Ind. Eng. Chem. Res. 2000, 39, 1437-1442
1437
GENERAL RESEARCH Analysis of Concentration Multiplicity Patterns of Continuous Isothermal Mixed Suspension-Mixed Product Removal Reactive Precipitators Qiuxiang Yin,* Jingkang Wang, Zhao Xu, and Guizhi Li Department of Chemical Engineering, Tianjin University, Tianjin 300072, People’s Republic of China
The behavior of concentration multiplicity has been explored for the case of a reactive precipitation process in a continuous isothermal mixed suspension-mixed product removal crystallizer. The process involves homogeneous chemical reaction in first-order reaction kinetics and subsequent crystallization of the product. The number of steady states in the reactive precipitator for various multiplicity regions is determined and exact uniqueness and multiplicity criteria, which are only dependent on the kinetic properties and operational parameters of the process, are developed by use of bifurcation theory. The linear stability of these steady states is analyzed by using the Routh criterion approach. 1. Introduction
2. Model Development
Continuous crystallizers and reactive precipitators are known to show multiplicities of steady states in solute concentration and temperature over a certain range of kinetic and process parameters. Tavare et al.1,2 applied the theories of multiplicity and stability to a continuous mixed suspension-mixed product removal (MSMPR) crystallizer described by macroscopic lumped parameter models. Afterward, Tavere3 extended the analysis to an adiabatic reactive precipitation system and derived conditions for multiplicity and stability using dimensionless variables which include physical properties, operation parameters, and steady-state parameters in the process rather than just physical properties and operation parameters. We tend to think that this may create confusion concerning the effect of the operation parameters on the multiplicity behavior of the system. In our previous work,4 the features of concentration multiplicity in continuous MSMPR crystallizers were analyzed. The crystallization kinetics were described by power law growth and power law magma dependent nucleation kinetic models. Exact uniqueness and multiplicity criteria were developed by using variables which include only physical properties and operation parameters. Recently, Padia and Bhatia5 developed multiplicity and stability criteria for precipitation processes with agglomeration-controlled growth. The present study is aimed at investigating the concentration multiplicity in a continuous isothermal MSPMR precipitator involving chemical reaction between two reactants and subsequent crystallization of the product. The specific objective of this work is to determine the parameter regions in which multiple steady states exist and to analyze the stability of these steady states.
Let us consider a continuous isothermal reactive MSMPR precipitator with two feed streams, each containing a single reactant and premixed at the entry, as shown in Figure 1. The two reactants, A and B, react together homogeneously with first-order reaction kinetics with respect to each of the reactive species as follows:
A+BfC rC ) krcAcB
(1)
Product C being sparingly soluble, a majority of the resultant product C would tend to be precipitated from the reactant mixture. The precipitated product together with small amounts of the product and unconverted reactants remaining in the liquid phase will then be discharged from the precipitator through a common outlet. The crystallization kinetics are represented by power law growth and magma dependent nucleation kinetic models as
G ) kg(cC - c/C)g
(2)
B ) kb(cC - c/C)bMTj
(3)
The dynamic mass balances for reactant A, which is taken to be the key component because of its low concentration, and product C may be written as
dcA 1 ) (cAf - cA) - rC dt τ
(4)
d(cC + MT) 1 ) rC - (cC + MT) dt τ
(5)
Here, we consider an isothermal, constant-volume MSM-
10.1021/ie990461x CCC: $19.00 © 2000 American Chemical Society Published on Web 04/13/2000
1438
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000
Figure 1. Schematic representation of a continuous reactive MSMPR precipitator.
PR precipitator with clear feed streams. Furthermore, we assume that breakage and agglomeration are considered insignificant and that precipitated particles are nucleated in negligibly small sizes. Under these conditions the population balance may be written as
∂n n ∂ ) - - (Gn) ∂t τ ∂L
(6)
Magma density, MT, is related to n, L, and F as
MT ) kvF
∫
∞ 3 Ln 0
dL
(7)
Because of the appearance of the third moment of n in eq 7, it is more convenient to transform eqs 6 and 7 in moment form by defining
mi )
∫0∞Lin dL
(9) i ) 1, 2,...
MT ) kvFm3
(10) (11)
The mass-balance equations contain moments of order of 3 or less and the moment equations of order of 4 or higher do not affect those of lower order. Therefore, we will analyze the series of equations consisting of mass balances and the first four moment equations. For convenience, the following set of dimensionless variables and parameters are introduced:
β ) cBf/cAf ;
Da ) krCAfτ; θ ) t/τ; x1 ) 6kvkg3Fτ3cAf3g-1m0; x2 ) 6kvkg2Fτ2cAf2g-1m1;
dx2 ) -x2 + x1(x5 - x/5)g ) F2(x,P) dθ
(13)
dx3 ) -x3 + x2(x5 - x/5)g ) F3(x,P) dθ
(14)
dx4 ) -x4 + x3(x5 - x/5)g ) F4(x,P) dθ
(15)
dx5 ) -x5 + Dax6(x6 + β - 1)g - x3(x5 - x/5)g ) dθ F5(x,P) (16) dx6 ) 1 - x6 - Dax6(x6 + β - 1) ) F6(x,P) (17) dθ where x is the state variable vector comprised of x1, x2, ..., x6 and P is the parameter vector comprised of R, β, Da, g, b, j, and x/5. The combined effect of physical properties and operation parameters on the dynamics of the state variables can be determined by simultaneously solving the model equations (12)-(17) with appropriate initial conditions. At steady state, these equations yield
Fi(xs,P) ) 0
m0 dm0 )+B dt τ
R ) 6kvkg3kbFτ4cAf3g+b+j-1;
(12)
(8)
Thus, we obtain
mi dmi )+ iGmi-1 dt τ
dx1 ) -x1 + Rx4j(x5 - x/5)b ) F1(x,P) dθ
x3 ) 3kvkgFτcAfg-1m2;
x4 ) 6kvFcAf-1m3; x5 ) cC/cAf; x6 ) cA/cAf Notice that the dimensionless reaction group Da, that is, the Damkohler number, defines the size of the system relative to the reaction rate; it is the ratio of the mean residence time to the characteristic reaction time and determines the reaction performance. The typical range of the kinetic parameters g, b, and j can be taken as 1-2, 1-4, and 0-2, respectively, on the basis of experimental evidence reported in the literature.6 A typical set of actual dimensionless parameters used by Tavare3 are employed in the following analysis of ours. Recasting the series of equations in dimensionless form yields
i ) 1, 2, ..., 6
(18)
which, along with the given physical properties, reaction and crystallization kinetics, and operation parameters, can be solved simultaneously to yield steady-state solutions. 3. Multiplicity and Stability Analysis 3.1. Bifurcation Behavior. The nonlinearity of the expressions for reaction and crystallization kinetics discussed above suggests the probable existence of multiple steady states in the precipitation system. That is to say, the steady-state model, eq 18, may have multiple steady state solutions for some parameter regions which are physically possible. To show the bifurcation behavior of the system, the series of algebraic equations (18) are solved by using the continuation technique improved by Kubicek and Marek.7 Following Kubicek and Marek’s procedure,7 we can obtain the whole relationship between the steady-state solutions of the model equations and the variable parameters in the parameter space which are of interest with appropriate initial values. Thus, the number of steady states in the reactive precipitator for any given set of parameters can be determined. As examples, the schematic bifurcation diagrams for dimensionless magma density x4 versus the operational parameter R for magma order j ) 0.75, 1, and 1.85 are shown in Figure 2. It is seen from these plots that the bifurcation behaviors for the present case are similar to those for the continuous MSMPR crystallizer.4 When j e 1, the solution diagram consists of two solution branches which intersect at the point with R ) 0 for j < 1 and at a certain point with R > 0 for j ) 1. That is to say, for the cases of j e 1, there exist two steady states if R is greater than its value at the intersect point; otherwise,
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1439
Figure 2. Bifurcation diagram of dimensionless magma density x4 versus operational parameter R (g )1, b ) 2, Da ) 10, β ) 1.5, x/5 ) 0.5). (s) stable; (---) unstable.
there is only one. While on the bifurcation diagram for j > 1, there exist three solution branches and two of them intersect at a limit point smoothly and continuously. This means that for this case there exist three steady states when R is greater than its value at the limit point; otherwise, there is only one. It is important to note that there always exists a common steady state with x4 ) 0 (no precipitation) which is independent of the system parameters in the reactive precipitation processes with a secondary nucleation mechanism. In addition, parts a-c in Figure 2, illustrating the bifurcation diagrams of x4 versus j, can transform from one to the other by changing the value of j. When j increases slightly from 1 to 1, branch GH in 2c will draw nearer to PQ; thus, a transition from 2c to 2b will occur. 3.2. Uniqueness and Multiplicity Criteria. From the bifurcation diagrams discussed previously, it is seen that the multiplicity behavior is mainly determined by the value of magma order j and the operational param-
eter R, and that the number of the steady states would change at the intersecting points of the solution branches. Thus, to find the uniqueness and multiplicity criteria means to locate the intersecting points (“bifurcation set”). Here, conventional bifurcation analysis7 can be used. The “bifurcation set” of parameter P can be determined by setting
F8(x,P) ) det J ) 0
(19)
where J is the Jacobian matrix of eqs 12-17 with elements Ji j ) ∂fi/∂xj. At steady state, eq 19 reduces to
F8(xs,P) ) [-Da(2xs6 + β - 1) - 1][jR(xs4)j-1(xs5 x/5)3g+b - (3g + b)xs4/(xs5 - x/5) - 1] ) 0 (20) Thus, simultaneously solving eqs 18 and 20 can give the bifurcation set which classify the parameter space into regions with a different number of steady-state solutions. For different values of magma order j, the bifurcation sets are given as follows:
1440
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000
R˜ ) 0
for 0 < j < 1
R j ) (1 - xs6 - x/5)-3g-b Rˆ )
j-1
(j - 1)
(21)
for j )1
(
(22)
)
3g + b + j - 1 1 (3g + b)3g+b 1 - xs6 - x/5
The stability of the steady states was examined by the previous stability condition and labeled by solid lines for stability and dashed lines for instability in Figure 2.
3g+b+j-1
4. Discussion and Conclusions
for j >1 (23)
where
xs6 ) 0.5[Da - Daβ - 1 + x(Daβ-Da+1)2+4Da]/Da Figure 3 illustrates the bifurcation sets as a plot of R versus j. It is shown that the “boundary set”8 (dashed line in Figure 3) which exists at j ) 1 in the parameter space plots is independent of β, Da, g, and b, and that the multiplicity boundary (solid lines in Figure 3) determined by eq 23 exists only for magma order >1. The multiplicity boundary and the “boundary set” always intercepts at a certain point in the parameter space plots where the values of j and R are determined by eq 22. According to the bifurcation theory,7 when the parameter vector passes through the boundary set from one multiplicity region to another, the number of steady states would increase or decrease by “1”, whereas when it passes through the multiplicity boundary, an increase or decrease by “2” would be expected. 3.3. Linear Stability Analysis. To assess the stability of the steady-state solutions for the series of equations, we perform the commonly employed local stability analysis7 through the linearization of the model eqs 1217 around the steady state and examine the eigenvalues of the Jacobian matrix. The characteristic equation of the Jacobian determinant is
det(J-λI) ) (λ + 1)[λ + 1 + Da(2xs6 + β - 1)](λ4 + a1λ3 + a2λ2 + a3λ + a4) ) 0 (24) where
a1 ) 4 + gxs4/(xs5 - x/5)
(25)
a2 ) 6 + 4gxs4/(xs5 - x/5)
(26)
a3 ) 4 + 6gxs4/(xs5 - x/5)
(27)
a4 ) 1 + (3g + b)xs4/(xs5 - x/5) - jR(xs4)j-1(xs5 - x/5)3g+b (28) According to the stability theory,7 to keep the system stable, real parts of the eigenvalues of the characteristic equation (24) must be negative. Applying the classical Routh criteria, we find that for stability the following conditions must hold: (i) All the coefficients of the characteristic equation must be of the same sign; that is, a1, a2, a3, and a4 should be positive. (ii) a1a2 - a3 > 0. (iii) a1a2a3 - a12a4 - a32 > 0. The Routh criteria provide a necessary and sufficient condition so that absolute stability is guaranteed.
The various dimensionless parameters of the model based on power law kinetics with a secondary nucleation mechanism are R, β, Da, g, b, and j. Among these, R, β, and Da are operating parameters which can be varied by changing the inlet concentration and resident time, respectively. The parameters g and b, corresponding to the order of the growth and nucleation kinetics with respect to supersaturation, and j, which is the order of the magma dependence of the nucleation kinetics, can be measured independently. Types of multiplicity behavior possible and the regions of multiple steady states are determined in these parameters’ space. It is evident from Figure 3 that the parameter space is classified into three types of multiplicity regions by the multiplicity boundary and the boundary set. As a conclusion from the prevous bifurcation and stability analysis, the number of steady states as well as their nature in the different regions of multiplicity indicated in Figure 3 are outlined in Table 1. In region I of Figure 3 or when j ) 1 and R > R j , there exist one stable and one unstable steady states, and the system will attain the stable one. However, in region III or when j ) 1 and R < R j , there is only one stable steady state corresponding to MT ) 0, indicating that no precipitation would take place. It is noticeable that when no precipitation occurs, the behavior of the precipitator is identical to that of a conventional homogeneous reactor. From Figure 3, it is shown that a decrease in kinetic order g or b or an increase in operational parameters β or Da will result in narrowing of region III and lead to lesser possibility for nonprecipitation. Region II, trapped between the multiplicity boundary and the boundary set, exhibits three steady states: the low production one is unstable; the other two corresponding to high production and nonprecipitation respectively are stable. Which of the two stable steady states the system would attain is dependent mainly on the start-up conditions. From the practical point of view, when j < 1, precipitation occurs over the whole range of R, and with gradual increases in R, the production rate increases steadily and continuously as shown in Figure 2a. That is to say, in this kinetic situation, we can always obtain a solid product, regardless of the start-up conditions and the production rate increases with the inlet concentration and/or the resident time. When j ) 1, however, no precipitation would occur until R > R j ; then, a transition from no precipitation to continuous precipitation is initiated, as can be seen in Figure 2b. Practically, only when the inlet concentration is so high or the resident time is so long that the operating parameter R is greater than the critical value, R j , can we obtain solid product. Otherwise, no precipitation would occur. As for the case of j > 1, no precipitation might take place over the whole range of R, but when R > Rˆ , the process can dramatically shift from nonprecipitation to precipitation and vice versa, depending on the start-up conditions. Namely, if the operating parameter R is less than the critical value, Rˆ , we cannot obtain any solid product. Contrarily, if R >R j , the steady-state behavior of the system is depend-
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1441
Figure 3. Multiplicity boundary as a plot of R versus j for different kinetic and operational parameters. (---) boundary set; (s) multiplicity boundary. Table 1. Number of Stable and Unstable Steady States for Various Multiplicity Regions as Indicated in Figure 3 j)1 region
I
RR j
II
III
stable unstable
1 1a
1a 0
1 1a
2a 1
1a 0
a
One of the steady states is MT ) 0.
ent on the start-up condition. For a start-up condition corresponding to a low magma level, the system is likely to attain the nonprecipitation steady state. However, for a start-up condition corresponding to a high magma level, the system attains the precipitation steady state. The initial magma level can be adjusted by adding seeds into the precipitator directly at the start-up of the process. As stated in our previous study,4 the analysis solely based on steady-state balances indicates the possible existence of multiplicity in a continuous reactive MSMPR precipitator. The order term of magma density in the nucleation model appears to be the crucial parameter exerting a significant influence on the multiplicity
behavior. As the model describing a reactive precipitation system is generally represented by a set of nonlinear equations, the strong interaction between them is the main reason leading to the multiplicity of steady states. Although the parameter ranges within which the multiplicity might occur are narrow and the phenomenon of the multiplicity is limited to some special drastic cases or to localized situations, prior knowledge of the multiplicity regions is useful in the design, startup, and control of a precipitation process. Extreme parametric sensitivity in the regions close to multiplicity is another important feature. Designers should be aware of these regions to avoid some false choices of operation parameters that might cause the precipitator to operate under conditions quite different from those intended. Nomenclature a1, a2, a3, a4 ) coefficients in the characteristic equation (24) b ) nucleation order with respect to supersaturation
1442
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000
B ) nucleation rate, no.(s ‚ g of solvent)-1 c ) concentration, mol ‚ g-1, g of solute ‚ (g of solvent)-1 Da ) Damkohler number Fi(x,P ) ) dimensionless function defined by eqs 12-17 g ) growth rate order G ) overall linear growth rate, m ‚ s-1 I ) identity matrix j ) order of magma density in nucleation rate J ) Jacobian matrix kb ) nucleation rate constant, no. ‚ (s ‚ g of solvent)-1 kg ) growth rate constant, m ‚ (s ‚ g of solvent)-1 kv ) volume shape factor L ) crystal size, m mi ) ith moment of n(t,L) MT ) magma density, g of crystal ‚ (g of solvent)-1 n(t,L) ) population density function, no. ‚ (m ‚ g of solvent)-1 P ) parameter vector rC ) reaction rate, mol ‚ (s ‚ g)-1 t ) time, s x1, x2, x3 ) dimensionless m1, m2, m3 x4 ) dimensionless magma density x5 ) dimensionless concentration of product C x6 ) dimensionless concentration of limiting reactant A x ) variable vector R ) operational parameter R˜ , R j , Rˆ ) boundary sets defined by eqs 21-23 β ) dimensionless inlet concentration of reactant B λ ) eigenvalue of the characteristic matrix θ ) dimensionless time F ) crystal density, g ‚ m-3 τ ) mean residence time, s
Superscripts * ) saturation s ) steady state Subscripts A, B, C ) species A, B, C f ) feed
Literature Cited (1) Tavare, N. S.; Garside, J. Multiplicity in Continuous MSMPR Crystallizers, Part I: Concentration Multiplicity in an Isothermal Crystallizer. AIChE J. 1985, 31, 1121. (2) Tavare, N. S.; Garside, J.; Akoglu, K. Multiplicity in Continuous MSMPR Crystallizers, Part II: Temperature Multiplicity in a Cooling Crystallizer. AIChE J. 1985, 31, 1128. (3) Tavare, N. S. Multiplicity in Continuous Crystallizers: Adiabatic Reactive Precipitation. Chem. Eng. Commun. 1989, 80, 135. (4) Yin, Q. X.; Wang, J. K.; Wang Y. L. Multiplicity in Continuous MSMPR Crystallizers. J. Chem. Ind. Eng. (China) 1997, 48, 692. (5) Padia, B. K.; Bhatia S. K. Multiplicity and Stability Analysis of Agglomeration Controlled Precipitation. Chem. Eng. Commun. 1991, 104, 227. (6) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes, 2nd ed.; Academic Press: New York, 1988. (7) Kubicek, M.; Marek, M. Computational Methods in Bifurcation Theory and Dissipative Structures; Spring-Verlag: New York, 1983. (8) Balakotaiah, V. Steady State Multiplicity Features of Open Chemically Reacting Systems. Lect. Appl. Math. 1986, 24, 129.
Received for review June 25, 1999 Revised manuscript received December 21, 1999 Accepted January 11, 2000 IE990461X
