Trans. Inst. Chem. Eng. 1938, 76,1681. Kunii, D.; Levenspiei, 0. "Fluidization Engineering", Wiley: New York, 1969. Michell, S. J. "Fluid and Particle Mechanics", Pergamon Press: London, 1970. Needham, L. W.; Hili N. W. FuelSci. Pract. 1947, 26, 101. Richardson, J. F.; Zaki, W. N. Trans. Inst. Chem. Eng. 1954, 32, 35. Waddell, H. J. Franklin Inst. 1934, 277, 459. Wen, C. Y.; Yu, Y. H. AIChE J. 1988, 72, 610.
Department of Industrial Chemistry T h e Queen's University o f Belfast Belfast, Northern Ireland
Greek L e t t e r s = voidage, dimensionless t, = voidage at minimum fluidization, dimensionless
Engineering Department Loughry College of Agriculture and Food Technology Cookstown, Northern Ireland
t
& = sphericity factor, (dimensionless p = liquid density, kg p, = particle density, k.g m-3 p = liquid viscosity, N s m-2
Gordon McKay* Henry D. McLain
Received for review March 28, 1979 Accepted March 24, 1980
+,r= constants in the Ergun equation
Pressure Drop and Heat Transfer Prediction for Total Condensation inside a Horizontal Tube Pressure drop and heat transfer coefficient prediction equations for the total condensation of dry saturated steam inside horizontal tubes have been obtained from the results of experiments on tubes 12.7 to 50.8 mm internal diameter and 1.22 to 4.88 m long. The pressure drop results were correlated by
and the heat transfer results by
-h a-4 k
1.1
x io-" ' I
Introduction Visual studies made by the authors on the total condensation of steam inside a horizontal glass tube showed the simultaneous existence of more than one type of two-phase flow. At the tube entrance, an annular flow was observed which was maintained by high vapor shear. This was followed by a region in which the reduced vapor rate allowed the condensate to drain by gravity at an oblique angle to the tube axis (observed by rippling) and resulted in stratified flow. This continued to the outlet where drainage from the tube surface was at right-angles to the tube axis, because the vapor shear was zero. At higher condensation rates this mechanism was accompanied by slug flow. Because of experimental difficulties, the higher condensation rates which are used in some industrial applications (and which were obtained in most of the runs in the present work) could not be achieved. However, in such cases also, it is likely that several flow regimes would exist simultaneously in the tube. This conclusion is supported by the work of Palen et al. (1979), who consider that proper flow regime prediction is necessary before satisfactory tube 0196-4305/80/1119-0715$01.00/0
side condensation heat transfer coefficients can be estimated. Heat transfer coefficient prediction inside a tube for condensation with stratified flow which accompanies low vapor shear has been analyzed by Chaddock (1957) and Chato (1962). The case of high vapor shear resulting in annular flow was first analyzed by Carpenter and Colburn (1951) and subsequently by others including Soliman et al. (1968) and Bae et al. (1970). The use of these prediction methods for the design of total condensers as is sometimes recommended is questionable because both these flow regimes and others may exist in the tube simultaneously. In view of this, the empirical correlation approach of the present investigation may be considered preferable. Description of Apparatus A diagram of the apparatus is shown in Figure 1. Deaerated steam from a boiler is dried in a vapor-liquid separator before passing through the condenser tube. This is cooled by distilled water which runs outside the tube in a cylindrical jacket and boils a t atmospheric pressure (100 "C). The vapor from the jacket is condensed and 0 1980 American
Chemical Society
