-
3
0 17.3
05 1.o 2.0 CONTROL PARAMETER, R
3.0
Figure 1. A summary of ’the USBM data from the 150-ton retdrt tests as fractional Fischer assay shale oil recovery as a fiinction of a
proposed control parameter. portional to the square root of mi, the increase in H, is proportional on mg1/2, and a large increase in 1 would not be easily obtained as a result of changing H,. The increase in 1 by increasing R as R approaches 1should be very effective. However, the deflagration velocity, -ms/ps, is coupled to m, by the combustion process, and the best method for increasing riz, without also increasing the deflagration velocity would be to reduce the oxygen content of the retorting gas. Obviously, this approach has its limits. If the oxygen content is too low, h, goes to zero and retorting stops. It appears that an operating value of the control parameter R of 0.90 to 1should be sought.
Discussion Figure 1 is a plot of the fractional oil recovery as a function of the control parameter R from data of the 150-ton
retort for the ten runs reported by Harrak et al. (1974). The prediction, based upon the arguments presented above, appears to be confirmed. Here, fractional recovery is the volume fraction of Fischer assay. To calculate R , the values of the deflagration velocity, -m,/p,, and the gas mass velocity, m,, were obtained from Table I of Harrak et al. (1974). A value of 1280 kg/m3 (80 lb/ft3) was used for the “dumped” density, ps, of the shale, and over the temperature range of interest, it was assumed that the solid-gas heat capacity ratio C,/C, was 1. The data presented in Figure 1represent a wide range of operating conditions, and the validity of the proposed control parameter is rendered plausible. In the range of values of R between 0.9 and 1.0, very high oil recoveries might be obtained. In practice, one could select the retorting advance rate desired by setting oxygen (air) flux density, and then by measurement of the actual retorting rate, the recycle rate of burned gas would be adjusted to ensure an R value near 1 to ensure maximum recovery.
Literature Cited Gamson. B. W., G. Thodos. Hougen. 0. A,, Trans. Am. Inst. Chem. Eng., 39, 1-21 (1943). Harrak, A. E., Dockter, L., Long, A., Sohns, H. W.. “Oil Shale Retorting in a 150-Ton Batch Type Pilot Plant,” U.S. Bur. Mines Rep. Invest., No. 7995
(1974).
Department of Chemical Engineering University of Utah Salt Lake City, Utah 84112
Alva D. Baer* Norman W. Ryan
Received for review June 9,1975 Accepted November 10, 1975
Diffusion in Mixed Polymer Solutions
Mass transport in solutions containing various polystyrenes in toluene was investigated in order to determine the effects of the polymers on the diffusivity. Although the addition of a polymer of a different molecular weight than those already in solution sometimes does, and sometimes does not, change the diffusivity of a third component through the solution, the results can be predicted a priori.
Introductionand Background Diffusion through polymer solutions is encountered many times in the polymer processing industry. Since diffusion is always a slow process, many times it is the ratecontrolling step in a diffusion-reaction sequence. The effect on the diffusivity of altering the polymer concentrations has been studied previously (Navari et al., 1971; Osmers and Metzner, 1972) and equations have been presented for the a priori prediction of the effect. Frequently the solutions contain polymers having widely different molecular weights. This will obviously alter the molecular weights and viscosities of the solutions. Thus, one might expect there to be different effects on the diffusion when varying the concentration of single polymer in solution than when varying the concentration of that polymer in the presence of other polymers having different molecular weights. Indeed such a result is apparently present in protein solutions. Varying the concentration of the pro-
tein albumin alone in solution appears to have little effect on the diffusivity, but the same variation results in rather large changes in the diffusivity if other proteins are present (Navari et al., 1970). Later studies (Gainer and Chisolm, 1973) have shown that this effect appears to be dependent on the types of proteins present in the solutions. In the work presented here, we have investigated the effect of changing the concentrations of polystyrene in toluene on the diffusion of cyclohexane in order to see if an effect similar to that seen in protein solutions exists. In addition, we have applied the equations of Navari et al. (1971) to determine if a priori predictions of the behavior can be made.
Theory The complete theory for diffusion in protein and polymer solutions proposed by Navari et al. (1971) is available elsewhere in the literature and will be discussed only briefInd. Eng. Chem., Fundam., Vol. 15, No. 1, 1976 83
ly here. Their theory has its basis in absolute rate theory (Glasstone et al., 1941) with subsequent modifications by Li and Gainer (1968). The equation is written as
I ‘.Ot
where (-,“>
AE = AE,,, exp
q,,,is called the “polymer constant” and is found by knowing the bond lengths and molecular weight of the polymer, and k , the “solvent constant”, is related to the relative viscosity of the polymer solution a t infinite dilution (Navari et al., 1971). Since all of the quantities in eq 1 and 2 can be calculated, a priori prediction of the diffusion behavior in polymer solutions is possible. Although the authors did not use their equations for calculating the behavior in mixed polymer solutions, there is nothing in the theoretical derivation that should prohibit this. Experimental Section Apparatus. A microinterferometric method was used to measure the diffusion coefficients of cyclohexane in polystyrene solutions. This procedure cannot only be used for diffusion studies but is also useful in determining the concentration profiles in crystal growth (Ambrose, 1948; Berg, 1938) and in determining the thermal conductivity of liquids (Jones, 1973). The method, in principle, monitors the refractive index change during the diffusion process. The refractive index reflects the concentration profile in the system, and from the concentration profile data, the Boltzmann method (Jost, 1960) is used to calculate the diffusion coefficient. The microinterferometric method does have its limitations, however. The difference in refraction indices of the solute and solvent must be at least 0.01. Temperature control is also more difficult than with other methods of measuring the liquid diffusion coefficient. The apparatus has been described elsewhere (Nishijima and Oster, 1956; Li and Gainer, 1968). Our apparatus consisted of a Bausch and Lomb metallograph, a sodium light source, a collimator, and a Polaroid camera. The procedure is quite simple and a typical experimental run requires only a few drops of liquid and several minutes of time. Solutions a n d Procedure. Three polystyrene-toluene solutions were made using Dow Chemical polystyrenes having number-average molecular weights of 239 000, 530 000, and 1 400 000, with a ratio of the weight-average molecular weight to the number-average molecular weight approximately equal to 1.14 for each polymer. Solutions of eight different concentrations, ranging from 0.05 to 11 g/dl, of the lowest molecular weight polystyrene in toluene were prepared. Two sets of similar solutions were also prepared. One set contained 0.5 g/dl of the 530 000 molecular weight polymer in each of the eight previously mentioned solutions. The other set of solutions contained 0.5 g/dl each of both of the other molecular weight polymers (530 000 and 1400 000) in each of the previously mentioned eight solutions. Diffusion coefficients for cyclohexane diffusing into these solutions were determined using the microinterferometer apparatus. At the beginning of an experimental run, a few drops of polymer solution were placed in a wedge made from silvered microscope slides and placed on the stage of the metallograph. A small amount of cyclohexane was then inserted in the wedge with a hypodermic needle placed at the edge, allowing the surface tension to move the solute toward the solvent. When the solutions first touch there is 84
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
’
20
40
60
en
100
120
GRAMS POLYSTYRENE (MW- 239,00O)/DL
Figure 1. Diffusivity of cyclohexane in solutions of polystyrene and toluene a t 24.2’ C: -, predicted; 0,measured.
08 lo:
O‘
io
4b
6b
ab
Id0
12’0
GRAMS POLYSTYRENE (MW= 239,000)/DL
Figure 2. Diffusivity of cyclohexane in solutions of two polystyrenes in toluene at 24.2’ C (0.5 g/dl of polystyrene of MW = 530 000 present in all solutions): -, predicted; 0,measured.
a step change in concentration a t the interface producing a sharp change in the observed index of refraction profile. With increasing time of diffusion, the concentration profile is reflected as the index of refraction lines become a smooth S-shaped curve. These curves are then used to calculate the diffusivity (Nishijima and Oster 1956). Viscosity measurements needed for the theoretical predictive equation were found with a capillary viscometer. Results a n d Conclusions The results of diffusing cyclohexane into the solutions containing only the lower molecular weight polymer are shown in Figure 1, those for the solutions containing the two lower molecular weight polymers in Figure 2 , and those for the solutions containing all three polystyrenes in Figure 3. The results obtained using eq 1 are also plotted on each graph. As can be seen, the equation of Navari et al. (1971) appears to result in reasonably good predictions of the diffusivity, especially in Figures 2 and 3. It should be remembered that the equation of Navari et al. (1971) assumes that the system is a pseudo-binary one, with the polymer solution being treated as one entity. Such an assumption, though, does not appear to lessen the accuracy of the predictions. Figure 4 is a representation of the combination of Figures 1, 2, and 3, and is intended to better point out the relative changes in the diffusivity in the various solutions. As can be seen, the addition of the 530 000 molecular weight polymer caused a decrease in the diffusivity, as one might intuitively expect. However, the additional increment of polystyrene of molecular weight 1400 000 had little effect. The error associated with the method is on the order of lo%, so the lower two curves are essentially identical. The solutions of Figure 3 had much larger viscosities than the
dicts values to the contrary and agrees with the experimental results. OS
0
02
210
4b 6b 810 Id0 O ;I GRAMS POLYSTYRENE(MW=239,OOO)IDL
Figure 3. Diffusivity of cyclohexane in solutions of three polystyrenes in toluene at 24.2O C (each solution contains 0.5 g/dl each of polystyrenes having molecular weights of 530 OOO and 1400 OOO):
-, predicted; 0,measured.
Nomenclature A = solute B = solvent c = concentration of polymer in solution DAB = diffusivity of solute through solvent DAS = diffusivity of solute through the polymer solution AE! = activation energy difference between diffusion of solute in solvent and the diffusion of solute in the polymer solution Urn,,= “polymer constant”, related to structure of the polymer 12 = “solvent constant”, related to the relative viscosity at infinite dilution of the polymer solution M = molecular weight R = easconstant S = polymer solution (S = solvent B + polymer) T = absolute temperature V = molar volume Literature Cited
20
40
60
80
100
120
GRAMS POLYSTYRENEIMW=239,WVOL
Figure 4. Diffusivity of cyclohexane in polystyrene solutions.
solutions of Figure 2, which might lead one to speculate that the overall mass transport in the solutions ‘containing all three polymers should be lower. But eq 1, which evaluates both the polymer and the solution characteristics, pre-
Ambrose. E. J.. J. Sci. Insfum.. 25, 134 (1948). Berg, W. F., Proc. Roy. SOC.London, Ser. A, 164, 79 (1938). Gainer, J. L., Chisolm, G. M., Adw. Exptl. Med. Biol., 378, 729 (1973). Glasstone, S., Laidler, K. J., Eyring. H., “The Theory of Rate Processes”. Chapter 9. McGraw-Hill, New York, N.Y., 1941. Jones, R. C.. 8. S.Thesis, University of Virginia, Charlottesville,Va.. 1973. Jost, W., “Diffusion in Solids. Liquids and Gases”, Academic Press, New York, N.Y., 1960. Li. S. U., Gainer, J. L., Ind. Eng. Chem., Fundam., 7, 433 (1968). Navari, R. M., Gainer, J. L.. Hall, K. R., in “Blood Oxygenation”, p 243, D. Hershey, Ed., Plenum Press, New York, N.Y., 1970. Navari, R. M., Gainer, J. L., Hall, K. R., AIChEJ.. 17, 1028(1971). Nishijima, Y., Oster, G. J., J. Polym. Sci., 19, 337 (1956). Osmers, H. R., Metzner, A. B., Ind. Eng. Chem., Fundam., 11, 161 (1972).
Department of Chemical Engineering University of Virginia Charlottesuille, Virginia 22901
Randall C. J o n e s John L. Gainer*
Received for review July 10,1975 Accepted November 13,1975 The financial support for this project was provided by the National Science Foundation.
Ind. Eng. Chem., Fundam.. Vol. 15, No. 1, 1976
85
