obtaincd from those just used by adding only men t.
d
single incre-
GREEKLETTERS
V
roots of J1 ( a ) = 0 In ( 1 6). predicted 3 thickness of annular region surrounding fiber in which fluid has sig-nificant temperature. (concentration). ( vrlocity parallel to fiber ). defined more precisely by Equations 10 and 11, (similar t o 1 0 and 1 1 ) . ( 7 and 8 ) . 6.8 like 6 except for region extending from hand = thermal diffusivity = value of x at which step change in surface teniperature dccurs in .\ppendiu .A = dynamic x-iscosity = kinematic viscosity
E
= 4d17Re
T
= =
a4
@,BO
Nomenclature
at, a,,
fiber radius concentration (of bath if not otherwise designated by superscript). (C* - CO*)/(C'o* - CO*) specific heat mass diffusivity tensile force heal transfer coefficient in Appendiv A , annulus thickness elsewhere Ressel functions thermal conductivity effrctii-e diameter of test tank (see Figure 3 ) u I - / 1) a17i'R U C ' " 'k L'rdr) 'C,[(h
(2
+ 1 ) * - 11.
lB
, Q B + Q
radial coordinate measured from center of fiber in units of a fl [
:v
.Yl.,/
'u
6 L , :u 2 a 1 7j
I,
temperature (of bath if not otherwise desigliated by superscript). ( T * - TU*) ( T'o* - T 0 ) fiber velocity linear velocity of fluid at exit of spinnerette velocity in Y direction divided by Cj radial rlocity- divided by 1 'j radial \.-elocitythrough free area in periphery of filler bundle avial coordinate measured from spinnerette in units of a
K
x
P
9
=
= 6,6 8 =
+
shear stress at fiber surface heat f l u x , @ *a , ' k i T,J*' - T U * )
S VP E R SG R I PT S
*
=
,
=
quantity measured in absolute units-e.g.. units for fiber
c.g.s.
SUBSCRIPTS 0 = initially MISCELLANEOUS s Y M B O L O( ) = order of magnitude of bracketed quantity given literature Cited (1) Carslaw. H. S., .Jaeger; J. C., "Conduction of Heat i n Solids." p. 346, Oxford University Press, London. 1959. (2) Eckert. E. R. G., "Introduction to Heat and Mass Transfer." 1st ed.. p. 89. McCrabv-Hill, New York, 1950. (3) Fage, A , Proc. Roy. Sot. A144, 3 8 1 (1934). (4) Sakiadis. B. C.: '4.I.Ch.E.J. 7 , 221, 46' (1961).
RECEIVED for review Soveinber 15. 1963 .ICCEPTED May 14, 1 9 6 4
BATCH AND CONTINUOUS THICKENING Pyediction of Batch Settlirig Behavior from Initial Rate Data with Results for Rigid Spheres PAUL T. S H A N N O N , ' ROBERT D. DEHAAS,ZAND ELWOOD P. STROUPE3 Puidut 1-nzwrsitj. L a f q r t t e . Ind.
E L M E R M . T O R Y 4
\IC tlaste, I ' n n i r t i j i / j . Hmiilton Ontaiio, Cariada A general method i s presented for predicting batch settling for slurries of initially uniform concentration from the solids flux plot based on initial rate data. It i s assumed that settling velocity i s a function of local solids concentration only. Experimental batch settling curves for rigid glass spheres in water are presented. The experimental results are in complete qualitative and close quantitative agreement with the predicted behavior and confirm that the flux plot i s effectively doubly concave. A rising concentration gradient immediately above the fixed bed was observed for initially uniform slurries of intermediate concentrations. The intersection of this gradient with the fluid-slurry interface accounts for the nonlinearity of settling curves when the initial volume fraction of solids is between 0.1 5 and 0.45.
Is
A S earlier paper (9) the basic equations for the movement of planes of constant concentration were derived from continuit)- considerations. Fundamental to the approach is the anal)-sis of batch and continuous thickening in terms of the solids flux. T h e method for experimentally determining the
' Present address. 'Ihayer School of Engineering, Dartmouth College. Hariover. N. H. 2 Prcsent address. Plastics Division. E. I. du Pant de Nemours Pr C o . , \l'ilminqton. Del. 3 Ptrsent addrrss. General Electric Co.. APED. San Jose. Calif. Prcscnt address. Brookhaven National Laboratory. Upton. L. I . . s.Y. 250
l&EC
FUNDAMENTALS
flux plot for a given slurry was presented. Settling velocities for rigid spheres in water were given and compared to the results of previous workers. T h e flux plot for rigid spheres was found to be doubly concave. This paper presents a general method of predicting the entire batch settling behavior of "ideal" slurries from the solids flux plot determined from initial settling rate data. T h e method of analysis is then applied to the results obtained for rigid spheres in water. T h e detailed method for computing the settling curves from initial rate data expressed as a po\ver series in reduced concentration was presented by
Shannon, DeHaas, and Tory (8) and is given in the Appendix. .4 very complete description of the method and a digital computer program for performing the calculations are given by DeHaas ( 3 ) . Theory
This presentation is restricted to the case of batch settling of slurries having a n initially uniform concentration in vessels of constant cross-sectional area. While other cases can be analyzed in a similar fashion, these restrictions greatly simplify the analysis (5,9 ) and they apply to the usual batch test. A s sho\vn by Shannon, Stroupe, and Tory (9),the velocity of propagation. p*, of a plane of constant concentration through a region where a continuous concentration gradient exists is given by
8*
= -
1 =) : (
\\’here there are sudden changes in concentration, the velocity of the discontinuity. 6*. is given by
La
-
cb
If the settling velocity is a function of the local solids concentration only, the velocity of fall of the fluid-slurry interface is determined by the concentration of the solids immediately belobv. Thus the problem of predicting a given batch settling curve is basically the problem of determining the solids concentration a t the interface. As long as the concentration just below the interface remains constant, the interface will settle a t a constant rate. T h e rate of settling will change as other concentrations, propagated upward from the bottom of the bed. intersect the interface. T h e basic question is what concentration \vi11 be propagated upward and with what velocity. It is assumed that immediately following the start of a batch settling test. higher concentrations are formed a t the bottom of the vessel. If no intermediate concentrations are formed, the solids will undergo a step change in concentration to that in the final fixed bed. For this case, the total time for the slurry to settle from the initial height, zi, to the final height, z , , is given by
t, =
-2,/6*
= 2d ( U i
- 6*)
(3)
\vhere 6* is calculated from Equation 2, in which e, is the initial concentration, Sa is the corresponding flux, c b is the final concentration of the fixed bed, and So= 0. Here, the velocity of fall of the fluid-slurry interface is constant during the entire batch settling test--i.e., t , = t,, where tcris the length of
the constant-rate period. If, for a given flux plot such as Figure 1, a chord connecting the initial and final concentrations does not intersect the flux curve? the slurry !vi11 undergo a step change in concentration from the initial to the final value, A more complicated situation is that in which a stable concentration gradient is formed a t the bottom of the bed and propagated upward. If (dS-Lvith increasing 2 , . Step change in concentration from e , to c m . l‘otal settling time decreases as Concentration increases.
For the singly concave flux plot (Figure 2) two ranges of initial concentration exist in which the batch settling curves have distinctly different forms, For all concentrations within a region the hatch settling curves have the same general shape. Region 1 is the range of concentration less than the concentration a t the inflection point of the flux curve: C Q . Region 2 is the range of concentration equal to or greater than this concentration. Predicted batch settling curves for selected concrntrations in each raiige are shown in Figure 4. I n Figure 4, the curve for c i = L : ! is central to the set. T h e first parts of the other curves appear as a spray of lines from this central concrntration. T h e batch settling curves predicted from the singly concave flux plots are very simple compared to those predicted from the doubly concave flux curve. These settling curves will have the following characteristics for a constant weight of solids :
I N REGION 1 ci < c2. Fast initial constant settling rates. T h e rate decreases monotonically as c, increases. ‘Time of the constant rate period decreases as the initial concentration increases. Step change in concentration from c, to a concentration greater than cz as a concentration gradient is propagated uphvard from the bottom of the bed. Sharp break in z-t curve when the gradient reaches the fluid-slurry interface. Magnitude of the finite concentration change decreases with increasing initial concentration, so that the sharpness of the break in the r-t curve decreases. Gradual decrease in the settling velocity as the gradient is “propagated through the fluid-slurry interface.” Infinite total settling time if (dS/dc), = 0 a t S = 0. ~
I N REGION 2: c, 2 C Z . Slokver initial constant settling rates than for Region 1. Length of constant rate period increases monotonically with increasing c,. Concentration a t the fluid-slurry interface changes continuously from c, to c a . Infinite total settling time if (dS,’bc), = 0 a t S = 0 . Experimentally it may be difficult to determine whether or not the flux plot. determined by measuring the initial settling velocities for slurries of initially uniform concentration over the entire concentration range for a given weight of solids, is singly or doubly concave. However. if the complete batch settling curve for each initial concentration is determined, the distinction can be made on the basis of the shape of the hatch settling curves. I n addition, the accurate determination of the complete batch settling curves should give a great deal of insight into the basic mechanism of the settling process ,411 of the above discussion is predicated on the assumption of an initially uniform concentration and no particle size
X
Temperature.76- F d,:67 m i c r o n s
~-
0
02
-
04
REDUCED SOLIDS. CONCENTRATION,( I - € )
Figure 5. Flux plot determined for micron spherical glass beads in water
67-
segregation during settling. If these conditions are not fulfilled, the experimental batch settling curves will have the same basic shape as those described above, but one would expect the experimental constant settling rate period to be shorter than that predicted on the basis of ideal slurry behavior. and the breaks in the z-t curve to be less sharp. (This is not the only possible explanation if this behavior is observed.) If a known initial concentration gradient is present in the slurry, the batch settling curves may be predicted from the theory, but the calculations will be more complex . theory is whether T h e test of the basic assumption. u = ~ ( c ) and or not one can predict from the flux plot, based on initial settling rate data. the complete batch settling curve for an initial concentration. It is the measurement and analysis of the settling curves after the initial constant rate period which provide a test of the theory and give insight into the basic fluid-solids dynamics of the system. Experimental Batch Settling Curves
T h e initial settling velocity as a function of initial concentration and the resulting flux plot for Set X I 1 (66.9-micron spherical glass beads in water at 76’ F., 11.2 grams per s q . cm.) were presented previously ( 9 ) . T h e solids flux plot for Set X I I . shown in Figure 5, was doubly concave and hence corresponds to the general form shown in Figure 1. Thus: assuming “ideal” slurry behavior, the batch settling curves for rigid spheres are predicted by the theory to have the characteristics shown in Figure 3 and described above. T h e averaged height and time readings: obtained from observations of the fall of the fluid-slurry interface and the rise of the final solids concentration, are given by Stroupe ( 7 1 ) . These are the averages of three or more runs for each initial concentration in each set. Figure 6 is the family of batch settling curves obtained by plotting these points for Sct S I I . Figure 7 is a n expansion of the curve for the latter portion of the settling period. Figure 8 shows the rise of the final solids concentration for all initial concentrations of Set XII. T h e curves shown in Figures 6. 7. and 8 were drawn by eye through the experimental points. T h e corresponding curves for Set X (67 microns, 33.6 grams per s q . cm.). which \yere very similar t o those for Set XII. are given by Stroupe ( I I ) . D r H a a ~( 3 ) presents the results for Set XXI (22.5 grams per s q . c m . ) . \vhich are in very close agreement Lvith those of Set X I I . VOL. 3
NO. 3
AUGUST
1964
253
6
-
245 p/CC
d e . 67)
,w
I
I1
- I I P v43qcrn.
2 3 4 6 6
0
Figure
I
1
30
60
6.
I
I
90 120 T I M E , (8e.c.)
150
I
I35
7
040 046
e
066
I50
165
180
Figure 7. Expanded view of batch settling behavior of 67-micron spherical glass beads in water
Experimental batch settling behavior
For doubly concave flux plots, the batch settling curves fall into four distinct types depending upon the limits between which the initial concentration falls. The results are discussed by regions for this reason. T h e critical values defining these regions of Sets X, X I , XII, and XXI are given in Table I. These values were calculated from the correlation of initial rates in terms of a power series in initial concentrations. Region 1. Very hazy fluid-slurry interfaces caused by particle size segregation and inability to mix to a uniform initial concentration made it impossible to obtain readings of the fall of the interface in this region. Therefore. it was impossible to determine the actual shape of the settling curves for slurries in this range of concentrations. However, this can be experimentally determined if the solids are very closely sized ( 6 , 73). Verhoeven (73) found that very dilute slurries settled at a constant rate until settling was complete, as expected from the theory. For the reduced concentrations of 0.10, 0.075, and 0.05 of Set XII, observations were made of the rise of the fixed-bed region from the bottom of the bed. T h e height-time readings
I20
036
T I M E . (sec.1
of 67-micron spherical glass beads in water
Qualitative Comparison of Experimental and Predicted Settling Behavior
105
%O
180
020 025 030
for these concentrations, plotted in Figure 8, clearly demonstrate an increasing total settling time as the concentration decreased. This behavior is predicted by the doubly concave flux plot but can be explained by the singly concave flux curve only if it has However, the experimental a strongly negative slope a t 6., height-time curves show a marked departure from a linear rise. In this reduced concentration range, both particle segregation and the presence of an initial concentration gradient with higher concentrations near the bottom of the vessel would cause the fixed bed portion to grow faster than predicted. The increasing deviation from linearity with decreasing concentration is believed to be caused by the increasing effect of particle segregation. The deviation was especially pronounced in thc latter portion of the settling period. However, without knowing the actual initial concentration gradients, if any, within the bed it was not possible to isolate these two effects. Addii ional studies with very closely sized particles are required. Region 2. The experimental batch settling curves of this region were very similar to those predicted from the doubly concave flux plot. T h e characteristics common to both the
-
de = 67 )I 2.45 g./CC.
pa w
11.2 g.hq.crn. T e m p e r a t u r e =76'F.
4
Table I. Critical Reduced Concentrations of Experirnenta I Flux Plots for 67-Micron Spherical Glass Beads
0 485 0 641 33 6 0 488 \O 635 10 9 XI 0 120 0 325, 0 542 0 497 (1 642 11 2 XI1 0 130 0 339, 0 545 0 469 0 641 22 5 XXI 0 111 0 311, 0 534 a Reduced concentration at intersection of flux curve with, chord from c m which is tangent toflux curve. Inflectionpoints. Reduced concentration at point of tangency toflux curve. d Final or maximum reducedconcentration. X ._
254
0 097
0 311. 0 546
l&EC FUNDAMENTALS
w
-3
r
I
eo
1
120
I
180
I
TIME,(sGC.)
Figure 8. Rise of fixed bed during batch settling of 67micron spherical glass beads in water
b
0
Figure 9. Photographs of concentration gradient during botch settling of 67-micron spherical glass b e a d s in w a t e r Initio1 solid. concentration ( 1
predicted and experimental curves were a n initial constant settling rate changing discontinuously to a higher concentration, cT, followed by a decreasing rate period with a finite settling velocity to the final bed height. A concentration gradient was visible as a well defined band directly above the final packed bed as shown in Figure 9. This hand expanded in thickness as it moved u p the slurry. T h e predicted concentrations a t the upper and lower interface are + and ca, respectively. A 16-mm. film was made showing the rising gradient and its subsequent collapse into the packed bed. T h e presence of this concentration gradient during hatch settling of rigid spheres apparently has not been observed previously. Yet it is the intersection of this rising gradient with the slurry-fluid interface which accounts for the nonlinearity of batch settling curves after the initial constant-rate period. T h e total settling times for all slurries in this range of concentration were essentially the same, as can be seen from the settling curves of Figure 6 or the rise of the packed bed in Figure 8. This is explained by the theory for the doubly concave flux plot, in that slurries in this initial concentration range must always pass through a region of concentration 68 immediately before reaching the final concentration. A singly )~ concave flux plot with a negative value of ( ? l S / ? l ~evaluated a t cm would also explain the constant settling time but could not account for the second discontinuity nor the linear behavior of Region 4. T h e experimental curves of this region differ slightly from those predicted by the doubly concave flux plat in two respects. The experimental I-t curves show a slight deviation from linearity immediately before the first discontinuity. Also, Figure 7 shows a slight banding (for reduced concentrations less than 0.3) as the slurry approached the final bed height. This band was only a few millimeters wide and hence no definite conclusions may b e reached. Region 3. T h e experimental curves are identical in form to those predicted by the doubly concave flux plot. There was a constant rate period, followed by a period of gradually decreasing settling velocity. T h e first discontinuity in the z-1 curves disappeared and there was a decreasing rate period followed by a discontinuous change in the settling curve as the final concentration was reached by a step change from 6 1 to 6 , . As in Region 2, the concentrations of this region must reach the final concentration by first passing through an interface concentration, ca. Therefore, the settling times far all con-
-
e,)
= 0.15
centrations in Region 3 are predicted to be equal to those for Region 2, as was found experimentally. Region 4. This region is very important because of the very significant differences between settling curves predicted for the two types of flux plots. T h e singly concave flux plot shown in Figure 2 predicts a continuously decreasing settling rate with an infinite time required to reach the final concentration. T h e doubly concave flux plot predicts a slow constant settling rate followed by a step change to the final concentration, as was found experimentally. T h e constant settling rate observed during the entire hatch settling period confirms the existence of the second concave region of the flux plot. T h e settling velocity of the slurries decreased with increasing concentration. However, as the concentration increased, the velocity of rise of the final concentration increased; hence, the settling time decreased as predicted. Quantitative Comparison of Experimental and Predicted Settling Behavior
T h e predicted batch settling curves and the experimental d a t a are shpwn in Figures 10 and 11. T h e curves shown in Figures 5, 10, and 11 were calculated from the equation u(cm./sec.) = 0.338433 1.37672 (1 - e)
+ 1.62275 (1 - e)* +
0.11264 (1 -
- 0.902235 (1 -
e)‘
(10)
T h e agreement is excellent for all concentrations. I n some cases, a straight line through the experimental points did not extrapolate to exactly the initial height, so that the experimental points lie very slightly below the theoretical line. T h e detailed calculation of that part of the batch settling curves following the constant rate period is given in the Appendix. Initial concentrations from 0.15 to 0.45 correspond to Regions 2 and 3 of Figure 1. For all initial concentrations in these regions, a concentration gradient is formed a t the bottom of the bed and is propagated upward immediately above the rising fixed bed. Since the solids flux curve is doubly concave, continuity requires that the rate of rise of the discontinuity between the fixed bed a t concentration to, and the concentration a t the bottom of the gradient, c3, must equal the raie of rise of concentration ca within the gradient. This is expressed mathematically in Equation 4. All initial concentrations within Regions 2 and 3 result in the formation of a conceptraVOL. 3
NO. 3
AUGUST
1964
255
pa .2
45
o/cc
Temp - 7 e 0 ~ -
-w
-112 oQpcm
6YMBOL
(I-€)
0 15 0 25 0 35 0 45 I
30
\
,
i
I 120
eo
60
\
I
150
30
TIME,(sec.)
Figure 10. Predicted total batch settling curves for 67-micron spherical glass beads in water
I
eo eo T I M E , (sec.)
120
I50
180
Figure 1 1 . Predicted total batch settling curves for 67-micron spherical glass beads in water
20
tion gradient in which the concentration at the lower boundary of the gradient is 63. Since c 3 and c, uniquely determine the total settling time, it is independent of c I in these regions. 4 s the reduced concentration a t the point of tangency of the chord from c m is 0.50 (Table I ) , the total settling time for this initial concentration is the same as that for the initial concentrations of Regions 2 and 3. Thus, in Table 11: the calculated settling time for initial reduced concentrations of 0.15 to 0.50 is constant. The experimental settling time is also constant and in close agreement with the calculated value. T h e calculated value for 0.55 is appreciably less and here too the experimental and calculated values check. Considering all the values in Table 11, the experimental total settling times agree to 1.27, with the calculated times-i.e., ( l / S ) Z [(tealcc*. t e x p t l . ) / t e x p t * . 1= 2 (0.012)2.
Table II.
Calculated and Experimental Settling Times and Heights Set XII.
Initial Reduced
Concn., 1
- ~i
0.55 0.50 0.45 0.40 0 35 0 30 0 25 0 20 0 15 a
256
~
67-micron spherical glass beads in water
Time,Sharp Break from ct, tcT.Sec. Calcd. Exptl.
Height, Sharp Break from ci, I,,,
Calcd.
147 155
...
... . .
...
115
8 8 7 7
109 113 122 141
116 128 144
Time,
Cm. ~
146 154 . .
Total Settling
7.3 7.3
Exptl.
7.2 7.3 ... .
3
.
2 9
8 2 8 3 7 9
5
7 5
Based on rise ofj'xed bed (extrapolated to 7.3 cm.).
l&EC FUNDAMENTALS
t,, Sec. Calcd.
146 154 154 154 154 154 154 154 154
147 155 152 154 154 153 156 158 153
- 6 1 ) i 0. I 5 2 1 = 62.9c m . 1
(I
I
p s = 2.450.ICC. d,.
I
I
67p
TEMP. z 7 7 . F
TIME,(sec.)
Figure 12. Dimensions and position of concentration gradient during batch settling of 67-micron spherical glass beads in water
Table I1 also gives the height and time at the end of the constant rate period. The calculated and experimental times and heights of the first break agree within 3.5 and 170. respectivelv. T h e calculated values in Table I1 \\;ere obtained from the flux plot determined for Set XI1 using Equations 5. 6, and 7. .4 further check on the theory results from the prediction of the formation of a concentration gradient during settling tests l+ith initial uniform reduced concentrations from 0.1 5 to 0.45, .4s the position of the top of the gradient can be observed only if there is a n appreciable discontinuity there, experimental confirmation \vas sought Lvith (1 - e,) = 0.15. Not only was the concentration gradient observed (Figure 9 ) , but its dimensions and position were in complete accord with predictions. as can be seen in Figure 12.
Final Solids Concentration, Figures 6 and 7 show that the final bed height of Set X I 1 varied slightly as the concentration varied. This variation was not systematic with concentration variations and appeared to be a random experimental effect. Any small vibration of the column caused the bed to assume a more closely packed configuration. T h e final solids concrntration of (1 - E ) = 0.64 was based on the 105 tests summarized in Table 111
Table 111. Sets X, XI, and XII.
Final Solids Concentration 67-micron spherical glass beads in woter
LVt. of
Solids, Set
Grams
x
59- 7
XI XI1
193 5 199 2
‘VO. of Tests
.Mean ( 7 - em)
35 27 43
0 64131 0 63596 0 64202
Std. Deb.
0 00502 0 00569
0 00402
These results are in close agreement with each other and with results obtained by DeHaas ( 3 )with 400.0 grams of solids. T h e rate of rise of the packed bed for Set XI1 was calculated using a best fir equation which led to a value of (1 - e,) = 0.6416. T h e experimental and theoretical results are in very close agreement. indicating that the results are internally consistent, Inasmuch as a very small error in (1 - e,) would lead to a large error in 6, *: the rate of rise of the packed bed. this is a very good check. Benenati and Brosilow (2) report (1 - E , ) = 0.61 for uniform spheres randomly packed. If the true value with the present size distribution had been would have 0.61 instead of 0.64. the calculated value for a,* been 50% higher than the experimental in the case of a slurry of (1 - e l ) = 0.55. Discussion
Continuity of Flux Curve at High Solids Concentrations. Although the flux plot is shoivn in Figure 5 as a continuous c., curve, it is conceivable that it is discontinuous as c If only fluid-particle interaction occurs for all initial solids concentrations less than the packed bed. there may indeed be a discontinuity. O n the other hand, if interparticle contacts occur throughout the slurry, the curve may be continuous and fall below that based on the hydrodynamic action of the fluid. Verhoeven (73) has shown that a single large particle is “boxed in” by small ones and forced to travel a t their speed \.\.hen the solids concentration is very high. There is some evidence ( 7 3 ) of interparticle contacrs in very- closely sized systems: but it is not conclusive. The question can be resolved only by future experiments in the range 0.55 < (1 - e l ) < 0.64. A s stated earlier, unless (@SI b ~ *2) ~0 for c b < G 5 T , . a finite concentration change will occur a the lower boundary of the gradient. This indicates that the discontinuous curve is effe(:tively doubly concave. Hence, the same analysis applies to both the continuous and discontinuous curves, except that for the continuous case lim = dS;dcic,. whcreas in
-
c-c,
the discontinuous case lim AS;‘Ac
= - m.
Expressed another
c -+L,
way. rhe continuous curve specifies a finite setrling time for c, = ( c d - dc) lvhile the discontinuous case calls for an infinitesimal settling time. However, inertial and end elfects have to be considered Lvhen t , -+ 0. Particle Segregation. If particle segregation occurred to a n appreciable extent, the settling velocitl, of the solids tyould no longer be a funcrion of the local solids concentration only.
I t would also be a function of the particle size and size distribution. I n a qualitative argument, it can be said that if particle segregation does occur, the larger and! or more dense particles wi 1 settle more rapidly through the slurry, leaving the smaller and ’or lighter solids behind. However, the solids concentration will be reduced, and this will increase the settling rate of the smaller a n d i o r lighter solids. ,For slurries of reduced initial concentration of 0.30 to 0.55 the’ slurry-hvater interface was distinct. This indicated that particle size segregation did not occur for slurries of this concentration range. If particle size segregation were to occur to an appreciable extent in the upper portion of the bed of initial uniform concentration, it would also occur at thc slurrywater interface and cause a n increasingly hazy interface as settling proceeded. This was observed during the initial settling period for slurries of reduced initial concentration of 0.25 and less. For slurries of reduced initial concentrations from 0.15 to 0.25, the slurry-water interface became very distinct ar the first sharp break in the settling curve and remained distinct thereafter. After the first break in the z-t curve, the concentration a t the interface was observed to be intermediate between the initial and final concentrations. Thus, the clearing u p of the interface was caused by the rapidly settling dilute solids entering the slower-settling zone of intermediate concentration Lvhich had been propagated u p to the interface from the bottom of the bed. This effect is clearly shown in the photographic sequence of Figure 9. Orientation of Particles. Adler and Happel ( 7 ) attribute differences in U;U, values to “agglomeration and segregational effects“ and state that the applicatLon of a proper orientation factor should unify all solid-liquid data. Theoretical calculations by Stimson and Jeffrey (70) and experimental work by Happel and Pfeffer ( 4 )indicate reduced viscous drag when one sphere follows another. T h e theoretical treatment by Richardson and Zaki (7) also indicates the importance of orientation. However, if for a given slurry and method of dispersion, a given concentration has a given orientation-i.e., orientation is a function of local solids concentration only-the effect of particle orientation \vi11 automatically be incorporated in the analysis in the experimental determination of the solids flux relationship based on initial rate data. T h e agreement between theory and experiment indicates that the local solids concentration was indeed the important variable. As noted above, the total settling time was the same for all initial concentrations from 0.15 t o 0.50: inclusive, showing that the settling velocity a t a local solids concentration (1 - e) = 0.50 )vas the same regardless of the initial concentration, and that the settling velocity was independent of the concentration gradient. Also, the agreement between the predicted (based on initial rate d a t a only) and experimental times to the sharp break from the initial concentration indicates that orientation \vas not important at solids concentrations greater than 0.35. Either that, or else the preferred orientation was formed almost instantaneously and was itself a function only of local solids concentration for a given slurry. Further. in the curved portion of a settling curve: the solids concentrntion and hence the Orientation change continuously, yer the settling rates \.\.ere essentially the same as the initial rate> for these concentrations. This seems to be adequate Justificarion for not introducing a n Orientation factor a t this stage of development. I n actual slurries, the initial concentration is uniform only iii a sratisrical sense (73). Individual particles d o not setrle uniformly a t the mean rate but are sometimes accelerated by VOL. 3
NO. 3
AUGUST
1964
257
the close proximity of other particles and sometimes held stationary in upward-flowing liquid ( 6 , 1 3 ) . Despite these local irregularities. the durry-supernate interface falls uniformly a t a reproducible rate. This transient differential s e t t m g very likely results in a more rapid over-all rate than that which would be obtained with a steady orientation of particles such that a1 particles settle a t the same velocity. Acceleration Effects. T h e theoretical curves in Figures 10 and 11 were calculated by assuming that the effect of acceleration forces could be ignored. At the sharp break from the initial settling velocity, the particles are decelerated rapidly. Further deceleration occurs in the curved portion of the experimental settling curves. T h e agreement between the theoretical and experimental values indicates that a momentum balance is unnecessary for the system studied. Thus, the analysis presented should also be valid for slo\versettling systems. Iktermining the complete settling behavior for a slurry of closely sized, rigid, 67-inicron spheres is primarily a matter of finding the velocity-concentration relationship. T h e analysis of systems of compressible solids and the use of the batch velocity-concentration data to explain and predict the behavior of continuous thickeners w 11 be given in subsequent papers.
t Figure 13. Rationale of method for computing curved section of settling curves
plot. provided that u can be represented by a fourth-order polynomial in concentration, ‘The initial settling velocity of a slurry of uniform. rigid spheres can be expressed ( 9 ) as n
A,(1 -
u =
€)’
J=O
The reduced solids flux ( 9 ) for batch settling is n
Conclusions
The experimental batch settling behavior for rigid spheres in water was found to be in complete qualitative and very close quantitative agreement with the behavior predicted from the solids flux plot by assuming that the settling velocity of the solids is a function of local solids concentration only. The small observed deviations are believed to be due to slightly nonuniform initial concentrations and particle segregation effects a t dilute concentrations. The settling behavior a t intermediate and high initial concentrations confirmed that the solids Nux plot for rigid spheres is effectively doubly concave and justified the assumption of uniform particle size. A rising concentration gradient immediately above the fixed bed was observed for initially uniform slurries of intermrdiate concentrations. The intersection of this gradient with the fluid-slurry interface accounts for the nonlinear settling curves \\hen 0.15 5 (1 - e t ) 5 0.45.
As shown in Figure 5%the flux plot for rigid spheres is doubly concave. so that there are four concentration regions and three distinct tvpes of settling curves. The critical concentrations, cl. c2, c3. and c,. marking the divisions between regions are calculated first. The maximum reduced solids concentration, (1 - e,), is obtained from 4,(1 - ea)’
=
j=O
+I =
0
(‘43)
T h e final height, z,? is calculated from the fact that p,(l E,)L, is the mass of solidstper unit area. Reduced concentrations (1 - e l ) and (1 - € 3 ) are given by the equations
-SI :Ps -~ (1 - e , ) - (1 - e , ) (1
~
Acknowledgment
Because of Equation A2
An XEC fellowship (E.P.S.) and the financial support of the Purdue School of Chemical Engineering are gratefully acknowledged.
-
-
-s3/1ps - E,)
- (1 -
€3)
n
A,(1
-
€3)7+1
=
j=O
Appendix
Calculation of Batch Settling Curves from Initial Rate Data Expressed as a Power Series in Reduced Concentration. .4unique flux plot may be determined froni the initial settling rates of sliirries if tlie settling rate is a function of the local solids concentration only. ‘This flux plot may then be used to determine the complete batch settling curves for all concentrations b y calculating the movement of planes of constant concentration. An eqiiation relating the initial settling velocity. u . to the solids concentration. c. is necessary to perform the calculations. T h r method is illustrated by presenting the calculations for initially uniform slurries of glass spheres in \vatex-. but the basic approach is applicable to any flux plot and the calculation of batch curves is done in essentially the same \Yay. T h e computer program given by DeHaas ( 3 ) can be applied directly to any singly or doubly concave flux 258
l&EC
FUNDAMENTALS
[(I
1) A , (1
-
6,)
- (1
-
1) ~4j-11(1
-
€3)’
= 0
]=I
(‘46) Since (1 - E,) and the coefficients A , are known, Equation A6 is simply a n equation in (1 - € 3 ) which is easily solved numeiically using the Xewton-Raphson technique. LVhen (1 - E ? ) is known. (1 - e l ) is easily found. The reduced concentration, ( 1 - e ? ) , is the value a t the first inflection point, so
2
j-0
(j
+ 1) jAj(1 -
4 7 - 1
=
0
(A?)
If the initial solids concentration is constant or decreases Lvith height, it is possible to calculate the complete settling curve. T h e latter case is more complicated a n d is not dealt \vith here. I t has been given a brief qualitative treatment by Kynch (5),and is discussed under “Predicted Batch Settling Curves” in the present paper. For initial, uniform concentrations in the ranges 0 < (1 - E ~ (1 - E ~ ) ,and (1 - € 3 ) 5 (1 - e t ) < (1 - E - ) > Regions 1 and 4 in Figure 1, the settling curve is a straight line with a sharp break a t the final height, This type of settling curve is designated Type 1. T h e total settling time, t , , is given by
where 6 ,
*
=
velocity of discontinuity =
T h e curves for (1 - E % ) = (1 - el) a n d (1 - E , ) = ( 1 - € 3 ) are especially important because the total settling time in these two cases is the same as that for Types I1 a n d 111. For initial concentrations in the range (1 - e l ) < (1 - e t ) < ( 1 - e l ) the settling curve is initially a straight line. A sharp break occurs when concentration (1 - e,) reaches the interface. Graphically, the curve joins that for (1 - E ~ )= (1 - E ? ) . Thereafter, the settling rate decreases monotonically until the end of the settling period. This is a Type I1 curve. T h e time, t c r , a n d height. z c r , of the break in the settling curve are found from the fact that
give the time a n d height a t which the first concentration change occurs a t the slurry-fluid interface and hence the first change in velocity takes place. However, these are both infinitesimal changes. For a n initial concentration of (1 - 62): the remainder of the settling curve [until (1 - € 3 ) is reached] is determined from a numerical solution of the equations )
and
T h e significance of these equations can be seen from Figure 13. T h e quantities in the curved section are denoted by a prime (’) to avoid confusion of subscripts with those previously used. Because u varies with (1 - E ) , the solution of Equations A16 and A17 requires an iterative procedure. T h e starting point for the calculation is z?. t2, (1 - € 2 ) . For m = 1, t’, = t2, ( 1 - e’,n) = (1 - E * ) , a n d z’, = z?. Subsequent values are obtained from the equations below by setting m = 2, 3, 4: . . , . until concentration (1 - € 3 ) is reached. A first estimate of z’, + 1 can be obtained from =
Z’,+l
z’, - u ’ , (t‘,+1 - t’m)
(A1 8)
I n view of Equation ‘42, Equation A17 becomes
If a small section of the settling curve is taken, it can be approximated by the parabola or 2’ =
z’,
+ a ( t ’ - t’,) + b ( t ’ - t’,)*
(A201
Then and
where or
This equation is solved for (1 - c b ) by the method used for Equation A4. T h e remainder of the curve is discussed under Type 111. Initial concentrations in the range ( 1 - € 2 ) 5 ( 1 - e t ) < ( 1 - e 3 ) lead to Type I11 curves. T h e only case which needs be worked out in detail is that for a n initial concentration of (1 - E ? ) , the inflection point of Figure 1. I n all cases. however, there is a slight decrease in rate as the interface d(l - E). concentration increases from (1 - e t ) t o (1 - e t ) T h e equations
+
z’,+~
= z’,
-
I -
2
(u’,
+ u’,+I)
(t’,+l
- t’,)
(A422)
T h e value of z’,+l obtained from Equation A18 is used in Equation A19 to obtain an estimate of (1 - E ’ , -1) a n d hence u’, 1. T h e latter is used in Equation .422 to obtain a new estimate of z’, T h e solution converges very rapidly. T h e time increments are continued until the solids concentration reaches (1 - E A ) ? a t which there is a sharp break to a concentration of (1 - E - ) . This point (t,, ze) has already been located (Equation A8). For concentrations benveen (1 - e * ) and (1 - € 3 ) . Equations A14 a n d A15 indicate where the settling curves join that for (1 - e ? ) . Nomenclature
c c’p,
and
= = =
= =
=
where
=
= =
solids concentration, grams/cc. reduced concentration = (1 - E ) , dimensionless particle diameter. microns batch solids flux = cu, g r a m d s q . cm.-sec. time. seconds doivnward velocity of solids relative to slurry. cm./ second Stokes velocity. cm.,’second lveight of solids per unit cross-sectional area, grams,’ sq. c m . height from bottom of vessel, cm VOL. 3
NO. 3
AUGUST
1964
259
of plane of constant concentration, cm. 'second = propagation velocity of concentration discontinuity, cm./'second = partial differential operator = voidage, dimensionless = solids density, grams/cc. = propagation velocity
/3
6
b t
p8
literature Cited
(1) Adler, I. L., Happel, J., Cheni. Eng. Progr. Symp. Ser. 58, KO. 35, 98 (1962). (2) Benenati, K. F.: Brosilow, C . B., A.I.Ch.E. J . 8, 359 (1962). (3) DeHaas: K.D.. M S .thesis, Purdue University, 1963. (4) Happel, ,J,: Pfeffer, K.! d.I.Ch.E. J . 6 , 129 (1960). (5) Kynch, G. J., Trans. Faraday SOC.48, 166 (1952). (6) Oliver, D. R., Chem. Eng. Sci. 15, 230 (1961). (7) Richardson. J . F., Zaki, \V.N.,Ibid.. 3, 65 (1954). (8) Shannon, P. 'r.. DeHaas, R. D., Tory, E. M . , Chemical Engineering Symposium. Division of Industrial and Engineering Chemistry, ACS, University of Maryland, College Park, Md., NO\.. 14-15. 1963. (9) Shannon. P. T.. Strouue. E. P.. Torv. E. M.. IND. E s c . CHEM.FUSDAMENT'.UA 2, i o 3 (1963): (10) Stimson, M.. Jeffrey, G. B.,Proc. Roy. Soc. A 111, 110 (1926). (11) Stroupe, E. P., h1.S.. thesis, Purdue University, 1962. (12) Tory, E. M., Ph.D. thesis, Purdue University, 1961. (13) Verhoeven, J., B. Eng. thesis, McMaster University, 1963.
SUBSCRIPTS a = above concentration discontinuity 6 = below concentration discontinuity cr = pertaining to constant rate period i = initial value-i.e., a t time zero t = tangent point on flux curve of chord from initial flux 1 = intersection of flux curve and chord from final flux tangent to curve 2 = a t first inflection point of flux curve 3 = tangent point on flux curve of chord from final flux m = final value-i.e., a t infinite time
\
/
,
I
RECEIVED for review November 9, 1962 RESUBMITTED h-ovember 5, 1963 ACCEPTED March 2, 1964
SUPERSCRIPT * = pertaining to batch settling ( u = 0)
HOMOGENEOUS FLUIDIZATION E
.
R UC KENST E I N
,
Polytechnical Institute, Bucharest, Romania
A physical model is proposed for a homogeneous fluidized bed. An equation of motion is established for one of an ensemble of particles in interaction with a fluid. By means of this equation an expression is established for the variance, ut2, of the void fraction, a quantity connected with the mixing process taking place in the fluidizing agent. An expression for the axial diffusion coefficient is established starting from the model suggested here and using the equation derived for ue. This equation i s in satisfactory agreement with experimental results obtained b y Kramers.
M
a fluidized bed behaves a t low velocity like a liquid, a t somewhat greater velocities like a liquid containing gas bubbles, and finally, a t high velocities, like a boiling liquid. T h e first case corresponds to homogeneous fluidization. I n the last two cases part of the fluid travels through the bed as bubbles (8, 72, 20, 24, 35, 37). If the flow rate of that part of the gas which travels through the bed as bubbles does not increase along the height of the bed (36), the bed behaves like a liquid traversed by gas bubbles; if it increases ( 3 ) ,the bed behaves like a boiling liquid. An attempt has recently been made (28) to explain the appearance of these structures. T h e macroscopic analogy between a homogeneous fluidized bed and a liquid has led Furukawa and Ohmae (70) to propose a "microscopic" theory of the fluidized bed, based on an analogy with the microscopic theories of the liquid state. IVithin this theory the attractive forces bet\veen molecules have as a counterpart the weight of the solid particles, while the repulsive forces correspond to the forces of friction between the fluid and particles. Assuming that the volume of the bed is pulsating. they express the potential of these forces as a function of rhis volume. The transition from global to microscopic is made by equating the hydraulic diameter to the distance bet\\.een rlvo particles. This assumption yields a relation bet\\.een the void fraction and the distance between two parrides. ACROSCOPICALLY,
260
I&EC FUNDAMENTALS
Furukawa and Ohmae assume further that the solid particles are subject to a harmonic oscillatory motion; they then establish an equation for the mean kinetic energy of the particles and consider that this quantity represents in a fluidized bed Xvhat temperature represents for a liquid. .\ similar idea has been suggested by Todes ( 3 4 ) . By using the mean kinetic energy of the solid particles for the temperature in a number of equations valid for the physical constants of a liquid, FurukaLva and Ohmae have obtained equations for certain physical properties of the homogeneous fluidized bed-e.g., viscosity and surface tension. T h e aim of the present paper is to obtain information as to the behavior of a homogeneous fluidized bed without using any analogy, but starting with a certain model and an equation of motion for a particle which is one of an ensemble of particles interacting with a fluid. The model as well as the equation of motion is used to study axial mixing in the fluidizing agent. The Physical Model
The bed becomes fluidized when the pressure drop through the fixed bed multiplied by the cross section of the fluidization column equals the Lveight of the solid particles. For fluid velocities higher than rhe minimum fluidization velocity the bed expands and solid particle motion sets in. The difference in behavior of a fixed and a fluidized bed is due to this motion
