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C = total concentration of electrolyte solution, mol/cm3 C!i = concentration of species i in bulk solution, mol/cm3 C = total concentration of electrolyte in membrane phase, mol/cm3 ci = concentration of species i in membrane phase, mol/cm3 Di = diffusion coefficient of species i, cm2/s D m = average interdiffusion coefficient of ions A and B in liquid film, cm2/s Dm = integral interdiffusion coefficient in membrane, cmz/s D ' B = differential interdiffusion coefficient in membrane, cmz/s d, = XA DAB, apparent interdiffusion coefficient, cm2/s E , = activation energy, kcal/mol = selectivity coefficient, dimensionless ki = Donnan equilibrium constant of ion i, dimensionless 1 = thickness of ion exchange membrane, cm Ni = mass flux of species i, mol/cm2.s T = temperature, K V = volume of electrolyte solution in each compartment, cm3 x = coordinate, cm Greek Letters 6 = thickness of diffusion liquid film, cm hi = distribution coefficient of species i, dimensionless Subscripts 0 = at time t = 0 t = at time t Superscripts 0 = at coordinate x = 0 1 = at coordinate x = 1 ' = in compartment 1 " = in compartment 2
Literature Cited Bird, R. B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena", Wiley, New York, 1960. Bird, R. B., Stewart, W. E., Lightfoot, E. N., Chapman, T. W., "Lecture in Transpat Phenomena", AIChE Continuing Education Series, 1969. Crank, J., Park, G. S.,"Diffusion in Polymers", Academlc Press, New Ywk, 1968. George, J. H. B., Courant, R. A., J. Phys. Chem., 71, 246 (1967). Glueckauf, E., Proc. R. Soc. London, Ser. A , 268, 350 (1962). Glueckauf, E., Watts, R. E., Proc. R. SOC. London, Ser. A , 268, 339 (1962). Heifferich, F., Ion Exchange", McGraw-Hill, New York, 1962. Helfferich, F., Discuss. Faraday SOC.,21, 83 (1956). Heifferich, F., J . Phys. Chem., 67, 1157 (1963). Helfferich, F., Plesset, M. S.,J. Chem. Phys., 28, 418 (1958). Hills, G. J., Jacobs, P. W. M., Lakshiminarayanakh, N., Proc. R. Soc.London, Ser. A , 262, 246, 257 (1961). Huang, T. C., J. Chem. Eng. Data, 22, 422 (1977). Huang, T. C., Tsai, F. N., Yu, I.Y., J . Chinese Chem. Soc. Ser. II, 20, 151 (1973). Huang, T. C., Wang, T. T., Desalination, 21, 327 (1977). Kitamoto, A., Takashima, T., J. Chem. Eng. Jpn., 3 , 54 (1970). Kobatake, Y.. J. Chem. Phys., 26, 146, 442 (1958). Kobatake, Y., J . Chem. Phys., 40, 2212, 2219 (1964). Kobatake, Y., Takeguchi, N., Toyoshima, Y., Fujita, H., J . Phys. Chem., 69, 3981 (1965). Mackay, D., Meares, P., Kolloid Z.,167, 31 (1959). Mackay, D.. Meares, P., KolloidZ., 171, 139 (1960); 176, 23 (1961). Nightingale, E. R., Jr., J . Phys. Chem., 63, 1383 (1959). Peterson, M. A., Gregor, H. P., J . Electrochem. Soc., 106, 1051 (1959). Scattgood, E. M., Lightfoot, E. N., Trans. Faraday Soc., 63, 1135 (1967). Schloge, R., Z . Elektrochem., 57, 195 (1953). Spigler, K. S.,Trans. Faraday Soc., 54, 1408 (1958). Steward, R. J., Graydon, W. F., J. Phys. Chem., 60, 750 (1956). Tanaka, Y., Denki Kagaku, 45, 630 (1977). Toyoshima, Y., Fujita, H., Trans. Faraday SOC.,63, 2828 (1967). Toyoshima, Y., Kobatake. Y., Fujita. H.. Trans. FaradaySoc., 63, 2814(1967a). Toyoshima, Y., Kobatake, Y., Fujita, H., Trans. Faraday Soc.,63, 2803 (1967b). Wallace, R. A., Ampaya, J. P., Desalination, 14, 121 (1974). Wyiiie, M. R. J., Kannaan, S. L., J . Phys. Chem., 58, 73 (1954).
Received for review March 6 , 1978 Accepted March 8, 1979
Mixing in Intermeshing Twin Screw Extruder Chambers: Streamlines and Strain for Down-Channel Circulation Jagdish C. Maheshri and Charles E. Wyman' Department of Chemical Engineering, University of New Hampshire, Durham, New Hampshire 03824
A flat plate model was developed to describe the down-channel and channel depth velocity components in an idealized intermeshing twin screw extruder chamber. The model's predictions were tested against hypothetical zero helix angle results obtained in cylindrical coordinates. Although the streamline patterns agreed well, the absolute strain calculated was in some error due to "unwinding" the channel. The strain prediction was improved when the flat plate velocities were transferred to the original cylindrical coordinate geometry, but the reliability of this method was uncertain. The flat plate model was used to obtain stream functions and velocities numerically for 15, 30, and 45' helix angle channels in both co- and counter-rotating operation assuming a leakproof chamber, and a generally uniform absolute strain profile was predicted for both modes of screw rotation in the limiting case studied. The absolute strain was also found to increase with helix angle.
Introduction
Twin-screw extruders are often used instead of single screw devices because the mixing is considered superior (Prause, 1967), and a number of investigators have examined the residence time distribution in twin screw extruders to characterize their mixing capabilities (Janssen, *Address correspondence to this author at SERI, 1536 Cole Boulevard, Golden, Colo. 80401. 0019-7874/79/1018-0226$01 .OO/O
1976; Todd, 1975). Such studies are quite useful for materials which require closely controlled melt histories, but the residence time distribution by itself will not predict whether elements of fluid with the same residence time can be considered well mixed. In single screw extruders, strain has been used as a tool to account for this local mixing with some success (Bigg and Middleman, 1974; McKelvey, 1962; Tadmor and Klein, 1970) since the mixing action occurs mainly in the single spiral channel formed by the extruder flights. However, in a twin-screw extruder, 0 1979 American Chemical Society
Ind. Eng. Chern. Fundarn., Vol. 18, No. 3, 1979 BARREL WALL JUST T H E TOP OF FLI
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Figure 1. Cross-sectional view of a co-rotating twin screw extruder perpendicular to the screw axis with the barrel and the second screw rotating around the stationary screw and with the second screw rotating on its axis.
the chambers containing the majority of the fluid are geometrically complex with additional strain imposed on the material by the intermeshing second screw lands. In addition, material leaks from one chamber to other chambers in either screw so that a macroscopic mixing or blending action is achieved which is beyond that normally predicted by strain. Fairly complete analyses of mixing on the basis of strain have been made for single screw extruders (McKelvey, 1962; Tadmor and Klein, 1970) with a recent paper examining the strain distribution produced as material traverses the entire length of a plasticating single screw extruder (Lidor and Tadmor, 1976). However, less quantitative pictures have been developed for twin screw machines. For the center of a chamber, Wyman (1975) compared the shear rate distribution in a rectangular representation of a twin screw extruder channel to that for a single screw extruder and found that the shear rate profile due to the down channel velocity in a perfect positive displacement twin screw extruder is the inverted image of that in a single screw extruder with no net flow. Kim et al. (1973) predicted the three components of velocity far from the intermeshing second screw, calculated the shear rate components, and averaged them over the channel height to analyze mixing in the chambers. None of the twin screw extruder studies to date has examined the interaction of the intermeshing second screw lands with the material contained in the chamber formed between the first screw flights. This investigation was undertaken to determine whether the screw interaction could be significant in mixing. The problem is complex and the available knowledge of twin screw extruders is quite primitive. Therefore, a number of simplifications were required in order to analyze these extruders, including an approximation of the geometry of the intermeshing second screw lands. However, the picture that results shows that twin screw extruders are capable of a unique mixing action in the limiting case studied, and the mixing would probably be magnified if a more detailed analysis could be carried out. Furthermore, this investigation reveals some interesting limitations of flat plate models for twin screw extruders not generally appreciated. Flat Plate Chamber Model In order to obtain velocity profiles in the twin screw extruder chamber, a model is developed in rectangular coordinates since the helical nature of the screw flights combined with the curved boundaries of the second screw lands would make it impractical to solve the problem in cylindrical coordinates (Tung and Laurence, 1975). First, the assumption of a rotating barrel and stationary screw root is made so that the second screw revolves around the stationary first screw as shown in Figure 1 while simul-
Figure 2. Cross-sectional view of the chamber of a co-rotating twin screw extruder: a, the helical channel; b, the unwound flat plate model. vR Y f*,iz&,H,,
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