UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
DETERMINATION OF THE CRYSTAL GROWTH, BIRTH OF CRYSTAL NUCLEI, DYNAMICS AND STABILITY OF THE CRYSTALLIZER By CHARLES UMEJEI ONWUEGBUNEM OKONKWO A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1982 Copyright 1982 by Charles Umejei Onwuegbunem Okonkwo This project is dedicated to my parents for their continual love, encouragement, and support throughout the years ACKNOWLEDGMENTS Glory be to God for seeing me through the many years of my educational experience. I wish to thank the chairman of my doctoral super visory committee, Dr. Hong Lee, for his guidance in this research, and Dr. Robert Coldwell for his assistance with the computer subroutine (simple). Thank you to Dr. Robert Gould and Dr. Ranganatha Narayanan for their encouragement. Special thanks go to Dr. Charles Burnap and Dr. Ulrich Kurzweg for the many fruitful discussions I have had with them during the course of my studies at this university. Thank you to Linda McClintic and Elaine Everett for their encouragement. Also sincere thanks go to Vita Zamorano for her diligence and cooperation in typing the manuscript. In conclusion, a special word of thanks is extended to my family for their moral support, prayers, love and patience. Thank you Mom, Dad, Thomas, Elizabeth, Paul, Rosaline, Francis and Fidelis for your understanding, encouragement and assistance throughout my educational experience. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ................................... iv KEY TO SYMBOLS ............ ... .................... .. vii ABSTRACT ............................................. xiii CHAPTER I INTRODUCTION ..... .......................... 1 Dynamics and Stability ..................... 6 Crystallizer Control ....................... 10 II LITERATURE REVIEW .......................... 12 Nucleation and Growth ...................... 12 Dynamics and Stability ..................... 19 Control ..................................... 30 III THEORY AND METHODOLOGY ..................... 33 Growth and Birth Rates in Steady State Crystallizer ................. .... .......... 34 Growth and Birth Rates in Unsteady State Crystallizer ...... ........................ 40 Dynamics and Stability of the Complex Crystallizer .............................. 48 Complex Crystallizer Balance Equations ..... 49 Solute and Solid Balance ................... 53 Dynamics and Stability ..................... 56 Nucleation Models .......................... 73 IV RESULTS .. ................................. 81 Steady State Determination of Growth and Birth Functions ........................... 81 Unsteady State Determination of Growth and Birth Functions ............................ 93 Stability Criteria for the Complex Crystallizer ...... ....................... 107 Discussion ............................... .. 141 V SUMMARY, IMPLICATIONS, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH ....... 153 Summary ...................................... 153 Implications ................................. 154 Conclusions .... ........................... 156 Recommendations for Further Research ....... 157 Page APPENDICES A DERIVATION OF THE CRYSTALLIZER POPULATION BALANCE ................ ................. 159 B ERROR ANALYSIS FOR THE DETERMINATION OF GROWTH AND BIRTH FUNCTIONS ................ 166 C THE DETERMINATION OF SEPARABLE GROWTH AND BIRTH FUNCTIONS .......................... 176 D COMPUTER PROGRAM FOR THE ILLUSTRATION OF THE SUBROUTINE (SIMPLE) AND FOR THE MONITORING OF THE STABILITY OF THE COMPLEX CRYSTALLIZER .............................. 181 BIBLIOGRAPHY ................. ..... ..... .. ........ 218 BIOGRAPHICAL SKETCH ................................ 222 KEY TO SYMBOLS a Withdrawal rate of intermediate size product a. Dimensional constants in balance equations 1 (353) and (354) a. Constants in the fit for n(,t) a Coefficients in characteristic equation n Al Constant in W(z), example data A2 Constant in r (z), example data A3 Constant in R(z), example data A Constants in series for N(z) n AT Total crystal area b Size dependent birth function b Size dependent part of birth function in equation (345) b Net birth function in equation (A20), Appendix A b* Dimensionless derivative of concentration dependent part of birth function B1 Constant in W(z), example data B Birth function of crystals in equation (333) B Birth rate at zero size B Cummulative birth function measured in steady state crystallizer and defined in equation (317a) c Concentration c Constant in ASL model, equation (210) c Saturation concentration c c ,C2,C Cl C. 1 CS1,CS2,CS3 C* D EM EM2 EMH ENINT fn FAi Fi F(z) g g(x) g* h. 1 h(z) i Steady state concentration Reference concentration Constants in the fit for number size distribu tion, steady state case Constants in r3(z), example data Absolute error in n, Appendix B Concentration of the ith stream Metastable concentration Inlet concentration in equation (337) Constants in rl(z), example data Dimensionless inlet concentration Death rate of entities, Appendix A Constant in W(z), example data Constant in R(z), example data Constant in no(z), example data Integral of size distribution for seed crystals Appropriate set of orthonormal functions Fitted values for steady state size distribution Experimental data for steady state number size distribution Integral of h(z) Concentration dependent part of growth rate in equation (344) Vector function in equation (222) Dimension derivative of the concentration dependent part of growth rate Classification function Quantity occurring in equation (365a) Ratio of nucleation to growth rate (nucleationgrowth sensitivity parameter) viii I Quantities occurring in equation (390) j Exponent to which suspension density is raised J(z) Quantity occurring in equation (369a) k Volumetric shape factor K2,K3 Constants in B(c), equation (13) K Constant in equation (395) c K Constant in B equation (11) n KN Constant in equation (216) L Critical size of crystal LD Dominant size L* Maximum size of crystal fines Lf Dimensionless maximum size of crystal fines LK Discrete size L Quantities occurring in equation (390) L Maximum largest size measured max Lm Associate Laguerre polynomials n m Modified orthonomal set of functions, example n data L Lower limit of integration, equation (24) L Maximum size for intermediate size crystals p L ,L1 Lowest size measured M Matrix m Amplitude of dimensionless time dependent concentration mk kth moment of number size distribution MT Suspension density n Number size distribution n. Inlet number size distribution for seed 1 crystals in inlet feed n. Dimensionless number size distribution for in seeds n Size distribution for the mth stage m no Number of zero size crystals per unit volume N Groups of quantities defined in equation (317c) N(z) Size dependent part of dimensionless number size distribution P Nucleationgrowth sensitivity parameter occurring in equation (395) P Quantities occurring in equation (390) Q Volumetric flow rate Q* Quantity defined in equation (375) r Fraction change in concentration due to dissolving and recycling fines R Recycle ratio of dissolved fines R(z) Dimensionless size dependent part of birth rate S Sum of square error t Time t Time of particle growth after birth g t Discrete time m t ,tl Lowest time measured T Reference time u Dimensionless concentration dependent part of birth function v Dimensionless concentration dependent part of growth function v1 Growth rate, Appendix A ve External velocity, Appendix A v. Internal velocity, Appendix A V Volume of crystallized's contents V Volume of crystallizer's contents 1 V Crystallizer's volume, Appendix A VL Volume of clear liquor volume VS Slurry volume W Dimensionless size dependent part of growth rate x Spatial coordinate, Appendix A X Quantity defined in equation (317b) y Spatial coordinate, Appendix A y Dimensionless concentration y Time independent dimensionless concentration in perturbation analysis z Spatial coordinate, Appendix A Z Dimensionless size Z* Withdrawal rate of oversize products Greek Symbols a Fractional error in n, Appendix B a. Constant in expression in steady state birth 1 function, equation (38) a. Constants relating residence times of crystals 1 to their corresponding flow rate in different size ranges a Quantities in equation (379) mn B Product recycle rate n Quantities in equation (379) mn X Square root of chisquare error Xn Quantities in equation (379) 6 Quantities in equation (379) a Quantities in equation (390) m n Dimensionless numbersize distribution e Perturbation error E Error in cumulative birth rate, Appendix B Ec Ratio of clear liquor volume to slurry volume Y Constant in equation (210) y1,Y2 Constants in fit for n(), steady state case yn Quantities in equation (379) F1 Group of terms defined in (347) I2 Group of terms defined in (349) F3 Group of terms defined in (350) X Eigen value in stability analysis An Quantities in equation (390) 1m Microns V5 Dimensionless residence time R Quantity in equation (374) ( Size dependent part of growth rate (Some entities, Appendix A *mn Quantities in equation (390) p Density of crystal a Concentration dependent part of birth rate T Residence time T4 Average residence time of crystals in crystallizers Ti Residence time of seed crystals T. Residence times of crystals in different size 1 ranges in equation (334) 9 Solid fines recycle rate 8 Dimension time xii Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DETERMINATION OF THE CRYSTAL GROWTH, BIRTH OF CRYSTAL NUCLEI, DYNAMICS AND STABILITY OF THE CRYSTALLIZER By Charles Umejei Onwuegbunem Okonkwo August 1982 Chairman: Hong H. Lee Major Department: Chemical Engineering The task of quantifying growth and nucleation rates in a crystallizer grows increasingly difficult as one demands more accuracy in the calculation of these qualities, growth and nucleation rates. The associated stability problem of the crystallizer becomes very complex as more efficient and useful criteria are sought. The search for very good criteria necessitates that the research work with more complete balance equations. Novel methods for quantifying growth and nucleation rates in steady and unsteady state crystallizers are developed. There were no assumptions made in the develop ment of these methods which utilize some minimization procedure. The only approximation was in the minimization procedure. When applied to some literature data,the methods gave good results. In addition, the critical size, Lc, at which the growth rate is zero could be determined by these methods. xiii Very comprehensive and meaningful stability criteria are developed for the complex crystallizer via perturba tion techniques. The complex crystallizer was a well mixed suspension crystallizer equipped with dissolving and recycling of undissolved crystals and product clas sification. The criteria were derived from the complete crystallizer's balance equations. The only approximation was that the concentration dependent parts of the birth and growth functions were linear in concentration. In addition, the fourth order characteristic equation, from which the criteria are derived, is used in monitoring the effects of various crystallizer's parameters on the system's stability. These effects were simulated via the RootLocus method. When applied to some example data,the charac teristic equation gave results that are in agreement with experimental observations. The computer simulation and the criteria agree in all cases. New ways of stably operating a crystallizer under conditions in which the crystallizer may be inherently unstable were obtained. xiv CHAPTER I INTRODUCTION The preparation and handling of solids areinfluenced not only by gross chemical composition, but also by particle size distribution, morphology and surface chemistry. These parameters are in turn often fixed by processes that occur as the solid forms and is separated from its mother liquor. Crystallization products include food materials, pharma ceuticals, fertilizers and commodity chemicals. A few uses of crystallization include the following: (1) in process development, (2) in separation techniques, (3) in poly merization processes, and (4) in controlling product quality and shape. Thus if one wants to know about behavior of solidsreaction rates, caking, blending, toxicology, storage, dusting, bioavailability, etceteraan adequate knowledge of the crystallization process will prove worthy. Sometimes a problem that is not initially viewed as a crystallization problem becomes one in the end. Unlike most chemical processes, particle conservation balance is required in addition to mass, momentum, and energy balances to determine process yields and energy requirements. The population balance is an added complication which makes solution of the balance equations extremely difficult, if not impossible. Primary and secondary nucleation are the two forms of crystallization. Primary crystallization is further sub divided into homogeneous and heterogeneous nucleation. Classical homogeneous nucleation is that which occurs only because of supersaturation as the driving force. In other words, it is conceived as the coming together of sufficient solute molecules to form a critical mass capable of sur viving as a solid phase. An example of this is found in simple cooling and temperature reduction of a solution or in the evaporation of a portion of the solvent of a solu tion. Heterogeneous primary nucleation is similar to homo geneous nucleation except for the presence of a foreign solid which acts as an active site for nucleation. This foreign substance is of a different chemical composition. Again heterogeneous nucleation has supersaturation as its driving force. In general primary nucleation occurs at high supersaturation levels and nucleation and growth kinetics have a high order dependence on supersaturation. On the other hand, secondary nucleation is one which can occur at very low supersaturation provided there is some form energy and a seed crystal of the same chemical com position as material being crystallized. Secondary nuclei can form in saturated and undersaturated solution. For the nuclei formed to survive the solution must be slightly supersaturated. Secondary nucleation is not a strong func tion of nucleation and growth kinetics. The driving force is the presence of the seed crystal and energy. This energy might be one of several forms such as crystal im peller collisions, ultrasonic waves or fluid shear on the crystal surface. Crystallizers which operate at high and low super saturation levels irrespective of the throughput or process conditions are classified as classI and class II systems respectively. Class II systems are usually high yield and fast growing while the contrary is true for class I sys tems. Primary and secondary nucleation are phenomena with high probability of occurrence in class I and class II systems respectively. Though primary crystallization has been for a long time a topic of interdisciplinary interest among various scien tists and engineers and has been extensively researched, a better understanding of the phenomenon of secondary nucleation awaits still much work to be at par with its counterpart. It was no more than thirty years ago when chemical engineers began work in secondary nucleation when they realized that it was the dominant form of nucleation in industrial crystallizers which usually operate at low supersaturation. Since then, several authors (Miller and Saeman, 1951; Murray and Larson, 1965; Timn and Larson, 1968; Clontz and McCabe, 1971; Randolph and Cise, 1972; StricklandConstable, 1972; Ottens and de Jong, 1973; Sung et al., 1973; Bauer et al., 1974; Garside and Jancic, 1976; Lee, 1978; Garside and Jancic, 1979; Randolph and Puri, 1981) have addressed one aspect or the other of secondary nucleation. Some of the aspects of secondary nucleation that have been of interest to chemical engineers are the following: (a) factors such as the crystallizer environment affecting the crystallizer (these may include effects of impurities, crystallizer hydrodynamics, impeller speed, surface regeneration time, hardness of contacting surfacethe effects of these factors are usually treated via empirical correlations); (b) mechanisms involved in the crystallization process such as that responsible for nucleation and growth or the effect of impurities; (c) dynamics, stability and control of crystallizer; and (d) quantification of nucleation (birth of secondary nuclei) and crystal growth. Of the aforementioned areas the last is most critical in understanding the phenomenon of secondary nucleation. Knowledge about accurate quantifica tion of nucleation and growth is important in understanding other areas as well. The mixedsuspensionmixedproductremoval (MSMPR) pioneered by Randolph and Larson (1971) is a simple continuous crystallizer with inlet and outlet flow and in many ways is analogous to the continuous stirred tank reactor (CSTR). The main difference is the simultaneous occurrence of nuclea tion and growth in the MSMPR crystallizer. The name MSMPR crystallizer as used in the literature is a steady state crystallizer of the above description but obeying certain constraints that are rarely satisfied in practice. Though the MSMPR crystallizer is valid for the large size region of constant growth rate (that is the McCabe AL law), researchers have used the MSMPR theory for the small size region. Because of the complexity introduced by the simultaneous occurrence of the two fundamental quantities birth and growth ratesauthors working with the steady state crystallizer have simply ignored the birth function in the population balance equation even when there is nuclei generation by contact nucleation. Contact nuclea tion is a form of secondary nucleation where energy is provided by contacting the seed crystal. Khambaty and Larson (1978) failed to account for nuclei generation in their steady state crystallizer experiment involving contact nucleation in addition to assuming that nuclei are born at zero size. Garside and Jancic (1979), like their predeces sors, found it necessary to invoke the "zero size nuclei" assumption in their attempt to determine birth and growth rates in a steady state crystallizer. Authors have had more problems in determining the two fundamental quantities in an unsteady state crystallizer. To alleviate the problem Randolph and Cise (1972), while working with an unsteady state crystallizer,assumed zero birth rate in the population balance in order to calculate growth rate and to calculate birth rate they assumed zero growth. Similarly, Garside and Jancic (1976) obtained birth and growth rates in an unsteady state crystallizer via a differential technique, a method which not only gives gross errors, but also is very impractical considering the type of experimental data that one works with. While the above list is by no means exhaustive, it is representative of the general trend adopted by most researchers. The above mentioned problems are addressed in the first part of my work. Dynamics and Stability MSMPR crystallizers inherently produce a wide crystal size distribution with a large coefficient of variation (C.V.) and a dominant size LD = 3GT, where G and T are constant growth rate and residence time respectively. This wide distribution is industrially undesirable. Design specifications for commercial products might require a narrow or large distribution or somewhere between the two. Sometimes one might desire some nucleation when there is little or none, while at other times there may be un desirable excessive nucleation. These and other problems dictate the need for controlling CSD. One way to change the CSD is to incorporate some selective removal whereby crystals of a certain size range are preferentially removed. Preferential removal, dissolution and recycle of fines generally lead to an increase in characteristic dimension (such as dominant or average crystal size) and spread of the distribution,while preferential removal of coarse products leads to a narrower distribution with a concomitant reduction in dominant size. Incorporation of clear liquor advance has the effect of increasing crystal residence time which in turn has little or no effect in increasing dominant size. While it should be remembered that the purpose of designing selective crystal removal into a crystallizer is to force the CSD to a larger dominant size or narrow the distribution, actual incorporation of selec tive removal often leads to sustained oscillations. This action results in no improvement in product quality. It would be desirable to be able to operate the crystallizer without the incursion of instability even in regions where the system is inherently unstable by choosing appropriate control strategy. Ability to do this requires very meaningful stability criteria. Several researchers have investigated various aspects of stability and control of one form of crystallizer or the other. Some of these are complex in nature. Unlike the MSMPR crystallizer which is analogous to the CSTR reactor as used in reaction engineering, complex crystallizers have no such counterpart. Complex crystallizers do not easily lend themselves to mathematical description and solution. The type of crystallization considered varied from slow growth rate kinetics (class I systems), where residual supersaturation is appreciable, to fast growth systems (class II systems) with negligible residual supersaturation. Miller and Saeman (1947) while working with an industrial ammonium nitrate producing crystallizer observed cyclic fluctuations in particle size distribution. Similar observa tions were made by Finn and Wilson (1954) during a fermentation process and by Thomas and Mallison (1961) in the field of continuous polymerization process. Yu and Douglas (1975) showed in their paper that a class I MSMPR crystallizer could be operated to give oscillations in the size distribu tion. These oscillations were the results of interactions within the system and these interactions were in turn the direct result of disturbances. Randolph (1962) quantified stability of class II MSMPR dL B in the following manner: dL < 21. The logarithms of n nuclei birth rate and grow rate are Z B and Z G respectively. By assuming the empirical powerlaw nucleation growth kinetics as expressed in equation (11), B = K Gi (11) n he obtained the following constraint: dL Bo n = i < 21. (12) dL G In equation (11) B is the nuclei birth rate defined via the assumption of zero size nuclei; G is the size independent growth rate and "i" is the kinetic order of birth to growth. Oscillations generated by kinetic values as high as 21 are referred to as highorder cycling. Sherwin, Shinnar, and Katz (1967) derived stability criteria for an isothermal class I MSMPR crystallizer using Volmer's nucleation model in equation (12) and assuming nuclei birth rate is size independent. B(c) = K2exp(K3/(Znc/s) 2) (13) In equation (13), B(c) represents the concentration dependence of birth function, K2 and K3 are constants, and c and cs are the concentration and saturation concentration, respectively. They claimed that instability was due to the nonlinear nature of the dependence of nucleation on super saturation and that size dependency of the growth rate has a stabilizing effect. Attempts to treat size dependence of growth rate up to second order resulted in nonlinear moment equations which were not closed. Similar attempts not to linearize B(c) resulted in a closed set of nonlinear moment equations which were solved numerically. The authors concluded for the case of size independent growth rate and B(c) as defined by equation (13) that the tendency towards cycling increased with increasing particle size. Considering the assumptions in their model these conclusions are questionable. Hulburt and Stefango (1969) modeled a double drawoff crystallizer by assuming size independent growth and birth rates and Volmer's nucleation model. They numerically solved the coupled nonlinear population and mass balance equations. The assumption of zero nuclei size was made in deriving these equations. They concluded that one of the drawoff streams, the clear liquid overflow,seemed to enhance stability. They also concluded that both increased seed addition and increased seed size in feed stream enhanced stability for high growth rate conditions. Randolph, Beer and Keener (1973) showed the birth growth stability constraint of a complex RZ crystallizer to be a function of "i," fines exponential decay factor, X, product classification size, L and normalized product classification rate, z. Their model assumes that the birth and growth rates were size independent and that nuclei were of zero size. Beckman (1976) experimentally demonstrated sustained limitcycle behavior in an RZ type crystallizer equipped with clear liquor advance and product classification. He concluded that cycling behavior was chiefly induced by product classification. Crystallizer Control Timn and Gupta (1970) investigated the stability control of class II MSMPR crystallizer using fines seeding and fines destruction plus recycle as manipulated variables. They concluded that controlling the zero moment tended to stabilize the system while controlling the second moment tended to destabilize a normally stable system. Lei, Shinnar, and Katz (1971a) investigated the stability of a class I crystallizer with a point fines trap via spectral method. The control variable was fines crystal area. They were able to stabilize the system by manipulating the throughput rate while by manipulating fines destruction and recycle rate the system's stability was not enhanced. The idea of a point fines trap (zero mass of fines in trap) has no practical value since industrial crystallizers operate with appreciable amounts of crystal mass in fines trap. In order to avoid the excruciating mathematical com plexity that would result, the aforementioned researchers found it necessary to make one assumption or the other even in situations when the conditions of their assumptions are never realized in practice. In consideration of the above problems, the second part of this work is devoted to obtaining more general stability criteria from which meaning ful control strategy can be obtained. The crystallizer in this study will be discussed in detail in a later chapter. While it is more general, it is in many ways physically similar to the RZ crystallizer modeled by Randolph, Beer, and Keener (1973), and will henceforth be referred to as "the Complex Crystallizer." CHAPTER II LITERATURE REVIEW The concept of the population balance is mainly responsible for the progress and development witnessed in crystallization thus far. The study of crystallization has shifted from what was originally considered an art to what is becoming, more and more, an engineering science. The development of crystallization science started with nucleationgrowth kinetics and steady state system analysis and progressed to dynamic system analysis. The various aspects of crystallization have not been equally embraced by the many works published on this subject. More attention is now being directed to the very important but least addressed areas of crystallization. Nucleation and Growth Saeman (1956) investigated the simplest continuous crystallizerthe mixedsuspensionmixedproductremoval (MSMPR) crystallizer. For this he essentially derived equations for the steady state crystal size distribution (CSD) in a mixed suspension. By first generating nuclei he no longer had to consider the birth function as part of the particle balance. Then he obtained the balance equation (21) with respect to time of growth after particle birth. dn + 0 (21) dtg T The number of particles per unit volume per unit size and the residence time of these particles are n and T, respectively. The variable t represents time (time of particle growth g after birth). The solution to equation (21) is given by n(t ) = n(t =0) exp(t /T). (22) By assuming that growth was not a function of particle size, this allowed him to relate time of growth, tg,to growth rate, G, and crystal size, L, in the following manner: L = Gt (23) Substituting (23) in (22) and assuming that initial dis tribution of particles are nuclei of size L he obtained. n(L) = n(Lo) exp ( ) (24) In particular he chose Lo = 0 and obtained the standard MSMPR crystallizer equation, n(L) = n(0)exp ( ) = n exp(L) .(25) Saeman then compared his theoretical distribution with that obtained from an OsloKrystal type crystallizer producing ammonium nitrate crystals under conditions of hindered settling and found good agreement. Similar work was done by Robinson and Roberts (1957) who obtained the theoretical crystal size distribution resulting from a cascade of MSMPR crystallizers with nucleation in the first stage only, but same residence time in all stages. The size distribution for the mth stage could be represented by m1 n1(0) (G) L n (L) = (ml exp( L) ; m = 1,2,... m (m1) Gr (26) Randolph and Larson (1962) later extended the work of Robinson and Roberts (1957) to include nucleation in all stages (and equal residence time for each unit). By so doing, they were able to extend application to more indus trial crystallizers. The size distribution of the mth stage for this type of crystallizer is k1 Sn(0) (L/GT)k L m+lk n (L) = () r ml(27) m GT (k1)2 k=l Abegg and Balakrishnan (1971) obtained the distribution in equations (26) and (27) in their attempt to model the mixing in different crystallizers. They found good agree ment between the theoretical distributions, equations (26) and (27), and data taken from Draft Tube Battle (DTB) and Forced Circulation (FC) crystallizers respectively. Hulbert and Katz (1964) derived with great generality the distribution of some entities over any associated property. This entity and associated property can be, for instance, particulate entity and particle size respectively. The associated property can also be age or any other quantity that varies with time. Their model was then applied to particle nucleation and agglomeration, referred to as net birth (birth and death) and growth in crystallization terminology. Randolph (1964) in a "Note to the Editor" derived essentially the same population balance and noted that it would be of major importance in advancing the theoretical understanding of continuous crystallization. Murray and Larson (1965) designed and constructed a continuous mixed suspension salting out crystallizer to test both the MSMPR model as well as the unsteady state model given by equation (28), 3n r3n n (28) at a C T * They obtained nucleation and growth rates for the MSMPR model and some of the first transient data in the literature. They experienced great difficulty in the description of the very small crystals which dominate the size distribution behavior in terms of numbers. As a result determinations of nuclea tion and growth rates for this small size range were in error. Timm and Larson (1968) obtained nucleation and growth kinetics for three materials using steady state and transient data. They found that unsteady state experiments have some advantages over steady state experiments in determining the kinetic order of nucleation rate to growth rate. McCabe and Stevens (1951) found the growth rate of copper sulfate in a suspension maintained by means of an agitator to be dependent on crystal size if the relative velocities of the crystals to solution velocity differ and if these relative velocities are in the effective low range. They found that larger crystals grew faster than small ones because larger crystalshave higher relative velocity. Youngquist and Randolph (1972) while working with a class II system involving ammonium sulfate obtained with great difficulty the nucleation rate after making several assump tions. They remarked that a rigorous quantitative defini tion of and use of secondary nucleation kinetics must await quantitative separation of size dependent growth and birth rates G(C), b(C) in the small size range. They said, "the increased detail and accuracy of secondary nucleation measurements made in this study have indicated the near impossibility of quantitatively characterizing secondary nucleation using the MSMPR technique as well as casting doubt on the general applicability of gross secondary nuclea tion kinetis so obtained" (p. 429). It will become obvious from this study, outlined in a later chapter, that one can go much further with the quantitative determination of birth and growth functions than they had thought. Randolph and Cise (1972), while working with an unsteady state class II system involving potassium sulfate, thought that it was patently impossible to uniquely specify both size dependent functions G(C) and b(C) using only the single size dependent measurement n(C), without further simplifying assumptions or other independent measurements. Faced with this dilemma they assumed zero birth rate in the population balance in order to calculate the growth rate and to calculate birth rate they assumed zero growth rate, thus suppressing popula tion changes due to convective number flux. Similarly, Garside and Jancic (1976) while working with potash alum in an unsteady state crystallizer used a differential technique to obtain birth and growth rates. This method is not only in gross error, but also impractical as regards typical experimental data. Garside and Jancic (1979) found it necessary to invoke the "zero size" assumption in their attempt to determine birth and growth rates in a steady state crystallizer. They rearranged the steady state population balance as shown in equation (29): dl n(S) 1 d G( ) b(C) = n(C)G() d + + d (29) d G()T dC They said that b() can be calculated if only G(C) is known. Then they used surface integration rate and mass transfer correlations to estimate the terms in equation (29) that involve G(C). Their semilog plot of particle number versus size showed some curvature. They agreed that for small sizes, particularly below 15pm, overall growth was strongly size dependent. Khambaty and Larson (1978) in their experiment with magnesium sulfate heptahydrate crystals obtained a curved line on a semilog plot of particle number versus size and attributed the curvature to size dependency of the small crystals. They ignored the birth function in the population balance despite the fact that there was crystal generation by contact. They also invoked the'zero size assumption. Rousseau and Parks (1981) similarly studied size dependency of the growth rate of magnesium sulphate heptahydrate crystal and obtained some curvature in the semilog plot of particle number versus size. They similarly concluded that growth rate was strongly dependent on crystal size for the 44 to 10OOm size measured in the class II MSMPR crystallizer. The size dependent growth rate was obtained via the two parameter reduced version of the AbeggStevensLarson (ASL) model given in equations (210) and (211): G = go(l+yL)c ; c y = 1/(GT) (211) The parameter, G, is the growth rate of nuclei and the parameters y and c are simply constants. The purpose of studying nucleation and growth in crystal lizers is to be able to control crystal size distribution in these crystallizers, which are often plagued by oscillatory crystal size distribution dynamics. The associated dynamic behavior of crystallizers presents serious industrial problems. Dynamics and Stability Many investigators have dealt with the dynamic behavior of crystallizers. Some have addressed problems associated with class I crystallizers while others have directed attention to class II systems. In addition, some of the works have approached crystallizer instability from the nucleationgrowthkinetics viewpoint while others have taken the viewpoint that the mode of operation is the primary cause of crystallizer instability. Cycling of crystal size distribution has been reported in class I crystallizers. Bennett (1962) observed cycling with an industrial crystallizer with a 15hour period and a large amplitude swing of particle weight on a 12 SSM screen. Song and Douglas (1975) designed a laboratory scale sodium shloride crystallizer in which they produced oscillatory out puts using constant inputs. The experimental values of the oscillatory output agreed slightly with their theoretical predictions based on steady state measurements of growth rate and nucleation rate kinetic parameters of an isothermal MSMPR class I system. Only one cycle was observed in the experiment. Their theoretical work for this system is described in an earlier paper. Sherwin, Shinnar and Katz (1967) carried out a theo retical investigation of the effect of feed seeding and fines dissolving and recycling on the stability of a class I system. These authors have expressed their stability criterion in terms of ratio b/g which they defined as dB dc1 b/g i (212) dG In equation (212) "bar" represents steady state values; G is a linear function of supersaturation, G = kl(ccs) (213) In equation (213) k1 is a constant. The quantity B was approximated with Volmer's nucleation model defined earlier in equation (12), and in which the quantity (c/c 1) is assumed very small. By expanding B up to first order in c/cs and utilizing the above assumption, equation (212) can be written as 2K2 b/g 2 = i (214) ( 1) cs For a clear feed, it was found that the system becomes un stable when b/g > 21. Seeded feed was found to improve stability strongly while volume fraction (void fraction) of solution had an insignificant effect. Another observation made was that the system stability increases significantly if nuclei had a finite size and that the region of stable operation enlarges with increasing nuclei size. The kinetic order of nucleation rate to growth rate played a significant role in determining the system stability. Sherwin et al. (1969) were the first to uncover the extreme importance of product classification in causing crystallizer instability, in spite of the fact that their model was too idealized to be of practical value for engineering simulation. It was assumed in the model that as soon as a particle reached product size it was instantly removed thus producing monosized product crystals. It was found that extreme classification enhanced the sensitivity of the crystallizer to disturbances and increased the tendency toward cyclic fluctuations. Hulbert and Stenfango (1969) investigated a double drawoff crystallizer by assuming size independent growth and birth rates, zero size nuclei and Volmer's nucleation model. These assumptions resulted in a coupled nonlinear population and mass balance equations which were numerically solved. They found that one of the drawoff streams, the clear liquor overflow, tended to increase particle size and crystal solid content due to the longer particle retention time. As Bennett and Van Buren (1969) had already pointed out, increasing particle retention time does not always result in larger product crystals. In addition, the clear liquor overflow seemed to enhance the system stability. Another conclusion reached by Hulburt and Stefango was that increased feed seeding and increased seed size in feed stream enlarged the stability region for high growth conditions. Nyvlt and Mullin (1970) observed damped oscillatory behavior while experimenting with a 200liter drafttube crystallizer fitted with an elutriating leg and containing sodium thiosulfate as the specimen under investigation. The data obtained were too scanty to conclusively demonstrate sustained limit cycle behavior. Lei et al. (1971a) carried out a theoretical study of the stability and dynamic behavior of a continuous crystal lizer equipped with a fines trap. The fines had a fixed size. Nucleation was assumed to occur at a fixed size and growth rate was assumed independent of size. Their study showed that cycling in crystallizers can be reduced or eliminated by adjusting the operation of a fines trap, especially when the size of the fines is increased slightly. They also noted that fines dissolution and recycle do not always stabilize a crystallizer system, and can, in fact, destabilize the sys tem depending on the conditions of operation. Using perturbation methods Yu and Douglas (1975) carried out a theoretical investigation of the stability of a class I MSMPR crystallizer, and from which they concluded that, in some cases, oscillatory behavior can produce yields that exceed the predicted steady state value. The study enter tained many assumptions in the derivation of the balance equations and process parameters used do not correspond to typical industrial conditions. Attention has been directed to class II crystallizers by some investigators. Oscillatory behavior has been observed in some class II systems. Miller and Saeman (1947) observed oscillatory behavior of the crystal size distribution while working with and industrial ammonium nitrate producing crystallizer (an Oslo type crystallizer). KerrMcGee (in Beckman, 1976) observed cyclic fluctuations in an industrial potassium chloride crystallizer as shown in Figure 1. The severity of the fluctuations caused continuous rippling of the weight distribution of product crystals. Cycling behaviors have been observed in a continuous polymerization process by Finn and Wilson (1954) and in a fermentation process by Thomas and Mallison (1961). Randolph and Larson (1965) investigated the problem of stability in a class II MSMPR. The dynamic population balance equation (215) in conjunction with boundary condi tion equation (216) and initial condition (217) were trans formed into moment equations, which were then solved on an analog computer. an Gan n 3n + = 0 (215) n(=0) = KNGi (216) n() = n(0) exp(C/T) (217) The coefficient, KN, of Gi is a constant and "T" is the crystallizer residence time. All other symbols in equation (215) are as defined in previous sections. The bar re presents steady state values. Step changes in production rate and in nuclei dissolving rate were introduced into the system, and the resulting disturbances in the zeroth, 0 CCO cd r*H 0c 40 4) *r OQ CO 4l i1 w a^s + (3 F C 0 SCoO 0 0 0 0 0 0 N3 NO T a3N N33HOS NO .3NIV.136 LN363ad LH913M first, and second moments were plotted as functions of time. The plots showed cyclic fluctuations. Randolph and Larson (1971) modeled a class II MSMPR crystallizer with equation (215) which was first trans formed into a closed set of moment equations in time. The characteristic equation of this closed set of ordinary differential equations was obtained via Laplace trans formation technique. RouthHorwitz method was used to establish stability criterion which guaranteed stability of a crystallizer obeying equation (215), whenever the slope of the loglog plot of nucleation rate versus growth rate was less than 21. Randolph and Larson (1969) included product classification and fines dissolution and recycle in their crystallization. They concluded that the net effect of fines dissolution and recycle is to force growth rate to a higher level, producing the same production on larger average size crystals having less total area. They noted that if the higher supersaturation produced from the fines dissolution and recycle resulted in a greater than propor tionate increase in nucleation rate (as would be the case if nucleaction rate were a stronger function of supersatura tion than growth rate), then size improvement would be some what less than expected. They also noted that this internal feedback (that is, an increase in supersaturation causing an increase in nucleation rate) might affect the system in such a way that disturbances to the system might not damp out, thereby causing sustained oscillation of the crystal size distribution. They further remarked that imperfect classification of larger product crystals is almost always accompanied by a reduction in average size, unless nuclea tion is independently controlled. Using the concept of a classification function, Randolph et al. (1973) modeled an RZ crystallizer and showed the stability of such a crystallizer to be a function of "i," fines dissolution and recycle, product classification size, L and normalized product classification rate, z. Beckman (1976) experimentally demonstrated sustained limit cycle behavior in an RZ type crystallizer equipped with clear liquor advance and product classification. He then solved the population balance equa tion (with a perturbed steady state solution as initial condi tion) via Laplace transform technique to obtain the dynamic behavior of the crystallizer. He then compared his semianalytic solution with a regularfalsi method of numerical solution. Saeman (1956) in his study of the simplest continuous MSMPR crystallizer noted that regardless of the classification device used, a crystallizer cannot put out large crystals unless conditions which are conducive to growth of large crystals are maintained in the suspension. They also re marked that positive and direct means for size control lie largely in the provision of effective means of segregation and elimination of excess fines. He recommended that fines should be eliminated at an average age that is less than or equal to onetenth that of the product crystals, and that whatever the classification device used for removing fines, it must not inadvertently overload with large intermediate size crystals. Saeman further showed that classification could yield a narrow distribution as well as one with relatively larger dominant size. Cohen and Keener (1975) used a multiple time scale perturbation technique on a third order system of nonlinear ordinary differential equations to predict the bifurcation of time periodic solution of a class II MSMPR crystallizer. Their model is essentially represented by equations (218), (219) and (220). 3n + 1 an + n = 0 >0, t>O (218) S20 n(C,t)d 2( n(0,t) = [ t > 0 (219) [c0 n(C,t)dC]1 n(S,0) = f(C) = e5 (220) The function, f(C), is the initial steady state distribution of the CSD. By defining the Kth moment as in equation (221), they transformed their equations into a set of nonlinear ordinary differential equations of the moments. m(t) = k n(,t)dC (221) k 0 In order to study their moment equations they investigated a general system of nonlinear ordinary differential equa tions represented by equation (222): dX = PX + 2AX + g(X) (222) Here X is a threecomponent vector, P and A are constant matri ces, and g(X) is a smooth nonlinear vector function con taining no linear terms near the equilibrium point. In other words, derivatives with respect to each component of X varnish at the equilibrium point. In addition, the func tion g itself varnishes at the equilibrium point. If we re present XT = (x,y,z), then this condition is represented by equation (223). gi(0,0,0) = gix(0,0,0) = giy(0,0,0) = giz(0,0,0) = 0 i = 1,2,3 (223) The parameter, E, is very small (0<<<1). The subscripts x,y,z represent first order partial with respect to x, y, and z respectively. By casting the nonlinear ordinary dif ferential equations of the moment into the form of equation (222), they were able to show oscillation in the crystal size distribution,n, and the growth rate, G, defined by equation (224). G(t) = 1/i( 2n( ,t)dl (224) Their analysis predicted both amplitude and period of oscil lation. The authors noted that including the mass balance would present a formidable task. In addition, they eliminated the birth function from the population balance, equation (218), and by representing it as a size independent func tion and assuming a power law type of nucleationgrowth kinetics, they included it as a boundary condition. Because of the many assumptions made their whole analysis is questionable. Nyvlt and Mullin (1970) carried out a numerical study of a class II MSMPR crystallizer equipped with an elutriator via a Monte Carlo simulation technique. They used their study to explain experimental data obtained from a pilot plant. They concluded that crystallizers. with or without product classification can exhibit periodic changes in pro duction rate, product crystal size, supersaturation, magma density and other related parameters, and in some cases, the steady state may not be reached. The cyclic period, which is comparatively long in most cases, depends on the supersaturation rate and on the fraction of product crystals removed during classification. They noted that the stability of the system increases with increasing growth rate, in creasing magma density, decreasing nucleation order, i, decreasing minimum product size, and decreasing quantity of crystals withdrawn per unit time. They also remarked that seeding would have the same effect as decreasing the effec tive nucleation rate and should lead to a stabilization of the system. Good stability criteria is a prerequisite for an effective control scheme for dynamic crystallizers. Control Publications in control of crystallizer dynamics are very few. A good predictive model in terms of good stability criteria is a prerequisite for effective control. Simple proportional control ofsome crystallizer variable has been used by most of the investigators. Bollinger and Lamb (1962) and Luyben and Lamb (1963) have outlined the basic design of feed forward control. Han (1967) investigated a feed forward control of class I MSMPR crystallizers by using feed concentration as the disturbance, supersaturation as the controlled variable and flow rate as the manipulated variable. By controlling the supersaturation, he intended to control production rate. Han's control scheme worked better when the system was operated in the stable instead of the unstable region.of birthgrowth kinetics. Lei, Shinnar and Katz (1971b) investigated a feedback control of a class I MSMPR crystallizer equipped with a fines trap. Because of its relatively easy accessibility by light transmission measurement, the total surface area of fines in the fines trap was used as the controlled variable, with flow rate through the crystallizer as the manipulated variable. This scheme showed good control even when the crystallizer was operated in the unstable region. However, attempts to use fines recirculation rate (that is, the amount of fines destroyed) as the manipulated variable did not readily stabilize an unstable operation. The above results were derived from a linearized stability analysis of a crystallizer equipped with point fines trap and in which a simple proportional control was incorporated. Timmand Gupta (1970) investigated a feedforward/feed back control of a class II MSMPR by using flow rate out of the crystallizer as the disturbance, seeding and fines destruction and recirculation rate as manipulated variables and cumulative number of crystals as the controlled variable. The control scheme monitoring the cumulative number of crystals was demonstrated to be superior to the one monitoring the total area of crystals within the crystallizer. The control scheme with total area of suspended crystal magma as the controlled variable was worse than no control. The simple proportional control was capable of stabilizing the system within an inherently unstable region. The feedforward/ feedback scheme showed remarkable improvement over conven tional feedback control. Beckman (1976), using proportional control of nuclei density with fines destruction and recirculation rate as the manipulated variable, studied the control of a class I MSMPR crystallizer equipped with fines destruction, clear liquor advance and product classification. He theoretically pre dicted the proportional control constant necessary for stability. The system was amenable to control. From a com puter simulation utilizing experimentally determined nuclea tion and process parameters, he obtained a value for the pro portional control constant which agreed with that predicted theoretically by his model. Beckman assumed nuclei were of 32 zero size and growth rate was size independent. By neglecting the birth function in the population balance, he included it as a size independent function in the boundary condition. .CHAPTER III THEORY AND METHODOLOGY As pointed out in earlier chapters, previous investiga tors encountered tremendous difficulty in their endeavor to simultaneously quantify the two fundamental quantities, growth and nucleation rates for all sizes. Apparently it was realized by these researchers that the most important tool towards understanding crystallization phenomenon in steady state crystallizers is the ability to quantify growth and nucleation rates. The MSMPR technique pioneered by Randolph and Larson (1962, 1971) was not capable of explain ing experimental steady state data for the small size range where most of the nucleation occurs. It was designed for the large size region where growth rate is independent of size and for the size region where particle birth is negligible. Youngquist and Randolph (1972) in their work with a class II system remarked that a rigorous quantitative definition of and use of secondary nucleation kinetics must await quantitative separation of size dependent growth and birth rates G(C), b(i) in the small size range. They said that their work indicated the near impossibility of quanti tatively characterizing secondary nucleation using the MSMPR technique as well as casting doubt on the general applica bility of gross secondary nucleation so obtained. It is now obvious that empirical correlations involving crystallizer process parameters and environment must await the rigorous quantification of growth and nucleation rates. It is the unraveling of the simultaneous quantification of growth and nucleation rates to which the next section is devoted. Growth and Birth Rates in Steady State Crystallizer Consider the following crystal population balance equa tion (31), the derivation of which is included in detail in the appendix: 1 3 3(Gn) V t (Vn) + b + (nQi n Q ) (31) where t = time n = number of crystals per unit volume per unit size in crystallizer, #/MZIm) = crystal characteristic size, um G = crystal growth rate, um/min b = net birth rate of crystal nuclei, #/(mjimmin) n. = number of crystal seeds in inlet (feed) stream 1 per unit volume per unit size, #/mipm) no = number of crystals in outlet stream per unit volume per unit size, #/(mi.m) Qi = inlet volumetric flow rate, (mi/min) Qo = outlet volumetric flow rate, (mi/min). For a constant volume, well mixed, steady state crystallizer with no seeds in inlet feed,the above equation reduces to equation (32).: a 1 (Gn) = b(S) n( ) (32) V where T = is the average crystal residence time. Most Qo researchers usually assume the McCabe AL Law to hold and take growth rate, G, to be indendent of size and in addition assume zero birth rate even in situations where secondary nuclei are being generated by contacting the seed crystal. When these assumptions are made, equation (33) is obtained; Gnn n G5 + T = (33) The solution to equation (33) is given by equation (34): n(0) = n(Lo)exp((CL )/(GT)) (34) Further they assume nuclei are born at size zero (that is L =0) so that equation (34) becomes n(C) = noexp(C/(GT)) (35) where n is an abbreviation for n(0). A curved population balance on a semilog plot of n versus size is of the form shown in Figure 2. Since equa tion (33) is not capable of explaining experimental data of the form represented in Figure 2, it suggests that one t Z ;4 0 Figure 2. Lf size + Plot of Population Density Versus Size. + Figure 3. Plot of N Versus X. c=L size  Figure 5. Growth Rate Versus Size. G(L1) Figure 4. Plot of Total Chi Square Error Versus G(L1) Values. size  Figure 6. Birth Rate Versus Size. should work with the more complete equation (32) for the steady state crystallizer. Before proceeding, it is necessary to define a critical size, L ,as that at which the growth rate is zero (that is G(Lc)=0). This definition is motivated by the fact that the growth rate, G, is a monotone increasing function of size in the interval, I = (0,c). The shape of it is well documented. Integrating equation (32) with respect to size and applying the condition in equation (36) yields equation (37). G(Lc) = 0 (36) G(C)n() = b(A)dA n(A)dA (37) L L c c The shapes or functional forms of the functions b(C) and n() are very well established from experimental as well as theoretical considerations. The shapes of these func tions are exponential. For this reason, these functions are represented by equations (38) and (39). ac2(5L ) b(C) = ale ) (38) n(i) = clexp(yl,) + c2exp(y2C) (39) The parameters al, a2' cl, c2' Y1, and Y2 are constants. Usually the parameters cl, c2, y1, and y2 can be determined from a nonlinear least squares fit of number size distribution data. Substitution of equations (38) and (39) into equation (37) gives G(C)n() = c exp(a2(AL ))dA L [clexp(yIA) C c + c2exp(y2A)]dA (310) We make the following definitions: Fi = G(i)n(Ci) i = 1,2,...,n (311) FA = lexp(a2(AL ))dA [clexp(ylA) Ai fL LT L c C + c2exp(y2A)]dA , i = 1,2,...,n (312) Experimental data are always reported for the size interval, I = [Lw,Lmax], where L, and Lmax denote the respective lowest and maximum experimentally measured sizes. The size denoted as L in Figure 2 is that beyond which growth rate is constant as indicated by the straight line portion of the plot in the interval I = [L,Lmax]. It is for this same size interval that the MSMPR technique usually applies. Once n(C) is obtained by fitting experimental data as represented by equation (39), it is used to interpolate for more values so that n(.i) is available in much finer par tition; in other words, the interval, [L ,L ma], is sub divided into much finer partitions than the original experimental data. The only unknown values al, a2, and L can be obtained by minimizing the sum, S, defined by equa tion (313) over al, a2, and L . N 2 S= (FiFAi) i (313) i=l Another version constitutes confining the minimization to the interval, 12 = [L*,Lmax], for which Fi is known for 2 f maJ 1 each "i." The minimization procedure is carried out via an appropriate nonlinear least square method. The determina tion of al, a2, and Le immediately yields the function, b() and G(i) can be calculated from equation (310). The integration in equation (310) are to be carried out analytically. This concludes the analysis for the quantifica tion of the two fundamental quantities, G() and b(). It should be noted that no assumption was made in the popula tion balance equation. The only approximation made is in the minimization procedures. This approximation is in general negligible since minimization procedures usually 6 have builtin tolerances, some smaller than 106, that must be satisfied before convergence is attained. I proceed to the determination of G(C) and b(C) for the unsteady state crystallizer. Growth and Birth Rates in Unsteady State Crystallizer The difficulty in quantifying the two fundamental quantities in an unsteady state crystallizer is compounded by the transient nature, which is an added dimension to the steady state case. Randolph and Cise (1972) attested to this difficulty in their experiment with an unsteady state Class II crystallizer involving potassium sulphate. They remarked that it was patently impossible to uniquely specify both size dependent functions G(C) and b(C) using only the single size dependent measurement n(C), without further simplifying assumptions or other independent measurements. Lee (1978) was the first to conceptualize a method of attack that set the direction for determining the two fundamental quantities in his study of the Single Seeded Batch Crystallizer (SSBCR). The following study will proceed along the same lines of thought. Equation (31) shall form the basis of our analysis for a constant volume, well mixed crystallizer with seeds in inlet feed; equation (31) reduces to equation (314) n. n(C,t) + [G(C)n(C,t)] = b(c) +  Tt Ti T (314) where T. is the residence time of seed crystals in the feed. Despite the fact that it is easier to start integration of equation (314) at t=O and at =0 because of initial and boundary conditions expressed by equations (315) and (316), practical considerations dictate that we use the lowest experimentally measured time, tw and size, Lw . n(0,0) = 0 n(O,t) = 0 = Uim n(C,t) G(O) = 0 (315) (316) First integrating with respect to time yields t n(C,tm) n(C,t ) +  G() m n(C,T)dt = (tmtw)b(C) w t t 1 1 rm n(lT)dT + 1 n(C,T)dT. T t i t w w A second integration with respect to size results in the following expression: Lk Lk tm k n(,tm)d f I n(C,tw)d; + G(Lk) n(Lk,T)dT L L t w w w t m G(L ) tm w Lk t 1 Lw w w w L n(Lw,T)dT = (tmt ) Lw S Lk n(,T)dTdi + 1 i Lw b( )dC ni(C,T)dTdg. After rearrangement and denoting tw and Lw by tl and L1 respectively, the following equation is obtained: 42 N(Lk'tm) = G(Lk)X(Lktm) + B(Lk) Lk B = L1 b( )dg t S1 ft m =  n(L T)dT Lk Lk L n(C,tm)d I n(C,tl)dC t L1 1 = tt t tl G(L Lk t T(t t l L1 t1 n(t ,T)dTdC L t T (ti L n(,T)dTdc. Ti~mI)1 tlI1 (317b) n(Ll, )dT (317c) Cognizant of the discrete nature of experimental data, we let m = 2,3,...,M and k = 2,3,W. For each value of k, N and X are Mdimensional vectors. A plot of equation (317) in the form N(Lk,tm) versus X(Lk,tm) for each value of k results in a straight line with a slope equal to G(Lk) and intercept, B(Lk), provided N(Lk,tm) and X(Lk,tm) can be completely determined. Thus G and B can be generated at corresponding sizes, Lk 's by repeating the above plot for various values of k. Finally, G and B can each be plotted against size. where (317) (317a) If we had started the integration at zero size (that is L1=0), G(L1) would be zero and N would be completely determined without much ado. Unfortunately N is yet un determined because G(L1)f 0. However, by following the procedure enumerated below, N and hence G and B can be determined. Step 1. First fit particle size data via least squares regression to obtain a functional form for n(,t). One does not need to perform a fitting for ni.(,t) since this is usually given. Step 2. Calculate the various integrals that make up N and X analytically, if possible. Step 3. Because N cannot be calculated until G(L1)isknown, guess a value for G(L1) and calculate N and X. For each value of k>2, plot N(Lk,tm) versus X(Lk,tm). For example, data for k=2 would be as shown below. Data for k=2 N(L2,t2) X(L2,t2) N(L2,t3) X(L2,t3) N(L2,t4) X(L2,t4) N(L2,t5) X(L2,t5) N(L2,tM) X(L2,tM) A plot of N versus X data is illustrated in Figure 2 for the case k=2. The plot in Figure 3 is typical since every value of k yields a straight line with negative slope and positive intercept. An alternative approach is performing a least squares linear regression on the data to determine the slope and the intercept. For a given G(L1) record the sum of square error or the chisquare error for each k and the total chisquare error for all k's. Total chisquare S2 2 error = I Xk where Xk is the chisquare error for a k=2 particular value of k. For perfect data the coefficient of determination of the fit for each k is one; in other words, the chisquare error is zero. Step 4. Repeat step 3 for different guesses of G(L1). Step 5. Plot total chisquare error against G(L1) values and record the G(L1) corresponding to the lowest chisquare error. The value of G(L1) so obtained is more accurate than other values of G(L1). A plot of total chi square error versus G(L1) values might look like that in Figure 4. Step 6. Using the value of G(L1) corresponding to the minimum error, plot B(C) and G(C) each against C, where takes discrete values, Lk, k = 2,3,...,W. Obtain G(Lc) by extrapolation. Typical plots for B(C) and G(C) might look like those in FiguIrs 5 and 6. The first guess in step 3 can be generated by minimizing equation (318), a rearranged version of equation (317), over unknowns, G(L1), G(L2) and B(L2) when k=2. N(L2,t ) + G(L2)X(L2,tm) B(L2) = 0, m = 2,3,...,M (318) For a crystallizer with no seeds in inlet feed equation (317) is still valid with N no longer containing the term in n.(C,T). Similarly,for a batch crystallizer N no longer contains the term in n. as well as the double integral of 1 n(C,T). Thus for the batch crystallizer N is represented by equation (319): Lk Lk n(,t m)d n(c,t )dc t 1 1 = G(L) n(L1,T)dm t t t t 1 l ^ (319) Every other term in equation (317) remains the same. This concludes our analysis for the determination of growth and birth functions in an unsteady state crystallizer. It should be noted that no assumptions were made in the above analysis. The only approximations made were in the minimiza tion procedure used for data treatment. The methods treated above for the determination of growth and birth functions in steady and unsteady state crystallizers also allow for the determination of these functions to be made in terms of process conditions, such as supersaturation, crystallizer impeller speed, etcetera. The growth and birth functions are simply determined for each fixed value of a process condition, and correlations could be obtained, if desired. Good empirical correlations can only come after an accurate method of quantification of growth and birth rates has been established. 46 u rC i1Y "? 1 3 c; N rcu w c) 1 L.il,,L ? CU C' r 0 .Lt I i j I I I j j I I I i I / I ~I ~9 (D c Q ~I LC   _. _i r _  47 MII+ + N Z 4 4, ___ *S, 1I 4 "0 L 0 0 . O 0 0 0 T 4 H 4 S 0 0 II J d HH .,.q 4) 4) 0 C I i ^ (A* =~~~~~ ~ J^FLrrqT .^^ 48 A stability criteria would not be very practical unless a method for the accurate quantification of growth and birth rates was available. This is so because the very equations from which stability criteria are derived contain the two fundamental quantities, growth and birth rates. Dynamics and Stability of the Complex Crystallizer The problem associated with obtaining a good stability criteria is much more complex mathematically than that associated with the quantification of growth and birth rates. Because of the associated mathematical intricacy of crystallizers' stability problems, previous researchers have oftentimes found it necessary to make oversimplifying assumptions, thereby making the criteria so obtained of less practical value. The crystallizer under study is very similar to the RZ crystallizer studied by Randolph, Beer and Keener (1973), and will be referred to as "the complex crystallizer." The complex crystallizer, the schematic diagram of which is shown in Figure 7, is that which is equipped with fines destruction and recycle, seeding, clear liquor advance, and product classification. It is obvious from Figure 7 that, in addition to the population balance equation, other material balance equations are required to completely establish the balance equations of the complex crystallizer. Because the conditions are isothermal and there is no significant energy changes in the system, the balance equations do not include an energy balance equation. Complex Crystallizer Balance Equations The various system boundaries, A, B, D, E, in Figure 7 are those around which material balances are made. The flows P. and Qi (i = 0,1,2,...,6) represent solid and liquid flow rates respectively. The classification functions, h.(C),are size dependent removal functions representing the ratio of crystals removed in a certain size range during classification to that expected with ordinary mixed with drawal. The classification functions, hi.(),can also be viewed as removal probability functions. As an example, a classification function h1(5) is defined by equation (320). Rl (O,Lf) h1() = (320) 0 other sizes Equation (320) indicates that crystals in the size range (O,Lf) have a probability of removal that is (Rl) times that in a mixed withdrawal case while the probability of removal of crystals larger than size Lf is zero. The functions h.(C) are represented below: Rl [O,Lf] hl() = (321) 0 otherwise for all sizes (322) h2(0) = 0 , h3() = h1(?) (323) 1 [O,Lf] h4() = a (Lf,L (324) z (Lp O) 16 [O,Lf] h5(c) = a (Lf'Lp] (325) z(l8) (L p,) where 0 < e < 1 0 < 8 < 1 6 [0,Lf] h6(o) = 0 (Lf'Lp] (326) z6 (Lp' 0) In the above equations Lf and L represent maximum sizes for fines and intermediate size product. Sizes larger than L represent the oversize product. The advantage of the classification functions is that differences in crystal distribution are completely accounted for, thereby eliminating the need to separately indicate the different distributions, ni's. It would be desirable that the volume on which the birth function and crystal size distribution (CSD) is based remains constant, even during transient operation. To achieve this it is necessary to define the void fraction within the crystallizer, Ec, in order to reexpress all volume based quantities on a slurry volume basis. The slurry volume remains constant during transient operations. The void fraction, ec, is defined as the ratio of clear liquor volume, VL, to slurry volume, VS. c = VL/VS (327) It suffices to specify four of the seven volumetric flow rates in Figure 7. From Figure 7 the following relation ships among the various flow rates can be established: Qo = Q3 + Q5 (328) Q1 = Q3 + Q2 (329) Qo = Q1 Q2 + 5 (330) Q4 = 5 + Q6 (331) Q = Ql Q2 + 4 Q6 (332) The flow rates Q Q1' Q2' and Q5 are feed, fines removal, dissolved fines recycle and product volumetric flow rates, respectively. The clear liquor volumetric flow rate is represented by Q3. Applying the general population balance derived in the appendix to Figure 7, the following particle number balance equation results: T (Vsn) + (GVsn) = VBEc + Qhl( 4h4 + Q6h(C)ln(C) + Q no() Since Vs is constant the above equation reduces to n + a (Gn) = BE: + 1_ + Q6 h (,t) + (G) = B~c V h1h h h6(c) n( ,t) s s s Qo + no(W (333) s Let Ti = V /Qi (334) Ti = ai 5. (335) By combining equations (333), (334) and (335), the following equation results: an + r 1 1 1 S+ (Gn) = BE + hl(C) h() + h6(') n(r) t c 1T5 a4 t5 b+6 T5 j + n (336) aOT5 The terms of equation (336) represent, in order, the accumulation of crystals at size C, the net flux of crystals away from size C due to growth, the input of particles due to nucleation, the input or withdrawal of particles as a result of classification, and the input of particles due to solids in feed. Solute and Solid Balance As mentioned earlier the solute and solid balance is required as it complements the crystal population balance equation, both of which, together, form the governing equa tion for the dynamics of the complex crystallizer. The solute and solid balance is derived by setting all inputs in both dissolved and solid form equal to all outputs plus accumulation. A solute balance around boundary E of Figure 7 results in the following expression: dt (Vs cC) = QoC + QC [C+Q5] (k/T)Vs (,t)d 0 (337) where T4 is taken to be the average residence time of crystals in crystallizers. The quantities, Ci, i = 0,2, denote the supersaturation of the ith stream. The volu metric shape factor, crystal density and the supersaturation of the well mixed suspension inside the crystallizer are denoted by k p, and C, respectively. Combining equations (335) and (337) yields the following equation after rearrangement: dE E C C kp c c dc +o + C2 r1 rvp) v 3) S d+ + + n(Ct)3 d dt C dt ToC C2 T 1 5J J o0 (338) A solute plus solid balance around boundary E of Figure 7 yields dCd V c dt = Vs(cp) dt + QoCo + Q2C2 + (kvVs/T o) Vs Ca dt000 I1 n (c)3 dC  Q1C + Q5C + (kvVsp/T5) h5(C)n(C,t) 3f + (k V P/T1) hl(C)n(C,t)C3d where crystals in the feed have sizes in the range 1 to C2' Upon rearrangement the above equation yields dE dc (cp) c dt Ec dt C + 0 T c C 2 + + 2EC (k P/To) ) 2 Ec ~ n o()C ds O h5(C)n(C,t) C3 d 0 S(kv/T5) E (k P/T ) f 31 + (kv hl(s)n(s,t)c3d c 0 1 de Substituting the expression for dT in the following equation: (339) in (cp) in c c de results dt dE c dt S(cp) E c c dC + C dt Tr o C2 +C2 1 1 + + T T5 1 5 [ O n(Ct) 3de Upon rearrangement, the following equation results: cp) c (cp) dC cc dt C dt k p(cp) EC T4C (cp)Co cCTo0 (cp)C2 1(cp) 1 Ec 2 c (1 n(c,t) 3dg 0 C + 1 c C T5:c (cp) E C 1 Substituting the above equation into equation (339) yields dC (c) dC (p)Co (cp)C2 (cp) 1 ,1 dt c dt E CT E cCT2 E+ T k p(cp) n(, 3 C C SC n( T,t)S o T + Ec 4 0o + C2 n (W)3dC c + C E T c 5 c 1 (k P/T5) c 3 (k vp/ l) h (C)n(C,t)C3dI + 0 c h0 l()n( Ct)3d which after a slight algebraic manipulation becomes p dC o C2 pL f + 1 1 (cdt ECo + 2 dt E CT E CT E Tl 5 co c2c 15 + (kvP/o) I2 c 51 k p(cp) 3 + ETC 4 n(,t)3 d c 4 0 3 kvP/T5 no0 de E c c h0 h5()n( r t). 3d3 f0c k= VEP/ hl()n(C,t)3d .C c 0 dC Solving the above equation for d results in equation (340). dC o  + dt E T c 0 C2 C f kv(cp) C2 C 1 + 1 cT4 ECT2 c 1 5 c4 kvC n n (O)C3d h5(C)n(C,t)C 3d 0 c T5 0 Shl()n(c,t) 3d 0 (340) n(c,t)c3dC k C + v E T C 0 k C v Ec T Substituting equation (335) into (340) yields equation (341): dC Co C2 C 1 11 k () 3 dt aT + + n( X,t) dT dt Ecao 5 c2 T5 Ec 15 5 c4T 5 0 k C 2 k C +C n(O) i EC5 0 h5(C)n(c,t)C3dC Ec o o 0 c T 5 0 kvC m 3 S hl( )n( ,t) dC (341) Ec T10 1 Combining equations (333), (334) and (335) results in equation (342): n + (Gn) = B + hl() h4(c) + 1 h6( n(C,t) 1t U ( 5 BX a4 5 a6 5 + n ( ) (342) a oT5 o This completes the derivation of the balance equations (341) and (342) of the complex crystallizer. It must be noted that there were no assumptions in the above derivation. Dynamics and Stability Most of the stability criteria for crystallizers have been derived by making oversimplifying assumptions in the balance equations because of the mathematical complexity. Previous authors have found it very difficult if not im possible to deal with the size dependence of the birth and growth functions. For this reason, they often assume that birth occurs at one size and describe this as the "lumped birth function." Moreover, this size (at which birth occurs) is usually taken as the zero size. The criteria derived with these assumptions, therefore, have little or no practical value. In order to carry out a thorough analysis one must use more complete equations such as (341) and (342) as the basic equations to which stability methods are applied. We shall apply our stability method to equa tions (341a) and (342), which are recapitulated below for easy reference. Equation (341a) is a slightly modified version of equation (341). dC k_ (c) dC + C(l+r) 1 kv(p n(,t) dt Ecoa T5 C25 Ec al5 5 45 kC 2 kC 3 + k v C2n (C)C3dC c5 0C h5()n(C,t)C3dC Ecao 5 (1 oC T5 k C 3 CT I hl(S)n(s,t) 3d( (341a) 1 1 1n(t) + (Gn) = BE + hl(C) h4(C) + h6(, n(,t) t B c al 4) + a6 T5 6T6 5 + no(() (342) Equation (341a) was obtained by combining equations (341) and (343). C2 = (l+r)C 0 < r < 1 (343) Because the mass of dissolved fines is usually very small, C2 is often approximated by C, in which case, r equals zero. However, the form of equation (343) above is chosen to allow for conditions under which mass of dissolved fines is not negligible. The growth and birth functions not only depend on size but also depend on the crystallizer environment as represented by equations (344) and (345). G = g(c)() (344) B = a(c)b(C) (345) where g(c) and (c) denote the environmental dependence of the growth and birth rates respectively, while c(C) and b(C) similarly represent the size dependence. The crystal lizer environment has been taken to be the effective con centration that surrounds the various crystals. Incorporating equations (344) and (345) into equation (342) yields the following expression: an + )1 3n + g(c) (n) = Ec(c)b(C)+rl()n + no () (346) 3t 3 o o 5 where 1 ( 1 1 1 h() h4() + h6(5) (347) a1 4 6 With the appropriate definitions of F2 and r3(C), equation (341a) is rewritten as equation (348): C C CT kC r kp r dC o 2 k v v 3d + + r (5)nC d n4 d c o 5 c 5 c 5 0 c a4 T5 '0 k C 2 + v :caoT5 5 1 n (C)3d d l+r 1 2 CL2 1 1 1 3 4 h5(4 I h5a l()1 (350) Before applying our stability method equations (346) and (348) are made dimensionless by defining the following dimensionless variables: c c v(y) = g(c) ; R(z) = b( b C C c u(y)  t I n no() 6 = T ; n = n ; ni (z)  n n :(c) z = W(z) = a T5 S T 5 T where the "bars" denote some reference variables. The variable, T, represents an appropriate reference time. Upon substitution of the above dimensionless variables, equations (346) and (349) can be transformed into the following equations: where (348) (349) 60 d T v(y) (Wn) = b u(y)R(z) + l() de dz c 5c 1 nin + 0 5 0o 5 d c + r2y + [k n().4] y 3(z)nz3dz de cao5 a 5 [k v n E c V5 0 33 pn(C) 41 1 3 c ca4 5 0 y z2 + [k n(c4] E:cN j ca o5 z 1 terms in square brackets dimensionless quantities Let n. (z)z dz in equations (351) and defined below. SbaT n 4 a3 = k n( ) 4 k vn()4 a4 =  Using the above definitions, equations (351) and (352) are recasted into equations (353) and (354) respectively. n S+ alv(Y) (Wn) = a2u(y)R(z) + Fl(z) 5 + (353) de 1 dz (351) The are (352) (352) 61 dr c* 2y ay 3 a4 3 dy c* + 2 + a3Y r3(z)nz dz 4V5 nz3dz de cao 5 c 5 c 5 0 45 0 S a 3y 2 Ecov5 z1 nin(z)z dz in Without loss of generality the a.'s, i = 1,2,3,4, are set equal to one. dn 8(wn) rl(z)n 1 + v(y ) = u(y)R(z) + + in. d 9z V5 O+ o I5 (355) c + y + y r3(z)nz3dz nz3dz de ECao 5 c 5 Ec 5 J0 3c 4 c 5 J0 2 + 1 co5 2 + c ao Eco5 z 1a V 1 n. (z)z3dz in The boundary and initial conditions for equations (355) and (356) are given below: n(z,O) = no(Z) y(O) = yo lim n(z,e) = 0 Zom n(o,e) = 0 (357) (358) (359) (360) As a reminder, we indicate below the change from the original variables to the dimensionless variables: (354) (356) n + # of particles g v concentration dependent part of growth rate W size dependent part of growth rate a u concentration dependent part of birth rate b R size dependent part of birth rate C z particle size c y supersaturation c c* inlet concentration t e 6 time T5 * 5 residence time Equations (355) through (360) constitute the final forms of the crystallizer governing equations to which our stability method will be applied. We proceed by taking a Taylor's series expansion of some of the quantities in equations (355) and (356): y(6) = y0 + EyI(6) v(y) = vo + evo1(6) u(y) = uo + cuoy1(6) n(z,6) = n (z) + Enl(Z,6) where n (z) is a time independent solution and E is a small parameter. The quantities g' and a' are defined below; O O0 v = dv(y) (361) o dy y=Y = du(y) (362) o dy Y= y=y y O Upon substitution of the above Taylor expansions into equations (355) and (356), the following equations result: ae (no+Ey ) + (vo+EYlV ' (W(no+el)) = E R()(Uo+el) F1(z) 1 + (n +El) + n. (363) o a 5 in d c* 2 1 de (Y o+EY) =c + (Y +EYl) + (y +Ey) S 0 ca V5 :cV5 1 Ec V5 Sr3(z)(n +E l)z3dz 1 ( +E z3dz 0 o Ec a45 0 + o1 3z2 + 1 (y +y ) nin(z)z dz (364) ca V5 in Equating coefficients of equal powers of E on both left and right hand sides of equations (363) and (364) results in the following equations: d rl(z) 1 v d (Wn ) = ER(z)u + nr +  ni (365) o dz o c o v5 o av in 5 o 5 0 + y + 5 y 0 3(z)n z dz zdz Coa 5 c 5 c 5 o 0 4 5 00 z + y 1 nin(z)z dz (366) Eca V5 z in ( W d 1(  I+ V 2 (W 1) + ylvo d (WTIo uCoylR(z) + "1 S1 1 o dZ (367) (367) dyl F2Y1 1 d6 Vc5 EV5 3 1 45 0 3(z)(Ylo0+yn 1)z 3dz = E 0 3 E c L 4V 0 Tlz dz ^0 + 1 z 2 + caov 1 nin(z)z3dz in Rewriting equations (365) and (366) one obtains Ec R(Z)Uo c 0o 1 V + n. voW a05VoW in where h(z) dW h(z) = dz TV d(z Yo c 51 YO aco 5 T 3 nl z dz OQ o C*1 1 + 1 Eco5 z 2 1 2 + 1 n + c5 nc 5 E cV5 nin(z)z 3dz . F3(z)n z3dz (366a) It should be observed that y is simply a constant. The solution to equation (365a) is given by no(z) = e no(z ) + eF() Sz z 0 + 1 + avW n. (x) dx oV5 o n where dno dz + h(z)no dz 0 (368) (365a) (369) I~cuoR(x) eF(x) EcuovRx) [ ~V0o Vrl W V5 vow z F(z) = h(x)dx By choosing zo equal to zero, equation (367) reduces to equation (369): o (z) = eF(z) J(z) (369a) where z F(x) coR(x) 1 J(z) = e +j c (x) dx. 0 v o co 5 W in Equations (366a) and (369a) constitute the pair of time independent solutions, (no,Yo). Considering equations (367) and (368) we make the following substitutions to obtain equations (370) and (371), nl(z,6) = N(z)e yl(e) = moe d XT d d (N(z)e ) + v dz = u'm coo d X _2 d (me = 2 m e de o c 5 o XO AG d (WN(z)e ) + moe v' (Wdo) 0X r0 (z) N , e XR(z) + N(z)e x 05 (370)  _6 1 Xe 3 Xe + 1 T3(z)[m n +y N(z)]e z dz c v5 f0 3 o o 0 1  3 + m c XN(z)e z dz e oe coV5 oe sca4 5 0 co 5 22 *z nin(z)z3dz. z1 Xe (371) 66 Equation (371) reduces to equation (372): 3 1 N(z)z dz  0 5c 5 1 Ec 5 f r 3() 3 S3(z)n z3dz 0 o 0 3(z)y N(z)z3dz 0 o z2 a+ 1 I 2 E c ov5 fz 1 1 3 n. (z)z dz (372) Substitution of equation (372) into (370)results in the following equation:  A S(W()) + N(z) + v (WN(z)) = v5 o dz 1 Ec4 5 N(z)z3dz 1 1 (z) 5I 0rP3(z)y N(z)z dz z) c' 5 0 0(A+Q*) where d Q(z) = E u'R(z) v (W o) c o o dz o 1 1 3(z)noz3dz + 1 0 c o 5 z2 1 nin (z)z dz (375) Upon rearranging equation (373) one obtains equation 21 1I N1 d d  2 + + Q + Q* N(z) + Av d (WN(z))+ Q*v d (WN(z)) V5 1 5 = 1 N(z)z3dz 1 cL4 5 0 c5 0P3(z)y N(z)z3dz 2(z). (376) (376) 1 Ec4V5 m = o 2 cV 5 (373) 2 Q* = c v5 1 + 1 EcV5 (374) Since any piecewise continuous function with finite left and right hand limits and with a square integrable first derivative may be expanded in terms of a complete set of functions, and since N(z) satisfies these restrictions, we choose to expand N(z) as in equation (377): N(z) = I Anf (z) (377) n=0 where f (z) is a member of a complete and orthonomal set of functions. Combining equations (376) and (377) one obtains equation (378): + Q + + Q* = Af(z) + Xv d'z W Af (Z) X v 5 v 5 j n0 o dz n0n n 5o 5 n n=0 o dz n1 5 O n n n=0 I 45 0 n=0n J 03(z)yo (A nf(z))z3dz 2(z) (378) c 5 0 0 Multiplying equation (378) by f (z), integrating with respect to z from zero to infinity, and applying orthogonality principle, one obtains equation (379) after a slight algebraic manipulation. ( X2 AQ*)A (A+Q*) Aa + (A+Q*)v A B m n=O mn mn 1 0 YO CO 0 c45 6m ~I A y + 6 A X 0 c 4 5 .n=0 c 5 n=O n where a = 1; 5mn = m = lo Y6 = o xn = nX, = Fl(z)f (z)f (z)dz (W'(z)f (z) + f'(z)W(z))fm(z)dz f m(z)dz m f (z)z3dz S3(z)fn(z)z3dz It should be noted that stability is guaranteed when the real part of X is positive and the system is said to be asymptotically stable even if the imaginary part of A is nonzero. A nonzero imaginary part of X gives rise to an oscillatory system which may or may not be damped. The expression representedby equation (379) is an infinite dimensional system for the A 's and is represent able in matrix form by (379) MA = 0 , (380) where M represents the matrix of coefficients, and I denotes a vector of A 's. Obviously for nontriavial A 's n n the determinant of M must be zero, that is, det M = 0 . (381) For a finite dimensional matrix, M equation (381) becomes the characteristic equation which is a polynomial equation in X, and of the form, a + a n + ...+a =0. n n1 o (382) Suppose Ao O, then the resulting characteristic equation is a quadratic in A and is denoted below by the a 's. n a2 = 1 (383) (384) a = Q* + oo /V oo 1 00o 5 000o a (Ca o V5 00 5 (Yoa4yoxo)  v oo ) + 6 45 5 o oo' o sc4 5 (385) For A Al 0, the resulting fourth order polynomial equation is similarly denoted below. a = 1 a3 = 2Q* + i0o (386) (387) + 11 a2 = Q* + 2Q*(+oo+Pll) + Aoo  %01 0l (388) + A11 + 1ooPll a = oo 11 + Q* oo+A11) + 2Q*( ool110 + 11o00 + 00oAll OA10 I10A01 (389) a0 = Q*2 (oo110 10 + Q*(QpljAVo+oo I A10o10oA1) + AoAll 01A10 (390) where amn mn v 0o mn A =6P mn mn Pn =(na4YoXn)/( ca 5) 6 = u'I v'L m om om Im = Ec fmR(z)dz 0 S= d Lm fm(z) [no(z)W(z)]dz. For stability purposes, the quadratic equation usually suffices. However, one can also utilize the quartic equa tion, which will show similar behavior as does the qua dratic, since higher and lower order characteristic equations show similar trends with regard to stability. By judiciously choosing f (z) to closely resemble N(z) and by satisfying the boundary conditions of the system, the similarity behavior mentioned above is made even better. Applying RouthHurwitz criterion to the polynomial equation (382) reveals that, in general, a necessary condition for stability is for the a' s to have alternating signs. An additional equation relating the various a 's is usually required to obtain a sufficiency n condition. However, for the quadratic case, the alter nating signs are both a necessary and sufficient condi tion. Illustrating this for the quadratic case, one obtains a2>0, al<0, a >0. Since equation (382) is in such a form that a =1 always, a is therefore always positive. The stability criteria for the quadratic case are a2 = 1>0 (383a) a1 = [Q* + o/o5 Voo < 0 (384a) 0 5(Y0a4Y0X0) a = (a v5Vo ) + 60 o4 J5 > 0.(385a) 0 v5 OO 5 ooo0 Ea S 5 Similarly for the quartic case the stability criteria are given by a4 > 0 (386a) (387a) a3 < 0 a2 > 0 (388a) a1 < 0 (389a) a > 0 (390a) al(a3a2+al) aa > 0 (399) It should be noted, however, that embedded in the a 's n are all the parameters of the complex crystallizer. As a result the effects of the parameters on the stability of the crystallizer can be monitored efficiently. It is pos sible that for some of the parameters, the effects are reflective of the degree of positivity or negativity of the expressions in the stability criteria. In other words, if an increasing value of such a parameter shows an in creasing stabilizing trend on the system, this effect will be reflected by a corresponding increase in the positivity or negativity of the expressions. A better way to monitor the stabilizing or destabilizing effect of a parameter is to apply the RootLocus method directly to the charac teristic polynomial equation (382). The usual data available in the literature are not suitable for a stability analysis such as that described above. In addition to the time dependent number size distribution, one needs to obtain the growth and birth functions in the separable forms shown in equation (344) and (345). G = g(c)(() (344) B = a(c)b(). (345) How one might obtain these separable functions is discussed in the appendix. It should be noted that the above method of analysis does not presume the existence of nucleation models. Instead it requires that the analyst experimentally determine b(c) and g(c). It accommodates the total size dependent functions (() and b(C) and the complex crystal lizer operating parameters. The only assumption is that, like its predecessors, it approximates g(c) and b(c) up to first order in concentration, c. Therefore, in comparison with others, this method is more thorough and is based on very practical considerations. This ends the discussion of our stability analysis. In Chapter IV we devise example data that closely approximate reality to illustrate how to utilize this method. Before proceeding to Chapter IV some discussion about nucleation models is appropriate inasmuch as these models form an integral part of other stability methods. Nucleation Models The three main nucleation models that have been used extensively are the (i) Volmers model, (ii) Mier's model, and (iii) powerlaw model. (i) The Volmer's model, based on thermodynamic considerations, has an Arrehenius type of concentration dependence as evidencedby equation (13) which is recapitulated below for easy reference: B(c) = K2exp(K3/(nc/Cs)2) (13) The concentration dependent part of the birth function, B(c) represented by equation (13), decreases to zero exponentially as the concentration decreases. In general, the birth function depends both on particle size as well as the environment which is represented by concentration. Oftentimes researchers would simply ignore the size dependence and represent the complete birth function by B(c). Volmer's model was based on clear solution and should, therefore, apply only to homogeneous nucleation. To this effect Sherwin et al. (1967) mentioned that the use of such a model in the design of continuous crystal lizers might seem questionable. However, they used the same model in their study entitled "Dynamic Behavior of the WellMixed Isothermal Crystallizer" because they felt that in a bed of large amounts of crystals the dependence of nucleation on total crystal surface is small compared to the nonlinear dependence on saturation. To support the use of the model they cited the work of Rumford and Bain (1960) who claimed the dependence of nucleation on supersaturation in a bed of large crystals of sodium chloride was similar to that of homogeneous nucleation, Rumford and Bain found the lower metastable concentration, cm, for sodium chloride system to be 1.5 gm/liter. The metastable concentration, cm, is that higher than the saturation concentration, cs, and below which all secondary nucleation ceases but above which nucleation increases rapidly. Another version of equation (13) often used consists of a oneterm Taylor series expansion of Zn(c/c ) and is represented by equation (392): B(c) = K2exp(K3/(c/csl)) (392) The above equation assumes (c/csl) is small. (ii) Mier's metastable model is represented by equa tion (393): B(c) = Kn(CCm) c>cm (393) 0 c Mier's model shows the same discontinuity as Volmer's model. Beckman(1976) claimed that both models are qualitatively the same for small deviations about cm, that both are valid for the lower metastable region and that Mier's model is essentially a oneterm Taylor expansion of Volmer's model. It would be interesting to see some justification for this claim. (iii) The powerlaw model was obtained when Randolph and Larson (1962) approximated cm by cs in equation (393), thereby obtaining equation (394), by virtue of which B can be expressed as a function of G defined in equation (213) B = K (cc )P (394) G = K1(cCs) (213) Combining equations (394) and (213), one obtains equa tion (395): B(G) = KNGP = k (ccs) (395) In support of the power law model, they argued that cm is very close to cs for most inorganic systems. On the contrary, however, Rumford and Bain (1960) argued that the approximation, that (cmcs) is very small compared to (ccs) might be valid only for precipitation of insoluble compounds but hardly for crystallizers which, in most cases, operate close to the metastable range. Equation (3 95 ) thus leads to a significant simplification of the mathematics that would be involved in any analysis involving B and G. Another version of the power law model is represented by equation (396) in which MT denotes the suspension density, B(c) = Kc(cc )P Mi = K'GPM (396) cc T (396) The incorporation of MT accounts for the fact that nucleation is by secondary mechanism. Unlike Mier's model and Volmer's model, the powerlaw model is nicely behaved and does not possess any discontinuity. Researchers often use these models in defining steady state dimensionless parameters, b* and g* used in stability analysis. The quantities b* and g* are defined essentially to be the steady state first derivatives with respect to concentration of the birth and growth rates with appropriate dimensionalizing quantities as follows: (c c) c c = o dB(c) b* = dc (397) G d c g* = c) dG(c) (398) Sdc Hence b*/g* is defined by equation (212) : dc c_ b*/g dc) (212) dc c The quantity, co, is inlet concentration as defined earlier. However, any other appropriate concentration parameter could be used in place of c The quantities, b*/g*, for the various models are represented by the expressions below. Volmer's model (second version): 2k b*/g* = i (214) (C 2 Mier's model: cc ) b*/g* = P ( C c>cm (399) (cc ) m Powerlaw model (both versions): din B(c)I b*/g* = P = dn G( (3100) 1 c Randolph and Larson (1971) indicated that a value of 21 for the nucleation to growth sensitivity parameter, i, would represent such an extreme kinetic order causing discontinuity in nucleation rate, that the corresponding higher concentration could be described as an upper meta stable threshold of homogeneous nucleation. Such a value has never been observed in most kinetic studies. Of the three nucleation models above Beckman (1976) claimed that the powerlaw model is the most versatile inasmuch as it is the only model good for both class I and class II systems. However, attemptsbySong and Douglas (1975) to explain their cyclic data with this model failed, thus contradicting Beckman's claim. They were able to explain the same data with Volmer's model by using long retention times thereby approaching very low supersaturations. Similarly Lei et al. (1971a claimed that Mier's model is capable of approximating the behavior of crystallizers with fine traps with either low or high b*/g* values. A look at the above models reveals some striking differences in the expressions for b*/g* as we let c approach cs' Volmer's model (second version): Rim b*/g* + (3101) s Mier's model: Rim b*/g* = 0 (3102) ccs Powerlaw model (both versions): Rim b*/g* = P. (3103) C+C s The resulting b*/g* values show inconsistencies and hence the models are not equivalent for all crystallizer condi tions. Researchers usually show the region of stability on the graphs of the steady state parameters, g* and b*/g*. Yu and Douglas (1975) showed plots with g* and b*/g* taking values in the intervals (0,) and (11,i ) respectively. In practice, of course, one operates at one fixed point (g*, b*/g*) which one hopes is in the stable region. Because the parameters, g* and b*/g*, in theory, can only be controlled inferentially and cannot easily be set to some desired values, especially via the models discussed above, 80 their usefulness as stability parameters is very limited. As a result of the inconsistencies arising in b*/g* for the various models, it is not surprising that one model can explain cyclic data while another cannot. It would be useful to establish stability criteria which depend on more readily accessible parameters. In addition, there are some algebraic errors in the derivations of the stability equations in the studies by Sherwin et al. (1967) and Yu and Douglas (1975) which might invalidate their results. .CHAPTER IV RESULTS Steady State Determination of Growth and Birth Functions One of the purposes of this study is to devise a method for accurately quantifying growth and birth rates in a steady state crystallizer. This method developed in the first section of Chapter III will be illustrated with data taken from the literature. Kambaty and Larson (1978) showed plots of particle number size distribution for magnesium sulphate heptahydrate (MgSO4 7H20) crystals as depicted in figures 9 and 10 Some of the plots were for nuclei generation from the faces or the edges of the crystals, while other plots were for dif ferent impurity concentration, equivalent to different solute concentration. In all these plots nuclei genera tion was by contact. Repetitive contacting was carried out for an interval of approximately ten residence times after reaching steady state and sampling was done at one residence time intervals (one residence time is 9 minutes). The size distribution of the various samples was measured with a model TA II Coulter Counter. The concentration of each experimental run was fixed. Since the purpose of this study is simply to use these data in exemplifying the + 2 E 3 a, S E Z e o (U ^2 Size * Figure 9. Experimental and Fitted Curves for Particle Number Versus Size for Both Edge and Face of Magnesium Sulfate. 83 100.~ 10 t B 1.o oA, Original Dat A, Fitted Curve S 1 B, Original Data Z B, Fitted Curve a *C, Fitted Curve o *C, Original Data "D, Fitted Curve d OD, Original Data o. I I I I 10 20 30 40 50 60 70 Size  Figure 10. Experimental and Fitted Curves for Crystal Size Distribution for Magnesium Sulfate at Four Different Concentrations. A,B,C,D Have Concentration in Descending Order of Magnitude. Table 1 Data for Plot C1=5407.625 y1=45.79739 E in Figure 10 C2=0.6569098 Y2=0.03146495 Original Data Fitted Date 0.25000E 0.850000E 0.4700001 0.350000E 0.280000E 0.250000E 0.230000E 0.1900000 0.150000E 0.115000E 0.750000E 0.500000E Data for Plot F C1=890.0442 yl=20.33548 0.735000E 0.300000E 0.185000E 0.169500E 0.129000E 0.105000E 0.940000E 0.664000E 0.600000E 0.422000E 0.277000E 0.177000E 0.241687E 0.911892E 0.410214E 0.347251E 0.300283E 0.260334E 0.225640E 0.190965E 0.145233E 0.111151E 0.761961E 0.506159E in Figure 10 C2=0.5652139 Y2=0.03509961 02 0.697932E 02 0.329321E 02 0.174400E 02 0.149199E 02 0.127664E 02 0.108451E 01 0.923421E 01 0.766462E 01 0.564760E 01 0.419077E 01 0.275024E 01 0.174263E Size 5.0 7.0 10.0 11.5 14.0 18.0 22.5 27.8 36.5 45.0 57.0 70.0 5.0 7.0 10.0 11.5 14.0 18.0 22.5 27.8 36.5 45.0 57.0 70.0 Data Date Table 2 Data for Plot Ci=26044.52 Y1=34.03978 Original Data Size 7.2 9.2 10.09 14.2 18.3 23.3 28.8 36.2 44.4 56.8 71.7 A in Figure 11 C2=0.8731303 Y2=0.03112719 Fitted Data 0.756792E 02 0.340184E 02 0.287520E 02 0.219863E 02 0.192605E 02 0.164820E 02 0.138886E 02 0.110312E 02 0.854617E 01 0.580956E 01 0.365359E 01 Sin Figure 11 C2=0.7708762 Y2=0.03808283 0.750000E 0.37000E 0.265000E 0.214300E 0.191700E 0.166700E 0.150000E 0.118500E 0.800000E 0.567000E 0.370000E Data for Plot B C1=5378.973 Y1=19.04890 0.35000E 0.200000E 0.137500E 0.118800E 0.106000E 0.800000E 0.625000E 0.413000E 0.350000E 0.229000E 0.129000E 0.353847E 02 0.178923E 02 0.152241E 02 0.111869E 02 0.949264E 01 0.784354E 01 0.636124E 01 0.479902E 01 0.351181E 01 0.219001E 01 0.124169E 01 7.2 9.2 10.09 14.2 18.3 23.3 28.8 36.2 44.4 56.8 71.7 Table 3 Data for Plot C1=1621.810 Y1=15.34124 C in Figure 11 C2=0.7644494 Y2=0.05621562 Original Size Data Fitted Data 0.166700E 0.118800E 0.850000E 0.700000E 0.567000E 0.413000E 0.300000E 0.200000E 0.129000E 0.330000E 0.270000E 0.168360E 02 0.105774E 02 0.942475E 01 0.693656E 01 0.548517E 01 0.414013E 01 0.303902E 01 0.200479E 01 0.129437E 01 0.329706E 00 0.272500E 00 Data for Plot D in Figure 11 C1=6608237. Y1=13.46922 0.138000E 0.800000E 0.680000E 0.583000E 0.430000E 0.336000E 0.236000E 0.174000E 0.930000E 0.430000E 0.170000E C2=1.955293 Y2=0.06041371 0.138004E 0.782786E 0.733949E 0.571179E 0.445860E 0.329619E 0.236432E 0.151200E 0.901307E 0.435574E 0.177062E 7.2 9.2 10.09 14.2 18.3 23.3 28.8 36.2 44.4 56.8 71.7 7.2 9.2 10.09 14.2 18.3 23.3 28.8 36.2 44.4 56.8 71.7 