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A non-parametric method for particle settling velocity determination in a slurry bubble column
The Chemical Engineering
Jownal,
54 (1994)
1
l-6
A non-parametric method for particle settling velocity determination a slurry bubble column MkuT. Western
in
Ityokumbul* Research
Centre,
CHIVMET, 1 Oil Patch Drive,
PO Bag 1280, Devon, Alb. TOC lE0
(Canada)
(Received June 27, 1991; in final form May 1, 1993)
Abstract The use of an EisenthabCornish-Bowden-type plot to determine the particle settling velocity in a continuous slurry bubble column under steady state conditions is presented. The procedure permits the identification of improperly run experiments, since the plots do not pass through the common intersection point. In the range 0.4 < Re, < 500 the particle settling velocity is given by V,, = [O.Ol 7ESg2@,- h)2/~#%,. It is shown that the solid dispersion coefficient cannot be determined from a steady state experiment. All dispersion data determined in supposedly steady state experiments are therefore meaningless.
1. Introduction Three-phase (slurry) bubble columns are employed in such diverse applications as coal liquefaction and recovery of fine mineral particles by flotation. In all these applications knowledge of the flow regime and the particle residence time is important, because the conversion or achievable recovery depends on both factors. In spite of the variety of these applications, the design and scaleup of shrrry bubble columns remain a difficult task. The problem areas are in the estimation of nonadjustable parameters (e.g. phase hold-ups, mixing coefficients, etc.). For an accurate estimation of the particle settling velocity and dispersion coefficient, a mathematical model of the process is often required. For particle one-dimensional sedimentadispersion the tion-dispersion model remains the most widely used [l-6]. While a considerable amount of data on solid dispersion exists, no comprehensive solid mixing theory is currently available. This is due in part to the wide variance in the reported values. As with liquid dispersion studies, thii variance arises from, among other things, inadequacies in experimental methods and the application of wrong boundary conditions [ 7, 81. Experimental errors are by far the most difficult to identify from reported data. The main purpose
of the present paper is to provide a non-parametric method for the determination of the particle settling velocity in a slurry bubble column from steady state measurements which allows for ready identification of wrong data points. It is also shown that it is impossible to determine both the particle settling velocity and the solid dispersion coefficient from the continuous procedure under steady state conditions.
*Present address: 115 MineralSciences Building, The Pennsylvania State University, University Park, PA 16802, USA.
0923-0467/94/$07.00 0 1994 Elsevier Sequoia. All rights reserved SSDI 0923-0467(93)02805-7
2. Theory The behaviour of solids in a three-phase slurry bubble column is frequently described using the sedimentation-dispersion model. By analogy with liquid dispersion, the flux of solids is described using a Fickian equation of the form
(1) The (+) and (-) signs in eqn. (1) refer to the countercurrent and cocurrent flow of slung and gas respectively. At steady state eqn. (1) reduces to
The solution of eqn. (2) is given by
C,=A, +A2 exp
(3)
M.T. Ityokumbul
2
/ Determinutian
Two boundary conditions are required for the evaluation ofA, and A,. The correct boundary conditions are
PI
atz=L
of particle
settling
(determined graphically) with the experimental quantity serves as a check on the accuracy of the experiments as well.
dG
==O 3. Experimental
atz=O
(4b)
where L is the column length. The assumption for the condition at z = 0 is that there is no slip in the shury fed to the cohimn, i.e. the liquid and solid have the same velocity. This is indeed a reasonable assumption for feed entering the column through small tubing. Since A2 = 0 (by application of eqn. (4a)), the value of ES cannot be determined from steady state measurement of the axial solid concentration alone. Application of the boundary condition at x= 0 results in the solution (5) Equation (5) is similar in appearance to the Michael&Menten rate expression in biochemical engineering [lo]. Accordingly, it is proposed to use the Eisenthal-Cornish-Bowden [ 111 non-statistical method for the determination of the particle settling velocity from eqn. (5). 2.1. Graphical method for the determination of particle settling velocity The use of the Eisenthal-Cornish-Bowden-type plot requires rearrangement of eqn. (5) to yield
q=c,f+
v,cs v,
(6)
and cs=cs’--
velocity
v,cS u,
(7)
for the cocurrent and countercurrent flow of gas and slurry respectively. At a given feed solid concentration the procedure involves determining the solid concentration in the reactor as a function of slurry velocity. Thus each run is represented by only two points on the graph: U, is plotted on the ordinate and C, is plotted on the abscissa. These points are joined by a straight line. When several pairs of points are plotted and connected by straight lines, the lines should intersect at the point (C,‘, VP>. As with the procedure in biochemical engineering, bad experimental points are readily identifled from their non-convergence at the common intersection point. A comparison of the value of Csf
verification
The best data currently available in the literature to test this procedure are those of Bhaga and Weber [ 121 because they satisfy the following conditions: (i) the feed solid concentration as well as the average solid concentration in the slung bubble column under cocurrent and countercurrent conditions are provided, and probably of more importance as will be shown later, (ii) the authors reported that the amount of solids in their slurry bubble column was small in comparison with that in the reservoir and thus the feed solid concentration may be assumed to be constant. This assumption is indeed reasonable, since their reservoir was 24 times the size of the slurry bubble column. With the use of the method being proposed here, this assumption may be independently verified by comparing the solid concentration at the common intersection point with the experimental quantity. It is this second assumption that is usually not valid in all other studies reported in the literature! Figure 1 shows a typical plot of the data for both the cocurrent and countercurrent flows of shn-ry and gas. In this plot the feed solid concentration and gas velocity were fixed while the shnry velocity and relative directions of flow between the shury and gas varied. Both series had the same feed particle concentration and the same particle size. In either case the plots have a common intersection point and bad data points are readily identified from their non-convergence at the feed solid concentration.
o’20 : 0.15
0.10
0.05
0.00
-0.05
0
100
200
300
C, kg/m3 slurry Fig. 1. Direct linear plot for the determination of particle settling velocity: effect of slurry velocity. Runs D3 and DD3 [ 121.
M.T. Ityokumbul
/ Determination
The convergence of the two plots at a common intersection point and the close agreement between the experimental (150 kg rne3) and predicted (155 kg mw3) values of the feed solid concentration are proof of the validity of this approach. In general there was comparatively more scatter with the countercurrent than the cocurrent plots. This may be due to magnillcation of the errors arising from the rather smaller range of slurry velocities used by these authors in the countercurrent experiments. 3.1. Eflect of gas velocity cm particle settling velocity The effect of gas velocity on the estimates of the particle settling velocity is shown in F’ig. 2. In these experiments the particle size and feed solid concentration were fixed while the gas and liquid velocities were varied. With the exception of the few bad experimental data, all the lines have a common intersection point, thus confirming that the particle settling velocity in a shnry bubble column is not a function of the gas velocity or hold-up. 3.2. Eflect of solid concenwatim on particle settling velocity The effect of feed solid concentration on the particle settling velocity determined by this method is shown in Fig. 3. In these plots the feed solid concentration was varied from 30 to 155 kg me3 while the gas velocity was kept constant. As a result of the changing feed solid concentration, the average solid concentration in the column varied from 31 to 230 kg rnb3. Since the common intersection points give approximately the same value for the particle settling velocities, the effect of average solid hold-up appears to be minimal.
of particle
settling
velocity
3
B OA
i .t: 8 5 >
0.2
e 2 u) 0.1
0.0 0
100
200
G , , kg/m3
slurry
Fig. 3. Effect of solid concentration.
Runs Dl, D2 and D3 1121.
2 r” 20
0.20
s" c z
0.10
0.20 0
100
200
G , , kg/m 3 slurry Fig. 4. Effect of particle size. Run Gl
[ 121.
3.3. Eflect of particle diameter on particle settling velocity F’igure 4 shows the plot for a particle diameter of 300 pm. For clarity, only the line corresponding to the average settling velocity for the 150 pm particle is shown. There is an almost linear dependence of the two quantities. 4. Discussion
50 =.
100 '
krJm3
IS0
slurry
Fig. 2. Effect of gas velocity on partkle settling velocity. Runs Dl, El and E2 (121.
All the trends observed with the data of Bhaga and Weber [ 121and the analytical solution obtained here are at variance with those reported by other authors.As an example, eqn. (5) predicts a constant value for the axial solid concentration at steady state while most authors report axial concentration gradients. The major differences may be summarized as follows. (i) The analytical solution for the axial solid concentration (i.e. eqn. (5)) is not a function of the solid dispersion coefficient.
4
M.T. Ityokumbd
/ Determination
(ii) The solid concentration at steady state is constant. (iii) For the range of conditions used by Bhaga and Weber [ 121 the particle settling velocity is not a function of gas velocity, liquid hold-up, etc. (iv) The particle settling velocity varies linearly with particle diameter in the intermediate range of Reynolds numbers. It will be shown here how these differences may be explained on the basis of wrong mathematical treatment of the sedimentation-dispersion model and poorly run experiments. 4.1. Analytical solution fw axial solid concentration Equation (5) is the only correct analytical solution for the axial solid concentration at steady state. To fully understand this, it should be recalled that the continuous procedure for solid input in the column is the same as the steady state method for studying liquid mixing using a step change in concentration or the F(t) curve [ 13 1. At steady state the concentration in the column is constant for all values of the Peclet number. It is thus impossible to determine the liquid dispersion coefficient from any concentration gradients in the column (since none exists) and it is instead the transient response in the system that is used for the estimation of this parameter. The fact that several authors [2-61 have reported concentration gradients in their shnry bubble columns at steady state is explained later as being due deficiencies in their experimental procedures. 4.2. Existence of axial solid concentration at steady state It is the existence of an axial solid concentration that is used in the curve-fitting exercise to determine the particle settling velocity and dispersion coefficient [3-6, 141. The question that arises is why the divergence between theory and practice? In actual fact there is no divergence - it is the experimental procedures that are deficient. As explained earlier, all these authors have used the cocurrent upflow of shit-r-yand gas with slurry recirculation. with this method and the known solid slip phenomenon, deposition of solids in the slurry bubble column will result. Naturally this will be at the expense of solids in the reservoir. Since none of the authors who have used this procedure indicate that their reservoir was continuously charged with solids (in order to maintain a constant feed solid concentration), it is obvious that the feed solid concentration will become time dependent (vari-
of particle
settling
velocity
able). This effect will be minimal if the size ratio of the reservoir to the slurry bubble column is very large as in the case of Bhaga and Weber [ 12 1. Unfortunately, some of these authors report that the reservoir was only as large as the slung bubble column [6]. When this happens, the boundary condition at z = 0 becomes a time-dependent variable. Unfortunately, the mathematical treatments by these authors all ignore this important point as if the solid concentration were constant! Thus it is this continuous drop in the amount of solids in the reservoir (and hence the changing feed conditions) that is responsible for the concentration gradient at the so-called steady state. Build-up of solids in the column naturally starts at the base of the column. Because the amount of solids in the reservoir is not enough to maintain the solid build-up and since it is impossible to redistribute the already settled solids at the base of the column over the entire column, the existence of an axial solid concentration is primarily due to improperly run experiments. For the cocurrent upflow of slurry and gas, eqn. (5) predicts a positive value when U]> VP. When the converse is true, eqn. (5) predicts a negative quantity for the solid concentration. While this is not physically possible, it merely suggests that the column acts as a sink for solids entering it. In most of these experiments the particle settling velocity is usually larger than the slung velocity. As a result the depletion of solids from the reservoir is accelerated and the so-called steady state attained in a shorter time. It should be recalled that Cova [ 1 ] who made the first mistake used a once-through flow of slurry for the cocurrent upflow of slurry and gas and reported that attainment of steady state was difllcult (required over 3 h). The fact that some authors report attainment of steady state in 30 min [4, 51 under similar operating conditions is consistent with the explanation provided here. Furthermore, in most of these studies [2-6, 141 the experimental solid concentration assumes a flatter profile with axial distance as the particle terminal velocity decreases (at a given slung velocity) or as the slurry velocity increases (for a given particle size), which is also consistent with the explanation provided here. If eqn. (5) does indeed represent the correct analytical solution for the sedimentation-dispersion model, the estimates for the solid dispersion coefficient should be very sensitive to the values of the axial solid concentration. Probably the best proof for this may be found in the work of Brian [6]. This author used the wrong analytical solution of Cova [l] and reported that the estimates for VP and ES were very sensitive to the value assumed
MT. Ityokumbul / Lletermitiia
for the solid concentration at the reactor exit. He reported that small changes in C,’ caused E, to vary by several orders of magnitude. Clearly this effect is expected, since ES does not appear in the solution for the axial solid concentration. 4.3. Variables ~ecting particle settling velocity in slurry bubble columns The results obtained here show that the particle settling velocity in a shury bubble column does not depend on the gas velocity and liquid hold-up. As explained earlier, the trends currently found in the literature are the products of wrong analytical solutions and improperly run experiments. For the 150 and 300 pm particles the particle settling velocities determined by this procedure (0.025 and 0.051 m s- ’ respectively) agreed rather well with the intercepts reported by Bhaga and Weber [ 121in their study (0.023 and 0.049 m s-l). However, there are slight differences in the theory, since the intercepts reported by Bhaga and’weber [ 121 also contain a hindered settling velocity component. As shown earlier, the particle settling velocity determined from the present method is not a function of the solid hold-up in the slurry bubble column. The linear dependence of the particle settling velocity on diameter is in agreement with the recent correlation provided by Ramachandranand Chaudhari [15] for the intermediate regime as shown below: v
@P2(PP--d
=
P
(8)
W-h
and o.0178g2@p- fi)2 1’3d
v =
(9)
P
P (
API
)
The limiting values for the application of eqns. (8) and (9) were given as Rep < 0.4 and 0.4
of particle settling velocity
5
TABLE 2. Comparison of predicted and experimental slopes for solid hold-up Slope m
Sand
Weldron 705 Weldron 505
U,=O.OOSS m s-l
U,=O.O043ms-’
Exp.
csk.
Exp.
talc.
0.33 0.20
0.33 0.20
0.26 0.16
0.26 0.15
bubble column is higher than that in a two-phase (liquid-solid system) and is reasonably well predicted by eqn. (9). 4.4. Extension to a polgdisperse particle fqy&%n The author [S] measured average solid concentrations in a 0.06 m bubble column using the countercurrentflow arrangementunder various solid feed concentrations and gas and slurry velocities. The solids used were Ottawa sand (weldron Plant designations 505 and 705) with a wide particle size distribution. Details of the experimental set-up and particle size distribution may be found elsewhere [ 8 1. For each sand the solid concentration in the reactor was plotted against the feed solid concentration. The slopes of these plots were determined and the results are shown in Table 2. The predicted slopes were estimated from eqn. (5) using the equation m=C
(_Ls_ 1 u,+vfi
(10)
where m is the slope and xif and V, are respectively the weight ratio of fraction i in the feed and its settling velocity calculated using either eqn. (8) or (9). The comparison of the predicted values with those determined experimentally is also shown in Table 2. The close agreement in the two quantities suggests that this method may be extended to a polydisperse solid system.
6. Conclusions TABLE 1. Comparison of experimental and predicted particle settling velocities in a slurry bubble column
4 (pm)
150 300
Particle settling velocity (cm s-l)
Re,
Exp.
Eqn. (8)
Liquid solid’
2.4 4.9
2.4 4.8
1.3 3.3
‘Particle settling velocity for solid-liquid
flow
3.6 14.7
[ 121.
It has been shown that the steady state method for studying particle dispersion in slurry bubble columns predicts a constant solid hold-up in the bubble column at steady state. The reported existence of a significant axial solid concentration gradient in the column at steady state by several authors is attributed to the variable nature of the feed solid concentration. It is shown that it is
6
M.T. Ityokumtrul / Determination
impossible to determine the solid dispersion coefficient from the steady state axial concentration profile. For this reason, all published data on solid dispersion coefficients which are based on this procedure are meaningless. However, the particle settling velocity may be determined from the steady state measurements using an Eisenthal-cornishBowden-type plot. As with the original plot, this method permits the ready identification of wrong experimental data, since such plots do not pass through the common intersection point. It is shown that the particle settling velocity in a slurry bubble column is well predicted by eqns. (8) and (9). It is also shown that eqn. (5) can be extended to a polydisperse particle system.
References 1 D.R. Cova, Ind. Eng. Ch.em Process Design Develop., 5 (1966) 20. 2 K. Imafuku, T. Wang, K. Koide and H. Kubota, J. Chem. Eng. Jpn., I (1968) 153. 3 Y. Kato, A. Nishiwaki, T. Fukuda and S. Tanaka, J. C;zem. Eng. Jpn., 5 (1972) 112. 4 D.N. Smith and JA. Ruether, Chem. Eng. Sci., 40 (1985) 741. 5 D.N. Smith, Jh. Ruether, Y.T. Shah and M.N. Badgqjar, AIChE J., 32 (1986) 426. 6 B.W. Brian, Hydrodynamics of a three phase bubble column, Ph.D. Thesis, Lehigh University, 1986. 7 M.T. Ityokumbul, N. Kosaric and W. Bulani, C/rem. Eng. Sci., 43 (1988) 2457. 8 M.T. Ityokumbul, Hydrodynamic study of bubble column flotation, Ph.D. Thesis, University of Western Ontario, London, 1986. 9 P.V. Danckwerts, Chem. Eng. Sci., 2 (1953) 1. 10 L. Michaelis and M.L. Menten, Biochem. Z., 49 (1913) 333.
of particle settling velocity 11 R. Eisenthal and A. Comish-Bowden, Biochem. J., 139 (1974) 715. 12 D. Bhaga and M.E. Weber, Can. J. Chem. Eng., 50 (1972) 329. 13 0. Levenspiel, ChemicalReaction Engineering, Wiley, New York, 1972, Chap. 9, pp. 253-315. 14 Y. Kato, S. Morooka, T. Kago, T. Saruwatari and S. Yang, J. Chem. Eng. Jpn., 18 (1985) 308. 15 P.A. Ramachandran and R.V. Chaudhari, Three-phase Catalytic Reactors, Gordon and Breach, New York, 1983, p. 313.
Appendix A,, & G 4 ES 9 L u, v, Xi x
A: Nomenclature constants defined in eqn. (3) (kg rnm3) solid concentration in reactor (kg m-“) particle diameter (m) solid dispersion coefficient (m2 s- ‘) acceleration due to gravity (m s-‘) column height (m) slurry superficial velocity (m s-i) particle settling velocity in presence of gas (m s- ‘) weight ratio of fraction i axial distance (m)
Greek letters viscosity (kg m- ’ h-‘) P density (kg m-3) P Subscripts 1 liquid particle P sup@rscripts
f 1 0
feed stream column exit, z =L column entrance, z=O
Jownal,
54 (1994)
1
l-6
A non-parametric method for particle settling velocity determination a slurry bubble column MkuT. Western
in
Ityokumbul* Research
Centre,
CHIVMET, 1 Oil Patch Drive,
PO Bag 1280, Devon, Alb. TOC lE0
(Canada)
(Received June 27, 1991; in final form May 1, 1993)
Abstract The use of an EisenthabCornish-Bowden-type plot to determine the particle settling velocity in a continuous slurry bubble column under steady state conditions is presented. The procedure permits the identification of improperly run experiments, since the plots do not pass through the common intersection point. In the range 0.4 < Re, < 500 the particle settling velocity is given by V,, = [O.Ol 7ESg2@,- h)2/~#%,. It is shown that the solid dispersion coefficient cannot be determined from a steady state experiment. All dispersion data determined in supposedly steady state experiments are therefore meaningless.
1. Introduction Three-phase (slurry) bubble columns are employed in such diverse applications as coal liquefaction and recovery of fine mineral particles by flotation. In all these applications knowledge of the flow regime and the particle residence time is important, because the conversion or achievable recovery depends on both factors. In spite of the variety of these applications, the design and scaleup of shrrry bubble columns remain a difficult task. The problem areas are in the estimation of nonadjustable parameters (e.g. phase hold-ups, mixing coefficients, etc.). For an accurate estimation of the particle settling velocity and dispersion coefficient, a mathematical model of the process is often required. For particle one-dimensional sedimentadispersion the tion-dispersion model remains the most widely used [l-6]. While a considerable amount of data on solid dispersion exists, no comprehensive solid mixing theory is currently available. This is due in part to the wide variance in the reported values. As with liquid dispersion studies, thii variance arises from, among other things, inadequacies in experimental methods and the application of wrong boundary conditions [ 7, 81. Experimental errors are by far the most difficult to identify from reported data. The main purpose
of the present paper is to provide a non-parametric method for the determination of the particle settling velocity in a slurry bubble column from steady state measurements which allows for ready identification of wrong data points. It is also shown that it is impossible to determine both the particle settling velocity and the solid dispersion coefficient from the continuous procedure under steady state conditions.
*Present address: 115 MineralSciences Building, The Pennsylvania State University, University Park, PA 16802, USA.
0923-0467/94/$07.00 0 1994 Elsevier Sequoia. All rights reserved SSDI 0923-0467(93)02805-7
2. Theory The behaviour of solids in a three-phase slurry bubble column is frequently described using the sedimentation-dispersion model. By analogy with liquid dispersion, the flux of solids is described using a Fickian equation of the form
(1) The (+) and (-) signs in eqn. (1) refer to the countercurrent and cocurrent flow of slung and gas respectively. At steady state eqn. (1) reduces to
The solution of eqn. (2) is given by
C,=A, +A2 exp
(3)
M.T. Ityokumbul
2
/ Determinutian
Two boundary conditions are required for the evaluation ofA, and A,. The correct boundary conditions are
PI
atz=L
of particle
settling
(determined graphically) with the experimental quantity serves as a check on the accuracy of the experiments as well.
dG
==O 3. Experimental
atz=O
(4b)
where L is the column length. The assumption for the condition at z = 0 is that there is no slip in the shury fed to the cohimn, i.e. the liquid and solid have the same velocity. This is indeed a reasonable assumption for feed entering the column through small tubing. Since A2 = 0 (by application of eqn. (4a)), the value of ES cannot be determined from steady state measurement of the axial solid concentration alone. Application of the boundary condition at x= 0 results in the solution (5) Equation (5) is similar in appearance to the Michael&Menten rate expression in biochemical engineering [lo]. Accordingly, it is proposed to use the Eisenthal-Cornish-Bowden [ 111 non-statistical method for the determination of the particle settling velocity from eqn. (5). 2.1. Graphical method for the determination of particle settling velocity The use of the Eisenthal-Cornish-Bowden-type plot requires rearrangement of eqn. (5) to yield
q=c,f+
v,cs v,
(6)
and cs=cs’--
velocity
v,cS u,
(7)
for the cocurrent and countercurrent flow of gas and slurry respectively. At a given feed solid concentration the procedure involves determining the solid concentration in the reactor as a function of slurry velocity. Thus each run is represented by only two points on the graph: U, is plotted on the ordinate and C, is plotted on the abscissa. These points are joined by a straight line. When several pairs of points are plotted and connected by straight lines, the lines should intersect at the point (C,‘, VP>. As with the procedure in biochemical engineering, bad experimental points are readily identifled from their non-convergence at the common intersection point. A comparison of the value of Csf
verification
The best data currently available in the literature to test this procedure are those of Bhaga and Weber [ 121 because they satisfy the following conditions: (i) the feed solid concentration as well as the average solid concentration in the slung bubble column under cocurrent and countercurrent conditions are provided, and probably of more importance as will be shown later, (ii) the authors reported that the amount of solids in their slurry bubble column was small in comparison with that in the reservoir and thus the feed solid concentration may be assumed to be constant. This assumption is indeed reasonable, since their reservoir was 24 times the size of the slurry bubble column. With the use of the method being proposed here, this assumption may be independently verified by comparing the solid concentration at the common intersection point with the experimental quantity. It is this second assumption that is usually not valid in all other studies reported in the literature! Figure 1 shows a typical plot of the data for both the cocurrent and countercurrent flows of shn-ry and gas. In this plot the feed solid concentration and gas velocity were fixed while the shnry velocity and relative directions of flow between the shury and gas varied. Both series had the same feed particle concentration and the same particle size. In either case the plots have a common intersection point and bad data points are readily identified from their non-convergence at the feed solid concentration.
o’20 : 0.15
0.10
0.05
0.00
-0.05
0
100
200
300
C, kg/m3 slurry Fig. 1. Direct linear plot for the determination of particle settling velocity: effect of slurry velocity. Runs D3 and DD3 [ 121.
M.T. Ityokumbul
/ Determination
The convergence of the two plots at a common intersection point and the close agreement between the experimental (150 kg rne3) and predicted (155 kg mw3) values of the feed solid concentration are proof of the validity of this approach. In general there was comparatively more scatter with the countercurrent than the cocurrent plots. This may be due to magnillcation of the errors arising from the rather smaller range of slurry velocities used by these authors in the countercurrent experiments. 3.1. Eflect of gas velocity cm particle settling velocity The effect of gas velocity on the estimates of the particle settling velocity is shown in F’ig. 2. In these experiments the particle size and feed solid concentration were fixed while the gas and liquid velocities were varied. With the exception of the few bad experimental data, all the lines have a common intersection point, thus confirming that the particle settling velocity in a shnry bubble column is not a function of the gas velocity or hold-up. 3.2. Eflect of solid concenwatim on particle settling velocity The effect of feed solid concentration on the particle settling velocity determined by this method is shown in Fig. 3. In these plots the feed solid concentration was varied from 30 to 155 kg me3 while the gas velocity was kept constant. As a result of the changing feed solid concentration, the average solid concentration in the column varied from 31 to 230 kg rnb3. Since the common intersection points give approximately the same value for the particle settling velocities, the effect of average solid hold-up appears to be minimal.
of particle
settling
velocity
3
B OA
i .t: 8 5 >
0.2
e 2 u) 0.1
0.0 0
100
200
G , , kg/m3
slurry
Fig. 3. Effect of solid concentration.
Runs Dl, D2 and D3 1121.
2 r” 20
0.20
s" c z
0.10
0.20 0
100
200
G , , kg/m 3 slurry Fig. 4. Effect of particle size. Run Gl
[ 121.
3.3. Eflect of particle diameter on particle settling velocity F’igure 4 shows the plot for a particle diameter of 300 pm. For clarity, only the line corresponding to the average settling velocity for the 150 pm particle is shown. There is an almost linear dependence of the two quantities. 4. Discussion
50 =.
100 '
krJm3
IS0
slurry
Fig. 2. Effect of gas velocity on partkle settling velocity. Runs Dl, El and E2 (121.
All the trends observed with the data of Bhaga and Weber [ 121and the analytical solution obtained here are at variance with those reported by other authors.As an example, eqn. (5) predicts a constant value for the axial solid concentration at steady state while most authors report axial concentration gradients. The major differences may be summarized as follows. (i) The analytical solution for the axial solid concentration (i.e. eqn. (5)) is not a function of the solid dispersion coefficient.
4
M.T. Ityokumbd
/ Determination
(ii) The solid concentration at steady state is constant. (iii) For the range of conditions used by Bhaga and Weber [ 121 the particle settling velocity is not a function of gas velocity, liquid hold-up, etc. (iv) The particle settling velocity varies linearly with particle diameter in the intermediate range of Reynolds numbers. It will be shown here how these differences may be explained on the basis of wrong mathematical treatment of the sedimentation-dispersion model and poorly run experiments. 4.1. Analytical solution fw axial solid concentration Equation (5) is the only correct analytical solution for the axial solid concentration at steady state. To fully understand this, it should be recalled that the continuous procedure for solid input in the column is the same as the steady state method for studying liquid mixing using a step change in concentration or the F(t) curve [ 13 1. At steady state the concentration in the column is constant for all values of the Peclet number. It is thus impossible to determine the liquid dispersion coefficient from any concentration gradients in the column (since none exists) and it is instead the transient response in the system that is used for the estimation of this parameter. The fact that several authors [2-61 have reported concentration gradients in their shnry bubble columns at steady state is explained later as being due deficiencies in their experimental procedures. 4.2. Existence of axial solid concentration at steady state It is the existence of an axial solid concentration that is used in the curve-fitting exercise to determine the particle settling velocity and dispersion coefficient [3-6, 141. The question that arises is why the divergence between theory and practice? In actual fact there is no divergence - it is the experimental procedures that are deficient. As explained earlier, all these authors have used the cocurrent upflow of shit-r-yand gas with slurry recirculation. with this method and the known solid slip phenomenon, deposition of solids in the slurry bubble column will result. Naturally this will be at the expense of solids in the reservoir. Since none of the authors who have used this procedure indicate that their reservoir was continuously charged with solids (in order to maintain a constant feed solid concentration), it is obvious that the feed solid concentration will become time dependent (vari-
of particle
settling
velocity
able). This effect will be minimal if the size ratio of the reservoir to the slurry bubble column is very large as in the case of Bhaga and Weber [ 12 1. Unfortunately, some of these authors report that the reservoir was only as large as the slung bubble column [6]. When this happens, the boundary condition at z = 0 becomes a time-dependent variable. Unfortunately, the mathematical treatments by these authors all ignore this important point as if the solid concentration were constant! Thus it is this continuous drop in the amount of solids in the reservoir (and hence the changing feed conditions) that is responsible for the concentration gradient at the so-called steady state. Build-up of solids in the column naturally starts at the base of the column. Because the amount of solids in the reservoir is not enough to maintain the solid build-up and since it is impossible to redistribute the already settled solids at the base of the column over the entire column, the existence of an axial solid concentration is primarily due to improperly run experiments. For the cocurrent upflow of slurry and gas, eqn. (5) predicts a positive value when U]> VP. When the converse is true, eqn. (5) predicts a negative quantity for the solid concentration. While this is not physically possible, it merely suggests that the column acts as a sink for solids entering it. In most of these experiments the particle settling velocity is usually larger than the slung velocity. As a result the depletion of solids from the reservoir is accelerated and the so-called steady state attained in a shorter time. It should be recalled that Cova [ 1 ] who made the first mistake used a once-through flow of slurry for the cocurrent upflow of slurry and gas and reported that attainment of steady state was difllcult (required over 3 h). The fact that some authors report attainment of steady state in 30 min [4, 51 under similar operating conditions is consistent with the explanation provided here. Furthermore, in most of these studies [2-6, 141 the experimental solid concentration assumes a flatter profile with axial distance as the particle terminal velocity decreases (at a given slung velocity) or as the slurry velocity increases (for a given particle size), which is also consistent with the explanation provided here. If eqn. (5) does indeed represent the correct analytical solution for the sedimentation-dispersion model, the estimates for the solid dispersion coefficient should be very sensitive to the values of the axial solid concentration. Probably the best proof for this may be found in the work of Brian [6]. This author used the wrong analytical solution of Cova [l] and reported that the estimates for VP and ES were very sensitive to the value assumed
MT. Ityokumbul / Lletermitiia
for the solid concentration at the reactor exit. He reported that small changes in C,’ caused E, to vary by several orders of magnitude. Clearly this effect is expected, since ES does not appear in the solution for the axial solid concentration. 4.3. Variables ~ecting particle settling velocity in slurry bubble columns The results obtained here show that the particle settling velocity in a shury bubble column does not depend on the gas velocity and liquid hold-up. As explained earlier, the trends currently found in the literature are the products of wrong analytical solutions and improperly run experiments. For the 150 and 300 pm particles the particle settling velocities determined by this procedure (0.025 and 0.051 m s- ’ respectively) agreed rather well with the intercepts reported by Bhaga and Weber [ 121in their study (0.023 and 0.049 m s-l). However, there are slight differences in the theory, since the intercepts reported by Bhaga and’weber [ 121 also contain a hindered settling velocity component. As shown earlier, the particle settling velocity determined from the present method is not a function of the solid hold-up in the slurry bubble column. The linear dependence of the particle settling velocity on diameter is in agreement with the recent correlation provided by Ramachandranand Chaudhari [15] for the intermediate regime as shown below: v
@P2(PP--d
=
P
(8)
W-h
and o.0178g2@p- fi)2 1’3d
v =
(9)
P
P (
API
)
The limiting values for the application of eqns. (8) and (9) were given as Rep < 0.4 and 0.4
of particle settling velocity
5
TABLE 2. Comparison of predicted and experimental slopes for solid hold-up Slope m
Sand
Weldron 705 Weldron 505
U,=O.OOSS m s-l
U,=O.O043ms-’
Exp.
csk.
Exp.
talc.
0.33 0.20
0.33 0.20
0.26 0.16
0.26 0.15
bubble column is higher than that in a two-phase (liquid-solid system) and is reasonably well predicted by eqn. (9). 4.4. Extension to a polgdisperse particle fqy&%n The author [S] measured average solid concentrations in a 0.06 m bubble column using the countercurrentflow arrangementunder various solid feed concentrations and gas and slurry velocities. The solids used were Ottawa sand (weldron Plant designations 505 and 705) with a wide particle size distribution. Details of the experimental set-up and particle size distribution may be found elsewhere [ 8 1. For each sand the solid concentration in the reactor was plotted against the feed solid concentration. The slopes of these plots were determined and the results are shown in Table 2. The predicted slopes were estimated from eqn. (5) using the equation m=C
(_Ls_ 1 u,+vfi
(10)
where m is the slope and xif and V, are respectively the weight ratio of fraction i in the feed and its settling velocity calculated using either eqn. (8) or (9). The comparison of the predicted values with those determined experimentally is also shown in Table 2. The close agreement in the two quantities suggests that this method may be extended to a polydisperse solid system.
6. Conclusions TABLE 1. Comparison of experimental and predicted particle settling velocities in a slurry bubble column
4 (pm)
150 300
Particle settling velocity (cm s-l)
Re,
Exp.
Eqn. (8)
Liquid solid’
2.4 4.9
2.4 4.8
1.3 3.3
‘Particle settling velocity for solid-liquid
flow
3.6 14.7
[ 121.
It has been shown that the steady state method for studying particle dispersion in slurry bubble columns predicts a constant solid hold-up in the bubble column at steady state. The reported existence of a significant axial solid concentration gradient in the column at steady state by several authors is attributed to the variable nature of the feed solid concentration. It is shown that it is
6
M.T. Ityokumtrul / Determination
impossible to determine the solid dispersion coefficient from the steady state axial concentration profile. For this reason, all published data on solid dispersion coefficients which are based on this procedure are meaningless. However, the particle settling velocity may be determined from the steady state measurements using an Eisenthal-cornishBowden-type plot. As with the original plot, this method permits the ready identification of wrong experimental data, since such plots do not pass through the common intersection point. It is shown that the particle settling velocity in a slurry bubble column is well predicted by eqns. (8) and (9). It is also shown that eqn. (5) can be extended to a polydisperse particle system.
References 1 D.R. Cova, Ind. Eng. Ch.em Process Design Develop., 5 (1966) 20. 2 K. Imafuku, T. Wang, K. Koide and H. Kubota, J. Chem. Eng. Jpn., I (1968) 153. 3 Y. Kato, A. Nishiwaki, T. Fukuda and S. Tanaka, J. C;zem. Eng. Jpn., 5 (1972) 112. 4 D.N. Smith and JA. Ruether, Chem. Eng. Sci., 40 (1985) 741. 5 D.N. Smith, Jh. Ruether, Y.T. Shah and M.N. Badgqjar, AIChE J., 32 (1986) 426. 6 B.W. Brian, Hydrodynamics of a three phase bubble column, Ph.D. Thesis, Lehigh University, 1986. 7 M.T. Ityokumbul, N. Kosaric and W. Bulani, C/rem. Eng. Sci., 43 (1988) 2457. 8 M.T. Ityokumbul, Hydrodynamic study of bubble column flotation, Ph.D. Thesis, University of Western Ontario, London, 1986. 9 P.V. Danckwerts, Chem. Eng. Sci., 2 (1953) 1. 10 L. Michaelis and M.L. Menten, Biochem. Z., 49 (1913) 333.
of particle settling velocity 11 R. Eisenthal and A. Comish-Bowden, Biochem. J., 139 (1974) 715. 12 D. Bhaga and M.E. Weber, Can. J. Chem. Eng., 50 (1972) 329. 13 0. Levenspiel, ChemicalReaction Engineering, Wiley, New York, 1972, Chap. 9, pp. 253-315. 14 Y. Kato, S. Morooka, T. Kago, T. Saruwatari and S. Yang, J. Chem. Eng. Jpn., 18 (1985) 308. 15 P.A. Ramachandran and R.V. Chaudhari, Three-phase Catalytic Reactors, Gordon and Breach, New York, 1983, p. 313.
Appendix A,, & G 4 ES 9 L u, v, Xi x
A: Nomenclature constants defined in eqn. (3) (kg rnm3) solid concentration in reactor (kg m-“) particle diameter (m) solid dispersion coefficient (m2 s- ‘) acceleration due to gravity (m s-‘) column height (m) slurry superficial velocity (m s-i) particle settling velocity in presence of gas (m s- ‘) weight ratio of fraction i axial distance (m)
Greek letters viscosity (kg m- ’ h-‘) P density (kg m-3) P Subscripts 1 liquid particle P sup@rscripts
f 1 0
feed stream column exit, z =L column entrance, z=O