Low Reynolds Number Mass Transfer in Packed Beds of Cylindrical Particles Surendra Kumar, S. N. Upadhyay,' and V. K. Mathur Department of Chemical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-22 1005, India
Mass transfer coefficients from dumped packed beds of cylindrical pellets of benzoic acid to water and 60 % aqueous propylene glycol solution are measured in the N R range ~ of 0.01 to 600. Conventional correlations in terms of mass transfer factor and Reynolds number as well as asymptotic relations (Karabelas et al., 1971) in terms of Grashof, Reynolds, Schmidt, and Sherwood numbers are presented.
Introduction Mass transfer between particles and fluid in packed beds has been widely studied because of its fundamental importance in many chemical engineering operations and considerable published information on the subject is available in the literature (Upadhyay and Tripathi, 1975b). Most of the reported data are confined to moderate and high flow rates. Low flow rate studies are made mostly with gaseous systems and relatively few data are available for liquids. Recently such information has assumed a greater importance in the development of processes such as ion exchange, chromatography, etc. In the present investigation, therefore, new mass transfer rate data are obtained by measuring the rate of dissolution of compressed benzoic acid pellets into water and 60% aqueous propylene glycol solution. Data obtained in the Reynolds number range of 0.01 to 600 are correlated and compared with the published data. Experimental Section The apparatus and experimental procedure employed were very similar to these used by Williamson et al. (1963) and Upadhyay and Tripathi (1975a). Measurements were made with test columns of diameters 4.27,5.60, and 8.00 cm. Each of these was made of Pyrex tubing and was about 50 cm in length. The inlet flow of the test fluid in each case was upflow through a packed bed of known weight of benzoic acid pellets (weighed to the nearest 0.05 mg) sandwiched between two layers of inert particles. Glass beads of 6-mm diameter were used as inert particles in all the measurements with a 4.27-cm diameter column, and glass pellets of dimensions identical with the test pellets were used with 5.60 and 8.0-cm diameter columns. At low rates, the flow through the bed was a gravity-induced upward flow and the rate was measured by collecting the existing fluid in a measuring vessel. At higher rates, the flow was maintained by means of a centrifugal pump and was metered by means of a precalibrated rotameter. The average rate of dissolution of benzoic acid pellets was determined by finding their weight loss after drying them to a constant weight in a desiccator a t the end of the run. For runs using aqueous propylene glycol solution, the pellets were washed with saturated aqueous benzoic acid solution before drying in the desiccator. Measurements of the inlet and outlet temperatures were also made during the runs. Depending upon the flow rate, the runs lasted for a period of 20 to 30 min. Void fraction of the bed was determined from the weight of the pellets in the active bed and its height which was obtained from at least ten measurements taken at different radial positions. Fresh pellets were used for each run. In a separate set of blank runs, the loss in weight during charging of the pellets into the test column, during their re-
moval from the same, and during their washing with saturated aqueous benzoic acid solution (only for runs using aqueous propylene glycol) was determined. At least ten such measurements were made for each pellet size and the mean of these in each case was used as the correction factor. The blank run weight loss depended on pellet size and ranged from 2.3 to 20% of the total weight loss during the test runs. All the test run readings were corrected by subtracting the appropriate blank run weight losses. Those runs where the blank run correction was more than 20% of the total weight loss were rejected. The loss in weight by sublimation during drying in the desiccator was found to be negligibly small and, therefore, no correction was considered necessary. Measurements were made with seven cylindrical pellets of benzoic acid. All the pellets were made from Sarabhai-Merck chemically pure benzoic acid. The pellet characteristics are listed in Table I. The pellet diameter and thickness for a particular size are the mean of respective measurements made on 50 randomly chosen samples. The density of the pellets was determined by a liquid displacement method using saturated aqueous benzoic acid solution. The measured dissolution rate for a run was converted to the mass transfer coefficient, k.,, defined by
where
Equation 1 can be easily obtained by applying the material balance across a differential element of the active bed and by integrating the resulting expression assuming k , to be independent of concentration change because of the low mass transfer rate. This in turn was converted to the Chilton--Colburn (1933) and Gaffney-Drew (1950) mass transfer factors defined by
and
respectively. The Reynolds number was calculated using the superficial flow velocity and equivalent particle diameter, D,. The relevant physical properties a t the average temperature of a run were evaluated from the respective graphs prepared on the basis of the reported values. Solubility data for benzoic Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
1
Table I. Characteristics of the Pellets ~
~
~~~~~~~
No.
~~
~
Thickness, h, cm
Diameter, d , cm
Surface area, A,," cm2
Volume,
4.508 4.103 3.828 2.4410 2.2750 2.1540 0.9097
0.4876 1.276 1.283 2 0.3770 3 0.3080 1.283 0.8760 4 0.4494 0.9598 5 0.2750 0.8780 6 0.3415 0.5537 7 0.2464 a A, = (Irdh + nd2/2). V , = (rd2h/4). Dp = VZ$. 1
V,,b
D,,C
Density,
cm3
cm
p4,g/cm3
0.6228 0.4872 0.3983 0.2707 0.1989 0.2068 0.0593
1.198 1.143 1.104 0.8817 0.8512 0.8278 0.5383
1.285 1.338 1.262 1.265 1.290 1.251 1.252
Table 11. Diffusivity of Benzoic Acid in Water --__
-
Temp, "C D , X lo5, cm2/s
10 0.5786
15 0.6757
20 0.7796
25 0.8915
30 1.012
35 1.140
40 1.276
46 1.420
50 1.571
0.85 I 2 n
Y
u,( crn/sec.)
Figure 1. Effect of particle size on k,. Table 111. Range of the Parameters Covered Particle shape Particle diameter, D,, cm Column diameter, D,, cm Red height, L , cm Void fraction NRt? N S C
Flat-end cylindrical pellets 0.5383-1.198 4.27, 5.60, 8.00 (2.81-5.09) f 0.05 0.3360-0.4560 0.01-600 767-42400
acid for this purpose were those reported by Seidel (1941), Seidel and Linke (1952), Steele and Geankoplis (1959), and Stephen and Stephen (1963). Molecular diffusivities were calculated by Wilke-Chang's (1955) relation and are listed in Table 11. Viscosity data for water were taken from the work of Perry (1963). All relevant physical properties for 60% aqueous propylene glycol solution were those of Steinberger and Treybal(l960). The range of the various variables covered in the present work is given in Table I11 and the detailed tabular listing of the complete data is given in Table S1 (deposited as supplementary material). See the paragraph at the end of the paper regarding supplementary material. Discussion 1. Effect of Particle and Column Diameter. As observed by previous investigators (McCune and Wilhelm, 1949; Upadhyay and Tripathi, 1972; 1975),the mass transfer coefficient, k increased with decreasing particle diameter. A graphical presentation of this effect is shown in Figure 1. At 2
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
lower flow rates ( u < 0.2 cm/s), the k , values for various pellet sizes were nearly the same. This is probably due to the increased predominance of the natural convection over forced convection a t low flow rates. Comparison of the mass transfer coefficient values for the same particle size obtained with different column diameter revealed no effect of the latter on the mass transfer coefficient. 2. Correlation. In correlating their results, previous investigators have used many varied approaches (Upadhyay and Tripathi, 1975). Most of the reported correlations are in terms of mass transfer factor and particle Reynolds number. The following Reynolds numbers
N R =~DpG//l
(5)
(6)
N R ~ "= DpG/p(l - €1
(7)
have been frequently used in the previous correlations. The exponent of the Schmidt group in the mass transfer factor is still open to question. The majority of the investigators have, however, used the Chilton-Colburn (1933) Jd factor with the Schmidt group exponent as T?. Some have used the mass transfer factor with 0.58 as the Schmidt group exponent which was first used by Gaffney and Drew (1950). Carberry (1960), on the other hand, has shown that the Schmidt group exponent depends upon the magnitude of the former and is /'z for N s , < 1 and y3 for N s c > 1. Pfeffer (1964) also arrived a t a similar conclusion. Some workers have tried to improve the
100
.
I
D p , CM
Dc
CM
WATER
AQ.PROF GLYCOL
IO
~10. The regression results for various situations are listed in Table S2 (deposited as supplementary material). These results indicated that i89~’
(NR,”
< 20)
(8)
1.5006(N~,”)-’ 4099
(NRe”
> 20)
(9)
J,j = 4.2258(N~,”)-’ and Jd =
correlate present data with lowest deviations of 21.84 and 10.0%, respectively. These deviations are only marginally better than those for d d =
1 . 1 2 9 9 ( N ~ , ) - ~ ’ * ~ ~( N R p
< 10)
(10)
d d =
0.4422(N~&”~’*’
> 10)
(11)
and (NRe
which correlate the data with 21.98 and 10.43% deviations, respectively. Equations 8 through 11are compared with the corresponding experimental data in Figures 2 and 3. The deviations for the corresponding correlations involving J d ’ are found to be smaller, but the improvement is only marginal and is not helpful in finalizing the Schmidt group exponent. In order to have a clearer picture of the effects of void fraction and Schmidt group, correlations have also been developed using the published literature data which covers a void frac-
tion range of 0.26 and 0.632 and a Schmidt group range of 123 to 70 600. Regression analysis of the entire data made in the same manner as above gave d d =
1.110(NF 10)
(I: 20)
(15)
give, however, larger deviations. A comparison of eq 12 through 15 with the experimental data is shown in Figures 4 and 5. From the larger calculated deviations and from Figure 5, it is clear that eq 14 and 15 are not suitable for correlating the ordered and distended packed bed data because the data for beds having high void fractions (Rowe and Claxton, 1965; Jolls and Hanratty, 1969; Karabelas et ai., 1971) deviate too much from the correlating line. As observed above here again the deviations for the correlations involving the J d ’ factor were consistently better than those for the corresponding Jd factor correlations and all of the data can be successfully correlated by
tJa’ = 0 . 5 6 4 3 ( N ~ , ) -“ I~
~ N ~ 10
for
< 10,123 < Nsc < 42 000 and
(30)
for 0.001 < N R
