SEPARATBCN OF POLAR AND NON-POLAR MOLECULES IN A NON-UNSFflkM ELECTRIC HELD Them for {he Degree 0‘ pl]. D. MICHEGAN STATE UNIVERSITY Charles D. Beals 1967 THESIS L I B R A R Y Méchigan State University _ fflifl ‘. ABSTRACT SEPARATION OF POLAR AND NON-POLAR MOLECULES IN A NON-UNIFORM ELECTRIC FIELD by Charles D. Beals A theoretical and experimental investigation of the dielectrophoresis of dilute aqueous solutions of polymetha- crylic acid in a separation cell with cylindrical geometry was undertaken in this study in order to examine the feasi- bility of using the dielectrophoretic effect as a separation technique. Analysis of the forces acting on a dipolar molecule in an A.C. or D.C. non-uniform electric field shows that the molecule will migrate in the direction of highest field strength. Solution of the transport equations describing the system yields equilibrium temperature, velocity, and concentration profiles as a function of radial position. The equilibrium separation factor or ratio of top to bottom cell concentration is found from the radial concentration profile. The theoretical separation factor is found to in- crease with increasing cell length, applied voltage, and molecular polarizability and to decrease with increasing values of the ratio of the outer to inner cylinder radii. Charles D. Beals Separations were obtained experimentally at various values of the solute concentration, cell length, and applied power. The use of radioactive tracer and resistance tech- niques enabled accurate concentration measurement. Both cells used in the experimental investigation were found to have an optimum operating power at which the observed sepa- rations were maximized. Operating at optimum power, a cell 24 inches long yielded a separation of 24 percent at a polymer concentration of 0.01 gm./£. and a 12 inch cell, at the same concentration, gave a 3.5 percent separation. Increasing separations were found as the polymer concentra- tion approached zero. This is attributed to increased molecular polarizabilities caused by polyion elongation in extremely dilute solutions. The experimental results obtained using radioactively tagged polymer for concentration determination, verify the values obtained from resistance measurements for runs made at or below the optimum cell power. For these conditions, favorable agreement between the experimental and predicted results is also found. For runs made above the optimum cell power, the resistance results show greater separations than the counted results and seem to be in error. SEPARATION OF POLAR AND NON-POLAR MOLECULES IN A NON-UNIFORM ELECTRIC FIELD BY Charles Du Beals A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1967 To Carrol ACKNOWLEDGMENT The author wishes to express his appreciation to Dr. Donald K. Anderson for his guidance during the course of this work. The author is indebted to the Division of Engineer- ing Research of the College of Engineering at Michigan State University for providing financial support. Appreciation is extended to William B. Clippinger for his advice and assistance in the construction of the laboratory apparatus and also to Dr. J. Sutherland Frame- for his consultation pertaining to theoretical derivations included in this research. The understanding and patience of the author's wife, Carrol, is sincerely appreciated. iii TABLE OF CONTENTS Page ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . vi LIST OF TABLES . . . . . . . . . . . . . . . . . . . viii INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 BACKGROUND WORK . . . . . . . . . . . . . . . . . . 4 Dielectrophoresis . . . . . . . . . . . . . . . . 4 Polyelectrolytes . . . . . . . . . . . . . . . . 10 THEORY . . . . . . . . . . . . . . . . . . . . . . . 12 Previous Theoretical Considerations . . . . . . . 12 Statement of the Problem . . . . . . . . . . . . 14 Temperature Distribution . . . . . . . . . . . . l6 Velocity Distribution . . . . . . . . . . . . . . 20 Concentration Distribution . . . . . . . . . . . 26 EXPERIMENTAL METHOD . . . . . . . . . . . . . . . . 58 Apparatus . . . . . . . . . . . . . . . . . . . 58 Electrical System . . . . . . . . . . . 61 Synthesis of Polymethacrylic Acid . . . . . . . . 64 Solution Preparation . . 66 Procedure for Experimental Run of DielectrOphoresis Cell . . . . . . . . . 68 Determination of. the Separation Factor . . . . . 70 The Isotopic Exchange Effect . . . . . . . . . . 73 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . 76 Theoretical . . . . . . . . . . . . . . . . . 76 Experimental Results . . . . . . . . . . . . . . 80 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . 96 iv FUTURE WORK Experimental Theoretical APPENDIX I APPENDIX II - APPENDIX III - APPENDIX IV - APPENDIX IV-A - APPENDIX V - APPENDIX VI - APPENDIX VII - APPENDIX VIII NOMENCLATURE BIBLIOGRAPHY . The Solution to Equation (45) in Terms of T . . . . . . . Fortran Program for Velocity Pro- files The General Procedure for Solving Equation (81) Fortran Program for Radial Concen- tration Ratio Calculations Fortran Program for Separation Factor Calculations Theoretical Data . Experimental Resistance and Radio- activity Data . Experimental Separation Factor Curves Sample Polarizability and Separation Factor Calculations . Page 98 98 99 100 104 106 113 116 119 124 144 160 167 173 Figure 10. 11. 12. 13. 14. Variation of Variation of y for K = Variation of y for K = Variation of y for K = Variation of y for K - Variation of y for K = Variation of y for K = LIST OF FIGURES power per unit volume with y dimensionless velocity, V*, S dimensionless 10 dimensionless 25 dimensionless SO dimensionless 7S dimensionless 100 velocity, velocity, velocity, velocity, velocity, v*, v*, v*, V*, v*, Comparison of several velocity profiles with with with with with with Variation of the dimensionless concentration, C*, with y Variatiqn * ci/co. Variation K = 35 Variation 8' = 0. Variation L = 30. Schematic with 8' S cm. vi K. of the radial concentration ratio, for various values of K of the separation factor with L at for several values of 8' of the separation factor with L at 025 for several values of of the separation factor with K at for several values of B' diagram of dielectrophoresis cell Page 19 27 28 29 30 31 32 33 51 52 S4 55 56 59 Figure Page 15. Schematic diagram of electrical system in relation to the dielectrophoresis cell . . 62 16. Variation of reciprocal resistance with con- centration for polymethacrylic acid, molecular weight = 453,000, temperature = 25°C.. . . . . . . . . . . . . . 71 17. Variation of radioactivity in glass sample bottles with time . . . . . . . . . . . . 73 18. Variation of resistance with time for Run 32, C = 0.000225 gm./£., a' = 20%, cell 1, and Pavg = 2.8 watts . . . . . . . . . 31 19. Variation of separation factor with time for Run 32 . . . . . . . . . . . . . . . . . . 83 20. Effect of power on separation factor for tagged data. at C = 0.01 gm./£. in cell I with forced feed . . . . . . . . . 84 21. Effect of power on separation factor for re- sistance data at C = 0.000225 gm./£. in cell II . . . . . . . . . . . . . . . . . 86 22. Effect of power on separation factor for tagged data at C = 0.01 gm./£. in cellII. 87 23. Effect of concentration on separation factor for runs of approximately equal power in cell I . . . . . . . . . . . . . . . . . . 90 vii 'Table II III IV VI LIST OF TABLES Summary of experimental dielectrophoresis results Summary of forced feed results Radial concentration ratio data, from Equation (111) . . . . . . . . . . . . . . . . . Separation factor data, varying L, from Equa- tion (115) . . . . . . . . . Separation factor data, varying K, from.Equa- tion (115) . . . . . . . . . . . . . Experimental resistance and radioactivity data . viii Page 93 95 . 120 . 122 . 123 125 INTRODUCTION The term "dielectrophoresis” was introduced by 1H. A. Pohl50 in 1951 and is defined as the motion of matter caused by polarization effects in a non-uniform electric field. This subject has received very little attention in the past, whereas its counterpart in magnetic studies has provided one of the most powerful tools to structural in- organic chemistry. Some possible uses of the dielectrophor- etic technique include; chemical separations, an alternate technique for determining dipole moments, and a method of fractionating polymers by molecular weight. Another use of 'this method may be found in separating chemically similar :substances, such as cis- and trans-isomers, on which the <:ommon methods of separation are quite often ineffective. The criteria for a separation to be effected by the (dielectrophoretic technique is that the solute and solvent lrave different electric dipole moments. A dipolar molecule lias a.finite separation of equal amounts of plus and minus Cfllarge. If this charge separation is caused by the molecu- liir structure of the molecule, as in water or p-nitroaniline, tile molecule is said to have a permanent dipole moment. The nlagnitude of this moment is given by the product of the quan- 1liity of its charge times the distance between the charges. A molecule is said to have an induced dipole moment if a charge separation is caused by the influence of an electric field. This induced moment is caused in some mole- cules because their charges are relatively loosely bound and can be forced to migrate due to the force exerted on them by the electric field. The magnitude of the dipole moment of a polarizable molecule is given by the product of a constant, called the polarizabilfty, times the electric field strength operating at the site of the molecule. When a dipolar molecule is placed in an electric field it will experience a torque, which will tend to orient it in a direction parallel to the field direction. If the field is spatially uniform, no translational movement will result. If the field is spatially non-uniform, one pole of the dipole will be in a stronger field than the other and the force on it will accordingly be greater. This results in a net translational movement in the direction of highest field strength. It will be shown that the direction of di- polar movement is independent of the polarity of the field, making the method applicable in either A.C. or D.C. fields. Comparing dielectrophoresis to the more common pheno- mena "electrophoresis," it is seen that dielectroPhresis: 1. Produces motion of the particle which is not af- fected by the direction of the applied field. 2. Requires highly divergent fields such as are produced by concentric cylinders or spheres. 3. Requires relatively high field strengths. 4. Is, in general, a relatively weak effect and will be easily observable only in systems Which have a strong field and high electric moment. On the other hand electrophoresis: l. Produces motion of the particles in which the di- rection of motion is dependent on the direction of the field. Reversal of the field reverses the direction of travel. 2. Is observable with particles of any size. 3. Operates in either divergent or uniform fields. 4. Requires relatively low voltages. 5. Requires relatively small charges per unit volume of the particle. Pohl51 has used the dielectrophoretic technique to produce some interesting phenomena which include; selective precipitation, mixing, separation of course suspensions, and pumping of nonconducting liquids. In this study, the dielectrophoresis of a polyelec- trolyte solution in a concentric wire-cylinder electrode system is investigated. A theoretical analysis of the steady state temperature, velocity, and concentration profiles for this system is presented. Experimental measurements are ob- tained by resistance measurements and are verified by a radioactive tracer technique. BACKGROUND WORK Dielectrophoresis The theoretical and experimental aspects of dielec- trophoresis have been, in a limited sense, treated in the past. Mueller,46 50 38 and Pohl, and Loesche and Hultschig independently studied the size and direction of the dielec- trophoretic effect. Mueller in a theoretical analysis con- cluded that the effect would be small for particles of molecular size. Loesche and Hultschig also concluded this from their study of the theory and their experiments. Other investigator557’27 have, however, using very high field strengths and extremely non-uniform geometries, obtained appreciable separations of small molecules with this method. Loesche and Hultschig38 and Debye et_gl.9 have shown that measurable concentration changes are observed in the dielec- trophoresis of polymers. The first experimental evidence of the movement of polar molecules in solution was given by Karagouni527’28 in 1948. He placed a solution of nitrobenzene in benzene in the annular volume of a concentric tungsten wire and metallic cylinder. The apparatus was equipped with a continuous feed system and product was taken from a glass capillary which surrounded the wire. Upon applying an electric field to this system, he observed that after some time the concentration of nitrobenzene close to the wire was greater than that of the bulk solution. An applied voltage of 10,000 volts gave con- centration changes of up to 12 percent. In a second appara- tus the mixture was passed through a net, with horizontal wires forming the negative electrode and vertical ones the positive electrode. After passing through the net, the liquid was less concentrated in polar molecules than the liquid re- maining in the apparatus, demonstrating that the net acted as a dipole filter. A third apparatus consisted of a porous metal tube concentric with a larger outer tube. Using these two tubes as electrodes and applying up to 16,000 volts D.C. across them, separations of up to 39.5 percent were measured with tetraethylammomium picrate in benzene (dipole moment = 18.0 Debye). This method was also used to obtain a 4.5 per- cent separation at 16,000 volts of a mixture of cis-and trans-azobenzene, where transazobenzene has a dipole moment of zero. Debye g£_al.,9 in 1954, reported that the dielectro- phoretic effect could be observed in 1 percent solutions of polystyrene, a highly polarizable molecule, with an applied voltage of 7,000 volts. The apparatus consisted of a cylin- der coaxial with a fine central wire. It was proposed that since there was a concentration rearrangement within the cell, an accompanying change in capacitance of the system should also occur. The increase in capacity was determined as a measure of the concentration change in the cell. It was observed that after applying the voltage the establish- ment of the new capacity required about three minutes. A diffraction technique was alternately used to obtain more accurate concentration measurements. The possibility of using dielectrophresis for polymer fractionation was sug- gested by this work. Loesche and Hultschig presented a theoretical and experimental work on dielectrophoresis in 1955. In the theoretical treatment, the dielectrophoretic force was equated to the sum of the osmotic and frictional forces acting on a dipolar molecule. The following basic differential equation was obtained. H Q) C MF 5? - kT C —? = v St where: M - a + £32. = Total solute dipole moment 3kT St = 6nur = Stoke's resisting force and: a = polarizability u‘ = dipole moment of polar molecule = temperature P k = Boltzman's constant T F electric field strength r = direction of molecular movement C = concentration v = molecular velocity in the r direction u = viscosity rm = effective radius of dipolar molecule and all quantities are expressed in a consistent set of units. This equation considers only the dipolar molecules in the solution, and when extended to polymer solutions, the Debye inner field approximation8 is used to obtain the dipole moment. The time dependence of the concentration rearrangement in a cylindrical geometry was expressed by BC 1 3(CvAr) 8? a -'A; _—_§F—_ where: A = anL r r = radial position from the center of the cylinder. L - length of the cylinder. The general solution to the differential equation was not obtained. A relationship was obtained for the variation of concentration with time at very small times and an approx- imate steady state radial concentration distribution was determined. The experimental work consisted of studying the dielectrophoresis of nitrobenzene in carbon tetrachloride and polyvinylacetate in nitrobenzene. The apparatus and measur- ing technique are similar to those used by Debye g£_al.9 A separation of 0.04 percent was obtained at 750 volts for nitrobenzene in carbon tetrachloride, and separations of almost 1 percent were measured at 120 volts for the polyvinylacetate solutions. It was shown that for the polymer solutions, the maximum concentration change was proportional to the degree of polymerization. It is well to note here that although these separations are very small, this can be attributed to the low voltages used and the short length of the separation cell (4 cm.) Pohl,51 in 1858, obtained expressions for the dielec- trOphoretic particle velocity in cylindrical and spherical geometries. The work also includes the derivation of the ex- pression for the dielectrOphoretic force on a spherical parti- cle for the above geometries. Experimentally, it was shown that enrichment factors* of up to 2.5 could be obtained when separating polyvinylchloride suspended in an equal volume mixture of benzene and carbon tetrachloride. The process was termed "dielectro-precipitation." Perhaps the most notable work done on the dielectro- phoresis of small molecules was reported by Swinkels and S7 in 1961. Their theoretical analysis yielded the Sullivan expression, previously stated by Karogounis, for the ratio of polar to non-polar molecules at any given position in a non-uniform field. This was accomplished by applying the Boltzman Distribution Law to the expression for the net force *The enrichment factor was the ratio of the concen- tration of polyvinylchloride taken from close to the central cylinder of a coaxial cylinder apparatus to the concentration at the outer cylinder. on a polar molecule in the field. This expression is appli- cable for the steady state concentration distribution in a system with no external forces acting other than the electric field. A more detailed description of this derivation is presented later. The experimental work consisted of a study of the dielectrophoresis of nitrobenzene in carbon tetra- chloride. The apparatus was constructed to approximate a point electrode with a spherical outer electrode. This was equipped with a dropwise feed system and product was removed from a capillary surrounding the wire electrode. Concentra- tions were measured spectrophotometrically. For nitrobenzene in carbon tetrachloride, the solute concentration increased about 5 percent with an applied potential of 30,000 volts, and for p—nitroanaline with 50,000 volts applied, a separa— tion of 25 percent was observed. Analyzing the previous work on dielectrophoresis, it is seen that in order to obtain readily measurable sepa- rations with this technique either a very large electric field or a very large molecular dipole moment is required. Most molecules have permanent dipole moments of less than S or 6 Debye, thus a molecule with a very large polariza- bility is desired. This suggests the possibility of using a polyelectrolyte molecule which is capable of having an extremely large induced dipole moment in a moderate electric field. 10 Polyelectrolytes A polyelectrolyte is a macromolecule carrying a large number of ionic charges with small counterions sur- rounding it, rendering the total system electrically neutral. The unique properties possessed by polyelectrolytes are attri- buted to the configuration of the polymeric chain and the distribution of counterions associated with it. Whenever an uncharged polymer chain is converted to one carrying a large number of ionized groups, the mutual repulsion of fixed charges may lead to a very large chain expansion. Since the molecular polarizability is proportional to the cube of the end to end length of the polyion chain,10 this expansion will greatly increase the induced moment of a polyelectrolyte in an electric field. For a weak polyacid, such as polymethacrylic acid, the degree of ionization can be controlled by the addition of a 63,64 has strong base to the aqueous polyacid solution. Wall shown that increasing the degree of neutralization correspond- ingly increases the degree of ionization of a weak polyacid. 18’29’49 have observed large increases in Several investigators the viscosity of polymethacrylic acid solutions as the degree of neutralization increases from O to 50 percent. The in- creased viscosity is attributed to polyion expansion caused by the mutual repulsion of the ionized groups. Viscosity results have also shown that decreasing the polymer concentration leads to increased chain expansions. This results from 11 decreased shielding of the fixed charges since the counterions are distributed further from the polyion chain in increasingly dilute solutions. The molecular polarizability is also dependent on the freedom of the counterions to move along the direction 10 have shown that of the extended chain. Eigen and Schwartz polyelectrolytes in an electric field exhibit enormous dipole moments which are attributed to this freedom of counterion movement. O'Konski47 termed this effect surface conductivity. Dielectric constant studies by Mandel and Jenard41 support the view that polyelectrolytes exhibit a longitudinal polari- zation due to the mobility of bound counterions. Polymethacrylic acid has been the subject of many of the experimental investigations of polyelectrolyte be- havior. The availability of information about this polymer as well as its large polarizability have been the primary considerations leading to its use in this study of dielectrophoresis. THEORY Previous Theoretical Considerations Several attempts have been made to describe the concentration changes observed when a dipolar particle is subjected to an electric field. These have been qualita- tively discussed in the previous section. The basic con- siderations and assumptions used by Swinkels and Sullivan57 in their derivation of the steady state concentration dis- tribution as a function of field strength will be given here. Consider a dipolar molecule in an electric field. Assuming that the applied field is non-uniform in a direc- tion, r, and decreases with increasing r, the net transla- tional force acting on a dipole of moment M is: _ dF f-MaT (l) where F is the electric field intensity acting at the site of a molecule. M, the total moment, is the combination of contributions due to the polarizability of the molecule, a, and its permanent dipole moment. The induced moment is given by mi = aF. (2) The contribution of permanently polar molecules is given by 12 13 an average moment, uL(x), where u is the absolute value of the dipole moment, L(X) is the Langevin function and x is uF/kT, with k and T being the Boltzman constant and tempera- ture respectively. For a detailed description of the concept 7 Thus, of an average moment, the reader is referred to Debye. the total moment M is M = aF + uL(x). Considering a solution of polar and non-polar mole- cules and introducing subscripts p and n for them respectively, the difference in force on the polar and non-polar molecules is Af = fp - fn = [uL(x) + F(ap - an)] 55- (4) Applying the Boltzman distribution law to the system under consideration, the ratio of polar to non-polar molecules is given by N N NR] = N2 exp c-Er/kT.) (5) n F =F nI~‘=0 r where Er is the energy difference between polar and non-polar molecules at a point where the electric field intensity is Fr' If this energy difference is zero at r = m, then Fr Er = -/£ [uL(x) + (up - an)F] dF. (6) Introducing the approximate form of the Langevin function for small values of x, uF/3kT, and integrating, 14 l “2 Er='7 3121*“p'0‘n F.» (7) Equation (5) then takes the [orm N N u2 2 NE. = N2' exp SET + up - an Fr/ZkT .(8) F = Fr Fr = 0 The assumptions included in the previous derivation are: l. The field is continuous but non-uniform in the r direction. 2. Interactions between neighboring molecules are neglected. 3. Only the mean polarizability of a molecule is considered in determining its induced moment. 4. The variation of field intensity over a molecu- lar distance is neglected in the calculation of the induced moment. It is immediately evident from Equation (8) that separations of polar mixtures may be obtained in either an A.C. or D.C. field, since the concentration ratio is depen- dent on the square of the field strength.* An analogous expression has been obtained by Frank15 from a thermodynamic analysis of polar mixtures. Statement of the Problem A more fundamental approach to the problem of ob- taining equilibrium concentration distributions for a *This requires that when the u2/3kT terms in Equa- tion (8) is important, the molecules can re-orient within the time represented by one cycle of an A.C. field, and when the a term is important, the mobile charge associated with a molgcule can shift within one cycle of an A.C. field. 15 particular geometry is to solve the transport equations for the system. This is accomplished by first solving for the temperature and velocity profiles for the given system. They are then used in the solution of the equation of con- tinuity of species. The system under consideration is a very dilute polyelectrolyte-water solution in the annular space between two concentric cylinders connected to reservoirs at both ends. Assume that the cylinder is long enough, compared to its radius, such that end effects may be neglected. The inner cylinder, in this case, is a fine wire. Further, as- sume that in the limit of extreme dilution, the concentra— tion dependences of density, viscosity, coefficient of volume expansion, and diffusivity are negligible. The com- peting effects of sedimentation and thermal diffusion are also neglected in this treatment. On applying a potential across the wire and outer cylinder of the above system, three effects begin to occur simultaneously. First, the dipolar molecules are oriented and attracted to the wire. Second, back diffusion starts to occur due to the concentration change caused by dielec- trophoresis. Third, the applied voltage produces a current in the solution causing Joule heating. This establishes a temperature gradient through the solution, with the inner electrode at a higher temperature than the outer one, and natural convection takes place. The separations obtained 16 are caused by this combination of effects. It may be stated that no loss in generality is incurred by saying dipolar "molecules" rather than "particles" as the following treat- ment is applicable in either case, as long as the system conform to the stated assumptions. Temperature Distribution For the system under consideration, define the radius of the inner cylinder, or wire, as R1 and the outer radious R0. The ratio of outer to inner radius is then a constant, K. From Bird, Stewart, and Lightfoot,4 the follow- ing simplified energy equation is obtained _ 1 3 3T O‘klf'a‘?(r'5‘f)+se (9) where: k1 = thermal conductivity of solution r = radial dimension T - temperature 8 = power produced/unit volume due to electrical dissipation. The simplification of the basic energy equation includes neglecting: a. viscous heating b. all variables with angular (6) dependences c. all velocities except those in the length (2) direction d. variation of temperature with time 17 e. the temperature variation in the z direction. The power generated by Joule heating is obtained from examining the radial variation of resistance in a cylin- drical geometry: pldr where: dR = a differential increment of resistance pl = the resistivity of the medium dr = a differential increment of distance in the r direction L = the length of the cylinder. From basic electrical relationships it can be shown that _ AVI r where: P(r) = total power generated by Joule heating as a function of r AV applied voltage across the cell I = current across the cell. Similarly the electrical source term, Se’ is s = AVI ( 1), (12) e ZnL £n K .;2 Introducing the dimensionless variable y = _£, (13) R. 1 the eXpression for Se becomes 5 = AVI ( 1) (14) e 2nLR§ tn K ;7 18 Figure 1 shows the variation of —% as a function Y of y. This is directly proportional to the power produced per unit volume as a function of radial position. Examina- tion of Figure 1 shows that Se decreases very rapidly over a very short radial increment. This indicates that for geom- etries of practical interest (i.e. K > 20), a reasonable as- sumption is that all the heat is produced at or very near to the wire. This simplifies the mathematics by reducing Equation (9) to _ l 8 8T with the boundary conditions that T = T. at r = R., 1 1 r = R , T = T . o 0 Making use of Equation (13) and letting T ‘ Ti (16) 2 9.129. +192=0, (17) dy ydy The boundary conditions on Equation (17) are also made di- mensionless and are: at r = Ri’ y = l; T = T. O = 0 II 75 '-3 II '-3 C) II H r = R0, y It is seen that by the substitution 19 0 1 ‘1 1 l l r I I l 5 10 15 20 25 30 35 40 Y Figure 1. Variation of power per unit volume with y. 20 3% = Q. (18) Equation (17) may be reduced to a readily soluble first order differential equation in Q, d 1 = 5*28 “ (w) the solution of which is C1 Q = 7- = 3;. (20) where c1 is a constant of integration. Equation (20) solved in terms of 0 becomes 0 = d in y + .2 (21) l where c2 is the second constant of integration. The con- stants c1 and c2 are then eliminated by applying the dimen- sionless boundary conditions to Equation (21), and the radial temperature distribution in terms of the characteristic parameters of the cell is _ + _ in T - T1 (T0 Ti) IH‘K . (22) Velocity Distribution Having obtained a mathematical relationship for the temperature as a function of radial position, the problem of the velocity profile for the cell may be considered. From Bird, Stewart, and Lightfoot,3 the generalized equation of motion, simplified to satisfy the system under consideration, is 21 3V _ 3 1 8 . Z 0'§E+u?fi(r_5—r+pgz’ (23) where p = pressure 2 = length variable 0 = viscosity V = fluid velocity in the z direction 9 = density gz = acceleration of gravity in the z direction In obtaining Equation (23), the simplification of the equa- tion of motion included neglecting the variation of the 2 component of the velocity with time, terms resulting from bulk flow, terms with an angular dependence, and the second order viscous term. Expanding the density, p, in a Taylor series in T about a reference temperature T: o=olT+%%|T (T-T) + ........ , (24) and expressing the volume coefficient of expansion as H3) P Equation (24) may be rewritten as p = 6 - 68(T - T) + ........ , (26) where 5 is the density evaluated at T and similarly B is the volume coefficient of expansion evaluated at T. Noting that 82 = '8 (27) and that the pressure gradient in the fluid is due only to the weight of the fluid 22 3 _ _- 5E ' pg: (28) Equation (23) may be rewritten 2 d V dV _- 3 + i __“'z = - MB (T - T). (29) dr r dr u Inserting the dimensionless radial variable, y, and the temperature from Equation (22), Equation (29) takes the form 2 -- 2 -- 2 d V2 1 de BpgRi(Tb - Ti) BpgRi ___7 + — ———- (in y) = - (Ti - T3- (30) dy y dy u in K H Letting 56gRi(TO - Ti) A = - u in K , (31) and -_ 2 _ oBgRi(Ti - T) B = - 9 (32) u Equation (30) becomes 2 d V2 + l dvz = A in + B (33) -——7 ——— Y , dy y dy subject to the boundary conditions: at y = l, V = 0 z (34) y = K, VZ = 0 Again it is seen that with the proper substitution, de ——— = Z, (35) dy 23 Equation (33) may be reduced to the readily integrable first order differential equation 3% + % Z = A Kn y + B. (36) Rearrangement of Equation (36) to an exact form and subse- quent integration yields 2 2 2 yZ = flé— £n y - 5%— + §§— + c3, (37) where c3 is a constant of integration. Reinserting Equation (35), in terms of the velocity gradient, into Equation (37) and integrating again gives A 2 A 2 B 2 VZ = —%— Kn y - —%— + —%— + c3 in y + c4 (38) where c4 is the second constant of integration. Applying the boundary conditions to Equation (38), the constants of integration are 2 and = ($71.19., (40) C4 and Equation (38) becomes 2 2 A A-B 2 A-B 2 AK £n ,, A-B Vz = ‘6‘ Z” Y ‘ L‘Z‘ly * 4 n l (K '1) z” y ‘ 4 *y L 4 )° (41) The velocity profile is now defined except for ob- taining a relationship which determines the average temperature, 24 T, about which the Taylor series expansion was made. The expression which defines T is obtained from requiring that the net volume flow in the z direction be zero, which ex- pressed mathematically is Ro 211VZ r dr = 0, R. 1 or equivalently K I V'ydy=0. (42) l _where V' is a dimensionless velocity defined by 4 V2 1: '- V A _ B' (43) Equation (41) expressed in terms of the dimensionless velo- city is 2 V' = (TS—gin y(y2 - K2) - (y2 - 1) + Win y. (44) Inserting Equation (44) into Equation (42) and rearranging, the expression which defines T is K K K 3 A 3 0=1ydy-fydy+A_Bly£nydy 1 K 2 +[KTET1‘K-i—E)KZ]1 yinydy. (45) 25 Inspection shows that the result obtained from solving Equation (45) will be a relationship for B in terms of the parameters A and K. This relates the reference tem- perature T to the AT across the cell and the characteristic constant of the cell, K. The reader is referred to Appendix I for the details of this manipulation, the result of which is B = A 1 _ -K4 2n K +(3/4))<4 - K2 + 1/4 , (46) 1 4 1 2 2 ‘K “‘27:?“ '1) which in terms of T is 2 — _ i 0 -K4 in K +(3/4)K4 - K + 1/4 T - Ti + Kn K 1 - l - K4 + 1 (K2 - 1)2 In K (47) Equation (41) may now be written, using Equation (46) to eliminate the unknown B, as -K4 in K 4(3/4)K4 - K2 4 1 2 2 1‘K+m(K‘-1) 2 4 1 2 + 2/4 ‘£——;—l 44 Y. (48) l " K + m(K " 1) in K + 1/4 4vz = A(l - yz) + A(y2 - K2) 2n y + A -K4-£n K +(3/4)K4 - K2 or in terms of a new dimensionless velocity, 4 V2 * = I V ‘7T—" (49) 26 as -K4 [n K +(3/4)K4 - K2 + 1/4 2 2 2 V* = 4 1 (1 _ y ) + (y - K ) in y 1'K ”'27.?“ ‘1) 4 4 2 2 + -K in K4+(3/:)K - K +21/4 £__;_1 in y. (50) 1'K*m("'1) ”K Thus,Equation (50) is the complete velocity expression in terms of the radial variable y and the cell's characteristic value K. The velocity equation was programmed in Fortran computer language and profiles were obtained with the aid of a Control Data 3600 computer. The program used is shown in Appendix II. Figures 2-7 show calculated velocity profiles for various values of K from S to 100. Figure 8 demonstrates the relative magnitude of several calculated velocity profiles in relation to the cell parameter K. Concentration Distribution Having expressed the temperature and velocity distri- butions in mathematical form, the radial concentration as well as the equilibrium reservoir concentrations may now be obtained. From Bird, Stewart, and Lightfoot,5 the generalized equation of continuity of species in cylindrical coordinates is 8C 1 3 1 BNAG aNAz - 31—* a? 33““) *‘5‘33‘* 3z “RA (51) where CA = concentration of species A, the polyelectrolyte, in a binary mixture, hereafter referred to as C 0.6- 0.4_ Ln K 2M Z BBgR. -0. b Figure 2. 27 «II-N Variation of dimensionless velocity, 5. V*,with y for K qty—m uni-.p pBgRgAT 4qu £n K V* -1. 28 5 Figure 3. Variation of dimensionless velocity, V*, with y for K = 10. £nK 4uV pBgRiAT 21- 29 —10' Figure 4. Variation of dimensionless velocity, V*, with y for K 25. QT pBgR 30 -40’ Figure 5. Variation of dimensionless velocity, V*, with y for K = 50. 4qu in K 53gR41T v* 31 180 160 140 120 100 80 6o ' 4o 20- -20 -4o -60 Figure 6. Variation of dimensionless velocity V*, with y for K = 75. 2871K 68gRgAT 4uV Vt 300 280 260 240 220 200 180 160 140 120 100 80' 60 40 20 -20 -40 -60 —80 "100 b Figure 7. Variation of dimensionless velocity, V*, with y for K = 100. 33 Variation of V* with K and y -- 2 V* = (4uV’ZKnK)/p8g(Ti—TO)Ri Ro/Ri r/Ri OUTER / / WALL / / //// / //// .. / / > x / x / / //// / Y=K / //// / //// / //// , WIRE / / //// ’ 4 I l I I I I I l I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Y/K Figure 8. Comparison of several velocity profiles. 34 - flux of A in the r direction 2 2 <1: I l flux of A in the 0 direction N - flux of A in the z direction RA = rate of production of A per unit volume by chemical reaction. Equation (51), with the assumptions of steady state, no chemical reaction, and no G dependencies, may be simplified to 1 3 SN Az ? W” NAr) = ' 82 (52) Since NAr is the net flux of the polyelectrolyte in the r direction, it is seen that this term will be the com- bination of contributions due to dielectrophoresis and dif- fusion. The expression for NAr is obtained from considering the chemical potential of the polyelectrolyte in an electric field, m N up 2 UC=U0+CV+1)RT£nm‘2-m+ap Fr(53) where “c = total chemical potential of a dipole in an electric field3 :57 1.10 = constant R = the gas constant m = the solute molality N = Avogadro's number v = number of ionized groups per polyion axui electrostatic interactions between ions have been 35 neglected for infinitely dilute solutions. The term m . . “0 + (v + 1) RT 8n T000 represents the portion of the chem1- cal potential due to the ideal behavior of a dilute solute which is dissociated into (0 + 1) particles per molecule.32 Assuming the polyelectrolyte to have a very small dipole moment compared to the magnitude of its induced moment, and 1,)! further assuming that the number of ionized groups per polyion ‘ is much greater than one, Equation (53) may be reduced to m Na 2 “V UC=H0+VRT£nm‘—2—Epr. (54) The driving force for molecular movement in the r direction is obtained by taking the derivation of “c with respect to r, du d 8n m Na dFr ‘ 71% = ‘ VRTT * 7E (”H 71? (55) Noting that at infinite dilution the molality may be approxi- mated by the concentration C, Equation (55) may be written duC d in C dC dF dC .. _ . r o ‘ “a? " “RT—at— a? * NapFr ‘37: a? (56) The driving force may then be equated to the resisting force as d 8n C dF dC __ _ r OTIIJNrmVr - VRT—a-C-— + NapFr W a}? (57) where u = solution viscosity. Rearranging Equation (57) the velocity in the r direction is VRT d 8n C No Fr dFr dC V3 = 6n0Nrm ' dC + vRT —dC d?‘ (58) 36 The flux of polyelectrolyte in the r direction is given by NAr - C vr (59) which, combined with Equation (58), yields vRT Na Fr dFr dC NAr = SnuNrm 1 + VET d In C d?‘ (60) With dc — c, (61) after some simplification, the flux in terms of an approxi- mate form of the diffusion coefficient, DAB’ may be written N = _ D dC + DABqPFrC dFr, (62) Ar AB 3? va dr where DAB = vRT/6nNurm For the system of coaxial cylinders, the field strength as a function of radial position is AV Fr = _____1T' (63) r in R2 i and dFr = _ AV ,v_1 (64) ——— 2. 37 Substituting Equations (63) and (64) into Equation (62), the expression for the net flux in the r direction takes the form dC D (AV)2a 0 AB p . (65) N =-D _. 0— AR AB dr vKT £n2(RO/Ri) r3 Equation (65) is then simplified to dC N = - D __ + B (66) AR AB (dr _3) by letting vKT an Ro/Ri The left-hand side of Equation (52) then becomes 2 1 8 _ 3 C 1 3 28C 8 8C . (68) r 3r Ar AB 3r2 r 8r r r3 3r The flux of polyelectrolyte in the z direction, NAz’ is due to mass flow in the z direction and is NAZ = CV2. (69) Since the velocity in the z direction is a function of radial position only, the right-hand side of Equation (52) may be written BNAZ 3C az - - V2 82' (70) Eqiurting Equation (68) and Equation (70) yields _ 1 B 28 6‘) Y 38 which is the differential equation describing the concentra- tion as a function of radial position and length in the cell. Equation (71) is separable but application of this technique did not render it soluble. The first simplification employed in solving Equation (71) is the assumption that ac : —Z _ K. (72) This substitution reduces it to a second order differential equation in r only. Letting y = r/Ri (73) and making use of Equation (72), the dimensionless form of Equation (71) is 2 K11 =DAB dc. i. B E- ”C (74) z —_2 _—2 __2_3 ——2_4 ' Ri dy y Ri y dy Ri y An irregular singular point at r = 0 in Equation (71) has now been shifted to y = l. The singularity may be removed, facilitating a series solution, by changing the variable y with the substitution _ 1 S - 7. (75) Y In terms of the new variable 5 dC = ZS3/2 dC (76) 2137 ' 3's— 39 and 2 2 d C 3 d C 2 dC ——7 = 4s -——7 + 65 + —-. (77) dy ds ds Substituting Equations (75), (76), and (77) into Equation (74), reduces it, after simplification, to J dZC 1 8 dc BC RiZK +---———2-—--—-2—=———V. (78) ds2 5 2Ri ds 2Ri s DAB 2 The velocity, V is that defined by Equation (50), expressed z’ in its dimensional form. Equation (50) may be inserted into Equation (78) in terms of the dimensionless profile if the coefficient of V2 in Equation (78) is multiplied 0/4)A,where A is defined by Equation (31). The terms in Equation (50) can be rearranged to V* = c 2 8n + c 8n + c 2 + c (79) z 11 Y Y 22 Y 33 y 44 where O (N (N I A I 1—3 I 5 5.; 7K and 4o 4 l 4 2 1 2 2 “K + m (1.75K -' K - 3/4) ' z 2 K (K - l) T' = 4 I 2 2 n 1 ‘ K + m (K " 1) Letting 8' = B 2. (80) 2R. 1 R.2KA multiplying cll’ CZZ’ css, and C44 by the factor —%U——, AB denoting them as ci, cé, c3, C4’ respectively, and transforming Equation (79) into terms in 5, Equation (78) may be written as dZC 1 dC B'C ci cé c3 7+_-B'—-——=-—£ns-——£ns+-—-+C,'1-(31) ds 5 ds s 25 2 s It is interesting to note that the term, 8', in Equa- tion (81) is just the exponential portion of Equation (8) applied to a polyelectrolyte and evaluated at the inner cylin- der. This similarity leads one to suspect that the general solution to Equation (81) may be some combination of exponen- tial functions. The boundary conditions on the concentration distribution, in terms of r, are dC C N - - D ——-+ B = 0, at r = R AR AB(dr :3) 1 and r = R0. (82) lkl‘terms of y they become d5 R1 Y and y = K, (83) 41 and in terms of 5 they are further transformed to dC B —— = ——7 C = B'C; at s 1 ds 2R. _2 1 and s K (34) The general procedure for solving Equation (81) is to first set the right-hand side of the equation equal to zero and obtain a complimentary solution by the series method .. of Frobenius. The particular solution may then be obtained by the method of undetermined coefficients. The sblution by u. this method is given in Appendix III. A simpler technique, 14 involves inserting an appropriate sub-. suggested by Frame, stitution into Equation (81). This reduces the problem to the solution of two first order differential equations. As both procedures yield the same result, the latter is presented here. A new variable, ¢, is introduced, where (b =%.%- B'C, (85) and 2 d6 d C dC __ = __7 - 3' __. (36) ds ds ds Substituting Equations (85) and (86) into Equation (81), it is reduced to do 1 c' c' c3 ag+§¢=-.2%£ns-—:—£ns+—§+C&. (87) )Nith the boundary condition ¢ = 0; at s = l. (88) 42 The problem is now reduced to that of a first order differ- ential equation with a variable coefficient and is readily soluble by the methods of elementary differential equations. The complimentary solution, 6C, to Equation (87) is obtained by setting the right—hand side of the equation equal to zero, and solving the left side. Thus, as s c ’ or d¢> .c _ ds —$E - ‘ —§, (90) which yields on integration in 6C = - Kn s + C5 (91) where c5 = constant of integration. Simplification of Equation (91) gives C _ 5 4’6 ‘ ’5" (92) The calculations of Appendix III lead to the assump- tion of a particular solution for Equation (87) of the form op = A s (n s + B 8n 5 + Cs + D (93) where A,B,C,D, = arbitrary constants to be evaluated by the method of undermined coefficients. 43 The values of the coefficients are found by putting the assumed solution, Equation (93), into the differential equa- tion, Equation (87), and equating coefficients of like terms in s. The manipulation is straightforward, the results being C' _ _ 2 A‘ ‘7) C! _ _ l B" ‘2 I 1 c=c4+°2 —2 _8 Ci =' __ and D c3 + 2 . Thus the particular solution is _ cé ci C4 cé ' ci ¢p — _4 5 8n 5 - —7 8n 5 + —7 + ‘3 s + (c3 + —7 , (94) and the total solution, ¢, by linear combination of solutions, is c cé c' s 1 C2 Ci ¢=¢+¢=—§-—-IS£VLS+—2-£ns- +—3- S' CPS-*7. (95) The constant of integration, CS, is determined by applying 0 N45" the boundary condition, Equation (88), c' c' I .))_) The expressions for c5 and ¢ as a function of con- centration, Equation (85), are then reintroduced into 0 Nb‘ 44 Equation (95) giving dC 1 C4 cé ci cé ci a? - B C = - §._7 + _E + cs + —7 - _4 s in s +—7 in S c' c" c' -(é + _§. 5 - (cg + —%—) . (97) e'B'S. (98) Letting the right side of Equation (97) again be called ¢. and multiplying both sides by the integrating factor, Equa- tion (97) reduces to -B's ar21(e’B'SC) = e ¢, (99) which yields (3 = 38'S [fe’e'sqms + c6] (100) = a constant of integration. where C6 The function Je-B'S¢ ds, (101) in terms of s is I -B's c' c' c' e C' _ I - (_%+_%+c3+_%) s ds}-—}Jesss£nsds Ci _E C' . C' _ . - -—% je-B 5 8n 5 ds + (—% + ) e B S 5 ds c' , + c3 + _%)5e’8 5 ds . (102) 45 The bracketed terms in Equation (102) are labeled F1, F2, F3, T4, and P5 respectively and are evaluated separately. The results of these integrations are: I I 1“? °°_n Inn“. p=-(c_4.cz+cv+.c_1_ gns+2(1)(8)s (103) l 2 _8 3 2 I l n - n! where n = summation index C2 ; -B's -B's P2 = ——_T_7 in s e (B's + l) + e 4(8 ) ~ 1 J m n n n-“i V (-l) 8' s - [in s + i n S n1 1 L. J.) ) I" 00 I c' . ' e c-l)“ (3')“ sn T — ——l e'8 5 Zn 5 in s + 4 3 28' 1 n - n! T (c1 Ci ) e'B'S ( ) =- ——+— B's-+1 4 2 8.1(8')2 and l c' _ , 1’5“?" (CH—11685 The relations; F1, T2, F3, T4, and P5, are inserted into Equation (100). The right-hand Slde of Equation (100), after I .multiplication by e8 S and combination of terms with like powers of s, is 46 , c' c' c' c' c' , w (-l)n(8')nsn C=C6eB S_ _4+_.2..+c:'5+_l+ 2 42+ 1 e8 S [n 5 + Z 2 8 2 4(8') 28' l n - n! c' c' c' c' c' c' c' c' c' + 2 s Zn 5+ 2 2+ 1 in s— 4+ 2)s-(—§+ 1+ 4 2 2 2 4BI 4(BI) 28v ZBI 88! B! 28! 2(BI) 8(BI) (104) In order to evaluate c6, the boundary condition at the wall must be examined. The flux at each wall is zero, thus the net driving force at each surface must also be zero. It will be shown that if the driving force at the outside wall is zero, the radial concentration gradient evaluated at the out- side wall is also very close to zero. Modification of Equation (56) relates the concentration gradient to the known variables for a given experimental situation. The defining relationship, evaluated at r = R is O, Na (AV)2 RT dC 3 P7 = —— — . (105) vr in Ro/Ri r = R0 C dr r = R0 where, for a typical experimental run, AV = 100 volts = 0.333 e.s.u. a = 0.6 x 10-13 cm.3 P £n2 R /R = 12 8 o i ' r3|Ro = 0.0118 cm.3 C = 0.01 gm./£. 47 T 3000K. 0 2,350 The concentration gradient at the wall obtained from using the above values for the variables is about 10'4 gm./£./cm. Comparing the gradient at the wire to that at the wall shows that for the above geometry the value at the wire is about 46,000 times that at the wall and essentially is independent of all the variables except Ri and R0. This justifies assuming that dC ~ ' a? - 0, (106) r = R 0 which transformed into the s coordinate is dC a? _ _2 = 0. (107) S-K The constants in Equation (104) are redefined by I °° _ n n n C = c6eB S - A'eB's in s + E:( 1) (3') s ‘]+ A's in s 1 l n - n! 2 _J + Ag Ln 5 - AAs - Ag (108) where c' c' c' c' c' Ai = _% + _% + C3 + _% + EYE??? + EET 48' cé ci '5‘ 48 I I A = C4 + C2 4 28' 88‘ and I I I A, = :3 + C1 C4 c2 8' 28' 2(6')2 - 8(6')2 ' Applying the boundary condition, Equation (107), to Equa- tion (108) yields Eg' 1% m (-1)n (B'%“ n AiKzeK + AiB'eK -2 in K + f’ n ° n! K C6: §% 4- B'eK - A' (l - 2 in K) - K A' + A' 2 6T1 3 4. (109) .7 B'eK Inspection of Equations (108) and (109) shows that ~very term includes one of the prime constants; Ai, Aé, A3, 4’ or Ag, ither ci, cé, c3, or C4' The primes on these constants sig- and that every term in these constants contains ify that each term in Equations (108) and (109) is multiplied 2 R. KA / the factor'-%fi——. Dividing both sides of Equation (108) AB r this group defines a dimensionless concentration, 4CDAB Ri KA I I I I d returns c1, c2, c3, and c4 to the constants C11’ C22’ 49 c33,andcu4respectively, defined by Equation (79). Equa- tim1(108)in terms of C* takes the form II: - C c6e Ale in s + E. n . nT’ + A2 3 Ln 5 + A3 DIS - A45 - A5 (111) where 2 I I m n —§ E7 (-1)n (B!) AleeK + AIB'eK -2 in K + 1 n ' nIK2n c6 = E% + B'eK -A (l - in K) - KZA + A 2 3 4 BI :7 B'e .nd c c c c c 44 22 ll 22 11 A = + + C + + + 1 2 3 33 2 4(6')2 28' c _ 22 -A2 ' 167 c c 22 11 .A == + 3 2(6')2 28' C44 C22 c c c c 33 11 44 22 A. = + + — —————7 5 8' 28' 2(8')2 8(8') 50 Albrtran computer program for Equation (111) was developed and run on a Control Data 3600 computer. The pro- gmmn shmwlin Appendix IV, yielded values of C* as a function of y finrvarious values of the parameters 8' and K. The shape of the dimensionless concentration profile, shown in Figure EL shows an extremely large concentration gradient for very small radii. “This is to be expected as the electrical por- 3 tion of the driving force is inversely proportional to r’. The variation of the radial concentration ratio, C*i/C*o, with B' at several values of K is shown in Figure 10. The equilibrium top and bottom reservoir concentra- tions are found by relating C* to the length variable z, and evaluating 2 at L, the length of the separation cell. From the relation _ dC K - d? (72) ‘ith the condition that at z = 0, C = CB, 1e constant K is defined as C - C B. (112) K = ______ z bstituting Equation (112) into Equation (110) and rearrang- g yields _12 = C* (113) CB CT - 0' In K (z)’ 're O a- 51 j c... (4p2nKDABLC)/EBS(T1'TO)RimT-CB) / / / / / / / / / / / , / , / / / / / / / / / / / / / Z WIRE Y Figure 9. Variation of the dimensionless concentration, C*, with y. i: O * Ci/C 52 1.12T 1.08” 1.04‘ T l r . . I O 1.0 0.2 0.3 0.4 ' 0.5 2 (AV) OE B'= vaZRiZnZK Figure 10. Variation of the radial concentration ratio, CE/C; with B' for various values of K. 53 4D )' = -7r-4饣n (114) RiATpBg Equation (113) is seen to be the relationship between the concentration, C, at some point in the cell relative to the concentration in the bottom reservoir as a function of verti— cal position in the cell. The separation factor, CT/CB, is obtained from evaluating Equation (113) at z = L, as C T _ C* (115) E-C’T-O'KKKQY Separation factors were obtained as functions of K, B', and L from the computer program shown in Appendix IV-A. The variation of separation factor with length at a constant K for several values of B' is shown in Figure 11. The effect of length on separation factor for several values of K at a constant value of B' is shown in Figure 12. Figure 13 shows the effect of K on separation factor for several values of B'. For a discussion of these curves the reader is referred to page 76. The expressions for the dimensionless velocity and concentration profiles may be expressed in terms of dimen- sionless groups. For the velocity expression, 4p in K Vz Re v1 = __ = 16—— 2n K (116) pBgRiAT Gr 54 )0 — (AV)2 a BI = O 2 Z a vaZRi£n K K = Ro/ Ri )0" 30 r B' = 0.100 a 00 - a 00 — D 75 ~ .50 — D 8' = 0.025 .25 — a !_Gr————__Gr______cr______€y—————43 O a .00‘ r1 1. l l 1 I !~ 0 15.25 30.50 45.75 61.00 76.25 91.50 106.75 122.00 Length, Cm. igure 11. 'Variation of the separation factor with L at K = 35 for several values of B'. 55 L40— Dotted lines indicate an extension of the limiting SIOpe of the origin. .35 — 30 — C 25 - O // 0 — o / / / K = 35 / / .- ° / / / o / / c; / I " / a o / / ’/ / ,I ’ / D ’ I / ’ ’ _ ° '3 ’ K = 45 / ’/ ’ O n I l i L J I I I 1 15.25 30.50 45.75 61.00 76.25 91.50 106.75 122.00 Length, Cm. 12. Variation of separation factor with L at B' = 0.025 for several values of K. 2 AV ( )‘gp ‘ 8' = 0 VkTZRiznZK 1.80 ~ K = RO/Ri 8' 0.025 1.0 I I I l I 0 5 10 15 20 25 K Figure 13. Variation of the separation factor with K at L = 30.5 cm. for several values of B'. .__l 57 where Reynolds number Re Gr Grashof number. The concentration expression is 4L DABuLn K C L/D C in K c*= = 64 R?(CT - CB) 68gAT (Gr)(Sc) CT - CB where L/D - the length to diameter ratio Sc = Schmidt number. (117) EXPERIMENTAL METHOD Apparatus Experimental measurement of the dielectrophoretic separations was accomplished with a coaxial tube and wire separation cell. A schematic diagram of the separation cell is shown in Figure 14. Two cells of this same general de- sign were used in the experimental program. The first cell was constructed of Type 304 Stainless Steel. The tube, or working portion of the cell, was made from 0.25 inch O.D. tubing with 0.035 inch walls and was 12 inches long. It was equipped with an electrical connection, spot-welded equidistant from the ends of the tube. Teflon sleeves, which screwed onto the tube, insulated the working portion of the cell from the reservoirs. The central wire was 0.005 inch diameter platinum wire. The wire was fas- tened with a plug at the bottom of the cell and was attached to a spring loaded clamp, equipped with an electrical lead, at the t0p of the cell. The reservoirs were made of 0.875 inch O.D. stainless tubing with 0.156 inch walls and were 2 inches long. They also were equipped with electrical con— nections. Caps, made from 1.5 inch diameter extruded teflon rod covered both bottom and top reservoirs. Extreme care was taken to ensure that the holes drilled in the caps 58 59 J ELECTRICAL TERMINAL —:_) LOADING SYRINGES -~<\\\d_f UPPER ,4 j; RESERVOIR / / EZ/ Q 6: 54' f 3 / / . 2 / v/NP / _____‘/I\/\ SEPARATION ; A TEFLON TUBE "’h h SLEEVES / H ELECTRICAL / . TERMINALS / , / A w --E-w0¢"-u—'—' C . ) ’ / [T /’ €——w1RE // LOWER ,/ RESERVOIR /’ F U :ure l4. Schematic diagram of dielectrOphoresis cell. 60 enabled accurate positioning of the wire in the center of the wqfldng tube. The cap at the top of cell I was threaded swfllthatthere was a space of 0.5 inches from the top of the reservoir to the bottom of the cap. This ensured that when'Uuecell was filled, the liquid level was always above the meUfl.reservoir, enabling both the upper and lower re- servoirs to be used as conductivity cells. Both reservoirs were fitted with capped hypodermic needles, located at the bottom of the reservoir, for loading and sampling. Data taken using cell I indicated that the polymer degraded in the cell. This was eliminated by the use of a second cell made of tantalum. Cell 11 was of the same general design as cell I with the following modifications. The separation tube was 0.25 inch O.D. seamless tantalum tubing with 0.020 inch walls and was 24 inches long. The small wall thickness prohibited the use of screw connections to the teflon sleeves. Slip fittings were used to make these This required that the teflon be machined to :onnections. Minute irregularities flit tightly over the tantalum tubing. n the surface of the tantalum caused the connections to leak hen.tflue apparatus was full. This was eliminated by sealing T1e connections with high vacuum stopcock grease. The reser- Jirs, £1150 tantalum, were made of 0.75 inch O.D. tubing with Ie same wire, filling syringes, electrical connections, and re clamps as werez-used'in- cell I. 61 Both cells could be modified with a forced feed This enabled the collection system and capillary device. A glass capillary of solution which was close to the wire. This was placed tube was inserted into a thin teflon disc. In this between the tOp reservoir and the separation tube. The position, the wire was coaxial with the capillary tube. forced feed system consisted of a piece of tygon tubing con- nected to the bottom hypodermic needle. The tubing was L shaped so that the open end of the tubing was above the top of the capillary tube. Feed could then be introduced into the cell by controlled addition of polymer solution at the open end of the tygon tubing. This forced the solution close to the wire through the capillary tube and into the (previously empty) top reservoir. Electrical System A schematic diagram of the electrical apparatus used 0 produce the electric field is shown in Figure 15. The lectrical equipment consisted of two parts; (a) the circuit squired to produce a potential difference across the wire d tube of the separation cell, and (b) the circuit necessary measure the solution resistance in the cell reservoirs dur- an experimental run . , The applied voltage to the experimental cell was ’ duced by amplification of the output signal of a Hewlett- kard, model 200 C.D., wide range oscillator. A slide—wire 62 .HHQO mwmouonmouuooHOHw may on cofiumfiou a“ Eoumxm Hwofipuooao mo Empmmww ofiumEonom .mH opsmfim 44mg mHmmmozmomeumqua “a moaHmm >HH>Heu=azou . mmoumongumo L e A , zoeHzm _ mmHqumz< moe<44Humo mmem2140> .IIIIH 1 v . mmemzz< N moemHmmm H moemHmma 63“ resisuM'was placed in parallel between the output of the oscillator and the input terminals of a General Radio Com- pamy,nwdel 1233aA, amplifier. This permitted very fine control of the input voltage to the amplifier. The ampli- fiercnuput was connected to an "on-off" switch which controlled the input signal to the cell. A variable resistor in series with the amplifier output enabled accurate voltage control and current was measured with a Weston, model 425, milliammeter. Thus, with the switch "on", the amplifier out- put was applied to the separation cell, whereas the circuit could be broken, enabling resistance measurement of the soluv tion in the reservoirs, without having to turn off the oscillator and amplifier. The input voltage to the cell was measured with a Simpson, model 260, variable scale voltmeter at the output of the amplifier on the cell side of the switch. This gave an accurate measure of the voltage across the separation cell as the only sources of error were due to the resistances of the leads and electrical connections. Although the volt— age loss due to dissipation in the leads and connections is extremely low compared to the voltage drop across the cell, care was taken to keep all connections corrosion free and the leads In) the cell as short as possible. The input signal to the cell was monitored with a Dumont, model 304-H. cathode- ray oscilliscope in order to correct any fluctuations in the oscillator or amplifier outputs. Either a sine or square 64 wave input to the cell could be obtained by varying the oscillator gain control. The solution resistances in the top and bottom cell reservoirs were measured during an experimental run with an Industrial Instruments Inc. RC-18 conductivity bridge. Dur- ing a conductivity measurement,the input voltage to the cell was turned "off" and each reservoir was used as a conductivity cell with the platinum wire serving as one electrode and the reservoir the other. This enabled the measurement of concentration changes without sampling the reservoirs. For experimental runs in which natural convection controlled the solution flow, the input voltage to the ex- perimental cell was also applied across the wire and lower reservoir. This was necessary to ensure that the solution in the bottom reservoir was warmer than the downflow from the separation tube, otherwise the downflow would not have entered the lower reservoir. Synthesis of Polymethacrylic Acid Polymethacrylic acid was prepared from a free radi- cal polymerization of the monomer, methacrylic acid, obtained from Eastman Organic Chemical Company. The monomer was vacuum distilled to remove the inhibitor and 100 gm. of the product were charged to a 2 8. flask. Added to this were 300 ml. of Reagent Grade methanol, 300 ml. of distilled water, and varying amounts of benzoyl peroxide. The benzoyl peroxide 65 is the initiator for the polymerization, and its concentra- tion controls the average molecular weight of the polymer product. For product molecular weights of about one million, about 0.2 gm. of initiator is required. The reaction mixture was stirred by bubbling nitrogen through the system and was maintained at 700 C. by immersing the reaction vessel in a constant temperature bath. The reaction proceeded for about four hours when the appearance of a cloudy, viscous solution indicated completion.60 The swollen polymer product was dissolved in l 8. of methanol and was precipitated with 3 2. of diethyl ether. This purification procedure was repeated several times to ensure the complete removal of monomer and catalyst. The polymer has the formula 66 where 11= the degree of polymerization. The purified polymer was fractioned by the method of Runy.lz The product was dissolved and then precipitated Inrdrop-wise addition of diethyl ether to collect products of different molecular weights. The polymer fractions were dried under vacuum for 48 hours and then ground to a powder giving a yield of about 35 percent usable polymer. Polymer molecular weights were determined viscome- trically by the method of Katchalsky and Eisenberg.29 Polymer of molecular weight 9.6 x 104 to 1.18 x 106 was obtained by this method of preparation. Solution Preparation Polymethacrylic acid is soluble in water. However, dissolution becomes increasingly difficult as the polymer molecular weight is increased. The experimental polymer solutions were prepared by adding a weighed amount of polymer powder to»a.l.£u volumetric flask. The contents were diluted Adifli.500 ml. of triply de-ionized water with a conductivity )f approximately 10'6 (ohm-cm.)' To this was added the mount of sodium hydroxide necessary to obtain the desired .egree of neutralization. The mixture was gently agitated ith a magnetic stirrer until all the polymer was dissolved. r1 the case of very high molecular weight polymer, this 67 required several days. The concentration of the stock solu- tion was l.0,gm./£. Prior to an experimental run, a solution (fifthe desired concentration was prepared by dilution of the stock solution. Radioactively tagged polymer solutions were prepared by addition of a known amount of NaZZOH to the experimental solutions during their preparation from the stock material. In all cases the concentration of NaZZOH was too low to sig- nificantly change the degree of neutralization of the polyion, yet was high enough to give reliable counting statistics. The first tagged polymer solutions were prepared with radioactive sodium obtained from the Nuclear Chicago 3301 in Corporation. In its initial form, the tracer was NA 1 M. HCL with an activity of lOuc./ml. of solution. The tracer was carrier free (i.e., all of the sodium was radio- active). The NaZZCl was converted to NaZZOH in H20 by ion exchange of the original material and was diluted to the desired concentration. A second batch of radioactive sodium was obtained from Volk Radio Chemical Company. The Na22 came as ZOOIJc. 22C1 in 1.0 ml. aqueous solution with a specific acti- Calcula- of Na vity'cxf 69 mc./mmole of total sodium in solution. tion showed that about 0.05 percent of the sodium is radio- active and the rest is carrier. The same preparation procedure as described above was used for this material. 68 Procedure for Experimental Run of DielectrophoresisICell 1) An aqueous polymethacrylic acid solution of the desired concentration and neutralization was prepared by dilution of the stock solution. For runs in which the con- centration was to be determined by the solution activity, the volume of NaZZOH solution added was included in the dilution volume. 2) The separation cell was rinsed several times with conductivity water and then twice with the polymer solution. 3) The cell was placed in a horizontal position with the syringe needles up. Polymer solution was injected into the bottom needle with a hypodermic syringe until solution flowed from the upper needle. Entrapped air was eliminated by forcing solution into the bottom needle while holding the cell at a 450 angle from the horizontal. The procedure was then reversed using the top needle. This was continued un- til no air bubbles were seen in the overflow. 4) The loaded cell was weighed to check for evapora- tion losses later. 5) The cell was clamped to a ring stand. Care was taken to ensure vertical alignment of the separation tube. 6) The resistances in the tOp and bottom reservoirs as well as the separation tube were measured. This was pri- marily to ensure that the central wire was properly aligned so there was no short circuit. 69 7) The apparatus was placed in an incubator main- tained at 100 C., and the electrical leads were connected. 8) The resistance of the solution in the top and bottom reservoirs was measured until it was constant, indi- cating that the cell and its contents had come to tempera- ture equilibrium. 9) The oscillator, amplifier, and oscilloscope were turned on and the desired frequency was set on the oscilla- tor. With the conductivity bridge leads disconnected at the instrument, the switch was turned "on", applying an A.C. potential to the separation cell. The desired voltage or power was obtained by adjustment of the second variable resistor. 10) The resistance of the solution in the top and bottom reservoirs was measured as a function of time. This was accomplished by turning the switch "off" and rapidly measuring the resistances. The voltage was then reapplied to the cell, making sure that the conductivity bridge leads were disconnected. The resistances were measured with the bridge operating at 1000 cps. 11) After a run was completed, the cell was re-_ weighed to ensure no loss of solution due to evaporation. When tagged polymer solutions were used, 10 m1. samples were withdrawn from the top and then the bottom cell reservoirs. 70 These were stored in plastic vials for further con- centration measurement. The cell was then dismantled, thoroughly cleaned with a dilute HCl solution, rinsed several times with de-ionized water, reassembled, and filled with deaflnfized water. The cell was then ready for the next run. Determination of the Separation Factor A measure of the separations obtained in the dielec- trophoresis runs is given by the separation factor. This is defined as the ratio of top to bottom reservoir concentration. As is seen from the linearity of Figure 16, the reciprocal of the solution resistance may be used as a measure of its concentration. The separation factor is then, in terms of resistance, C R __T=B T ‘8 Equation (118) implies that the reservoirs have the (118) same cell constants. To eliminate any cell constant effects, the resistances in Equation (118) are divided by their ini- tial values yielding C [C0 R /R0 T 1T B 1B = (1193a) CB7 CiB RT; iiiT Noting that the initial concentration is the same in both reservoirs and subtracting "one" from both side of Equation (119) , the concentration change may be written as 71 0 0.025 0.05 0.075 0.10 0.3125 0.150 0.175 0.200 0.225 Concentration, gm./£. Figure 16. Variation of reciprocal resistance with con- centration for polymethacrylic acid, molecular weight - 453,000, temperature - 25°C. 72 C ' CE = RB/RiB ‘ RT/RiT (120) CB RT/RiT The concentration change represented by Equation (120) is termed the fractional separation, and is the separation factor minus one. The percent separation is calculated from the experimental resistance data by substituting the appropriate measured resistance values into the right-hand side of Equation (120) and multiplying by 100. The deter- mined values may then be plotted as a function of time. 22 For the runs made with tagged polymer, the Na activity in the top and bottom samples is a measure of the polymer concentration. The solution activity is determined as follows. Five ml. of each sample taken at the comple- tion of a run, as well as five ml. of feed are measured into plastic scintillation bottles with a five ml. burrette. To each of these are added 20 ml. of scintillation liquid.* The number of radioactive counts per unit time is measured for each sample in a Packard Tri-Carb Liquid Scintillation Spectrophotemeter. The solution activity is directly pro- portional to the polymer concentration. Thus the separation factor is CT counts from top sample/unit time CT— = (119-b) B counts from bottom sample/unit time, *The scintillation liquid was a mixture of 1.4 gm. 111.0 m . P.O.P.O.P., 125 m1. anisole, 750 m1. p- P.P.O., g and 125 ml. 1,2 o-dimethoxyethane. dioxane, ea w v 73 .M4m. .0536 abs: moaupon oHaEmm mmwam :fi xufi>wuomoflpmu mo eofiumfihm> when .6539 N .KH oeemfim d ueomopa hos» anemone u Hog oe epmofleew -u 6530a 3335 -O 162 o OH ON om cc .1 om ”w 9 I so .3 m OK "1 1. S om m” cm a. en T: m 2: u o: u. m 9 oma._ ”M omH. D ova omH 74 and may be calculated by direct substitution of the values Obtained from the scintillation counter. The Isotopic Exchange Effect Some of the first samples measured with the radio— active tracer technique were stored in glass sample bottles at the completion of the run. The feed sample had also been placed in a glass container, usually prior to the start of the run. Subsequent determination of the top, bottom, and feed activities invariably showed the feed to have the lowest value. This erroneously indicated that the feed was less concentrated than either the top or bottom products. Investigation of this abnormality showed that the erroneous measurements were due to isotopic exchange of the Na22 with the Na in the glass sample bottles. Tests were run simultaneously in which one set of glass sample bottles was filled with Na 2OH in H20 and another identical set was filled with tagged polymer solution. active sodium used was the same in both cases. The amount of radio- The samples were allowed to stand for varying lengths of time after which each bottle was emptied and thoroughly rinsed with de-ionized water, methanol, and acetone. The sample bottles were then filled with scintillating liquid and counted. Figure 17 shows the variation of the bottle activity with time for both tests. Examination of Figure 17 shows not only that isotopic exchange occurs but also that the amount 75 of exchange is greater for NaZZOH in water than for the equivalent tagged polymer solution. Background counts were subtracted out of this data, and the counting time in each case was 150 minutes. The ratio of the correspond— ing values from the tests show that with no polymer present about 50 percent more isotOpic exchange occurs than does for the tagged polymer. This indicates that some fraction of radioactive sodium is effectively bound to the polyion decreasing the amount of Na 2 which is free in solution for isotopic exchange. A more elaborate investigation of this phenomena could possibly yield information about the nature and amount of counterion binding in very dilute solutions. The erroneous measurements due to isotopic exchange were eliminated in later experiments by using plastic sam- ple bottles. RESULTS AND DISCUSSION Theoretical The equation describing the dimensionless concentra- tion distribution for dielectrophoresis in a cylindrical geometry, Equation (111), is given on page 49. The variation of C* with radial position is shown in Figure 9. The extreme gradient, noted at very low values of y, is attributed to the inverse proportionality of the dielectrophoretic driving force to the cube of the inner cylinder radius. This suggests that the inner cylinder radius will seriously affect the mag- nitude of the radial concentration ratio and should be as small as possible. Figure 10 shows the variation of the radial concen- tration ratio with the factor B' for several values of the cell constant, K. The curves show a slightly greater than linear dependence of the concentration ratio on 8'. It is seen that for values of 8' less than 0.1 the concentration ratio is independent of K. At larger values of B', the con- centration ratio increases with increasing K. The concen- tration ratios merge to a common curve for values of K larger than 25. A comparison of the values obtained from Equation (111) to the values obtained from Swinkles and Sullivan's 76 77 expression for the concentration ratio, Equation (8), is informative. Equation (8), applied to the system under consideration, is modified to cg qE(AV)Z 1 1 1 __ =-eXp - (121) 0* ZOkT 2nz R6 R? R7 0 fi— 1 O i where 2 2 Ri<< when: N HONHOz .N\.EN Honesz womwme oocmummmom 56:6366Hm > Hozom :oHu HmHDUOHoz :oHu cam comumhmmom pcoohom -mNmHmhuaoz -mhucoocou .muHSmOH mHmOHocQOHuooHoHp HmpcoEHHomxo mo preezm .H OHQOH 94 .Houmpu xcmmEou HmoNEogu xHo> nu“: mums mash mmumuwwcHg OO.O O.O OOO.O HH OOH OO.O ON OOO.OOH.H HO.O NO O0.0 O.O OO0.0 HH OOH O.O ON OOO.OOH.H H0.0 «ON O.O- O.O- OOO.O HH OHH O.O ON OOO.OOH.H HO.O «ON NO.H H.H- OOO.O HH NNH OO.O ON OOO.OOH.H HO.O OON OO.O- N.H OOO.O HH OO O.O Om OOO.OOH.H HO.O OON OO.O- H.N- OOO.O HH OO OO.O Om OOO.OOH.H HO.O ONN OO.H- O.N- OOO.O HH OO OO.O Om OOO.OOH.H HO.O OHN OO.O O.N OOO.O HH O.NO OO.O Om OOO.OOH.H HO.O OON OO.H O.H OOO.O HH OHH OO.O ON OOO.OOH.H HO.O OOO OO.O- O.O- OO0.0 HH OHH OO.O ON OOO.OOH.H H0.0 OOO ON.H O.N OOO.O HH NHH NO.O ON OOO.OOH.H HO.O ONO ON.O 0.0 OO0.0 HH OO H.O ON OOO.OOH.H HO.O OO OO.ON O.NH OOO.O HH OO OO.O ON OOO.OOH.H HO.O OO OH.H O.OO OOO.O HH OO O.O ON OOO.OOH.H HO.O OO NO.O O.ON OOO.O HH OOH NO.O ON OOO.OOH.H HO.O OO OO.O O.O OOO.O HH NO NO.O ON OOO.OOH.H HO.O HO O.O OOO.O H Om OH.N ON OOO.OOH.H OO.O HO 0.0 OOO.O H OO O.H ON OOO.OOH.H OO.O OO mumm mama .m.m.u Haou muao> mupmz w unmfioz .N\.Em Hmnszz wmmmmh mucmumMMmm xucoscmnm > Hozom cofip HmHsumHoz sown cam :ofiumummom udmupmm -mNmHmHusoz‘ -mNpcmucoo .OoOOHOOOO .H OHOOH 95 ON.O NO.H OO N.O HH OO0.0 ON OOO.OOH.H HO.O OO OO.N O0.0 ONH ON.N H OO0.0 ON OO0.00H.H HO.O OO OO.N OO.H ONH NOON H OO0.0 ON OOO.OOH.H HO.O NO HN.O OO.H OHH ON.N H OOO.O ON OOO.OOH.H HO.O OO ON.H NO.H ONH O.N H OOO.O ON OOO.OOH.H HO.O OO mama wmmmmh mhzo: muHo> muumz HHmu .m.m.u O “swam: .N\.Em Honssz cowuwnmmom oswe vomm > umzom zucmscmhm soap HmHsuoHoz :ofipwpuamucou cam ucmuuoml -mNOHmHHsz .mpHSOoH wmmm uouuom mo thessm .HH manmh CONCLUSIONS The theoretical investigation of dielectrophoresis of polyelectrolytes in a system with cylindrical geometry _ shows that the predicted separation factors increase with i increasing 8', which is directly proportional to the molecu- L.“ lar polarizability and applied voltage and inversely propor- tional to the inner cylinder radius. The predicted separations were also found to increase with increasing tube length, L, and decrease with increasing values of the ratio of the outer to inner cylinder radii, K. The results show that the radial concentration ratio for a system with flow is only slightly less than the static equilibrium value. This enables the radial concentration ratio obtained from Equation (111) to be approximated by the form obtained from considering only dielectrOphoresis; the exponential of 8'. The radius of the inner cylinder is seen to critically affect the radial con- centration ratio and should be as small as possible. The experimental investigation demonstrated that dielectrophoresis may be used to separate polymethacrylic acid from water. An optimum applied power, at which the observed separations were maximized, was found for both of the experimental cells investigated. For cell I (12 inches long), the optimum operating power was 2.6 watts, and for 96 97 cell II (24 inches long) it was found to be 4.5 watts. The optimum power is interpreted as being the ideal balance be- tween the natural convection forces (which increases the separation) and the thermal mixing forces (which decrease the separation). The same dependence was noted in equili- brium radial concentration ratio measurements. The separation factor increases greatly with de- creasing solute concentrations for extremely dilute solutions. This is attributed to increased molecular polarizabilities resulting from polyion chain extension in increasingly dilute solutions. The experimental results obtained using radioactive sodium for concentration determination verify the values obtained from resistance measurements for runs made at or below the optimum cell power. For runs made above the opti- mum power, it is felt that the radioactivity results are the most reliable. The experimental separation factors, obtained at optimum cell powers, compare favorably to the predicted values calculated from experimental values of B'. This substantiates the theoretically predicted affect of length on the separation factor and indicates that the assumption of a.constant concentration gradient in the z direction was reasonable. FUTURE WORK It is suggested that future work continue along the following lines: Experimental 1. Runs should be made with longer cells since the cell length greatly affects the equilibrium separation. 2. Consideration should be given to the continuous operation of a long cell. The present design could be modi- fied with the addition of feed and product streams and if the cell was long enough, appreciable staging would result. A study of this nature might greatly enhance the practicality of dielectrophoresis. 3. Experimental runs should be made with radioactive polymethacrylic acid which is tagged on the polymer chain. This would eliminate any uncertainty in the measurements due to impurities in the radioactive sodium. 4. For the present system, a device should be con- structed such that the wire could be pulled through the apparatus. This would impart an upward velocity, equal to the wire velocity, to the material immediately adjacent to the wire which is at the highest radial polymer concentration. 'The modification could readily be incorporated in the theoreti- caJ.velocity and concentration expressions. 98 99 Theoretical 1. The power generation term could be included in the solution of the temperature profile. This would affect only the particular solution of the energy equation and would extend its applicability to low values of K. 2. The equation of continuity of species could be solved without assuming that the concentration gradient in the z direction is constant since the equation is separable. A numerical method would probably be required. APPENDIX I The Solution to Equation (45) in Terms of T APPENDIX I The Solution to Equation (45) in Terms of T. K' 'K K 3 3 0=£yd>~fly dy+r¥gfiy 11'1de K2 — 1 A 2 K + 'EZE—E_ - (I—T—F-K y in y dy (45) The procedure necessary to obtain an expression for T from Equation (45) is to integrate each term in Equation (45) and then solve for the group I—é—F, which can be re- arranged to give B as a function of A and K. Denoting the individual integrals in Equation (45) as l, 2, 3, and 4, .Equation (45) takes the form 0=fi+j2+f3+fl (I-1) K 2K _ __1 2 Lydy-E—l-gw-l) (I-Z) where: \W\ - 1) (1-3) \N.'\ II I 7§ ~< (N 3‘ II I “f1 —— 7x II I A] r—I ,\ 7K A 101 102 1 4 4 = A 5 B (I‘ z" K ’ 16 + 1%) (I 4) and Z I K S4=[—z—K n-K1- m1} )Kz] betydy 1 _ K 2 2 Z _ K - l A 2 ’LW'(R—-—FK] §—£"Y'§—) 1 2 Z 2 TH“) Adding the individual integrals and factoring all terms multiplied by A—é—E’ Equation (45) becomes .A l(1 - K2) + (K4 - 1) - (K2 - 1} K2 in K - K2 + 1) = 2 ‘1 In K j 7— 7_ ‘I 4 2 A - B K 3 4 K 1 ’T£”K+EK ’T+1‘6 (1-6) vfliich on inversion and further simplification is A - B -K4 in K + % K4 - K2 + % A = 4 2 (1'7) l-K «km—l—KRZ—l) 103 Rearrangement of Equation (I-7) gives the following rela- tionship for B as a function of A and K, I ) ~K4 in K f”; K4 - K2 + %' B = A $1 - . (46) 4 1 2 2 1'K+m(K'1) APPENDIX II Fortran Program for Velocity Profiles 105 ozm azm moam O N 09 oo amzoo.m«am>.Omozsm O.H.Omozsm aHnoarmmamum¢9m> NH NHAOOmOmoOHNNNHOOOOoOHONH OOmOHOOmu.HnHHOw«NHO»+HNnvmvmooq\HNHvamooqva amzoou BHQQN HH NNOOmOmooq\.H+HHHOOOmoOHONNHNOOmOmoquNNevmvmoOHVNNNOOmONOOmIHO+HH Nnvmvmooq\NnOmONOOmOnHNOOmOOoOH\HHuNHHOwOmwoquNHOOONHOwu «New NNNnOmOmooq\¢«ONOOm+NNOOmOmoOHNHOOmONOOmO.NINNnvmvmooq\.H+O OOOHevmu.HO\HN«ONOOmuHNevm«NOOmIO\OuO«OHOOmOON.HOOHNNnvmvmoog\.HO+O NOOONOOmn.HuNOOmaNOOm«.NO«NHNHOOmOmoOH«HNnvmvmooqv\.HOOu amzoo O.O.ONH«>mm9OmH NHOOINOOmuHN>mms H+HuH OuH Hun HN.HuO.AOOmO.HONrHuH.NHOwO.No4mm HNOm.HOOOw onmzmzHo NOH.OH.>mHO942mom H0.0HmOauwoo~o> how Emhwoum :muuuom APPENDIX I I I The General Procedure for Solving Equation (81) APPENDIX III The General Procedure for Solving Equation (81) dZC 1 dC B'C ci c2 c3 ._7 + _ - B' —— - ——— = - —— in s - —— £n s + —— + CA ds 5 ds 5 25 2 s (81) The general method for solving Equation (81) is to obtain a complimentary solution by applying a series tech- nique, such as the method of Frobenius, and then use the method of undermined coefficients or variations of para- meters to obtain a particular solution. The sum of the complimentary and particular solutions is the total solu- tion of the equation, and the constants of integration may be eliminated by application of the boundary conditions of the problem. Applying the method of Frobenius to Equation (81), let the right hand side of the equation equal zero, and assume a solution of the form, c = Z A sn + Z (III-l) n 0 where n = summation index K = dummy variable 107 108 An = constant determined by n. Differentiating Equation (III-l), substituting into Equa- tion (81), and rearranging yields P A :z(z - 1) + z]s£'2 + ZA (n + z)2 smb2 0| n L 1 -ZAn B' (n + 1: + 1) 3’1”“1 = o 0 (III-2) where A0 = An evaluated at n = 0. The indicial equation, evaluating the possible values of £, is obtained from the first term in Equation (III-2) as £2 - o - , (III-3) yielding z = 0, 0 The second and third terms in Equation (III-2) may be com- bined by letting n = n + l in the second term. This relates the constant An+1 to AD as An 8' (III-4) An+1 = n + I + 1’ and if£= 0, = _E——— (III-5) 109 or A 3' An = —E;%——— . (III-6) Relating the constant An to A0 yields (8')n A0 An = ———fiT——— . (III-7) Substituting Equation (III-7) into Equation (III-1) and- noting that z = 0, the first part of the complimentary solution to Equation (81) is 8'5. (III-8) Another solution is seen to be required as the indi- cial equation yielded two values of 1. Since the values of K are equal, the procedure for obtaining the second portion of the solution, CII’ is to take the derivative of CI with respect to £ and evaluate it at £ = 0. An alternate proce- dure is to use the method of variation of parameters which generates both CI and CII’ Using the latter method, let G = U A eB'S (III-9) C11 = U(s) I (s) where U(s) = a function of s to be determined. lflie differential equation describing U(s) is found by sub- stxituting Equation (III-9) into Equation (81), as 110 d U U ._EO§§1 + l + 3') €3I22.= o. (III—10) s s - s The solution of Equation (III-10) is . m _ n , n n 0(5) = a1 2n 5 + E:( 1)n Sen} 5 + b1 (III-ll) where a1, b1 = constants of integration. The second portion of the complimentary solution is then / 00 8'5 2(4)n (8')“ sn C = U C = A e a fin s + + b II (S) I o 1 1 n , n! 1 ’ (III-12) and the total solution is - = 8'5 _ 8'5 Z (-1)n (8')n snl C - CI + CII c6e Ale Zn 5 + 1 n . n! (III-13) where C6 = A0 + blAO A1 = "A0 31’ and c6 and A1 are the same as in Equation (lll). Examination of Equation (III-12) shows that the Inethod of variation of parameters generates both parts of 'the complimentary solution from the first solution, CI‘ 'The particular solution to Equation (81) is obtained by the 111 method of undetermined coefficients. Assume that Cp = A s [n s + B £n s + Cs + D (III-14) is a solution of Equation (81), where Cp = particular solu— tion of Equation (81) and A, B, C, and D are arbitrary constants to be evaluated. Substituting Equation (III-l4) into Equation (81) and equating like powers of 5 yields A = ZET (III-15) Ci cé B = + —————7 28' 4(8') and t I 1 I D = — .C_3 - C2 + C1 + C4 81 8(8')2 28' 2(Bv)2 The total solution of Equation (81) is the sum of the com— plimentary and particular solutions. Thus . . °°_n.nn C = c6eB S - Ale8 5 £n s + E:( 1)n SSH} s + A s [n s + B Zn 5 + Cs + D (III-16) Comparison of Equation (III-16) with Equation (108) shows that both methods of solution yield the same dependence of the concentration profile on the variable 5. Evaluation 112 of the constants c6 and A1 in Equation (III-l6) is accom- plished by applying the boundary conditions, Equation (84). The solution obtained is identical to Equation (lll). APPENDIX IV Fortran Program for Radial Concentration Ratio Calcultations 114 Nm«mao.OO\No u HOOOOO.NO\OU + NmOO.NO\Ho + m\mo u OK NOOO.OO\NO + NOOO.NO\Oo "ON NOOO.NO\Ho + NOOOOO.OO\NU u Om NOOO.OO\NO u Na NNOO.NO\Ho +NOOAOO.OO\NO + O.NNHo + me + 0.0\No + O.N\Oo u HO mo- u Oo xNO.Huau u Oo Nx«xO\NmOmuo\HOux\HmOmOuNO.HuOOOOOOONxNO.HO u No ANNO.HO u Ho NNHO.OO«OO+ N NOOOOO.NO-O.HO«HxNO.HO+NO.O«OmOuO.HO\NNO.OOOmOnHON.uNmOmOu H HHOOOmOOON.HOOOHx\O.HO+HHO.OOOOOIO.HIAOOOOO.NOO«NHxOxO\O.HOO u a vamooq u x m . OOH eszm OOH aszm NON O.OH O\ .O u A O n ON ON .H n O O on m .OOH aszH NOH aszm mamflm OH .H u x N on 0440 zmmoomm mm.Ozem meoOpOHSUHmu ofiumm cofiumhucoucou Hmflomm Now Emumopm ampuuom 115 OOQMNMNZDM ozm HO OH m>Hmm Nemm mo mOH<> mma OO O a¢zmom OOH NO.Nm . xN . «OO O a¢zmom OOH N O OH quzOHHom may mom oHaam OOHOOO may OOO BNSOOH NOH NH0.0Hm.xOON .«OOO amazon HOH AA m.mamtxhvv.«oxv B<2m0m OOH szHazoo N mozHezoo O x .m .O¢ .ON .ON .N< .H« .HOH aszm oHa .o .w .OOH aszm H m¢ I m«v¢ I 3am¢ + 3xmKN¢ + Ao.mH\w¥m*m«m«m«m H I o.v\m«m«m«m + mum I 3V#Am«mvmme¥ ad I Awsmvhmxm«m H U «o + NOOOOU + NOOHOOHONo + HOOOOOHOOOOOHo u > vamooq u z NOOOO\O.H u m H u w 2 .H u H H on m u z 2OO\HHO.O«OOOO.OHO\HOHONOI HO.OOOmOO.OHO\HOmOm u HO.OOOmOO.OO\HOm + m\Hu u 25m Aaam«mv\mvmmxm«mv\fiv¢ + m<¥m«MI N NxOO.NnO.HO «NO I HNO.OO«OOO.OHO\O.OOOOI HO.O«OOOO.OO\AON + H m\mn xOO.NuOONNOOmO\HOmmmem«HN + HNm«mv\mgo.NvmmmemOm«H may OO O a¢zmom NO.Nm . xN . OO« O e .o .w .OOH aszm N . OHHNmo . HOH aszm HNO HmO muoq + N.OOH uoO \ o u oHeOmo qma mN.mH H N Hu Hm O.H u H O on ON I OOOO u 3OO< + zOOONN + HO.OHNOOOOOOOONOO H I o.v\m«w¥m«m + mam I KOOAmkmOmmxma Hfi I Am«mvmmxm«m H U «U + Ntwamu + vamwoqimo + ANOmUOA«N¥N«HU H > NmOHooq u 3 HNONONO.H u m H n w z . z . H u H H on N\z u z m u z zsm\NHO.OOOmOO.OHO\H«HOmOn u oHaOm HO.O«ONOO.OHO\H«O«H I HO.O«OOOO.OO\OOO + ONO: u zom HANOOmO\mOOmxm«mO\HON + OmOmOmn N Ax«o.NI0.HO «Nd I Afio.m«*m«o.mHO\o.m*«ml Ho.v««m«o.vO\m¥m + H m\mn NOO.NIOOHNOONONNOOONMONOHO + NNOOOONHOO.NOmemOm«mOHNO u m oomw NMNZDm voa moa NOH HOH ooa Q‘I—IMN mooscHucouO :oHumHsono Houomm :oHHmHmmom How EOHmoum :OHHHom APPENDIX V Theoretical Data 120 Theoretical Data Table III. Radial concentration ratio data, from Equation (111) * t K B. Ci/Co s 0.5 1.5823 s 0.25 1.2618 5 0.125 1.1249 5 0.100 1.099 5 0.05 1.0487 5 0.025 1.0242 10 0.5 1.6151 10 0.25 1.2717 10 0.125 1.1233 10 0.100 1.1016 10 0.05 1.0499 0.025 1.0248 15 0.5 1.6302 15 0.25 ‘ 1.2769 15 0.125 1.1302 15 0.100 1.1029 15 0.05 1.0503 15 0.025 1.0250 20 0.5 1.6370 20 0.25 1.2794 20 0.125 1.1312 20 0.100 1.1037 20 0.05 1.0506 20 0.025 1.0250 121 Table III. (Continued) * t K B' Ci/CO 25 0.5 1.6406 25 0.25 1.2808 25 0.125 1.1318 25 0.100 1.1041 25 0.05 1.0508 0.025 1.0251 30 0.5 1.6427 30 0.25 1.2816 30 0.125 1.1321 30 0.100 1.1044 30 0.05 1.0509 30 0.025 1.0251 35 0.5 1.6441 35 0.25 1.2822 35 0.125 1.1323 35 0.100 1.1045 35 0.05 1.0510 35 0.025 1.0252 122 Table IV. Separation factor data varying L, from Equation (115) I B K L (cm.) CT/CB 0.025 35 15.25 1.027 0.025 35 30.5 1.056 0.025 35 45.75 1.087 0.025 35 61.0 1.119 0.025 35 76.25 1.153 0.025 35 91.5 1.190 0.025 35 106.75 1.229 0.025 35 122.0 1.270 0.100 35 15.25 1.133 0.100 35 30.5 1.307 0.100 35 45.75 1.545 0.100 35 61.0 1.887 0.100 35 76.25 2.425 0.100 35 91.5 3.391 0.100 35 106.75 5.637 0.100 35 122.0 16.70 0.025 45 15.25 1.013 0.025 45 30.5 1.026 0.025 45 45.75 1.039 0.025 45 61.0 1.053 0.025 45 76.25 1.067 0.025 45 91.5 1.081 0.025 45 106.75 1.096 0.025 45 122.0 1.111 123 Table V. Separation factor data varying K, from Equation (115) 8' L (cm.) K CT/CB 0.025 30.5 15 1.847 0.025 30.5 20 1.320 0.025 30.5 25 1.159 0.025 30.5 30 1.091 0.025 30.5 35 1.056 0.025 30.5 40 1.037 0.025 30.5 45 1.026 0.025 30.5 50 1.018 0.025 30.5 55 1.014 0.075 30.5 20 15.16 0.075 30.5 25 1.948 0.075 30.5 30 1.390 0.075 30.5 35 1.211 0.075 30.5 40 1.130 0.075 30.5 45 1.086 0.075 30.5 50 1.060 (1.075 30.5 55 1.043 APPENDIX VI Experimental Resistance and Radioactivity Data 125 Table VI. Experimental resistance and radioactivity data. Run 11 Cell I, C = 0.1 gm./£., a' = 15%, M = 96,000, V an = 21 volts, P avg = 0.7 watts t (hr.) R (ohm) R (ohm) Separation T B Factor 0.0 5,450 8,850 1.000 0.5 5,300 8,650 1.041 1.0 5,250 8,500 1.032 1.5 5,175 8,450 1.041 2.5 5,125 8,300 1.034 3.0 5,050 8,400 1.061 3.5 5,100 8,400 1.053 4.5 5,050 8,500 1.073 6.5 5,000 8,250 1.051 10.5 4,900 8,250 1.072 14.5 5,000 8,450 1.077 Run 15 Cell I, C = 0.01 gm./£., a' = 50%, M = 860,000, V avg = 50 volts, P avg = 1.5 watts ‘t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 13,000 11,400 1.000 0.25 12,500 10,900 0.995 0.75 12,400 11,000 1.013 2.75 12,000 11,200 1.064 3.50 11,800 11,200 1.082 4.50 11,550 11,400 1.126 8.75 11,900 11,900 1.141 11.00 11,600 11,600 1.141 12.25 11,500 11,500 1.141 126 Run 16 Cell I, C = 0.01 gm./£., a' = 50%, M = 860,000, V avg = 48 volts, P avg = 1.7 watts t (hr.) RT (ohm) RB (hm) Separation Factor 0.00 10,400 11,100 1.000 0.50 9,700 10,500 1.015 1.75 9,500 10,400 1.026 2.25 9,500 10,600 1.045 2.75 9,500 10,600 1.045 3.75 9,400 10,550 1.051 4.75 9.300 10,600 1.067 5.75 9,200 10,600 1.079 6.75 9,100. 10,500 1.081 9.75 9,200 10,600 1.079 13.25 9,400 10,800 1.075 14.25 9,400 10,600 1.057 Run 17 Cell I, C = 0.005 gm./£., a' = 50%, M = 860,000, V avg = 80 volts, P avg = 1.8 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 26,000 28,700 1.000 0.25 25,400 28,000 1.000 0.75 24,800 27,800 1.016 1.25 24,600 27,800 1.023 1.75 24,200 28,000 1.050 2.25 24,200 27,900 1.045 2.75 23,800 27,700 1.055 4.75 23,300 26,900 1.048 127 Run 23A Cell I, C = 0.07 gm./£., a' = 15%, M = 860,000, V avg = 75 volts, P avg = 1.09 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 97,400 109,930 1.000 0.25 95,250 107,950 1.004 0.50 94,730 107,270 1.005 1.00 93.300 106,200 1.010 1.50 92.950 105,600 1.009 2.00 92,700 105,400 1.008 2.50 92,600 105,150 1.007 3.00 92,200 105,100 1.012 5.00 92,150 104,710 1.006 9.00 91,700 105,070 1.005 10.00 91,150 105,150 1.013 Run 23B Cell I, C = 0.07 gm./£., a' = 15%, M 860,000, V an = 102 volts, P avg = 1.65 watts t (hr.) R (ohm) R (ohm) Separation T B Factor 0.00 97,400 109,930 1.000 11.00 86,500 105,400 1.081 15.00 85,240 105,900 1.102 16.75 84,400 105,850 1.112 22.50 83,280 105,030 1.121 23.50 82,290 105,020 1.122 128 Run 25 Cell I, C = 0.225 gm./£., a' = 20%, M = 453,000, V avg = 35 volts, P avg = 6.2 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 4,800 5,778 1.000 0.50 4,300 4,910 0.952 1.00 3,920 4,500 0.952 1.50 3,800 4,355 0.951 3.00 3,605 4,010 0.948 4.25 3,590 4,115 0.953 5.00 3,615. 4,140 0.951 6.50 3,510 4,055 0.959 9.50 3,535 4,060 0.957 9.75 3,485 4,060 0.968 13.00 3,440 4,035 0.976 21.50 3,365 4,010 0.996 Run 28 Cell I, C = 0.000225 gm./£.,a‘ = 20%, M = 453,000, V avg = 100 volts, P avg = 2.1 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 70,850 64,920 1.000 0.05 67,720 59,650 0.961 0.75 67,600 59,400 0.957 4.00 65,150 60,100 1.005 7.25 64,000 60,000 1.021 8.00 63,900 60,300 1.029 20.25 59,850 60,350 1.099 23.25 58,820 59,500 1.035 25.50 58,250 59,550 1.114 43.00 53,650 60,250 1.227 44.50 53,250 60,800 1.244 47.00 52,250 60,375 1.259 50.00 51,300 59,750 1.279 67.50 47,400 59,500 1.372 71.25 46,200 57,900 1.368 129 Run 30 Cell I, C = 0.000225 gm./£., a' = 20%, M = 453,000, V avg = 100 volts, P an = 4.2 watts t (hr.) R (ohm) R (ohm) Separation T B Factor 0.00 34,120 45,450 1.000 0.50 31,100 40,850 1.000 1.00 29,950 39,600 1.016 1.50 29,550 39,050 1.016 2.00 29,100 38,500 1.016 3.75 28,100 37,650 1.030 5.00 28,050 37,700 1.034 12.50 24,650 35,350 1.102 23.00 23,900 33,500 1.077 24.75 23,500 33,300 1.093 26.50 23,200 33,100 1.100 Run 31 Cell I, C = 0.000225 gm./£., a' = 20%, M = 453,000, V avg = 100 volts, P avg = 1.3 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 79,050 101,200 1.000 0.25 76,540 95,700 0.978 1.25 74,000 92,300 0.973 2.75 73,250 90,775 0.969 3.50 72,850 91,100 0.974 11.25 71,200 92,050 1.011 14.25 66,500 86,800 1.019 16.75 70,300 90,350 1.005 19.75 69,200 89,600 1.011 29.75 67,450 87,950 1.021 130 Run 32 Cell I, C = 0.000225 gm./£., a‘ = 20%, M = 453,000, V avg = 100 volts, P an = 2.8 watts t (hr.) R (ohm) R (ohm) Separation T B Factor 0.00 57,660 50,040 1.000 0.50 55,200 46,800 0.978 0.75 54,000 46,150 0.983 1.75 49,100 44,325 1.037 4.75 46,400 41,875 1.041 7.00 46,850 43,850 1.092 8.75 45,350 42,920 1.093 9.75 45,200 43,000 1.098 10.75 45,100 43,100 1.101 11.75 44,000 42,950 1.126 20,75 44,000 43,800 1.149 22.25 44,075 43,800 1.146 Run 34 Cell I, C = 0.000225 gm./£., a' = 20%, M = 453,000, V avg = 100 volts, P avg = 2.6 watts t (hr.) R (ohm) R (ohm) Separation T B Factor 0.00 70,700 50,900 1.000 0.75 64,850 45,150 0.972 1.50 63,450 43,110 0.949 2.75 60,750 42,150 0.950 4.75 51,500 43,030 1.164 8.00 47,410 44,450 1.303 19.00 45,380 45,480 1.392 21.00 45,100 46,015 1.421 22.50 44,950 45,750 1.417 24.00 44,800 45,750 1.419 26.00 44,500 45,400 1.420 131 Run 38 Cell I, C = 0.005 gm./£., a' = 20%, M = 1,180,000, V avg = 121 volts, P avg = 1.75 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 72,470 67,270 1.000 0.50 69,100 61,100 0.953 1.00 68,650 60,850 0.955 1.75 68,400 61,500 0.969 2.25 68,650 62,100 0.975 2.75 68,550 62,200 0.972 3.25 68,550 62,350 0.979 3.75 68,500 63,950 0.988 4.25 68,400 64,000 0.992 4.75 68,350 63,950 0.992 Run 39 Cell I, C = 0.005 gm./£., a‘ = 20%, M = 1,180,000, V avg = 115 volts, P avg = 1.85 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 63,400 62,800 1.000 0.50 60,600 57,900 0.965 1.00 59,900 56,600 0.955 2.00 59,400 55,200 0.938 2.50 59,400 55,550 0.946 6.00 58,700 55,000 0.945 6.50 58,600 55,500 0.955 7.00 58,600 55,550 0.957 9.50 57,750 54,400 0.957 132 Run 40 Cell I, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg = 95 volts, P avg = 1.8 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 43,500 36,460 1.000 0.50 42,675 39,575 0.848 1.25 41,560 38,125 0.839 4.00 40,850 38,625 0.867 9.50 37,700 37,270 0.906 11.00 37,075 36,550 0.902 Run 41 Cell I, C - 0.01 gm./£., a‘ 20%, M = 1,180,000, V avg = 104 volts, P an = 1.5 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 55,500 66,100 1.000 0.50 54,000 62,550 0.974 1.00 53,000 61,300 0.971 1.50 52,400 60,900 0.976 3.50 51,200 59,700 0.978 4.75 50,800 59,350 0.981 6.50 50,400 59,150 0.987 7.75 50,300 58,900 0.984 133 Run 43 Cell I, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg = 100 volts, P avg = 1.48 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 17,020 58,000 1.000 0.50 16,900 54,350 0.944 1.00 16,670 52,835 0.932 2.00 16,550 51,800 0.919 4.50 16,575 51,850 0.919 7.00 16,600 51,550 0.913 8.75 16,450 51,300 0.916 18.50 16,300 51,760 0.932 19.25 16,350 51,900 0.934 20.25 16,300 52,150 0.941 21.25 16,335 51,950 0.936 22.75 16,250 52,600 0.952 25.00 16,260 52,200 0.945 31.75 16,270 52,050 0.943 Run 44 Cell I, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V an = 98 volts, P avg = 1.36 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 18,000 58,000 1.000 1.25 17,490 51,950 0.923 2.00 17,960 51,750 0.922 4.00 17,415 51,600 0.921 5.75 17,445 51,650 0.921 16.75 17,415 52,775 0.940 18.50 17,405 52,700 0.940 19.75 17,325 52,850 0.947 21.75 17,315 52,600 0.944 134 Run 45 Ce11_I, c = 0.05 gm./£., 6' = 20%, M = 1,180,000, V avg - 52 volts, P avg - 1.87 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 13,380 17,000 1.000 0.50 12,750 15,640 0.966 1.00 12,575 15,370 0.963 1.75 12,490 15,260 0.963 2.50 12,435 15,205 0.963 3.50 12,415 15,195 0.964 5.50 12,400 15,185 0.965 6.50 12,370 15,250 0.973 7.75 12,300 15,240 0.978 9.00 12,280 15,250 0.980 21.75 12,185 15,305 0.992 22.25 12,170 15,370 0.996 24.00 12,125 15,295 0.993 25.75 12,100 15,220 0.993 26.00 12,135 15,370 0.997 27.75 12,840 15,240 1.000 Run 47 Cell I, C = 0.05 gm./£., a' = 20%, M = 1,180,000, V avg = 53 volts, P avg = 2.05 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 13,140 18,080 1.000 0.50 12,555 16,760 0.971 1.00 12,400 16,395 0.961 2.00 12,000 16,150 0.979 2.75 11,855 16,100 0.987 5.75 11,630 15,915 0.994 6.50 11,710 15,835 0.985 6.75 11,715 15,880 0.986 7.50 11,600 15,790 0.990 135 Run 48 Cell I, C = 0.05 gm./£., a' = 20%, M = 1,180,000, V avg = 45 volts, P avg = 1.9 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 12,660 4,676 1.000 0.50 12,150 4,316 0.959 1.00 11,770 4,260 0.981 1.75 11,950 4,200 0.949 3.75 11,760 4,135 0.947 5.25 11,735 4,115 0.946 7.25 11,700 4,116 0.949 9.75 11,500 4,123 0.970 11.25 11,310 4,138 0.990 18.50 11,350 4,190 0.999 Run 51 Cell I, C = 0.05 gm./£., a' = 20%, M = 1,180,000, V avg = 50 volts, P avg = 2.15 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 14,750 4,780 1.000 0.50 13,750 4,412 0.991 1.25 14,100 4,285 0.949 2.00 14,150 4,220 0.935 2.50 14,125 4,245 0.938 13.75 14,150 4,432 0.975 14.50 14,150 4,454 0.986 136 Run 61 Cell 11, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg = 82 volts, P avg = 3.67 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 47,300 45,400 1.000 0.75 42,300 40,100 0.989 1.75 42,400 38,955 0.958 2.75 42,225 38,015 0.935 3.75 41,485 37,305 0.934 4.75 39,950 36,615 0.956 6.25 39,175 35,725 0.951 9.00 38,250 34,600 0.942 16.75 37,200 33,275 0.934 18,25 36,870 32,970 0.932 19.00 36,800 32,930 0.934 20.25 36,600 32,725 0.932 21.00 36,450 32,700 0.934 22.25 36,425 32,500 0.930 WT (c.) WF (c.) WB (c.) Separation Factor * 1,028,780 973,468 976,353 1.054 Run 63 Cell 11, C = 0.01 gm/£., a‘ = 20%, M = 1,180,000, V avg = 93 volts, P avg = 5.8 watts ‘t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 62,400 47,500 1.000 0.50 57,100 42,300 0.972 1.75 56,900 42,450 0.981 2.75 57,500 41,500 0.949 4.75 52,600 42,650 1.067 14.75 47,050 42,665 1.193 16.75 46,700 42,150 1.186 19.00 46,250 42,435 1.190 20.50 46,300 42,515 1.240 21.75 45,800 41,850 1.205 Nd. (c) WF (c.) WB (c) Separation Factor 844,205 697,594’ 809,352 1.043 137 Run 64 Cell II, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg = 93 volts, P avg = 5.8 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 67,000 40,000 1.000 0.50 62,700 34,850 0.930 1.25 56,150 36,130 1.075 2.75 46,700 36,260 1.296 6.25 43,950 35,920 1.366 18.00 37,370 35,550 1.590 18.50 37,515 35,630 1.587 W (c.) w (c.) W (c.) Separation T F B F actor * 955,674 942,585 945,412 1.011 Run 65 Cell 11, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg: 60 volts, P avg = 4.55 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 49,200 48,700 1.000 1.00 51,000 52,000 1.062 1.75 51,500 55,250 1.086 2.75 51,200 58,000 1.149 5.50 50,400 58,000 1.169 N (c.) W (c) W (c.) Separation T F B F actor 1,274,690 1,209,489 1,025,020 1.243 138 Run 66 Cell 11, C = 0.01 gm./£., a‘ = 20%, M = 1,180,000, V avg = 93 volts, P avg = 4.1 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 61,300 51,000 1.000 0.75 55,950 47,900 1.030 1.50 57,515 48,835 1.022 2.75 57,710 50,450 1.050 4.00 57,650 50,600 1.055 5.00 57,600 50,420 1.051 5.50 57,500 50,420 1.053 11.00 57,350 50,200 1.055 11.75 57,100 50,120 1.055 22.00 56,500 50,000 1.064 22.50 56,550 50,135 1.065 W (c.) W (c.) W (c.) Separation T F B F actor 967,501 959,873 918,727 1.053 Run 67 Cell 11, C = 0.01 gm./£., a' = 20%, M = 1,180,000 V avg = 117 volts, P avg = 5.02 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 64,400 59,200 1.000 0.50 65,000 61,550 1.029 1.00 64,000 62,410 1.057 2.00 63,700 62,600 1.069 6.25 62,500 61,400 1.067 17.50 61,100 60,425 1.074 22.25 60,875 60,825 1.084 WT (c ) WF (c.) WB (c.) Separation Factor 2,018,588 2,030,858 1,994,262 1.012 139 Run 68 Cell 11, c = 0.01 gm./£., 6' = 20%, M = 1,180,000, V avg = 115 volts, P avg = 4.53 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 67,700 66,600 1.000 0.25 65,850 66,375 1.025 2.50 63,500 64,750 1.036 4.00 63,150 64,150 1.033 5.75 62,750 63,200 1.025 17.25 63,000 62,300 1.005 18.25 62,650 61,565 0.998 20.25 62,800 61,450 0.995 22.25 62,450 60,900 0.992 WT (c.) WF (c.) WB (c.) Separation Factor 2,018,042 2,121,454 2,111,602 0.956 Run 69 Cell 11, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg = 118 volts, P avg = 4.53 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 72,700 78,600 1.000 1.25 67,600 71,570 0.980 2.25 67,050 70,000 0.967 3.00 66,850 69,700 0.965 4.00 66,550 69,200 0.962 8.25 66,300 69,200 0.966 10.00 65,900 68,815 0.966 20.75 65,250 68,800 0.974 22.50 65,000 68,800 0.979 23.75 64,935 68,800 0.981 WT (c.) WF (c.) WB(c.) Separation Factor 2,102,373 2,136,349 2,074,904 1.013 140 Run 70 Cell 11, C = 0.01 gm./£., a' = 35%, M = 1,180,000, V avg = 92.5 volts, P avg = 4.48 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 44,500 48,230 1.000 0.50 42,200 45,640 0.997 1.25 41,800 45,025 0.993 2.25 41,200 44,900 1.003 4.00 40,650 44,960 1.020 4.75 40,650 44,900 1.019 12.50 40,120 44,210 1.017 23.75 . 39,800 43,800 1.016 W (c.) W (c.) W (c.) Separation T F B F actor 1,610,249 1,623,270 1,547,994 1.040 Run 71 Cell 11, C = 0.01 gm./£., a‘ = 35%, M = 1,180,000, V avg = 96 volts, P avg = 4.95 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 44,900 46,400 1.000 1.25 41,450 41,750 0.975 2.50 41,275 41,250 0.967 3.50 41,370 41,060 0.960 5.00 41,050 40,435 0.951 20.50 40,700 39,475 0.938 22.25 40,800 39,600 0.938 W (c.) W (c.) W (c.) Separation T F B Factor 2,157,968 2,186,102 2,188,923 0.986 141 Run 72 Cell 11, C = 0.01 gm./£., a' = 35%, M = 1,180,000, an = 94 volts, P avg = 5.05 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 42,630 41,200 1.000 1.25 42,000 39,800 0.981 2.25 41,550 39,450 0.983 3.75 40,900 38,800 0.982 5.50. 40,500 37,900 0.968 9.75 40,000 36,500 0.945 22.50 39,700 34,880 0.911 W (c.) W (c.) W (c.) Separation T F B Factor 736,547 744,660 742,819 0.991 Run 73 Cell 11, C = 0.01 gm./£., o' = 35%, M = 1,180,000, avg = 89 volts, P avg = 4.5 watts t (hru) RT (ohm) RB (ohm) Separation Factor 0.00 44,300 38,200 1.000 3.00 41,500 37,190 0.973 5.75 41,250 36,625 0.964 15.75 40,700 36,190 0.965 17.50 40,650 36,365 0.970 19.25 40,400 36,425 0.972 21.00 40,450 36,460 0.977 ‘WT (cu) WF (c.) WB (c.) Separation . Factor 1,466,764 1q483,119 1,477,048 0.993 142 Run 75 Cell 11, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg = 122 volts, P avg = 4.45 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 64,850 74,500 1.000 1.25 60,650 45,720 0.656 3.00 60,400 43,950 0.634 4.00 60,100 43,600 0.631 6.50 59,700 42,425 0.618 17.75 58,400 41,750 0.624 20.25 57,350 41,300 0.627 23.00 57,250 41,375 0.630 23.50 57,200 41,175 0.628 25.50 56,800 40,300 0.620 WT (c.) WP (c.) WB (c.) Separation Factor 980,763 977.514 970.192 1.010 Run 76 Cell 11, C = 0.01 gm./£., a' = 20%, M = 1,180,000, V avg = 110 volts, P avg = 4.5 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 57,400 55,000 1.000 1.00 52,650 32,600 0.666 2.00 52,350 31,200 0.613 3.50 52,050 30,250 0.616 12.00 52,550 28,945 0.576 13.25 52,400 28,815 0.575 16.25 52,400 28,645 0.573 18.25 52,400 28,430 0.567 20.25 52,050 28,000 0.562 23.50 52,250 28,050 0.561 W (c.) W (c.) W (c.) Separation T F B . Factor 842,937 854,090 875,728 0.961 143 Run 78 Cell II, C = 0.01 gm./£., a‘ = 20%, M = 1,180,000, V avg = 108 volts, P avg-= 4.5 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 58,100 18,325 1.000 1.25 57,600 16,220 0.892 3.00 57,300 16,155 0.894 12.50 57,000 16,120 0.896 14.75 56,200 15,990 0.902 17.50 56,200 15,920 0.900 20.25 56,250 15,860 0.894 23.75 56,050 15,790 0.893 WT (c.) WF (c.) WB (c.) Separation Factor 1,198,185 1,194,367 1,187,437 1.009 Run 82 Cell 11, C = 0.0015 gm./£., d' = 20%, M = 1,180,000, V avg = 140 volts, P avg = 4.45 watts t (hr.) RT (ohm) RB (ohm) Separation Factor 0.00 140,200 98,500 1.000 0.50 125.200 84,700 0.964 1.00 122,700 82,800 0.962 1.75 119,800 81,500 0.972 3.50 116,700 79,700 0.974 5.50 108,200 77,300 1.015 7.00 104,700 77,000 1.050 17.00 99,400 73,300 1.051 19.00 97,600 72,700 1.063 23.00 96,800 71,000 1.052 WT (c.) WF (c.) WB (c.) Separation Factor 321,860 293,449* 298,213 1.097 *Indicates feed sample was in glass sample bottle during run. APPENDIX VII Experimental Separation Factor Curves Separation Factor, CT/CB 1. l. 17 16 .15 .14 H H H I-' I'-' 5.: p...» I.» H 1..» H O .09 .08 O \l O O\ 0 U1 0 h 145 I Variation of Separation Factor with Time I- O= Run 11, 0= Run 15, A= Run 16 4J1Run 15 E _ 5 Run 16 o N a A Run 11 H (3 4A (3 I A I I 4 I I I I I I I J 3 4 5 6 7 8 9 10 Time, hr. Separation Facter, CT/CB H H H .09 .08 .06 .05 .04 .03 .02 .01 .00 .12r .11- .10? .07— #‘ - 12 146 Variation of Separation Factor with Time Runs 23 A and B O, 16 Time, hr. 20 2'4 147 1.061- Run 17 0 <9 <9 0 1.04- Q) 1.02 b O m U \ E-' U 3 1.001 0 8 m n 0 LL. .9 Run 38 E 0.981- I: 8. E’ E, 8 E 0.96 - D 3 Variation of Separation Factor with Time O= Run 17, D= Run 38 0.94 - I I I I #1 0 1 2 3 4 5 Separation Factor, CT/CB O O O .00 .99 .98 .97 .96 .95 .94 .93 .92 .91 Run 48 Run 25 Variation of Separation Factor with Time O= Run 25, 0= Run 44, A= Run 48 " o (D A o 16 O 0 A 0 A z. E A Run 44 — E] [3 I— E I I I I l I_ L I l I I I 0 2 4 6 8 10 12 14 16 18 20 22 24 Time, hr. 149 cm .H: .oEHe on on om ow om om mm cam Now oEHH :HHS Hepomm :oHumHmmom mo :oHHOHHO> A em.o Om.o voo.H mo.H oo.H ao.H NH.H mH.H mH.H HN.H ON.H ON.H om.H mm.H om.H 3/13 ‘101395 notieiedes H .40 .35 .30 .25 .20 .15 .10 O= Run 30, 0= Run 32, A= Run 34 Variation of Separation Factor with Time Run 32 4g :99 9 E’ 6) Run 30 1 1O 1 I I I I 4 8 12 16 20 24 28 Separation Factor, CT/CB .02 .01 .00 .99 .98 .97 .96 .95 151 G . Run 31 Variation of Separation Factor with Time O= Run 31, [:1= Run 43, A= Run 45 Time, hr. Separation Factor, CT/CB O .00 .99 .98 .97 .96 .95 .94 .93 .92 152 Variation of Separation Factor with Time O= Run 39, D= Run 41, A= Run 47 I I I I I L I I I I 1 2 3 4 5 6 7 8 9 10 Time, hr. Separation Factor, CT/CB O O O O .92 .90 .88 .86 .84 .82 153 G) Variation of Separation with Time 0!- Run 40, 0= Run 51 Separation Factor, CT/CB .60 .50 .40 .30 .20 .10 .90 154 Run 64 ha Run 63 Variation of Separation Factor with Time O= Run 63, [3= Run 64 I I I I 8 12 16 20 Time, hr. Separation Factor, CT/CB H H I—l I—I H 1. .17- .16- .14v .12L .10- .08 .09- T .07’- .05r .04. .03— .02- .01- .00 06— 155 Variation of Separation Factor with Time for Run 65 Time, hr. Separation Factor, CT/CB 156 Variation of Separation Factor with Time O= Run 66, 0= Run 67, A= Run 82 1. I I I 12 16 20 24 Time, hr. Separation Factor, CT/CB I—' H O .04 .01 .00 .99 .98 .97 .96 9 157 Run 70 A A G <9 Run 68 C) El Run 69 g B a n E] B Variation of Separation Factor with Time - O= Run 68, CF Run 69, A= Run 70 I I I I OI I 0 4 8 12 16 20 24 Time, hr. Separation Factor, CT/CB .00 F .99 - .98— .97 - .96 - .95 - .94 - .93 - .92 P .91 o 158 Variation of Separation Factor with Time O= Run 71, 0= Run 72 Run 71 ~—G>-——C> Run 72 4 I L l O1 0 8 12 16 20 24 Time, hr. Separation Factor, CT/CB O O O .00 .95 .90 .85 .80 .75 .70 .65 .60 .55 159 Variation of Separation Factor with Time Run 78 O= Run 73, D= Run 75, A= Run 76, 9= Run 78 L. Run 75 fl 312.— 13 W .A I I l I 0 12 16 20 24 ‘Time, hr. APPENDIX VIII Sample Polarizability and Separation Factor Calculations APPENDIX VIII Sample Polarizability and Separation Factor Calculations Theoretical Polarizabilities: The theoretically predicted polarizability, up, is calculated from the approximate expression10 as £3 0‘1. = m (“I“) where as = the dielectric constant of the solvent 8 = end to end length of the molecule p a ratio of molecular length to diameter. The end to end length of the molecule is obtained from an expression given by Krause36 and alternately from molecular bond calculations. Considering the former method, the rela- tionship for the root mean square end to end length for un- neutralized polymer is 1 (£2) 7 = I '= GMd (VIII-2). where G = 0.69 for P.M.A. 9.. II 0.49 for P.M.A. 161 162 M = polymer molecular weight. 6 For polymer of molecular weight 1.18 x 10 , the length calculated from Equation (VIII-2) is 678 A0. The expansion ratio for polymethacrylic acid at 49 20 percent neutralization is 4.935. Multiplying this times the polymer's unneutralized length gives 3.34 x 10‘5 cm. as the theoretical length of the 20 per- cent neutralized P.M.A. molecule. Substituting this in Equation (VIII-l), using as = 80 for water at 200 C. and and p 2 104, the theoretical polarizability is found to be 1.26 x 10’14 cm.3. The second method of calculating 2 is to deter- mine the end to end length of a completely extended P.M.A. molecule from bond length and angle considerations. For P.M.A. the carbon-carbon bond distance along the polymer chain should be 1.54 A0. Since the angle between neighboring bonds is about 105°, the length of a repeat— ing unit is 2.46 A0. Multiplying this by the degree of polymerization (2 1500) gives a fully extended polyion length of 3.72 x 10'4 cm. At 20 percent neutralization, the polyion end to end distance is 29.25 percent of its fully extended length.49 4 Multiplying 0.2925 times 3.72 x 10' cm. gives the polymer end to end distance equal to 1.09 x 10"4 cm. Using this in Equation (VIII-l) as before, the molecular polarizability is 4.38 x 10'13 cm.3. 163 Experimental Polarizabilities: The molecular polarizability may be determined ex- perimentally by calculation from Equation (122) using the equilibrium radial concentration ratios obtained from the forced feed runs for cells I and 11. Equation (122) in its appropriate form is C. a A Cl = exp , (VIII-3) O 2va RE £n2 K where for the experimental conditions using cell 1: AV 100 volts 2 0.333 e.s.u. I2 16 k 1.38 x 10' ergs/OC. 3000K '—3 I (for 206percent neutralized polymer, .18 x 10 , from reference 32) 76 II C O I—‘ n 5 (taken as the radius of the capillary Using the above values in Equation (VIII-3) with the separation factor, 1.0265, obtained at optimum power for cell 1, the polarizability is 0.59 x 10-13 cm.3. The same calculation performed for cell 11 where K = 42 and the sepa- ration factor is 1.0875 yields up = 2.06 x 10-13 cm.3. Theoretical Separation Factors: The theoretical separation factors for experimental cells I and 11 operating at Optimum power are calculated from 164 CT C* 98 = 0* - 01w: 8) L (“5) where 0’ - 4DAB u REATBBg (114) and C* = dimensionless concentration L length of separation tube 8 = average volume coefficient of expansion. Electrical considerations show that at optimum cell power about two watts is dissipated in the separation tube. The AT is determined by equating the heat produced from electrical dissipation as a function of r, (Equation 11), to the heat transfer by conduction, (Fourier's Law). Solu- tion of the ensuing differential equation gives _ AVI in K - (Ti - To) — 713-177, (VIII 4) which, since _ AVZNL I - 3I_ZW—E’ (VIII-5) reduces to (41112 AT = (VIII-6) 191 where 165 k 1 thermal conductivity of solution pl electrical resistivity of solution. Examination of Equation (VIII—6) shows that the theoretical AT is dependent only on the solution thermal conductivity and electrical resistivity as well as the applied voltage to the cell. This lack of dependence on cell geometry enables the calculation of a value of 0', which is applicable to both of the experimental cells. The AT obtained from Equation (VIII-4) using two watts power, K = 36, L 30.5 cm., and k1 = 0.35 Btu/ft. hr. 0C. is 3.080C. The values of 6, 8, and u are calculated at the average solution temperature. An approximate value of the average solution temperature was found to be 35°C. from heat transfer considerations. The calculation in- cluded assuming that the heat transfer from the cell wall to the ambient air at 10°C. was due to natural convection. At the approximate average solution temperature, 0 = 0.007225 gm./cm. sec. 8 = 2.87 x 10‘4 °c‘1 - _ 3 p - 0.994 gm./cm. Using these values h1Equation (114) with R. 1 0.00635 cm., 6 D 5 x 10’ cm.2/sec. (from reference 32), I2 AB 166 and AT = 3.08°c., the value of 0' is 103.2 cm.-1. The calculation of the theoretical separation fac- tor then is accomplished by direct substitution of the cell K and L as well as the dimensionless concentration from Equation (111) into Equation (115). Using the values for cell 1; K = 36, L = 30.5 cm., and 0* = 2.3624 x 105, the predicted separation factor is 1.049. For cell 11 with K . 42, L - 61 cm., and c* = 1.1574 x 105, the theoretical separation factor is 1.255. A A,B,C,D,E,F Ck CiT,CiB C P C C I’ II C1’C2’53'C4’C5'C6 C11'C22’C33’C44 5143.58.52. D DAB NOMENCLATURE constant defined by Equation (31) arbitrary constants determined by the method of undetermined coefficients, Equations (93) and (III-14) constants defined by Equation (108) constant determined by the value of n in an infinite series, Equation (III-1) cylindrical area at a given value of r activity of dipolar species, Equation (53) constants of integration, Equation (III-ll) constant defined by Equation (32) polymer concentration, gm./£. dimensionless concentration defined by Equation (110) initial reservoir concentrations, gm./£. particular solution of Equation (81) solutions to Equation (81) obtained by the "Method of Frobenius" constants of integration constants defined by Equation (79) constants defined by Equation (81) diameter of outer cylinder, cm. mutual diffusion coefficient, cm.z/sec. 167 Ar A0 A2 168 energy difference between polar and non-polar molecules, Equation (5) electric field strength translational force on a dipolar mole- cule in an electric field acceleration of gravity, 980.665 cm./ sec. electrical current constant defined by Equation (72) 16 Boltzman's constant, 1.380 x 10‘ erg/0k. thermal conducsivity of solution, cal./sec. cm. C length of separation tube Langevin function average end to end length of polymer molecule, Equation (VIII-1) dummy variable, Equation (III-1) average polymer molecular weight total dipole moment, Equation (1) molality of solute magnitude of an induced dipole moment 23 Avogadro's number, 6.023 x 10 molecules/ mole number of molecules of a particular type, Equation (5) molar flux of component A in the r di- rection molar flux of component A in the 0 di- rection molar flux of component A in the z di- rection T! 169 degree of polymerization summation index for infinite series, Equation (103) constant defined by Equation (114) power produced by Joule heating, watts ratio of molecular length to diameter, Equation (VIII-1) variable defined by Equation (18) solution resistance, Equation (10) universal gas constant, 8.31 x 107 ergs/ mole K, Equation (53) rate of production of species A per unit volume by chemical reaction initial solution resistance in bottom Cell reservoir, ohm initial solution resistance in top cell reservoir, ohm radial variable equivalent radius of a dipolar molecule frictional force on a moving particle defined by Stoke's Law power per unit volume produced from Joule heating variable defined by Equation (75) temperature average solution temperature, Equation (47) constant defined by Equation (79) time V! V* Cr Re Sc mm B! 170 variation of parameters variable, Equation (III—9) electrical voltage fluid velocity in the z direction dimensionless velocity defined by Equation (43) dimensionless velocity defined by Equation (49) molecular velocity in the direction of molecular movement number of radioactive counts in a given time mole fraction of polymer in solution dimensionless radial variable, Equa- tion (13) variable defined by Equation (35) length variable Grashof number Reynolds number Schmidt number molecular polarizability degree of neutralization, percent volume coefficient of expansion, OC-l variable defined by Equation (67) volume coefficient of expansion evaluated at T variable defined by Equation (80) r1’P2'P3'F4’P5 A Subscripts P n 171 variables defined by Equation (103) indicates difference dielectric constant of solvent dimensionless temperature defined by Equation (16) angular variable ratio of outer to inner cylinder radii permanent dipole moment, Equation (3) solution viscosity, Equation (23) total molecular chemical potential in an electric field constant, Equation (53) number of ionized groups per polyion mathematical constant, 3.14159 density of solution, gm./£. density of solution at temperature T resistivity of solution variable defined by Equation (85) complimentary solution of Equation (87) particular solution of Equation (87) refers to polar molecule refers to non-polar molecule refers to inner cylinder radius 'fl w H avg min max refers refers refers refers refers refers refers 172 t0 to to t0 t0 to to parameter refers refers t0 to outer cylinder radius radial direction vertical direction top reservoir bottom reservoir feed solution the average value of a minimum value of a parameter maximum value of a parameter 10. 11. 12. 13. 14. 15. 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