Holdup and Axial Mixing in Bubble Columns Containing Screen Cylinders Bih H. Chen Nova Scotia Technical College, Department of Chemical Engineering,Halifax, Nova Scotia, Canada
The residence-time-distribution technique coupled with the one-dimensional dispersion model was used to obtain the axial mixing coefficient and gas holdup in a gas-liquid bubble column containing screen cylinders. The primary factors considered were gas flow, liquid flow, and screen packing properties. Data show that the screen cylinders can significantly reduce the liquid-phase axial mixing, increase the gas-phase holdup, and evenly distribute the nearly uniform-size bubbles throughout the entire column. The screen cylinder also makes it possible to show the effect of liquid flow on both axial mixing coefficient and gas holdup.
The bubble column is widely used in industry as a simple relatively inexpensive means of achieving intimate contact between a gas and a liquid. However, the bubble column suffers from two important disadvantages: the bubble coalescence and the serious back mixing of the liquid phase; both may significantly reduce its contacting efficiency. Many investigators have used various types of column packing to minimize these drawbacks in a bubble column (Carleton et al., 1967; Hofman, 1961). Recently, Sahay and Sharma (1973) and Mashelkar and Sharma (1970) have studied the mass transfer characteristics of bubble column containing packings of various shapes and sizes. They have observed some improvement in the mass transfer coefficient over that of an unpacked bubble column. It appears more advantageous to use screen cylinders than conventional ring-type packing for improving the performance of a bubble column. The screen cylinder is highly porous; it occupies less than 5% of the column volume. Its mesh opening can be so chosen as to give a uniform dispersion of the dispersed phase, eliminating the dispersedphase channeling which frequently occurs in a bubble column containing conventional column packings. This channeling phenomenon undoubtedly contributes to the unusually low value of gas holdup obtained by Hoogendoorn and Lips (1965) in a Raschig-ring packed bubble column. Voyer and Miller (1969), Chen et al. (1971), and Chen and Vallabh (1970) have investigated mass transfer characteristics of a screen-packed bubble column. The values of the mass transfer coefficient can be accurately obtained in those studies when the residence time distribution of the liquid phase is available. It is the purpose of this study to measure the gas holdup and axial mixing in the liquid phase of a bubble column as a function of gas and liquid flow rates and packing dimensions. A transient response technique using 0.1 N KCl solution as tracer was used.
Experimental Section Equipment. Figure 1 schematically represents the experimental apparatus. The column, 190 cm long, was made from Plexiglas tubing of 7 cm i.d. At its bottom end is a gas distributor consisting of six syringes of diameter approximately 0.1 mm and a 1.9 cm central down-comer for the drainage of liquid. The test section of the column was carefully packed with screen cylinders up to a height of 129.5 cm. The packing was made from wire screen cloths of different mesh numbers. Their physical properties were given elsewhere (Chen et al., 1970). 20
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
Room air was fed to the distributor and discharged to the atmosphere a t the top. Water was taken from the city mains and introduced to the top of the packing from a hollow cylinder, 6.35 cm in diameter and 2.54 cm long, with 21 perforations of equal size (0.278 cm in diameter) at the bottom face. The hollow cylinder was internally baffled to achieve a good mixing of tracer and the main water stream. The step change of tracer concentration was generated at the water inlet by the instantaneous opening and closing of a solenoid valve located very near the liquid distributor. An electrical conductivity cell was used in conjunction with a fast response recording system to monitor the tracer concentration of the effluent. Details of the conductivity probe were given elsewhere (Chen and Douglas, 1969). The recording system consisted of a single channel carrier-amplifier (HP8805A), a Honeywell strip chart recorder, and accessories. The carrier-amplifier unit received a signal from the probe and delivered it to the recorder along with a starting pip which was electronically placed on the recorder chart when the solenoid valve was turned off. This determined the point t = 0. Procedure and Data Treatment. The instruments were allowed to warm up. The column was filled with water to a predetermined height, and the screen cylinder was then dropped, one at a time, into the column. The gas and liquid flows were then turned on at the required rates, followed by adjusting the liquid drain valve so that the aerated liquid height was maintained at a level equal to the packed height. The flow of tracer was started and set at a rate equal to 1%of the main water flow. When steady state was reached the solenoid valve was deenergized, automatically placing a spike on the recording chart. This spike signaled the beginning of the recording of a break-through curve for the purging step. The curve was converted to a dimensionless plot of C/Co vs. t / 8 with 8 evaluated from 8=
Lrn
(C/C,) dt
(1)
The dimensionless curve was then transferred to a semilogarithmic plot on which also plotted for several values of M is the solution of the diffusion model (Brenner. 1962) given by =2
c =
,=I
62,
6, sin 6, M 2 2M
+
+
Figure 1. Flow diagram: A, bubble column; B, solenoid valve; C, water distributor; D, air distributor; E, conductivity cell; F, rotameters.
Here, 6’s are the roots of the transcendental equations
-6,t a n - =6,2 2
M
v,,d
Cm/ I
Figure 2. Gas holdup versus gas superficial velocity: 0; empty column (Singh, 1969); 0 , 1.9 cm, 10 mesh screen cylinder; V, 1.27 cm, 14 mesh screen cylinder; 0,1.27 cm, 10 mesh screen cylinder; X, 1.27 cm. 8 mesh screen cvlinder: v b, Chen and Vallabh’s data, 1970.
( n = 1 , 3 , 5 , . . .)
2
6, 6, -M -cot=2 2 2
( n = 2, 4, 6 , . . .)
In this process, Brenner’s numerical values for the solution (1962) were directly used (see Figure 4). The value of the mixing parameter M that gave best fit to the experimental line was designated as the mixing parameter characteristic of the experimental system. The best fit was determined using least squares. The error involved in this procedure was believed to be less than 10%. The static and aerated column heights were used in the following equation to calculate the average gas holdup
L-Lo (3) L Holdup was also estimated from the response curve using the following equation 6G
=-
1 1
I .4
1
I
I
I
I
I
1
.6
.8
1
2
4
6
8
cm/s
(4)
Figure 3. Gas holdup versus liquid superficial velocity; packing dimension = 1.27 cm X 1.27 cm; mesh number = 14 mesh; +, 6000 ml/min; 0,2000 ml/min; A , 1000 ml/min; 0, 500 ml/min.
Results and Discussion Gas Holdup. Data on gas holdup were obtained in this study as it was closely related to the axial mixing characteristics of a bubble column. Typical data are shown in Figures 2 and 3 as a function of gas and liquid flows and packing geometry. Data of Chen and Vallabh (1971) are included which may support the validity of the present results. For comparison holdup data for unpacked bubble columns presently obtained and those obtained by Singh (1969) are also included. The addition of screen cylinders is seen to have caused a two- to three-fold increase in CG as compared with unpacked columns. This increase may be attributed to the improved dispersion of bubbles throughout the column and to the reduction of bubble rise velocity due to the presence of screen packing. Figure 2 indicates that CG is strongly affected by gas flow rate. In general the variation of 6~ with Vs,d is represented by q; c: Vs,dnwith n approximately equal to unity at low V,,J. It should be noted that this particular relationship is also pertinent to unpacked bubble columns. For bubble columns containing conventional ring-type packing, n has
the value of approximately 0.3 (Weber, 1961). Therefore, a screen-packed column has behaved identically as an unpacked one. Figure 2 also indicates that both packing size and mesh . decrease of number have considerable effects on t ~ The packing size from 1.9 cm to 1.27 cm. results in the increase of t~ of approximately 30% for a given mesh number and flow rate. The variation of t~ with packing diameter is of course directly related to bubble size, but the bubble size is determined solely by packing dimensions (Ishii and Osberg, 1965). Large screen cylinders produce large bubbles and hence less gas holdup than small ones. Similar observations were reported by Sutherland et al. (1963) and Chen and Osberg (1967) in connection with studies on the characteristics of a screen-packed fluidized bed. The effect of mesh number on c~ as indicated in Figure 2 is believed to be the consequence of channeling of the bubble phase which becomes increasingly obvious as the mesh number is increased. With close-mesh packings, most of the rising bubbles were observed to follow the paths created by the void spaces between neighboring cylinders, rather than passing through the mesh opening. This situation
The two values of
t~
checked very well.
Ind. Eng. Chem., Process Des. Dev., Vol. 1 5 . No. 1, 1976
21
worsens as the gas flow rate increases. Obviously, channeling always reduces the gas holdup for a given gas flow rate. One unexpected result of this study is the discovery of the effect of liquid flow on t ~ Most . of the investigators in this field have reported no effect due to liquid flow. (Hoogendoorn and Lips, 1965; Tadaki and Maeda, 1965). Figure 3 shows the typical results obtained in a screenpacked bubble column. This variation of t~ with Vs,d cannot be attributed to experimental error as the error involved is less than 6%. It is of interest therefore to examine this particular behavior. In two-phase fluidization systems, Bridge et al. (1964) have shown that t~ is a unique function of slip velocity for a given set of experimental conditions. Taking velocities as positive downward, the slip velocity in a countercurrent flow gas-liquid bubble column is, according to the slip-velocity theory
This equation clearly defines the effect of liquid flow on t ~ However, in previous studies of gas-liquid bubble columns, t~ is small and the average bubble rise velocity ( U b = Vs,d/ t ~ is) about 30 cm/sec, the term u = Vs,c/(l - t ~ is) always negligibly small as compared with the term Vs,d/t~.Therefore one concludes that the effect of Vs,con t~ has not been studied. On addition of screen cylinders to the column, the growth of bubble is limited and the gas holdup increases greatly. Consequently, the average bubble rise velocity should decrease. The gauze-type packing is reported to be able to reduce the bubble rise velocity to approximately one-third of its value in a normal unpacked column (Chen, 1971). It can be expected that the variation of V, in a screen packed bubble column is far greater than that which is possible in an unpacked column under normal operating conditions. For example, in the present investigation a v b ) from 10.5 to 15.4 cm/sec has been variation of ( u achieved for a gas flow rate of 0.868 cm/sec. If it were not for the use of screen cylinder, one would have obtained essentially no variation in the same variable. The result of this study should therefore be more representative of the behavior of a bubble column than results obtained in earlier studies.
+
Axial Mixing of the Liquid Phase Axial Mixing Curves and the Diffusion Model. In the liquid phase of tall bubble columns, significant circulation patterns are generated by the rising bubbles. Superimposed on these circulation patterns are random eddies. These are the two factors contributing to the fluid phase mixing. In the present experiments with screen cylinders in place, the overall circulation of liquid was effectively baffled and broken up into a large number of localized mixing zones, about the size of a bubble. Since the behavior of eddies has been treated successfully as a statistical process, the diffusion model should be expected to apply to the present case. Typical experimental response curves are shown in Figure 4 for both screen packed and unpacked columns. The experimental curve for the packed case is seen to be in excellent agreement with that predicted by the diffusion model whereas for the unpacked case, the curve is not so well defined. The possibility of the existence of two different mixing zones in a bubble column (Deckwer et al., 1973) is therefore entirely eliminated as a result of the use of screen cylinders. Axial Mixing Coefficient and Gas Flow. The effect of the superficial gas velocity Vs,d on the axial mixing coeffi22
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
.
2
.6
1.0
1.4
1.8
2.2
119
Figure 4. RTD and dispersion model
+ 0 0
' S A
VSA
M
Packed
cm/sec
cm/sec
-
-
Yes no Yes theoretical curves representing eq 2 .
4.36 0.597 25.9
0.43 4.93 4.93
3.7 6.5 2.6
cient D L is shown in Figure 5. The data a t zero gas flow were obtained in a previous study (Chen 'et al., 1971). A comparison of these two sets of data points out that the fluid phase mixing is entirely due to gaseous bubbles. The fluid-phase flow pattern is suddenly disrupted to a great extent when the gas is first introduced to the column, but as more and more gas flow is allowed, no large further change is evident. The present data may also be compared with data obtained in an unpacked column to appreciate a large reduction in the mixing coefficient. Using the same equipment, Singh (1969) measured the axial mixing coefficient under virtually identical conditions and found D L to be in the order of 50 cm2/sec. This reduction, as inferred in previous sections, is believed to be a direct consequence of the baffling action of the screen cylinder. The bubble growth is limited, the overall circulation is destroyed, and the bubbles are dispersed relatively evenly throughout the entire column. All these phenomena contribute to the reduction of axial mixing of the liquid phase. Axial Mixing Coefficient and Packing Properties. The effect of screen mesh number on D L is discernible from Figure 5. The increase in D L with a decreasing mesh number is much larger than previously reported for single phase flow (Chen et al., 1971). As pointed out in foregoing sections on gas holdup, channeling of the dispersed phase occurs in a bed of closely mesh screen cylinders resulting in the reduction of gas holdup as compared with a bed of open-mesh screen packing. Since it is the gaseous bubble that is responsible for axial mixing in the liquid phase of a gas-liquid bubble column whether it is packed or unpacked, gas holdup should be a prime factor affecting axial mixing. In fact, Kat0 and Nishiwaki (1972) have reported a nearly direct correlation between D L and t ~ Therefore . the present results are con-
4
6
8
"=
0
1.27
IO
"
8
14
0.95 l i t e i s i s
t
I4
.6
.4e
0
6
Figure 6. Axial mixing coefficient vs. liquid interstitial velocity; packing: 1.9 cm, 14 mesh cylinder: V$,d, 0, 0.22 cm/sec; A, 0.44 c d s e c ; +, 2.61 cm/sec.
8
A liquid (low
4
V S , C / ( I- e G ) , c m i s
I O mesh
1.9crn
0
2
1.0
-
-.
5
1.5
1
2
2.5
3
2A1 2 n
V5,* cm/s
Figure 5. Axial mixing coefficient vs. gas superficial velocity. 10-
sistent with the results on gas holdup as reported earlier in this paper. Figure 5 indicates that the effect of packing size on D L is considerable. This is quite obvious in view of the fact that the bubble size which determines the amount of axial mixing is determined solely by the nominal dimensions of the screen cylinder (Ishii and Osberg, 1965). A large bubble yields a large wake and hence larger axial mixing than that is possible for a small bubble. Axial Mixing Coefficient and Liquid Flow Rate. The axial mixing coefficients plotted against the liquid interstitial velocity u are shown in Figure 6. It is seen that D L is virtually independent of u when u is less than about 0.90 cm/sec but increases consistently when u becomes greater than 0.90 cm/sec. Similar results were reported by Chen (1972) for 8 mesh/in. screen cylinders. At high liquid flows, the increase of D L can be expressed as
DL a
~0.24
which is quite surprising in view of the findings reported in the literature. The results of other investigators in this field indicate that over a wide range of liquid flow rate D L is independent of u (Hoogendoorn and Lips, 1965; Kat0 and Nishiwaki, 1972). The explanation for the present finding probably lies in that it is the liquid flow velocity relative to bubble rise velocity which should be the deciding factor. Taking velocities as positive downward, the relative velocity in a countercurrent gas-liquid bubble column is (u + U b ) . As mentioned previously under normal operating conditions the bubble in a bubble column rises at an average velocity of about 30 cm/sec and therefore the variation of (u U b ) in previous studies are too small to cause any noticeable effect on DL. However, in the present study the average bubble rise velocity has been greatly reduced because of screen cylinders. Chen (1971) reported that the bubbles in a screen-
+
8-
6. I
./.
I
I
I
13
15
1
I
1
17
1
packed column rise a t an average velocity only one-third of the value normally expected in an unpacked column. This would therefore enable one to vary the relative velocity ( u U b ) over a significantly wide range such that any relationship between the relative velocity and D L could be detected. The same data are also plotted against (u + Ub) as shown in Figure 7. Ub is assumed to be constant at 10.1 cm/sec which is the rising velocity at the lowest liquid and gas flow rates in this investigation. The variation of D L can be represented by
+
DL a
(U
+ ub)1,21
+
over the narrow range of (u U b ) studied. This form of relationship is in line with the relationship for packed beds which is reported to be
DL a un with n varying between 0.93 and 1.17. It is noted that in packed beds Ub represents the velocity of the fixed packing and is therefore zero. The exponent 1.21 for the present data as compared with the range of n for spheres and other conventional types of packing may indicate there is more mixing in the same length because of the presence of screen cylinders. Thus, the present method of presentation of data has brought the description of mixing behavior in a screenpacked bubble column to a close analogy with that generally accepted for packed and fluidized beds. Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
23
This plot also supports the argument advanced previously that it is the relative velocity V, which determines the axial mixing characteristic of a bubble column, provided that the gas-phase flow velocity is sufficiently high. The experimental data indicate that a t low gas flow velocities D L is dependent on V, as well as on Vs,d, the absolute gas velocity. However, the importance of the latter as a factor diminishes as the flow rate is increased, it finally disappears when Vs,d becomes greater than 0.43 cm/sec, leaving the relative velocity as the only factor controlling DL.This phenomenon is also true for 10 mesh/in. screen cylinders used in this study.
L = aerated column height, cm Lo = static water level, cm M = uL/PDL, dimensionless Q = volumetric flow rate, ml/sec t = time, sec u = V,,J(l - t ~ =) interstitial liquid velocity, cm/sec v b = Vs,d/tG = average bubble rise velocity, cm/sec V, = slip or relative velocity, cm/sec V,,c = superficial liquid velocity, cm/sec Vs,d = superficial gas velocity, cm/sec z = axial distance, cm t~ = gas holdup, fractional 0 = average residence time, sec
Summary a n d Conclusions
Literature Cited
Axial mixing and gas holdup were determined in a bubble column containing screen cylinders of different size and mesh number. The data show that the screen cylinder can reduce axial mixing, increase gas holdup, and effect an even distribution of bubbles within the column. It is possible to investigate the effect of liquid phase flow on axial mixing and gas holdup by using screen cylinders in a bubble column. This study confirms that the relative velocity of the two phases, rather than the liquid phase velocity, should be one of the controlling factors for axial mixing in unpacked bubble columns. Acknowledgment The author acknowledges the assistance of J. W. Hines in the experimental work. Nomenclature A = column cross section, cm2 C = tracer concentration, g/ml Co = initial concentration, g/ml D L = axial dispersion coefficient, cm2/sec
Brenner, H.. Chem. Eng. Sci., 17, 229 (1962). Bridge, A. G., Lapidus, L., Elgin, J. C.. A./.Ch.E. J., I O , 619 (1964). Carleton, R . J., Flain, R. J., Rennie, J., Valentine, H. H., Chem. Eng. Sci., 22, 1839 (1967). Chen, B. H., Vallabh, R., ind. Eng. Chem.. Process Des. Dev., 9, 121 (1970). Chen, B. H., Manna, B. B., Hines, J. W.. ind. Eng. Chem., Process Des. Dev., 10, 341 (1971). Chen, B. H., Douglas, W. J. M., Can. J. Chem. Eng., 47, 113 (1969). Chen, B. H., Osberg, G. L., Can. J. Chem. Eng., 45, 90 (1967). Chen, B. H., Can. J. Chem. Eng., 50, 436 (1972). Chen, B. H., Brit. Chem. Eng., 16, 197 (1971). Deckwer, W., Graeser, V., Langernann, H., Serpernen, Y. Chem. Eng. Sci,, 28, 1223 (1973). Hofman, H., Chem. Eng. Sci., 14, 193 (1961). Hoogendoorn, C. J., Lips, J., Can. J. Chem. Eng., 43, 125 (1965). ishii, T., Osberg, G. L., A.i.Ch.E.J., 11, 279 (1965). Kato, Y., Nishiwaki, A,, Int. Chem. Eng., 12, 182 (1972). Mashelkar, R. A., Sharma, M. M., Trans. inst. Chem. Eng., 15, 162'(1970). Sahay, 8. N., Sharma, M. M., Chem. Eng. Sci., 28, 2245 (1973). Singh, S., B. Eng. Thesis, Nova Scotia Technical College, Halifax, N. S., Canada, 1969. Sutherland, J. P., Vassilatos. G., Kubota, H., Osberg, G. L., A.i.Ch.E. J., 9, 437 (1963). Tadaki, T., Maeda. S., Chem. Eng. (Japan),2 , 195 (1965). Voyer, R. D., Miller, A. I., Can. J. Chem. Eng., 46, 335 (1968). Webber, H. H., Adv. Chem. Eng., 7, 105 (1968).
Received for reuiew June 12, 1974 Accepted May 27,1975
Linear Feedback vs. Time Optimal Contol. I. The Servo Problem Alan H. Bohl' and Thomas J. McAvoy' Deparfment of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1002
The effectiveness of linear feedback controllers containing proportional, integral, and derivative modes is compared to time optimal control for the servo response of second-order dead-time processes. A proportional derivative controller with ideal preload is shown to be nearly time optimal. Such a controller represents a practical and economic solution to the time optimal servo problem. Because of the undesirable features of the integral mode a standard PID controller is shown to be poor for the servo problem.
Introduction Linear feedback control (see Figure 1) using various combinations of the proportional, integral, and derivative modes is commonly used in the chemical process industries. I t is straightforward in its implementation and requires a minimum of information about the process. A more recent development is time optimal control, which gets the process from some given initial conditions, to the 'Present address: McNeil Labs, Camp Hill Rd., Fort Washington, Pa. 19034. 24
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
desired final conditions, in minimum time. According to Pontryagin's Maximum Principle (Pontryagin, 1962), this can be accomplished (for a linear, single-input, single-output system) by using a combination of full-on, full-off forcing. This study uses linear feedback control to obtain approximate solutions to the time optimal problem. In Part I the servo mechanism problem is considered and in Part I1 the regulator problem is treated. To solve the time optimal problem analytically, the limits on available control action and the mathematical model of the process must be known. For linear systems, the pro-
