These calculated CV’s were then empirically correlated by the relationship
A(CV) = 19.5
[&I
0.7 m
where A(CV) = 51.7 - CV. Figure 5 illustrates the excellent fit of this empirical correlation to the calculated CV values. Equation 29 enables a consistent set of parameters q and m to be chosen to simulate a given amount of breakage, as given by the CV from an experimental CSD. Thus, if the level of breakage, y, were independently determined, Equation 29 would permit the selection of a consistent value of m giving the same effect on the CV as measured experimentally. Such a simple breakage model, calibrated in this way, could be inserted in the complete equations describing CSD-Le., Equation 3 together with a mass balance and appropriate nucleationgrowth rate kinetics. Crystal breakage, classified product removal, and decrease in linear growth rate with crystal size all skew CSD, producing a narrower size range of crystals than would be expected from Equation 1. In practice it might be difficult to distinguish among these three possible factors. The presence of solids classification can be determined by careful sampling of both the crystallizer suspension and discharge, comparing both CSD and total solids content between suspension and discharge samples. Changes in linear growth rate with size can be determined by studying the growth of single crystals in the laboratory or inferred by the absence of the other two factors. Crystal breakage can usually be established from microscopic examination of various screen fractions and from experience with handling the solid phase material-e.g., amount of breakage encountered in screening and/or in standardized crushing
tests. Recent work (Austin, 1967) in the area of crushing and grinding indicates that a promising way of independently measuring breakage in a real crystal suspension would be to follow the history of a narrow size range of radioactively tagged crystals. Such direct measurement has obvious advantages over the indirect method of inferring birth and death functions from the over-all affect on CSD, and would allow the formulation of more realistic breakage models. Such refined models, as they become available, can be used to predict CSD using Equation 3, as illustrated in this paper. I t appears to be of interest to extend these CSD calculations, using such experimental or postulated breakage models. An example of the latter case would be the random breakage model, outlined above. Acknowledgment
The writer thanks H. Wengrow and B. Fairchild for their help and patience in assisting the writer to adapt Program AMOS to this problem. Free computer time was made available through the University of Florida Computing Center. literature Cited
Austin, L. G., “Equations of Grinding,” ACS Christmas Symposium, “Characterization of Dispersed System,” Cambridge, Mass., December 1967. Bennett, R. C., Chem. Eng. Progr. 58,76 (1962). Bransom, S. H., Brit. Chem. Eng. 5,838 (1960). Bransom, S. H., Dunning, W. J., Millard, B., Discussions Faraday SOC. 5. 83 119491. Canning, T.‘F., Randolph, A. D., A.Z.Ch.E. J. 13, 5 (1966). Randolph, A. D., A.Z.CI2.E. J . 11, 424 (1965). Randolph; A. D.; Can. J . Chem. Eng. 42,280 (1964). Randolph, A. D., Larson, M. A., A.I.Ch.E. J . 8, 639 (1962). Saeman, W. C., A.I.Ch.E. 5 . 2 , 107 (1956). - 2
RECEIVED for review January 2, 1968 ACCEPTED October 23, 1968
TWO-PHASE COCURRENT DOWNFLOW IN PACKED BEDS JACK M. HOCHMAN AND EDWARD EFFRON
Chemical Engineering Technology Division, Esso Research and Engineering Co., Florham Park, N . J . 07932 Cocurrent trickle flow was investigated in a 6-inch diameter column packed with 3/~e-inchglass beads using Nz and MeOH. Liquid- and gas-phase residence-time distribution measurements were made, and for comparison with previous work were characterized by an effective axial dispersion coefficient and mean residence time. Radial variations in both liquid and gas velocity were small, and liquid holdup was in fair agreement with a literature correlation for countercurrent flow. Axial dispersion in both phases was considerably greater than in single-phase flow. Increased gas-phase dispersion was attributed to bridging of liquid on the packing. Liquid-phase behavior was satisfactorily described by a model which assumed exchange between free flowing and stagnant liquid.
packed-bed catalytic reactors have become increasingly important in recent years. Hydrocrackers and heavy fuel hydrogenation and hydrodesulfurization reactors, for example, generally operate in the trickle flow regime. Better understanding of the fluid dynamics of cocurrent gasliquid downflow in packed beds is required to design these reactors most efficiently (Ross, 1965). Of particular importance is knowledge of the gas and liquid holdup and axial dispersion. IXED-PHASE,
Axial dispersion for single-phase flow through packed beds has been extensively studied. Recently, Bischoff (1966) reviewed this field. However, there have been relatively few data reported for gas-liquid flow, and the great majority of these concern countercurrent air-water flow through Raschig rings. Examples include work by Sater and Levenspiel (1966), Ruszkay (1962), Hoogendoorn and Lips (1965), De Waal and Van Mameren (1965), and De Maria and White (1960). VOL 8
NO. 1 F E B R U A R Y 1 9 6 9
63
Data on holdup and axial mixing in cocurrent downflow are scanty and include the work of Glaser and Lichtenstein (1963), Schiesser and Lapidus (1961), and Reiss (1967). In view of the limited information available on axial dispersion and holdup, cocurrent trickle flow operation was investigated in a 6-inch diameter column packed with 3/1~-inchglass beads using methanol and nitrogen. Methanol was used, since its surface tension (22 dynes per cm.) approaches that of hydrocarbons in commercial operation (-5 to 10 dynes per cm.).
Liquid-Phase Analysis
Liquid tests were made using the two-probe technique in which tracer is injected into the bed in an arbitrary manner and the response patterns are measured at two positions in the bed. As discussed by Sater and Levenspiel (1966), under the proper conditions first and second moments of the bed between two probes can be obtained from the difference in first and second moments seen by the probes:
(4)
Models Used to Describe Flow in Packed Beds
Several models have been used to describe flow and mixing in a mixed-phase packed bed. The first, the “cells in series” model, considers a packed bed to consist of a series of mixing cells in the interstices of the packing. Flow is then characterized by the number of cells in series and the mean residence time (or holdup). The other model often used to describe mixing, the axial dispersion plug-flow model, assumes that spreading can be described by plug flow (uniform velocity) with superimposed axial dispersion. In this case, flow is characterized by two parameters: an effective axial dispersion coefficient and the mean residence time. There is some experimental evidence that considerable liquid stagnancy exists in these systems which is not properly accounted for by either the cells in series or axial dispersion models (Shulman et al., 1955; Smith and Bretton, 1967). Hence, several recent papers (Hoogendoorn and Lips, 1965; Ruszkay, 1962) have suggested a crossflow model to account for this capacitance effect of the liquid. This model assumes that the liquid holdup can be split into two parts: stagnant pockets or films and liquid in plug flow, with exchange between the two. This crossflow model requires three parameters: the fraction plug flow, the exchange coefficient, k , and the mean residence time. Method of Data Analysis
ac
where
L
C,t2dt
ut =
(;)2
bC +% =
b2C
(7)
Ct dt
Equations 4 to 7 were used to obtain the mean residence time and axial dispersion coefficient, D. The parameters of the crossflow model can be obtained in several ways. Ruszkay estimated them from calculated values of the first three moments. However, because of the uncertainty in measuring third moments, the technique used by Hoogendoorn and Lips is probably more accurate. They show that if the flowing liquid is assumed to be in plug flow $ can be determined from the initial breakthrough of tracer: $ =
I n the present investigation both liquid- and gas-phase residence-time distribution (RTD) measurements were made. Liquid R T D data were analyzed with the axial dispersion plug-flow model and the crossflow model in an attempt to establish the superiority of one of them. Since the responses of the “cells-in-series’’ and axial dispersion plug-flow models are essentially equivalent for reasonable ratios of bed length to particle diameter, only the latter model was used. The axial dispersion plug-flow model was also used to interpret gas RTD data. Conventionally, therefore, the axial dispersion plugflow model is expressed by: dt
(5)
tdt,
(8)
and k can be calculated from :
(9) The output response for the two liquid-phase models used was compared directly: 1. The response of the bed between probes to an impulse was calculated for the given model. For the axial dispersionplug flow model, the response was obtained from:
For the crossflow model, an approximate solution to the crossflow equations by Klinkenberg (1948) was used:
dX2 -
and the crossflow model by:
dC
$5
UL
bC
dX
+
[“ $1
+ k(C - c) = 0
1
for the flowing liquid, and
ac
(1 - $ ) - + k ( c - C ) = O at
(3)
for the stagnant zones, where + represents the fraction of liquid in plug flow and k is the exchange coefficient between flowing streams and stagnant zones. 64
I&EC FUNDAMENTALS
(11)
2. Using the experimental response of the top probe as the input, h, the response of the bottom probe (output) was calculated using the convolution integral:
0 -
c;
Regulator
pi
u
I
1
I
Slop Tank
Water Bath
Figure 1 .
Flow schematic of mixed-phase packed bed pilot unit
3. The bottom probe response calculated for each model was then compared to the experimental data. Gas-Phase Analysis
For the gas phase a tracer input step was used, and the response obtained by collecting samples at the bottom of the bed and analyzing in external equipment. The moments were obtained, after correcting for lag in the collecting tubes, from:
1.11 =
rf)
P m
co
I*'[
r n~
g2
=
2
(f>?
tm = PI/(UQ/L)
for downstep (elution) and
for upstep (saturation)
(13)
i
Ct dt
- 1 for downstep
(elution) and
(saturation)
For the values of P obtained, the correction to 1.1 is
