I
Time (mid
Figure 1. Experimental values of platelet deposition determined at various periods of time and for flow rates of 3.82 ( O ) , 7.64 (A), and 15.3 ( X ) ml/min plotted together with the theoretical correlation from eq 4 and that from Grabowski et al. (dotted lines). Each experimental point is the mean of a t least five exposures with the blood of different dogs.
from eq 4 for a diffusion-controlled reaction rate, and the linear theory of Grabowski et al. A value for A , = 15 rZ/platelet was calculated from values of average maximum surface densities of platelets on glass. The theory shows an improved agreement with experimental values for surface coverages as high as 80%. This modification of the theory of Grabowski and coworkers in no way alters the conclusions made by them that the platelet transport can be described by diffusion-controlled kinetics and that the magnitude of platelet diffusivity can be correlated with a power law function of shear rate. In fact, it lends credence to these conclusions since additional experimental data can be included. It should be noted, however, that the linear theory used by Grabowski et al. yields initial fluxes, j o , which are lower than those predicted by the revised theory as illustrated by the dotted lines in Figure 1. Thus, the values of diffusivity inferred from these fluxes would be slightly underestimated in their analysis.
be derived from a simple particle balance. Thus, the rate of change of covered surface is equal to the arrival of surface area (platelets) less the departure of surface area (platelets which collide with occupied areas), or
Acknowledgment
where S is the fraction of surface covered, A , is the effective surface area covered by a single platelet, j o is the initial platelet flux calculated from Solbrig and Gidaspow, and t is time. Solution of eq 1gives
Nomenclature A , = effective surface area of a platelet, cm*/platelet j = platelet flux per unit time, platelet/cmz-sec JO = initial platelet flux per unit time, platelet/cmz-sec J = total platelet flux, platelets/cm2 S = fraction surface area covered with platelets t = time, sec
S = 1 - exp(-A,jot)
( 2)
The probability that a platelet will reach an unoccupied site and will adhere if at an unoccupied site can be expressed as the product of the two independent probabilities. Thus, the platelet flux is
I would like to thank L. I. Friedman for making available his experimental results.
Literature Cited Friedman, L. I . , Doctoral dissertation, School of Engineering and Applied Science, Columbia University, New York, N . Y . , 1972. Grabowski, E. F., Friedman, L. I . , Leonard, E. F., Ind. Eng. Chem., Fundam., 11, 224 (1972).
Solbrig,C. W., Gidaspow, D., Can. J . Chem. Eng., 45, 35 (1967).
and the total platelet flux to the surface is
Department of Experimental Medicine F. Hoffmann-La Roche & Co., Ltd. CH-4002 Basel, Switzerland
The original experimental data of Friedman (1972) are plotted in Figure 1, together with the curves calculated
Vincent T. Turitto
Received for review September 6,1974 Accepted March 5, 1975
Concentration Profiles in Ternary Gaseous Diffusion
Experimental fluxes and concentration profiles from the steady-state diffusion of methanol and acetone through air are compared with those predicted by the Stefan-Maxwell relationships and an approximate model. Little deviation is found for the comparison of fluxes, but concentration profiles predicted by the approximate model show deviations.
Mass transfer and simple molecular diffusion in multicomponent systems continue to be subjects of theoretical, experimental, and applied research. Much of the theoretical work has dealt with the reduction of the mass transfer and diffusion equations to sets of equations uncoupled in the diffusion terms (Stewart and Prober, 1964; Toor, 1964a,b). This permits solution of the multicomponent problems in equivalent binary form. These solutions are 276
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
valid for both liquid and gas systems provided the physical and diffusive properties of the systems are reasonably averaged. The validity of the Stefan-Maxwell relationships, given for unidirectional diffusion by
dy,= dz
y," j f i
- YjNi CDi,
( ~ i= 1, 2 ,
*
.
. )
n)
(1)
Table I. Parameters of Experimental Run Shown in Figure 2 ~~~
Parameter
~
Experimental
Calculated
~~
( D a c ) T , pcm2/sec' , (Dbc)T,p,
cm2/sec cm2/sec
0.1372 0.1991
0.0848 24.25 23.8 1.779 X lo-' 1.781 X 3.121 x lo-' 3.186 x lo-?* 328.5 745.2 >'ai 0.3173' 0.31gd Y bl 0.5601' 0.528d a Subscripts a, b, and c refer to acetone, methanol, and air, respectively. Toor's (1964a) approximate solution using z = 23.8 cm. From VLE data of Freshwater and Pike (1969). Calculated from eq 2 and 3 using z = 23.8 cm and experimental fluxes. (Dab)T,fi,
cm Na, mol/(cm2 sec) N,, mol/(cm2 sec) T , "K P , mm Hg z,
..........
Tki$ 4
Figure 1. Detail of Stefan tube assembly.
has been experimentally investigated by Hoopes (1951), Duncan and Toor (1962), and Getzinger and Wilke (1967). In these investigations, experimentally measured fluxes were compared with those predicted by the inverted, integrated form of eq 1. The only experimental examination of the concentration profiles was done by Hoopes, who determined the concentrations a t various points along the diffusion path for the equimolar countercurrent diffusion of nitrogen and hydrogen through ammonia. It should be noted that in this system the hydrogen behaves as if it is diffusing in a single gas because the diffusivities of hydrogen in nitrogen and hydrogen in ammonia are nearly equal. Here we compare experimental and theoretical concentration profiles for a ternary gas system in which all binary diffusivities are dissimilar. The system is acetone and methanol diffusing cocurrently through stagnant air. For this n = 3 case, we integrated eq 1 in terms of the concentrations to yield
1
Subscript c refers to the stagnant component and subscript 1 refers to a known value at any arbitrary point along the diffusion path. Equations 2 and 3 can also be obtained by inversion of Gilliland's well known solution for the fluxes as presented by Sherwood (1949). To use eq 2 and 3 for predicting the axial concentration profiles, the fluxes, binary diffusivities, and concentrations a t a point are needed.
The fluxes and concentrations were measured experimentally.
Experimental Section A modified Stefan tube assembly consisting of a liquid cup, a diffusion tube, 5.08 cm i.d., and an air distributor all enclosed in a constant-temperature air bath was used to obtain the desired fluxes and concentrations. One unusual characteristic of the tube assembly, which is shown in Figure 1, is the feature that permits isolation and sampling of gas along the diffusion path. Additional details of the apparatus have been presented by Carty (1973). A liquid mixture of acetone and methanol was pumped at a very low but constant rate into the bottom of the cup attached to the lower end of the diffusion tube. The mixture continuously rose, formed a very smooth surface film, and then overflowed into the outer chamber from which i t was drained to the outside through a trap that maintained the pressure inside the tube. The flow rate of the liquid was such that there was no detectable difference in the concentrations of the inlet and outlet streams. By this method, a constant-composition liquid interface was maintained a t the bottom of the diffusion tube. Vapors of acetone and methanol from this liquid interface diffused up the Stefan tube and were swept away at the top of the tube by a stream of pure dried air. Turbulence in this air stream was minimized by passing it through the screens and calming vanes in the air distributor prior to its sweeping across the top of the diffusion tube. The modified construction of the tube permitted gas and vapors in the seven sections along the tube's axis to be isolated in sealed compartments. Samples of the contents of these compartments and the effluent stream of air plus vapors from the top of the tube were analyzed for composition with a gas chromatograph. Sampling of the effluent stream commenced after the system had been running for a t least 1hr, and when several successive samples showed no change in composition, the system was taken to be a steady state and the components in the seven sections of the tube were isolated. From the air flow rate and the composition of the effluent stream, the fluxes of acetone and methanol were calculated. The chromatographic analyses gave the desired experimental axial concentration profiles. Values of the diffusivity of acetone in air and methanol in air at 55°C and 1 atm were taken from the work of Richardson (1959) and Mrazek et al. (1968). Air may be considered a pure component, since there is essentially no difference between the diffusivities of acetone and methanol into oxygen and nitrogen. The difInd. Eng. Chem., Fundam.. Vol. 14. No. 3, 1975
277
I
.o 1
i
Gcetone 7 Methanol
Gir
-S t e f a n - Maxwell Solution --a p p r o x i m a t e Solution. 0.7
1
J: /
/
0.0
I8 0
9.0 2,
27
1
0
cn-
Figure 2. Air concentrations along d i f f u s i o n p a t h in t h e Stefan tube.
2 , cm
Figure 3. Comparison o f experimental a n d predicted concentration profiles in t h e Stefan tube.
fusivity of acetone in methanol was not available and had to be calculated; the method of Bae and Reed (1971) was selected and used. All diffusivities were corrected to the experimental temperature and pressure.
Results The parameters for a sample experiment are summarized in Table I. Using the experimental fluxes and the binary diffusivities presented in this table, the remaining unknowns in eq 2 and 3 are the concentrations ycl and Ybl at one point on the diffusion path. Initially, the interfacial concentration as determined from the vapor-liquid equilibrium data of Freshwater and Pike (1967) were chosen as the end point. However, it was found that relatively small errors in these values caused large variations in the calculated profiles. Using these values for the end point, the slope of the predicted methanol profile was positive and the predicted concentrations of the other two components were in substantial disagreement with the experimental values. I t should be noted that these interfacial concentrations are nearly equal to those obtained by the method presented below. Since the concentrations at any arbitrary but specified point in the system were suitable for use in the equations, an alternate end point at the top of the diffusion tube was selected, where the vapor concentrations were essentially zero. Furthermore, since the turbulence induced by the crossflow of air at the top of the tube disturbed the unidirectional character of the simple diffusion, the geometrical parameter z may in effect be an incorrect or unrealistic value of the true length of the diffusion path. According to eq 2, a plot of In yair vs. z should follow a straight line. When the experimental point from cell 1 is neglected, the plotted values of the air concentrations follow this relationship exactly as shown in Figure 2, and a true value of z = 23.8 cm a t ycl = 1.0 and Y b l = yal = 0.0 is obtained. Using this end point and the experimental fluxes, the interfacial concentrations presented in the second column of Table I were calculated from eq 2 and 3. As noted in Table I, these calculated interfacial values are very close to the values obtained from the vapor-liquid data. The concen-
278
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Ind. Eng. Chem., Fundam., Vol. 14, No. 3. 1975
tration profiles predicted by eq 2 and 3 and the experimental values are plotted in Figure 3. This figure shows that the experimental values for methanol fall very close to the predicted profile while the points for acetone and air show some deviation. The average percent deviations of the experimental and theoretical concentrations for this run are 12.1% for acetone, 2.4% for methanol, and 4.7% for air (neglecting the one point). Also shown in Table I and Figure 3 are the fluxes and concentration profiles which were calculated by the approximate method of Toor (1964a) using the terminal concentrations of yal = 0.319, Y b l = 0.528, yaz = 0.0, and Yb2 = 0.0. The approximate values of the fluxes are in close agreement with the exact values. However, as seen in Figure 3, the approximate concentration profiles deviate considerably from the profiles predicted by eq 2 and 3. Solutions to the Stefan-Maxwell relationships give very good comparisons with experimental fluxes and concentration profiles in simple ternary gaseous systems. Approximate methods predict reasonable fluxes, but care must be taken when such methods are used to predict profile curves.
Literature Cited Bae, J. H., Reed, T. M., iil, Ind. Eng. Chern., Fundam., 10, 36-41 (1971). Carty, R. H., Ph.D. Thesis, University of Kentucky, Lexington, Ky., 1973. Duncan, J. B., Toor. H. L., A.I.Ch.E. J., 8, 38-41 (1962). Freshwater, D. C., Pike, K. A,, J. Chem. Eng. Data. 12, 179-82 (1967). Getzinger, R. W.,Wiike, C. R., A./.Ch.€. J., 13, 577-580 (1967). Hoopes, J. W., Jr.. Ph.D. Thesis, Columbia University, N.Y., 1951. Mrazek, R. V., Wicks, C. E.. Prabhu, K. N. S., J. Chern. Eng. Data, 13, 508510 (1968). Richardson, J. F., Chem. Eng. Sci., 10, 234-242 (1959). Sherwood, T. K., "Absorption and Extraction," 1st ed. McGraw-Hill Book Company, Inc., New York, N.Y., 1949. Stewart, W. E., Prober, R., hd. Eng. Chem., Fundarn., 3, 224-235 (1964). Toor, H. L., A./.Ch.E. J., 10, 448-455 (1964a). Torr, H. L.. A.I.Ch.€. J., 10, 460-465 (1964b).
Department of Chemical Engineering R. Carty Uniuersity of Kentucky T.Schrodt* Lexington, Kentucky 40506 Receiued for reuiew December 5, 1974 Accepted April 21,1975
