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THE STABILITY OF A MIXED SUSPENSION CRYSTALLIZER WITH CLASSIFIED PRODUCT WITHDRAWAL By ASHISH JAYANT MEHTA A THESIS PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ENGINEERING UNIVERSITY OF FLORIDA 1969 ACKNOWLEDGMENTS The author wishes to thank Dr. A.D. Randolph of the University of Arizona, Tuscon, Arizona, for help in developing the basic equations used for analyzing the problem presented in this work. He would also like to thank Dr. Thomas M. Reed of the Department of Chemical Engi neering of the University of Florida for assisting him in completing the present thesis. TABLE OF CONTENTS Page ii v vii xi ACKNOWLEDGMENTS LIST OF FIGURES NOTATIONS ABSTRACT CHAPTER I INTRODUCTION TO CRYSTALLIZATION Theoretical Approach Dynamic Behavior Problem of Stability The Classified Product Withdrawal Crystallizer II THE POPULATION BALANCE AND THE MIXED SUSPENSION MIXED PRODUCT WITHDRAWAL CRYSTALLIZER Conservation of Particulate Entities Nucleation and Growth Kinetic Growth Models Kinetic Nucleation Models Nucleation and Growth Rate Interaction: The Power Law The Mixed Suspension Mixed Product Withdrawal Crystallizer Steady State Size Distribution Stability Analysis Page III THE MIXED SUSPENSION CRYSTALLIZER WITH 21 CLASSIFIED PRODUCT WITHDRAWAL Description of the Model 21 Population Balance 23 Steady State Population Density 23 Dimensionless Equations 26 Moment Transformation 28 Steady State Moments 32 Constraint on the Growth Rate 35 Expressing the Quantities y and y as 36 Functions of the Moments IV STABILITY ANALYSIS 39 Linearization of Moment Equations 39 Stability Analysis of the Matrix 42 V RESULTS OF THE ANALYSIS 44 RootLoci 44 Effect of Suspension Void 53 Discussion 62 Summary 65 REFERENCES 67 APPENDIX I LEIBNITZ RULE 70 APPENDIX II CRYSTAL PRODUCTION RATE 71 APPENDIX III TRANSFORMATION MATRIX [A] 73 BIOGRAPHICAL SKETCH 74 LIST OF FIGURES Figure 1 Population Density Plot for a Mixed Suspension Mixed Product Withdrawal Crystallizer 2 The Mixed Suspension Crystallizer with Classified Product Withdrawal 3 Population Density Plot for 0 < a < 1 Population Population RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus RootLocus Density Plot Density Plot for a = 0 for a > 1 Plot for a = 1.0 Plot Plot Plot Plot Plot Plot Plot Plot Plot Plot Plot Plot Plot Plot for for for for for for for for for for for for for for = 0.5, B = 1 = 0.1, B = 1 = 0.05, B = 1 = 0.01, B = 1 = 0.005, 8 = 1 = 0.001, B = 1 = 0.0, B = 1 = 1.2, = 1 = 0.005, B = 2 = 0.005, 8 = 3 = 0.005, B = 4 = 0.001, B = 2 = 0.001, B = 3 = 0.001, B = 4 Page 17 Figure Page 21 Block Diagram Depicting the Interrelationship 63 Between the Variables in a Mixed Suspension Crystallizer with Classified Product With d awal NOTATIONS a Constant relating growth rate to size Ag, ..., A4 Constant coefficients in the fifth order characteristic equation [A] Nonsymmetric transformation matrix Co Solute concentration in feed, lb./cu.ft. Cl Solute concentration in outflow, Ib./cu.ft. C Solute concentration in crystallizer, Ib./cu.ft. s D Dissipation function f nt undersize moment (normalized) n f Zero thundersize moment (normalized) fZ First undersize moment (normalized) f2 Second undersize moment (normalized) f3 Third undersize moment (normalized) gn n oversize moment (normalized) go Zero othment (normalized) g1 First oversize moment (normalized) g Second oversize moment (normalized) g3 Third oversize moment (normalized) g3 Third oversize moment (normalized) G Generation function i Sensitivity parameter in the power law, p 1 kI Constant relating growth rate to supersaturation k2, k3 Constants in Volmer's nucleation model vii ,K2,K3,K4,K5 Constant in Mier's metastable model Crystal area shape factor, sq.ft./crystal/sq.ft. Constant relatingthe void fraction to the third moments Constant relating nucleation rate to growth rate Crystal volume shape factor cu.ft./crystal/cu.ft. Steady state parameter relating the growth rate to the second moments Steady state parameters in the matrix [A] Steady state parameter relating the perturbation in dimensionless void fraction to perturbation in the third moments Characteristic crystal size, ft. Mean crystal size, ft. Crystal classification size, ft. Total crystal mass, lb. Nuclei population density, number/ftcu.ft. of mother liquor Undersize population density, number/ftcu.ft. of mother liquor Oversize population density, number/ftcu.ft. of mother liquor Crystal population density, number/ftcu.ft. of mother liquor Number of nuclei per unit volume, number/cu.ft. of mother liquor Sensitivity ratio of nucleation rate to growth rate in the power law Volumetric outflow rate in mixed product withdrawal crystallizer, cu.ft./sec, Volumetric feed rate in mixed suspension crystal lizer with classified product withdrawal, cu.ft./sec. Volumetric underflow rate in mixed suspension crystal lizer with classified product withdrawal, cu.ft./sec. viii s s m S t u v v xi v V W1 W2 X1, ..., X n x, y, z y o y Y Y2 a B Y Volumetric overflow rate, cu.ft./sec. Crystal growth rate, ft./sec. Region of ndimensional space and mdimensional property space Supersaturation in solution, lb./cu.ft. Critical supersaturation below which the solution is metastable Total crystal surface area, sq.ft. Time, sec. Region of space Velocity vector in the spatial region, ft./sec. Velocity component in spatial region along xi direction, ft./sec. Velocity component in property space along i direction, ft./sec. Crystallizer volume, cu.ft. Undersize product weight distribution, lb. Oversize product weight distribution, lb. Coordinates of ndimensional spatial region Coordinates of threedimensional space Dimensionless crystal population density, n/n0 o Dimensionless nuclei population density, n /n0 O Dimensionless undersize crystal population density, n1/no Dimensionless oversize crystal population density, n2/n0 Ratio of undersize to oversize outflow rates, QI/Q2 Dimensionless classification size, L /r r c oo Dimensionless crystal population density of the classification size r Average residence time of oversize particles, V/Q2, sec. r sAverage residence time ofsolids, sec. E Crystallizer void fraction y Point population density of particulate entities 8 Dimensionless time coordinate, t/r X Root of the characteristic equation th Pn n moment of distribution in a classified product withdrawal crystallizer th y0 Zeroth moment of distribution in a classified product withdrawal crystallizer pI First moment of distribution in a classified product withdrawal crystallizer U2 Second moment of distribution in a classified product withdrawal crystallizer 13 Third moment of distribution in a classified product withdrawal crystallizer v Dimensionless void fraction, E/E p crystal density, Ib./cu.ft. a Variance of weight distribution Dimensionless growth rate, r/r X Mean value A1, *...' m Coordinates of the mdimensional property space Superscript Perturbation Subscript o Steady state Abstract of Thesis Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering THE STABILITY OF A MIXED SUSPENSION CRYSTALLIZER WITH CLASSIFIED PRODUCT WITHDRAWAL By Ashish Jayant Mehta March, 1969 Chairman: Thomas M. Reed Major Department: Chemical Engineering The effect of size classification on the dynamic behavior of a mixed suspension crystallizer with classified product withdrawal is investigated by analyzing a simple model of such a crystallizing unit. It is observed that the stability of the crystallizer depends on the relative kinetic rates of nucleation and growth. The results show that classification lowers the sensitivity of the crystallizer to dis turbances and reduces the tendency toward cyclic fluctuations. CHAPTER I INTRODUCTION TO CRYSTALLIZATION Theoretical Approach In continuous crystallization processes, a considerable importance is attached to the technically complex problem of controlling the crystal size distribution. The form of the size distribution depends on the physicochemical characteristics of the crystallizing material and on the constraints imposed upon the crystallizing unit itself. Industrially and otherwise, the study and control of crystallization processes and, hence, of crystal size distribution have been carried out for several years [1]. However, even up to the present time, most of the process techniques are still something of an art and depend to a considerable extent on the ex perience and intuitive skill of the operator. Theoretical approach to industrial crystallization has been late in coming, and papers written on the matter deal mainly with the steady state behavior of operating systems. In 1956, Saeman [2] derived equations for the steady state size distribution in a mixed suspension. He compared the theoretical distribution with that obtained from an OsloKrystal type unit producing ammonium nitrate crystals under conditions of hindered settling, and found a fairly good correlation between the theoretical prediction and experimental results. A similar work was done by Robinson and Roberts [3] who obtained the theoretical crystal size distribution resulting from a series of wellmixed units with nucleation in the first stage only. A comparison of the size distribution obtained from a single unit with that obtained from an industrial ammonium sulfate producing crystallizer showed good quantitative agreement. The special feature that distinguishes continuous crystallization processes from other continuous reactors is the simultaneous occurrence of nucleation and growth. This point was brought out clearly in the work of Saeman and in that of Robinson and Roberts; and, indeed, it has been observed that the control of nucleation rate is the key to con trolling the size distribution in a crystallizer. Saeman [2] has dis cussed the problem of controlling the nucleation rate by removal of crystal nuclei in order to control the size distribution for the case of a crystallizer with classified product withdrawal. Dynamic Behavior Papers dealing with the dynamic behavior of crystallization processes are yet fewer in number. This is not surprising in view of the fact that the processes of nucleation and growth in a crystal lizing system are of a complex nature and do not lend themselves easily to mathematical description. In 1962, Randolph and Larson [4] derived time dependent equations relating crystal population density to size in a mixed suspension of arbitrary volume. These equations are based on the concept of conservation of particle population, and it was pointed out that a particulate system must conserve particles in addition to satisfying laws describing the conservation of mass and energy. Ana lytically, the concept of population balance is of importance inasmuch as it provides a way of characterizing the size distribution by taking into account the effect of nucleation on the size distribution. The steady state form of the population balance equation was solved for the case of a series of crystallizers with nucleation in each unit. The size distribution thus obtained was reduced to that obtained by Robinson and Roberts [3] when nucleation was assumed to occur in the first stage only. Further, for the case of a single unit, the size distribution was found to be identical to that derived by Saeman earlier [2]. In another study, Randolph and Larson [5] solved the transient population balance for the case of an ammonium sulfate producing mixed suspension crystallizer. The transient response of the size distri bution to upsets in nuclei dissolving rate and in production rate was studied by simulating the system equations on an analog computer. The results were interesting inasmuch as it was observed that the size distribution undergoes long term cyclic fluctuations due to pertur bations affecting the system. In an earlier study, Miller and Saeman [6] reported similar observations in an industrial ammonium nitrate producing crystallizer. In the field of continuous polymerization and in fermen tation processes, such observations have also been made [7, 8]. Problem of Stability In view of the observed dynamic characteristics of crystallizing systems involving long term cyclic fluctuations in particle size distri bution, the question of the stability of such systems becomes one of interest inasmuch as it has a bearing on the problem of controlling the size distribution. Randolph and Larson [5] have investigated the sta bility limits of a mixed suspension crystallizer with mixed product withdrawal. These authors point out the importance of the relative kinetic rates of nucleation and growth in defining the stability limits 4 of the system. The stability criterion was expressed in terms of the ratio of nucleation rate to growth rate which, as the authors point out, would be transgressed in the event of a discontinuous jump in the nucleation rate. Such a situation would arise for instance when the supersaturation becomes so high that sudden mass nucleation occurs as the metastable solubility boundary is crossed. The Classified Product Withdrawal Crystallizer Many crystallizers are operated with some kind of a classification device, either in the crystallizer bed itself or in the outlet of the crystallizer. The types of classifiers in general use are either those that segregate particles by gravity separation or those that use centri fugal action. An OsloKrystal type unit may be cited as an example of the first type wherein only the larger particles that settle down at the bottom of the crystallizer bed are withdrawn. Devices using the elutri ative principle belong to the same class. On the other hand, devices such as hydroclones, which make use of the centrifugal principle, fall into the second category. Theoretically, an ideal classifying device would cause a perfect separation of oversize and undersize particles. However, in most of the known classifying devices in which a certain amount of mother liquor is withdrawn to maintain overall material balance, a proportionate amount of undersize particles cannot be prevented from being withdrawn with the mother liquor. Further, a factor that prevents the product from being perfectly uniform is that product withdrawal takes a finite amount of time, and, therefore, the oversize particles exhibit a certain degree of size distribution. An ideal classifier would require an infinite recycle rate for an instantaneous withdrawal of oversize crystal product. 5 The steady state aspects of a crystallizer with classified product withdrawal have been studied previously [2, 9, 10]. Miller and Saeman [6] observed that classification tends to produce crystals of a poor quality. They found that it was impossible to operate acompletely classified ammonium nitrate producingunit continuously, but that it was necessary to withdraw the product periodically in a cyclic manner. SSherwin et al. [11] have analyzed the dynamic behavior and the stability limits of a crystallizer with ideal classifying action. These authors found that,for the classified case, the stability criterion is considerably relaxed in relation to the case of mixed product withdrawal, or in other words, as was pointed out, size classification introduces greater instability into the system. However, the validity of their re sults is questionable inasmuch as, for reasons given earlier, the concept of an ideally classified crystallizer is not realistic when applied to practice. The mathematical description of such a system, wherein there is no withdrawal of undersize particles and the withdrawal rate of over size particles is infinite, is not suitable for a realistic evaluation of the actual situation. The present work attempts to analyze the stability problem of a classified product withdrawal crystallizer with a more real istic model wherein both the undersize and the oversize particles are removed at a finite rate, and there is.,preferential withdrawal of oversize particles. A linearized stability analysis of the system e quations is made, and the analytical approach follows closely that for a crystallizer with mixed product withdrawal as given by Randolph and Larson [5]. The theoretical background, including the concept of population 6 conservation, is developed in Chapter II. The stability problem of the mixed withdrawal crystallizer is also discussed, and,in Chapter III, the classified withdrawal case is analyzed. In Chapter IV, the linearized system equations are investigated for the stability criterion, and the results are discussed subsequently in Chapter V. CHAPTER II THE POPULATION BALANCE AND THE MIXED SUSPENSION MIXED PRODUCT WITHDRAWAL CRYSTALLIZER Conservation of Particulate Entities In Chapter I, it was mentioned that the primary factor controlling the particle size distribution in continuous crystallization processes is the kinetics of nucleation and growth. Now, inasmuch as the product crystal size distribution is also dependent on the residence time proba bilities of particles within the system, a theoretical development at tempting to describe the size distribution in various systems must take into account the interaction between the residence time distribution of particles and the kinetics of nucleation and growth. Now, a particulate system must conserve numbers in addition to con serving mass, momentum, and energy if the size distribution is to be fully defined; and, therefore, equations of the conservation of the last three quantities are not sufficient for formulating the problem of particle size distribution. In general, one can speak of the distri bution of particulate entities over any associated property. This property can be, for instance, the size of a particle, or age, or any quantity that varies as a function of time. A population balance of particulate entities derived with sufficient generality can be used to describe the property distribution in various constrained systems [12, 13, 14]. Such a general derivation of a population balance due to Randolph [15] is given in the sequel. 7 Consider the distribution of particulate entities that are continu ously distributed through a region U of ndimensional space Xn(xi, X2, ..., xn) and through a region V of mdimensional property space '(V l 2, ..., m) A point number density in the region R = U + V can, therefore, be de fined as n = n(x1, x2, ..., Xn, 1i, P2, 2, ), Next, the velocities of propagation of the entities along the ith spatial axis and along the ith property axis may be expressed as axi .i V = = t x. at at Further, consider the generation and dissipation functions operating throughout the region R Generation: G = G(X1, X2, ..., x I, 2>2, ..**, n' t) Dissipation: D = D(x1, x2, ..., xn, 1i, P2, ., ** ', t) which, of course, are functions of time. Now, a particulate balance may be written as {Accumulation} = {Input} {Output} (21) If a Lagrangian approach is taken, that is, if it is assumed that the region R translates with local velocities v the particulate balance Equation (21) becomes d ndR = (G D) dR (22) R R where dR = dxI dx2 ... dxn dil dp2 ... dmi or equivalently, dt // /// dxm ... d*1 dxn "' dxl R X1 x2 ... xn i 2 ...*** m (23) Equation (23) may be differentiated by repeated application of the Leibnitz rule (See Appendix I) to give n m (a + a (Vx n) + (a p + D G} dR = 0 (24) at ax a 8 R i=l i=l which further reduces to n m S)+ D G = 0 (25) 7t ax S+ xi i i=l i=l inasmuch as R is an arbitrary region. Equation (25) is a population balance for a particulate system but is of a form that is too general for application to the size distribution problem in a crystallizer. However, before Equation (25) is constrained to represent population density distribution in a crystallizer, it is essential to discuss the role of nucleation and growth in a crystallizing system. Nucleations and Growth The driving force in the process of crystallization is the super saturation of the solute material in suspension. Further, inasmuch as the nucleation rate and the growth rate both are functions of super saturation, one would expect a functional relationship of the following type, expressing the dependence of nucleation rate and growth rate on supersaturation. Growth rate: dNo dt = f(s) (26) Nucleation rate: r = g(s) (27) where No is the total number of nuclei per unit suspension volume. The functions f(s) and g(s) must, of course, be determined experimentally. Kinetic Growth Models Miller and Saeman [6] found a linear function of the form r = kls (28) describing the dependence of growth rate on supersaturation. The work of Saeman [2] and of Van Hook [16] has supported this kind of relation ship. Kinetic Nucleation Models Volmer found an Arrhenius type kinetic law [17] describing nu cleation behavior. His model is based on thermodynamic considerations and may be expressed as k3 En(s + C ) dN k2 e .s (29) where k2 and k3 are constants, and C is the solubility concentration. Another nucleation model, which is a simplified representation of Equation (29), is Mier's metastable model [17] dN_ d = k4(s s )P (210) where k4 is a constant, and sm is the critical supersaturation below which the solution is metastable. There is some experimental evidence supporting this model [4]. At this point, it should be mentioned that the process of nu cleation in a mixed suspension crystallizer is a rather complex phe nomenon involving both homogeneous and heterogeneous nucleation, the latter process being the formation of nuclei on a crystal surface. Volmer's model was derived for homogeneous nucleation only, and, therefore, its applicability to a mixed suspension crystallizer is questionable. In a heterogeneous nucleation process, the nucleation rate should depend both on concentration and total crystal surface. In the presence of a large number of crystals, the dependence on surface area is relatively less important than the dependence on supersaturation. Rumford and Bain [18] have investigated the nucleation of sodium chloride in a bed with large crystals. These authors found that the dependence of nucleation rate on supersaturation is very similar to that of homo geneous nucleation. For such a system, Volmer's model would well express the process of nucleation in a crystallizing system. The ad vantage of using Equation (29) is that it is smooth over the entire concentration range. Nucleation experiments on an ammonium sulfate system have indi cated that the metastable supersaturation level in such a system is almost nonexistent [4]. In this event, Mier's nucleation model would reduce to dN = k4SP (211) where the value of p has been given as 4 for the ammonium sulfate system. Sherwin et al. [19] have offered criticism for the use of Equation (211) in dynamic studies on the grounds that in a large number of systems metastable supersaturation cannot be neglected. It will, however, be shown in the sequel that this problem does not arise in determining the interactive relationship between nucleation and growth since in awellmixed crystallizer both the nucleation rate and growth rate depend on the same level of supersaturation. Nucleation and GrowthRate Interaction: The Power Law The nucleation rate can be related to the nuclei density n (which is actually the population density of particles of "zero" size) as follows dN= L ) = rn0 (212) dt at L where r = 3L/3t is the crystal growth rate expressed as the rate of change of the characteristic size L of the particle; the characteristic size being any one spatial dimension of a collection of geometrically similar particles that can be used to characterize the size of the particles. Combining Equations (28) and(211) results in the power law dN= kNr (213) dt  where kN is another constant. In Equation (213), inasmuch as the nucleation rate is expressed as a function of the growth rate, the level of supersaturation on which the former quantity depends does not enter into the expression. The advantage of using the nucleationgrowth power model over that of Volmer is clearly seen from the form of Equation (213) which is mathematical simplification of Equation (29). The ex ponent p in Equation (29) is actually the sensitivity ratio of nu cleation rate to growth rate when both are expressed as functions ofsupersaturation. As was mentioned in Chapter I, this ratio is im portant in defining the stability limits of a mixed suspension crystal lizer. Substituting Equation (212) into Equation (213) results in the expression n = kNr (214) which is another form of the power law and where i = p 1. Equation (214) expresses the nuclei density in terms of the growth rate and is a convenient boundary condition for solving the population balance equation as will be shown later, and it is known to hold over a wide rangeof concentrations for many inorganic systems [20]. The Mixed Suspension Mixed Product Withdrawal Crystallizer Prior to investigating a crystallizer with classified product re moval, the case of a crystallizer with mixed product withdrawal must be reviewed since the problem ofthe stability of the former is ana lyzed along lines similar to the latter. The general population balance Equation (25) may now be reduced further to suit the description of a mixed suspension mixed product withdrawal crystallizer. In a crystallizer of arbitrary volume V, the processes of particle nucleation and growth can be characterized by three orthogonal spatial coordinates x, y, and z, and a size coordinate L defining the particle size. Hence, in this case, Equation (25) reduces to an a__(rn) an + V.(,n) + ( + D G = 0 (215) at dL where n is the population density in a mixed suspension crystallizer. v is the vector velocity of the particles in the spatial region, and r = 3L/8t is the growth rate. If it is further assumed that the generation and dissipation terms are absent, Equation (215) reduces to n + V. ) + (rn) = (216) at 3L This equation can now be integrated over the volume V of the crystal lizer, giving S+ (rn)} V + V.(n) dV = 0 (217) at 3L V V Then, by the divergence theorem, the volume integral in Equation (217) can be converted to a surface integral I V (n) dV = f ndS (218) where the velocity v is orthogonal to the surface S. The following two contributions to this integral are important. a. The term due to the change in the volume of the system, which is ndV (219) dt b. The integral over an arbitrary number of inflows and out flows, which yields a term involving the flow rates n (220)  Qini i=1 where by convention inflows are considered positive. Equation (217) thus becomes n n + _(rn) at L i=l d(anV) = 0 Qini + dt The following constraints may now be applied without much loss of generality. 1. The feed is clear, that is, there are no nuclei in the feed. 2. The crystallizer volume is constant. (221) Under these conditions, Equation (221) reduces to Dn + () + qn = 0 (222) at aL V where Q is the volumetric withdrawal rate. In order to uncouple the second term in Equation (222), it is necessary to assume a certain functional dependence of the growth rate on the size. Sherwin et al. [19] have assumed a growth rate model of the form r = 1 + aL (223) where a is a constant. Canning and Randolph [21] have used the same growth model to derive expressions for the size distribution in a mixed suspension crystallizer under various constraints. Steady State Size Distribution Under conditions of a steady state and assuming that McCabe's AL law [24] holds, that is, the growth rate is independent of size, Equation (222) can be integrated to give *QL n = n e V (224) o o where the subscript o refers to the steady state condition. Equation (224) suggests that in a mixed suspension crystallizer with mixed product withdrawal the population density decays exponentially with size. In Figure 1, the logarithm of the population density is plotted as a function of size. The slope of the straight line is QL/V, and the intercept on the yaxis is, of course, the nuclei density n Several systems are known to exhibit this kind of behavior [6, 22]. O 0 O O FIGURE  0 a. a 0 0 FIGURE I: CRYSTAL SIZE POPULATION DENSITY PLOT FOR A MIXED SUS PENSION MIXED PRODUCT WITHDRAWAL CRYSTALLIZER SStability Analysis The form of Equation (222) suggests that even under conditions of steady state energy and material balance transients in the size distribution are expected. Randolph and Larson [5] investigated the problem of the stability of size distribution in a mixed suspension mixed product withdrawal crystallizer. The linearized moment trans forms of the system equations, obtained from the population balance Equation (222) in conjunction with the initial condition Equation (224) and the boundary condition Equation (214), were solved on an analog computer. Step changes in production rate and in nuclei dissolving rate were introduced into the system, and the resulting disturbances in total suspension population, crystal length, and area were plotted as functions of time. The plots showed wavelike cyclic fluctuations in these quanti ties propagating with time, and the periods of the disturbances were found to be three or four times the order of magnitude of residence times of particles in industrial units. Transients have been observed in commercial crystallizers [6]; and, in view of this observation, the question of the stability of an operating crystallizer becomes one of interest. The same authors analyzed the system equations of a mixed product withdrawal crystallizer by application of the RouthHurwitz stability criterion and found that the system exhibits instability when p > 21, that is, when the nucleation rate varies with a power of growth rate greater than twentyone. Sherwin et al. [19] have also investigated the same problem. Their treatment of the situation is more general since they have considered the effects of seeded feed and of nuclei dissolving on the system stability. These authors have expressed the stability criterion in terms of a ratio b/g which they have defined as d dNo ds dt o o (225) b/g = dr dN ds o dt 0 Substituting Equation (212) into Equation (225) gives b/g = kNprop do ro p(226) dr ()o kN ro For a clear feed, it was found that the system becomes unstable when b/g > 21, an observation which is in agreement with that of Randolph and Larson. A seeded feed was found to improve the stability in certain cases. An interesting observation in connection with the effect of liquid voidage was that change in the liquid void fraction does not have a significant bearing on the stability criterion. In a qualitative sense, this observation appears to be correct. As it will be shown in Chapter V, a fluctuation in the voidage causes a perturbation in the rate of nucleation in the opposite sense, such that the two effects tend to nullify each other, so that the dynamic behavior of the system is not altered significantly. Another observation made was that the system sta bility increases significantly if the nuclei have a finite size and that increasing the nuclei size enlarges the region of stable operation. The important point brought out in these studies is that the rela tive kinetic order of nucleation rate to growth rate plays a significant role in determining the stability limits in a mixed product withdrawal 20 crystallizer; and that,for unstable operation, the criterion of p > 21 establishes the stability limit of the system wherein the feed is clear, and the nuclei have a size of "zero" dimension. CHAPTER III THE MIXED SUSPENSION CRYSTALLIZER WITH CLASSIFIED PRODUCT WITHDRAWAL Description of the Model The concept of population conservation, developed in Chapter II, may be applied to a mixed suspension crystallizer with classified product withdrawal. Any model describing such a system should conform to simple mathematical formulation and, at the same time, must be flexible enough to fit a realistic situation. Consider a mixed sus pension crystallizer with a classifying device as shown in Figure 2, wherein particles below the classification size Le (undersize) are withdrawn at a rate Q1, and particles above the classification size (oversize) are removed at a rate Q2 such that Qi = aQ2 where a is the ratio of the two flow rates, and Qi is also the volumetric withdrawal rate of the clear liquor for maintaining material balance within the system. The volumetric feed rate is Q0, and the crystal lizer has a constant volume V. The classifying device C with a recycle is shown as an external component of the crystallizer; however, size classification may occur internally as in an OsloKrystal type unit. In either case, the basic mechanism of size classification remains un altered and, thus, lends itself to the same mathematical description. FIGURE 2: THE MIXED CLASSIFIED QI ,C SUSPENSION CRYSTALLIZER WITH PRODUCT WITHDRAWAL It isfurther assumed that thecrystallizer is operatingunder conditions of steady energy throughput; that is, the process is isothermal, and is under a steady state mass balance with constant material densities. Population Balance Two population balances are needed to describe the particle popu lation distribution within such a system, one for the undersize and one for the oversize. These are 9an S3n Qin1 + r + = 0 L in [0, L ] (31) at aL V c and an2 an2 Q2n2  + r + = +0 L in [L m] (32) at aL V c where nj and n2 are the population densities of the undersize and over sizeparticles, respectively, and where the growth rate is assumed to be independent of size. Further, let F = V/Q2 be the average residence time of the particles based on the drawdown of oversize particles. Equations (31) and (32) then become 3nI Sn1 an1 + r  0 L in [0, L ] (33) at aL r c and an2 an2 n2 + r +  + 0 L in [Lc, =] (34) at aL r c Steady State Population Density At this point, it is of interest to analyze the steady state behavior of Equations (33) and (34). At steady state, these equations become dnio anlo dL r r oo dn2o n2o dL r r oo L in [0, Lc] L in [Lc, m] (35) (36) which may be integrated using the following boundary conditions o B.C. 1 at L = 0, nlo = nlo B.C. 2 at L = L C, nlo = n2o Thus, 0 n10 = no  aL r r 0o e (37) L (la)L rr n20 = n20 e (38) Several cases of these expressions are of interest. Case 1 (a < 1) In this case, as shown in Figure 3, the drawdown rate of oversize particles is greater than that of undersize particles. In other words, there is size classification,and the slope of Equation (38) is greater than the slope of Equation (37). In Case a = 0, there would be no withdrawal of undersize particles so that Equation (37) would be repre sented by a horizontal line. This idealized situation is shown in Figure 4. z 0 O 0 0 0 0 0 0 i CRYSTAL FIGURE 3: POPULATION DENSITY PLOT FOR O< < I. SI  0 0  Lc CRYSTAL SIZE FIGURE 4: POPULATION DENSITY PLOT FOR SIZE C= 0 Case 2 (a = 1) This, of course, is the same as the case of mixed product with drawal inasmuch as the rates of removal of oversize and undersize particles are equal. Figure 1 depicts this case. Case 3 (a > 1) This case would arise if the undersize particles were prefer entially removed from the system. As is shown in Figure 5, in such an event,the slope of Equation (38) would be less than that of Equation (37). Case 4 (L = 0, L c+ o) In both of these cases, there is really no size classification, and, therefore, Equations (37) and (38) are reduced to those for a mixed product withdrawal. Dimensionless Equations The following dimensionless groups may be defined in order to ex press Equations (33) and (34) in dimensionless forms t n L r S y = x and =  r no rr r 0 00 0 Hence, Equations (33) and (34) become 3yl yil  + + ayl = 0 x in [0, B] (39) and ay2 aY2 _ + X + Y2 = 0 x in [8, ] (310) where 8 = L /r r is the dimensionless classification size. c oo no z w o z o x 03 JI 0 0 Lc CRYSTAL SIZE FIGURE 5: POPULATION DENSITY PLOT FOR C( > I MomentTransformation The nth moment of the population distribution is defined as the integral I (Population density) LndL, n = 0, 1, 2, ... (311) which for a classified product withdrawal crystallizer may be written as the sum of two integrals L 0 pn f niLndL + f n2L dL, n = 0, 1, 2, ... The physical significance of the moments is as follows L c 1'O = f 0 L c .1 = njdL + f L c 00 njLdL + L c L kA12 = kA f nL2dL + kA 0 (313) n2dL = The total number of particles in the system n2LdL = The total length all the crystals the system (314) n2L2dL = The total crystal area in suspension (315) where kA is the area shape factor. L c pk v3 = pk f nL3dL + pk f n2L3dL = The total mass of crystals in 0 L suspension c (316) (312) where k is the crystal volume shape factor, and p is the density. Now the weight distributions of the product crystals may be written as Wi = pk nlL3 (317) W2 = pkn2L (318) so that the mean crystal size, with respect to the weight distribution, is L c SWILdL + WLdL 0 L c L =  (319) L c fWdL + / W2dL 0 L c which may be written in terms of the moments as L =  (320) P3 Similarly, the variance of the weight distribution is L c f (LL)2 WIdL + / (LL)2 W2dL 0 0 a = (321) c SWdL + WdL 0 L L L 00 L CO c C c f W1L2dL + f W2L2dL 27 { WIdL + W2dL} + 2 {j WdL + W2dL} 0 L 0 L 0 L c C C L 00 C f WidL + f W2dL 0 L c or in terms of the moments P5 V4 2 0 =  M(322) Finally, the coefficient of variation may be written as V5 42 2 { } C.V. (323) 11I4 13 The moments in Equations (320), (322), and (323) can be deter mined by a sieve analysis of the product, and, conversely, a knowledge of these moments is sufficient to characterize the performance of a crystallizer. Further, as it will be shown, the partial differential Equations (39) and (310) can be transformed into a set of ordinary differential equations involving the moments. These moments, which represent the total properties of crystal number, length, area, and mass through Equations (313), (314), (315), and (316), respectively, de fine the stability of the system. Normalizing Equation (312) about the steady state gives B 0 normalizedd ylxndx + y2xndx fn + g (324) Next, multiplying Equation (39) by x" and integrating over the size range x in [0, 8] gives nayl ayl {x +x  + axny} dx = 0 ae ax (325) 0 which reduces to a set of ordinary differential equations involving the undersize moments fn This set of equations is dfo o  + 4(yyY) de df d + 0(n  de + afo0 0 nf )n + af = 0 n1 n where 0 n o yl " Y 1x0 n o Yx=8 Similarly, Equation (310) may be moment transformed to yield a set of ordinary differential equations involving the oversize moments gn. Co f yY2 aY2 {x ;0 + axn 7 + xny2} dx = 0 86 n Ya yax (328) which gives dg0 d6 *Y + go = 0 (n = 0) (n = 0) (n # 0) (326) (327) (329) dgn + ng dg a (n Y + ngn1)+ g = 0 (n # 0) Steady State Moments Under conditions of a steady state, Equation (324) becomes 8 (u ) yf x dx + fyx dx = f + g (no)normalized = f xndx + f2xndx = fno + no 0 8 (330) (331) The steady state normalized population distributions yl and Y2 are the dimensionless forms of Equations (37) and (38). Substituting these into Equation (331) gives (nonormalized no normalized = eax ndx+ e (1a)x xndx 0 8 Expanding the integrals on the right of Equation (332) gives f e [, + 2+n n(n1)8 +n2 no n+1 a a2 a3 a+ + n n ]}, n n+l a a n = 0, 1, 2, ... (333) gno ea [Sn + n8nl + n(nl) 8n2 + ... + n!8 + n!], no n = 0, 1, 2, ... (334) so that the first few moments may be written as foo = (1 eu ) f 0 o = (332) flo 1= 72 e". (I.+E as)l f2o l [1 ( + a0 + 2a")] f3o = e6 (1i + ac + 2i + 2 ( )] (335) and aB go80o = e glo = ea (B + 1) g2o = ea8 (82 + 28 + 2) g3o = ea (83 + 382 + 60 + 6) (336) At this point, it is interesting to note the relation between the solids residence time rs and the liquor residence time in a crystal lizer with classified product withdrawal. The average solids residence time is defined as r = crystal holdup volume s crystal volumetric production rate L = c k { nloL3dL +f n2oL3dL} 0 L s "L 'c S  k r v aIQ2 noL3dL + Q2 n2oL3dL} (337) 0 L which upon normalization of the involved quantities becomes fo + 83of y io + g3o r=f30 + 930 I [. r (338) af30 + 30 Q2 af30 + g83 For the case of mixed product withdrawal, a is unity, hence, r r (339) or in other words, the solids residence time and the liquor residence time are equal, which is expected. Alternatively, the production rate can be written in terms of the growth rate and the total surface area as (See Appendix II for the derivation) Production rate = ( k) r kA 2oP (340) so that the solids residence time may be expressed as L c k{ nloL3dL + fn2oL3dL} p 0 L r = c (341) s L c kA nloL2dL + n2oL2dL} (o) p 0 L c Further, if the shape factors kA and k are defined in terms of a unit cube then k = 6k (342) A v so that Equation (341) is reduced to 1 f3o + g30o r I2o +g2 (343) s 3ry tf2o + 820 Now, for the case of a crystallizer with an ideal classifying action, Equation (343) is further simplified, giving 1 f3o L r = c (344) s 3ro f2o 4r o or L 4F r (345) C s Thus, the classification size is four times the growth rate per solids residence time, which was derived by Saeman [2]. Constraint on the Growth Rate The growth rate can be expressed in terms of the moments inasmuch as it is constrained by material balance. An overall material conser vation for the crystallizer may be expressed as {mass in) {mass out} = {accumulation} (346) Here, the accumulation term involves two factors, namely, the accumu lation of mass in supersaturation and the accumulation of mass on the crystals. The former factor is small compared to the latter and, therefore, may be neglected. Thus, the material balance becomes QoCo (a + 1) Q1C1 = ( ) kA2VP (347) and Equation (347) reduces to QoC (a + 1) Q1C1 () kAn (r ro)3 (f2 + g2) Vp (348) A QoC 00 or Kg r = 2 + 2 (349) f2 + 82 where Q0CO (a + 1) Q1CI Ko (350) () kAno (roro)3 Vp Thus, the growth rate is inversely proportional to the total crystal area. The normalized growth rate becomes f20 + 820 r f2 + g20 (351) r 0 2 +982 Expressing the Quantities y and yo as Functions of the Moments Now that the growth rate is a known function of the moments, Equations (326), (327), ,329), and (330) can be used to solve the stability problem. These equations form a closed set at n = 2 inasmuch as the growth rate involves the second moments of distribution. However, the quantities y and y or the values of the normalized population density at the classification size and the nuclei density, respectively, are still unknowns in these equations; and their behavior as functions of the moments must be known. Hong's approximation The following approximation expressing the relationship between y and the moments is due to Hong and Larson [23]. Consider the integral 8 8 8 x dx = xyi ydx Y fo (352) 0 0 0 wherein the righthand side is obtained by the method of integration by 37 parts. Further, changing the limits of the function on the left gives f Yl f x dx = xdyl o y0 (353) so that / xdyI o y = BY fo (354) The mean value theorem may and yl is bounded. Thus, now be used since x is continuous in [0, 8], Y Y fo f xdyl (Y yO) X 0 y x may now be approximated as fl x =  so that S fo Y fl By fo = (y yO) and, therefore, 2 0 fo y fi Y afo f (358) which expresses y in terms of the zeroth and the first moment of distri bution. Alternatively, y may be expressed as a function of the oversize moments inasmuch as Y = y1(B) =" y2(B) (355) (356) (357) (359) and, therefore, following the same approach 00 Jx T By go Yx (360) and letting X = gl/g0 gives 2 go ( ag go ) (361) (B80 gl) One way to establish the validity of Equations (358) and (361) is to obtain the steady state value of y by substituting the corresponding values of the undersize and oversize moments in these equations. In both cases, yo ea8 (362) which is correct. yv and the power law In order to express the normalized nuclei density yO in terms of the moments, the power law of Equation (214) is used since it is assumed to relate correctly the nuclei density to the growth rate. Normalizing Equation (214) gives 0 i y =i (363) and the value of 0 may be substituted from Equation (351) to give f20 + g2o0i S= 2 +2 2 ) (364) The stability problem can now be analyzed since all the quanti ties are now known in terms of the moments of distribution. CHAPTER IV STABILITY ANALYSIS Linearization of Moment Equations In Chapter III, the basic population balance equations, describing the distribution of crystal entities within a mixed suspension crystal lizer with classified product withdrawal, were converted into a set of linear differential equations in terms of the moments of distribution. It was also mentioned that a set of these equations involving up to the second moments close satisfactorily, and that this set is essentially sufficient for a stability analysis. This set of equations is df0 de + (Y y0) + afo = 0 (41) dfI d + (Byo foo) + afl = 0 (42). df2 d + (82y 2flo) + af2 = 0 (43) dgo d yo + go = 0 (44) dgl do 4(8o + goo) + gl = 0 (45) dg2 d (82yo + 2glo) + g2 = 0 (46) 39 where f02 y0fo S= fo f (47) and Sf20 + 820 i S f2 + g2 (48) The set of Equations (41) through (48) are now linearized by introducing linear perturbations in the system variables. A linearly perturbed quantity may be expressed as the sum of the steady state value and the perturbation. Thus, fn (t) fno + n(t) (49) gn(t) gno + g(t) (410) y(t) = Yo + Y(t) (411) y(t) = 1 + y(t) (412) 4(t) = 1 + $(t) (413) where the quantities with ~ are the perturbations, and the steady state values of the normalized (about the steady state) nuclei density and growth rate, of course, are unity. The linear equations therefore be come dfo de + (1 yo) y + y auf (414) 41 dfl  d y + (2flo 8Yo) + f ofl (415) df2 d 82 + (2flo 82y) 0 + 2f1 af2 (416)  = y + y 80 (417) dgi d = 8 + (8o + goo) + g0 g1 (418) dg2 d 2 27 + (2yo + 2glo) 4 + 2g g2 (419) S= K1(f2 + g2) (420) yo i (421) y = K2fo + K3Y0 + K1fI (422) where K 1 (423) f2o + g20 2foo (424) K2 o (424) K2 = foo flo flo K3 = (425) 8foo flo and y 1 K4 = (426) Bfoo flo It is interesting to note that all the quantities involved in Equations (414) through (426) are dimensionless, and, therefore, the stability of the system can be defined entirely in terms of dimensionless quantities. 42 Equations (414) through (419) are linearly dependent inasmuch as the moments fo and fl are related to each other through Equation (422). Thus, the five independent equations may be expressed in the vector form as fl f1 f2 f2 d = [A] (427) de go go gl gl 82 82 where the transformation matrix [A] is obtained by substituting the o values of y and y from Equations (421) and (422) into Equations (415) through (419). The matrix elements are shown in Appendix III. The rank of the non symmetric matrix is five. In the matrix, the d6oeffic~int K5 is the ratio flo/f0o. Stability Analysis of the Matrix The matrix [A] has a'fifth order characteristic equation of the form A5 + AAX4 + A3X3 + A2A2 + AjX + A0 0 (428) in which the coefficients of the terms are functions of the dimensionless steady state parameters. A stability test can be made by applying the RouthHurwitz criterion to Equation (428). However, inasmuch as the process of evaluating the coefficients of Equation (428) is somewhat tedious, an alternative method consisting of evaluating the roots of the characteristic equation, or the eigenvalues, was used in the present study. As will be seen in the sequel, 43 an investigation of the rootlocus plots obtained from the eigenvalues is sufficient in defining the stability limits of the system. The digital analysis was made on an IBM 360/50 computer. CHAPTER V RESULTS OF THE ANALYSIS RootLoci The rootlocus plots obtained for various chosen values of the para meters a and B are shown in Figures 6 through 20. Only the complex roots are plotted since they are important in determining the stability limit. In a stable system, all the roots lie on the lefthand side of the com plex plane, and a crossover to the righthand side implies that the system is exhibiting instability. In the present analysis, only the com plex roots crossover to the righthand plane at the stability limit. The following cases are of particular interest: Mixed product withdrawal (a = 1) Figure 6 shows that when the value of a is unity, the observed be havior is that of a mixed product withdrawal crystallizer; that is, the system exhibits instability at p > 21. The value of the dimensionless classification size B is, of course, arbitrary in this case. Size classification (a < 1, 0 = 1) Figures 7 through 13 show plots for a values ranging from 0.5 to 0.0. In all of these cases, the system is found to be more stable than when the withdrawal is mixed. It is seen from these figures that all the rootloci are nearly identical. From this observation, it may be concluded that the system stability increases rapidly when a takes on a value that is less than unity, that is, when the oversize withdrawal rate is larger than the undersize withdrawal rate. However, little 44 45 IMAGINARY AXIS 0* 0 c o  m w 0 w O I I 0 II. D 0 UI D CL 4 0L U I I 4 IdO 0 I) II 16 O 0 Co 0 I LL 0 o OL 0 0 U: 0 Sa 0 U 0 J LL 0 0 0 0J 0L 0 Q, 0 0 U. 0 j CL U O 0 o 0 0 CD 0t 0 g 0 IO .0 0 o 0 I 0 0 0L 3 CD ti 0 0 0 I 0 U 0 0. C, O 0 I O I w LL O 0 0 0 LL I 0 a. 0 O 0 0o 0 O ..I O 0: 0 IL I cu 0 0 0 U 1 0. a0 =3 change in the stability criterion occurs once a is less than unity. Another point of interest is the case of noundersize withdrawal (a = 0). Figure 13 shows that even for this idealized case, the system continues to exhibit a high degree of stability. Preferential removal of undersize particle (a > 1. 8 = 1) This case would arise if the classifying action was reversed, that is, if the underflow was made higher than the overflow. Figure 14 shows that, inthis event, the stability criterion is relaxed, and the cross over occurs at P = 17 for a = 1.2. Effect of varying the dimensionless classification size The following cases were investigated: For a 0.005; 6 = 2, 3, and 4 For a = 0.001; 8 = 2, 3, and 4 The rootloci for these cases are shown in Figures 15 through 20. As expected from the previous results for values of a less than unity, the stability criterion is found to be independent of the value of a, but changes with increasing value of the dimensionless classification size. A comparison of Figures 15 and 18 with Figures 11 and 12 shows that for a given value of a the system stability increases greatly when 8 changes from a value of 1 to 2. For higher values of B, there is little change in the stability criterion. Effect of Suspension Void It would be interesting to investigate the effect of fluctuations in the suspension void on the stability of the system. The void fraction e in the crystallizer may be expressed in terms of the moments as e = 1 kM(f3 + g3) (51) 0 II I 0 * oC C) O 0 h 0 J 0 0 IL UF II 0 oj 0 LL 0 O 0 0 I 0 Q: w a: 0 arj 0 in Z o U o 0 J 0 cr :3 LL O 1 O LJ L0. 37 IMAGINARY AXIS 0 0 0 O N c; O to a:4 0 _\_.___ CL = 0 c. ___ Ix z u 2 .J < Oa r 0 CLj of it 0 LL I 0 i ao CO 0 3 at 0 z w h: w 41 2C a oL 0 w J $ 0 rr 0 0 II 0 0 o 0 0 1 0 0 0 Ow O LJ r. D, 0 II 0 It Z5 O O U U) j C) 0 J L 0 0 CD Oh) w? cr where k = k nrr (52) M V 0 0 Further, the nucleationgrowth rate law of Equation (214) must be re defined inasmuch as the nuclei density is based on the free volume. Thus, nO = e i (53) Introducing the dimensionless void fraction V (54) EO Equations (51) and (53) may be made dimensionless to give 1 kM(f3 + g3) 1 kM(f3o + g30) and y = i (56) The linearized forms of Equations (55) and (56) are v = K6(f3 + 83) (57) where kM Kk (58) K6 1 M(f3 + g3o)58) and y = + i0 (59) So that, substituting Equation (57) into Equation (59) gives yo i K6(f3 + g3) (510) If Equation (510) were used instead of Equation (421) in stability analysis, a matrix of rank seven would result inasmuch as the third moments of distribution are now required to close the set of moment equations. Discussion Figure 21 is a block diagram depicting the interrelationship be tween the various quantities involved in defining the size distribution. The level of supersaturation s is determined by the feed rate Qo and the total crystal surface area A. The rate of nucleation and growth, in turn, depend on the supersaturation. Finally, as is seen on the right, the size distribution is defined by nucleation and growth through the population balances. However, since the total surface area is a function of the size distribution, there is an internal feed back in the system as shown. This feed back is the cause of the cyclic insta bility in the crystallizer [4]. The important point brought out in the present analysis of the stability problem is that size classification tends to stabilize a crystallizer. The residence time probabilities of the undersize and of the oversize particles are important in defining the stability limit of the system. If the residence time of the oversize particles is re duced relative to that of the undersize, the system shows increase stability. On the other hand, if the smaller particles are drawn down at a relatively lower rate, the stability criterion is relaxed. Sherwin et al. [11] with their model of an ideal classifying device having no drawdown of undersize particles and with an infinitely high rate of withdrawal of oversize particles find that classification 63 OLJ W (0 a 0 ZW z z SIz 0  u, z 0 w I X CO :3 w sZ <0 a O CO 0 actually introduces a high degree of instability into the system. They give a value of 2.3 for the sensitivity ratio of nucleation rate to growth rate as compared to a value greater than 21 found in the present analysis as the criterion for stability. How ever, for reasons given in Chapter I, the validity of their work is in question inasmuch as their model does not describe the function of an actual classifying device in a realistic manner. It has been shown [4] that in a single mixed suspension crystal lizer, the dominant crystal size, with respect to the weight distri bution, occurs at 0 = 3. From the present analysis, it appears that the classified product withdrawal crystallizer operates with a high degree of stability when size classification occurs near the region of dominant crystal size. The stability criterion defined in terms of the sensitivity ratio p would be violated, as it was mentioned earlier, in the event of a discontinuous jump in the nucleation rate. This would happen, as was pointed out, at high supersaturations when a fluctuation in the feed concentration causes the supersaturation to cross the metastable limit, resulting in mass nucleation. Industrial crystallizers do operate near this region of mass nucleation in order to achieve a maximum growth rate consistent with low nucleation rate. Mass nucleation is therefore likely to occur when there are upsets in the feed or in the temperature of the system. Miller and Saeman [6] have made such observations in a crystallizer with a classified bed producing ammonium nitrate crystals. Referring again to the block diagram of Figure 21, it is apparent that a positive perturbation in the suspension void in the system will cause an increase in the level of supersaturation inasmuch as the suspension area is reduced. This, in turn, will increase the nu cleation rate so that the surface area will tend to increase again. Qualitatively, it therefore appears that the inclusion of the effect of suspension void in the analysis will not have a significant bearing on the stability criterion. Summary A realistic model of a mixed suspension crystallizer with classi fied product withdrawal was analyzed for stability limits. The moment transforms of the population balance equations for the model were solved for their characteristic roots on a digital computer for a range of classification sizes and at different residence times of undersize and oversize particles. The stability criterion was expressed in terms of the sensitivity ratio of nucleation rate to growth rate, and it was found that, in general, size classification tends to stabilize the classified product crystallizer. The following points are relevant to the current discussion: 1. The form of the nucleationgrowth rate law used for the analy sis is of considerable importance in predicting the dynamic behavior of the system inasmuch as the stability limit of the system is defined in terms of the relative order of nucleation rate to growth rate. Un fortunately, as was pointed out earlier, experimental information on this important interactive effect is scarce, and the power law used in the present study is just an approximation. 2. The residence time was arbitrarily based on the drawdown rate of oversize particles. It is likely that a more definitive way of parameterizing the problem exists. 66 3. Hong's approximation for correlating the population density at the classification size with the moments of distribution may not correctly represent the actual dynamic behavior. In view of the above considerations, it is clear that the present work needs further elaboration. Also lacking, among other things, are essential experimental results for gaining insight into the dynamic behavior of the classified product crystallizer. More information in this area is necessary for a greater clarification of the problem. REFERENCES 1. Perry, J.H., Chemical Engineers' Handbook, 4th Ed., p. 177, McGrawHill, New York, 1963. 2. Saeman, W.C., A.I.Ch.E. Journal, 2, p. 107, 1956. 3. Robinson, J.N. and R.C. Roberts, Can. J. Chem Eng., 35, p. 105, 1957. 4. Randolph, A.D. and M.A. Larson, A.I.Ch.E. Journal, 8, p. 639, 1962. 5. Randolph, A.D. and M.A. Larson, Chem. Eng. Progr. Symp. Series, 61, p. 147, 1965. 6. Miller, P. and W.C. Saeman, Chem. Eng. Progr., 43, p. 667, 1947. 7. Finn, R.K. and R.E. Wilson, Agr. Food Chem., _, p. 66, 1954. 8. Thomas, W.M. and W.C. Mallison, Petrol. Refiner, p. 211, 1961. 9. Bransom, S.H., Brit. Chem. Eng., p. 838, 1960. 10. Han, H.D. and R. Shinnar, Paper presented at the 60th National Meeting of the A.I.Ch.E., April, 1967. 11. Sherwin, M.B., R. Shinnar, and S. Katz, Paper presented at the A.I.Ch.E. Symposium on Crystallization, Houston, February, 1967. 12. Cha, L.C. and L.T. Fan, Can. J. Chem. Eng., 41, p. 62, 1963. 13. Frederickson, A.G. and H.M. Tsuchiya, A.I.Ch.E. Journal, 9, p. 459, 1963. 14. Hulburt, H.M. and S. Katz, Chem. Eng. Sci., 19, p. 555, 1964. 15. Randolph, A.D., Note to the Editor, Can. J. Chem. Eng., p. 280, 1964. 16. Van Hook, A., Crystallization: Theory and Practice, ACS Mono graph 152, p. 94, Reinhold, New York, 1961. 17. Shinnar, R., J. Fluid Mech., 10, p. 259, 1961. 18. Rumford, F. and J. Bain, Trans. Inst. Chem. Engrs., 38, p. 10, 1960. 19. Sherwin, M.B., R. Shinnar, and S. Katz, A.I.Ch.E. Journal, 13, p. 1141, 1967. 20. Randolph, A.D., A.I.Ch.E. Journal, 11, p. 424, 1965. 21. Canning, T.F. and A.D. Randolph, A.I.Ch.E. Journal, 13, p. 5, 1967. 22. Murray, D.C. and M.A. Larson, A.I.Ch.E. Journal, 11, p. 728, 1965. 23. Hong, K.C. and M.A. Larson, Paper presented at the A.I.Ch.E. Joint Automatic Control Conference, Minneapolis, May, 1963. 24. McCabe, W.L., Ind. Eng. Chem., 21, p. 30, 1929. APPENDICES APPENDIX I LEIBNITZ RULE Leibnitz's rule for the differentiation of definite integrals is given as bt(t) b(t) f(x,t) dx = a(t) {(x.t) + d dx t l t "xd[ f(x,t)]A dx at dx dt (1A1) where the function f(x,t) is integrated over the variable limits a(t) to b(t).. a(t) APPENDIX II CRYSTAL PRODUCTION RATE The total mass in the crystallizer is L L L M WIdL + / W2dL = k n1L3dL + k p n2L3dL (3A1) 0 L 0 L c c Now at specified drawdown rates and at a given classification size, the total mass is constant. Therefore, L O c dt dt nL3dL + n2L3dL 0 (3A2) 0 L c or L anL an2 SL3 dL + L3  dL = 0 (3A3) 0 L Substituting the values of the partial differential quantities within .. the integral from Equations (33) and(34) gives L an1 can1 p an2 n2 L3{ r  } dL + L3{ r } dL = 0 4) 0 Lc These integrals upon evaluationbythemethodofintegration by parts give Lc a f nlL3dL +f n2LdL 0 L r = L (3A5) c 3r f njL2dL + n2L2dL 0 L c So that the production rate may be expressed as L Production rate a nL3dL + n23dL rk (3A6) 0 L where the shape factors kA and k are expressed in terms of a unit cube so that kA = 6kv A v (3A7) r1 o o o 0 0 SI + 0 I 0 0 N o N _ + SboN + 1 4 + ,a ca N m crv m m CY * H  M M I I E44 00 0 0 0 0 N H + I O Iy g~ ,6 O <0 0 N + 44 1 + 0 0 0 C ca + 0N % + (M% '. + E4 Scn cN c ar e( a t t^ m (nl 1t b H Ln N1 Ni N I I txi I M MG 3r BIOGRAPHICAL SKETCH Ashish Jayant Mehta was born July, 1944, at Bombay, India. He was graduated from Fellowship High School, Bombay, in June, 1960. In June, 1964, he received the degree of Bachelor of Science in Chemistry from the University of Bombay. He then studied at the University of California at Berkeley, and there received the degree of Bachelor of Science with a major in Chemical Engineering in December, 1960. In January, 1967, he enrolled in the Graduate School of the University of Florida where he had a departmental assistant ship. From June, 1967, until the present time, he has been working for the Department of Coastal and Oceanographic Engineering of the University of Florida. During this period, he also completed the final phase of the present work. Ashish Jayant Mehta is a member of the student chapter of the American Institute of Chemical Engineers. This thesis was prepared under the direction of the chairman of the candidate's supervisory committeeandhasbeenapproved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Master of Science in Engineering. March, 1969 Dean, Graduate School Supervisory Committee:  UNIVERSITY OF FLORIDA 11111lll I 11111i11111ill l1111 ll ll l11111 1 1111 3 1262 08554 0341 