Ind. Eng. Chem. Fundam. 1983, 22, 500-502
500
COMMUNICATIONS Mass Transfer from a Solid Sphere to Power Law Fluids in Creeping Flow Mass transfer from spheres of benzoic acid to power-law fluids, consisting of aqueous solutions of carboxymethyl cellulose, was examined in the creeptng flow regime. The rate of transfer Increased with Peclet number, but it tended to vary by as much as a factor of 4 from theoretical predictions. The influence of fluid pseudoplasticity could not be determined possibly because of an increasing effect of natural convection on the transfer process as the continuous phase approached Newtonian behavior. Introduction Mass transfer from a single sphere to a flowing fluid has received considerable attention because of ita importance in many chemical engineering processes. Until recently, much of the previous work has focused on mass transfer from fluid and solid spheres to Newtonian fluids. For example, Friedlander (1957,19611, Bowman et al. (1961), and Ward et al. (1962) have examined transfer in the creeping flow regime. On the other hand, Steele and Geankoplis (1959) investigated transfer in highly turbulent flow while Garner and Suckling (1958),Garner and Keey (1958,19591, Garner and Hoffman (1960,1961), and Steinberger and Treybal(1960) have proposed correlations covering a wide range of Reynolds numbers and including both free and forced convection transfer. Because the utilization of non-Newtonian fluids has now become widespread in the chemical industry in areas such as polymer and paper processing, fermentation, waste disposal, and many other biological operations, research efforts have been increased in order to elucidate the mass transfer mechanisms between an isolated sphere and a non-Newtonian continuous fluid phase. Hirose and Moo-Young (1969, 1972), Wellek and Huang (1970), Gurkan and Wellek (1976), Bhavaraju et al. (1978), and Kawase and Ulbrecht (1981) have developed models for the transfer between fluid or solid spheres and non-Newtonian fluids in the creeping flow regime. Shirotsuka and Kawase (1973) and Wellek and Gurkan (1976) have also examined mass transfer between fluid spheres and a non-Newtonian continuous phase in the intermediate Reynolds number region. In general, these studies indicated that mass transfer increases with increasing pseudoplasticity of the continuous phase. Recently, Kumar et al. (1980) experimentally examined mass transfer between a solid sphere and pseudoplastic fluids over a wide range of Reynolds number. Using an effective viscosity for the fluid, the results were correlated by an expression showing that the natural and forced convection mechanisms can be adequately represented by an additive relationship, as has been recommended for Newtonian fluids by Steinberger and Treybal (1960). This preliminary investigation attempts to explore experimentally the mass transfer phenomena that occur between a soluble solid sphere and a power-law fluid in the creeping flow regime. The effect of the non-Newtonian character of the fluid phase on the transfer process will be examined and compared to theoretical predictions. Experimental Section A diagram of the test loop is illustrated in Figure 1. The 0196-4313/83/1022-0500$01.50/0
fluid was drawn from a supply tank and fed into a horizontal 2.74 m long X 7.62 cm i.d. pipe. A by-pass line before the test section allowed the flow rate to be reduced to the desired flow regime. The flow rate was regulated by means of a rotameter located prior to the test section and a valve at the end of the section. A benzoic acid sphere, having a nominal diameter of 2.3 cm, was placed along the axis of the pipe 24 cm from the end of the test section. The temperature during all of the runs was maintained at 298 f 1K. Two types of carboxymethyl cellulose (CMC) obtained from Hercules Co. were used to prepare the aqueous solutions. A medium molecular weight 7MF CMC was used to make solutions having concentrations of 1.5 and 3.0% by weight, while a low molecular weight 7LF CMC was used to obtain a 4.5% solution. The densities of the CMC solutions and the solubilities of benzoic acid in the solutions were determined experimentally at 298 K. Diffusivities were estimated from data published by Kumar and Upadhyay (1980). The rheological properties of the solutions were obtained with a Brookfield viscometer, Model LTV, and the data were fit to a power-law model. The power-law constants were evaluated by methods developed by Runikis et al. (1958) and Sikdar and Ore (1979). The properties of the solutions used in this work are listed in Table I. The mass transfer behavior was determined by calculating the S h e r w d number at the various flow rates from the following analysis. The mass transfer coefficient k, can be related to the mass flux N of benzoic acid by the empirical equation N = k&C, - c b ) (1) where C, is the solubility of benzoic acid in the solution at the solid-fluid interface, and Cbis the solute concentration in the bulk fluid. For this work, C b = 0 because the solution entering the test section was solute-free. The Sherwood number for mass transfer with a sphere is defined as
where D, is the sphere diameter and Dv is the diffusivity of the solute in the solution. Combining eq 1and 2 yields (3) The mass flux was computed from the weight loss of the sphere, the average sphere surface area during a run, and @ 1983 American Chemical Soclety
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983 501
Table I. Properties of Solutions at 298 K
CMC concn,
wt % 0 1.5 3.0 4.5
density, kdm3 997 1002 1013 1017
K , g/(cm s2-") 0.008937 0.821 16.94 14.26
n
sol of benzoic acid, kdm3
1.0 0.9935
3.2 5.6
D, X 1O'O m2/s 9.1 7.4
PVC Pipe
H20
C o o l i n g Water t o Drain Feed Tank, B= Moyno Pump, C= By-Pass L i n e , D= Globe Valve E- Rotorneter, F- Pressure Gage, G= Thermometer, H= Gate Valve I= R e c e i v i n g Tank, J= Cast Sphere
A-
0 0
1.5%CMC 3.0%CMC 4.5%CMC
-
Figure 1. Diagram of test loop.
the duration of the run. D, represents the average sphere diameter during the course of a run, and C, and D,were obtained from Table I for runs with a particular solution. Finally, since the sphere cross section was about 10% of the flow area in the test section, the experimental data were examined on the basis of a mean velocity as defined by Vliet and Leppert (1961). Results and Discussion Analysis of mass transfer behavior in the creeping flow regime has shown that the Sherwood number is commonly represented as a function of the Peclet number (Lochiel and Calderbank, 1964). For transfer between a solid sphere and a power-law fluid, Kawase and Ulbrecht (1981) derived the following equation
0
1.5% CMC
0
3.0% CMC
1
1
4 5 % CMC
162
I
16~
8 1 1 1 ' ' "
Id'
'
' ' 1 1 1 ' 1 1 '
" l l t 1 8 '
Id
I
1
8 1 1 1 1 1 1 '
I
10
1 1 1 1
102
NRe
huation 4 simplifies to the result obtained by Friedlander (1961) for mass transfer with a Newtonian continuous phase. Figure 2 shows the relationship between the Sherwood and Peclet numbers from our experimental data along with the prediction by eq 4. Although mass transfer increased with increasing Peclet number for a particular fluid, the Sherwood number tended to vary by as much as a factor of 4 in comparison with theoretical predictions. However, as the fluid phase became more non-Newtonian, the agreement with eq 4 seemed to improve. This result could be attributed to a decrease in the effect of the natural convection mechanism on the transfer process as the continuous phase was made more non-Newtonian. The magnitude of the Grashof numbers for the experiments with the different fluids tend to support this hypothesis, because they were on the order of for the 3.0 and 4.5% CMC solutions, about 20 for the 1.5% CMC solution, and approximately lo5 for water. Moreover, Garner and Keey (1958) indicated that forced convection would be the dominant transfer mechanism when the Reynolds number exceeds a limiting value given by Nh > 0.4NGR1/2Ns~1/6 (5) The Reynolds numbers for the experiments using the 3.0 and 4.5% CMC solutions were either near or above the limiting value given by eq 5, while the Reynolds numbers
Figure 3. Effect of Reynolds number on mass transfer.
using the 1.5% CMC solution were either near or below the limiting value. Thus, it seems that the effect of pseudoplasticity on mass transfer in the creeping flow region may have been concealed by the influence of the natural convection mechanism, which appeared to increase as the continuous phase approached Newtonian behavior. In addition, the results when using water were expected to be greater than predictions by eq 4 because the Reynolds numbers for those tests ranged between 11 and 120, which is well beyond the creeping flow region. Figure 3 compares our data to the following correlation proposed by Kumar et al. (1980)
Nsh = Nsho + 0.8869NRe1/2Ns:/3
(6)
where NReand Nh are evaluated using an effective viscosity for a power-law fluid and Nsh0 represents the contribution of molecular diffusion and natural convection to the overall transfer process. There are significant differences between eq 6 and our results, which could be attributed to the following factors. First, the bulk of the data used to develop eq 6 appeared to be in a higher Reynolds number region. Most of our results were obtained in the creeping flow regime where the Sherwood number is a function of Nh1f3instead of Nh1f2.Second, there seem
502
Ind. Eng. Chem. Fundam. 1983, 22, 502-503
to be some differences in the values of the diffusion coefficient used by Kumar et al. (1980) and by us. This factor also may account for a portion of the discrepancy and indicates the need for reliable and accurate ways for measuring or calculating diffusion coefficients for nonNewtonian fluids because of their importance in the development of a successful model. Conclusions Mass transfer from benzoic acid spheres to pseudoplastic fluids has been investigated in the creeping flow region. The Sherwood number increased with Peclet number but tended to vary by as much as a factor of 4 in comparison with theoretical predictions. The effect of pseudoplasticity on transfer may have been concealed by the contribution of the natural convection mechanism, which seemed to increase as the continuous phase approached Newtonian behavior. Nomenclature Cb = solute concentration in bulk fluid (ML-3) C, = solubility of solute in fluid D, = sphere diameter ( L ) D, = diffusivity ( L 2 T 1 ) k , = mass transfer coefficient ( L T l ) K = power law consistency index (ML-"P-2) n = power law index of determination N = mass flux (ML-2T1) Npe = Peclet number = ( V D J D ) N R =~ Reyonolds number = (P%-~D,"/K)= (pVDs/we) NS, = Schmidt number = (we/pDv) Nsh = Sherwood number = (k$JD,) Nsb = S h e r w d number for diffusion and natural convection V = mean velocity (LT')
p = density (ML-9 ke = effective viscosity
(ML-lT') = K ( D , / V)l-n
Literature Cited Bhavaraju, S. M.; Mashelkar, R. A.; Blanch, H. W. AIChE J . 1978, 2 4 , 1063. Bowman, C. W.; Ward, D. M.;Johnson, A. I.; Trass, 0. Can. J . Chem . Eng , 1961. 3 9 , 9. Friedlander, S. K. AIChE J . 1957, 3 , 43. Friedlander, S. K. AIChE J . 1961, 7 , 347. Garner, F. H.; Hoffman, J. M. AIChE J . 1960, 6 , 579. Garner, F. H.; Hoffman, J. M. A I C M J . 1961, 7 , 148. Garner, F. H.; Keey, R. B. Chem. Eng. Sc/. 1958, 9 , 119. Garner, F. H.; Keey, R. B. Chem. Eng. Sci. 1959, 9 , 218. Garner, F. H.; Suckling, R . D. AIJ . 1958, 4 , 114. Gurkan, T.; Wellek, R. M. Ind. Eng. Chem. Fundam. 1976, 15, 45. Hirose, T.; Moo-Young, M. Can. J . Chem. Eng. 1969, 4 7 , 265. Hlrose, T.; Moo-Young, M. Ind. Eng. Chem. Fundam. 1972, 1 1 , 281. Kawase, Y.; Ulbrecht, J. J. Chem. Eng. Commun. 1981, 8 , 213. Kumar, S.; Tripathi, P. K.; Upadhyay. S. N. Lett. Heat Mess Transfer, 1980, 7, 43. Kumar, S.; Upadhyay, S. N. Ind. Eng. Chem. Fundam. 1980. 19, 75. Lochiel, A. C.; CaMerbank, P. H. Chem. Eng. Sci. 1964, 19, 471. Runikis, J. 0.;Hall, N. A.; Rising, L. W. J . Am. Pharm. ASSOC.1958, XLVII, 758. Shirotsuka, T.; Kawase, Y. J . Chem. Eng. Jpn. 1973, 6.432. Slkdar, S. K.; Ore', F. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 722. Steele, L. R.; Geankoplls. C. J. AIChEJ. 1959, 5, 178. Steinberger, R. L.; Treybal, R. E. A I C M J . 1960, 6 , 227. Wet, G. C.; Leppert, G. J . Heat Transfer 1961, 83. 163. Ward, D. M.; Trass, 0.; Johnson, A. I. Can. J . Chem. Eng. 1962, 4 0 , 164. Wellek, R. M.; Gurkan, T. A I C M J . 1976, 22, 484. Wellek, R. M; Huang, C-C. Ind. Eng. Chem. Fundam. 1970, 9 . 480.
Chemical Engineering Department University of Lowell Lowell, Massachusetts 01854
Michael A. Hydet Alfred A. D o n a t e l l i *
Receiued for review October 10,1981 Revised manuscript received June 13, 1983 Accepted July 8,1983 'General Foods Corp., Woburn, MA 01801.
Neutron Attenuation: A Novel Approach to Residence Time Studfes in Coal Hydrogenation Reactors A novel approach to the measurement of resldence time distributions (RTD) in chemical reactors is described. The technique involves the attenuation of neutrons by tracers and is particularly sukable for obtaining residence time measurements in coal hydrogenation reactors.
Introduction The successful scale-up and optimization of continuous reactors for coal hydrogenation will rely heavily on generalized kinetic data obtained in laboratory-scale and pilot-scale reactors. Residence time distribution (RTD)data are essential for such generalizations to be made with confidence. Previous attempts to measure RTD's in coal hydrogenation reactors have involved the use of techniques which are not applicable to studies of reaction kinetics (Barreto et al., 1977; Vasalos et al., 1979). Standard optical and spectrophotometric methods are also unsuitablemainly because of the thickness of reactor walls, the difficulties and hazards associated with having optically transparent windows in high pressure-high temperature reactors, and the opaqueness of reaction products. We have developed a technique which involves the attenuation and scattering of neutrons by tracers and which is particularly suitable for the measurement of RTD's in coal hydrogenation reactors. Several elements, including boron, cadmium, and gadolinium, have markedly different neutron absorption cross sections to carbon and hydrogen which are the major reaction species in coal hydrogenation.
Other atoms such as deuterium and helium have different neutron scattering cross sections to hydrogen. Hence any of these elements is distinguishablefrom the reactants and is thus a potential tracer in coal hydrogenation reactors. The technique is readily adaptable to reaction processes other than coal hydrogenation and also has potential in the area of powder technology where determinations of material homogeneity and effectiveness of mixing are required. Experimental Section The results presented here were obtained by passing thermal (very low energy) neutrons through a section of a simulated three-phase reactor. Since little obstruction to neutron flux is provided by the stainless steel reactor walls or the surrounding heaters and insulation, the degree of attenuation of neutrons by the reaction components can be monitored by the detection and counting of emergent neutrons. The apparatus used in this study is shown schematically in Figure 1. The energy of neutrons radiating from the h / B e source is reduced (thermalized)by the surrounding wax. The howitzer produces a directed flux of thermalized 0 1983 American Chemical Society
