literature Cited
(1) Cooke, B. A , , Electrochim. Acta 3, 307 (1961). (2) Kirkham, T . A., “Report on Design, Installation, Start Up and Operation of Ionics 650,000 GPD ElectrodialysisPlant, Buckeye, Arizona,” Division of Water and Waste Chemistry, 144th Meeting, ACS, Los Angeles, Calif., 1963. (3) Mason, E. A., Kirkham, T. A., Chem. Eng. Progr., Symp. Ser. 5 5 , 173 (1959). (4) Phelps, R. O., Water Works Eng., 111, 752 (1958).
(5) Rosenberg, N. W., Tirrell, C. E., Znd. Eng. Chem. 49, 780 (1957). (6) Seko, M., Dechema Monographien, Band 47, No. 805-834, 575 (1963). (7) Weiner, S. A., Chcm. Eng. 70, 238 (1963). (8) Weiner, S . A., Rapier, P. M., Baker, W. K., IND. ENG.CHEM. PROCESS DESIGN DEVELOP. 3,126 (1964). RECEIVED for review February 26, 1965 ACCEPTED May 17, 1965
LONGITUDINAL MIXING IN GAS-SPARGED TUBULAR VESSELS W . B. A R G O A N D D . R . C O V A , Organic Research Department, Monsanto Go., St. Louis, M o .
The degree of longitudinal mixing in gas-sparged vessels has received scant attention, other than the observation that such vessels appear “well mixed.” For reactor design, particularly where it is desired to drive a reaction essentially to completion, the departure from plug flow can b e very important. Even when complete mixing is desired for a proposed commercial vessel, a knowledge of mixing encountered in pilot plant equipment i s needed for proper scale-up. Mixing studies for water in vessels of various sizes are reported in the form of longitudinal dispersion coefficients.
As-sparged vessels have been in use in the process industries for a number of years. Certain obvious advantages in using gas bubbles to obtain agitation, particularly in pressure reactors, indicate that this use will continue to grow. The authors have studied this type of system with regard to the degree of mixing obtained in the liquid phase as a function of vessel diameter and gas flow rate. Partial results of this study are presented here. The term “mixing” is used here to denote movement, distribution, or diffusion of a component through a reaction (or other) vessel, tending to make fluid composition uniform throughout. By way of illustration, in a flow reactor the presence of a chemical reaction tends to produce concentration gradients throughout the reactor; mixing tends to eliminate these gradients. As used here, mixing is not to be confused with agitation (or contact, as between trio phases), which may be purely local in nature. I n spite of the extensive use of gas-sparged vessels in chemical processing, relatively little has been published concerning the mixing or longitudinal dispersion encountered in such systems. Siemes and Weiss (77) have presented data for a nitrogen-water system in a 1.65-inch-diameter tube. The current paper presents additional data for gas-water systems, giving an indication of the effect of changes in vessel diameter, gas density, and gas flow rate. Introduction of a gas into a column of liquid results in the formation of bubbles whose size depends on the gas flow rate, orifice size, and physical properties of the fluids. At low gas rates, the bubbles rise discretely through the liquid a t almost constant velocity, the buoyant force of the bubble being balanced by viscous drag from the liquid. This viscous drag, in turn, leads to turbulent eddies and movement of the liquid, which constitutes longitudinal mixing. Siemes and Borchers (76) show that in the laminar flow range, at superficial gas velocities higher than about 3 feet per minute, bubble size becomes independent of gas floiv rate. From this it follows 352
I&EC PROCESS DESIGN A N D DEVELOPMENT
that the ascending velocity of the individual bubbles is also roughly independent of gas flow rate, and that increasing gas flows must be accomplished by an increased gas holdup. As the gas rate is increased, the tube reaches a condition where it appears “filled” with bubbles. At still higher rates, collision between bubbles results in coalescence into large bubbles which rise faster, and eventually to slugging of the gas phase and violent turbulence in the liquid. The onset of coalescrnce and slugging starts a t superficial gas velocities on the order of 10 feet per minute, depending somewhat on tube diameter as well as the physical properties of the fluid. This latter phase has been denoted the turbulent regime. Finally, the gas rate may be made so high as to blow the liquid through the tube whereupon the gas becomes the continuous phase. This paper is concerned with both the laminar and turbulent regimes a t gas rates such that the liquid is the continuous phase. Phenomenological Theory
The experimental method consisted of the injection a t a constant rate of flow of a tracer into a continuously flowing reactor and measurement of the steady-state concentration gradient of tracer upstream of the injection point. Figure 1 presents this method schrmatically. Longitudinal mixing of the liquid phase passing through the reactor tends to bring tracer upstream of the injection point, while the bulk continuous flow of liquid through the vessel tends to sweep the tracer out of the vessel. This procedure is similar to that reported by Gilliland and Mason (6) for some of their studies in fluidized beds. It has also been employed by Westerterp and Meyberg (79) for rotating disk contactor studies. This technique has the obvious advantage of permitting steady-state (rather than transient) measurements. Dispersion coefficients were then calculated from these data through application of the following model :
A steady-state material balance on the tracer component is taken over a thin croiss section of the reactor, resulting in the familiar equation:
The assumptions inherent in this equation are: T h e flows caused by mixing can be represented by a diffusion-type mechanism, diffusivity does not vary along the tube axis, and radial variations in traicer concentration are absent or negligible. Siemes and \Veiss (77) in their transient state measurements also used a model based on the diffusion mechanism to interpret their data. The general solution of Equation 1 is:
In applying Equation 2 to that portion of the reactor upstream of the tracer injection point the following boundary conditions are appropriate: C = C, a t y = 0 (see Figure 1)
(3)
Application of these boundary conditions to Equation 2 gives the specific solution : (5)
- C,) - UL/DL
I t is evident from Equation 5 that a plot of log (C us. y should yield a straight line with slope equal to
when applied to experimental measurements. The assumptions listed in connection with Equation 1 were found, in general, to be justified by the experimental data, with Equation 5 providing a test of the applicability of the use of a diffusion-type mechanism, and radial concentration traverses indicating thNE absence of radial variations. Figure 3, a plot of typical data, illustrates the validity of these assumptions. Two exceptions should be noted. I n the immediate vicinity of the gas-sparging device, Equation 1 would not be expected to be valid. Secondly, under extreme conditions in columns of large di.ameter the diffusion model may not be applicable. With these exceptions, the diffusion model appeared to represent the data very well. Equation 5 was used in calculating dispersion coefficients
#
Experimental Apparatus and Procedure
Experiments were carried out on the systems: (1) waternitrogen and (2) water-ammonia synthesis gas in tubes of 1.8-inch (A), 4-inch (B), and 17j/~-inch(C) inner diameter. Superficial gas velocities were varied over the range of 0.8 to 39.9 feet per minute. Superficial liquid velocities ranged from 0.74 to 3.19 feet per minute. I n the largest tube gas pressure was varied from 100 to 1500 p.s.i.g. and temperature from 80’ to 200’ F. I n the smaller tubes, conditions were atmospheric pressure and room temperature. The apparatus and procedure used for all three tubes was in essence the same. T h a t for tube C is described in detail.
Tube C. The experimental apparatus and flow diagram are depicted in Figure 1 (the countercurrent flow apparatus). The tube was 175/8 inches in inside diameter and 27 feet long and contained a number of hairpin cooling tubes running the length of the tube. For high temperature runs, steam was passed through the hairpin tubes to bring the reactor to the desired temperature level. \Vater was fed in at the top of the tube a t a rate of 13 to 18 gallons per minute by two Aldrich high-pressure pumps. For high temperature runs, the feed water was heated. \Yater flowed out of the bottom of the reactor through two letdown valves actuated by a level controller. it’ater flow rate was measured by a n orifice in the feed line. The orifice calibration was checked by weighing effluent water collected over a measured time interval. Ammonia synthesis gas was fed into the bottom of the tube through a small pipe (see Figure 2 for details). The configuration of the feed pipe was such as to impart a tangential swirling motion to the gas stream. Gas leaving the top of the reactor passed through a cooler and thence through a letdown valve actuated by a pressure controller. Gas flow rate was measured by orifices in the inlet and outlet gas lines. Concentrated solutions (just below the saturation point) of potassium chloride or sodium chloride were used as tracers. T h e tracer was fed into the bottom of the tube a t a rate of about 0.5 gallon per minute by a duplex Philadelphia pump. The tracer entered the tube a t the center of the cross section and approximately 1 foot above the gas inlet (see Figure 2 for details). Sampling taps were provided at five levels upstream of the tracer feed point. Samples could be taken from the center of the tube a t all five levels. Two radial sampling taps were
AND t GAS LIQUID EXIT
E:XIT
WATER FEED
from experimental data. In the calculations the superficial liquid velocity, Us, was used. Therefore, a superficial dispersion coefficient, Ds, was obtained. The relations between “actual” and “superficial” quantities are given in the following equations, where Ho is the gas fraction:
4
FEED
LI Q U I t) EFFLUENT
FEED
CONCURRENT
TUBE
COUNTERCURRENT
LJQ~ID EXIT
TUBE Figure 1.
Flow diagrams
Figure 2.
Configuration of exit end of tube C VOL. 4
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353
provided at four levels 6'/8 and 7l/4 inches from the center. The radial sampling taps were located on opposite sides of the tube. The sample lines were passed through a cold water bath to cool the samples and thus prevent evaporation of water in the samples. In each experiment, gas flow rate, liquid level in the tube, and tube temperature were brought to constant levels before the tracer feed pump was started. The rate of depletion of the tracer feed reservoir was measured throughout the run to verify a constant tracer feed rate. One hour after the tracer feed was started, a complete set of samples was taken. I t was found that 1 hour was sufficient time to ensure steady-state conditions in the tube. A duplicate set of center samples was then taken 1 hour after the first set. All the samples in a set were taken simultaneously over about 20 minutes, in order to cancel out the effects of minor fluctuations in concentration. Large samples of 1 to 2 liters were taken (however. the amount of sample was small in comparison with the total liquid flow). The samples were analyzed for chloride ion by potentiometric titration with silver nitrate using a silver-silver chloride, glass electrode system. The use of a weight buret in the titration enhanced accuracy. The mixing experiments covered a range of superficial gas velocities of from 0.8 to 39.9 feet per minute. The pressure range covered was 100 to 1500 p.s.i.g. Series of experiments were carried out at both 80' and 200' F. Liquid holdup in the reactor \vas measured a t 750 p.s.i.g. and 65' F.: and for superficial gas velocities of from 6.1 to 37.3 feet per minute. These measurements were made by allo\ring the system to come to steady state. At this point, the tube was isolated and drained and the volume of liquid holdup determined directly. T u b e A. The tube was 1.84 inches in inside diameter and 3.0 feet long. T h e general arrangement was as shown in "concurrent tube," Figure 1. Water and nitrogen were fed in at the bottom, the latter through a porous sintered stainless steel sleeve. A 470 solution of potassium chloride (KCI) in water \vas used as the tracer. This was fed in a t the top of the tube in the same direction as the \rater flow. LYater and gas flow rates Ivere controlled manually and were observed by rotameters. Tracer flow rate was controlled by a metering Pump. Tracer concentrations a t different levels in the tube were measured in situ by means of Ag-AgC1 electrodes using the electrode potential to determine the chloride ion concentration. T h e Ag-AgC1 electrode was connected through a porous plug (an asbestos fiber) to a saturated KC1 bridge leading to a standard Hg-HgZCl? reference electrode. The potential of this electrode pair \vas fed to a modified Leeds S: Northrup pH amplifier and recorded on a high speed recorder. Water velocities ranged from 0.74 to 3.2 feet per minute and gas velocities from 1.74 to 13.1 feet per minute. Experiments were carried out with the tube baffled and unbaffled. Baffles were constructed of brass strips to give an area reduction of 66y0. They were spaced 9 inches apart. In t\ro experimental series, a vibrating disk agitator was mounted in the unbaffled tube. The agitator consisted of a long shaft with 7 disks ( l 9 / 1 6 inches in diameter) mounted on it, spaced 41:,' inches apart. Each disk was perforated with 18 holes, inch in diameter. A Fisher Vibro-mixer provided the driving power a t 3600 cycles per minute and small amplitude. l I l 2 Liquid holdup was measured for the tube without baffles or disk assembly. T u b e B. The tube was 4 inches in inside diameter and 48 inches long from the gas sparger to the top of the tube. Water was fed in near the bottom of the tube 3 inches above the gas sparger and allowed to overflow a t the top of the tube. Nitrogen gas was fed in a t the bottom of the tube through a sintered stainless steel disk approximately 3'/2 inches in diameter. Tracer was injected into the tube at the center of the cross section l ' / g inches from the top of the tube, and was metered through a Sigma pump. \Yater and gas flow rates were manually controlled and were measured by the use of rotarneters. Samples were taken from the center of the tube cross section at eight different levels, spaced 4 to 6 inches apart. The samples were withdrawn through a length of '/B-inch stainless steel tubing inserted into the tube from the top. The tracers used were concentrated solutions of potassium 354
l&EC PROCESS DESIGN A N D DEVELOPMENT
z 0.50
$-44.0
I-
Z W
0
0.30
a 0
I
1
4
8
HEIGHT
I
I
IN
12 1 6 2 TUBE,FT.
Figure 3. Tracer concentration in tube C as a function of height in tube 200' F., 1 5 0 0 p.s.i.g., 33.5 feet/minute superficial gas velocity
-/-ADuplicate center samples 00
Radial samples
CI
kl80I60 3 ~140-
-
*
L
-120-
40080 60 40 20 -
-
O'
4
h
c,
Ib
112 SUPERFICIAL GAS VEL,, FTJMIN.
Figure 4. Superficial dispersion coefficient in water in tube A at atmospheric pressure as a function of superficial gas velocity
-k
Opentube Baffled tube Tube baffled by disk Vibrating disk in tube
0 X
A h
2 1 d
CY
I
1
1
I
1
15 20 25 30 .35 SUPERFICIAL GAS VEL,FT/MIN.
I
40
Figure 5. Superficial dispersion coefficient in water a t in tube C as a function of superficial gas velocity
90" F.
+0 0
100-p.s.i.g. pressure 750-p.s.i.g. pressure 1500-p.s.i.g. pressure
permanganate in watw and ferric chloride in water. Samples containing potassium permanganate were analyzed by observation of light transmittance with a Spectronic 20 spectrophotometer. Ferric chloride samples were analyzed in the same fashion after color was developed by the addition of a thiocyanate compound. The gas velocity range covered by the experiments was 4.58 to 13.7 feet per minute. Water flow ratio was held a t 1.6 and 1.9 feet per minute. Liquid holdup was measured as a function of gas rate. Sample Calculation
The follo\ving is a sample calculation of a dispersion coefficient from the experimental data given in Figure 3 (for the large tube). By the method of least squares the data of Figure 3 give the following linear equation : loglo c' = -0.3860
- 0.00311~
From this equation it was calculated that
C,=, = 0.4111 weight 76 KC1 = 0.3563 rveight yoKCI
TVater flow rate was 186.7 gallons per minute and the tube cross section was 1.4 sq. feet. Also: for these experiments CF = 0. Substitution of these values into Equation 5 (after solving for D s ) gives
DS =
.
U S ( J > !-
y1)
-
CF\
IC,
= 13,600 sq. ft./hr.
Discussion of Results
Since the data obtained were all concerned with backmixing in Lvater, little can be said concerning the effects of physical properties of the liquid. Holyever, the data show the effects of reactor geometry and are useful in estimating dispersion coefficients in scale-up and design calculations. Tube A. Dispersion coefficient data for tube A are plotted in Figure 4 as a function of gas flow rate. The values of D, reported are for the upper portion of the tube-Le., a t least 12 inches above the sparger. For the unbaffled tube a smooth increase in axial mixing occurs in the range of 5 to 10 feet per minute superficial vrlocity. Slugging of the gas phase was noted a t 11 feet per minute and D, increased markedly. The results approximate those found by Siemes and LVeiss (77) in the upper portion of an unbaffled 1.65-inch i.d. tube. LVith baffling in the tube Ds was reduced by a factor of 2. The vibrating disk agitator gave a drastic reduction in Ds a t the higher gas floiv rates. T h e agitator in operation gave
a
rise to lower Ds, smaller bubble size, and a more even gas distribution than did the stopped agitator. Experiments in which the water flow rate was varied while gas rate was maintained constant showed that liquid velocity had no noticeable effect on the dispersion coefficient. Tube B. The data for tube B are presented in Figure 8. T h e gas flow was essentially laminar u p to a gas velocity of 11 feet per minute. In this region, D , decreased slowly as gas velocity was increased. This result was unexpected and cannot be explained a t this time. At gas velocities above 11 feet per minute the gas flow became turbulent and Ds rose rapidly with increasing gas velocity. Tube C. These studies were made in a commercial reactor during a brief "down" period in which the reactor could be used for experimental work. Time limitations required that a preplanned series of runs be conducted, and the results could not be analyzed until after the experimental program was completed. The data obtained from each run were plotted on semilog coordinates to establish : A.
Reproducibility.
B. Lack of radial gradient. C. Linearity as required by Equation 5 The first of these conditions verified that the data represented steady-state conditions. Requirements B and C are necessary criteria which must be satisfied if Equation 5 is to be considered a suitable mathematical model of the physical situation. A typical plot of this type is shown in Figure 3. Generally, the data taken a t higher gas velocities satisfied the above criteria very well. Unfortunately, attempted measurements a t low gas flow rates generally failed to satisfy the criteria for lack of radial gradient. This failure was attributed to the design and location of the sparger (as shown in Figure 2). At low gas flows this sparger configuration gave nonuniform gas distribution across the diameter of the tube, resulting in large scale circulatory f l o s~of liquid up and down the tube. High Gas Flow Rates (Tube C). Over the range of gas velocity 14 to 40 feet per minute, Ds increased slightly as the gas flow increased, as shown in Figures 5 and 6. A pronounced increase in dispersion coefficient was observed with a n increase in temperature as indicated in Figure 7. However, the present data are not sufficient to pin down the temperature-dependent physical properties responsible. These figures also show that gas density-i.e., pressure-had essentially no effect on D,.
-
16
a 14
I
I
\
\
N
N
t'
12
IO
v
Y
8 6 X 4 ni 2
n
h
1
0
0
X
d O
IO
U
I
I
1
15 20 25 30 SUPERFICIAL GAS VEL., FTIMIN.
3!5
Figure 6. Superficial dispersion coefficient in water at 200" F. in tube C as a function of superficial gas velocity
+
CI
750-p.s.i.g. pressure 1500-p.s.i.g. pressure
VOL. 4
NO. 4
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355
Table 1.
Data and Calculated Values for Laminar and Turbulent Flow Regimes
LAMINAR REGIME 1.835
4
2.56 3.90 5.62 5.62 6.98 4.58 4.58 6.86 9.15 11.46
0.743 1.14 1.56 1.56 3.19 1.90 1.90 1.90 1.62 1.90
0.065 0.112 0,162 0.162 0.196 0.10 0.10 0.17 0,24 0.32
39.4 34.8 34.7 34.7 35.6 43.7 43.7 38.1 36.0 33.0
3.19 1.62 1.69 1.57 1.98 1.44 1.83 1.88 tube diameter.
0.296 0.39 0.29 0.35 0.46 0.28 0.35 0.40 * Data at 90"
24.6 27.4 51.8 52.6 117.8 256.7 303.3 251.8 214.5 217.6
0.0082 0.0092 0.0095 0.0095 0.0095 0.0078 0.0078 0,0082 0,0082 0,0082
0.77 0.68 0.36 0.36 0.15 0,080 0.067 0.074 0.083 0.075
0.12
...
...
...
... ... ... ... ...
...
4.7
... ... ...
1. o
... ...
0.35 0.42
...
0.23
*..
TURBULENT REGIMEa 1.835 4 17.63b
a
13.1 13.7 15.6 24.2 39.9 c 14.1 24.9 31.8 For turbulentjow, d,
=
dl
=
39.7 348.1 0.153 32.5 927.9 0.333 51.4 8,225 1.47 66.7 9,415 1.47 86.7 13; 593. 1.47 48.4 5,361 1.47 68.3 7.692 1.47 76.4 11,167 1.47 F., 750p.s.i.g. c Data at 90' F., 75OOp.s.i.g.
Effect of T u b e Diameter. Diffusivities increased rapidly with tube diameter, as shown in Figure 8. Unfortunately, the previously mentioned difficulty with low flow measurements in tube C prevents a comparison of dispersion coefficients at identical flow rates, but it is apparent that a 10-fold increase in tube diameter increased D, by a factor of a t least 20. Gas Holdup. Gas fraction in tubes A, B, and C is plotted in Figure 3 as a function of superficial gas velocity. Gas holdup rose rapidly in the lower range of gas velocities and tended to level off a t higher velocities. Generalization of Data. The limited number of experimental conditions militates against a generalized correlation. However, it is useful to cast portions of the data in as general a form as possible for comparison with other mixing studies. Laminar Flow Regime. Comparisons of the laminar data may be made with the mixing encountered in packed beds and in liquid-solid fluidized systems. Figure 10 is a plot of the data in the form of Peclet number us. superficial gas velocity. Peclet number here is defined as:
...
... ... ... ...
...
1.04 0.70 0.55 0.62 0.56 0.80 0.78 0.60
...
...
to much higher values. The mixing is influenced by tube diameter, and, in smaller tubes, by superficial gas velocity. Figure 11 shows the laminar mixing data for tubes A and B in the form of a modified Peclet number proposed by Wilhelm (20)3
where rn is the hydraulic radius. (Pe)B3fis plotted against gas fraction. Also shown in Figure 11 are some of the values obtained by Cairns and Prausnitz ( 7 ) for 1.33-mm. and 3.0mm. diameter lead spheres fluidized in water in 2-inch and 4-inch diameter beds.
4,000 2,000-
l0,OOO
7,000-
1,000-
where dB is the bubble diameter, lJ, the relative velocity of bubble to liquid, and DL the true longitudinal dispersion coefficient (DL= D s / L ) . Bubble diameters were not measured in the present work, but may be estimated from the correlation of Siemes and Borchers (76). UR is calculated from gas and liquid flows and holdup data as:
700-
d
70-
-
40 Figure 10 also shows, for comparison, ranges of Peclet numbers calculated by McHenry and Wilhelm (74) for axial mixing of binary gas mixtures in packed beds and values calculated by those authors from the data of Kramers and Alberda (72) and Danckwerts ( 3 ) for liquid flow in packed beds. Figure 10 shows pronounced effect of tube diameter upon axial mixing in gas-sparged systems. I n general, it might be concluded that axial mixing in gas-sparged systems varies from a lower limit, comparable to liquid packed-bed values, 356
I&EC PROCESS DESIGN AND DEVELOPMENT
lob
5 Ib Ik io
;5 i o 3k
10
SUPERFICIAL GAS VEL., FT./MIN. Figure 8. Superficial dispersion coefficients in water at ambient temperature as a function of superficial gas velocity
+0 0
A
Unbaffled tube A Tube B Tube C under 1500-p,s.i.g. pressure and 90' C. Tube C under 750-p.s.i.g. pressure and 90' C.
SUPERFICIAL GAS VEL,FT/MIN, Figure
9.
Gas holdup as a function of superficial gas velocity
0
A
+
Tube A Tube B Tube C at 750 p.r.i.g. and 6 5 " F.
I n view of the previously mentioned effect of tube diameter o n gas-sparged diffusivities, it is obvious that this system is more nearly analogous to fluidized beds than to packed beds. Turbulent Regime. For gas-water systems this regime begins a t superficial gas velocities of about 10 feet per minute and higher. I t is characterized by agglomeration of bubbles. The larger bubbles behave more erratically and capture other bubbles. with further increase in size and ascending velocity. T h e eventual limit on lateral bubble dimension is the tube diameter, by which time the gas phase has formed slugs which move very rapidly up the tube. In this flow pattern, a pronounced circulation of liquid around the slugs is noted. Smaller fragmentary Iiubbles also move randomly with extensive vertical circulation. I n such a flow pattern it is no longer feasible to estimate bubble size Iyith any degree of precision. Rather, the characteristic length term of a modified Peclet number might be taken as the tube diameter:
T h e highest ve1ocit.j runs for tubes A and B were in the turbulent region. as \\ere all values reported for tube C. Such Peclet numbers for turbulent runs are plotted against gas superficial velocity in Figure 12. An order of magnitude agreement for all tube sizes is noted, with values of (Pe)BT ranging from 0.55 to 1.0. Data and calculated values used in Figures 10 through 12 are summarized in Table I. Applications to Reactor Design. In general, three types of information are nevded for design of a gas-liquid reactor: heat transfer, mass transfer and/or kinetic data, and degree of mixing. The correlations of Kolbel et al. (70, 77) for heat transfer coefficients in gas-liquid systems appear satisfactory, with due precautions (as pointed out by Kolbel) in interpreting the geometry of the system in terms of Kolbel's dimensionless correlations Mass transfer of gas-liquid systems has been treated by Johnson et ill. ( 9 ) , Calderbank ( 2 ) , and Gibbs (5). Calderbank and. in pax ticular, Gibbs show mass transfer to be a function of vessel geometry and gas flow rate, with added mechanical agitation having little or no effect. However, since the mass transfer between gas and liquid may not be a rate-limiting step in the reaction, and since kinetics, per se, are needed in the reactor design, this field is frequently explored experimentally in bench scale or pilot plant equipment. T h e
translation from pilot plant to large scale requires a knowledge of the degree of mixing encountered in both sizes of equipment. Classically, the treatment of flow reactors has until recently been limited to the two extreme cases: (1) in which complete mixing of the fluid occurs. and (2) in which no mixing-i.e., piston or plug flo\v--occurs. The simplicity of the extreme conditions-complete mixing or no mixing-is obvious. All that is required is a knowledge of the reaction kinetics plus a laboratory evaluation of appropriate reaction velocity constants. Consideration of partial mixing, however, introduces added mathematical complexity, and also requires the assignment of definite numerical values to the degree of mixing encountered. Assuming that such numerical values are on hand, the design problem may be approached along either of two general lines : (1) that of a residence time distribution as proposed by Gilliland (6, 7, 8),Danckuxrts (3, d ) , Sherwood (75). or Zweitering (27) ; or ( 2 ) that of a diffusion model solution also suggested by Danckwerts (3, p. 10) and since elaborated by Wehner and Wilhelm (78) and Levenspiel and Bischoff (73). T h e latter
0.1 0.8
0,s
-
A
A 1
1
VOL. 4
I
NO. 4
1
1
1
OCTOBER 1965
357
a
u 0.3
I
1
1
1
IS
20
25
30
0.2
IO
@ 35
1 40
SUPERFICIAL GAS VEL., FT/MIN. Figure 12. Peclet number based on tube diameter as a function of superficial gas velocity for turbulent flow regime
0
A 0
0.1
0
0.2
0.3
0
0.1
0.2
0
GAS FRACTl,ON, i G Figure 1 1. Peclet number based on hydraulic radius of bubble as a function of gas fraction in laminar flow regime
A 0
______ __
----
---
Tube A Tube B 2-inch diameter particles ( 1 ) 2-inch diameter particles ( I ) 4-inch diameter particles ( I ) 4-inch diameter particles ( I )
tube, fluidized b e d with 3.0-mm. diameter tube, fluidized b e d with 1.3-mm. diameter tube, fluidized b e d with 3.0-mm. diameter tube, fluidized b e d with 1.3-mm. diameter
Tube Tube Tube Tube
A B C a t 90’ C a t 90’
F. and 1500 p.s.i.g. F. and 750 p.s.i.g.
Table II. Application of Mathematical Model Reactor dimensions, feet 1 . 5 I.D. X 2 4 . 5 long Gas feed (superficial selocity), ft.,/min. 17.7 Gas exhaust (superficial velocity), ft./ min. 9.2 Av. gas flow (superficial), ft./min. 13.5 Gas fraction 0.32 Average pseudo-first-order reaction velocity constant (from lab. and pilot plant data) k = 260/hr. Superficial liquid velocity, C,S.ft.jhr. 47.6 Actual liquid velocity, U L ,ft./hr. 70.1 Viscosity of reaction mass, cps. 0.30 Average superficial longitudinal dispersion coefficient, Il,q, estimated from 9,000 Figure 6, sq. ft.,/hr. Actual dispersion coefficient, L ) L , sq. ft./ 13,300 hr. 0.13 Axial Peclet number = U L I * / D ~ . Process requirement for conversion of 99.7-t key component, $c
2 = 0 a t reactor liquid entrance
approach employs a longitudinal dispersion coefficient and is considered by the present authors as more suitable both for studying longitudinal mixing and for use in reactor design of gas-sparged systems. Values of longitudinal mixing coefficients for a single fluid phase in pipe lines and packed beds have been published by a number of investigators, and are summarized by Levenspiel (73). For gas-sparged flow reactor design, the values of longitudinal diffusivity reported here are not sufficiently broad in scope to permit a generalized correlation ; however, they may be used to estimate order of magnitude diffusivities. T h e case of a continuous first-order reaction a t steady state may be represented by the follolving differential equation :
As a n illustration of the application of this mathematical model and dispersion coefficients. a n example (based o n d a t a from a commercial reactor) is cited (Table 11). T h e “process requirement” of 99.7+% conversion represents a realistic degree of completion needed to eliminate a subsequent purification step. I n the actual commercial reactor this may be achieved by varying the catalyst concentration (and thus the reaction velocity constant, k) to obtain the required conversion. Substitution of the above data in Equation 14 gives the calculated fraction of key component reacting as:
For the similar case of a packed tubular reactor, Danckwerts (3, p. 10) has presented the following solution to Equation 13:
or 39.87, conversion, which may be compared with actual process conversions of 99.9+70.
y =
Y a t reactor liquid exit
-l
where a =
358
l&EC
dl
+ (~~DL/UL’)
(15 )
PROCESS DESIGN A N D DEVELOPMENT
T h e value of the above calculation (considering partial mixing) may be illustrated by comparing the reactor lengths (keeping the same reactor diameter) needed for the limiting cases of plug and completely mixed flow. For the plug flow case and 99.8y0 conversion, the required length would be:
(16) \ -
0.998 = 1
- exp
L = Y
from which
=
~I
(-2’) ~
1.67 feet
Since the heat release for this reactor is over 2,000,000 B.t.u. per hour, it is evident that a true ‘.plug” flow reactor would present a n impossible heat transfer problem and require different reactor geometry, Indeed, in pilot plant studies in a 2.5inch-I.D. reactor (which was not plug flow but had D s values estimated a t 300 to 500 sq. feet per hour) very intensive local cooling was required in the vicinity of the feed point, a shorter reactor was employed, etc. For the case of complete mixing a t 99.87, conversion, the required length Lvould be:
0.998 =
260 Y 70.1 260 Y
=
Nomenclature
+
41 4 k D r u?L cross-sectional area reactor constants of integration concentration of tracer or reacting component concentration of tracer or reacting component in effluent stream from reactor = concentration of tracer or reacting component in feed to reactor = bubble diameter = dispersion coefficient = superficial dispersion coefficient. D , = D L L = tube diameter = gas fraction = 1 - L = reaction velocity constant = fraction of reactor occupied by liquid = natural logarithm = hydraulic radius-volume of liquid \ \ etted area = Peclet number based on bubble diameter = d B C R / m .uI’ (Pe)Bli = Modified Peclet number based on hydiaulic radius of gas bubble = 4 mC;.’DL (Pe)BT = Peclet number based on tube diameter = d t C R / D L t = time VGy = superficial mean gas velocity = actual mean liquid velocity C;, = actual mean gas velocity relative to actual mean L‘* liquid velocity = C ‘ G , / H G - C,’ = superficial mean liquid velocity, L‘s = GLL U8 = longitudinal distance in reactor L Y = length of reactor = = = = =
a
~
+
from which
L
drive a reaction almost to completion, even small departures from “complete mixing” can be very important.
Y = 130feet
literature Cited
T h e results for this example are summarized as:
(1) Cairns, E. J., Prausnitz, J. M., A.1.Ch.E. J . 6 , 400-5 (1960). (2) Calderbank, P. H., Trans. Inst. Chem. Engrs. (London) 37,173-85 (19591. \-.-.,.
Plug flow Partial mixing (realistic) Complete mixmg
Calcd. Required Length f o r 99.870 Conuersion, Feet 1.67 24.5 130
Conclusions
From the values of diffusivity presented in this paper, plus further examination of Equation 14, several important conclusions can be drawn. First, in planning, interpreting, and scaling u p pilot plant gas-liquid reaction studies, due regard must be given to the values of diffusivity likely to be encountered as a function of reactor diameter. For example, Equation 14 (cvith appropriate values of D s and a computer program) has been used to back-calculate reaction velocity constants from partially mixed pilot plant reactors. Secondly, in any f3ow reactor, there is a complicated relationship among reaction velocity, diffusivity, fluid velocity, and degree of converijion. IVhere a n effort is being made to
(3) Danckwerts, P. V., Chem. Eric. Sci. 2, 1-13 (1953). (4) Ibid.,8, 93 (1958).
(5) Gibbs. M. E.. “Mass Transfer in Gas SDarred Reactors.” AIChE Svmuosiuni. St. Louis. Mo.. ADril 14.’1961 (6) Gilliland, 8. K.: M a s o n , E. i., Znd. kng.C h h . 41, 1191 (1949). (7) Ibid., 44, 218 (1952). (8) Gilliland, C. R., Mason, E. A., Oliver, R.C.: 16id., 45, 1177 \
I
/.I
nz,,
(1 Y J J ) .
(9) Johnson, D. L., Saito, H.: Polejes, J . D., Hougen, 0. A , , A.1.Ch.E. J . 3. 411-17 (19571. (10) Kolbel, H.’, Borchek, E.’, Muller, K., Chem.-hg. Tech. 30,
729-34 (1958). (1 1) Kolbel. H., Sienies, \$-.> Maas, IChem.-Inc Tech. 29, 727 (1957). (18) \Yehner, J: F.: IVilhelin, K. H., Chem. Eng. Sci. S,’ 89-93 (1956). (19) IVesterterp, K. R.? Meyberg, IV. H., Ibtd., 17, 373-7 (1962). (20) \Vilhelni; R. H., Chern. En#. Prnqr. 49, 150-4 (1953). (21) Zweitering: T. N., Chern. Eng. Sri. 11, 1 (1959). RECEIVED for review October, 5, 1964 ACCEPTED April 26, 1965
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OCTOBER 1 9 6 5
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