I n d . Eng. Chem. Res 1987,26, 1472-1475
1472
/"
\
FeHL
7
5
,
C = concentration in liquid phase, mol/dm3 E = activation energy, kJ/mol k = reaction rate constant appearing in eq 2, dm3"/(mol"-s) k,, k , , k,', k , = reaction rate constants appearing in reactions b and c kLou = physical liquid-side volumetric mass-transfer coefficient, sel n = order of reaction with respect to Fe"-edta or Fe"-nta R = gas constant, kJ/mol r = overall reaction rate, mol/(dm3.s) T = reaction temperature, K t = reaction time, s
1
/ A '
9
Greek Symbols overall effectiveness factor defined in eq 2 4 = enhancement factor defined in eq 2
II
PH
7=
Figure 12. State of Fez+in Fel*-nta solutions.
(Sada et al., 1986) for Fen-edta and Figure 12 for Fen-nta. Conclusion The reaction kinetics for oxidation of Fe"-edta and Fe"-nta by dissolved oxygen were established. The oxidation reaction of Fe"-edta was found to be first-order with respect to dissolved oxygen and about half-order with respect to Fel'-edta. The oxidation was suppressed by about 30% by adding 20% excess EDTA. The oxidation rate of Fen-nta was shown to be 1st-order with respect to the dissolved oxygen concentration and approximately 0.7-order with respect to the Fe"-nta concentration. The oxidation could not be suppressed at all by adding an excess of NTA. Nomenclature A = frequency factor defined in eq 6, dm3"/(mol".s)
Subscripts
A = solute gas (oxy en) B = FeII-edta or FJI-nta i = value at gas-liquid interface 0 = initial value
Literature Cited Kurimura, Y.; Ochiai, R.; Matsuura, N. Bull. Chem. SOC. Jpn. 1968, 41, 2234.
Lin, N.; Littlejohn, D.; Chang, S.-G. Ind. Eng. Chem. Process Des. Deu. 1982, 21, 725. Sada, E.; Kumazawa, H.; Takada, Y. Znd. Eng. Chen. Fundan. 1984, 23, 60. Sada, E.; Kumazawa, H.; Hikosaka, H. Znd. Eng. Chem. Fundam. 1986, 25, 386. Received f o r review December 16, 1985 Accepted March 4, 1987
Relation between Wave Characteristics and Mass-Transfer Coefficient in Gas-Liquid, Two-Phase, Cocurrant Upward Flow Tahei Tomida Department of Chemical Engineering, The University of Tokushima, Tokushima 770, Japan
Wave characteristics, such as the average frequency, f , the Eulerian-type micro time scale, T,, and the time scale of the apparent wave size, 7w,in two-phase upward flow in a tube of 10-mm diameter and 4-m length were examined by analyzing fluctuations in the sequential film thickness. These wave characteristics could be related to the mass-transfer coefficient, kL, irrespective of the velocities of gas and liquid. kL was proportional to 7, T , , , ~ . ~ and , T , ~ . ~ respectively. , Comparison of these relations with the models of Danckwerts and Higbie shows that T, corresponds to the time scale controlling the mass-transfer rate. Previous papers from this laboratory described the characteristics of wave motion (film thickness fluctuation) (Tomida and Okazaki, 1974) and mass transfer in upward, concentric, two-phase flow through a vertical tube (Tomida et al., 1976, 1978). In two-phase flow, the liquid phase is forced to flow in a thin, wavy film owing to intense shear caused by the gas phase flowing a t higher velocity. The waves may enhance the mass-transfer rate owing to renewal of the film surface as suggested by the model of Danckwerts (1951), which is expressed by kL
=
(DLBS)l/Z
(1)
Accordingly, deducing the surface renewal rate, 4, from the wave motions or the turbulent state of the liquid film 0888-5885/87/2626-1472$01.50/0
may be important for predicting the mass-transfer rate, as well as for understanding the mechanisms of mass transfer in two-phase flow. However, no successful means of determining BS from the turbulent state of a liquid film in two-phase flow has been reported, although many theoretical approaches have been proposed based on various assumptions about eddies responsible for the mass-transfer rate (Banerjee et al., 1968; Lamont and Scott, 1970; Prasher and Fricke, 1974; McCready and Hanratty, 1984). Banerjee et al. (1970) showed that the frequency calculated from the experimental value of the mass-transfer coefficient, kL, based on eq 1 was many times greater than the slug frequency observed in two-phase flow in a coiled tube. Davies and Lozano (1979) determined the predominant 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1473
1)
(hliquid
film
t
-I
- 0 0
50
f CHz]
Figure 1. Electric circuit for measurement of film thickness: (1) electrode, (2) strain meter, (3) amplifier, (4) synchroscope, (5) microcomputer, (6) x-Y plotter.
I 1
0.05r
t
t
100
,&; 0 0
0.05
01
Csl Figure 2. Examples of film thickness fluctuation for U, = 0.22 m time
-I
1
LsCk
OO
S-1.
eddy sizes responsible for mass transfer a t various turbulent surfaces in gas-liquid flow. In the present work, the relationship between the characteristic sizes of waves aed the liquid-side masstransfer coefficients was studied. For this, the characteristics of fluctuations in film thickness (wave motion) in two-phase upward flow in a tube of 10-mm diameter and 4-m length were measured. The data were combined with previous data on kL measured in a geometrically similar test tube (Tomida et al., 1978). The time scales of the film thickness fluctuation, determined by analyzing the autocorrelation function, were compared with the exposure time and the surface renewal rate, which were calculated from the experimental values of kL based on the models by Higbie (1935) and Danckwerts (1951), respectively.
Experimental Section Wave Motion. The experimental apparatus was geometrically similar to that used in previous studies (Tomida et al., 1976, 1978). A vertical, transparent, plastic tube of 10-mm inside diameter and 4-m length was used. Wave motion or film thickness fluctuation was measured by the electrical method described previously (Tomida and Okazaki, 1974). As shown in Figure 1,electrical probes of 1-mm-diameter platinum wire were set 7 mm apart in the axial direction at a position 2 m above the inlet of the gas and liquid. A 5-kI-h and 1-V alternating voltage was applied with a strain meter, and changes of resistance due to changes in film thickness were recorded. Values for the film thickness were stored as discrete values at 1-or 2-ms intervals in the memory of a microcomputer (Sord Mark I1 223 in Japan). Measurements were carried out in the ranges of froth flow and wavy annular flow for the air-water system. Mass-Transfer Coefficient. The data on the liquidside mass-transfer coefficient, kL, were obtained from a previous study (Tomida et al., 1978) by interpolation of flow rates of gas and liquid. Results and Discussion Frequency and Time Scales of Film Thickness Fluctuation. Some examples of wave tracing are depicted in Figure 2. Flow patterns shown in the figure were es-
50
100
f CHz3
Figure 3. Autocorrelation coefficient, r(7), and spectral density, S(f), for waves at U,, = 0.22 m s-'; (A) U, = 4.4m s-l, froth flow; (B) U, = 11 m s-l, froth-annular flow; (C) U, = 29 m s-l, annular flow.
timated from the results by Troniewski and Ulbrich (1984). The film thickness varied randomly with time. With increases in gas flow rate, the flow pattern changed frcm froth flow to wavy annular flow with concomitant decreases of the amplitude, the minimum film thickness, and the dimensions of apparent waves. Similar tendency was observed with decreases in liquid flow rate a t a constant gas flow rate. These statistical characteristics of film thickness fluctuation were similar to those observed previously (Tomida and Okazaki, 1974). In this study, attempts were made to characterize the time scales of film thickness fluctuations. For this, the autocorrelation function, R(itAt), and the spectral density, S c f ) , were calculated from 1 N-k R(kAt) = - e(iAt)t((i + k)At) (2) N - k,=1 and m
S ( f ) = 4CD(hAt)r(hAt) cos (2.lrfhAt)At h=O
(3)
respectively, where D(hAt) is a lag window of Hanning defined as D(hAt) = (1 cos (h7r/m))/2 (4) In these calculations, the total number of data sampled were N = 1000, m = N/4, and h, k = 0-N/4. Examples of the autocorrelation coefficient, r(7) = R(kAt)/R(O), and the spectral density, S ( f ) ,are shown in Figure 3. The figures for the autocorrelation coefficient show that only 10-20% of the components of fluctuation are periodic; the rest are random. The spectral density curves display one or more peaks in the range from 0 to 100 Hz, and the peak frequencies correspond to the predominant frequencies of the periodic components. With an increase in the rate of gas flow, the domain frequencies shift to higher values and they are sometimes separated into many peaks of frequency, due to superposition of the many differnt periodic components produced (see Scf) in
+
1474 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
t 0
b,o
010
0
0
'Y
1
A A R
1 6 IO
I
Ugo
[m
I
e*
s-I]
Figure 4. Relation between the average frequency and superficial gas velocity.
0 63 0 84
A
lh
!t
l:I'lil, pj
1"-
l
0 3
I 0'
IO
I
Cm
Ugo
s - I ~
Figure 6. Relation between the time scales and superficial gas velocity. 301
, , , 8 1 1
I
I
t
n
,
I
-I
Figure 5. Sketch of the autocorrelation coefficient and the time scales of the wave.
ULO 0
Figure 3B). As a result, it is difficult to decide the most probable frequency. Therefore, the average frequency of fluctuation was calculated from eq 5 in the range from 0 to 100 Hz.
f = XSV)fAf/ESV)Af (5) In Figure 4, values off are plotted against the superficial gas velocity, U,, with the superficial liquid velocity, ULo, as a parameter. The average frequency is approximately proportional to Ug00.5, irrespective of the liquid velocity, but data a t U , = 0.1 m s-l tend to deviate from the line a t lower gas velocities. Next, two criteria of the time scale of fluctuation were determined from the autocorrelation coefficient as sketched in Figure 5. One was the "Eulerian-type" micro time scale, 7,, which is given by the intercept on the T axis of a parabola that fits curve, r ( ~ at ) , small T (Hinze, 1975). The other criterion was the time scale, T, which is determined as the time until r(7) first reaches a minimum. Values of T~ may represent a time measure of the meansized waves. As seen from the tracings of film thickness fluctuation in Figure 2, waves arise intermittently on the substrate liquid film. Therefore, the wave dimension may not be connected directly to the wave frequency. The value of l l f may correspond to the average time of spans from peak to peak of waves, irrespective of the wave dimension. In Figure 6, T, and T, are plotted against the superficial gas velocity, Ugo,as a parameter of the superficial liquid velocity, U,. The liquid velocity has little effect on these correlations; both T, and T, are inversely proportional to Uq to the power of about 1.2. The values of 7w are about 4 times those of 7,. Relation between the Mass-Transfer Coefficient and Time Scales of the Film Thickness Fluctuation. The values for the liquid-side mass-transfer coefficient, kL, were obtained from data reported previously (Tomida et al., 1978) by interpolation for velocities of gas and liquid. The liquid velocity has little effect on kL. kL is propor-
Cm 5.'
I
i
0 10
0 0 0 0
0
A A
22 40
63 84
I
I
,
I l l ,
+>, - -
\-..
,
I
- _ _ Higbie
.
Danckwerts
+..
F:
,
O ..s',
%> %bo: itI
I
I
I
I
___ n
O. % ;
-&?...
k i-
uLOCm 0 0 % I% .-s\ O
\
J
0
A A 1
(
I
0
.
0 22 0 40 0 63 0 84
-
1
I
Higbie
- Danckwerts -
0
2l y ) I 0 T
liquid-phase mass-transfer coefficient, kL, irrespective of the velocities of gas and liquid. kL was found to be proportional to f , T , ~ , ~and , T , ~ . ~respectively. , The correlation between kL/DL1I2and T~ agreed well with the theoretical correlations of both Danckwerts and Higbie. This implies that the circulation of liquid within the waves is responsible for the renewal of surface elements in the liquid film.
I I I I I I
/
I
l
I
I
Acknowledgment 1
, ,
l/O, = C X T ,
I l l i l l
I am grateful to Masahiro Kubo and Noboru Suekuni, master-course students of The University of Tokushima, for their cooperation.
Nomenclature DL = diffusion coefficient, cm2s-l D ( h A t ) = lag window, defined by eq 4 f = frequency, s-l f =, average frequency, s-l h, z, k = integer multiplier k L = liquid-side mass-transfer coefficient, cm s-l R ( k A t ) = autocorrelation function, cm2 r(7) = autocorrelation coefficient defined by r(7) = R ( k A t ) /
(7)
R(0) S ( f ) = spectral density, s At = time interval, s = superficial gas velocity, cm
and 0
c27,
(8)
In this case, C1 = 4.3 and C2 = 5.6 are obtained. A still better correlation was obtained by plotting kL/ DL1I2against the time scale of the apparent wave size, T ~ as shown in Figure 9. The good agreement of the experimental data with the theoretical relationships means that T~ may correspond to the exposure time of the newly exposed liquid surface, 0, or to the time of surface renewal, 1/08. In the regions of froth flow and wavy annular flow, the liquid film is wavy turbulent owing to intense shear caused by the gas phase flowing. Waves on the substrate liquid film may be very effective in increasing mass-transfer coefficient, kL, while entrainment and deposition of droplets may be repeated with different time intervals from the wave motions. But contribution of the droplets to the mass-transfer coefficient seems to be small, since the surface of the droplet is rigid; besides the amount of the droplets is limited in this experimental flow regions. Thus, wave motions may be predominant for the mass-transfer coefficient, kL, in upward two-phase flow. The circulation of liquid within waves seems to be largely responsible for the renewal of surface elements in the liquid film.
Conclusions The average frequency, f , the Eulerian-type micro time scale, T,, and the time scale of apparent wave size, T,, were determined by analyzing sequential film thickness fluctuations observed in upward two-phase flow. These characteristics of wave motion could be related to the
P --
superficial liquid velocity, cm s-l Greek Symbols ~ ( i A t )= fluctuation term of film thickness, cm 8 = exposure time by penetration model, s es = surface renewal rate, T = delay time, s T , = Eulerian-type micro time scale, s T , = time scale of apparent wave size, s Lo
,
Literature Cited Banerjee, S.; Scott, D. S.; Rhodes, E. Znd. Eng. Chem. Fundam. 1968, 7, 22.
Banerjee, S.; Scott, D. S.; Rhodes, E. Can. J . Chem. Eng. 1970, 48, 542.
Danckwerts, P. V. Ind. Eng. Chem. 1951,43, 1460. Davies, J. T.; Lozano, F. J. AIChE J . 1979, 25, 405. Higbie, R. Trans. Am. Inst. Chem. Eng. 1935,31, 365. Hinze, J. 0. Turbulence, 2nd ed.; McGraw-Hill: New York, 1975. Lamont, J. C.; Scott, D. S. AZChE J . 1970, 16, 513. McCready, M. J.; Hanratty, T. J. AZChE J . 1984, 30, 816. Prasher, B. D.; Fricke, A. L. Ind. Eng. Chem. Process Des. Deu. 1974, 13, 336.
Tomida, T.; Okazaki, T. J . Chem. Eng. Jpn. 1974, 7, 329. Tomida, T.; Yoshida, M.; Okazaki, T. J . Chem. Eng. Jpn. 1976, 9, 464.
Tomida, T.; Yusa, F.; Okazaki, T. Chem. Eng. J . 1978, 16, 81. Troniewski, L.; Ulbrich, R. Chem. Eng. Sci. 1984, 39, 751.
Received for review April 9, 1985 Revised manuscript received November 14, 1986 Accepted April 11, 1987
