FERMENTATION RESEARCH & ENGINEERING Mass Transfer from Individual Gas Bubbles
I
FRED H. DEINDOERFER and ARTHUR E. HUMPHREY School of Chemical Engineering, University of Pennsylvania, Philadelphia 4, Pa.
Motion picture studies of ascending bubbles show that time-dependent factors are important in mass transfer
T H E PROBLEM
OF
OXYGEN
TRANSFER
in aerobic fermentation processes is chiefly one of dissolving gaseous oxygen in the liquid phase. Because of the low solubility of oxygen in water this problem is treated as a special type of gas absorption commonly referred to as a liquid film-controlled mass transfer operation. Such an operation is best accomplished by dispersing the gaseither oxygen in pure form or air-in either of two ways. The more common is direct gas sparging. The other is gas induction using a draft tube or high speed agitator. The bubbles formed dissolve in the liquid during their formation and rise. Once dissolved, oxygen is transported through the liquid by turbulent and molecular diffusion to the respiratory terminals of the suspended microorganisms. The various steps in this over-all transfer network are shown below. As in any series of consecutive steps, the step offering greatest resistance to transport is the rate-determining one. For aerobic fermentation processes, this step has been shown both theoretically and experimentally to be dissolution of gaseous oxygen into the liquid phase. Process performance as reflected by reaction rate, as well as by ultimate yield, is usually directly related to the rate a t which oxygen can be transported across the gas bubble-liquid interface. Many empirical correlations have been derived which relate over-all
---
oxygen transfer capacity coefficient of fermentation equipment with operating conditions such as superficial air velocity and agitation intensity. Although these correlations sometimes provide a useful means for estimating or scalingup process performance, they tell very little about the actual mechanisms involved in the transfer operation. The capacity coefficient, for example, is the product of the over-all liquid-film mass transfer coefficient and the surface area of bubbles per unit volume of reaction mixture. I n such complex systems as agitated tanks it is most difficult to separate the effects of physical and mechanical factors on the film coefficient and the bubble surface area. To separate factors affecting the transfer coefficient and bubble surface area, a number of investigators (Table I) have studied dissolution of gas bubbles in a column of quiescent liquid. Even here, the approaches taken have not lent themselves to fundamental analyses. Results often are buried in generalized empirical correlations involving the Sherwood, Reynolds, and Schmidt numbers, with little attempt at a mechanistic explanation being apparent. The observed rates of mass transfer were based, in general, on gross effects-i.e., changes in average bulk liquid concentration in a given period of time during which a known number of bubbles contacted the liquid, or the average rate of mass transfer from a bubble as determined by its total change in size as it
Liauid
z
Liquid
/
Gas Film
Film
\ Liquid
Film
Path of oxygen transfer in submerged aerobic fermentation encounters many diffusional resistances
passed through the column. None of the previous investigators considered that since a bubble changes size and character as it rises and dissolves, its mass transfer coefficient should also change. The studies reported here indicate that the most important time in mass transfer from a rising gas bubble to a liquid is the period immediately following release of the bubble, Thereafter, circulation within a bubble rapidly decays, resulting in decreasing rates of transfer. From a practical point of view, it would appear that bubble surface regeneration by multiple impellers, multiple sparging, and other such techniques that enhance new bubble surface formation have distinct advantages in fermentation practice.
Theoretical The method used in this study is based on the fact that the mass transfer rate expression, Equation 1, is instantaneously and continuously applicable to a dissolving gas bubble as it rises through a liquid phase: -dN/dt
= KLA(C*
- CL)
(1)
The following assumptions are made in rearranging and reorganizing this equation : The gas is pure The gas is ideal Henry’s law applies a Humidification efTects are negligible 0 The system is isothermal 0 The liquid is incompressible 0 0
The number of moles of an ideal gas is expressed by Equation 2 : N = pV/RT
(2)
Differentiating Equation 2 with respect to time yields Equation 3 : dN/dt = pdV/RT dt
+ Vd.’dp/RTdt
(3)
I n freshly distilled water the concentration of dissolved gases, especially COS VOL. 53, NO. 9
SEPTEMBER 1961
755
Plexiglas Back & Front
4 Individual gas bubbles were observed in a long column equipped with a special bubble release syringe
\
/ /
Aluminum Sides Painted Black
b Cross-section of column indicates location of translucent screen and transparent side for viewing
For a submerged bubble of moderate size, the total pressure experienced by the bubble is that equivalent to the liquid head above the bubble plus the barometric pressure above the liquid :
used in ihis study, is essentially zero : CL = 0
(4)
The equilibrium concentration of a sparingly soluble gas may be expressed by its saturation concentration: C* = Hp
Table 1.
(5)
b . =
+ PLZB
(6)
Several Investigators Have Studied Mass Transfer from Gas Bubbles to Water Bubble Diameter, Cm.
Bubbling Method
Con, N z , 0 2
0.05-0.56
Continuous
con coz
0.20-0.46 0.160.74 0.30-0.66
Individual Individual Continuous
COZ, C2H6r C2H4, Hz,02 Air
0.15-0.90
Individual
0.060.16
Continuous
Air
0.07-0.16
Continuous
Air, 02
0.32-0.66
Continuous
Air
0.14-0.16
Continuous
Gases
Air,
PLZ
0 2
Neasurement Method
Ref.
Change in liquid concentration with time Total change in bubble size Total change in bubble size Change in liquid concentration with time Meniscus motion in capillary attached t o sealed system Change in liquid concentration with time Change in liquid concentration with time Change in liquid concentration with time Change in liquid concentration with time
(14)
Position of camera, lamp, and platform in relation to column permits continuous record of bubble ascent
Strobulux
d
+rn-
1
(3) (1)
(6) (8)
(IO) (12) (5)
Guide Channel Bolex Camera
u
Platform
756
INDUSTRIAL AND ENGINEERING CHEMISTRY
Plexiglas Cover
Differentiating Equation 6 with respect to time yields Equation '7: d,b/dt = P L dZ/dt
(7)
When Equations 3 to '7 are substituted into Equation 1 and this latter equation is solved for the mass transfer coefficient, Equation 8 results:
Equation 8 shows that bubble volume, surface area, and liquid head must be determined as functions of time to calculate a n instantaneous mass transfer coefficient for a rising bubble.
(4)
Cable to Pulley & Counter Weight Column
Translucent Screen
Experimental The equipment was designed to permit determinations of bubble volume, surface area, and liquid head from a single gas bubble as it ascends along a long verticle path (see diagram above). Orifices of different sizes were used to release bubbles of different initial volumes. A Bolex HI6 motion picture camera, flash-synchronized to fire a Strobclux high intensity lamp, was mounted on a platform; then the entire platform was raised to photograph a bubble continually as it rose in the column. The developed film provided a record of size, location, and time during bubble ascent A crcss-section of the column and the location of the translucent screen on which liquid depth intervals were marked is shown (above). Photographs oj bubbles rising in the column were taken against this screen when the light was flashed through it (left). The system chosen for study, COn and water, was selected for a number of reasons. The most important of these
M A S S TRANSFER are that a pure gas has no gas film resistance; COZhas almost 30 times greater solubility in room temperature water than has oxygen, and thus dissolves more noticeably; and water is a convenient liquid to work with and its physical properties are well established. Results obtained using C 0 2 can be expected to be similar to those that oxygen would yield.
Results Typical excerpts from a motion picture history of the dissolution of a rising bubble are shown (below, left). Over 90% of this 0.48-cm. diameter bubble dissolved during its ascent. Measurements of major and minor axes and bubble location were made from enlarged projections of such photographic records. Bubble shapes during initial periods of rise were unusual (below, right). These photographs are excerpts from two similar runs, pieced together to show the characteristic changes in shape following bubble release. Upon release, the tail of the bubble snaps into the rear of the bubble and carries itself through the bubble and out the forward face. The forward surface then reacts and completely inverts the bubble into a saucer-like shape. The counterreaction to this inversion brings the bubble to
a flattened shape which is followed by rapidly dampened oscillations of the bubble between a narrow high prolate spheroid and a flattened one, To obtain an accurate estimate of initial bubble size, essentially insoluble helium bubbles were released from each orifice and measured after their shape and form stabilized. Stability occurred in slightly less than 1 second after release. Carbon dioxide bubbles were assumed to have the same size upon release as the stabilized helium bubbles. After the measurements were corrected for parallax and enlargement, the volume, area, and locatjon of the bubble were calculated. Figure 1 shows a typical plot of these as functions of time during the rise of a bubble. Plots such as this provide all the information needed to calculate instantaneous mass transfer coefficients at any point along the bubble’s ascent using Equation 8. The results of a number of such runs at five different times during the rise of a number of bubbles are summarized in Figure 2, which relates mass transfer coefficient with bubble diameter. There appears to be a bubble size of about 0.3cm. diameter below which mass transfer coefficients decrease with decreasing bubble size and above which a reasonably constant mass transfer coefficient exists. Also, these results clearly indicate that
the longer the time a bubble exists, the lower is its mass transfer coefficient. A cross plot from this figure taken at a bubble diameter of 0.35 cm. is shown in Figure 3. This plot indicates an exponentially decaying type of decrease in instantaneous mass transfer coefficient with age of bubble.
Discussion That internal circulation plays an important role in gas absorption from bubbles has already been suggested by Hammerton and Garner (6) and more recently by Timson and Dunn (73),based on mass transfer measurements. A similar observation also may be based on the results of this investigation. Such observations should not be surprising, since an analysis of the drag data of Rosenberg (77) for air bubbles in water indicates the existence of a mobile surface for bubbles exceeding 0.1 cm. in diameter. To date, however, this circulation has been assumed a steadystate phenomenon associated with an invariable surface mobility. This highly idealized condition defines a perfectly circulating gas bubble. I n attempting to compare the data obtained in this investigation with the results of earlier investigators, one is faced with their unfortunate oversight of the dynamic decaying character of gas bubbles.
Typical excerpts from motion picture history show dissolution c rising bubble (left) and unusual bubble shapes (right) that occur during initial period of rise VOL. 53, NO. 9
SEPTEMBER 1961
757
e I
0.010
I \ \ \
0.008 -
*, . ,
5.
2
\
3
t
0.006-
P
\l-l
I
Figure 1 . Typical plot relates bubble volume, surface, and location with time during bubble ascent
i i
?? co
, 2
0.004 -
80 m
0.002
100
-
-
3
N
I20
0-
t
U (0
\
E
V
*
e
e
c c 0.04 .-asU 0.05 .-
Figure 2. Instantaneous mass transfer coefficient for various bubble sizes d e pends on a g e of bubble
L
as
3
c
-1
Y
I
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I
0
0.1
0.3 0.4 D, Bubble Diameter, Cm.
I
0.2
0.5
0.6
0
W
0.05 c
c
.-0, 0.04 .-0
L
i 0,
0“
L
0.03
0
W
0
c cu
v)
14 0.02 v e.
J
v v-+-vv
0.01
e
01 0
758
6 Sec. Rigid Surface (Sc hl i cht ing Boundary
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1
0.1
0.2
0.3
D, Bubble
T
I
0.4
Diameter, crn.
CMDUSTRIAL AND ENGlNEEIlNG CHEMlSBRY
I
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0.5
0.6
Layer)
Hammerton and Garner, as well as ‘Timson and Dunn, were quick to grasp the perfectly circulating bubble as a basis for predicting mass transfer coefficients using the mass transfer penetration model theory of Higbie (7). This thcory assumes unsteady-state absorption of a gas by an element of fluid ad,jacent to the surface. The element moves at a uniform velocity from the front of the bubble to the rear as penetration into it occurs. For a perfectly circulating bubble, the total time the element is exposed a t the interface is conveniently chosen as the quotient of the bubble diameter divided by the ascension velocity of the bubble. Prediction based on the penetration theory appears fortuitous for bubbles approximately 1 second old (Figure 4). However, as indicated by this investigation, circulation within bubbles evidently decays, and by the time a bubble is about 6 seconds old, instantaneous mass transfer coefficients begin to approach values predicted for rigid spheres based o n laminar boundary laver considerations. The perfectly circulating gas bubble, therefore, appears to be a much oversimplified model for predicting mass transrer coefficients for dissolving gas bubbles, and for the main part, if used, will yield estimates higher than attainable in columns of even moderate liquid depth. If a mobile surface is involved during transfer, then its mobility apparently decreases with the age of a bubble. Another plausible explanation may lie in the turbulent penetration model for gas-to-liquid mass transfer suggested by Kishinevskil ( 9 ) . H e proposed as a transfer mechanism the actual breach of the liquid surface by minute turbulent gas eddies. This explanation appears every bit as reasonable as that of a mobile surface, especially in view of the exponential type of decay as evidenced by the mass transfer coefficient in Figure 3. T o date only one group of investigators, namely Eckenfelder and Barnhart ( 3 ) , has recognized the effect (but not the cause) that age o l a bubble has on mass transfer coefficients. They esplain certain discrepancies between their data for oxygen absorption during continuous bubbling and those of other investigators in similar esperiments as bring due to “cnd” effects in the experimental columiis. rather than as circulation decay within the bubble. The photographs shown (p. 7 5 7 , right) do show erratic behavior immediately following bubble release, so that the “end” effpct explanation is partly correct. T h e correlation proposed by these investigators included a
4 Figure 3.
Instantaneous mass transfer coefficients for bubble a t two different ages are compared with theoretically predicted va!ues
M A S S TRANSFER column height correction. Quite obviously, column height and bubble existence time are proportional. This, then, suggests a useful approach for comparing the results of this investigation with those of other workers. To determine an ‘(average” coefficient over an extended bubble contact period, results such as those in Figure 3 can be integrated over a particular time interval corresponding to the total contact time these bubbles have with the liquid. The integrand, when divided by the total contact time, as shown in Figure 3, represents an (‘average’’ mass transfer coefficient which can be compared with similar coefficients obtained by other investigators:
Table II.
Comparison o f Mass Transfer Rates from Individual Gas Bubbles Bubble Exposure “Average” Integrand Temp., Diameter, Time, Coefficient, Cm./Sec. Ref. See. Cm./Sec. Gas O c. Cm. 3.45 0.025 0 2 20 0.32 0.030 (12) con 16 0.36 3.36 0.033 0.031 (6) 0 2 18 0.36 2.gb 0.039 0.034 (6) 0.35-em.-diameterbubble, 25’ C. Estimated.
Some criticism may be raised over the fact that circulation decay in an oxygen or air bubble whose size changes at a rate I/aOth or less than the rate at which a COS bubble’s size changes may be different from the circulation decay in the more rapidly changing bubble. The correlation of Eckenfelder and Barnhart (3) for air bubbles in water indicated a variation in average mass transfer coefficient proportional to column height power. Figure 5 shows a to the -l/a reasonably similar relationship for this investigation when the results are plotted as a function of distance traveled. This would lead one to believe that despite the more rapid change in size of COz bubb!es, the decay of circulation within these bubbles resembles that in bubbles of even less soluble gases.
Table I1 compares “average” results of other investigators with the integrand mass transfer coefficient for a 0.35-cm.-diameter bubble as obtained using Equation 9 and Figure 3 for exposure times identical to those reported by the other investigators. The comparison is reasonably good.
Nomenclature
A = surface area of bubble, sq. cm. C* = concentration of dissolved gas in C,
H KL
RL N
p R t
T V 2 Z, pL
Figure 4. Bubble age has a pronounced effect on instantaneous mass transfer coefficient
In
1 CI
-I
‘M
0.01
I I
10
1
I
I
2
I
I
3 4 t Time Sec.
I
I
5
6
Figure 5. So-called “average” or integrated mass transfer coefficient agrees well with empirical relationship suggested b y Eckenfelder and Barnhart
(3) I
20
I
I
I
30 50 IO0 Z T O ~ A,LColumn Height, cm.
I
200
equilibrium with gas phase, moles/ cc. = concentration of dissolved gas in liquid bulk, moles/cc. = Henry’s law constant, moles/ (cc:j (atm.) = over-all liquid film mass transfer coefficient, cm./sec. = over-all liquid film mass transfer coefficient averaged over time interval, cm./sec. = number of moles of dissolving gas = pressure of gas phase, atm. = universal gas constant, (atm.) (cc.)/(mole) (” K.) = existence time of bubble, sec. = absolute temperature, K. = volume of bubble, cc. = liquid depth, cm. = liquid depth equivalent to barometric pressure, cm. = liquid density, grams/ml.
”
literature Cited (1) Coppock, P. D., Meikeljohn, G. T., Trans. Inst. Chem. Engrs. (London) 29, 75 (1951). (2) Ditta, ’R. L., Napier, D. H., Newitt, D. M., Zbid., 28, 14 (1950). (3) . , Eckenfelder, W. W., Barnhart, E. L., 42nd Natl. Meeting, .Am. Inst.. Chem; Enms.. Atlanta. Ga.. Februarv 1960. (4) Guyer, A., Pfister,’X., Helv,‘Chim. Acta 29. 1173 (1946). (5) Hamm;rton,’D., Ph.D. thesis, Univ. of Birmingham, 1953. (6) Hammerton, D., Garner, F. H., Trans. Znst. Chcm. Engrs. (London) 32, S18 (1954). (7)’ Higbie, R., Trans. Am. Znst. Chem . Engrs. 31, 365 (1935). (8) Ippen, A. T., Carver, E., Mass . Inst. Technol. Hydrodynamics Lab. Rept . No. 14, 1955. (9) Kishinevskii, M. Kh., J. AfiflL. Chcm . U. S.S. R.24. 593 (1951). (10) Pasveer, I., Sewage ‘and Z7td. Wastes 27. 1130 (1955). (11) ’Rosenberg. ’B., David Taylor Model Basin Rept. No. 727, 1950. (12) Tereshkevitch, W., Ph.D. thesis, Mass. Inst. Technol., 1956. (13) Timson, W. J., Dunn, C. G., IND. ENC.CHEM.52, 799 (1960). (14) Zdonik, S. B., M.S. thesis. Mass. Inst. Technol., 1942. RECEIVED for review December 28, 1960 ACCEPTED April 25, 1961 Division of Agricultural and Food Chemistry, 138th Meeting, ACS, New York, September 1960. Work supported by a Wilson S. Yerger Memorial Fellowship in chemical engineering and a National Science Foundation Cooperative Fellowship. ’
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