GAS RESIDENCE T I M E IN AGITATED GAS-LIQUID CONTACTOR Experimental Test
of Mass
Transfer Model B E N J A M I N G A L - O R ' A N D W I L L I A M
R E S N I C K
Department of Chemical Engineering, Israel Institute of Technology, Haifa, Israel Gas residence time in an agitated gas-liquid contactor was measured for various gas rates and impeller speeds in an (experimental system designed to be completely similar to systems used by other experimenters who reported on mass transfer rates. The average residence time results could be correlated as a function of (P,/V,)0.45. These results plus literature data for mass transfer rates were used to test a recently proposed theoretical mass transfer model. Experimental values correlated reasonably well with predicted ones. The results indicate that experimental volumetric mass transfer rates should b e correlatable as a function of
+
V / Q 0 ) ' I 3 - 1 ](k/D)If2or Qe/V, the gas holdup ratio. Additional experimental results for E = a[(l power requirements for various gas rates and impeller speeds are reported. authors have recently proposed a theoretical model that permits the c a h l a t i o n of over-all mass transfer rates from gas bubbles in a n agitated vessel with and without simultaneous chemical reaction (7). Knowledge of the average gas residence time or gas holdup is necessary to predict mass transfer rates with the aid of this model. Average gas residence time and holdup have been studied (7, 5, 78), but because the experimental equipment was not geometrically similar to that used by other investigators (3, 77) who determined mass transfer rates, the model could not be tested against experiment from data available in the literature. The present work \vas undertaken to provide the information necessary to permit a test of this theoretical model. Although the nominal residence time can be determined from a knoivledge of gas holdup and volumetric gas flow rate, the nominal residence time will equal the statistical average residence time for only certain cases ( 9 ) . I n this research the statistical average gas rsesidence time in an agitated gas-liquid contactor \vas determinmed from measurements of the residence time distribution. The experimental equipment was geometrically similar to that used by Cooper, Fernstron, and Miller ( 3 ) .so that their mass transfer rate measurements, along Lvith the average residence time results determined in this work, could be used to comp,are the experimental values with the model. Simultaneous measurements of power consumption \vere also made. During the experimental program Westerterp et al. (75) mentioned that the gas residence time distribution in an agitated gas-liquid contactor may be assumed to be equivalent to that expected for a perfectly mixed vessel. Later, Hanhart. Kramers, and Westerterp (8) reported that the gas residence time distribultion in their mixing vessel was intermediate bet\veen that to be expected from one perfectly mixed vessel and that from two perfectly mixed vessels of equal size in cascade. HE
after a pulse of helium had been introduced into the feed gas. T h e experimental system is shown schematically in Figure 1 Air was used as the gas in all runs. The vessel was 39 cm. in diameter and equipped with four vertical baffles, 3.9 cm. wide, a t the wall. Air-free liquid depth was maintained at 39 cm. A vaned-disk impeller with impeller-to-tank diameter ratio of 0.4 was used, similar to those used by Cooper, Fernstron: and Miller ( 3 )and Yoshida et al. (77). The liquid used was either water or an aqueous solution of 1.Y sodium sulfite and 0.001.Y copper sulfate. Thus, the system bore nor only complete geometric but also chemical similarity to that used by Cooper, Fernstron, and Miller (3)in their work on mass transfer rate and power requirements. Additional measurements were also made in water with an impeller-to-tank diameter ratio of 0.3. Air was introduced underneath the impeller through a 8inch-diameter copper tube. The helium pulse \vas simply generated by rapidly rotating a small cock valve 180' (over and beyond the point that connects the two streams). resulting in a very short injection time. T h e air leaving the vessel was sampled through an inverted funnel whose cross-sectional area was l , ' y q of the total cross-sectional area of the vessel. T h e funnel, located so that its lower edge just touched the dispersion surface. was supplied with an auxiliary stream of air whose function \vas to swetqj the gas leaving the dispersion with a minimum of lag to the analytical station, consisting of a thermal conductivity cell and appropriate transducing equipment. Part of this stream was bypassed to the thermal conductivity cell and then recombined with the main sweep-air stream. Flow rates for the sweep-air and bypass stream were
/
.
L
ANALYTICAL- AND RECORDING EQUIPMENT
Experimental
The gas residence time distribution was determined by measuring the composition of the gas leaving the contactor
'
Present address, Department of Chemical Engineering. T h e Johns Hopkins University. Baltimore. Md.
ROTA METE^ Figure 1 . ment
DISPERSION AIR SWEEP AIR
Schematic diagram of experimental equip-
VOL. 5
NO. 1
JANUARY
1966
15
I
A
0
5
io
20
15
2 5 sec.
0.17 4 L i t e r s / m i n . N . 5 2 3 RPM I
0
5
0
IO
I5
20
25
IO
I5
20
25
30
sec.
sec.
0 ; I O 0 Liters/min. N = 4 4 0 RPM
5
Figure 2.
I5
10
sec
Typical C ( t ) functions for gas after helium pulse
08
One Perfect
TWOPerfect Mixers
15% o f G a s in P l u g Flow 0
05
I O
'?
20
30
25
t/8
Figure 3.
Table 1.
Impeller Speed, AV, R.P.M.
78 105 130 185 234 292 304 304 292 234 185 130 105 78 340 322 247 195 136 108
16
Residence time distribution curves
maintained equal and constant for all runs. Preliminary measurements showed that the system's response was independent of the radial funnel location. This is in accordance with the results reported by Hanhart et ai. (a), who found that the gas behaves as though i t were perfectly mixed. The total time delay in the piping, funnel, and instruments was determined by injection of a pulse of helium directly into the sampling funnel, and was found to be essentially that of a pure delay of 1.75 seconds. This time was independent of gas flow rate through the contactor and should also be independent of vessel dimensions. This funnel technique could also be used, therefore, to determine gas residence time distribution in industrial scale reactors and contactors. I n the Hanhart work the correction for time lag and head volume was dependent on gas flow rate and would be a function of vessel size. T h e write-out equipment for the residence time distribution consisted of a Texas Instrument Rectiriter operated at a paper speed of 0 . 5 cm. per second. ' I h e results were recorded directly as a C ( t ) curve ( 4 ) . The stirrer drive was a Chemineer Model ELB dynamometer. Power requirements were calculated from simultaneous measurements of torque and shaft rotational speed. Gas holdup was estimated from measurements of the height of air-free liquid and of the air-liquid dispersion. Experimental Results
Several typical C ( t ) functions are shown in Figure 2. These residence time functions indicate that an increase in impeller rotation speed results in a broadening of the residence time distribution and an increase in the gas flow rate narrows the distribution. The distribution behavior approaches that to be expected for t\co perfectly mixed stages in series but with part of the gas moving through the vessel in plug flow. I n Figure 3 the residence time distribution obtained for a typical run is compared with that to be expected for one and two equal volume perfectly mixed stages with 15% of the gas moving in plug flow (76). These results are in accordance with those reported by Hanhart et al. (8) and Ivith photographic analyses of bubble velocities and flow patterns recently made by the authors (6). Residence time was evaluated from the C ( t ) curves on the basis of the following assumptions: T h e pulse injection time is negligible as compared to the average gas residence time.
Experimental and Calculated Results for Residence Time and Power Requirements for Sodium Sulfite Solution
A i r Flow Rate Volumetric Q . cc ./sec. Linear V,, x 10-3 min.
0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.19 0.19 0 19 0.19 0.19 0.19 0.19 2.0 2.0 2.0 2.0 2.0 2.0
0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.31 0.31 0.31 0.31 0.31 0.31 0.31 3.2 3.2 3.2 3.2 3.2 3.2
Power Requirements No !as: W i t h gas! Po, hP. p,, hP.
0.00119 0,00286 0.00628 0.0164 0.0317 0.0586 0.0633 0 0633 0 0586 0 0317 0 0164 0 00628 0 00286 0 00119 0.00633 0.0586 0.0317 0.0164 0.00628 0.00286
l&EC PROCESS DESIGN A N D DEVELOPMENT
00114 00253 00496 0130 0266 0 0521 0 0584 0 0597 0 0548 0 0282 0 0152 0 00586 0 00286 0 00117 0 0541 0 0456 0 0194 0 0102 0 00476 0 00264
0 0 0 0 0
P*lV,, L b . / C u . Ft. 30.5 68.6 132.8 347 708 1388 1550 3810 3500 1800 970 375 192 74.6 340 287 122 64.3 30 16.6
s,
Residence T i m e
sec. 0.9 1.3 1.9 2.8 3.7 4.9 5.2 7.4 7.3 5.7 4.1 2.7 2.2 I .4 2 6 2.5 1.7 1.1 0.9 0.7
8h
I
sec . ijt .
0.7
1.0 1.5
2 2 2 9 3 8 4 1 5 8 5 7 4.4 3 2 2.1 1 7 1, I 2.0 1.9 1.3 0.9 0.7 0.55
Helium molecular diffusion and gravitational convection in the gas inlet pipe are negligible, so that a true pulse is obtained. The small amount of helium tracer used has no effect on the gas bubble velocities and flow patterns in the vessel. Mass transfer durini: the bubble’s travel time is negligible. Hanhart et al. (8) also made and verified this assumption in their work. T h e average gas resi’dencetime was calculated from
sa
la’ la tC(t)dt
8
=
-2-
-
tE(t)dt
(11
C(t)dt
by carrying out two graphical integrations for each run. Four pulses have been recorded for each operating condition and the maximum deviations of 6 did not exceed 0.1 second for the same run. The experimental and calculated results are presented in Table I and shown graphically in Figure 4 . The data can be correlated as as a function of P z / V s . The correlating equation is
oh
=
0.153 [P0/Vs]0.4S
(2)
where
Oh
=
P, V,
= =
average gas residence time, seconds per foot of gas-free liquid power input, ft.-lb./min. cu. ft. superficial gas velocity, feet per minute
These results can be compared to those obtained by Foust et al. (5), who determined a n “average contact time” as the average bubble residence time per unit of dispersion depth from a knowledge of gas flow rate and experimentally determined holdup. They reported their results as a function of horsepower per cubic foot and gas velocity as feet per second. When their data are recalculated to the units of Equation 2, they can be expressed as
(3) Thus the same slope is; obtained as in Equation 2, even though their data were obtained for a specially designed impeller consisting of arrowhead-shaped blades mounted on a flat disk. The excellent agreement between Equations 2 and 3,
coupled with the residence time distribution behavior, indicates that the statistical average residence time is essentially equal to the nominal average residence time. T h e residence time can, therefore, be estimated by measuring the liquid level before the gas is introduced and while gas is being fed to determine the gas holdup, and the average residence time is then calculated from the volumetric gas flow rate and holdup. The correlation as obtained in Figure 4 shows that only two points are necessary to estimate the residence time behavior over the entire range of operating variables. The data obtained on power requirements (Table I) were correlated according to the empirical equation proposed by Michel and Miller (70). (4)
where Po is the power requirement at the no-gas condition, Pg is the power requirement under gassing conditions, S is impeller rotational speed, L is impeller diameter, and Q is the volumetric gas flow rate. The results are shown graphically in Figure 5, which includes data obtained by the authors for the vaned-disk impeller with impeller-to-tank diameter ratio of 0.3, data by Oyama and Endoh ( 7 7 ) , and the Michel-Miller correlating lines. Although this correlation has no theoretical basis, it may be useful for estimating purposes. Comparison of Experimental Results with Theoretical Model
Gal-Or and Resnick (7) recently proposed a theoretical model for total mass transfer in a gas-liquid agitated contactor, based on the gas residence time. They assumed that the number of bubbles in the vessel at any instant is constant and that the total liquid volume is subdivided into a number of equal-volume elements equivalent to the number of bubbles in the vessel. Each bubble is then enveloped by a spherical shell whose volume is equal to the volume of the liquid element. Each bubble is introduced into each element for a retention time equal to the average residence time of the bubbles. At time equal to the bubble was removed, the liquid element completely mixed, and another bubble introduced. Thus the contact and mixing occurred alternately as in the case of the penetration and film-penetration theories.
e
P , / V . [Ib./cu. ft.1
Figure 4.
Average gas residence time VOL. 5
NO. 1
JANUARY
1966
17
Figure 5.
Power requirements for impellers operating in gas dispersions
-- Data of Michel and Miller. _ _ _Data - of Michel and Miller.
D r = 1 2 inches, 1 = 4 inches DT = 12 inches, I = 3 inches
_ _ A_ _ _ _ Data of Oyarna and Endoh Data of Gal-Or. I / D T 0.30, 0.40 =
By making use of the average bubble residence time, or the gas holdup, it was possible with this model to calculate not only the radius of the spherical shell but also the mass transfer per unit area, total interfacial area, average concentration of dissolved gas in the liquid, and the total average mass transfer rate in the vessel. A change in agitation intensity or gas flow rate would affect 6 and consequently would change the model's radius, average dissolved gas concentration, total bubble surface, and the total average absorption rate. Thus, the effects of mixing and flow patterns in the vessel were depicted by a n indirect mechanism. The differential equation with appropriate boundary conditions was solved for average total rate of absorption in the vessel, lVT,for diffusion with a simultaneous first-order chemical reaction in a semibatch system. T h e result was:
a=o.ecm,ci.i.*e .EXPERIMENTAL 10-6
,:-e
'
',.,','
MTA
'
'
16'
6,
8,
(TABLE
n)
' , , , d
10
3
'
IO
"
j . ' , , , '
Io2
'
, J J
4-J
I l l L U
.,o'
IO
10'
=.[(I+&)$-'!&
Figure 6. Theoretical behavior and experimental results in mass transfer
a coth E
rr
m
-a + b__
-
b ~
sinh f
+ 2 Ee-80
1
-
= a[
(1
k
+ Qe
114;
Po
(a dimensionless variable)
=
b = a[l
k6 f
$]Ii3
(a dimensionless variable)
(model's outer radius)
(6) (7) (8)
Absorption rate data reported by Cooper et al. (3, Table 111) were used in the comparison of experimental rates with the predictions of Equation 5. The data obtained by Cooper as 18
+
they were used to test the The additional parameters obtained as follows: The most reliable value rate constant appears to be (75).
in which
€
exp ( -n z ~ 2 P o / t 2 ) -~ [ b ( - l ) n - a] f2 n%r2
n=l
l&EC PROCESS DESIGN AND DEVELOPMENT
-
equation are shown in T a ble I1 necessary for the evaluation were for the sulfite oxidation reaction that proposed by Westerterp et al.
7.7 X 1012exp(-l2,3000/RT)
sec.?
(9)
The value of 5800 set.-' for k is obtained from this equation a t 22' C. For the air-water system D equals 2.3 X 10-5 sq. cm. per second ( 2 ) and ci equals 2.81 X 10-7 gram mole per cc. (72). The value for c t decreases with increasing solute concentration and for 1.V sodium sulfate concentration it was 48Yo lower than for pure water ( 7 3 ) . L'sing this correction c, is 1.46 X 10-7 gram mole per cc. The bubble radius, a, was assumed to be 0.2 cm.?which appears to be the average bubble radius obtained in systems of this type (7, 74, 75, 78).
Table II.
Experimental Data by Cooper et a/.(3) as Used to Test Mass Transfer Model
NT/V,a L6.-Mole/ Hr.-Cu. Ft.
P J V,,. eh,b V/QB = Lb./,Cu. Ft. Sec./Ft. 60/V&, 5 X 70-3c 1.07 X 33 0.75 246 16.7 1.09 X 630 2.75 46 8.29 1 . 4 0 x 10-3 364 2.15 336.5 7.76 1.62 X ;!32 1.75 31.4 6.95 2 . 0 9 x 10-3 63.9 1.00 18.8 5.41 1 . 5 3 x 10-3 20.5 0.6 31 6.93 5.82 x 10-3 ;!87 1.95 9.64 3.81 1.44 x 10-3 ;!63 1.90 41.6 7.9 2.99 x 10-3 37.8 0.80 18.7 5.4 3.88 x 10-3 3.4 996 23 6.0 5.48 x 10-3 104 1.2 11.9 4.25 1.29 X 3570 6.0 13 4.48 1.44 X 332 2.1 6.96 3.16 5.78 x 10-3 20.8 0.6 9.26 3.72 1.18 X 53.8 0.9 6.03 2.9 2.57 X 95.5 1.2 4.66 2.48 5.91 x 10-3 9050 9.0 21 5.72 2.52 x 10-2 369 2.2 7.73 3 37 7.68 x 10-4 40 0.8 48.1 8.45 2.67 x 10-4 1.74 1.55 122 12.6 4.14 x 10-3 695 2.9 19.1 5.46 8.86 X 940 3.6 52.6 8.76 a From Table ZZZ ( 3 ) for vaned-disk impeller, H / D T = 7. b As estimated from Figure 4. For k = 5800 sec.-I, D = 2.3 X 70-F sq. cm.lsec., a = 0.2 cm., ci = 7.46 X 70-7 gram mole,fcc., and V/QO from preceding column.
transfer data could be correlated as L’?T,’l’ as a function of either 6 o r Qe/V, the gas holdup ratio. Nomenclature
bubble radius, cm.
a
=
b ct
= outer radius of spherical shell, cm. =
C(t)
=
D
= =
DT
E(t) = H = k = L =
A’ NT bo
= =
Po
=
P, Q
= =
t
=
V V,
= = =
8
eh
= =
6
=
Po
=
equilibrium concentration of gas in liquid a t interface, gram mole/cc. residence time distribution function diffusivity, sq. cm./sec. tank diameter, feet residence time distribution function gas-free liquid depth, feet first-order reaction rate constant, sec. -l impeller diameter, feet impeller rotational speed, r.p.m. average over-all absorption rate, lb. moles/hr. power requirement, gassing condition, hp. power requirement, no-gas condition, hp. power input gassing condition, ft.-lb./min. cu. ft. volumetric gas flow rate, cc./sec. (cu. ft./min. in Equation 4 and Figure 5 ) time, sec. liquid volume, cc. or cu. ft. superficial gas velocity, ft./min. parameter defined in equation 7, dimensionless average gas residence time, sec. average gas residence time per unit depth of gas-free liquid, sec./ft. parameter defined in Equation 6, dimensionless
literature Cited
T h e theoretical behavior of the model as calculated with the aid of a digital computer is shown graphically in Figure 6 in the form of full or broken lines. The data points represent the experimental values (Table 11) obtained by Cooper et al. ( 3 ) , which correlate reasonably well with the predicted ones. As indicated by Equation 5, the expression for ”VTis directly proportional to ci. Thus, by using a computer to calculate .TT for one value of ci, it is possible to predict for any desired partial pressure of oxygen. Consequently, a better agreement between the theory and experimental results is possible if the outlet partial pressure of oxygen is estimated (by using Figure 6) and the average partial pressure of oxygen is used to calculate the average ci. However, reliable values for the parameters, such as ci, D,and k , are still not known for this system. Conclusions
The proposed model predicts mass transfer rates that appear to correlate reasonably well with the experimental data now available. Further experimental work as well as accurate values for the reaction rate constants, diffusivities, and solubilities for other systems will be necessary if this model is to be checked over a broad range of operating and system variables. FUIther, it vlrould appear that experimental mass
(1) Calderbank, P. H., Trans. Znst. Chem. Engrs. (London) 36, 443
(1958). (2) Chiang, S. H., Toor, H. L., -4.Z.Ch.E. .I. 5 , 165 (1959). ( 3 ) Cooper, C. M., Fernstron, G. A,, Miller, S. A., Znd. Eng. Chem. 36, 504 (1944). (4) Danckwerts, P. V., Chzm. Eng. Sci. 2, 1 (1953). (5) Foust, H. C., Mack, D. E., Rushton, J. H., Znd. Eng. Chem. 36, 517 (1944). (6) Gal-Or, B., Resnick, W., A.f.CI2.E. J . 11, 740 (1965). (7) Gal-Or, B., Resnick, W., Chem. Eng. Sci. 19, 653 (1964). (8) Hanhart, J., Kramers, H., Westerterp, K. R., Zbid., 18, 503 (1963). (9) Levenspiel, O., Bischoff, K. B., Aduan. Chcm. Eng. 4, 103 (1963). (10) Michel, B. J., Miller, S. A., A.Z.Ch.E. J . 8, 262 (1962). (11) Oyama, Y . , Endoh, K., Chem. Eng. ( J a p a n ) 19, 2 (1955). (12) Perry, J. H., “Chemical Engineers’ Handbook,” 4th ed., pp. 14-6, McGraw-Hill, New York, 1964. (13) Seidel, A., “Solubilities of Inorganic and Metal Organic Compounds,” 3rd ed., Vol. 1, p. 1354, Van Nostrand, New York, 1940. (14) Westerterp, K. R., Chem. Eng. Sci. 18, 495 (1963). (15) , Westerterp, K. R., von Dierondonck, L. L., de Kraa, J. A., Ibid., 18, 157 (1963). (16) Wolf, David, Resnick, William, IND. ENG.CHEM.FUNDAMENTALS 2, 287 (1963). (17) Yoshida, F., Ikeda, A., Imakawa, S., Miura, Y., Znd. Eng. Chem. 52, 435 (1960). (18) Yoshida, F., Miura, Y., IND.ENC. CHEM.PROCESS. DESIGN DEVELOP. 2, 263 (1960). RECEIVED for review January 19, 1965 ACCEPTED August 24, 1965
VOL. 5
NO. 1
JANUARY 1966
19
