z = m, moles/liter solute concentration in moving phase a t z = -m moles /liter longitudinal dispersion coefficient for a solute in the moving phase, sq. cm./sec. rate function, sec-1 mass transfer coefficient, cm./sec. solute concentration in stationary phase, moles/ liter solute concentration in stationary phase a t z = m , moles/liter solute concentration in stationary phase a t z = - m , moles/liter interstitial moving-phase velocity, cm./sec. dimensionless moving-phase solute concentration dimensionless stationary-phase solute concentration dimensionless equilibrium distribution isotherm d y * ( x ) / d x , dimensionless @ j * ( x ) / d x * , dimensionless time, sec. axial distance measured from upstream end of stationary phase, c m .
c( m )
= solute concentration in moving phase a t
c(-m)
=
D
=
G k
= =
4
=
q( m )
=
q(-
m)
=
U
=
X
=
Y
=
y*(x)
Y*’
= =
y*”
=
t
=
z
=
z* e
= distance coordinate, cm. = stationary-phase porosity, dimensionless
Literature Cited (1) Courant, R., “Differential and Integral Calculus,” pp. 117-21,
Interscience, New York, 1947. (2) Glueckauf, E., Coates, J. I., J . Chem. SOC.1947, 1315. (3) Lapidus, L., Rosen, J. B., Chem. Eng. Progr. Symp. Ser. 5 0 , No. 14, 97 (1954). (4) Michaels, A. S., Ind. Eng. Chem. 44, 1922 (1952). (5) Rosen, J . B., Ph.D. thesis, Columbia University, 1952. (6) Sillen, L. G., Nature (London) 166, 723 (1950). (7) Smail, L. L., “Analytical Geometry and Calculus,” pp. 190-1, Appleton-Century-Crofts, New York, 1953. (8) Walter J. E., J . Chem. Phys. 13, 332 (1945). DAVID 0. C O O N E Y E D W I N N. L I G H T F O O T University of Wisconsin Madison, Wis. RECEIVED for review November 2, 1964 ACCEPTED January 28, 1965 Financial support was provided by Union Carbide Corp. and by the Wisconsin Alumni Research Foundation.
COM MUN I CAT1ON
DIFFUSIVITY IN N O N - N E W T O N I A N LIQUIDS
Measurement of gas absorption rates in laminar liquid jets provides an easy way for determining the diffvsivity of the absorbing gas in the liquid. The absorption equation for ideal rodlike jets is a good approximation also for nowNewtonian liquids. The method has been used to measure the diffusivity of COz and CzH4 in a variety of aqueous non-Newtonian solutions and suspensions. Although the apparent viscosity of these liquids is much higher than that of water, the measured values of diffusivity are of the same order of magnitude as in water, and, in the case of polymer solutions, they appear to increase with increasing polymer concentration.
equations such as the Wilke equation (10) for predicting molecular diffusivities in usual solvents break down for non-Newtonian liquids, because they contain the solvent viscosity as a parameter. It is known (2, 3, 6, 9) that in no case is the decrease of diffusivity as strong as the increase in appareut viscosity when. a solute or a suspended solid is added to a solvent resulting in a non-Newtonian solution or suspension. Furthermore, it appears that, contrary to apparent viscosity. the diffusivity is, if at all only a weak function of shear rate ( 3 ) . Conventional diffusion measurements in slurries and polymer solutions are difficult. For Newtonian fluids, the measurement of gas absorption rates into a laminar liquid jet provides a useful technique for determining the diffusivity. D ; the value of D thus determined is the value a t zero shear rate, inasmuch as the concentration boundary layer developing in the jet is narrower than the corresponding zero-shear-rate boundary layer. This technique can be extended to non-Newtonian liquids.
CONVENTIONAL
236
I&EC FUNDAMENTALS
T h e rate of absorption in an ideal rodlike cylindrical jet is given by :
v
=
4 A , (DLZ)”*
(11
where A t is the volumetric gas solubility, L is the liquid flow rate, Z is the jet length, and Vis the volumetric gas absorption rate. I n a real jet issuing from a circular nozzle, second-order effects may cause small deviations from Equation 1. These effects are: T h e surface velocity in the vicinity of the nozzle tip is smaller than the bulk velocity, because of the boundary layer which has developed in the nozzle. When the jet curvature is neglected. the velocity defect problem is formally analogous to the wake decay behind a flat plate ( 8 ) : the latter problem having been solved for Newtonian fluids by Goldstein (5). The solution can be extended to power-law non-Newtonian fluids! and it can be shown that this effect is minor for pseudoplastic fluids (7). T h e jet shrinks under the influence of gravity, so that the surface velocity and interface area are different from those in a
Table I.
Summary of Results
706 D , Gas
Soluenta
con
Solubility, L./L.b
Sq. Cm./
5 . lOY0 bentonite 0.900 5 . 0 1 % bentonite 0.962 3 . 80y0bentonite 0.980 1 .50Yc CMC 0.940 0 . 7 5 % CMC 0.935 0 . 3 7 5 % CMC 0.950 0 . 222Yb Carbopol 0.740 0 1724;, Carbopol 0.800 0 . 1 32y0 Carbopol 0,977 0.122% Carbopol 1 .oo 3 .OY0 ET597 0.890 2 . 0 % ET597 0,933 1 . O % ET597 0,955 2 . 0 % CMC 0.105 1 . 0 % CMC 0.113 0 . 5 y 0 CMC 0.120 0 . 222y0 Carbopol 0.0820 0 . 172Y0 Carbopol 0.0937 0.132% CarboDol 0.109 0.122% Carbopol 0.112 7 .by weight. CMC carboxymethylcellulose. ET597 a addittue. At 78" C. and 7 atm. At 18' C. ~
Q
- 1 "
~
~~
Sec.c 2.40 2.02 1.91 2.44 2.06 1.72 2.16 1.98 1.82 1.82 1.80 1.71 1.76 1.29 1 .OS 1.06 1. I 8 1.05 0.90 0.91 viscoelastic
rodlike jet, and a velocity component normal to the gas-liquid interface sets in, which contributes to absorption. Beek ( 7 ) has shown that the over-all effect is negligible when the group gr/U2 is much less than 1. T h e jet may expand because of the relaxation of elastic stresses. This effect should be minor in the case of axially short nozzles. Except for highly elastic liquids, this effect should be of the same order of magnitude as, and of direction opposite to, the shrinkage effect. T h e absence of the second-order effects discussed above can be ascertained by the fact that the data in a V us. (LZ)'I2 plot are well correlated by a straight line passing through the origin, and show no shift with either L or Z . T h e method of gas absorption has been used for determining the diffusivity of COz and CZH4 in a number of non-Newtonian aqueous solutions and suspensions. Both L and Z have been independently varied within each set of data, the absorbing medium being a laminar jet issuing from a 0.2-cm. I.D. glass nozzle. The rates of absorption have been determined with a soap-film flowmeter. T h e values of A , have been measured by direct determination of the volume of gas absorbed under 1-atm. partial pressure. Plots of V us. (LZ)"* have been made for all the gas-liquid couples, and the value of D has been calculated through Equation 1 from the slope of the straight
line correlating the data. Detailed data are available (4, 7) ; a summary of results is given in Table 1: The results show that the measured values of diffusivity are of the same order of magnitude as in water, and increase with increasing polymer concentration, or with increasing solid concentration in the case of slurries. This is rather surprising, and an alternative explanation of the high absorption rates observed could be thought of. Possible experimental errors leading to higher absorption rates (turbulence, end effects in the receiver) should result in a shift of data with either L or Z , while the data are randomly scattered around a straight line passing through the origin in the V us. (L2)1'2plots, and show small average deviations (about 5%). Chemical interaction may be excluded from consideration, mainly because measured values of .4, are always smaller than the corresponding values in water. T h e author believes that the values of D reported in Table I are substantially correct, although he has no explanation to offer for the fact that they are unexpectedly high. Nomenclature
A , = volumetric gas solubility, dimensionless D = diffusivity, sq. cm./sec. g = gravity acceleration, cm./sec.* L = liquid flowrate, cc./sec. r = nozzle radius, cm. U = jet velocity, cm./sec. V = gas absorption rate, cc./sec. Z = jet length, cm. Literature Cited
(1) Beek, U'. J., dissertation, Delft, 1962. (2) Biancheria, A., Kagles, G., J . Am. Chem. Soc. 75, 5908 (1957). (3) Clough, S. B., Read, H. E., Metzner, A. B., Behn, V. C . , A.1.Ch.E. J . 8, 346 (1962). (4) Fortino, G., chem. eng. thesis, Naples, 1964. (5) Goldstein, S., Proc. Cambridge Phil. SOC.26, 19 (1930); Proc. Roy. Soc. London A142, 545 (1933). (6) Hartlev. G. S.. Trans. Faraday SOC.45. 820 11949). ' (7) Palumbo, G., chem. eng. the&, Naplds, 1963. (8) Scriven, L. E., Pigford, R. L., A.Z.Ch.E. J . 5 , 397 (1959). (9) Tsvetkov, V. N., Klenin, S. I., Tech. Phys. URSS. (Engl. trans.) 4, 1283 (1960). (10) U'ilke, C. R., Chem. Eng. Progr. 45, 218 (1949).
GIAIVNI ASTARITA University of Naples Naples, Italy
RECEIVED for review February 17, 1964 ACCEPTED March 10, 1965 56th Annual Meeting, American Institute of Chemical Engineers, Houston, Tex., December 1963.
VOL. 4
NO. 2
MAY
1965
237
