Golovina, E. S.,Khaustovich, G. P., "Eighth lnternation Symposium on Combustion", p 784, 1962. Jenkins, R. G.. Nandi. S. P., Walker, P. L., Jr., Fuel, 52, 288 (1973). Nandi, S. P., Lo, R., Fischer, J., Prep. Pap. Natl. Meet. Div. FueiChem., Am. Chem. Soc., 88 (1975). Parker, A. S.,Hottel, ti. C., ind. Eng. Chem., 28, 1334 (1936). Satterfied. C. N., "Mass Transfer in Heterogeneous Catalysis", M.I.T. Press, p 41, 1970. Sergeant, G. D., Smith, I. W., Fuel, 52, 52 (1973). Tu. C. M., Davis, H., Hottel, H. C., ind. Eng. Chem., 26, 749 (1934). Turkdogan, E. T., Olson, R. G., Vinters, J. V., Carbon, 8, 545 (1970). Walls, J. R.. Strickland-Constable, R. F., Carbon, 1, 333 (1964).
Walker, P. L., Rusinko, F.,Austin, L. G., Adv. Catai., 11, 133 (1959). Wakao, N., Smith, J. M., Chem. Eng. Sci., 17, 825 (1962). Wen, C. Y., Dutta, S.,"Reaction Rates of Coals and Chars with Carbon Dioxide", report submitted to the Energy Research and Development Administration, Morgantown, W.Va.. 1975.
Receioed for reuiew October 31, 1975 Accepted J u l y 8, 1976 T h e work was supported by grants f r o m t h e U n i t e d States Energy Research and Development Administration.
Backmixing and Liquid Holdup in a Gas-Liquid Cocurrent Upflow Packed Column G. J. Stiegel and Y. T. Shah' Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 1526 1
An experimental investigation was carried out to determine the effects of gas and liquid flow rates and packing size on the liquid phase axial dispersion coefficent and the liquid holdup in a gas-liquid, cocurrent, upflow packed column. The experimental measurements were carried out in a 2-in. i.d. glass column using approximately '/*-in. diameter polyethylene extrudate with two different lengths. Sulfuric acid was used as a tracer and its concentration in the liquid phase was determined by measuring the electrical conductivity of the liquid phase. The data were analyzed by the method of moments and by the evaluation of the transfer function of the axial dispersion model. The measured liquid holdup was in excellent agreement with those reported by previous investigators and was dependent upon both gas and liquid velocities. Unlike in a bubble column, the liquid Peclet number was found to be dependent on both gas and liquid flow rates. The experimental data for liquid Peclet number are compared with the previously reported data of Heilmann and Hofmann (1971) obtained under different ranges of packing size and gas and liquid velocities.
Introduction During recent years a considerable amount of literature has been published on the dynamics of a packed column. A quantitative understanding of the fluid behavior, such as the liquid holdup and the backmixing in packed bubble columns, is of considerable importance for the proper design of packed, multiphase catalytic reactors. A vast amount of work has been published on countercurrent, gas-liquid packed columns. Michell and Furzer (1972) and Chung and Wen (1968) have summarized a number of these studies. More recent articles have been published by Hoogendoorn and Lips (1965), Sater and Levenspiel (1966), Mears (1971), Co and Bibaud (1971), and Chen (1975). For cocurrent downflow systems considerably less information is available. Hochman and Effron (1969) and Charpentier (1971) studied the effect of gas and liquid flow rates on the liquid holdup and the liquid backmixing coefficient. Bischoff (1966), Mashelkar (1970), and Wen and Fan (19751, have reviewed the literature in these areas along the work published for single phase flow. Recent developments in hydroprocessing (Montagna and Shah, 1975) have created a considerable interest in understanding the dynamics of a two-phase cocurrent upflow packed column. In particular, the quantitative correlations among the liquid and gas flow rates on the liquid holdup and the liquid phase backmixing coefficient are of considerable interest. l'urpin and Huntington (1967) have studied theholdup characteristics in cocurrent, upflow packed columns. Chen et ai. (1971) recently studied the dispersion of a liquid in a single phase, upflow packed column while Eissa et al.
(1971) and Kato and Nishiwaki (1972), have studied the liquid phase dispersion in a bubble column. To date, only Hofmann (1961) and Heilmann and Hofmann (1971) have published data concerning dispersion in a gas-liquid, cocurrent, upflow packed column. Results of these studies showed that the liquid phase dispersion coefficient is dependent on both gas and liquid flow rates. Heilmann and Hofmann (1971) presented a correlation for the liquid phase Peclet number as a function of liquid and gas holdup, liquid Reynolds number, and the particle size. Their experimental data were, however, obtained for the particle sizes larger than the ones commonly encountered in catalytic hydroprocessing reactors. The purpose of this paper is to present correlations for the liquid holdup and the liquid phase backmixing coefficient as functions of gas and liquid phase Reynolds numbers and the particle size for a two phase cocurrent upflow packed column. The results are obtained for the packing size similar to the ones encountered in catalytic hydroprocessing reactors. The empirical correlations for the liquid phase Peclet number presented in this paper are based on the experimental measurements obtained using standard tracer analysis.
Experimental Section Figure 1 presents a schematic diagram of the apparatus employed in this study. The packed column was constructed using a 2-in. i.d. glass pipe with a glass tee and a reducer located a t the ends of the pipe. The reducer a t the entrance served as a transition for the fluid between the piping and the packed column and also to hold the packing support plate in place. The tee located at the exit was used to hold the retaining Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1. 1977
37
Dra'n4 Solumeter
ar
Recorder
Flowmeter
Solumeter
D
r
Injection a i port
.
'
~Fl;meters ~ Water inlet
Figure 1. Schematic diagram of experimental apparatus.
screen in its proper position and to serve as a transition zone from the packed column to the discharge pipe. The support plate and retaining screen were constructed of 316 stainless steel with %2-in. perforations arranged in a triangular pitch on approximately I/4-in. centers. The 7-ft column was packed completely with (0.11 in. diameter X 0.22 in.) or (0.123 in. diameter X 0.123 in.) polyethylene extrudates supplied by Gulf Research, and Development, Harmarville. The column was packed by filling in 5-in. increments and tapping gently with a plunger. This was done in order to pack the bed as firm as possible so that the bed would not lift when subjected to high gas and liquid velocities. Tap water was used as the liquid phase and was taken directly from the main laboratory supply line. The liquid superficial mass velocities ranged from 2200 lb/h-ft2 to 18 400 lb/h-fP. The flow rates were measured with two calibrated rotameters. Air, the gas phase, was taken from the laboratory supply line and its flow rate was measured with a calibrated rotameter. A pressure gauge located a t the rotameter exit was used to measure the gas pressure so that a correction factor could be applied to the calibration curve. The gas flow rates employed in the present study varied from zero to approximately 500 lb/h-ft2. Sulfuric acid was used as the tracer element and was introduced into the system via the stainless steel injection port located immediately before the reducer at the base of the column. The injection system consisted of a straight portion of piping with a bypass where the acid was injected through a capped tee connection. Three valves were employed in this injection system: one on the main line (valve B) and two on the bypass line (valves A and C). All piping on this injection port was of the same size so that no change in the flow characteristics occurs when the flow is switched from the straight portion to the bypass for the purpose of injecting the tracer. Electrical conductivity was used to measure the concentration of tracer a t specified points in the system. Taps were installed along the tube for the purpose of inserting the conductivity probe into the bed. These taps were located 1,4, and 6 ft from the base of the column. The conductivity probes were connected to two solumeters supplied by Beckman and Industrial Instruments Co. The response was recorded on a two-channel Hewlett-Packard strip chart recorder and could be converted to concentration by a prior calibration. Startup procedure for this unit consisted of running the system a t a liquid velocity of approximately 13 000 lb/h-ft2 for at least 0.5 h. This was done in order to thoroughly wet the 38
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
band and to run enough water through the main line to achieve a relatively constant water temperature. Once a steady-state temperature was obtained, the solumeter temperature compensators were adjusted to that temperature. The desired flow conditions were then set on the rotameters and ample time was allowed for the system to equilibrate. After equilibrium was achieved, the recorder was manually started and zeroed. A small quantity of tracer was then injected into the bypass line. After this valve A was manually opened followed by a rapid simultaneous opening and closing of valves B and C, respectively. Sufficient time was allowed for the response of the system to return to its initial value before another injection of tracer was introduced.
Method of Analysis Several types of models have been used by previous investigators to describe the flow conditions and mixing patterns in packed columns. One of these is known as the cells in series model. This is a one-parameter model which views the system as a series of equal size ideal stirred tanks. The flow is represented by a parameter which represents the number of tanks in series. A second model which has been used extensively in the study of nonideal flow is the dispersed plug flow model. This model assumes that the backmixing occurring in a system can be described by plug flow which is superimposed with some degree of backmixing. This model implies that no stagnant pockets or gross bypassing exists in the system. The prediction of this model should then range from plug flow to mixed flow. A generalized expression for the material balance of this system is given by:
Because of experimental evidence that some liquid stagnancy exists in packed columns which is not accounted for in the above two models, Hoogendoorn and Lips (1965) have suggested the crossflow model to account for these stagnant pockets. This model assumes that the total liquid holdup can be split into two parts: a stagnant pocket and the flowing liquid, with mass transfer occurring between the two. In the present study the dispersed plug flow model was assumed applicable. This model has been used extensively in the past to describe the flow behavior in packed beds that do not deviate considerably from plug flow. Even when the flow behavior deviates considerably from plug flow, Wen and Fan (1975) state that this model can still be applied in each phase when two phases are involved. In this study the flow behavior of the liquid phase is observed and therefore eq 1 could be applied. Equation 1can be simplified by first assuming that the axial velocity is constant over the cross section of the bed. According to Sater and Levenspiel (1966) this assumption is reasonable if the ratio of the column diameter to the packing size is greater than 8. A second assumption is that no concentration gradients exist in the radial direction. This assumption is also reasonable if the axial velocity is uniform and if the tracer is introduced over the entire cross-sectional area. Introducing these assumptions into eq 1 reduces this model to what is known as the axial or longitudinal dispersion model. The following equation is then obtained for changes in one direction:
ac
at
ac
a2c
+ U ~ J I =
In dimensionless form this equation yields (3)
In the present study the two-probe technique described by Sater and Levenspiel (1966) was employed. This method permits the tracer to be injected in any arbitrary fashion upstream of the first probe. The boundary conditions for eq 3 for an open system with this tracer injection technique are presented and discussed in detail by Aris (1959) and Van der Laan (1958). According to Bischoff and Levenspiel (1962), the assumption of an open system (i.e., undisturbed flow at the boundary) is valid if the probes are located at least three particle diameters into the bed. In this study all probes were located at least 1 f t into the bed, hence validating the assumption of an open system. In previous studies the response curves obtained a t the two-probe position for an arbitrary tracer input have been analyzed by evaluating the first and second moments of the response curves. These moments are calculated from the equations below:
Wl
=
Jo
)(;
C;t d t '
(4)
tm Cid t
F ( s ) = exp
[(%) ($,
%) )] 1I 2
(1
- (1 +
(10)
Pe d,
This equation can then be rearranged into a linear equation of the form:
For a stable, one-dimensional, linear system, the transfer function can be evaluated from the experimental residence time distribution curves obtained by numerical integration of the transient response to any arbitrary function measured at two locations (Michelsen and Ostergaard, 1970.) Values of the transfer function for the linear system can then be calculated from the following equation for arbitrary values of s:
JO
J= C i t 2 dt J " C i dt Since the first moment of the response curve is essentially the mean of that curve, the average residence time of the tracer can be calculated by taking the difference of the first moments of the response curves. tm =
L
(ru2 - 11.1) U
(6)
The second moment of the response curves indicates the spread of the curves and the difference of the second moments is a measure of the amount of backmixing occurring between the two measuring points. For the open system, the second moment has been derived analytically by Levenspiel and Smith (1957). u2 = 2
(5)+ (z) 8
0
2
(7)
Aris (1959) has shown that for a one-shot injection of tracer eq 7 can be simplified without introducing any significant error to:
In terms of the dimensionless particle Peclet number, eq 8 becomes
From eq 9 the Peclet number and the dispersion coefficient can be calculated. Because the location of the cut-off point of the response curves have affected the analysis of previous investigators by the method of moments, an analysis technique involving the Laplace transform of the axial dispersion model and the evaluation of a linear transfer function as described by Ostergaard and Michelsen (1969,1970) was also employed. These investigators claim this method produces much more consistent results than the method of moments regardless of the severity of the tailing and the location of the cut-off point. Taking the Laplace transformation of the axial dispersion model, eq 3 yields:
Plotting [log (l/F'(s))]-l vs. s[log (l/F(s))]-? from eq 11 should yield a straight line if the axial dispersion model is applicable. From this line the slope and intercept yield the values of the average residence time and the Peclet number. In order to calculate the dispersion coefficient from the Peclet numbers obtained from the above techniques, B characteristic packing dimension is required. For cylindrical packings used in this study Mears (1971) characterized the equivalent spherical diameter by the equation:
dd&,+ $d 2
d, =
(13)
In this study, the Reynolds and Peclet numbers are based on ds.
Results and Discussion A. Liquid Holdup. From the evaluation of the mean residence time of the liquid phase in the test section, the fraction of the bed occupied by the liquid phase can be calculated. This value is commonly referred to as the total liquid holdup and is expressed as the volume of liquid in the bed to the bed volume. The total holdup is the sum of what is referred to as the static holdup and operating holdup. Operating holdup is defined as the amount the liquid flowing through the column when in operation while the static holdup is defined as the amount of liquid retained on the packing after the column is allowed to drain. For the two packings used in the present study, static holdups were found to be approximately 0.11 for the (0.11 in. diameter X 0.22 in.) packing and 0.14 for the (0.123 in. diameter X 0.123 in.) packing. Somewhat higher values of static holdups are probably due to the high surface area per unit volume of the packing used in the present study. The total liquid holdup was calculated from the following equation:
HtL=
(14) PtVO where the values for the void fraction e were found to be approximately 0.41 and 0.35, respectively. The mean residence time used to calculate the liquid holdup was obtained from Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
39
Table I. Comparison of Two Methods of Data Analysis Bed height, ReL
Ret,
ft
tm,
127
137
5
12.8 12.9 7.9 121.6 111.4 39.1 23.1 30.7 19.7 34.5 33.5 21.4 13.2
5
3 16
16
40
98 97 92 90 64
5
5 32 32 51
1.0
.8
I
-
5
3 5 3 5 5 5
71 73
3
--
Turpin and Huntlngton (1967)
A
d,:0173
o
d , = O 150 i n
Pe
s
-
0.410 0.479 0.496 0.382 0.394 0.238 0.226 0.476 0.519 0.236 0.363 0.455 0.437
12.8
12.9 7.9
121.7 111.4 39.1 23.1 30.7 19.7 34.5 33.5 21.4 13.2
t
6'
v)
.4
Pe
s
I *
in
D 0
d
tm,
0.407 0.476 0.480 0.380 0.319 0.262 0.211 0.483 0.518 0.221 0.332 0.445 0.429
V
E
Method of Ostergaard and Michelsen
Method of moments
I
-
0 10
20
30
(G,/G,
)O
40
50 0
''
02 H,,=l
04
06
4 7 Re,'
ReG.'
08 (ad,
IO
jo"
Figure 2. Comparison of liquid phase holdup data with previous in. vestigators.
Figure 3. Parity plot of the observed liquid holdup vs. those obtained
both the method of moments and by the evaluation of the transfer function. Table I presents some of the data obtained by using the two analysis techniques. As can be seen, excellent agreement is obtained for the mean residence time from both techniques. A comparison was made of the total liquid holdups obtained in this study under various operating conditions with those of Turpin and Huntington (1967). This comparison is illustrated in Figure 2. Turpin and Huntington obtained their data using 0.025 and 0.027-ft diameter alumina particles and correlated their data using the ratio of the liquid mass velocity to the gas mass velocity. The correlation they presented (their eq 15) is applicable only for the range 1 I ( G L / G G ) " ~ I * 6. Their correlation, however, exceeds one for values of ( G J G c ) ~2. 5.69. ~ ~ This is impossible since the total liquid holdup must be less than or equal to 1 and should approach 1 asymptotically for GI, >> GG. As shown in Figure 2, the slope of the present data is not as great as that of Turpin and *; at larger Huntington at low values of ( G L / G C ) ~ , ~however, values the present data seem to be starting an asymptotic approach to 1.The scatter in the data shown in Figure 2 is of the same order of magnitude as the one reported by Turpin and Huntington. Because fair agreement was obtained between the present liquid holdup data and those of Turpin and Huntington, it can be concluded that the present experimental procedure provided reliable results. The liquid holdup data were correlated by the expression:
The values for the constants in the above expression were determined by a nonlinear least-squares regression analysis. The following values for the constants along with their asymptotic standard deviation were obtained: Si = 1.47, u = 0.111; b = 0.11, u = 0.005563; c = -0.14, u = 0.00488; a = -0.41, u = 0.0405. Figure 3 presents a parity plot of the observed liquid holdup vs. those predicted from the above correlation. As can be seen, good agreement is obtained between the measured and predicted values of the liquid holdup. This good agreement is also evident from the magnitude of the standard deviation of each parameter shown above. A comparison of the total liquid holdup as calculated from the above correlation at various operating conditions is presented in Figure 4. As shown, eq 15 predicts an increase in the liquid holdup with increasing liquid flow rates and decreasing gas flow rates. The correlation also predicts an increase in the liquid holdup with an increase in the particle size or a decrease in the (a&,) term. B. Axial Dispersion Coefficient. Data for the evaluation of the liquid Peclet number were collected a t various gas and liquid flow rates, with two different packing sizes, and for bed heights of 3 and 5 ft. The results were obtained by using both the method of moments and the method proposed by Ostergaard and Michelsen (1969). A comparison of the liquid phase Peclet number predicted by these two techniques is illustrated
40
Ind. Eng. Chern., Process Des. Dev., Vol. 16,No. 1, 1977
from eq 15.
lor--
,
I
, , ,
I
,
IO
,
.e n
?
.6
ul
n 0
J
W
a
.* .I
t
---
d,
:oiso
4
8"
.2
I
I
I
,
100
IO
0
300
P e , : 0 128 Re:
Re,
Figure 4. Comparison of the liquid holdup from eq 15 a t various operating conditions.
-'i
100
0 2
04
245
06
08
10
ReGo16
Figure Parity plot of the liquid phase Peclet number vs. lose obtainec From eq 16.
Chen,Monno, ond Hines (1971) --&-d, :0 I73 i n
60
U
t al u
W Y)
,
n
N
0
r
2 W
n
n
0 X
0'
0
02
04
Pe,: 0 031 R e f ' * I
2
4
u x
6
8 IO
20
06
Re,"'
00
IO
( a S dI ) O
30
Figure 7. Parity plot of the Peclet number correlation obtained for the 3-ft packed section.
toz f t / s e c
Figure 5 . Comparison of the single phase axial dispersion coefficient with literature values.
in Table I. As shown, in general good agreement is obtained in the predictions by the two methods. An another check on the experimental procedure, data were collected and evaluated for a single liquid-phase system and compared with those of Chen and coworkers (1971) for %-in. packing. Figure 5 presents the results of Chen with those for the single-phase system obtained in the present study. The dispersion coefficient were somewhat smaller than those of Chen; however, the work of Chen showed a decrease in dispersion coefficient for smaller packings. Thus, the present results seem consistent with other reported single-phase data. The data for Peclet number obtained by the method of moments for bed heights of both 3 and 5 f t were correlated to the following expression. Per, = E'ReLb'Re~c'(acls)a'
(16)
Performing the same nonlinear least-squares regression analysis as was done for the liquid holdup, the following values for the constants along with their asymptotic standard deviation were obtained: E' = 0.128, u = 0.07; b' = 0.245, u = 0.0438; C' = -0.16, u = 0.0356; a' = 0.53, u = 0.29. Figure 6 presents a parity plot of the observed Peclet number vs. those obtained from the above correlation. Be-
cause of the large amount of data collected only the average Peclet number of each set of conditions is plotted in Figure 6. Part of the scatter in Figure 6 may be due to the random variation in the dispersion from point to point in the bed (Hochman and Effron, 1969). The data for both 3 and 5 f t bed length are included in Figure 6. There is no allowance for the effect of bed length in eq 16. Separate correlations for the Peclet number for 3 and 5 f t bed length were also obtained. The parity plots for these correlations are shown in Figures 7 and 8. As shown, the scatter in the points of these figures is smaller than the one shown in Figure 6. The scatter in Figures 7 and 8 is partly due to the random variation in the dispersion from day to day. This may be caused by the random variations in the pressure of the laboratory air supply line and small random variations in the laboratory water temperature. Part of the scatter can also be caused by the excessive tailing in the residence time distribution obtained in some experiments. The method of analysis for the Peclet number used in this was reasonable and it did not cause the scatter in the data shown in Figures 7 and 8 (Michell and Furzer, 1972). Using the method outlined by Hochman and Effron (1969) (eq 10 of this reference contains a sign error in the exponential term), output responses were calculated. Using the Peclet number calculated by the method of moments, a typical comparison between the measured and calculated residence time distriInd. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
41
10
.8
.6
I
tt
-
I
I
I
I
A
d s = 0 1 7 3 in
o
ds=0150in
I
I
I
I
1
/
-
I
0
0.2
0.8
0.6
0.4
1
I
1
I
I
/
I
!
,
,
I
Re,
1.0
Figure 11. Comparison of the Peclet number for the 3-fl Pe,: 0 . 0 2 2 Re:.'
section at various operating conditions.
R C , ~ . ' ~( ~ ~ d , ) ~ . "
Figure 8. Parity plot of the Peclet number correlation obtained for the 5-ft packed section. .e 14
w
0
I
,
,
,
,
,
,
,
,
,
,
.6 -
, ~ S D O ~c uSr i~e
-Actual ---PledlCfLd
ICSDOnSC
.4
-
.P
-
.I
I
c u r i e from convolution
.e
-
Lz
6 -
: e
\
I
3.
\\
4 -
D
P
0
u
e
'..
I I'
2 -
~
-.
I
,
10
IO
,
I
1
1
1
1
1
J
1
1
IO0
300
12 1 4 16 i e 20 22 24 26 28 30 T i m e , scc
Figure 12. Comparison of the Peclet number for the 5-ft packed
Figure 9. Typical comparison of the actual output response curve
section at various operating conditions.
with that predicted from the convolution integral: ReI. = 101, ReG = 132, d , = 0.150 in., PeL = 0.28, L = 5 ft.
IO
'
" " " y
1
1
:O 150
d,
II
I
,
I
,
,
,
100
10
,
1 300
Re,
Figure 10. Comparison of the liquid phase Peclet number from eq 16 a t various operating conditions.
bution is shown in Figure 9. Just as shown by Hochman and Effrom (1969), the predicted and measured response curves agreed reasonably well, indicating the reliability of the data analysis by the method of moments in this study. A plot of the Peclet number obtained from eq 16 as a function of the gas Reynolds number at a few typical liquid Reynolds numbers is shown in Figure 10. From this plot it can be seen that the Peclet number increases with decreasing gas flow rate, increasing liquid flow rate, and the value of as&. The Peclet numbers as functions of typical gas and liquid Reynolds numbers and packing size obtained from two separate correlations for 3 and 5 f t bed lengths are shown in Figures 11and 12, respectively. Although the effects are similar, it appears 42
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
1O.l
1
IO2
I
I
I
in
1
flow rate
I I I I I
1
lo3
1
1
I
I l l 1
IO4
ReL/H,odp3 ' ( c m )
Figure 13. Comparison of the present data for the liquid phase Peclet number with those of Heilmann and Hofmann (1971). that, in general, for a given set of conditions larger Peclet number is obtained at larger bed lengths. The present results are qualitatively in agreement with those reported by Hofmann (1961) and Heilmann and Hofmann (1971). The liquid velocities (0.25-3 cm/s) and the gas velocities (13-75 cm/s) used in the present study were considerably higher than the ones used by Heilmann and Hofmann (1971). Furthermore, the packing sizes used in the present study were considerably smaller than the ones used by them. As shown in Figure 13, the present data were fitted reasonably well by the correlation of Heilmann and Hofmann (1971), particularly at low liquid Reynolds number.
Conclusions From the present study the following conclusions are made. (1) Liquid holdup and the liquid Peclet number are dependent on both gas and liquid flow rates and the packing size. The dependence of the axial dispersion coefficient on both the liquid and gas flow rate is found to be mild. (2) Bed height appears to have some effect on the liquid Peclet number. An increase in bed height appears to somewhat increase the Peclet number. (3) The correlation of Heilmann and Hofmann (1971) fits the present data well at low liquid Reynolds numbers. At large liquid Reynolds numbers, the present data show somewhat larger values of (H,L/PeL) than the ones predicted by Heilmann and Hofmann (1971). Acknowledgment The help of Mr. B. Alexander of Gulf Research and Development for supplying some parts of equipment is very gratefully acknowledged.
Nomenclature a, - = external surface area per unit volume of particle a, E' = constant b, b' = constant c, c' = constant C = concentration of tracer C(s) = Laplace transformed distribution of tracer concentration C ( t ) = distribution of tracer D = axial dispersion coefficient d, = packing diameter d, = equivalent packing diameter F ( s ) = transfer function G = superficial mass velocity Htl, = total liquid holdup Htc = total gas holdup L = distance between measuring points L , = length of packing m = liquid mass flow rate PeL = liquid Peclet number uLd,lDL ReG = gas Reynolds number, d,Gc/pG ReL = liquid reynolds number, dsGLlwL r = reaction rate term s = Laplace transformation parameter S = source term t, = average residence time of the tracer u = real mean axial velocity of liquid
ui = velocity in ith direction of liquid V O = empty column volume x, = ith coordinate direction y = y coordinate direction z = dimensionless position Greek Letters = void fraction of the bed 8 = dimensionless time w = first moment p = liquiddensity a = standard deviation a* = variance a , a' = constants in eq 15 and 16, respectively t
Subscripts 1 = inlet 2 = outlet L = liquid G = gas i = position of response curve
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Received for review November 14,1975 Accepted August 9, 1976
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
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