DETERMINATION OF MECHANISMS CAUSING AND LIMITING SEPARATIONS IN COLUMN CRYSTALLIZATION Walter Collins Gates, Jr. A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1967 Doctoral Committee: Professor John E, Powers, Chairman Professor Edward E. Hucke Associate Professor Robert H. Kadlec Professor Edgar E. Westrum, Jr. Professor J. Louis York

~wo-) L) )) 4I E q

ACKNOWLEDGEMENTS This dissertation is the result of 2ontributLons from many sources. These contributors and contributions are acknowledged with gratitude. Professor J. E. Powers, chairman of the doctoral committee, for his continuing guidance and criticism. Professor R. H. Kadlec, committee member, for his generous assistance throughout the course of the study. Professors E. E. Hucke, E. E. Westrun, Jr., and J. L. York:, committee members for their beneficial assistance and helpful suggestions. Dr. R. Albertins and Mr. J. D. Henry, Jr. for their critical remarks and general assistance. The personnel of the shops of the Department' of Chemical and Metallurgical Engineering who were helpful beyond the call of duty. MIrL. F. Drogosz who assisted in the operation and maintenance of the analytical equipment. The National Science Foundation, Union Carbide Corporation, The Riggs Foundation, and the Chemistry Department of the University of Michigan for financial support. The Upjohn Company for the donation of the drive mechanism for the column crystallizer. Mr. Gary J. Powers for his assistance and stirmtlation during the early stages of the experimental work. Mr. Murray Player whose comments significantly improved the manus cript.

Mr. R. Stewart and Professor C. W. Peters for their gentlemanly conduct during trying periods. Mr. M. Newberger for his assistance with some of the computing. The staff of the Industry Program of the College of Engineering of the University of Michigan for its complete cooperation and efficient production of the final form of this dissertation. My wife, Lorna, for her encouragement, support, and assistance. Without these, this research would have been neither started nor completed.

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS...o.O.*. 0...... e....... n o............ i LIST OF.......TABLES.4 fo O 4 O O O g O O o... o vii LIST OF FIGURES.......... o....................... oo.......... viii LIST OF APPENDICES........... o o.o.. o.......... a* o o oeO** * *Oo e X NOMENCLATURE........................ *0 0 0 oooo* e*eo0 xi ABSTRACT.. @ @ ee es oe oob oX eoo oo ~ xiv I INTRODUCTION. 0 o............................................. 1 A. Definitions.......................................... 2 II REVIEW OF LITERATURE PERTAINING TO COLUMN CRYSTALLIZATION AND ITS ANALYSIS.................... 0000.. 4 A. Description of Process.............................. 1. Configurations........o.... o o........... 6 B. Applications................................ 6 C. Analysis............................... o.............. 7 1. Column Crystalliza tion....... 00 00.............o00 7 2. Other Processes.. O......e.... o eo o..... o.......o 8 D. Results of Previous Investigations..................... 8 1. Column Crystallization..... o.. o eo..o0. e ooe, 0 8 2, Other Processes...................,..........., 9 III INTRODUCTION TO EXPERIMENTAL i.NVESTIGATION e o..........o o 10 A. Equipment............................0..... o ooo e e o 10 B. Materials...........................o.............. 10 C. Procedures. o.oe..... o s..... o e..... o....... oo....... 14 D. Results...........O o....... o......................... o. 15 iv

TABLE OF CONTENTS (CONT'D) Page IV MATHEMATICAL DESCRIPTION OF COLUMN CRYSTALLIZATION........... 28 A. Mass-Transfer-Limiting - Model T..5................... 30 1. Description........................... 30 2, Limitations.................... 34 3. Solution to Differential Equation....... 35 4. Boundary Conditions..................... 36 5. Limitations......................5......... 39 6, Predictions..................................... 39 B. Heat-Transfer-Limiting - Model II.............. 4 1. Description.................................... 40 2 Solution. 46 2. Solution...................... 0...0 0......*. 0@6X 46 5. Predictions......................................... 47 C. Diffusion Within the Solid Phase - Model III. 48 1. Description................ 48 2. Solution to Concentration Profile in Solid.......... 49 3. Solution to Profile in Liquid.......... 52 4. Predictions.......... 54 D. Constant-Crystal-Composition - Model IV......... 54 1. Description.55 2. Solution................................. 58 5. P6edictin60 V EVALUATION OF MODELS..62 A. Mass-Trans fer-Limiting........... 62 1. Test 1........ o............e... *............. 62 2, Test 2......... o o ***....................... 64 30 Test 5........................................ 69 4. Test 4............. 0.0.e.*.*.*. 73 5. Test................... 74 6. Summary of Tests.................. 78 B. Heat Transfer-Limiting. 78 1. Effect of Composition...........................79 2. Influence of Crystal Rate............. 79 30 Effect of Liquid Diffusivity............ 79 4. Influence of Heat-Transfer Coefficient....... 83 5, Conclusion............................ 83 v

TABLE OF CONTENTS (CONT'D) Page C, Diffusion in Solid Phase...................... 84 Do Constant Crystal Composition......................... 84 VI ANALYSIS OF DATA FROM A SYSTEM FORMING A EUTECTIC....... 87 A. Determination of Diffusion and Mass-Transfer Factors. 87 1. Effects of Agitation............................. 88 2. Fallacy....................... 0..........*....*. 93 VII SUMMARY........................ e o..........Q..e**e oe** ee 98 VIII SUBJECTS FOR FUTURE INVESTIGATION.O..................... 100 BIBLIOGRAPHY. 102 APPENDICIES.....*......... 0.0.e o o ooooaeo o 0. 105 vi

LIST OF TABLES Table Page I Summary of Conditions Used in Reported Runs....o......... 27 II Influence of Difference in Phase Composition on Separation....................................... 63 III Comparison of Experimental and Literature Values of Diffusivity and Mass-Transfer Coefficient................ 72 IV Dependence of Slope of Phase Relation on Composition for Systems BNB-CNB and Azobenzene-Stilbene.............. 73 V Effect of Composition on Column Performance — HeatTransfer-Limiting Model.................................. 80 VI Effect of Crystal Rate on Separation —Heat.-TransferLimiting Model........................................... 81 VII Effect of Heat-Transfer-Coefficient on Separation........ 83 VIII Comparison of Experimental Results with Predictions of Constant-Crystal-Composition Model...................... 85 IX Profile of Liquid Composition for One Run................ 86 X Comparison of Analyses of Standard Samples............... 117 XI Effect of Slurry Segregation on Separation............. 128 XII Comparison of Data Reported by Powers,32) and Profile Calculated from Equation 22.......................... 133 XIII Phase Equilibrium Data of Hasselblatt(19) for System BNB-CNB o....................................... 156 XIV Constants Used in Correlation of Phase Equilibrium Data.. 140 vii

LIST OF FIGURES Figure Page Sections Used in Column Crystallization.e da,..........e.. 5 2 Diagram of Column Crystallizer............................ 11 3 Phase Diagram of m-chloronitrobenzene - m-bromonitrobenzene..........,1..............2................... 12 4 Phase Relation of m-chloronitrobenzene - m-bromonitrobenzene................................................... 15 5 Effect of Column Length on Column Performance............. 17 6 Effect of Column Length on Column Performance. 18 7 Effect of Stroke of Spiral Oscillation on Column Performance............................................... 19 8 Effect of Rate of Spiral Rotation on Column Performance... 20 9 Effect of Rate of Spiral Rotation on Column Performance... 21 10 Effect of Crystal Rate on Column Performance.............. 22 11 Effect of Crystal Rate on Column Performance............ 25 12 Composition Profile in Column.......................... 24 13 Composition Profile in Column............................. 25 14 Effect of Crystal Rate on Column Performance........... 26 15 Elemental Description of Column Crystallization-Mass - Transfer-Limiting Model.................................... 31 16 Elemental Description of Column Crystallization-Heat - Transfer-Limiting Model................................. 41 17 Illustration of Flows in Constant-Crystal-Composition Model..................................................... 56 18 Elemental Description of Column Crystallization-Constant - Crystal-Composition Model................................. 57 19 Effect of Charge Composition on Separation................ 65 20 Effect of Phase Separation on Separation................ 66 21 Effect of Crystal Rate on Column Performance............ 67 viii

LIST OF FIGURES (CONT'D) Figure Page 22 Effect of Crystal Rate on Columnn Performance........... 68 23 Determination of Diffusivity........................... 70 24 Determination of Diffusivity and Mass-Transfer-Coefficient.......................................... 71 25 Phase Diagram of Azobenzene-Stilbene................... 75 26 Phase Relation of Azobenzene-Stilbene.................. 76 27 Piecewise Application of Mass-Transfer-Limiting Model to Data Reported by PowersT32)........................ 77 28 Effect of Crystal Rate on Separation-Heat-TransferLimiting Model....................................... 82 29 Effect of Crystal Rate on Column Performance.. 89 30 Determination of Diffusivity and Mass-Transfer Coefficient.................. 90 31 Determination of Diffusivities and Mass-Transfer Coefficients........ o................................ 91 32 Effect of Agitation on Diffusivity................. 92 33 Profile of Liquid Composition for One Run.............. 94 34 Re-analysis of Previously Reported Data(1)............ 97 35 Cross-sectional View of Column with Insulation in Place 108 36 Cross-section of Wire from which Spiral was Wound...... 109 37 Sample Chromatogram............................ 116 38 Linearity of Calibration of Gas Chromatograph.......... 118 39 Illustration of Attainment of Steady-State............. 120 40 Analysis of Phase Equilibrium Data..........o...... 137 41 Analysis of Phase Equilibrium Data..o.............. 139 ix

LIST OF APPENDICES Appendix Page Al DETAILED DESCRIPTIONS................................. 105 a. Equipment........................................ 10 b. Operating Procedures.............................. 110 c. Analytical Procedures....1..... 1il A2 TESTS OF OPERATIONS................................ 1.. 119 a. Attainment of Steady-State........................ 11( b. Constancy of Crystal Rate......................... 121 A3 RESULTS NOT PERTAINING TO IDENTIFICATION OF MECHANISMS 1241 a. Maximum Crystal Rate...... 124 b. Effect of Agitation................ 127 c. Effect of Spiral........................... 128 A4 AREAS FOR FUTURE STUDY PERTAINING TO COLUMN OPERATION. 150 A5 CALCULATIONS............................... a. Analysis of Data from Azobenzene-Stilbene....... 152 b. Correlation of Phase-Equilibrium Data. 155 c. Correction of Diffusivities.158.................... 15 d. Crystal Rate at Minimum H...................... 141 A6 SAMPLE CALCULATIONS................................... 142 a. Determination of the Crystal Rate................. b. Determination of H l.........1.................... it4 c. Determination of D and K......................... 145 A7 DATA.................................................. L46

NOMENCLATURE A Cross-sectional area of column, measured perpendicularly to flow of liquid and solid phases, cm2. a Area available for interphase mass-transfer per unit volume of column, cm-1. B Slope of phase Equation (29), C b Intercept of phase relation (5), weight fraction. CunsubscriptedFlow of liquid adhering to crystals, g/sec. Csubscripted Constant in general solution to differential equation; or specific heat, cal/g-C. D Coefficient of diffusion, cm2/sec. E Diffusion faction (D FACTOR) defined by Equation (16), g-cm/sec. F Mass-transfer factor defined by Equation (16), g/cm-sec. f Frequency of spiral oscillation, sec- l; or notation for "function of". G Collection of terms defined by Equation (34), cal/g. g Notation for "function of". H Grouping of variables, defining the separating power of a column crystallizer operating at specified conditions, cm; or heat of fusion, cal/g. h Position in the column, measured from melting section, cm. ho Length of purification section, cm. J Thermal conduction in liquid phase, cal/sec. K Coefficient of mass-transfer between solid and liquid phases, cm/sec. k Rate at which concentration changes with time (see Equation (40b)), weight fraction/sec. L Mass flow of solid, g/sec. xi

NOMENCLATURE (CONT'D) m Slope of phase relation (3), dimensionless. N Diffusional flow of material, g/sec. n -Counting integer. P Collection of terms used in Equation (49), weight fraction/cm2. Q Symbol representing differentiation (d/dh), cm1'. q Constant in general solution to differential equation, defined by characteristic equation, cm'l. F. Constant defined by Equation (5); or collection of terms defined by Equation (35), cal/sec-C~-weight fraction. Ro Outside radius of spheres, cm. r Radius of sphere, cm. S Collection of terms defined by Equation (36), cal/g; orslope of composition profile, weight fraction/cm. s Stroke of spiral oscillation, cm. T Temperature of liquid phase, ~C; or reduced weight fraction in solid phase, weight fraction. t Temperature of solid phase, 0C. U Overall heat-transfer coefficient between liquid and solid phases, cal/cm2-sec- ~C. /V Mass flow of liquid, g/sec. W Collection of terms defined by Equation (47c), cm /sec. X Weight fraction in solid, g/g. The high melting component in solid solutions, and that excluded from the solid in eutectics. y Weight fraction in liquid, g/g (see definition of X). Z Weight f.raction cyclohexane in liquid adhering to crystals, z P& ition in column measured from freezing section, cm, xii

NOMENCLATURE (CONT'D) Greek Symbols Grouping of other constants, or ratio of adhering liquid to crystal. A~ SSmall increment. E Eddy-thermal diffusivity of liquid, cal/sec-cm-C ~. r) Volume fraction, dimensionless. p Density, g/cm3. ~ Summation symbol. Time, sec. Superscripts' B Refers to BNB C Refers to CNB L Refers to solid phase. o Refers to position, h=o. V Refers to liquid phase. Equilibrium value. Average value. xiii

ABSTRACT The process of column crystallization was analyzed mathematically and experimentally in order to determine the mechanisms by which separations are achieved and limited. The results of the study indicate that, at total reflux, separation is produced by the formation of crystals in the freezing section and by interphase mass-transfer in the adjacent purification section. The separation is mainly limited by eddy diffusion in the liquid phase. Data obtained from three binary chemical systems were used in the study. The first system, m-chloronitrobenzene - m-bromonitrobenzene, forms a solid solution with a small phase separation, less than 6 weight per cent. The second system, azobenzene-stilbene, forms a solid solution with a phase separation which exceeds 20 weight per cent. The third system,benezene-cyclohexane, has essentially no solid solubility in the range of compositions used (1/2 to 3 weight percent cyclohexane). Four mathematical models of column crystallization were developed and compared with experimental data. These models were based on different sets of assumptions as to the mechanisms which control the overall separation. The models which are consistant with the experimental results describe column crystallization as follows. Crystals form in a freezing section and undergo changes as they pass through an adiabatic purification section to a melting section. The changes manifest in the crystals depend on the type of crystals present. Crystals with considerable solid solubility undergo melting and recrystallization. Crystals which have little dissolved impurity are xiv

washed by the reflux liquid. In either case, mass-transfer between the countercurrently passing phases is controlled (limited) by the masstransfer coefficient, not by the heat-transfer coefficient. Eddy diffusion in the liquid phase opposes the separation being effected by interphase mass-transfer. The concentration of a component in the liquid phase, if a system which has a small phase separation is used, is described by the following equation: Y = YO - (Y-X*)o h/(E/L +:L/F). Y and X* are the liquid and equilibrium solid compositions, h is the distance from the melting section, L is the rate of crystal flow, and E and F are terms relating to diffusion in the liquid and interphase mass-transfer respectively. A system which exhibits little solid solubility is adequately described by Y = C1 + C2 exp[h/(E/L + L/F)]. C1 and C2 are experimental constants, C1 probably being related to the extent of the small solid solubility. Concurrent to the generation of data to evaluate the mathematical models, operational characteristics of a one inch by 24 inch column crystallizer were observed. Maximum crystal rates were influenced by: (a) the area of the heat-transfer surface in the freezing section (FS), (b) the local heat-transfer coefficient in the FS, and (c) probably the size of the crystals. Separation with respect to agitation had a maximum at about 30 RPM and 30 oscillations/min. Liquid phase diffusivities and mass-transfer coefficients, in satisfactory agreement with published values, were determined. Diffusivities increased linearly with the one-half power of xv

the agitation frequency, and varied between 1.3 cm2/sec and 4.6 cm2/sec. The values of the mass-transfer coefficient varied between 0.07 cm/sec and 0.64 cm/sec. xvi

CHAPTER I INTRODUCTION The study reported in this dissertation was undertaken with a two-fold purpose. The primary goal was the quantitative determination of the mechanisms involved in the process of column crystallization. Such a determination would confirm or invalidate the theoretical and qualitative predictions of Powers(32) which had not previously been subjected to a complete test. The secondary aim, the mathematical description of the concentration profiles produced in a column crystallizer, was established to accomplish the primary goal. The results of the study indicate that mass-transfer between phases and dispersion within the liquid phase are the two mechanisms involved. In order to obtain the data from which the above conclusions are drawn, a column crystallizer was built and operated at total reflux. A system which forms a continuous solid solution was used as the mixture to be separated. This type of system was chosen because it affords a severe test of the general applicability of this relatively new separation process. With preliminary data from this system in hand, several possible mathematical models of column crystallization were formulated. Additional data were taken which showed that only one of these models is consistent with experimental results. Also, previously published data concerning one solid solution and one eutectic system were reanalyzed with models derived from the one consistent model. These analyses indicated that the mechanisms which effect and limit the separation -1

-2achieved by column crystallization are independent of the type of system being separated. The dissertation is divided into two sections. Descriptions, the existing literature, theoretical models, and experimental results and interpretations which bear on the elucidation of the mechanisms just described are presented in the first section. This section is followed by a group of appendices which present in greater detail the descriptions of equipment, procedures, and analyses which are needed to repeat the experiments. Results which do not relate to the stated purposes of the research are also presented as an appendix. These several appendices support the first part of the dissertation but are not required for the full appreciation of the material presented therein. A. Definitions Several terms, used throughout this dissertation, have precise definitions which must be made clear. "Separation" is the difference in composition which occurs in the liquid phase between two points in a crystallization column. "Eutectic system", or "system which forms a eutectic" is a chemical mixture which exhibits little or no solid solubility and which has a eutectic point. "Mass-transfer" or "heat-transfer" is used exclusively to refer to the exchange of material or energy which occurs between the phases which are in countercurrent contact in a column crystallizer. "Diffusion" is used to describe that movement of material which occurs within a single phase either by molecular diffusion or by Taylor diffusion. "Diffusion" is distinguished from "backmixing,' defined below.

-3"Backmixing" is used to describe the movement of material within a single phase caused by the turbulence of that phase. Backmixing is synonomous with eddy-diffusion. "Dispersion" is the net effect of diffusion and backmixing as they are defined above. "Phase Separation" refers to the difference between the compositions of two phases existing at equilibrium. "Phase Relation" is the curve or equation describing the compositions of two phases in equilibrium. "Agitation" is the net effect of the motion of the spiral within the column. "Performance" is the separation per unit length of column. "Crystal Rate" is the rate of crystal flow across a given cross-section; if the column crystallizer is adiabatic, then the crystal rate is equal to the crystallization rate.

CHAPTER II REVIEW OF LITERATURE PERTAINING TO COLUMN CRYSTALLIZATION AND ITS ANALYSIS A. Description of Process Column crystallization was invented by Arnold(5) in 1951. This process incorporates the inherent theoretical attractiveness of crystallization in a separation process which eliminates some of the disadvantages previously associated with crystallization. Column crystallization occurs in three distinct sections which are illustrated in Figure 1. In the freezing section, two phases of differing composition are formed. One is a disperse solid phase and the other a continuous mother liquor. The solid phase passes through a purification section countercurrently to a liquid moving toward the freezing section. A melting section, adjacent to the purification section and opposite the freezing section, supplies energy to the system. This energy transforms the solid crystals to a liquid to produce reflux. Based on this brief description, simple analogies can be drawn for clarification. The melting and freezing sections are similar to parts of distillation equipment. The melting section is like a total condenser in which energy transfer causes a phase change with no change in composition. Contrarily, the energy transfer in the freezing section, like that in a reboiler, produces a phase change and a composition change. In addition, both distillation and column crystallization can be operated at total reflux or with continuous feed and removal of products. Figure 1 illustrates operation in the latter mode. -4

-5HIGH-MELTING PRODUCT HOT FLUID I MELTING SECTION FEED PURIFICATION SECTION SPIRAL REFRIGERANT FREEZING SECTION LOW- MELTING PRODUCT Figure 1. Sections Used in Column Crystallization.

-6The separation section of a column crystallizer is analagous to an extraction tower. Both contain a continuous phase and a disperse phase which has some degree of integrity. These two phases move countercurrently, with mass-transfer between phases and with dispersion in the continuous phase. 1. Configurations Subsequent to Arnold's invention, two significantly different modifications of column crystallization have evolved. The first, deriving from the invention of Arnold, is called the Phillips column crystallizer. This device is primarily used in the industrial scale production of materials, especially p-xylene from its isomers. References to the Phillips column by Weedman(44), Thomas(39), Findlay(9), and McKay(30) have dealt with improvements in its construction, operation, and control.(10)29'38'17) The other type of crystallizer is that developed by Schildknecht (Reference 36). It is used in the laboratory to produce materials of high purity. Modifications of the Schildknecht-type column have been described by Schildknecht(35) and Albertins.(l) The literature concerning both Phillips- and Schildknecht-type crystallizers has been recently reviewed by Albertins, Gates, and Powers (Reference 2) and by Albertins.(l) B. Applications The separation and purification by column crystallization of a great variety of materials has been described and reviewed.(l12) Aqueous systems(lll4,27) and organic mixtures have been processed.

-7These mixtures include light hydrocarbons, both aliphatic(l2,27) and aromatic,(14'43'44) and such heavy materials as fatty acids.(33) Binary(14'44) and multicomponent(l2'l4'16) separations have been described. C. Analysis Analyses applicable to column crystallization are found in two areas of the literature. The first deals specifically with column crystallization, and the second deals generally with processes involving countercurrent contacting of two phases. 1. Column Crystallization Early attempts to describe column crystallization mathematically were reported by Powers(32), Yagi et al. (45), and Anikin(3y4) and have been reviewed.(l,2) The analyses of Powers(32) and Yagi et al.(45) were tested experimentally and were in qualitative agreement with their data. Powers analyzed the separation of eutectic systems and systems which form solid solutions. He postulated that the mechanisms which are involved in column crystallization are dispersion and mass-transfer, but he did not determine the quantitative contributions of these mechanisms. Albertins(l) recently analyzed the column crystallization of a eutectic system, benzenze - cyclohexane. He predicted modified exponential profiles of the liquid composition based on a model including only dispersion within the liquid phase. The agreement between this model and the considerable amount of experimental data was satisfactory.

-8All of the analyses mentioned above consider dispersion within the liquid phase as one mechanism limiting the effectiveness of column crystallization. 2. Other Processes Many previous articles have dealt with the analyses of processes in which liquid-phase dispersion is considered. Li and Ziegler(26) recently reviewed much of this literature as it pertains to extraction in simple and in pulsed columns. Hartland and Mecklenburgh(l8) presented an analysis of processes involving countercurrent contact of two phases for which there is a linear equilibrium relation. Their analysis included dispersion in each phase and mass-transfer between phases. D. Results of Previous Investigations 1. Column Crystallization Albertins,(l) in his study of benzene-cyclohexane, indicated that a mathematical description of column crystallization which includes mass-transfer between phases may be incompatible with his data. He also showed that effective diffusivities in the liquid phase were in good agreement with values reported by Jones(21) and by Moon(31) who studied pulsed-column, liquid extraction. No attempt was made, however, to correlate diffusivity with agitation. Several operational limitations of column crystallization have been reported but not discussed. For example, a column capacity as high as 0.5 gm/cm2-sec has been described,(l'28) but capacities much lower than this are common.(8) No mention has been made of the factors which might limit these fluxes.

Previous reports of column crystallization have not correlated the size of the crystals with the maximum possible flux of the reflux liquid. 2. Other Processes Work by Hayford(20) and others,(31) who studied the effect of agitation on diffusivity (D) in fluid flow through packed beds, suggests that D is linearly related to the 1/2 power of the pulse frequency. These studies also suggest that D is proportional to the stroke (twice the amplitude) of the pulsation. The study of Smoot and Babb(77) indicated that effective diffusivities between 0.8 and 2.6 cm2/sec result in pulsed extraction columns when the pulse frequency varies between 30 and 100 oscillations per minute. The literature on packed beds indicates that the size of particles influences the maximum flow of liquid through a bed. Leva(24) shows that the maximum flux of a liquid increases as the square of the particle size.

CHAPTER III INTRODUCTION TO EXPERIMENTAL INVESTIGATION A. Equipment A small Schildknecht-type column crystallizer was designed and built to carry out the experiments described in this report. A sketch of the column appears in Figure 2. An insulated glass column, 2.60 cm ID and 38 cm long, was used. The freezing section was at the bottom and the melting section at the top. The column was equipped with 7 taps through which samples of the reflux liquid could be taken. A stainless-steel tube' which could be rotated and oscillated passed through the three sections of the column. This spiral was maintained concentric with the glass by a 1/2 inch diameter stainless-steel rod. A detailed description of the apparatus is included in Appendix Al-a. B. Materials Meta-bromonitrobenzene and meta-chloronitrobenzene (BNB and CNB) were used in this investigation. This system forms a continuous solid solution as is illustrated in Figures 3 and 4. This system was chosen for several reasons. First, equilibrium data from two sources(l9122) which were in agreement were available. Second, the liquidus and solidus lines, as seen in Figure 3, are very close to one another. A system with such a phase diagram offers a severe test of the general applicability of column crystallization to solid solutions. Third, the mixture is solid at room temperature and melts at slightly above room temperature. Such a mixture eliminates the need for an extensive refrigeration system or for excessive insulation or guard heaters. -10

AUXILIARY HEATER GLASS TUBE (3.2 CM. O.D.) SLURRY LEVEL IN COLUMN I CARTRIDGE HEATER MELTING SECTION 6.0 l_ SAMPLE TAP (I OF 7) 5.3 5.0 DIMENSIONS IN CM. 5.1 5.1 54.5 4.8 3.5 GLASS TUBE R I(4.8 CM. O.D.) FREEZING o Cs SECTION 70 JACKET END-SECTION 8.5 O-RING SEAL ANNULAR PLUG S.S. TUBE (0.497 IN. O.D.) Figure 2. Diagram of Column Crystallizer.

-1256 Data from International Critica4 Tables, V, IV, 53 - 0 0 j 50 - 47 44 0 20 50 80 100 MOL. % BROMONITROBENZENE Figure 3. Phase Diagram of m-chloronitrobenzene - m-bromonitrobenz ene.

-13100 80 0 0 _Z 50COMPOSITIONS X - WEIGHT *-MOLE 20 0 20 50 80 100 % BROMONITROBENZENE (BNB) IN LIQUID Figure 4. Phase Relation of m-chloronitrobenzene - m-bromonitrobenzene.

-14C. Procedures Summarized below are the procedures used in the operation and sampling of the equipment and in the analyses of samples and data. These procedures are spelled out in greater detail in Appendices Al-b and Al-c. Liquid of the desired composition was charged to the heated column. Warm water at a known temperature was fed to the freezing section to produce crystals in the charge. These crystals were agitated by the action of the rotating and oscillating spiral. About 90 minutes after the first crystals formed, the entire column contained a slurry of crystals and liquid. This slurry appeared to be uniform throughout. No material was fed to the column after the initial charge. The power input to the melter was adjusted to maintain a constant proportion of crystals. About 8 hours after the first crystals formed, and at least 2 hours subsequent to any significant adjustment in operating conditions, samples of the liquid in the column were withdrawn and analyzed chromatographically. Plots were constructed of the liquid composition vs. position in the column, and the slopes of these plots were determined. To accomplish the purpose of this study, the elucidation of the mechanisms involved in column crystallization, the influence of 6 operating variables on these slopes was determined. These variables were: 1. the charge composition, 2. the crystal rate through the column, 3. the length of the column, 4. the rate of rotation of the spiral, 5. the frequency of spiral oscillation, and 6. the stroke of the spiral oscillation.

-15D. Results The experimental results which bear on the elucidation of the mechanisms involved in column crystallization are summarized in Figures 5 to 14 and in Table I. These results are discussed in detail later in the dissertation. However, this summary collects in one place the important results. Figures 5 to 14 are plots of liquid composition vs. column position which is indicated by tap number. Tap number 7 was immediately above the freezing section, and the lowest numbered tap for a given run was immediately below the melting section. The variables listed on each figure, as well as charge composition, were varied among the runs included on each graph. Because the charge composition was changed only slightly between runs presented on each graph, each figure shows the effect of the variables listed on that graph. Table I summarizes the conditions used for each run. The line drawn on each figure is the linear regression line through the data points. The slope of this line is a measure of the effectiveness which a set of operating conditions has in causing a separation between two compounds. A large slope corresponds to a highly effective set of operating conditions. The variation of this slope with changes in several operating conditions was the object of investigation in this dissertation. Thus, the separation per unit length within the column is the important dependent variable resulting from each run. Runs made at the same conditions gave the same slope to within two per cent. (See Appendix A2-a).

As it was to be expected, data points scattered somewhat about the regression lines. However, only in 5 runs, Runs 7, 13, 31, 34, and 35, did the difference between the regression line and a data point exceed one weight per cent BNB for more than 1 data point. Eleven runs were made for which no points exceeded the difference just mentioned. Figures 5 and 6 indicate that the length of the purification section was not an important variable in determining the separation per unit length, the column performance. Rather, the performance was constant for a given set of operating conditions. Figures 7, 8 and 9 show that the performance decreased as the degree of agitation was increased, either by increasing the stroke of oscillation or by increasing the rate of spiral rotation. Figures 10 and 11 illustrate that the performance increased as the crystal rate increased. Each of the three figures represents a single charge composition and set of agitation conditions. Figures 12, 10, 9, 13, 11 and 14 show the effect of charge composition. Charges which were nearby pure (Figures 12 and 14) gave a low separation per unit length. Charges of about 50% BNB (Figures 9, 10, 11 and 13) resulted in much higher performances. Other experimental results are presented in Appendix A3. These relate to the design and operation of the column crystallizer rather than to the mechanisms involved in column crystallization. The data on which the experimental results are based are presented in Appendix A7.

1'S0.48 0.46 5 0.44 z'~ 0.42 z2 hD - w I Z 0 0 7 30.340 oA -18 25.1 0.36 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION,CM Figure 5. Effect of Column Length on Column Performance.

-180.72 0.70 0 0.68 A 0 z z z o 0.66 tic FI 0.64 _ RUN COLUMN NUMBER LENGTH 0 21 20.0 CM. 0.62 - A 24 30.4 0.60 1 I / I I I I I 1 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION,CM Figure 6. Effect of Column Length on Column Performance.

-190.70 0.68 0.66 A Z 0.64 A z z I — LLJ0.60 rY ~ \ 0.-: z 0.58 - - z Z z I cr U::) cc 0 22 20.O 9.0 0.56 0El 26 30.4 6.0 I 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION,CM Figure 7. Effect of Stroke of Spiral Oscillation on Column Performance.

-200. 72 0.?Z 03.70 _a F 0.68 8. a p o 0.66 c-) RATE OF.6 RUN SPIRAL A6 iWi ~ NUMBER ROTATION E 0 23 41 RPM A 24 41 0.62 0 25 78 0 0.60 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION, CM Figure 8. Effect of Rate of Spiral Rotation on Column Performance.

-210.52 0.50 - CO 0.48 UJ RATE OF RUN SPIRAL NUMBER ROTATION 0 14 67 RPM 0.42 A 0.40 4 I l NUMBER OFSAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FR-EEZING SECTION,CM Figure 9. Effect of Rate of Spiral Rotation on Column Performance.

-220.72 0.70 a 0.68- 0 z C 0.65 co _3 0.64 (' 0.62 A RUN CRYSTAL NUMBER RATE 0 29 0.038 GM/SEC 0.60 A 50 0.033 O 31 0.025 0.58 1. I I I I 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION,CM Figure 10. Effect of Crystal Rate on Column Performance.

-230.33 0] \-RUN CRYSTAL NUMBER RATE ol 33 0.026 GM/ \.0 SEC 0.32 34 0.017 0 35 0.008 a_ z 0 C 0.30 z CD C 0.29 0.28 0.27 I I I I I I I I 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION,CM Figure 11. Effect of Crystal Rate on Column Performance.

-24-0.96 a 0.95 RUN NO. 13 _J z 0.94 l/) 0to o c 0.93 IU-,92 0.91 1 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION,CM Figure 12. Composition Profile in Column.

0.48 0.46 CI -J 0.44 RUN NO. 16 0.3642 _LL I 0.40 I 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 0 DISTANCE ABOVE FREEZING SECTION,CM Figure 13. Composition Profile in Col_!u:,r

0.09 0.08 C a o 0 EL 0 0.05 - _O~ 0 RUN CRYSTAL W NUMBER RATE A 6 0.012 GM/SEC 0 7 0.009 0.04 0.03 0 2 3 4 5 6 7 NUMBER OF SAMPLE TAP 30 25 20 15 10 5 o DISTANCE ABOVE FREEZING SECTION,CM Figure 14. Effect of Crystal Rate on Column Performance.

-27TABLE I SUMMARY OF CONDITIONS USED IN REPORTED RUNS Agitation Nominal Charge Crystal Rate of Stroke of Rate of Column Run Composition Rate Oscillation Oscillation Rotation Length Weight g/sec OPM mm RPM cm Fraction BNB 17 0o,50 0.018 31 6.0o 29 30.3 18 0.50 0.018 31 6.o 29 25.1 21 o.65 0.034 43 6.o 45 20.0 24 o.65 0.035 45 6.0o 41 30.3 22 o.65 0,031 43 9.0 45 20.0 26. 65 0.032 43 6.o 48 30.3 23 0.65 0.033 45 6.o 41 30.3 24 o,65 0.035 45 6.o 41 30.3 25 o.65 0.033 43 6.0o 78 30.3 14 0.50 0.036 25 4.5 67 30.3 15 0.50 o.036 25 4.5 32 30.3 29 o.65 0.038 67 4.2 60o 30.3 30 o.65 0.033 67 4.2 60o 30.3 31 o.65 0.025 67 4.2 60 30.3 33 0.35 0.026 130 2.0 60o 30.3 34 0.35 0.017 130 2.0 60 30.3 35 0.35 0.008 130 2.0 60 30.3 13 0,95 o.o40 22 4.5 67 30.3 16 0.50 0.027 40 6.o 46 30.3 6 0.05 0.012 72 4.5 67 30.3 7 0.05 0.009 72 4.5 67 30.3

CHAPTER IV MATHEMATICAL DESCRIPTION OF COLUMN CRYSTALLIZATION Four mathematical models of column crystallization, each based on different assumptions as to what occurs physically, are developed in this section. These models are similar in that dispersion in the liquid phase is assumed. They are different in that several modes of masstransfer between phases are considered. Examination of the bromonitrobenzene - chloronitrobenzene phase diagram in Figure 3 indicates that BNB will concentrate in the melting section. Thus as a crystal moves toward this section, it moves into a region of increasingly high temperature. The crystal thereby becomes unstable. This instability must be relieved by a change in the composition of the crystal. The change can occur in one of several ways, each leading to a different mathematical model. Three such models are described. One possible mechanism for change is melting and recrystallization. The unstable crystal melts by absorbing energy. This energy is supplied by the formation of new crystals. These new crystals, relative to the ones formed in the freezing section, are enriched in BNB because they are formed from a liquid itself enriched in BNB. Melting and recrystallization involve the simultaneous interchange of both mass and energy. One would expect that these mechanisms would be governed by a mass-transfer coefficient and by a heat-transfer coefficient. In the general mathematical description both coefficients would be considered. Such a model would probably be very complex. Consequently, two simplified cases are described here. If neither of the -28

-29simple models is sufficient to describe the experimental data, then the general case should be developed and investigated., In the first simplified model, it is assumed that the heat transfer coefficient is large and that- the rate of mass-transfer is limited by a mass-transfer coefficient. The model describing this case is Model I or the mass-transfer-limiting model. This model is consistent with experimental results. The second case of melting and recrystallization which is considered assumes that the mass-transfer coefficient is large and that the interphase transfer is restricted by a heat-transfer coefficient. This model is referred to as Model II, or as the heat-transfer-limiting model. This model is not consistent with experimental results. Another possible mechanism for the relief of composition instability is diffusion within the solid itself. Rather than becoming enriched by recrystallization, the solid is enriched by diffusion of BNB from the liquid into the solid. This possibility and its mathematical description are discussed in Model III. This model is not consistent with experimental results. Another case which was evaluated, Model IV, assumes a constant crystal composition. This model can not be justified on physical grounds. However, since this model is said to describe eutectic systems, it might also describe data for a solid solution. As was expected, this model disagreed with experimental results. Each of the four models just described is developed in this section. Following each development is a description of the predictions of the model. The four models are then evaluated in Chapter V by comparing experimental results with these several predictions.

-30A. Mass-Transfer-Limiting - Model I The mathematical description of column crystallization based on this model follows the developments of Hartland and Mecklenburgh( l8) and Powers.32) 1. Description The physical description is as follows. Solid of weight fraction X moves through the column at rate L without backmixing. This solid, which is radially homogeneous, contacts liquid of weight fraction Y moving at rate V. There is dispersion in the liquid axially but not radially. Mass-transfer occurs between the two phases which pass countercurrently. The elemental description of the process is illustrated in Figure 15. The flows of liquid and solid respectively contribute VY and LX to the movement of BNB. The rate of dispersion of BNB within the liquid is assumed to follow Fick's Law, Equation (1). M b - g A r p a //d V (1) Here NB is the rate of dispersion of BNB in gm/sec, p is the density of the liquid in gm/cm3, 5is the volume fraction liquid, A is the area in cm2 through which the liquid and solid flow, and D is the effective diffusivity in cm2/sec containing contributions from molecular and Taylor diffusion and from backmixing. The last four factors, p, q, A and D, are assumed to be independent of position, h According to the model, the rate, J. of BNB mass-transfer from the solid to the liquid phase follows Equation (2). Y - K X A ahk( y- Y) (2)

- 31V,Y NB L,X __,'d —~ ~ ~ ---- h V Y+AY NB+ANS L,X +AX NB -D'71Ap dY/dh J KaAp Ah (Y-Y') Figure 15. Elemental Description of Column Crystallization- -Mass-TransferLimiting Model.

-32J is considered to be proportional to the difference between the existing liquid phase composition, Y, and the equilibrium compositionsY*. This latter composition is that ficticious value which is in equilibrium with the solid contacting the liquid Y. In Equation (2), h is the column position measured from the melting section in cm; a is the interfacial area between the two phases per unit volume, cm2/cm, and K is the mass-transfer coefficient, cm/sec. K and a are assumed to be constant. Based on the rate equations listed above and considering BNB, a mass balance on an element of the liquid phase yields Equation (3). Dalpdey Y _Vd/ - WHAM(y- i a)_ (3) The first term in this equation describes the dispersion within the liquid phase. The next two terms relate to the bulk flow of BNB,. and to the masstransfer between phases. V is assumed to be independent of h. A BNB balance on the upper part of the column, considering operation at total reflux,yields Equation (4). MA aY d'C - VY + * LX O (4) The first term relates to dispersion, and the last two terms to bulk flo; of the liquid and Of the solid, respectively. A total balance on the upper part of the column yields Equation (5). V W- L =O (5)

-33The solution to Equations (3) tc (5) requires another relationship between the dependent variables. The liquid-solid phase equilibriu;m furnishes the necessary relation. In general, the phase equilibrium will be such that an analog or numerical solution to the system of equations (Equations (3) to (5) and the equilibrium relation) will be required. However, BNB-CNB has a phase relation, as shown in Figure 4, which is linear over a wide range of compositions. Thus Equation (6) may be used as the required phase relation. AX- =,^- e-~ (6) Values of m and b can be determined from Figure 4 for the range of composition which applies in a given separation. The elimination of Y*, X, and V from Equations (3) to (6) produces Equation (7). At Y/,t ~- (./DA e -,. A9 /LA) A, //a + (I/m-1)1<4Y/D? > _i\<> /m a (7) This equation has as variables Y, L, and h which are all relatively easy to determine experimentally. Thus, the model can be tested for agreement with data. The definitions 8a to 8c are introduced into Equation (7) in order to simplify further manipulations. Equation (9), which is the differential equation describing the mass-transfer-limiting model of column crystallization, is the result. Rl = ~/Oant GALv A /LV (8a) P\= V /D q (8b) R 3 = t\/ \ -| \ (8c) ( 8d )

2. Limitations It is useful to restate the limitations inherent to Equation (9) before considering its solution. Equation (9) is strictly a material balance. Energy effects are involved, but only implicitly. That is, it is assumed that V and L are constant, independent of position. This assumes that heat capacities, heats of fusion and mixing, and heat leaks to ambient have a certain inter-relation. This relation is not set forth, but the constancy of V and L is considered in Appendix A2-b. Other factors are considered to be constant. These factors include all those which make up R1, R2, and R3. This assumption was not tested experimentally. However, the variations in R1 H R2, and R3 probably are second order effects and can be neglected for the purposes of this study. The success of the model incorporating these assumptions is the only justification of the assumptions. The given form of the differential equation explicitly assumes a phase relation which can be approximated by a linear equation. In a chemical system in which the liquidus and solidus lines are very far apart, such an assumption could only be applied to a very small range of compositions. Thus one would expect that Equation(9) would have to be applied piecewise along a column separating such a system. The constants m and b would vary markedly along the length of a column. If a piece-wise application were undesirable, then a numerical or analog solution to Equations (3) to (5), using an equation more closely describing the equilibrium relation than does a linear one, would be required.

-353. Solution to Differential Equation The solution to Equation (9) has the form given in Equation (10). Y=CCL eX(X) ( C 3 t (10) The last two terms of this equation are the general solution to the homogenous part of Equation (9). The constant C1 is the particular solution to the non-homogeneous part and is given by Equation (11). C - b/( l- a) (11) The constants q2 and q3 must satisfy the characteristic Equation (12). it- iR, Kit 0 (12) This is a quadratic equation whose roots are given by Equation (15).:q R. t\+(V-HRjK R3/R: )' i/2 (13) In many cases, the collection of terms R2R3/R1 will be less than 0.1 in which case the square root term in Equation (13) can be closely approximated as in Equation (14). (I- 4 ~R/ R " /R )";/ l- zR. R3 i/R ((14) The roots of Equation (12) thereby become Equations (15). - =, RP-j \ 3 /R\ I (15a) qGRL=R R R/R' (15b)

- 36The root q2 can be further simplified, subject to the above limitation on R2R/R1, to yield Equation (16), 2 I (16) Thus the solution to the differential Equat:ion (9) becomes Equatdior-: (17. y/ )/( - Q)t C e(,) + C3(x ( R /x) (1x7 In this equation, R1 and R4 are given by Equations (8a/ a:nd (8d;, and H by, Equation (18). Equation (18) defines E and F whiich w:ri, R7 /' -\ A p / L (8a) ( 8d. i H /3=D pL t u / K _ E / u + u/ 9 (18) be discussed later, The term H/R4 is a measure of the effectiveness tuat a given set of operating conditions has in achieving a separatl:n tetweelcomponents in a column, crystallizer. This term has the unilts of cm and therefore indicates the length of column requj.ired to produce a certainx separati.on. Throughout chemical engineering literat ure a ter:m o:f this typ,-:Is cal-led the height of a transfer unilt, HTUJ 4., Boundary Conditi ons The constants C2 and C5 in Equation (17) can be evaiuated iJsing appropri.ate boundary conditionso The first boundary co nd:lt1. rir is that the solution must be applicable at all condltions of operationSpecifically,, the value of Y must be between. zero and one for ali' valuies of L, The second boundary condition:is that th'e compostion: of

-37the liquid at h = 0 is known or can be determined experimentally. That is, Y = YO at h = 0. Examination of Equation (8a) indicates that R1 becomes very large as L increases or approaches zero. Because R1 appears in the argument of the exponential, the term C2 exp Rih would become exceedingly large under either of these conditions unless C2 were small. In fact C2 must be zero if this term is to remain finite as L approaches zero. A specific case may clarify this argument. As it will be demonstrated later, the terms in Equation (8a) have values near those given below. RI, L /0.2 e f0 /L (8a) The crystallizer can be operated, with some difficulty, with a crystal rate as low as 0.005 gm/sec. At this condition R1 is about 8 cm The second term in Equation (17) thereby becomes C2 exp 8h. For large h(h = 30) the exponential part of this term becomes very large. As neither of the other two terms in Equation (17) can counteract the influence of this large term, C2 must be very small, on the order of exp (-250). C2 can reasonably be assumed to be zero. Thus Equation (17) reduces to Equation (19). Y- b /R L C Q3 ey > (ViL / H) (19) The reduction of Equation (17) from two exponential terms to one can be based on an argument other than that just discussed. The factors R1 and R4/H from Equation (17) can be estimated by using typical or extreme values for D and K taken from the literature, and by using values for a, A, ~ and p approximated or determined from experimental

-38dat a. The result of such estimation is that R1 is always m(uch larger that R4/H, and for most cases, R1 is about equal to the reciprocal of R4/H Further, because of the particular dependence of R1 and R4 /H on the crystal rate, R4/HH is bounded regardless of the values esti:mrrated for D and K or the value of L, R1 on the other hand becomes very large:for some ranges of L regardless of the values of D a:nd K These two characteristics of R1, its large value:in relation to R4/H, and its unboundedness with regard to L, make it reasonable to require that C2 be zero. Application of the second boundary co:n.dition to Equation (19) indicates that C3 is given by Yo - b/:R4 The exponential in Equation (19) can be approximated by the linear form given in Equation (20) because, as it will: be shown later, H/R4 is large, Thus the solution to the differential equation becomes Equ.ation (21;. Y- \YQ + (? b/\L,) Vs/H By, the earlier assumption regarding phase equilibrium, mY',o, b is eq'ual to X*. Applying this to Equation (21) yields Equation (22/ th~.e f i:al form for the solution, \/ Yo,- (Y)o >XY )/

-395. Limitations The limitations discussed earlier were on the differential equation describing the mass-transfer-limiting model. Additional assumptions are applied to the general solution to the differential equation in order to produce Equation (22). These assumptions and their justification are reiterated below. If Equation (22) is to be applied to another chemical system then the validity of these approximations should be checked in view of the new experiments. The first two terms of the binomial expansion for the exponential in Equation (19) were used as an approximation of the whole expansion. Experimental evaluation of the variables making up this exponential indicates that the approximation is excellent. Similarly, the approximation of q2 by R1 (Equation (16)) is very good. Results of experiments used to evaluate terms in the general solution justify two other approximations. The determination that R1 is greater than 5 for some conditions of operation and greater than 1 for almost all conditions of operation, shows that C2 must be zero. The approximations used in obtaining Equation (21) from Equation (19) is justified by the large experimental values determined for H/R4. H was always greater than 10 cm, and R4 did not exceed 0.1. H/R4 was therefore always greater than 100 cm, and hR1/H was less than 0.3. 6. Predictions Equation (22) predicts that the composition of the liquid in the column will vary linearly with position. The magnitude of this variation, the concentration gradient, should increase as the phase separation, (X* - Y)0, increases. Thus, as the concentration of material charged to

-40the column approaches 0 or 100 per cent, the concentration gradient should approach zero. The model also predicts that the composition profile will be dependent on the crystal rate, and that this dependence will1. be determined by the magnitude of the operating parameters, especial:ly D and K (see Equation (18)). Another prediction of this model is that, if D and K are constant, the product HL will be a linear function of L2 This can be seen by multiplying the terms in Equation (18) by' L The final prediction of the mass-transfer-limiting model is that H, which is evaluated from the concentration gradient, is a function, of composition. The inclusion of m, the slope of the phase relation., Equation 4, in the second term in H is the basis of this prediction. The first three of these predictions were substantiated by the experimental results. The fourth prediction could not be tested. directly, but an indirect test seems to substantiate it, also. B. Heat-Transfer-Limiting - Model I: The heat-transfer-limiting model is similar to the mass-trasnsfer-limiting model in. its development although it is mathematicalEly more complex, Unless specifically noted to the contrary, the symbols used in Model II: have the same significance as in Model I. 1, Description The physical description for Model IT: i.s illustrated in Figure 16, Solid of weight fraction X at t~C moves through the colbumn crystallizer at L gm/sec a Countercurrently, a liquid of weight fraction Y at T~C flows at V gm/sec. Material moves axially within

-41No V,Y,T J Lx,t h qh + NB+ANB V+&V J+AJ L+AL V+AV X+ AX T+&T t+ A t NS -- DA77p dY/dh J -— EA7 dT/dh q - uaA (T-t) Figure 16. Elemental Description of Column Crystallation —Heat-Transfer-Limiting Model.

_42the liquid phase by dispersion at NB g/sec. There is no dispersion of material within the solid or radially in the liquid. The energy associated with the dispersion of materials in the liquid phase is assumed to be negligible. This assumption is probably justified because the dispersion of CNB will be about equal to that of BNB. Thus the associated energies will be about equal. There is heat conduction, J cal/sec, in the liquid but there is none in the solid because it is a disperse phase held in no fixed orientation. This conduction, given by Equation (23), is proportional to the cross-sectional area of the liquid phase Anr cm2, to the temperature gradient in the liquid, dT/dh Co/cm, and to the eddy-thermal diffusivity, ~ cal/cm-C~ -sec, which contains effects of simple conduction and eddy convection. 3= - & dr/\ vl jr /2 (23) Heat-transfer, q cal/sec, between phases is proportional to the heat-transfer coefficient, U cal/cm2-sec-Co, to the interfacial 2 area, aAAh cm, and to the temperature difference, T-t. q is given by Equation (24), q = A. (T- +) (24) The energy associated with mass-transfer is equal to the product of the change in V times the specific enthalpy of the solid which is produced, The third rate equation.us ed in the heat-transfer-limiting model, Equation (25), is the same as Equation (1) used in Model I. sI - A _ p e Y/dA (25) This equation gives the rate of axial dispersion in the liquid phase.

-43The three rate equations presented above permit energy balances to be written on an element of the liquid phase, Equation (26), and on the top end of the column, Equation (27). c~~~VN ~ ~ ~ ~ V C _ aV t a to 2 t ( a- X) CL C l t (26) c''ft rj, The first two terms in these equations represent the energy associated with the bulk flow of liquid, V. The third terms relate to heat conduction in the liquid. The last term in Equation (26) is the energy lost by the liquid resulting from a change in V. The last term of Equation (27) represents the energy associated with the bulk flow of solid, L In these equations, the subscripted C's are specific heats in cal/gm-C~. The first subscript, L or V, identifies the phase as solid or liquid, and the second subscript, B or C, identifies the material as BNB or CNB. It is assumed that specific heats are constant with respect to temperature and composition. The derivation of Equation (26) and (27) is based on the assumption that the enthalpies of solid BNB and CNB are zero at an arbitrary temperature To~C and that at this same temperature the liquid enthalpies are HB and HC, respectively. It is further assumed that there is no

heat of mixing within either phase. These assumptions are justified by the small separations achieved using BNB-CNB. Material balances on one end of the column yield Equations (28) and (29), just as in Model I. ADA 4W Y7\ X- \/YK - CD X (28) -- V +- L = 0 (29) Equation (28) is a BNB balance, and (29) is a total balance. The heat-transfer-limiting model is completed by using two further equations which relate the dependent variables, These equations are the phase equilibrium Equations (30) and (31). Tr- ~-, - e, c; s 6 o (,- Y) t- " o (o30) L34t \4\, 0 n(l-x)0o 32 K(1-)1 T -2o (31) These equations are incorporated into the model based on the assumptions that the liquid is at its freezing point and that the solid is at its melting point. The first assumption is justified by the condi.tions of agitation maintained in the crystallizer, Such conditions, as well as the presence of small crystals, make the possibility of supercooling very small. The numerical constants used in Equations (30) and (31) apply for X and Y between 0 and 0,7 and were determined by' fitting phase equilibrium data by regression lines (see Appendix A5-b). V is eliminated from the equations above to reduce the equations and dependent variables by 1. This produces a solution in terms of the crystal rate, L, which is easily measured at one point in the column.

To, which is arbitrary, is set equal to 44.6~. This eliminates two terms from each of Equations (30) and (31). This elimination simplifies the solution considerably. The solution of the heat-transfer-limiting model can be further simplified by converting the derivatives of T to derivatives of Y. This is accomplished by noting that dT/dh = (dT/dY)(dY/dh) and that dT/dY is equal to 4.056/(1-Y) as determined from Equation (30). Another simplification is possible based on experimental results. It was shown in Figures 5 to 14 that plots of Y vs. h were linear. Thus d2Y/dh2 is zero. If this is incorporated into the second derivatives of T, then d2T/dh2 is given by 4.056 dY/dh)/(1-,)2. Based on the simplifications listed above, Equations(26) to (31) reduce to Equations (52) to (36). (;h(LA/ t (C, (3 ~ v c, ( Y Y 7) -Cv, c d h\ ( T) ( t _ dUkXa( T)'I ICL, C ( gCLcL1 4J C (52) i- V}- 7 Lb~c + (H~- H-)LY, LTC v,,c vr' - 1.O L u \- ( 5)- (,,c.)

-46- 3,'O7 I(-x) O.382S C I 9' ) (36) The increase in the number of terms appearing between Equations (26) and (27), and Equations (32) and (33) is the result of collecting like terms subject to the previous assumption that heats of fusion and specific heats are constant. 2. Solution Equations (32) to (36) cannot be solved analytically. However, they can be solved numerically when put in proper form, Equations (37) to (42). Equations (37) to (42) derive from Equations (32) to (36) if all the derivatives of products are expanded, all like terms are collected, and simple rearrangements of terms made. There are no approximations made in these transformations other than those already made. Consequently, the simple algebra is omitted and only the result shown. T- -H.0sC 5G ('-Y) (37) X = t / X D? 3 ) CC~sC i- SC- HO Y (Cv,,c -,C v) /T -C V)C,, Pe I\-~ c~~c~Lc (38) -) (C -L,G } (39) ly ~~~~~~~ ~~~~~~~~(40'o) dV - [,a ^, A, _ /a,,,.. (41)

-47-IR ~ 6 An A a/ (42b' S = (Cv C - C9,)C -C,4cT ic I 42c)V Y (42a) _WIC CL/j t - CLA C L )X The method of solution is as follows. A set of values for U, E, and D is chosen, These parameters are respectively the heat,transfer coefficient, the effective eddy thermal-diffusivity of the liquid, and the effective mass-diffusivity of the liquid, As mentioned earlier, these two diffusivities contain contributions from the mixing action of the spiral within the column as well as from processes on a molecular scale, A set of conditions for Y and V at h - 0 is selected, These values correspond to conditions encountered during the experimen.tso T is calculated from Equation (37). X and t are determined by an iteration on Equations (38) and (39). The two derivatives, dY/dh and dV/dh, are evaluated from Equations (40) and (41). A fourth order Runga - Kutta integration is performed to determine new values for Y and V. The procedure is repeated until a length equal to the length of the column has been traversed. A table of Y vso h i.s prepared and compared with experimental results, 1o Predictions By proper selection of values for U, E, and D, profiles of Y vso h which are linear with small separations are predicted. The effect of increasing the crystal rate, L, at h - O is to increase the separation.

C, Diffusion Within the Solid Phase - Model III The two previously developed mode:ls each considered the liquid and solid phases to be radially homogeneous at all positions withi.n the co'lumn., Only axial variations were assumed. The model developed in. this section, which assumes diffusion within the solid phase, necessarily removes the limitation of radial homogeneity in the solid. This introduces considerable complexity to the model and to its mathemati.cal descrilption.. Such complexity is reduced by considering the crystal.s in the purification section to have a simple geometry. The general solution to a model which includes diffusion in the solid, dispersion in the liquid, and simultaneous heat- and mass-transfer between phases would indeed be complex. Such a solution was not attempted. Rather, one limiting case of such a solution, one which considers a single mechanism to be controlling, is developed. That mechanism is diffusion, wi.thin the crystal. It is therefore assumed that heat-transfe:r and masstransfer occur sufficiently rapidly to permit the diffusion. but..not so rapidly so as to destroy the crystals, L. Description The description of the model which embodies the conditions stated above is as follows. Spheres of radius Ro are formed in the freezing section and move toward the melting section, Initially these spheres are of uniform weight fraction Xo, and they are in equilibrium with the liquid in the freezing section. The surface of the spheres remains in equilibrium with the:liquid throughout the co:lumn, In order to describe the concentration profile in the crystals, it is assumed that the surface concentrati.on varies linearly with the

-49position in the column. This assumption is predicated on two previous assumptions and on one piece of experimental evidence. The previous assumptions are that: (1) the concentration at the surface of the solid is in equilibrium with that in the liquid, and (2) the phase relation is linear. The pertinent experimental observation is that the concentration in the liquid phase varied linearly with position (see Figures 5 to 14). Based on the assumption concerning the surface concentration, the spheres can be considered independently of the liquid. The concentration profile is then a function of the rate of variation at the surface, the radius Ro, the position in the column, and the diffusivity of the solid. 2. Solution to Concentration Profile in Solid The diffusion of BNB in the crystal is assumed to follow Fick's Law. With the spherical geometry assumed in this model, the rate of diffusion NB is therefore given by Equation (43). r\l,- q T P D \Ct At/X r (43) In this equation, X is the weight fraction BNB in the crystal, r is the distance from the center of the crystal in cm, p is the density of solid BNB in g/cm3, and D is the molecular diffusivity of solid BNB in cm2/sec. In the standard way, a BNB balance on a shell element of the crystal yields Equation (44).

-50The first term in this equation relates to the variation of X at a given position in the crystal with time, and the second term describes the movement of BNB within the crystal, The time, @ is related to the position in the col.umn, h by the crystal rate, L, the area through which the liquid and solid pass, A, the length of the column, ho, and the crystal holdup, Ahops(l1-',) Thus the time is given by Equation (45). - =A -o (0 - )/ LPo (45) The boundary conditions which apply to Equation (45) are Equations(46). X - X,) 0A ~ -; o = (46a) X=- xO +A@ e<+ Y - (46bj Here k is the rate of variation of the surface concentration of the crystal, weight fraction/sec. By defining a new variable T equal to X-Xo, the problem reduces to Equations (47). us &, a~(<X)//a~~= a( = < (< )/(47a) T O- iS G O - (47b) T=& AS- <+ -- = (47c) The solution to Equation (47) is given by Carslaw and Jaeger(6) as E;quations (48). T- t@ - ~ = - S R 3 (48a) X D D 73 1 u =\ (\^)~ A\) gpt ) Si ( Rc) (48bj

-51Integration of these equations from r = 0 to r = Ro leads to Equations (49). These give the average concentration, T, of a sphere at any time (position). Q.'; U..... 2..'..T Li (49 a) -,i ( -D >-l (49b) If it is assumed that Ds is less than 10-8cm2/sec, which is a high diffusivity for the solid phase, and that Ro is greater than 0.05 mm, which is a very small radius, then the g(n) for n greater than 1 can be collected into a constant term. The single exponential given in Equation (50) is produced..~ + TTL ) ( (50) This equation gives the average composition of the solid to which is imposed a linearly varying liquid composition, From Equation (49), it can be shown that the derivative of the solid concentration at the surface of the sphere is given by Equation (51). ( i, LR K-t(, j'(51) This derivative governs the rate at which BNB enters the solid. The preceding equations complete the first part of the overall problem of the solid-phase - diffusion model. The second part, the description of the liquid, is presented below.

-525, Solution to Profile in Liquid A BNB balance on an element of the liquid yields Equat:lon (52). pnYt e A L kh - ( )R (52) The first term relates to the diffusion within the liquid, and th.e second. to bulk flow in the liquid. The term on the right-hand side of the equa-, tion is the rate of diffusion of BNB into the solid and is given by Equation (5:1). I:ntroducing Equations (45) and (5:)j into Equat;ion (52) produces Equation (53). In this equation W1, R1 and R2 are given. by Equations (54)0 R, =L L/D R A Yj (54a) n, -- z a R a, o. Ps /DCD (54b3 9'/.~ = S -/(54c;) The solution to Equation (53) is given by Equation. (55), /C- C l t H(R>A) - kRV, /\R? R2 rW(- vh) /LTrw(R\-w)] (55)j Two boundary conditions are used to evaluate C1 and C2. Let the concentration in the liquid, and the derivative of this concentratil.on, be determined experimentally at h - O Thus at h:= 0, Y - Y, and. dY/dh = So. Application of these conditions defines C1 and C2 as

-53shown in Equations (56). c /= C)/ R- /L R.( R. -w)3 SV/ iR (56a) C i- ~ - cY -C R\/CL WT ~I - w)A (5Ib) The form of the composition profile can be rearranged if the restrictions on Ro and Ds which were imposed in the first part of the problem,are applied. Namely that Ro > 0.05 mm and Ds < 10-8 cm2/sec. This application permits the expansion of exp(-Wh) to its linear form l-Wh giving Equation (57). TT ( - \N) )1 Q ((57) _ x —.. 1 / x) If Ds is larger than 10-8, or Ro less than 0.05 mm, then this linearization is not valid and a more complicated form for Equation (57) would result. Equation (57) can be further reduced if Rlh is such that exp(-Rlh) can be expanded to its linear form 1-Rlh. If this expansion is valid, then the terms in Equation (57) which are enclosed in parentheses reduce to -h, and those enclosed in square brackets reduce to zero. Under this condition, Equation (57) becomes Equation (58). \/, (58)

If the two term expansion of exp (-R1h) is invalid, but the three term expansion is valid, then Equation (57) reduces to Equation (59)~. Y=~ YD cSO [~ I C 11 ( 1 \__ (59_ 4, Predictions The model incorporating di.ffusion in. the solid phase, in. its si.mpliest form, predicts that the concentration gradient withi.n t~he liq-'aid will be linear~ This prediction is based on the restri.cttions placed on. the values of Rlh If these restrictions are loosened slightly, t;hen the model predicts a parabolic concentration profile. The boundary conditions which were applied to Equati.on (55) permit no further predictionsto be made. The constants Yo and So are entirely empirical. The dependence of these constants on operational variables can not be predicted, Only correlations with experimental data will give information as to the possible functional forms of Y' and So Such a solution is not altogether satisfying especially because the variation of So with operating conditions was studied in this in.vestigation. If the model in the form presented here cannot be evaluat.ed. satisfactorily, then boundary conditions which are not empir:i.cal shoul.d be used and the new form of the diffusion model tested, D, Constant-Crystal-Composition - Model'I The model which assumes that the crystals pass unchanged thbroa.gh the purification section of the column is presented because this model was used successfully by Albertins(l) to interpret data for a e';:tectilc system. The development which is presented below will:faci.litate a lat;er d.scussion of the differences between the conclusions drawn by Albert;ins and those extracted from the current results.

-551. Description In this model, crystals of weight fraction Xo are considered to form in the freezing section and pass unchanged to the melting section at L gm/sec. An adhering liquid of weight fraction Z accompanies the crystals at C gm/sec. Mass-transfer occurs at J gm/sec between this adhering liquid and the free liquid which moves countercurrently at V gm/sec and weight fraction Y. Dispersion occurs in the liquid phase at N gm/sec. The model for this case is illustrated in Figures 17 and 18. The development of the constant-crystal-composition model closely parallels the development of the mass-transfer-limiting model. As in the earlier case, the rate of dispersion of BNB is proportional to the effective diffusivity, D, the area available for dispersion, Aq, the density of BNB, p, and the concentration gradient, dY/dh. That is, NB is given by Equation (60). l,- - - D A e Y/d A (6o) The rate of mass-transfer, J, is given by Equation (61). The factors K, a, A, and Ash are the same as in Model I. The driving force for mass-transfer is assumed to be the difference in concentration between the two liquid phases. A BNB balance on an element of the free liquid phase gives Equation (62). SDAtZY'VA - Y -a-A (7-i) (62) aftL d h

CRYSTALS -ADHERING LIQUID REFLUX LIQUID V,Y CZ LX CRYSTAL PHASE PHYSICAL SITUATION SIMPLIFICATION OF THE PHYSICAL SITUATION Figure 17. Illustration of Flows in Constant-CrystalComposition Model.

- 57V,Y NB, L, X-o h+Ah v,Y+AY NB+ANB C,Z+AZ L,Xo NBe=-D7Ap dY/dh J=KaAp Ah(Y-Z) Figure 18. Elemental Description of Column Crystallization — Constant- CrystalComposition Model.

-58As in Equation (3), the terms in this equation refer to dispersion, bulk flow, and mass-transfer, respectively. It is assumed that the flows V and L and the coefficients D, A, p., p, and K are independent of position, h. A BNB balance on one end of the coluxny gives Equation (63). A~ n JYd s E x o + C - VX/ =~ (63) Only the flow term, CZ is different from the terms in Equation (4). A total balance on one end of the column gives Equation (64). 1 + C-V = o (64) In order to solve Equations (62) to (64), it is assumed that C = aL. In general, a will be small because C and V are both liquid phases. A numerical value for alpha must be estimated, but data on "drained crystals" probably should not be used. The surface which forms on draining does not occur in the present situation. An approximation for alpha can be made on the basis of boundary-layer theory, or outright guesses can be made. When Z, which is very difficult to measure experimentally, is eliminated from Equations (62) and (63), Equation (65) is produced. A a-Y iR Y L (65) The constants Ri are defined by Equations (66). R,- o t Ap Lt K L (l Ck)/ D A p(6 (66a) R, = 2cR / U ~~1(66b) 2. Solution According to the model, Xo is constant. Thus the solution to Equation (65) is given by Equation (67).

-59= c,C, c xp(.,,) _ C3 x ( (67) In this equation, q2 and q3 are given by the characteristic Equation (68). t ct.'l i 2. = (68) C1 must be equal to Xo and by applying the same valid approximations which were used in Model I, the roots of Equation (68) are given by Equations (69). -, = (69 a) (2(3 He,-~~ "~/n, i(69b) C2 and C3 are defined by appropriate boundary conditions. As in the mass-transfer-limiting model, R1 becomes large as L becomes large or small. Therefore C2 must be zero if Equation (67) is to apply to all conditions of crystal flow. Equation (67) thus reduces to Equation (70). y- Xo t C3 J X(\Q ( i,/,K,) (70) The reduction of Equation (67) from three terms to two can be accomplished on a base similar to that suggested in Model I. Because the constant-crystal-composition model was said to describe a eutectic system(l), data from such a system are used as the basis for the reduction, Values of D and K can be extracted from Equation (67) by relating it to experimental data. In one method for doing this, it is assumed that the three terms in Equation(67) are non-zero. Then for a given set of data, the values determined for D and K depend on the value chosen for Xo. In any case, the resulting values of D agree

-6owith the values appearing in the literature. On the contrary, regardless of the value chosen for Xo, the resulting value of K is 3 or 4 orders of magnitude smaller than values presented in the literature, In the second method of applying Equation (67), C2 is assumed t;o be zero. The resulting equation is then used to eval.uate D an.d K: The values which result are in essential agreement, with published. vahlues. The second of these two methods seems preferable, although i.t is by no means mathematically exact, Thus, C2 is assum.ed to be zero, thereby yielding Equation (70>) 30 Prediction This model predicts that a plot of ln (Y-X) vs, h will be 1linear with slope R2/Rj1 This is the result found by Albertins(l) in. his study of cyclohexane-benzene, a eutectic system, Another prediction can be made on a purely physic-al basis although this prediction is i.n apparent contradiction to the one jj.st presented~ According to the model, the crystals formed in the freezing section pass unchanged through the purification section to the melti:n.g section'where they melt to form the reflux which is ana'lyzed~ Because the crystals melt completely, the composition of the liquid at h.= O0 should differ from that of the crystals only as the result of di lti.o:n. by, the small amount of impure clinging liquid~ Thus the composition of the liquid at h - 0 must be less than that of the crystals. The crystals which reach the melting section. are of the same composition as those in the freezing secti.on, and are approximate'ly il.n. equi.libri.um with the liquid in the freezing section. Thus the predicti.on

-61obtains. The difference in the composition of the liquid at the two ends of the column should be no more than the difference achieved by a single equilibrium phase separation. That is, if the liquidus and solidus compositions differ by 6 weight per cent at the temperature maintained in the freezing section, then the difference Y30-YO should not exceed 6 per cent.

CHAPTER V EVALUATION OF MODELS The four models developed in the previous section are evaluated in this section. The evaluations are made by comparing the predictions of each model with experimental results. As was already mentioned, only the mass-transfer-limiting model, Model I, is consist nt with the data. The models are evaluated in the same order as that in which they were presented. Thus five tests of the mass-transfer-limiting model are discussed in view of experimental data. Then pertinent results are compared with the predictions of the other models to indicate the inconsistencies of these models. A. Mass-Transfer-Limiting Five tests were made of the mass-transfer-limiting model. In four of the tests, the results presented below demonstrate that this model of column crystallization is consistent with experimental data. These data were presented earlier, on Figures 5 to 14. The result of the fifth test was inconclusive. 1. Test 1 The mass-transfer-limiting model includes the term (Y-X*)o which relates to the difference in composition between phases in equilibrium at the top of the column. The inclusion of this term, the phase separation, predicts that separations will decrease as the charge to the column is made more nearly pure. This prediction was fulfilled in the experiments as summarized in Table II. -62

-63TABLE II INFLUENCE OF DIFFERENCE IN PHASE COMPOSITIONS ON SEPARATION Run Charge Phase Separation Crystal Rotation Osci..'l.at,.i o:n Separation Y-Y 30 Rate Rat e St.'oke X*-Y Weight Fractions BNB g/sec RPM OPM mri 13 0o.95, 0 001o.o 0 o,040 67 22 4o: 29 o 65 0037 0o 089 0o 038 6o 67 4 2 30 o.65 0.039 0o089 0o033 60 67 467 31 o. 65 0o09 0.07Q 0.025 60 67 4,2 14 0o 50 0o 0057 Oo 055 o 036 67 25 o 5 15 o050 0o057 O 080 o 0036 32 2 4 16 0o35 0o.o8 0o.06 0.027 46 40 6o0 17 0.35 0,059 0.072 0o018 29 30 o 0 6 0,05 Oo 01 O o 005 0012 67 72 40 7 0o05 O0. 01 0.005 0o009 67 70 4o5

-64Table II presents the separations achieved in several runs for which the charge composition varied widely. It is clearly seen that as the nominal weight fraction m-bromonitrobenzene (BNB) in the charge was increased from 0.05 to 0.95, the separation attained passed through a maximum. For nearly pure charges (Runs 6, 7, and 13) the ultimate separation was only 0.015 weight fraction. For charges of intermediate concentration (Runs 14-17 and 29-31) the separation reached 0.106 weight fraction. Figures 19 and 20 summarize the data in Table II. It must be emphasized that variables other than charge composition varied among the several runs. 2. Test 2 The mass-transfer-limiting model also contains the expression E/L + L/F, which is referred to as H. In this expression, E and F are factors which are defined in Equations (71), as presented in Equation (18). E =' A 4 9 (71a) F _ < o A (7lb) They relate to dispersion and mass-transfer, respectively. The model thus predicts that the concentration gradient within the column is a function of the crystal rate, L. Data on Figures (21) and (22) show this variation. For the range of crystal rate which could be achieved, H decreased as L increased. Examination of the derivative dH/dL, as demonstrated in Appendix A5-d, indicates that H reaches

0.12 0.10 0 0.08 - o m0 z z 0.06 0.04 \ z 0, 0 \ a. / w0.02 _ // \'~/ / 0 I. 1 0 0.2 0.4 0.6 0.8 1.0 WEIGHT FRACTION BNB IN CHARGE Figure 19. Effect of Charge Composition on Separation.

-660.12 a3 z z 0.10 0 0.00 0.08 /.o 0.06 0 o / 0 0.04 0.02 0 0.0 1 0.02 0.03 0.04 0.05 0.06 PHASE SEPARATIOQN, X -Y WEIGHT FRACTION BNB Figure 20. Effect of PPhase Separation on Separat ion.

22 18 CHARGE COMPOSITION Extracted from 0- 70 WT. % 14 Equation (18) using Values of Data from Runs 29-31 ) | E and F from Figure 23 10 ~i Maximum Crystal Rate 6 0.01 0.03 0.05 0.07 CRYSTAL RATE - GM/SEC Figure 21. Effect of Crystal Rate on Column Performance.

-6880 70 CHARGE COMPOSITION 60 _ 0-35 WT.% DATA FROM RUNS 33-35 50 _ \ EXTRACTED FROM EQUATION 18 USING VALUES OF E AND F FROM FIG. 24 0 I I 40 MAXIMUM CRYSTAL RATE-a 30 I 20 0.01 0.02 0.03 0.04 CRYSTAL RATE- GM/SEC Figure 22. Effect of Crystal Rate on Column Performance.

-69a mi:nien'1mum when the term E/L is equal to L/", Relating this to t:he data shows that axial. dispersion within the liquid, not mass-t;ransfer between phases, is the dominant effect in column crystallizat-ion, Thi! s result is i.n agreement with that of Albertins. (,., Test 3 As it was menti:oned earlier, Equation (:18), when mul t:i.piied. by, L to produce Equation (72), predicts that the HL is a i.l:riear:function of L2 -\ " E/L + L/F (18 2_t E -\ ( 72 Thus plots of HL vs. L, as shown on Figures 25 and 24, have intterepui ii E and slopes 1. /F The data on Figures 23 and 24 were fit by linear regression lines in order t:o give an objective method of analyasis o A.l.t:h.ogh_ tL.ert is considerable scatter:in the data, especial:ly in:r tlhose:r..F:igu;.re 2,7 there is no trend, A parabolic fit of the data wou:ld have positiv-e curvature in one case and negative curvature:i.:n tNhe othe:r. It can be inferred from Figures 23 and 24 that E an:,d F arfe, i.L - dependent of L, as was assumed in. the deve'lopment of t;.e model, a:r.id that bothL dispersion and mass-transfer occur in the coltnn crystalii zati.on of a solid solaution. Such a conclusion mr!ay be oppcsed to t"Lrat rf Albertins, who studied a eutectic system, From. the slope and intercept of Figu.res 25 ard 24, valu.es o-. E and F were obt;ained. Substitution of these vai.tes together wlt'h esti.mated or calculated values for A,'q, p, and a y:i elded D and K

0.48 CHARGE COMPOSITION O - TO7 WT. % BMB DATA FROM RUNS 29 - 31 o: = 0.44 0 ~ LEAST SQUARES FIT 0.40 4 8 12 16 L2x 104- GM2/SEC' Figure 23. Determination of Diffusivity.

-71O 0.58 0.54 LEAST SQUARES FIT 0.50 cn 0.46 CHARGE COMPOSITION iz |/0-35 WT. % BNB DATA FROM RUNS 33-35 0.42 0.38 0.34 - O 3 5 7 9 L 104 -- GM/SECFigure 24. Determination of Diffusivity and Mass-TransferCoefficient.

-72(see Appendix A6). As shown in Table III, the experimental values determined by this procedure, especially those for D, are in agreement with values taken from the literature. TABLE III COMPARISON OF EXPERIMENTAL AND LITERATURE VALUES OF DIFFUSIVITY AND MASS-TRANSFER COEFFICIENT Value Source Diffusivity Mass-Transfer Coefficient cm2/sec 103 cm/sec 4.6 0.44 Figure 23 4.2 0.071 Figure 24 3.5 l.9xlO Albertins(l) 1.7 0.075 Figure 30 1.5, 1.3 0.26, 0.64 Figure 31 0.5-3.0 Jones(21) 2-5 Lewis(25) 2-3 Thorsen(40) 0.8-2.6 Smoot and Babb(37) *Considered to be incorrectly measured. The preceding paragraphs indicate three predictions of the mass-transfer-limiting model which were demonstrated by the experimental data. These are: (1) The concentration gradient within the column varies with composition, as indicated by the term (Y-X*)0. (2) The concentration varies with the crystal rate. The form of this variation identifies the dominant mechanism in the column crystallization of a solid solution. (3) Diffusivities and mass-transfer coefficients can be evaluated from appropriate cross plots of experimental data.

-734. Test 4 There is one prediction of the model which could not be directly verified with the system BNB-CNB. This relates to the second way in which the concentration gradient should vary with composition. As was mentioned previously, H, which is inversely proportional to the concentration gradient, contains the term m, which comes from the phase relation X* = mY + b. Table IV shows that m varies from 0.9 to 1.3 as the composition varies from 0 to 100 per cent BNB. IH should vary also. However, the influence of m on H is very small because the term L/F, which contains m, is a small part of H. At least this is true in the range of L which was studied. A twenty per cent change in m, the change which occurs between liquid compositions of 25 and 65 per cent BNB, produces a change in H at most equal to 10 per cent and generally less than 5 per cent. Such a small effect can not be determined by the current experiment. TABLE IV DEPENDENCE OF SLOPE OF EQUILIBRIUM RELATION ON COMPOSITION FOR SYSTEMS BNB-CNB AND AZOBENZENE-STILBENE - AVERAGE VALUES OF m OVER A COMPOSITION RANGE OF 0.1 WEIGHT FRACTION Li quid Composition mBNB-CNB mAzo.-Stil. Weight Fraction 0.05 1.3 2.7 0.15 1.2 1.7 0,.25 1.1 1.3 0.35 1.0 1.1 0.45 0.9 0.9 0.55 o.9 0.7 o.65 0.9 o.6 0.75 o.9 0.4 0.85 0.9 0.3 0.95 0.9 0.2

-745. Test 5 "One final te-st of Model I was made. Equation (22), which was derived for a linear phase relation, predicts that the concentration gradient within the liquid phase is related linearly to (Y-X*)o. If a system with a non-linear phase relation were separated by column crystallization, one would expect that Equation (22) would be applicable only over small sections of the column. And for each section, (Y-X*)o would have a different value. Equation (22) was hypothetically applied in this manner to the system azobenzene-stilbene, the phase diagram of which is shown in Figure 25. This system has a very non-linear phase relation as shown in Figure 26. Table IV presents the variation of m with composition for the system. These phase data were reported by Powers.(32) The profile predicted from the hypothetical application is shown in Figure 27. Also shown on this figure are experimental results of Roessler reported by Powers.(32) The agreement is excellent. The data reported by Powers are for the average composition at each cross-section. Because Equation (22) applies to the composition, Y. of the liquid phase, an estimate of Y was made from Roessler,'s data. It was assumed that the volume fraction liquid was 2/5 and that the liquid and solid were in equilibrium in order to make this estimate. The volume fraction which was used is the same as that which occurred in the present study. Values for E and F/m were estimated in order to make the calculations of the profile. The details of the calculation are presented in Appendix A5-a.

-75130 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~20~~ 12 0 - I10 - I00 - 0 OI 90 - 80 70TO 0 20 40 60 80 100 % STILBENE Figure 25. Phase Diagram of Azobenzene-Stilbene.

100 80 0 0 q C') I ~~z z 50 w m In -J u7 0 200 0 20 50 80 I00 /% STILBENE IN LIQUID Figure 26. Phase Relation of Azobenzene-Stilbene.

-77+, O-DATA (SEE APP A5-a) A -CORRECTED DATA (SEE APR A5-a) --- -ESTIMATION USING EQN. 22 1.0 0.8 o 0.6 ZI N z 0:: 0.6?I 0.4 0 10 20 30 40 50 POSITION - CM Figure 27. Piecewise Application of Mass-Trt.ier-Limiting Model to Data Reported by Powers

-78In applying Equation (22), m was allowed to vary with Y Better agreement than with m constant was obtained. It seems therefore that, although not tested directly, the inclusion of m in H is correct. 6. Summary of Tests The five tests presented above indicate that the mass-transferlimiting model is fully compatible with experimental results. This compatibility is demonstrated for two systems which form solid solutions with widely different characteristics. B. Heat-transfer-limiting In order to evaluate the heat-transfer-limiting model, Equations (37) to (42), which describe this case, were solved numerically. The results of several solutions are reported in this section. These results show that the model is contrary to experimental data. The heat capacities of liquid and solid BNB and CNB were estimated by the method of Sakiadis and Coates given by Reid and Sherwood.(34) The estimates are as follows: (1) CL C = 0.271 cal/g-Co (2) CLB = 0.209 C - B (3) CVC = 0.348 (4) CVB =0.268 Heats of fusion equal to 33.1 and 24.3 cal/g for CNB and BNB, respectively, were used. (42)

-791. Effect of Composition Table V shows that separations predicted by the heat-transfer-. limiting model were independent of composition for two widely different sets of operating conditions. The separations were essentially the same although the phase separations (the difference between liquid and sol.id compositions at equilibrium) varied by more than 20 per cent, Such results are in direct contradiction with the experimental results demonstrated in Table II, 2. Influence of Crystal Rate, A series of solutions was made for which the crystal rate was varied over an order of magnitude, from 0.005 g/sec to 0.05 g/sec, The results of this series are presented in Table VI and in Figure 28. It is clearly seen that the crystal rate divided by the separation, Lo/Y, which is directly analogous to HL in the mass-transfer-limiting model, increased linearly with Lo. Consequently, a plot of Lo/LY vs. Lo would be highly curved, especially near L=: 0. These reswilt. are in contradiction to the experimental results shown on Figures 21 to 24, 3. Effect of Liquid Diffusivity In the heat-transfer-limiting model, as well as in the other models which were evaluated, dispersion in the liquid was an important variable, However, in order to produce a profil Le at all similar to thces which were actually observed, values of E about four or five times t:hc.s, measured experimentally were required. In other words, if a value for E about equal to the observed value was used in the heat-transfer-limli ing model, then a separation three or four times that which occurred physically was predictede

-80TABLE V EFFECT OF COMPOSITION ON COLUMN PERFORMANCE HEAT-TRANSFER-LIMITING MODEL Run Charge Phase Predicted Composition Separation Separation Weight Weight Weight Fraction Fraction Fraction BNB BNB BNB A o.65 0o.o441 0.0986 B 0.40 0.0593 0.0928 C 0.65. 0441 O.0454 D 0.40 0.0593 0.0457 E.65 o.o0441 0.0637 F 0.50 o.o0569 o.o668 G 0.35 0.0585 0.0584 Variable Runs A-B C-D E-G E 1.2 0.3 1.5 Lo 0.025 0.025 0.025 U 5 x 106 5 x 10-6 5 x 10-6 MA 2.5 20. 3.

TABLE VI EFFEC7T OF 0RYESTAL RATE ON SEEjAFATOI ON HEAT -T.A:NSFER,- LIUMIT'TI:N MODEL Crystal. Separation C rys tal. Fa te 3r ys tal. a te Rate Separati on. Squa:ed. g/sec Weight g/sec per 1.0 g2/sC2 Fracti.on, Weigh.t Fraaction oo005 Oo012 0o410 0025 Oo01 0o024 0o41 10o0 002 0o047 0>424 400 003 0o.O9 0.435 900 oo 04 o, 091 0o 442 lfGo 0 o, o5 O. 11.1 o o452 2 o O0

-820.46 0.45 0.44 U.J c),:- 0.43 0.41 0.40. I l.'.I..I 0 0.01 0.02 0.03 0.04 0.05 CRYSTAL RATE, G/SEC,AT hO Figure 28. Effect of Crystal Rate on Separation —Heat-TransferLimiting Model.

-834, I:nflu.aence of heat-Transfer Coefficient One would expect; the heat-transfer coeffic:ent; betwee.r t, e t1iw, phases to have a major influ-lence of the predict ed profille inasm con as +;Y model is a heat-tranl.sfer model:,, Tab:Le VIT. irndicates ti..at t h.iis expe'tat Lo, is not borne onut, A varlatlcn of U of tlhree orders ofl r!agn:fltAde, fircm:1 to 1000 Bt:i/ft2-L.r-Fr~ (5xLO -7 to 5x10-4 cal/cmlssec-C',, prcd ced a var:iation in the predi.cted separation of only 15 pe:r cen.t, A c.:a:'lge.:. UI of one further order of magn.itude produced a changP.in separa+t'o.r — eq.aL to a:factor of t:w-o, TABLE VII EFFECT OF HEAT-TFANSFER CGEFFICIENT ON SEPARATIGON Heat-trans fer Pred c t ed. Run Coefficient Separation cal/cm2-sec-C~ weight fraction. BNB E 5x:10-7,, 666 F 5xlO 6 o0667 G 5x:.LO -5 0676 H l5xL0 o 0764,I. 5x:l0 1.L5 30 5. Conc.lusion From the discussions present;ed above,:n which cc:rtrad:ict,:' s betweean experime:ntta:l results andd predictions cf tr!e heat-traJsfer- L71 t - ir.ng model are demonstrat~ed, it is concliuded thati t',his mode'l is in.:cc:mpa.' A L wi th experi.mer'ts

-84C. Diffusion in Solid Phase The solution to this model indicates that the liquid composition is, in general, a complex function of position. Under special conditions, the solution can be simplified to a linear or a parabolic form. If Rlh is less than 0.4, then the model predicts profiles which are linear to within 10 per cent. This latter quantity is the maximum error introduced into the evaluation of exp(-Rlh) by using the linear approximation for Rlh less than 0.4. In order that Rlh be less than 0.4 for eighty per cent of the column, R must be less than 0.016. As it was shown in the evaluation of the mass-transfer-limiting model, D and the product ~Ap are less than 1 and 1/2, respectively. Together these terms make up the denominator of R1'(Equation 54a'). Consequently, L must be less than 0.006 in order for the approximation to hold. In other words, if the crystal rate exceeds 0.006 g/sec., then the linearization of exp(-Rlh) is invalid, and the model will predict a profile with curvature of at least the second power of h. The experimental run with the lowest crystal rate had L = 0.008 g/sec. Other runs had crystal rates between 0.015 and 0.040 g/sec. Under these conditions the model would not predict the linear composition profiles which were obtained (Figures 5 to 14). The diffusion model is therefore incompatible with experimental results. D. Constant-Crystal-Composition The prediction of this model that the separation achieved should be equal to or less than one phase separation did not occur. The data in Table VIII indicatethis. For each run in the two pairs, the phase separations in the freezing section were nearly the same. It is clearly

-85TABLE VI1:: OMPFAF:ISON?F3 EXPER.I:TVENWAL, EEST ~T.S WT.7.. FfL,, (... OF S ~NS. iS J T zN'v "SATSMALs vMPIS.'7:T: ON M.02L._, fun Phase Separati on (io!.>mn S( pa.r pa t i o;n Separa ti on - e ng t,'ieFe;'ni t-:00 0 50 We ight We i gh t cm'iTe i gh Fract ion Fraction F:rac t o pe:r c.m 17. 0 0589 0o073 3004 o 002 18 o 0. 591 o 0O58 25 o 1 0025 21 0o 046 o 0o6. 20oO.00'5. 24 O oo480 o o86 530 o002oo9

-86seen that the separations achieved had no relation to these equilibrium values. In fact, for each pair of runs, the separation per unit length was nearly constant. The data in Table IX refute the prediction of the mathematical development that a plot of ln(Y-X) vs. h will be linear. Examination of the composition data indicates that X cannot be greater than 0.625 if the function ln(Y-X) is to have meaning. But such a limitation is contrary to the phase equilibrium. The liquid in the freezing section was estimated to have the composition 0.609 weight fraction BNB by extrapolating the composition profile from the purification section. A liquid of this composition would produce crystals with a weight fraction 0.625. Or if the crystals were 0.625 weight fraction, then the liquid in the freezing section at equilibrium with those crystals would have the composition 0.576 weight fraction. Thus the two predictions of the constant-crystal-composition model are in disagreement with experimental data. This model can not be used to describe the column crystallization of mixtures which form solid solutions although it was used to describe data from a eutectic system, TABLE IX PROFILE OF LIQJID COMPOSITION FOR ONE RUN Tap No. Position Composition cm weight fraction FS 1.0 o.609'(X* = 0.658) 7 6.7 0.625 6 11.9 0.641 5 17.0 0.657 4 22.0 0.672 3 26.8 0. 687 2 32.1 0.703 1 37~.0 0.708

-87CHAPTER V!:I ANAL;YSElJS OF DATA FROM A SYSTEM FORM6ING A EUTECTi',C The applicability of the mass-transfer-: i-miting model, Model I:, to two systems which form solid solutions was demonstrated i.n the previous section. This demonstration. indicated that both di.spers.on. within the 1Liquid phase and mass-transfer between phases take piace I; column crystai.ulization with solid solutionso In this section i.t "Is shown that the same mechanlisms apply in the co.lumn crysta:,:,_iJzation of a eutectic systemo A. Determination of Diffusion and Mass-Transfer Factors Equation, (19), which is reproduced here, was generated in the description, of Model. I. Deveiopment of this equation indicated that the s.lope of p-lots of Y vs. h coul:d be used to evaluate H which is defiined by Equati.on (.8, - -L //F18 Further, plots of HL vso L2 could then be used to evaluate D ard K. A parallel procedure exists for eutectic systems, Equati.,on (70), taken from the model of constant-crystal-composition, i.s siam:, ar to Equation (19). Y Xo t sC X i( V (70' Thi s equation indicates that the slope, S, of a p:lot of In (Y-X) vs. r

-88can be used to evaluate R1/R2. R1/R2 differs from H only in that m is replaced by aC. Thus plots of LR1/R2 vs. L2 can be used to evaluate D and K. In order to prepare plots of LR1/R2 vs. L2 data presented by Albertins(l) were used. His data, which were values of S, were converted to values of R1/R2. These values are presented on Figure 29, plotted against L. The crossplots of LR1/R2 vs. L,2 which result from Albertins data are shown on Figures 30 and 31. Although there is scatter, the linearity of the curves is clear. The least-squares lines from Figures 30 and 31 were used to calculate the curves presented on Fiulrelr29. Table III, presented earlier, summarizes the results taken from Figures 30 and 31. This table also shows typical diffusivities and mass-transfer coefficients taken from the -literature. The arreeilmnt ainong the various values is satisfactory. 1. E-ffects of Agitation Values of D and K are not only in good agreement with literature values, the variations of these parameter with agitution can be adequately explained. Data extracted from Figures 30 and 31 are plotted on Figure 32 as E vs. fl/2, Such a plot is suggested by Hayford who studied the effects of' pulse frequency and stroke on diffusivity. The agreement between the data and the predicted variation is extraordinary. The above analyses indicate that a mathematical description of column crystallization which explicitly contains terms relating to

-8914 CHARGE - AA OSC/MIN % BENZENE 12 \ 0 290 99 0 ] 290 97 A\ Q140 99 V 67 99 10 V 8 0 Figure 29. Effect of Crystal Rate on Column Performance.290 140 2- 67 O0 0.02 0.04 CRYSTAL RATE - GM/SEC Figure 29. Effect of Crystal Rate on Column Performance.

-90CHARGE, % BENZENE 0 97 0.22 - 0 99 LEAST SQUARES w 0.18- FIT 3C 0.14 - 0.10 I I I 0 4 8 12 16 20 L2x I04-GM~SECz Figure 30. Determination of Diffusivity and Mass-Transfer Coefficient.

0.15 0.12 ~ LEAST SQUARES FIT w 006..... I 0'06 4 8 12 16 20 24 L2 x 104- GM2/SEC Figure 31. Determination of Diffusivities and Mass-Transfer Coefficients.

0.12 0.11 en 0.11 aI 0.10 0 Li.. 0.09 L I I I I. 0.086 8 10 12 14 16 18 (AGITATION FREQUENCY)"2 - (OSC/MIN)"2 Figure 52. Effect of Agitation on Diffusivity.

-93mass-transfer and dispersion is entirely compatible with experimental. data. This conclusion is true for a eutectic system as well. as for two systems which form solid solutions and confirms the predi.ction of Powers (32 ) Such a conclusion, and the agreement indicated i.n Tabie ILTJ are contrary to previously published statementso ) It was contended that the data in Figure 29 should be correlated with a model including only a diffusivity which is a function of crystal rate. It was al.so indicated that these data, if used to determine a mass-transfer coeffi.cient, give a value which differs from literature values, and thus from the results of the present analysis, by four orders of magnitude, The fallacy in the analysis which permitted such conclusions is indicated below. 2. Fallacy The previously published analysis indicated that the composi.-, tion of the liquid could be represented mathematica.l.ly by Equation (7?3. y-X +C, exp (9,) p Cep( ) (73) Here ql and q2 both contain factors relating to diffusion and masstransfer. This equation indicates that a plot of data in Y vso h wi.l be linear if Xo and one of the constants C are zero. Such a p:.ot reproduced from Albertins' paper, is shown in Figure 33~ Ci.earily tnle line is not linear. The curvature indicated on this plot was attributed to the second exponential, and Xo was assumed to be zero~ Th.e data were fit stat' sti.call.y to evaluate Cl, C2, ql, and q2o The linne drawn on Figure 33

-940 EXPERIMENTAL DATA -\ X O I0 0 CURVE GIVEN BY LL \ gEQUATION (74) _-0 164.I I....I 4 I 0 10 20 30 40 50 Z -CM Figure 33. Profile of Liquid Composition for One Run.

-95is given by Equation (74)o g = O c.c) 5 7 t 8 (-O. \q 3 t) O. OD0087 e/ p ( O O y04? i) (74) Here Z is the column position measured from the freezing sectiorn (z - 50-h)o Values for K and D were determined from ql and q2. The value of K was about four orders of magnitude di.fferent from the values determined by previous investigators in other types of studies.:It was therefore con'cluded that a model-.,nc.iudi.ng masstransfer and assuming Xo to be zero was incompatible with the data. Such a conclusion is not dictated by this resul.t The resu:lt does indicate that either: io The drawn conclusion is correct, or 2, The assumptions, that Xo =- O, and that the curvature of ln Y vso h is attributable to the second exponential.i, are incorrect. It appears that the second of these alternatives was not consi.dered previously. Detail.:ed examination of the form of ql and q2 in Equat-..,or. (62) (or in Equation (55)>, indicates that the seco.r.d of these alternatives must be correct. In fact, the constant in front of one of the exponentials must be zero. The same reasoning as app..ied in. the case of the mass-transfer-/lximiting model can be appli.ed here. For ei.t:ner very high or very ~L.ow crystal. rates, one of the q, becomes very:large,, The corresponding C must be zero to keep Y between zero and one for, al... crystal. rates.

-96Also, the second argument used to reduce Equation (67) from three terms to two, as presented in the development of the constantcrystal-composition model, indicates that C2 must be zero. Albertins' data are reanalyzed below applying the second alternative mentioned above, A plot of in (Y-X6) vs. h, with Xo = 0.000071, is shown in Figure 34. The linearity is clear. This indicates that one of the C's in Equation (73) is zero. The q accompanying the non-zero C thus retains effects of mass-transfer and of dispersion. Using data from several runs, as was done in Chapter VI, values of D and K were determined. The values determined in this way are in good agreement with those found in the literature. (See Table III). The results of this analysis indicate that a model which incorporates both mass-transfer and dispersion in column crystallization is not incompatible with data taken on a eutectic system as was previously postulated.

Y-XoWEIGHT FRACTION CYCLOHEXANE o 0 00 0D 0 CD -e l / O (D p3 H-I P~ OD 0 H 0o Oo- II ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I ~~~~~~~~~~0 ~~~~~~~~~0 CD0 cl — CD~~~~~~

CHAPTER VII SUMMARY The data and interpretations presented in this dissertation demonstrate that the mechanisms of mass-transfer between phases and dispersion within the liquid phase effect and limit separations by column crystallization. Data from three diverse binary systems were used in this demonstration. These systems were: 1) m-bromonitrobenzene - m-chloronitrobenzene which forms a solid solution with a nearly linear phase relation; 2) azobenzene - stilbene which forms a solid solution with a highly non-linear phase relation; and 3) benzene - cyclohexane which forms a eutectic. Four mathematical models, each based on different assumptions as to the physical nature of column crystallization, were formulated. These models were sequentially evaluated for agreement with pertinent data. Major contradictions between model and data existed for three of the models. The fourth model did not disagree with the data in any of the tested aspects. The model which is consistent with the data describes the nature of column crystallization as being dependent on the type of system under consideration. Column crystallization of a eutectic system is basically a washing operation. A nearly pure crystal forms in the freezing section and is surrounded by an impure adhering liquid. As the crystal moves through the purification section, this impure liquid is purified by mass-transfer to the free reflux liquid. Dispersion occurs in this reflux. -98

-99Column crystallization of a system which forms a sol.Lid so.ut..or, is essentially a process of melting and recrysta.liLzation,, As the crystal formed in. the freezing section moves through the pur.ification section, it becomes unstable. Heat-transfer from the warmer i.fquid melts the unstable crystal. This heat-transfer takes place rapidl..y, but the exchange of material. between the old crystal and the refi.ux l.qu:.d i.s limited by mass-transfer considerationso As in the eutectic system, cor,siderable dispersion occurs in the reflux liquid. In either type of system, however, both dispersion and masstransfer occuro In keeping with this result, values for effective diffusivity and mass-transfer coefficient, which were determined for the first and third chemical system mentioned above, were in good agreement with literature values taken from other types of studies. The effects of agitation on diffusivity were also in good agreement with predictions of literature correlations.

CHAPTER VIII SUBJECTS FOR FUTURE INVESTIGATION Data obtained during the present study, as well as previous data(, show that in the regime of operation thus far achieved, dispersion within the liquid phase is the dominant effect in column crystallization. The result of dispersion is of course to limit the separation otherwise being produced by the phase separation within the freezing section and by the mass-transfer which occurs within the purification section. Studies should be undertaken to find ways of reducing the effect of dispersion in relation to that of mass-transfer. One obvious way of achieving the desirable reduction at total reflux would be to produce a higher flux of crystals. This is seen by examination of Equation (16) which defines H, the effectiveness of the column as a separating device. A small H is desirable. This could be attained by increasing L to the point at which H is a minimum with respect to flux. Thus a study to determine the factors which limit the flux of liquid and solid through the column should be established. A second means of reducing the effect of dispersion is to reduce the diffusivity. This could be done by reducing the three variables which cause agitation; rate of rotation, rate of oscillation, and stroke of oscillation. Previously reported data indicate that there is an optimum set of these three variables. That is, reduction below certain values for each parameter is detrimental to separation. A method of reducing the. optimum value of D should be sought. Such -100

-101a method might include changing the configuration or the orientation of the crystallizer.

BIBLI OGRAPHY 1. Albertins, R., Ph.D. Dissertation, The University of Michigan, 1967. 2. Albertins, R., W. C. Gates and J. E. Powers, in Fractional Solidifica.tion (M. Zief and W. R. Wilcox, eds.) Marcel Dekker, Inc., New York (1967). 3. Anikin, A. G., Dokl. Akad. Nauk SSSR, 151, (5), 1139 (1963). 4. Anikin, A. G., Russ. J. Phys. Chem., 37 (3), 377 (1963). 5. Arnold, P. M., U.S. Patent 2, 540, 977 (1951). 6. Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, 2nd Ed., Oxford University Press, London, 1959. 7. Cohen, K. J., Chem. Phys., 8, 588 (1940). 8. Findlay, R. A. and D. L. McKay, Chem. Eng. Progr. Symp. Ser., 25, 163 (1959). 9. Findlay, R. A., U. S. Patent 2,683,178 (1954). 10. Findlay, R. A., U. S. Patent 2,676,167 (1954). 11! Findlay, R. A., U. S. Patent 2,855,100 (1958). 12. Findlay, R. A., U. S. Patent 2,898,271 (1959). 13. Findlay, R. A., and J. A. Weedman, in Advances in Petroleum Chemistry and Refining, Vol. 1, (Kobe, K. A., and John J. Mcketta, Jr., eds.), Interscience Publishers, Inc., New York (1958) p. 119. 14. Frevel, L. K., U. S. Patent 2,659,761 (1953). 15. Gates, L. J., unpublished data. 16. Green, R. M., U. S. Patent 2,765,921 (1956). 17. Hachmuth, K. H., U. S. Patent 2,894,997 (1959). 18. Hartland, S., and J. C. Mecklenburgh, Chem. Eng. Science, 21, 1209 (1966). 19. Hasselblatt, M. M., Zeit. f ur Physik. Chem., 83, 1 (1913). 20. Hayford, D. A., Ph.D. Dissertation, Virginia Polytechnic Institute, 1961. -102

-10321. Jones, S. C., Ph.D. Dissertation, The University of Michigan, 1962. 22. Kuster, F. W., Zeit. fW.r Physik. Chem., 8, 577 (1891). 23. Lange, N. A., ed., Handbook of Chemistry, 10th Ed., McGraw-Hill Book Co., New York, 1961. 24. Leva, M., Fluidization, McGraw-Hill Book Co., New York, 1959. 25. Lewis, J. B., Chem. Eng. Science, 3, 248 (1954). 26. Li, N. N., and E. N. Ziegler, Ind. Eng. Chem., 59 (3), 30 (1967). 27. McKay, D. L., G. H. Dale, and J. A. Weedman, Ind. Eng. Chem., 52, 197 (1960). 28. McKay, D. L., in Fractional Solidification, (M. Zief and W. R. Wilcox, eds.) Marcel Dekker, Inc., New York (1967) p. 427. 29. McKay, D. L., and H. W. Goard, Chem. Eng. Progr., 61 (11), 99 (1965). 30. McKay, D. L., U. S. Patent 2,823,242 (1958). 31. Moon, J. S.,Ph.D. Dissertation, University of California, 1964. 32. Powers, J. E., in Symposion Aber Zonenschmelzen und Kolonnen Kristallisieren (H. Schildknecht, ed.) Kernforschungszentrum, Karlsruhe, 1963, p. 57. 33. Quigg, D. J., U. S. Patent 2,890,239 (1959). 34. Reid, R. C., and T. K. Sherwood, The Properties of Gases and Liquids, 2nd Ed., McGraw-Hill Book Co., New York, 1966. 35. Schildknecht, H., presented at Symp. on Crystallization, 56th Nat'l. Meeting, A.I.Ch.E., San Francisco, 1965. 36. Schildknecht, H., Anal. Chem., 181, 254 (1961). 37. Smoot, L. D., and A. L. Babb, Ind. Eng. Chem. Fund., 1, 93 (1962). 38. Tarr, T. A., U. S. Patent 2,874,199 (1959). 39. Thomas, R. W., U. S. Patent 2,854,494 (1958). 40. Thorsen G., and S. G. Terjesen, Chem. Eng. Science, 17, 137 (1962). 41. Timmermans, J., Physico-Chemical Constants of Pure Organic Compounds, Elsevier Publishing Co., Amsterdam, 1950.

-lo442. Washburn, E. W., ed., International Critical Tables, 1st Ed., Vol. IV, McGraw-H5ill Book Co., New York, 1928. 43. Weedman, J. A., and R. A. Findlay, Petrol. Refiner, 37, 195 (1958). 44. Weedman, J. A., U. S. Patent 2,747,001 (1956). 45. Yagi, S., H. Inove, and H. Sakamoto, Kagaku Kogaku, 27 (6), 415 (1963).

APPEND'IX Al DETAI:LED DESCR:IPT IONS a Equipment The equipment used in this study was very smi'i.ar to that previously described by Albertins(1) Its main parts were t'e column itself, a drive mechanism, a refrigerant bath, electrical contro'i and measuring devices and an electronic temperature controi li ero Tnhe column used i.n the present study is illustrated in Figure 2, The column was formed from a 32 mm Pyrex glass tube 54~5 cm Tlong. This tube enclosed a 0~497 inch stainless-stee:l shaft and a stainless-steel. spiral.~ The spiral fit tightly around the central. shaft, and loosel.y along the gl.ass column. The bottom of the glass column consisted of the freezi.ng section (Fo S ) with a short "dead space" below it. Th.e F,,S. was defi.ned on. the inside by the gi.ass column and on the outside by a glass jacket of 48 mm Pyrex tubingo Two glass jackets were used in. t'h.is study,, One was 5o0 cm long, the other 700 cm. The top and bottom of the F S, were formed by nlyon rings wlich held the g.ass jacket in piacee Each. r.ing contained two taps through which refrigerant couild be passed,, One of th.ese taps was positioned radiall.y, the other tangentially.y The coon.tacts between the various pieces of the Fo S. were sealed wi.th neoprene O-r..rngs,, The sma:l1. "dead space" below the FoS. was uninsulated, and exposed to ambientO It was surrounded by approximateiy i.1:i ft,, of ti.ghtly wound resistance wireo Defini.ng thnis dead space at the bottaom, and forming the bottom of the column, was a nyil.on p'lugG Thi.s p..ug fi.t -105

-106tightly around the stainless-steel shaft and loosely into the glass column. The seals around this plug were formed by Viton 0-rings. There was a small drain through the bottom plug. This drain was capped with 1/16 inch stainless-steel tubing which was closed at one end and which could be pulled from the nylon plug. The purification section of the column crystallizer was of various lengths as determined by the position of the melter relative to that of the stationary freezing section. 3.5 cm above the F.S. and at 5 cm intervals thereafter there were sample taps in the wall of the glass column. Each consistedofanopening through the column wall. This opening was narrow (about 1/16 in diameter) at the spiral end and about 1/4 inch at the outside. Into each tap a rubber septum was placed to form a seal. An 18 gage hypodermic needle, with a 3/4 inch square-ended tube, passed through each septum toward the spiral. The end of the needle was within 1/8 inch of the spiral when there was no force on the needle. Each needle was wrapped with about 3 feet of resistance wire, and was capped with a rubber septum like the ones closing the taps. The resistance wire on the seven taps was connected in series to form a total resistance of 15 ohms. The purification and melting sections were nearly surrounded with close-fitting polyurethane insulation at least 2 cm thick. A gap of less than 2 cm, for the entire length of the column, permitted visual observation of the operation of the column. This polyurethane insulation was surrounded by an air space from 1 to 5 cm in thickness which was in turn defined on the outside by polyurethane foam 5 cm

-107thick~ The top and bottom of this air space was enclosed by polyuretnane foam 3 cm thick. A cross-section of the column and the layers of,nsuiation is illustrated in Figure 350 The melter in the melting section was a Chroma'.ox 75W cartr.idge'heater, Model. No. C301 INY. This heater'had a resi.stance of 1i.94 onms, was 1/2 inch in diameter and 6 cm long. I:t was welded t.o the stainl ess steel. shaft which passed from the bottom of the column t hrough tne rnylon. plug, the freezing section and the purification section. The position, of the me:lter within the co.lumrn. an.d therefore the length of the purification section, was varied,, The bottom of the melter was more than 23 cm and less than 35 cm from the top of the freezing sectiono The glass column forming the crystallizer extended above the mei. ter. The top of this column was wrapped with 9 ohms of resistance wire to form an auxi:l.liary heater. If the mel.ter could not mel.t all the crystals because insufficient energy was suppiied, the aux1la..i.ary iheater could be used to re-establish an energy bal.ance qui.ct..liy., The staini.ess-steel spiral., which extended from the freez:ng section through the glass column and out its top end, had a mod.i.f..i.ed,.l.enticular cross-section. This cross-section is sketched be:.ow, The spi.ral, which had 25 turns in 23.3 cm, was suppli.ed by Speci.alty Design Co,, Ann Arbor9 Michigan,, The spiral. was connected to a drive meclhani.sm wlh..ch permitted continuous variation of rate of rotation, and frequenc.y and stroke of oscillation. This drive was supplied by The Upjohn Co., Ka.iamazoo, Mienigan. o

INSULATIO N Figure 35. Cross-sectional View of Column with Insulation in Place.

-109RADIUS = 7/32 in., —0. 252 in. -'''0. 292 in. Figure 36. Cross-section of Wire from which Spiral was Wound.

-110Water baths, in which the temperature of the fluid could be regulated, served as sources of refrigerant. These baths had a continuous flow of tap water through them. Electrical power to the melter was adjusted by means of a Variac V20H variable transformer. This transformer was connected to the wall circuits through two rheostats connected in series. Large changes in the main transformers setting were thus required to effect small changes in the voltage applied to the melter. This voltage was measured by two meters which had been calibrated against a Siemens-Halske voltmeter previously calibrated in the Electrical Measurements lab of the University of Michigan by Dr. Mosher. Electrical power to the auxilliary heater at the top of the column, to the sample taps, and to the heater around the dead space below the freezing section was supplied by a 250 W Lionel transformer, model ZW, which was connected directly to the wall circuits. A 4-channel, electronic, phase-shift, proportional controller was used to control the temperature in the water baths, and thus to the annulus of the freezing section. This device maintained the temperature of the water to about + 0.05 Co~ This value is estimated from the fact that the reading of mercury thermometers, graduated in units of 1 C~, and reading 5.5 C~/cm, did not show any observable variation in temperature, The controller controlled on the resistance of a 500 ohm, disk thermister (Fenwal model JB25JI)which was immersed in the bath. b~ Operating Procedures The procedures followed in operating and sampling the column are described in this section.

-111The column, having been cleaned with benzene and dried, was assembled and put in place. The sample taps, with new septums at each end of each hypodermic needle, were set into the column. The outer layer of polyurethane foam, including the top and bottom pieces, was put in place~ The inner jacket of foam insulation was not wrapped around the column. Air, heated to 65~C or more, was circulated in the space between the column and the outer enclosure for two hours or more. While the column was being heated by the air, the material. to be charged to the column was heated in a steam bath. Also during this period the water baths, used as the source of refrigerant, were heated. The temperature to which the baths were raised was sufficiently high that, when introduced, the change would not freeze in the freezing section. Just before the charge was added to the column the refrigerant pump was turned on. Also power was supplied to the melter and to the auxilliary heater at the top of the col umn. In addition, the hot air was turned off and the front of the outer insulation was removedo The spiral. drive was turned on and the mixture of BNB and CNB was introduced The amount of charge varied with the position of the melter within the col.umn. However, for all. runs, the i quid Level on charging was between 2 and 3 cm, above the top of the melter. This prevented the ilevel of the slurry, which formed as the liquid was cooled, from dropping below the top of the melter. With the column operating, the inner jacket of insulation was put, in. place and the front of the outer jacket was kept off. The tem-, perature of the water (the refrigerant) was allowed to drop slowly to

-112the desired temperature. Crystals formed in the freezing section and filled the bottom of the column. As the level of the slurry-liquid interface rose, the front of the outer jacket of insulation was added. The power supplied to the melter was increased gradually as the interface between the slurry and the clear liquid rose. When this interface had reached the melter, or about the middle of it, maximum power for that given run was being supplied. At this same time, the entire front part of the outer insulation was put in place. This piece was then disturbed only occasionally to observe the level of the crystals. As the concentration gradient established itself within the column, the crystal rate changed. Consequently the power input to the melter decreased with time. A proper balance between the heat loads in the freezing and melting sections was maintained by observing the level of the crystals. After about 4 hours no changes in power input were required. After 2 to 3 hours of running with no changes in power the liquid in the column was sampled. While a run was in progress the syringes and needles to be used in sampling were washed in boiling water containing a phosphate soap, rinsed in distilled water and then in acetone, and finally dried. During this same time 3 cc, glass sample vials were weighed to + 0.0001 gm, and capped. About 15 minutes prior to sampling the dried syringes were assembled and placed in a steam bath. Ten minutes later the front part of the outer jacket of insulation was removed and power was applied to the sample taps. This energy melted any solid which had formed within the syringe needle which comprised the tap.

-113After the tap heaters had been on for five minutes the drive for the spiral, was turned off and the samples were taken. The top tap was al-ways sampled first and the bottom one last. This prevented the aet of samplhng from disturbing the i'Lquid at the taps yet to be sampled. As eahc'h, sample was taken, it was injected into the previously weighed sample vials. The entire sampling procedure usually took 3-1/2 minutes. When the last sample had been taken, either the column was draJned if no further runs were to'be made, or the drive was started again and a new run made. The vials containing the samples were weighed and the weight of each sample was determinedo Samples usually weighed about 0.1 gram. Each sample was then diluted with methyl chloroform (MC) (Eaton Chemical, Uninhibited trichloroethane) capped and placed in a freezer maintained at -23~C, The dilution with MC was performed as follows. MC at room temperature was measured with a O.5ml syringe which was graduated in units of 0o01 ml. A volume of MC in. ml equal to twice the weight of the samrple in grams was added to each sample. c, Analytical Procedures Samp:les of liquid extracted from the columr, were analyzed by gas chromatography. A model 1522-lB Aerograph chromatograph (Ser. No. 762-0068) was usedo Repeat analyses, made either sequentially or up to four months apart, indicated that the technique was highly reprodu. r i ble The entire analytical procedure is described in detail i n t+n s sec~ti on.

-114Chromatographic columns of 1/8 inch, thin-wall, stainlesssteel were used. These were 5-1/2 feet long and were filled with 2.1 + 0.2 grams of packing. The packing material was 30/60 Chromosorb A, AW (Johns Manville, lot 02746), seived to 40/45 onto which 10 weight percent of active liquid phase had been deposited. The liquid phase was Carbowax 20M (Wilkens 82-1115). It was deposited by evaporating at room temperature the methanol (Baker, Reagent 9070 lot 32776) from a 100o solution in which the Chromosorb was slurriedo Prepurified nitrogen (Matheson, FG-4908) was used as the carrier gas. This gas gave greater responses and shorter residence times than did helium. The carrier flow was 50 + 5 cc/min at an inlet pressure of 20 psig. Flame ionization detectors were used. These were cleaned regularly as the materials being analyzed produced considerable residue and corrosion, Matheson prepurified hydrogen (FG-4896) was burned in the detectors. Air, supplied from the room through a prepackaged molecular sieve by small fish-tank pumps was also used, The temperatures of the various parts of the chromatograph were as follows, The oven containing the columns was at 197~Co The injector was at 2300C and the detector block at 2250C. Output from the detectors, after amplification, was recorded on a Brown (Model 15312V-X-3OYlO, Ser. No, 324587) recorder. Simultaneously, an Infotroncis Model CRS-lOH (Sero No. 1728) integrator measured and printed out the peak areas, A sample analysis is shown on Figure 537 The integrated areas, together with the corresponding electrometer attenuations, are also presented on the figure. The quotient of the BNB

STANDARD ATTENUATION INTEGRATOR TIME COUNTS 1 ~ ~3582 7775 6 32 390 2:00. _ ~ ~ ~~~~~~1 4891 1 6562 lC 32 390 1.50 ~Iv.__..~_ 5750 [1 # 5650 7 32 371 1:40 5563 12 2717 7 32 380 5:25 - if —,rle Vhromr lI ~

peak area divided by the sum of the peak areas was determined to define a calculated mole per cent BNB. In analyzing samples, 0.8 p1 of material, prepared as described:in Appendix Al-b were injected. Samples were analyzed in the following order: 6, 1. 3, 7, 4, 2, 5. After a set of samples had been run, two or three standard samples, which had been prepared by weighing in quantities of BNB, CNB and solvent (methyl chloroform-MC), were analyzed. From the analyses of these known samples, a calibration of actual composition versus experimentally determined composition was prepared. The linearity of these calibrations is demonstrated in Figure 38 and in Table X. Such calibration curves were finally used, with the chromatographic analyses of the unknown samples, to determine actual composition. This latter composition used in all treatments of the data.

-117TABLE X COMPARISON OF ANALYSES OF STANDARD SAMPLES Sample Analysis of Standard Sample mole fraction BNB by weighing by GC 5 0.072. 065 6 o0.313 0.315 c 0.430 0.427 7 0.502 o.505 B o.600 0.595, o.6o8 D 0.742 0.743 A 0.880 0.879

1.00 080 0 /I: 0.601 — m zi E da- -— ~-.. DRAWN WITH SLOPE -1.0 0.20 - 0 0.20 0.40 0.60 0.80 1.0 MOL. FR. BNB BY G.C. ANALYSIS Figure 38. Linearity of Calibr tLot; G: C1i'orl' U!,orpho

APPENDIX A2 TESTS OF OPERATIONS a. Attainment of Steady-State Al.l t',e data reported in this dissertation were taken at steadystate. The procedure by which the time required to reach steady-state was determined is described in this section. The column was charged, operated and sampled as described in Appendix Al. The conditions of operation were chosen so that a severe test for steady-state would be made. These conditions included a low rate of agitation, a high crystal rate, and a slow approach to the ultimate crystal rate. When the first set of samples had been taken, after 6 hours of operation, the drive to the spiral was restarted and a second run was made at the same conditions as the first. This second run lasted 2 to 3 hours from the time that equilibrium was reestablished. A second set of samples was taken about 4 hours after the first set. The samples from the two runs were analyzed as described in Appendix Al-c,. To determine whether the two runs were the same a plot of composition - run!. vs. composition - run 2 was made. Each point on the plot thus represented two samples taken from a single tap. If there were nro difference between the runs, then the slope of the line drawn through the seven points would be one. Figure 39 is such a plot. The s:lope of the iine, as determined by a Least squares analysis, is 1.012. Cl: earl y there was no significant difference between the profiles at 6 and at. l0 hours for the test illustrated. A second test for steadystate, made with a lower crystal rate, also showed no significant -119

-1200.10 0.08 0 0 z 0.06 0 >- 0.04 0.02 0 0<. I3. I I 0.02 0.04 0.06 0.08 Yo-Y FOR RUN A Figure 39. Illustration of Attainment of Steady-State.

-121difference between the two runs (slope = 1.024). As a result of these two tests, all data were taken at least 6;houzis after. charging. b. Constancy of Crystal Rate The model which describes column crystallization assumes that the flows of crystals and reflux liquid are independent of position in the column. Three pieces of evidence which support this assumption are discussed in this section. However, no direct tests of constancy were made. The best evidence is that the model agrees with the data. Equation (22) presents the model as derived. Y - Y- ( /- X) o [/ A/ L /F] (22) For the most part, the term E/L was much greater than the term L/F. Thus, the concentration gradient in the liquid phase is very nearly proportional to the crystal rate (See Equation (75)). \/ ~ / - ( \ \ / / E (75) If L had varied appreciably, the concentration gradient would not have been linear. The crystal rate was measured at the top of the column by determining the power supplied to the melter. As the material in the column was more than 20C~ above room temperature, one would expect this measured value would be the maximum value. Thus, the "flooding" phenomenon (Appendix A3-a) which occurred at high crystal rates wodld have been most severe at the top of the column. In fact, there was no apparent trend in the flooding behavior.

-122When flooding occurred, it started essentially uniformly throughout the column. It seems reasonable, therefore, that there was no change in the crystal rate within the column. The last evidence confirming the constancy of crystal rate occurred during the start-up period. Consequently, it is a transient effect and should not be weighed heavily. However, it is consistent with the two previous results. Two runs were made in which the liquid charged to the column was cooled to the freezing point throughout the column. The insulation was put in place, the melter turned off, and the crystal bed allowed to form. The time required for the bed to form was determined as were the eventual crystal rate and the volume fraction solids. From this knowledge, the time required to generate the volume of crystals which were held in the column was determined. For both runs, this latter time was about 20% more than the actual time required for the bed to form. It can be concluded that crystals were formed more rapidly than the final rate would indicate. There was either an appreciable heat leak through the insulation, or the rate of formation within the freezing section decreased with time. The latter conclusion is certainly reasonable. When the run was started, the composition of the liquid in the freezing section was about 35% BNB. As a run progressed this value dropped to about 30%. As the temperature of the refrigerant was main, tained constant throughout a run, the temperature difference between the slurry and the refrigerant decreased with time. Thus a lower crystal rate was produced at the end of a run than at the beginning.

-123This last discussion may not be satisfying in that it does not directly indicate that there were no heat leaks from the column. However, the described difference in times can be explained qualitatively by the mentioned phenomenon, Together with the first two results described in this section, it is felt that an adequate case for the constancy of the crystal rate is presented.

APPENDIX A3 RESULTS NOT PERTAINING TO IDENTIFICATIONS OF MECHANISMS The previous sections of the report concern the determination of the mechanisms involved in the column crystallization of solid sol.utions. This section presents results which do not bear on that determination. These results concern: l. the maximum crystal rate, 2. the effect of agitation, and 3o the effect of the spiral. a. Maximum Crystal Rate The maximum obtainable crystal rate was influenced by hydro — dynamic factors within the column and by the construction and operation of the freezing section. The factors relating to the freezing section were the surface area and the heat-transfer coefficient between the refrigerant and the liquid being frozen. Hydrodynamic Factors Jackets which enclosed surface areas of 22~4 cm and 160o cm2 in the freezing section were used in this study. The maximum crystal rate with the larger jacket in place was 0o040 grams solid/sec. This is equivalent to 0.027 cm3/sec and to o0.0o6 cm3/cm2-sec. These values are low in comparison to those obtained by other workers. McKay reportsO.6 cm3/cm2-sec for a column without a spiral, operating on cyclohexane-isooctane. Albertins(1) reported a maximum crystal rate of 0.100 gram/sec.o for a spiral column of

-125similar dimensions to that used in the current study. This value i.s equivaient to 0.100 cm3/sec and 0.16 cm3/Cm2-sec, Attempts to increase the crystal rate beyond the maximum produced voids throughout the column. The crystals seemed to be more tightly packed than was usual~ The liquid which formed in the melting section could not percolate through the dense bed to the freezing section. This situation is similar to flooding in a gas absorption. At high gas rates, liquid is blown from the absorber. Similarly, at high crystal rates, the liquid is "blown from the purification section". The low value obtained for the maximum crystal rate in the present study can be explained only qualitatively. It is suggested that there were smaller crystals in BNB-CNB than in the benzene-cyclohexane used by Albertins. No measurements were taken to support this contention. However, three observers at various times suggested that the BNB solids appeared small. Leva(24) reported that the maximum flow of liquid through a packed bed rises with the square of the particle diameter. Thus the ratio of the maximum flows mentioned above could be explained by a ratio of particle diameters equal to i07@ Surface Area The maximum obtainable crystal rate when the small jacket was in place was 0.032 gram/sec. The limiting factor in this case was the surface area of the freezing section. At the maximum crystal rate, crystals began to appear on the inner wall of the freezing sectiono If the crystal rate were not reduced, this buildup of crystals soon choked the column~ In a choked column,

-126the spiral could not break the crystals from the wall and very quickly it no longer rotated freely. At such time, the whole column began to rotate. When the maximum crystal rate was obtained, the temperature difference between the freezing section and the liquid was 5.1 + 0.2C~. It is believed that at this temperature difference the temperature of the inner surface of the freezing section was low enough to permit rapid initiation and growth of crystals. The growth generated large crystals which collectively interferred with the rotation of the spiral. The alleviation of this difficulty was the use of a larger jacket, which provided an equal crystal rate from a higher inside surface tew- ure..fer Coefficient?.arly in the experimental work, crystal rates no higher than.ec could be achieved. The factor restricting the crystal time was a high, local heat-transfer coefficient. i the column was constructed, the taps for the refrigerant radially. Thus, when the refrigerant entered the fz I, it impinged directly on the inner wall. A high local heE efficient was produced. As a result, a heavy build-up of t als occurred on the wall inside the column at rather low t'ferences. These crystals soon restricted the rotation of the,tion of this problem was the construction of new taps for.nt. These taps were placed tangentially so that

-127liquid flowed around the column rather than directly against the inner wallo b. Effect of Agitation The effect of agitation on two variables, separation and density of slurry, is discussed in the following paragraphs. Separation It has been mentioned previously (Chapter V-A) that the separation increased as the agitation decreased. This is to be expected if dispersion in the liquid phase is the largest factor limiting separation. The results discussed before were, however, all taken from runs in which the slurry in the column appeared to be uniform. That is, there was no settling of the crystals between turns of the spiral. Such runs were therefore made with an agitation higher than some minimum allowable value. When a run was made with less than this minimum agitation, the separation decreased significantly. For such runs, the crystals settled measurably. There was in effect then two non-mixing bulk phases. One was the slurry moving upward toward the melting section. The other was the free liquid moving down. These two phases did not mix together so there was little opportunity for mass-transfer between phases. Table XI presents data which describe the effect of agitation on separation. In Run 20 segregation of the crystals occurred. It is evident that a lower separation occurred in this run.

-128TABLE XI EFFECT OF SLURRY SEGREGATION ON SEPARATION Run Agitation Separation per cm of column OPM RPM weight fraction per cm 21 43 45.0031 20 31 29. o0 4 Density of Slurry It was observed several times that the density of the slurry (the volume fraction solids) decreased as agitation increased. This observation is based on the fact that the interface between the crystal slurry and the liquid above it rose or fell rapidly as t aI.*' - t; in'_ui was increased or decreased, respectively. No data concerning the extent of the rise and f'all were taken, but the effect was real and reproducible. c. Effect of KSpif.al. Previous investigators have referred to the sp):i' a i in agitator and as a conveyor. The results of the present study indiirate that the latter description is not correct,. When a run was started, large crystals sometimes formed. These were generally conveyed to the me]lting section quil v-aiiiy:l. However, once an appreciable bed of crystals had been establi;shed, no large crystals were formed and a fairl]y sharp interface between slcurryand clear liquid was produced. The position of this interface could be

-129maintained any place in the column by balancing power input in the melting section and the heat load in the freezing section. The interface dropped when the melter power was too high, and it rose when the power was too low. If the two heat loads were dropped to zero, the interface remained stationary. It did not rise as one would expect if the spiral were acting as a conveyor.

APPEND IX A4 AREAS OF FUTURE WORK PERTAINITNG1 TO COLUMN OPERATION The resul'ts described in the previous appendix raise several questions wnich mignt form the basis for future studies,, The questi.or.s are discussed briefly in thiis section. The results presented in the previous section indicate that a crystal rate, greater than that corresponding to HnTinimum which is defined as Lmin, could not be produced. However, Albertins did attain a crystal rate equal to Lmin. It was also suggested in this report that the maximum crystal rate, Lmax, is fixed by the diameter of the particles. It would seem advantageous to determine the parameters which.imit the crystal sizec Then perhaps rates equal to Lmin could be achieved regularly~ Low separations accompanied the segregation which occurred at very low agitation. The adverse effect of segregation might be eiimin.ated if a column crystallizer were oriented, not vertically, but d:"agon. ally Diagonal oriertation would cause the crystals to settle a.l.ong the column wall rather than on the spiral. The settled slurry would thus form a porous seal through which thle reflux liquid would nave to pass to reach the freezing section~ Therefore, although there would still be segregation between. the slurry and the free -.iquid, tlhe by-passing present in a vertical column would be eliminated.,, The- result would be similar to the effect achieved by an Archimedes' screw which will deliver water when set diagonally2 but not when set vezt rtcll.y. -130

-131Diagonal operation might have another benefit. If, as described above, interphase contact were maintained by diagonal positioning, then the agitation currently used to effect such contact could be drastically reduced. This reduction would limit the diffusivity within the liquid. Because dispersion is currently the predominant effect limiting separations, a reduction of diffusivity would increase the separations almost proportionally. Reducing agitation would increase separations in another way. It was mentioned earlier that the crystal bed expanded as agitation increased. Thus the separation was affected adversely by increased dispersion (TL increased) and by decreased mass-transfer (a decreased).

APPENDIX A5 CALCULATIONS a. Analysis of Data from Azobenzen.e-Sti.lbene The model including mass-transfer and li.quiddispersion (Equati on (22)), which was developed in this report by assuming a linear equilibrium relationship, was applied stepwise to data for a system in which this relationship was not linear. The calculjated profile and the experimental one agreed very well, Data presented by Powers(32) represent the overall composition in a column crystallizer in which azobenzene was separated from stilbene. These data are the weighted average of the compositions in the liquid and in the solid phaseso As Equation (22) applies to the liquid phase, Powers' data cannot be analyzed directly. To determine a liquid profile from Powers' overall profile the following assumptions were made. 1) The volume fraction liquid was 2/3. This is about the value obtained in the present study and by Albertins. 2) The liquid and solid were in equilibrium. This gives the maximum reasonable difference between the compositions in the liquid and solid phases. PowersT application of an equation describing a liquid profile to data which represent overa.ll compositions implies that liquid and solid have the same compositior.o These two assumptions define a unique relation between overall composition and liquid composition~ This relation was used to prepare Table XII in which Powerst data are compared with equivalent liquid compositions. -132

-133TABLE XII COMPARISON OF DATA REPORTED BY POWERS(32) AND PROFILE CALCULATED FROM EQUATION 22 Position Powers' Data Equivalent Calculated of Overall Liquid Liquid Composition Composition Profile cm Weight Fractions 48 0.99 0.99 45 0.99 O. 990 42 0.97 0.98 4o0 0.97 o.968 39 o.94 0.97 36 o.91 0.94 35 0.93. 908 33 0.82 o.88 30 o.67 0.77 0.747 27 o.48 o.60o 25 o.49. 451 24 0.35 0.41 20 0. 08 0. 070 18 0.02 0.02 15 0.01 0.01 0.000 12 0.00

-134In order to apply Equation (22), values of L) (DrlAp) and (KaAP) must be known. Unfortunately, Powers does not give any clue to these values in his paper. The following procedure was used to make estimates. The present study indicated that in many cases the term DrTPA/L was 5 to 10 times greater than L/KaAp. Powers indicated in his paper that the sum of these two terms was about 4. These approximations were combined to give Equation (76). -\ 3. 5 + A /2 (76) Thus Equation (22) has the form given in Equation (77). -x V= /?- (77) A value of Yo equal to 0.990 at h = 45 cm was assumed. (Y-X*)o and m were determined from the phase equilibrium. These values were assumed to be constant for a column length of 5 cm. Then, fora LZh = 5,aAY was calculated, and Y at h = 40 was determined. This stepwise procedure was applied repeatedly for successive intervals of the column. The results of these calculations are summarized in TIable X1I. As it was mentioned earlier, the profile determined by tK'!Ls procedure was in good agreement with the liquid profile calculated from Powers' data. In fact, the determined concentrations lay between those presented by Powers (which assume equal compositions in both phases) and those assumed here (which assume equilibrium between phases).

-135b. Correlation of Phase Equilibrium Data Bromonitrobenzene-chloronitrobenzene Data of Hasselblatt(19) taken from International Critical Tables, Vol. IV, p. 122, were used in this study. These data are in substantial agreement with those presented by Kuster(22) and with tests made during the course of the present study(l5). Table XIII presents the temperature-composition data. Figure 3, presented earlier, is a second representation of these data. It was desired to convert these data to a more easily used form. This was done by statistically fitting curves through the data points. The statistical equations were then used to calculate temperature-composition data with a much smaller interval than given. As demonstrated on Fig. 26, the liquidusline, from 0 to 70 mole per cent BNB, was fit very well by Equation (78). -e r-. e. - r-i o p t e ( i- Y ) The solidus line required a second power term to give a good fit. Equation (79), again derived statistically, fit the data adequately from 0 to 70 mole per cent BNB. T.-.,~'-lD-\t1- )',':i, (lo (i-X))2 (79) This equation is shown on Figure 40. The liquidus and solidus lines at the BNB-rich end of the diagram were not statistically fit. However, as demonstrated on Figure 40, both lines are described very well by an equation of the form \ - A q N3 -t (Locke 4 Y~ti'\Ic7%)(80)

-136TABLE XIII PHASE EQUILIBRIUM DATA OF HASSELBLATT(19) FOR SYSTEM BNB-CNB Compos iti on Tempera ture Solidus Liquidus Mole Fraction ~C ~C BNB 0.0 o 0 44.6 44.6 0.05 44.7 44.8 0.10 44.9 45.0 0.20 45.3 45.5 0.30 45.7 46.0 o.40 46.3 46.7 0.50 47.0 47.4 o 60 47.8 48. 0.70 49.0 49.5 0,80 50.7 51.1 0.90 52.3 52.5 0o.95 53.1 53.2 1.00 54.o 54,o

54 BROMON I TROBENZENE 5I 0 CL 0 a: w 48 CH LORRONIT TRO BENZENE~ 45 I I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 30. 40 50 60 70 80 90 MOLE % Figure 40. Analysis of Phase Equilibrium Data.

-138Tables of X and Y vs. T and of X* vs. Y, were made from the equations which were derived. Figure 4 is based on such a table. The assumed linearitybetween X* and Y can be tested with the figure. It is certainly clear that for a narrow range of compositions, on the order of 0ol weight fraction, either m is in fact constant, or m can be approximated by a constant with only little error. Azobenzenzene-Stilbene Data presented by Powers(32) were used for this system. And, as in the case of BNB-CNB, the data were converted to a more useful form by fitting them with logarithmic equations. The data, and the resultant equations, are shown on Figure 41. It is clearly seen that the liquidus line at the stilbene end of the diagram and the solidus line at the other end were of the form given in Equation (80). The other ends of these lines were fit by quadratics of the form given in Equation (81). T = A + B(ln(mole fraction)) + C(ln(mole fraction))2 (81) The statistically evaluated constants for these several equations are presented in Table XIVo As in the case of BNB-CNB, the statistical equations were used to make tables of X and Y vs. T, and of X vs. Y. The data on Figure 26, which presents XX vs. Y, were taken from these tables. Clearly for this system X* vs. Y is highly non-linear. c. Correction of Diffusivities The diffusivities calculated from Equation (18) are based on the vertical distance between sample taps. Liquid does not follow this

-139STILBENE 120 - LIQUIDUS 110 - ) 100 0 a. 90 80 SOLIDUS AZOBENZENE 40 50 60 70 80 90 WEIGHT % Figure 41. Analysis of Phase Equilibrium Data.

TABLE XIV CONSTANTS USED IN CORRELATION OF PHASE EQUILIBRIUM DATA Argument of Logarithm A B C Xazobenzene 69.5 -31.7 Ystilbene 125.3 26.4 Xstilbene 124.7 89.6 69.8 Yazobenezene 69.6 -94.1 -64.1 path, however, but flows spirally around the central shaft of the column. Thus, the diffusivities calculated directly must be corrected. The correction factor is derived in this section. In order to determine the effective path length per cm of column, the dimensions of the spiral must be considered. As described in Appendix Al-a, the spiral was 2.58 cm OD by 1.26 cm ID. The pitch was 0.92 cm per turn and the solid volume of the spiral was 0.93 cm3/cm. Thus, the void volume of the spiral was 3.23 cm3/cm and the area through 2 which liquid flowed at right angles was 0.5 cm The void volume is equal to the area times the effective path Length. Thus the path length was 6.5 cm/cm. This number was used to correct diffusivities, The value for diffusivity determined from Equation (18) was multiplied by 6.5 cm/cm to give the corrected diffusivity.

-141d. Crystal Rate at Minimum H H, a measure of the separating power of a column crystallizer, has a minimum as a function of L. The values of H and L at this minimum are determined in this section. The functional form for H is given by Equation (18) H - E/L + L/F (18) In this equation E and F contain terms which relate to diffusion and mas-transfer, respectively. The derivative of H with respect to L is given by Equation (82). dH/dL = - E/L + 1/F (82) When H is a minimum, dH/dL is zero. Thus the value of L corresponding to Hmin is given by Equation (83). Lmin = (EF)1/2 (83) The value of Hmin is shown in Equation (84) which is determined by substituting Lmin for L in Equation (82). Hmin = (E/F)1/2 + (E/F)1/2 (84) Thus it is seen that at Hmin the effects of diffusion and mass-transfer are equal. That is, E/Lmin = Lmin/F

APPENDIX A6 SAMPLE CALCULATIONS A set of calculations is presented in this appendix to demonstrate the methods of calculation and to put forth the values used in the several computations. a. Determination of the Crystal Rate The crystal rate was determined from the power supplied to the melter and the heat of fusion of the crystals. This latter heat was assumed to be linearly related to the composition of the liquid in the melter, and to the pure component heats of fusion. The heats of fusion were assumed to be independent of temperature. The heats of fusion of BNB and CNB are 24.3 and 33.1 cal./g, respectively. (42) From these data and the assumptions listed above, the heat of fusion of a mixture is given by 33.1-8.8Xo cal./g. The values for XO which were used in determining heats of fusion were not raw data. Rather, the least squares line through all the data was used to generate a value for Xo. Thus, for example in Run 29, Xo was calculated to be 0.707 whereas the corresponding data point was 0.702. A value of 26.9 cal/g was determined as the heat of fusion for this run. The crystal rate was calculated by dividing the heat of fusion into the power input, V2/R. Here V is the voltage supplied to the melter, and R is its resistance (194 ohms). For Run 29, V was 28.7 volts. Thus the crystal rate was: (28.7)2/[194 x 4.19 watt-sec/cal x 26.9] = 0.058g/sec -142

-143b. Determination of H The slope of the concentration profile for Run 29 (see Figure 10) was 0.088 weight fraction per 6 taps or 30.4 cm. This was determined by a least squares fit of the concentration data. The concentration at the top tap was 0.688. From tables prepared from Equations (78) and (79) a value of 0.0355 was extracted as the equilibrium difference between the composition of the liquid and solid phases, That is, (X*-Y)o = 0.0355 for Run 29. According to Equation (22), the slope of the concentration profile is given by (X -Y)o/H. By rearrangement, H is given by (X*-Y)O/slope = 0.0355 x 30.4/0.088 = 12.2 cm. c. Determination of D and K Data for Runs 29 to 31 are shown in Figure 10, plotted as HL versus L2. The intercept and slope of the least squares line drawn through these data are 0.373 g-cm/sec and 64 cm/sec/g, respectively. The intercept is E, equal to DArp, and the slope is F, equal to KaAp/m. Thus D and K can be calculated from values of the slope, intercept and a, A, p, r and m. Ar e a The area A is that through which the liquid and solid flow. ZIt is defined by the glass column, by the stainless-steel shaft, and by adjacent turns in the spiral. The width of the annular space was o.66 cm, and the distance between turns was 0.76 cm. Thus the area was 0.50 cm2.

-144Density The density of the liquid was assumed to be a linear combiration of the pure component densities. These were: (1) BNB = 1.50 g/cm3 (2) CNB = 1.35 g/cm3 Therefore the liquid density was 1.46 g/cm3 Volume fraction The volume fraction, ~, was calculated by a mass balance on the column. The amount of material charged was known, and could be accounted for in the final liquid and solid. Thus, the sum of the products of volume times density for the two phases in the slurry equalled the same product for the charge. That is, Vol. Liq x PL + Vol. sol. x pS = Vol. chg. x pL On rearrangement this equation yields Equation ( 82 ) for S' s = aV PL/(P-pL)x(Vo-%) V) (82) For Run 29, the change in volume, AV, was 8 cm3 and the volume charged, V0, was 225 cm3. The densities were 1.46 and 1.65 g/cm3. Thus, qL' equal to 1-S, was 0.718. Interfacial area The interfacial area available for mass-transfer was estimated from an approximate particle diameter. A value of 0.04 cm was used. This value was derived from the approximation previously used by Albertins (Reference 1) and the observation previously discussed that the crystals

in the present study seemed smaller.'lThe saurface area of a particle with a diameter of 0.04 cm is n(0.04 ) = 5.04 x 10-3 cm2/crystal and the volume is n(0.04)3/6 = 3.35 x 10-5 cm3/crystal The number of crystals in 1 cm3 of slurry is (1 cm3) (1-L = 0.282)/3.35 x 10-5 = 8,430 crystals From the above data, the surface area per cm3 of slurry is 8430 x 5.04 x 10-3 = 42.5 cm2 /cm3 D and K The values for A, p, ~ and a presented above are substituted into the expressions for E and F to determine D and K D is given by D = 0.373/0.50 x 0.72 x 1.46 = 0.77 cm2/sec This value is corrected for path length by multiplying by 6.5 (see Appendix A5-C). Thus D is 5.0 cm 2/sec. F is equal to 64 cm-sec/g so that K is given by K = 0.9/64 x 43 x 0.50 x 1.46 = 0.44 x l0-3 cm/sec.

APPENDIX A7 DATA -146

Charge to the Column: 0.05 Weight Fraction BNB Integrator Counts Sample Run 6 Run 7 Tap Number CNB BNB CNiB BNB 1 9213 5852 9533 5294 2 8920 5786 9380 5945 3 8627 9718 10027 6271 4 8910 6026 9483 5658 5 9315 6166 6 10274 58X80 7 8759 6097 9681 6322 Attenuation 4 1 4 1 Operating Conditions Voltage to Meter-Volts 17.7 15.4 Stroke-mm 4.5 4.5 Oscillation Rate-OPM 72 72 Rotation-RPM 67 67 Co lumn length-cm 30.3 30.3

- 148Charge to the Column: 0.95 Weight Fraction BNB Integrator Counts Sample Run 13 Tap Number CNB BNB 1 7475 11029 2 7212 11266 4 6823 10302 5 8803 10322 6 7 7468 10273 Attenuation 1 8 Operating Conditions Voltage to Me ter-Volts 28. 3 Stroke-mm 4.5 Oscillation Rate-OPM 22 Rotati on ia'te-RPM 67 C olumn Length-cm 30.3

-149Charge to the Column: 0.50 Weight Fraction BNB Integrator Counts Sample Run 14 Run 15 Run 16 Tap Number CNB BNB CNB BNB CNB BNB 1 3530 2676 6700 5038 4751 3521 2 3421 2529 6555 4944 3 3689 2593 6509 4742 5097 3095 4 3603 2463 6652 4450 5589 3023 5 3532 2367 7072 4485 5661 2846 6 3670 2186 7040 4268 5635 2680 7 7369 4080 5244 3029 Attenuation 1 1 1 1 1 1 Operating Conditions Voltage to Meter-Volts 28.9 28.9 25.1 Stroke-mm 4.5 4.5 6.o Oscillation Rate-OPM 25 25 40 Rotation Rate-RPM 67 32 46 C olumn Length-cm 30.3 30.3 30.3

-150C'ha:rge to the Column: 0.50 Weight Fraction BNB Integrator Counts arnampl.e Run 1.7 Run 18 Tap Number CNB BNB CNB BNB 5068 311.6 2 6813 4268 5436 5047 3370 4063 4 5244 2801 6892 4085 5599 2715 3443 1875 5609 2579 3612 3650 5532 2622 3544 71.85 Attenu.ati.on. 1 1 1 1 Operating Conditions fVoltage to Me te:r-Volts 20.6 21.0 Stro ke -rrm 6.o 6.o:sci.. lati. on Ra t e - 0 PM 31 31. Rotati on Ra te,-RPHM 29 29 i 1 3.lumrn Lerg th..cm 30.3 25.1

-151Charge to the Column: 0.65 Weight Fraction BNB Integrator Counts Sample Run 20 Run 21 Run 22 Tap Number CNB BNB CNB BNB CNB BNB 2 3 5053 9312 5445 9327 4 4893 7586 4695 7978 5596 9205 5 5925 8877 4980 8234 5563 8815 6 6060o 9098 6176 9320 6677 10265 7 6344 8969 6560 9061 6061 8936 Attenuation 1 1 1 1 1 1 Operating Conditions Voltage to Meter-Volts 26.5 27.2 26.4 Stroke-mm 6.0 6.o 9.0 Oscillation Rate-OPM 31 43 43 Rotation Rate-RPM 29 45 45 Column Length-cm 20.0 20.0 20,0

-152Charge to the Column: 0.65 Weigrt Fraction BNB Integrator Counts Samrl e Run 23 Run 24 Run 25 Tap Numbe. CNNB BW3B CNB BNB CNB BNB 1 4434 7632 4529 8674 4516 7807 25074 8389 50113 9050 4259 71,15 5 _52 48 841.4 4662 8420 4592 7425 4 5225 7798 4111 6797 4811 7418 62356 8562 5312 7905 5318 74c3 6 5583 6998 5510 7313 5165 7199 5587 6781 5907 8111 5179 6825 Attenuation 1 1 1 1 1 1 Ope rating Conditions Voltage tto Me ter -Vo 1ts 27.0 27.5 27,1 S t,.oke-rmm 6.o 6.o 6,o s c i la ti. on aa +e0 PM 45 45 43 o ta tion -a t.e - RPM, 41 41 78 o.lumn,e rg th.- c m 3 0. 3 303 30 3

-153Charge to the Column: 0.65 Weight Fraction BNB Integrator Counts Sample Run 26 Tap Number CNB BNB 1 4737 9171 2 4887 9098 3 4972 8777 4 5414 8757 5 5360 7969 6 2960 3913 7 5573 7405 Attenuation 1 1 Operating Conditions Voltage to Meter-Volts 26.6 Stroke-mm 6.0 Oscillation Rate-OPM 43 Rotation Rate -RPM 48 Column Length-cm 30.3

-154~Charge to the Column: 0. 65 Weight F1raction BNB Integrator Counts Sample Run 29 Run 30 Run 31.:a p I\Numbe r CNB B B B CNB BNB CNB BNB 4265 8048 4205 8151 4364 8266 2 53933 7781 3991 7,90 5665 7153 4388 8141 3810 6875 4505 8369 4 4301 7949 4511 7995 4091 7337 4507 7941 4423 7580 4183 7317 4658 7537 4659 7705 4593 7455 4918 7762 4465 7410 At-tenuation 1 1 1 1 1 1 Operating Conditions Voltage to Me ter.Vo..ts 28.7 26.8 23.3 S t:r oke -rmm 4.2 4.2 4,2 Os c i 1.l.a tion rFa te —OPM 6y 67 67 Ro ta ti on ia te -RPM 60o 6o 60 Lerg th.- cm 30.3 303 30. 3

-155Charge to the Column: 0.35 Weight Fraction BNB Integrator Counts Sample Run 33 Run 34 Run 35 Tap Number CNB BNB CNB BNB CNB BNB 1 6339 4981 6861 6552 6523 4768 2 6345 4966 6611 6253 6958 5130 3 6742 5174 7287 7023 6932 5090 4 6527 4855 6871 6171 7192 4892 6687 5059 7488 7914 7383 5014 6 7 6620 4654 6921 6231 6788 4781 Attenuation 1 1 1 1 1 1 Operating Conditions Voltage to Me ter-Volts 25.2 20.4 14.0 Stroke-mm 2.0 2.0 2.0 Oscillation Rate-OPM 130 130 130 Rotati on Rate -RPM 60o 6 6o Column Length-cm 30.3 30.3 30.3

UNIVERSITY OF MICHIGAN i3 9015 02826 0266IlIl llll 3 9015 02826 0266