300
Ind. Eng. Chem. Fundam. 1986, 25, 300-303
Superscripts b = property before an event occurs c = property in the differential cell m = property on mesh point 0 = property immediately after Crank-Nicolson computation s = property immediately after shrinkage occurs ' = solute free base + = source - = sink
Literature Cited Bayens, C. A.; Laurence, R. L. Ind. Eng. Chem. Fundam. 1969, 8 , 71. Coulaloglou, C. A.; Tavlarides, L. L. Chem. Eng. Sci. 1977, 3 2 , 1289. Coulaloglou, C. A. Ph.D. Thesis, Illinois Institute of Technology, Chlcago, IL, 1975.
Cruz-Pinto, J. J. C.: Korchinsky, W.J. Chem. Eng. Sci. 1981, 3 6 , 667. Hsla, M. A.: Tavlarldes, L. L. Chem. Eng. J . 1980, 2 0 , 225. Hsia, M. A.; Tavlarides, L. L. Chem. Eng. J. 1983, 2 6 , 189. Hulbert, H. M.; Katz, S . S . Chem. Eng. Sci. 1964, 79, 555. Komasawa, I.; Ingham, J. Chem. Eng. Sci. 1978, 3 3 , 479. Jeon, Y. M. Ph.D. Thesis, Korea Advanced Institute of Science and Technology, Seoul, Korea, 1984. Shah, B. H.; Ramkrishna, D. Chem. Eng. Sci. 1973, 28, 389. Shah, 6. H.; Ramkrlshna, D.; Borwanker, J. D. Chem. Eng. Sci. 1977a, 32, 1419. Shah, B. H.; Rarnkrishna, D.; Borwanker, J. D. AIChE J. 1977b, 2 3 , 897. Sovova, H. Chem. Eng. Sci. 1981, 3 6 , 1567. Spielman, L. A.; Levenspiel, 0. Chem. Eng. Sci. 1965, 2 0 , 247. Zeitlin, M. A.; Tavlarides, L. L. Can. J. Chem. Eng. 1972, 50, 207.
Received for review March 23, 1984 Accepted June 6, 1985
Transient Flow Diffusometer Richard 0. Rice" and Mark A. Goodnert Department of Chemical Engineering, Louisiana State University. Baton Rouge, Louisiana 70803
A new flow dynamic apparatus for measuring liquid-phase diffusion coefficients is shown to reproduce literature
data. An important modification of the classic capillary diffusion tube is shown to give quick, reliable results using a minimum quantity of reagents. New data for the diffusion of m-xylene in o-xylene are also presented.
Introduction Expensive chemicals and protracted measurement times can often limit the number of experiments researchers undertake in the determination of liquid diffusion coefficients. This is especially troublesome for liquid systems, since the value of diffusivity is invariably concentration dependent. Stationary (steady-state) methods, such as the diaphragm cell (Mills and Woolf, 1968), are time-consuming and require sufficient capacitance (volume) to ensure development of steady composition profiles. We present details of a self-contained unit which is a modification of the nonstationary capillary tube method first developed by Anderson and Saddington (1949) and later used by Piret et al. (1951). Design Details Our motivation for the new apparatus arose from an effort to minimize wastage of highly purified xylene isomers. Thus, our attention focused on the capillary tube method. Typically, a sealed-end Capillary tube of precisely determined length is filled with solute mixtures and the open end is exposed to a flowing solvent sweeping past the open mouth. The mathematics for such an experiment is particularly simple and is outlined by Mickley et al. (1957). In the experiments of Piret et al. to measure salt diffusion, water was used as solvent, so there was little motive for minimizing effluent wastage. In the present effort, we set out to design a new apparatus (shown in Figure 1)which minimizes the amount of flowing solvent needed. The conditions to sustain a constant composition at the capillary mouth, when the solvent flow is transient, must first be addressed. The solvent velocity must be much greater than the solute rate diffusing from the capillary. *To whom all correspondence should be addressed. 'Present address: Rohm and Haas, Houston, TX.
Taking a high diffusion coefficient of 3 X cmz/s for the solute contained in a capillary tube measuring 0.1-cm i.d. by 5 cm long gives (by using the short-time solution given by eq 9) around 2.3% extracted in the first 6 min or 2.5 X lo4 cm3/s of solution exchanged by diffusion. Designating the flow past the capillary mouth to be 100 times as fast in order to dominate diffusion, one then estimates the minimum required channel flow to be 2.5 X 10" cm3/s. This value served as the lower limit for the transient flow device. The upper limit for flow can be taken as the onset of turbulence in the flow channel which holds the capillary tubes. For a channel of 0.15-cm radius, the upper limit is computed to be around 4 cm3/s. For the batch flow system shown in Figure 1, solvent flows by gravity into one side of the Teflon channel from a reservoir of cross section A2 and into another reservoir of cross section AI. The liquid height difference provides the driving force for flow through the channel, which can extract simultaneouslyup to three capillary tubes. By use of the Hagen-Poiseuille law for laminar flow, the liquid height differences corresponding to the upper and lower flow limits are 500 and 0.1 cm, respectively, with a total flow path length of 300 cm. Since exposure time is an important experimental quantity, we note that flow path length (number of coils) is a design variable and can be easily extended for longer experiments. Addition of fresh solvent to the feed reservoir can also expedite longer experiments. The fiial prototype shown in Figure 1 was designed for a run time of 3.5 h following the design principles outlined above. The unit was designed to process only 100 mL of solvent. The larger reservoir functions as the receiving volume, in order to maximize transient height difference. The coiled glass capillary tubing serves two purposes. Its length and small bore offer substantial flow resistance in order to allow long-time experiments, and secondly, the
0196-4313/86/1025-0300$01.50/00 1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25,
No. 2, 1986 301
E=l, 8=0
(4.1) (4.2)
E=O, { = 1 aE --
ar - 0, r = 0
T-
(4.3)
The Laplace transformed solution is convenient for defining short- and moderate-time solutions
39;
I It is the well-mixed average composition which is actually measured, so define the average as
which is simply the fraction unextracted. Thus, we obtain
As the dimensionless Laplace variable tends large, a short-time solution is obtained: E(s) g s-1 - s-312 (8) Hence, the classic result for exposure time te is
u LEGEND B = Teflon Block C = G l a s s Capillary Tube ECZExit Gloss Coil IC= I n l e t Gloss Coil
E R = Exit Reservoir I R = Inlet Reservoir S B : Support Brocas WB = Water Both
Figure 1. Schematic of diffusometer immersed in temperature bath.
coils are also needed to allow solvent to reach temperature equilibrium before exiting the block, since the coil and channel block are immersed in a constant-temperature bath. Also, the glass coil design allows lateral flexure in order to permit removal of the channel block for handling and cleaning. Ignoring entrance/exit losses for such long capillaries, one can estimate the variation of liquid level difference with time from H 2 - Hi = H2" exp(-t/d (1) 7=-(-)
8PL AlA2 rR4pg Ai + A 2
(2)
Had we made the receiving reservoir AI equal to the feed reservoir A2, then the time constant would be proportional to A2/2. However, by making the receiving vessel twice the diameter of the feed vessel, it is seen that the time constant is then proportional to 0.8A2. This small design detail increased the time constant by 60%. The final design is shown in Figure 1along with the major dimensions. The Teflon block, B, bored to i/s-in. diameter and mounted as shown, was press-fitted into place.
Mathematical Analysis The usual short-time solution (DLt/L2< 1)after Piret et al. (1951) is used to estimate diffusivity. In the mass balance equation, we take diffusivity to be constant, but for the short extraction times used, DLcan be viewed as a differential coefficient, especially if initial and final average compositions are not widely different. Thus, we write for the capillary tube, using the dimensionless quantity (3) so that the dimensionless diffusion equation is
aE = -a2E a0
a?
with 0 = DLt/L2and .(' = z / L and
(4)
-
which is generally valid for total exposure times te such that DLte/L2
