I n d . Eng. C h e m . Res. 1989, 28, 105-107
105
Mass Transfer by Natural Convection from a Solid Sphere to Power Law Fluids Torng-Lih Lee and Alfred A. Donatelli* Chemical Engineering Department, University of Lowell, Lowell, Massachusetts 01854
The mass-transfer behavior from solid spheres of benzoic acid t o non-Newtonian fluids, consisting of various types of (carboxymethy1)cellulose solutions, was examined experimentally in the natural convection flow regime and compared to theoretical models. Since the models were based on boundary layer theory, deviations between experimental results and theoretical predictions increased as the diffusion mechanism became more important in the overall transfer process. However, a recently proposed correlating equation appeared to be able t o predict the mass-transfer behavior over the entire laminar flow region. Mass transfer by natural convection between spheres and a continuous fluid phase has received attention because of its theoretical interest and practical significance in the field of chemical engineering. Although most practical operations involve multisphere systems, a thorough understanding of the single-sphere case is necessary before more complicated situations can be analyzed. Natural convection transfer between a sphere and Newtonian fluids has received more attention than transfer with non-Newtonian fluids even though the use of the latter types has increased in recent years. Garner and Suckling (19581, Garner and Keey (19589, and Garner and Hoffman (1960) investigated mass transfer between a solid sphere and Newtonian fluids in the laminar flow regime. Garner and Keey (1958) and Garner and Hoffman (1960) also examined the natural convection process in the turbulent flow region. A theoretical analysis of laminar free convection heat transfer between power law fluids and two-dimensional or axially symmetric surfaces has been performed by Acrivos (1960) using boundary layer theory. His analysis can be adapted easily to spherical surfaces and to the analogous mass-transfer problem. Stewart (19711, using boundary layer theory, examined laminar free convection in threedimensional systems. His analysis also can be extended to spherical surfaces and can be used to describe the mass-transfer process with power law fluids. Although the derivations of Acrivos and Stewart use different characteristic lengths and velocities, they yield identical predictions for the mass-transfer coefficient. Amato and Tien (1976) investigated experimentally natural convection heat transfer from isothermal spheres to aqueous polymer solutions in which the fluid behavior was described by the power law model. Their results corresponded well to an equation of the form derived by Acrivos (1960) for the thin boundary layer region but showed an upward deviation from theoretical predictions a t smaller rates of heat transfer. For fluids that have very small Grashof numbers, such as extremely viscous liquids, little motion would occur from the natural convection process. In this case, the diffusion mechanism contributes significantly to the transport process, and models using the thin boundary layer assumption do not apply. So, workers such as Singh and Hasan (19831, Geoola and Cornish (1981, 1982), and Fujii et al. (1981) have used numerical methods to solve the momentum and energy equations for heat transfer between a sphere and a Newtonian fluid. These investigations showed that the Nusselt number asymptotically approached a value of 2 as the Grashof number decreased. Thus, for the corresponding mass-transfer process, the Sherwood number should approach the same limiting 0888-5885/89/2628-0105$01.50/0
value. More recently, Churchill (1983) has proposed theoretically based correlating equations for either heat or mass transfer, which cover the entire range of the natural convection process. For laminar flow, the correlation was based on solutions for the thin boundary layer covering a range of Prandtl and Schmidt numbers from 0.7 to infinity. Then, a limiting value of 2 was added to the equation so that the final expression also could be used for small Grashof numbers. Churchill even showed that the correlating equation could be modified to encompass power law fluids. This work attempted to examine experimentally the natural convection mass-transfer process between a soluble solid sphere and non-Newtonian fluids within the laminar flow regime. The experiments were performed with benzoic acid spheres of different sizes that were immersed in pseudoplastic liquids consisting of aqueous solutions of (carboxymethy1)cellulose.
Experimental Section The mass-transfer experiments were performed in a jacketed column made from Plexiglas. The column had a height of 0.76 m and a 0.30-m square cross section. The column jacket was designed so that water could be pumped around it rapidly from a separate constant-temperature reservoir. Connections between the pump and jacket were made with rubber tubing in order to eliminate vibrations from the pump. The column was supported on a framework of light-angle iron and also set on a polystyrene foam pad to minimize the effect of any external vibrations. The temperature of the fluid in the column was controlled a t 298 f 0.5 K. The sphere was suspended at a distance of 0.23 m from the top of the column by a fine wire attached to a support at the top of the column. The test specimen were made by careful casting of molten benzoic acid using a technique similar to that of Steinberger and Treybal(1960). Spheres having nominal diameters of 0.0127,0.0191, and 0.0222 m were molded and used for the experiments. The clearance between the sphere surface and the sides of the column was considered to be larger than the thickness of the boundary layers, so that wall effects should be negligible. This contention can be supported by the work of Amato and Tien (1976) in which experimentally measured temperature and velocity profiles along with boundary layer thicknesses were reported. The fluids used in the experiments were demineralized water and aqueous (carboxymethy1)cellulose (CMC) solutions. Three types of CMC obtained from Hercules Co. were used to make the aqueous solutions. A high molecular weight 7HF CMC was used to make solutions having concentrations of 0.5%, LO%, and 1.5% by weight, and 0 1989 American Chemical Society
106 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 Table I. Properties of Solutions at 298 K type of CMC 7HF 7MF 7LF
CMC concn, wt 70 0 0.5 1.0 1.5 2.0 3.0 3.5 3.0 4.0 5.0
density, kg/m3 pure soin sat. soln 997 998 1000 999 1001 1002 1003 1005 1006 1007 1010 1011 1012 1014 1009 1011 1013 1015 1018 1020
a medium molecular weight 7MF CMC was used to obtain 2.0%, 3.0%, and 3.5% solutions, while a low molecular weight 7LF CMC was used to obtain 3.070,4.070,and 5.0% solutions. The density of the solutions and the solubility of benzoic acid in the solutions were determined experimentally a t 298 K. Diffusivities were determined by Lynch (1985) using a Taylor dispersion method (Taylor, 1953) which was extended to power law fluids (Fan and Hwang, 1965). The rheological properties of the solutions were obtained by measuring shear stress as a function of shear rate with a Brookfield cone and plate viscometer, Model HBT, and fitting the data to a power law model. The properties of the solutions used in this work are listed in Table I. The mass-transfer behavior was determined by calculating the Sherwood number 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)
sol of benzoic acid, kg/m3 3.2 4.1
5.6 6.2 6.4 7.1 7.3 6.9 7.3 7.5
lO'OD,, m2/s 8.9 11.8 8.7 7.1 7.3 5.9 5.9 6.0 5.5 5.7 I
K, kg/(m s2-")
n
0.000894 0.1236 1.1940 4.2134 0.1527 1.5925 3.1064 0.1149 0.6273 3.0504
1.0 0.9237 0.6814 0.6246 0.9736 0.7449 0.6960 0.8785 0.7925 0.5822
3
7 A 05% V loo/. m 15% 2 0% 0 3 0%
7HF
?HF ?HF
c
7MF 7MF 3 5 V 0 'MF
C
3 0 % 7-F
3 40V0 7-F
c 5c w ~ L F
J
I
1c
1
102
lo3
Figure 1. Comparison of experimental results with eq 4 and 7.
where C, is the solubility of benzoic acid in the solution a t the solid-fluid interface and C b is the solute concentration in the bulk fluid. Since the column contained a large volume of fluid, the bulk concentration was assumed to be small so that C b = 0. The Sherwood number for mass transfer with a sphere is defined as NSh'
=
k$ D,
where D is the sphere diameter and D, is the diffusivity of benzoic acid in the solution. Combining eq 1 and 2 yields
i"
v
3C"I. 7L'
4
(3)
The mass-transfer flux was calculated from the weight loss of the sphere, the average sphere surface area during a run, and the duration of the run. D represents the average sphere diameter during the run, and C, and D, were obtained from Table I for the runs with a particular solution. Three experimental runs were performed for each sphere size in every solution. The change in sphere diameter during the experiments ranged from 0.5% to 3.5%. For the runs with a particular fluid, larger changes in diameter occurred with smaller sphere sizes. Since the diameter change in all cases was small, the results were not affected significantly by using average values for the surface area and diameter. Results and Discussion Equations representing natural convection mass transfer between a solid sphere and a continuous fluid phase have shown that the Sherwood number can be expressed as a function of the Grashof and Schmidt numbers. For power law fluids, the transfer process also is dependent on the
Figure 2. Comparison of experimental results with eq 5 and 8.
non-Newtonian character of the fluid which is represented in terms of the flow behavior index, n. The analysis performed by Acrivos (1960) for laminar natural convection heat transfer can be used for the corresponding masstransfer process, yielding Also, the result derived by Stewart (1971) can be represented by (5)
The parameters C, and C2in eq 4 and 5 are weak functions of the flow behavior index (n),and they varied by less than 10% from the value for a Newtonian fluid for the solutions used in this work.
Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 107 Figures 1and 2 compare the experimental results with the predictions by eq 4 and 5, respectively. It is important to note that the works of Acrivos and Stewart used different characteristic lengths and velocities so that the Sherwood, Grashof, and Schmidt numbers were represented by different expressions and yielded different numerical values. Nevertheless, both equations gave identical predictions for the actual values of the mass-transfer coefficients. From Figure 1, there seems to be satisfactory agreement with eq 4 when (NG,1/[2(n+1)1N~n'(3n+1)) > 7. Likewise, there seems to be reasonable agreement with eq 5 when (NGiNsi)1/(3n+1) > 12 as shown in Figure 2. In this range, the experimental results were within 10% of the theoretical predictions. However, for (~G11'[z(n+1)1~~cn'(3"+1)) < 7 in Figure 1 and (NG;NSl)1/(3n+1) < 12 in Figure 2, the differences between experimental results and predictions by eq 4 and 5 increased, and in some cases, deviations as large as 90% occurred in which the experimental results generally yielded larger Sherwood numbers than the models. The apparent upward deviation from the theoretical expressions of the thin laminar boundary layer can be attributed to the increased thickness of the boundary layer. Amato and Tien (1976) studied the corresponding heat-transfer process from isothermal spheres to polymer solutions. Correlations developed in their work can be represented by the following equations for the analogous mass-transfer process. For
A comparison between the experimental results and eq 8 is shown in Figure 2. There seems to be satisfactory agreement especially in the region of small Grashof numbers where eq 8 is able to follow the general trend of the data. The average difference between the experimental results and eq 8 was 16.5%. Even when Cz was set equal to 0.589, the value for a Newtonian fluid, the average difference increased very slightly to 17.6%. Thus, it appears that the equation proposed by Churchill (1983) can be used to predict adequately the mass-transfer process by natural convection from solid spheres to power law fluids in the laminar flow region. Conclusions Mass transfer by natural convection from solid spheres to pseudoplastic fluids has been investigated in the laminar flow region. Deviations between experimental results and theoretical predictions based on boundary layer theory were significant only in the region of small Grashof numbers. However, a correlating equation proposed by Churchill (1983) appeared to be able to predict the mass-transfer behavior over the entire laminar flow region and seemed to be applicable for both Newtonian and power law fluids.
Nomenclature Cl = proportionality constant in eq 4 Cz = proportionality constant in eq 5 C b = solute concentration in bulk fluid (mass/length3) [NG,l~[2(~+1)1NS >c10 nl(and 3fl+1 5)
