On the influence of mixing on crystal precipitation processes—application of the segregated feed model

Chemical Engineering Science 57 (2002) 821 – 831

www.elsevier.com/locate/ces

On the in uence of mixing on crystal precipitation processes—application of the segregated feed model Rudolf Zauner1 , Alan G. Jones ∗ Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK Received 10 October 2000; received in revised form 8 November 2001; accepted 17 November 2001

Abstract The segregated feed model (SFM), a compartmental mixing model, is used to predict the in uence of mixing on crystal precipitation. In this method, the population balance is solved simultaneously with the mass balances using crystallisation kinetic, solubility and computational uid dynamics (CFD) mixing data. Mean properties are calculated for the three di7erent zones of the reactor (two feed zones and bulk zone). It is predicted that during continuous operation, the product particle size exhibits oscillating behaviour before reaching steady state after about ten residence times. In contrast, the second moment (surface area) sharply increases during the 9rst residence time and remains constant thereafter. Di7erent mixing conditions are modelled by varying the mesomixing and micromixing times, which can be regarded as convective and di7usive exchange parameters between the compartments of the reactor. The overall nucleation rate is found to strongly depend on the mixing conditions, as it depends in a highly non-linear manner on the level of supersaturation. In consequence, the nucleation rate varies over three orders of magnitude between ‘good’ and ‘poor’ mixing conditions. Using the SFM, the e7ect of di7erent feed points, feed rates, feed tube diameters, energy dissipation rates, impeller types and vessel sizes on the nucleation rate and the particle size during crystal precipitation is illuminated. Predictions of the model compare favourably with batch and continuous experimental data for calcium oxalate. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Population balance; Precipitation; Micromixing; Mesomixing

1. Introduction

2. Macro-, meso- and micromixing

The ows of reacting uids through agitated vessels widely used in industrial production can be very complex. For practical purposes, idealised models are necessary in order to describe the interaction of the resulting ow pattern with a chemical reaction or precipitation process. These interactions take place on di7erent scales, ranging from the macroscopic scale (macromixing) to the microscopic scale (micromixing). In this work, a compartmental mixing model, the segregated feed model (SFM), will be discussed in detail, coupled with the population balance and used to model the in uence of meso- and microscale vessel hydrodynamics on continuous and semibatch crystal precipitation characteristics.

Macromixing acts on the scale of the whole vessel and conveys uids through regions where the turbulent properties vary, while mesomixing re ects the inertial-convective disintegration of large eddies that contain partially segregated uid (Baldyga & Bourne, 1992). For the model used here, we will interpret the inverse of the time constant tmeso (mesomixing) as a transfer coeIcient for mass transfer by convection. Micromixing is regarded as turbulent mixing on the molecular level. It comprises the viscous-convective deformation of uid elements, followed by molecular di7usion (Baldyga & Pohorecki, 1995). Baldyga, Podgorska, and Pohorecki, 1995 have analysed the complex relation between the characteristic time scales for micromixing and the Kolmogorov time scale of eddy mixing. For simplicity, it is assumed here that the inverse of the time constant tmicro (micromixing) can be interpreted as a coeIcient for mass transfer by di7usion.

∗ Corresponding author. Tel.: +44-171-419-3828; fax: +44-171383-2348. E-mail address: [email protected] (A. G. Jones). 1 Current address: Austrian Research Centres. A-2444 Seibersdorf, Austria.

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 4 1 7 - 1

822

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

3. Meso- and micromixing models

3.2. Physical models

Over the last 50 years, many meso- and micromixing models have been proposed to describe the in uence of mixing on chemical reactions on the meso- and molecular scale. Most of them fall into one of the three categories discussed below (Villermaux & Falk, 1994). A brief description of the main ones is as follows.

As their name suggests, these models are based on the physical principles of di7usion and convection, which govern the mixing process. According to the ow pattern, the reactor is divided into di7erent zones with di7erent ow characteristics. Baldyga and Bourne (1984) developed comprehensive models based on the engulfment–deformation–di7usion (EDD) theory incorporating the e7ects of vorticity. Entering material is engulfed by bulk material forming vortices, subsequently deformed and stretched to form slabs and 9nally exchanges mass by molecular di7usion. Baldyga and Bourne (1989) show that under some conditions, engulfment becomes the rate-determining step in micromixing. They describe the molecular mixing based on the spectral interpretation of mixing in an isotropic turbulent 9eld. The concentration spectrum indicates that molecular di7usion starts between the viscous-convective and the viscous-di7usive subrange, and becomes dominant as the scale becomes smaller. Fluid elements in this subrange are laminar deformed by stretching and form slabs. van Leeuwen (1998) developed a compartmental mixing model for precipitation based on the engulfment theory mentioned above. The author obtains the mixing parameters between the feed and the bulk zone from the ow characteristics in the reactor and subsequently calculates moments and mean sizes of the precipitate.

3.1. Phenomenological models This type of model derives from the residence time distribution (RTD) concept of macromixing, which is applied on a microscopic level using idealised zones and exchange ows. The mixing parameters do not usually have any physical relevance and are determined experimentally. The coalescence–redispersion (CRD) model was originally proposed by Curl (1963) and is based on imagining a chemical reactor as a number population of droplets that behave as individual batch reactors. These droplets coalesce (mix) in pairs at random, homogenise their concentration and redisperse. The mixing parameter in this model is the average number of collisions that a droplet undergoes. Another popular phenomenological mixing model is the interaction by exchange with the mean (IEM) model, originally suggested by Harada, Arima, Eguchi, and Nagata (1962). Micromixing takes place by exchange between all points in the system i.e. between feed regions (well-mixed batch zones) and a mean environment (bulk) according to a mixing time constant. In the three and four environment (3E and 4E) models (Ritchie & Togby, 1979; Mehta & Tarbell, 1983), the reactor is divided into two segregated entering environments and one or two fully mixed leaving environments. The mixing parameter is the transfer coeIcient between the environments. Simple multi-environmental or ‘compartmental’ precipitation models employing mixing parameters based on turbulence theory appear to have been 9rst formulated by Pohorecki and Baldyga (1983, 1988) for batch and continuous operation, respectively, and used for interpretation of experimental data. Garside and Tavare (1985) used the IEM model to predict extreme cases of micromixing during precipitation. In the SFM (see later), the IEM model is reduced to exchange between the feed regions and a mean environment (bulk). Franck, David, Villermaux, and Klein (1988) developed a two-compartment mixing model with the exchange ow rate (recycle number) between the two compartments as the only adjustable parameter. They applied their model successfully to the precipitation of salicylic acid. Chang, Mehta, and Tarbell (1986) compare di7erent phenomenological mixing models and demonstrate their analogies and similarities to the theory of turbulence.

3.3. Numerical models Local information on the ow 9eld in a reactor is important and can be obtained using computational uid dynamics (CFD). Wei and Garside (1997) studied the precipitation of barium sulphate using CFD. Fluctuations of concentration and hence micromixing were neglected, and in order to account for the small-scale e7ects of mixing a very 9ne grid resolution had to be chosen. Consequently, it was found therefore that the computational demand often increases enormously, even for very simple reactor geometries. By coupling the ow problem with a mixing model, it should become possible to solve the micro- and mesomixing problem numerically. The model chosen to model mixing e7ects in precipitation processes in the present paper is the SFM referred to above. It was 9rst used to investigate micromixing e7ects of consecutive-competitive semibatch reactions (Villermaux, 1989), subsequently applied to predict the e7ects of mixing on semibatch polymerisation (Tosun, 1992) and used to model the semibatch precipitation of barium sulphate without accounting for agglomeration and disruption (Marcant, 1996). The SFM was found to be particularly suitable for modelling mixing e7ects, as it combines the advantages of both the compartmental IEM model and the physical models. It has been used recently to

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

predict mixing e7ects and scale-up in continuous and semibatch precipitation (Zauner & Jones, 2000a, b).

Qf 1 reaction plume f1

4. Segregated feed model In the Segregated feed model (SFM), the reactor is divided into three well-mixed zones: two feed zones f1 and f2 and the bulk b (Fig. 1). The feed zones exchange mass with each other and with the bulk as depicted with the ow rates u1; 2 , u1; 3 and u2; 3 , respectively, according to the time constants of micro- and mesomixing. As imperfect mixing leads to gradients of the concentrations in the reactor, di7erent supersaturation levels in di7erent compartments govern the precipitation rates, especially the rapid nucleation process. Using the SFM, the in uence of micro- and mesomixing on the precipitation process and properties of the precipitate can be investigated. Mass and population balances can be applied to the individual compartments and to the overall reactor accounting for di7erent levels of supersaturation in di7erent zones of the reactor. The individual volumes of the compartments feed plume 1 (f1 ), feed plume 2 (f2 ) and bulk (b) and the total volume of the precipitation reactor in steady-state operation can be written as

Qf 2 reaction plume f2

u1,2

u1,3

u2,3

bulk b

Qb Fig. 1. Segregated feed model (SFM).

with the exchange ows between the compartments uj; f1 b =

Vf1 cj; f1 Vf (cj; f1 − cj; b ) + 1 ; tmeso1 tmicro1

(9)

Vf2 cj; f2 Vf (cj; f2 − cj; b ) + 2 ; tmeso2 tmicro2

(10)

(Vf1 + Vf2 )(cj; f1 − cj; f2 ) : t12

(11)

dVf1 Vf1 = 0; = Qf1 − dt tmeso1

(1)

uj; f2 b =

Vf2 dVf2 = Qf2 − = 0; dt tmeso2

(2)

uj; f1 f2 =

dVb = Qf1 + Qf2 − Qb = 0; dt

(3)

dVtot =0 dt

(4)

4.1. SFM applied to continuous precipitation Using the following assumptions, the SFM will be applied to continuous precipitation: k

with Vtot = Vf1 + Vf2 + Vb :

823

(5)

The mass balances for a species j in the three zones give d(Vf1 cj; f1 ) = rj; f1 Vf1 + Qf1 cj;0 f1 − uj; f1 b − uj; f1 f2 ; dt

(6)

d(Vf2 cj; f2 ) = rj; f2 Vf2 + Qf2 cj;0 f2 − uj; f2 b + uj; f1 f2 ; dt

(7)

d(Vb cj; b ) = rj; b Vb − Qb cj; b + uj; f1 b + uj; f2 b dt

(8)

r • Instantaneous reaction: A + B→P. The reaction between the two feed solutions occurs instantaneously as soon as they are mixed. • Homogeneous conditions within the feed zones and the bulk. Within each compartment, there are no gradients in the 9eld of supersaturation. • The feed plumes are very small compared with the bulk volume (i.e. very short residence time in these zones). The nucleation process has the highest dependence on supersaturation. Since the local supersaturation will be at its highest in the feed plumes, it is assumed to be the only kinetic process occurring therein, the other processes being relatively slow. All four processes, viz. nucleation, growth, agglomeration and disruption, take place in the bulk zone.

824

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

• Direct di7usional mass exchange between the feed plumes is negligible as the feed points are too far apart and di7usion is too slow to play a signi9cant role in the di7usional mass exchange between the feed streams, leading to t12 → ∞. • No in uence of the particles on the ow 9eld is included as the suspension density is very low (approx. 0.1–1%). Consequently, the following model equations are obtained. 4.2. Volumes of compartments

d(Vf2 cB; f2 ) dt 0 =Qf2 cB; f2 −

−

Vf2 (cB; f2 − cB; b ) c − Bf0 2 kv L30 Vf = 0; tmicro2 MWc 2

(12)

Vf2 = Qf2 tmeso2 ;

(13)

(19)

Vf cB; f1 Vf (cB; f1 − cB; b ) d(Vb cB; b ) + 1 = −Qb cB; b + 1 dt tmeso1 tmicro1 +

Vf1 = Qf1 tmeso1 ;

Vf2 cB; f2 tmeso2

Vf2 cB; f2 Vf (cB; f2 − cB; b ) + 2 tmeso2 tmicro2

−Bb0 kv L30

c c 1 V b − Gb ka m2 Vb = 0: MWc 2 MWc (20)

4.4. Mixing times Vb = Vtot − (Vf1 + Vf2 ):

(14)

4.3. Mass balances 4.3.1. Species A Vf1 cA; f1 Vf (cA; f1 − cA; b ) d(Vf1 cA; f1 ) 0 = Qf1 cA; − 1 f1 − dt tmeso1 tmicro1 −Bf0 1 kv L30

c Vf = 0; MWc 1

(15)

Vf cA; f2 d(Vf2 cA; f2 ) Vf (cA; f2 − cA; b ) =− 2 − 2 dt tmeso2 tmicro2 −Bf0 2 kv L30

c Vf = 0; MWc 2

(16)

Vf2 cA; f2 Vf (cA; f2 − cA; b ) + 2 tmeso2 tmicro2

−Bb0 kv L30

c c 1 V b − G b ka m2 Vb = 0: MWc 2 MWc (17)

Vf1 cB; f1 tmeso1 Vf1 (cB; f1 − cB; b ) c − Bf0 1 kv L30 Vf = 0; tmicro1 MWc 1

@nP; b @nP; b = −Gb + Baggl + Bdisr @t @L nP; b Qb Vtot

(21)

with the birth terms for agglomeration and disruption
L2 L Kaggl n(Lu )n(Lv ) dLu Baggl + Bdisr = 2 0 L2v
∞ + Kdisr S  (Lu ; Lv )n(Lu )n(Lv ) dLu (22) Lv

0

d(Vf1 cB; f1 ) dt

−

One of the advantages of using a compartmental model is that the ‘full’ population balance, including terms for size-dependent agglomeration and disruption, can be implemented in the model. Therefore, the population balance for the bulk becomes

and the death terms for agglomeration and disruption
∞ Daggl + Ddisr = n(L) Kaggl n(Lu ) dLu + Kdisr n(L): (23)

4.3.2. Species B

=−

4.5. Population balance (Randolph & Larson, 1998)

−Daggl − Ddisr −

Vf cA; f1 Vf (cA; f1 − cA; b ) d(Vb cA; b ) + 1 = −Qb cA; b + 1 dt tmeso1 tmicro1 +

As mentioned above, the inverse of the time constants tmicro1 , tmicro2 (micromixing), tmeso1 and tmeso2 (mesomixing) can be interpreted as transfer coeIcients for mass transfer by di7usion and convection, respectively.

(18)

The expression for the nucleation rate Bj0 in the compartment j is derived from the theory of primary nucleation and found to be   163 2 0 Bj = A exp − (24) 3k 3 T 3 ln2 Sj

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

with the supersaturation de9ned as    1=(A +B ) cA;A j cB;B j Sj = = $j + 1: Ksp

(25)

The overall nucleation rate in the reactor becomes Bb0 Vb + Bf0 1 Vf1 + Bf0 2 Vf2 0 Btot = : (26) Vtot The dependence of the growth rate on supersaturation is modelled using the power law expression Gb =

kg $bg :

(27)

and xi as the representative volume of the ith size range and v as the particle volume. Moreover, according to Kumar and Ramkrishna (1996a, b), disruption can be accounted for with 

dNi dt

with ni; k =

Furthermore, the agglomeration and disruption kernels are also assumed to depend on the supersaturation in power law form: Kaggl = &aggl f(')$bg ;

(28)

Kdisr = &disr g(')$b−g :

(29)

The second moment of the particle size distribution used in the mass balances is obtained from
∞ m2 = nP; b (L)L2 dL: (30) 0

4.6. Numerical solution of the population balance The population balance represents a partial integrodi7erential equation that can only be solved analytically in speci9c cases (Ramkrishna, 1985). It is thus necessary to search for numerical solution strategies that solve the population balance rapidly and accurately in order to model precipitation processes.

xi−1 6(xj +xk )6xi+1

×,Kaggl Nj Nk − Ni

imax 

Kaggl Nk

(31)

k=1

with  x −v i+1    xi+1 − xi ; xi 6 v 6 xi+1 ; ,=  v − xi−1   ; xi−1 6 v 6 xi xi − xi−1

disr




=




ni; k Kdisr Nk − Kdisr Ni

(33)

k=i

xi+1

xi

imax 

xi+1 − v b(v; xk ) dv xi+1 − xi xi

xi−1

v − xi−1 b(v; xk ) dv: xi − xi−1

Using the parabolic attrition function       2 v 6 v b(v; xk ) = 4 +1 −4 xk xk xk

(34)

(35)

which was proposed by Hill and Ng (1995) and is based on an empirical form suggested by Austin, Shoji, Jindal, Savage, and Kimpel (1976) and Klimpel and Austin (1984), the weighting matrix becomes   xi xi ni; k = + xi+1 − xi xi − xi−1     2   3 xi xi xi × 6 −8 +3 xk xk xk  −

xi+1 xi−1 + xi+1 − xi xi − xi−1



    2   3 xi xi xi × 8 − 12 +6 xk xk xk   3 2    xi+1 xi+1 xi+1 xi+1 + 2 −4 +3 xi+1 − xi xk xk xk   3 2    xi−1 xi−1 xi−1 xi−1 + 2 −4 +3 xi − xi−1 xk xk xk (36) for the number of particles formed per breakage event. Nucleation and growth are modelled as described below:

  B0 i = 1; dNi (37) = dt nucl 0 i = 1; 

(32)



+

4.7. Discretization method of Kumar and Ramkrishna Kumar and Ramkrishna (1996a, b) present a solution to population balance problems of agglomeration processes without restricting the choice of the grid of the discretised length scale. Two arbitrarily chosen properties can be preserved. In the case of preservation of the zeroth and third moments (number and mass, respectively), the equations for a purely agglomerative process can be written in the following form:     j¿k  dNi 1 1 − +j; k = dt aggl 2 j; k

825



dNi dt dNi dt

 growth

=

G [(bg + cg r)N1 + cg N2 ]; L1

(38)

=

G (ag Ni−1 + bg Ni + cg Ni+1 ): Li

(39)



growth

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

+

dNi dt

 disr

−

Ni − Ni; in : -

(40)

The sets of non-linear ordinary di7erential equations ODEs, obtained by discretisation were solved using FORTRAN90 programming using the NAG subroutine D02EAF, which is particularly suitable for sti7 systems of 9rst-order ODEs. This variable-order, variable-step method implements the backward di7erentiation formulae (BDF) and is of an explicit type (Schuler, 1996). The ODEs are integrated over a time range of ten residence times, assuming that after that time, steady state has been achieved. Subroutines calculate values for the change of population due to nucleation, growth, agglomeration and disruption. The population density of each class is calculated from the number of particles in each size class. The equations for the population balance are solved within the compartmental mixing model, which delivers the 9eld of supersaturations and therefore the kinetic rates in the feed and bulk compartments. The exchange of mass and particles between the compartments is governed by meso- and micromixing and is characterised by the meso- and micromixing times. The micro- and mesomixing times were determined via CFD calculations (CFX 4.3, Harwell: AEA Technology) using the relationships given by Baldyga et al. (1995) and Baldyga, Bourne, and Hearn (1997), as described elsewhere (Zauner & Jones, 2000a). Typical run times of the SFM are of the order of minutes on a PC compared with hours or days for direct solution using CFD on a workstation.

5. Predictions of the SFM In order to illustrate the model, the following 9gures re ect a qualitative analysis of the precipitation process. The kinetic parameters used for the simulations are listed in Table 1. Initial conditions as per the experiments below (Zauner & Jones, 2000a) viz: Continuous case: cA = cB = 0 in the whole reactor, then feeding starts (i.e. solvent only in the vessel before feeding and withdrawal starts), no premixing of any streams, initially no particles at all in the reactor, particle size in 9rst size class, 0:5 m. Semibatch case: concentration cA in the whole reactor, then feeding of component B (cB ) starts, initially no particles at all in the reactor, particle size in 9rst size class, 0:5 m. In addition, the transient segregated volumes due to initial feeding are assumed to go from zero to

Parameter

Model input (units as listed in nomenclature)

Concentration cA ; cB Residence time Shape factor ka a Shape factor kv a Nucleation rate B0 Growth rate G Agglomeration rate &aggl

0.04 660 0.88 0.04 2 3:37536 × 1015 exp−52:09=ln S −10 2 5:879 × 10 $ 5:431 × 10−17 (1 + 2:296'1=2 −2:429')S 2:15 6:25 × 10−5 ' S −2:15 0.5

Disruption rate &disr Particle size of 9rst size class a Mersmann

(1994).

1.E+14

1.E+12 3



Table 1 Model parameters for computation (Zauner, 1999)

2

The discretised population balance can therefore be written as       dNi dNi dNi dNi = + + dt dt nucl dt growth dt aggl

m2 [µm /m ]

826

1.E+10

1.E+08

1.E+06 0

0.5

1

1.5

Time [h]

Fig. 2. Time dependence of the second moment of the distribution during continuous precipitation.

those given by Eqs. (12) and (13) instantaneously (rather than include the dynamic given by Eqs. (1) and (2)). Particle shape factors determined for the kinetics determination using zone sensing (Coulter Counter TM ) were taken as those of the sphere whilst for the semibatch experiments analysed via light scattering (Sympatec Helios), needle shape factors were used (Mersmann, 1994). In order to model the time-dependent behaviour of the continuous system, the mass balances were solved together with the population balance over an interval of ten residence times. After that time, it is assumed that the continuous system has achieved steady state. Being a measure of the crystal surface area available for growth, the second moment m2 is an important factor for crystal growth (Fig. 2). It increases sharply at the beginning and maintains a constant level after about four residence times. In contrast, the number mean size (Fig. 3) exhibits almost oscillating behaviour. It approaches a maximum after about two residence times (1.4 times larger than the steady state mean size) and then decreases before increasing again, eventually approaching steady state after nine residence times.

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

827

L10 [µm]

15

10

5

0 0

0.5

1

1.5

Time [h] Fig. 3. Time dependence of number mean size during continuous precipitation. Fig. 5. Dependence of the nucleation rate on the mixing conditions during continuous precipitation.

1.E+14

t/tau=8.40E-10 3.73E-07 3.72E-04 0.11 4.46 10.02

1.E+12

3

n [1/m µm]

1.E+10 1.E+08

1,024

Case B

Case C 0,256

1.E+06 10

13

1.E+04

t m icro [s]

0,064

1.E+02 100

10

0,032

0,016

0,008

0,004

0,002

10

t m eso [s]

Another interesting result of the SFM is shown in Fig. 4. The population density distribution in the reactor changes from the early moments dominated by high (primary) nucleation to a bimodal distribution after about 4.5 residence times and to the 9nal steady-state particle size distribution after ten residence times. Fig. 5 shows predictions for the overall nucleation rate in the continuous reactor. The micro- and mesomixing times determined via CFD vary between 0.004 and 2:05 s and between 0.002 and 1:02 s, respectively. Therefore, the volume of the feed zones, Vf1 and Vf2 , covers a range from only 0.0002% to 0.1% of the total volume. The fast reaction and nucleation processes take place in a very limited volume around the feed zones, while the slower kinetic processes occur in the bulk zone. This result clearly shows the importance of mixing in systems with very fast or instantaneous reactions. The nucleation rate shows a maximum under poor micro- and mesomixing conditions. Under very good micromixing conditions (di7usion is not limiting), the nucleation rates show a maximum at medium mesomix-

0,016

Case D

Case A Fig. 4. Time dependence of the particle size distribution during continuous precipitation.

14

13

1,024

80

0,512

60 L [µm]

0,256

40

0,128

20

0,064

0

0,004

0

3

B [1/m h]

Fig. 6. Two-dimensional projection of Fig. 5.

ing times. One explanation for this observation could be that, on the one hand, the level of supersaturation becomes uniform very quickly at very low mesomixing times, and therefore, does not cause excessive nucleation, and on the other, that the reaction zone remains very small at very high mesomixing times and therefore, restricts nucleation to a very limited zone and retards nucleation. Under conditions in between, however, the nucleation rate can increase to a maximum. The same applies to mean mesomixing times. In this case, the nucleation rate exhibits a maximum for mean micromixing times. David and Marcant (1994) observed similar behaviour in their model for double-jet semibatch precipitation. A two-dimensional plot of Fig. 5 gives Fig. 6 with iso-nucleation curves. Additionally, four di7erent mixing conditions show the in uence of micro- and mesomixing

828

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

on the nucleation rate in a qualitative way. The four curves represent di7erent feed points and mesomixing conditions. The rate of energy dissipation changes along the individual curves. Case A represents very intense micro- and mesomixing, obtained, for example, by a feed point close to the impeller and good distribution of the incoming reagents. The nucleation rates are relatively low as the level of supersaturation can be kept low due to the fast mixing. Case B represents poor micromixing but good mesomixing, obtained, for example, by a feed point remote from the impeller and good distribution of the incoming reagents. The nucleation rates are still relatively low as the volume for excessive nucleation can be kept low due to the slow micromixing. Intense mesomixing leads to a ‘dilution e7ect’ and therefore helps to keep supersaturation low. Case C represents very poor micro- and mesomixing, obtained, for example, by a feed point far away from the impeller and insuIcient distribution of the incoming reagents. The nucleation rates are very high as the local level of supersaturation in the reaction zone becomes very high due to the slow mixing. Case D represents good micromixing but poor mesomixing, obtained, for example, by a feed point close to the impeller and bad distribution of the incoming reagents. The nucleation rates are relatively low as the level of supersaturation can be kept low due to the fast micromixing. Poor mesomixing causes slower blending of the entering volume fractions and therefore supersaturation and nucleation rates increase to a higher level compared with Case A. In general, it can be assumed that the higher the nucleation rate, the lower the mean particle size becomes. Therefore, the in uence of the mixing conditions on the mean particle size can be derived in an analogous way. These qualitative considerations can be used to explain various e7ects that have been experimentally observed (Zauner, 1999).

6. Experimental The experimental apparatus and technique used for the continuous and semibatch precipitation of calcium oxalate crystals is described in detail elsewhere (Zauner & Jones, 2000a). The sets of experimental conditions of the chosen experiments reported here are shown for continuous precipitation in Table 2 and for semibatch operation in Table 3. The mean size L10 and the nucleation rate B0 during continuous precipitation are plotted in Figs. 7 and 8, respectively. Under one set of conditions (300 ml, 0:04 M, 7:5 min, id), the number mean size shows a minimum with stirrer

Table 2 Experimental conditions for continuous experiments (Zauner, 1999) Condition

Set 1

Set 2

Concentration Residence time Feed point position

0:04 M 7:5 min Inside DT (id)

0:01 M 11 min Outside DT (od)

speed (Fig. 7). Under the other set of conditions (300 ml, 0:04 M, 7:5 min, od), however, L10 decreases continually with stirrer speed. In both cases, the SFM predicts the trend correctly. The model also predicts correctly a maximum in nucleation rate during continuous precipitation (Fig. 8). Due to agglomeration, particle disruption and inhomogeneous mixing conditions, the number-density distribution deviates from the straight log-linear relationship obtained under ideal MSMPR conditions. In order to investigate the dependence of the particle size on the feed point position, one set of experiments was carried out with a feed point close to the impeller and another set with a feed point close to the surface. At low power inputs, the mean particle size obtained for the feed point in a highly turbulent zone near the Rushton turbine resulted in larger particles, giving mean particle sizes up to twice as large as those produced with a feed point close to the surface (Fig. 9). In comparison with the continuous mode of operation, the mean size is found to depend to a greater degree on the mixing conditions on all scales in the semibatch mode. In Fig. 10, the results for the reference conditions (Rushton turbine, 40 min feed time, feed point position close to the impeller, total concentration 0:008 M) for calcium oxalate con9rm this observation. By changing the energy input, the volume mean size varies over a wide range being from 7 to 26 m. For the two reactor scales of 1 and 5 l, scale-up with constant speci9c power input seems appropriate, while for the 25 l scale, smaller particle sizes are obtained in the industrially important range from 0.1–1 W=kg. The kinetic parameters, which were determined from laboratory-scale continuous experiments as a function of the energy input and=or supersaturation, are applied to the semibatch mode of operation without any adjustments or parameter 9tting. The SFM slightly underestimates the mean particle size in the range between 0.01 and 1 W=kg, but correctly predicts the smaller particle size obtained experimentally for the 25 l reactor. On the same scale, the model also predicts a lesser degree of dependence of the particle size on the speci9c power input due to the interactions of mixing and the precipitation kinetics. This behaviour is also been observed experimentally.

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

829

Table 3 Experimental conditions for semibatch experiments (Zauner, 1999)

Condition

From

To

Reference

End concentration Feed time Vfeed =Vreactor Speci9c energy input Feed point position

0:008 M 10 min 0.2 2:5 × 10−4 W kg−1 Impeller region

0:04 M 60 min 0.5 21:6 W kg−1 Surface region

0:008 M 40 min 0.2 — Impeller region

12 SFM, 7.5min, 0.04M, od Exp. 300 ml, 7.5min, 0.04M, od SFM, 7.5min, 0.04M, id

30 25

Exp. 300 ml, 7.5min, 0.04M, id

10

L43 [µm]

L10 [µm]

11

9

20 15 10

8

5 1E-3

7 0

500

1000

1500

2000

Fig. 7. Experimental and SFM results for the number mean size during continuous precipitation (300 ml, 500 –2000 rpm; 0:04 M; 7:5 min, id and od).

1.E+14

3

SFM, 7.5 min, 0.04M, od Exp. 300 ml, 7.5 min, 0.04M, od SFM, 7.5 min, 0.04M, id Exp. 300 ml, 7.5min, 0.04M, id

0

1.E+13

1.E+12 0

500

1000

1500

2000

2500

Stirrer speed [rpm]

Fig. 8. Experimental and SFM results for the nucleation rate during continuous precipitation (300 ml, 500 –2000 rpm; 0:04 M; 7:5 min, id and od).

7. Conclusions The population balance model was successfully implemented in the Segregated Feed Model (SFM), which

0.01

0.1

1

10

ε[W/kg]

2500

Stirrer speed [rpm]

B [1/m h]

f.p.p. close to impeller, exp. f.p.p. close to surface, exp. f.p.p. close to impeller, model f.p.p. close to surface, model

Fig. 9. Mean particle size vs. speci9c energy input for di7erent feed point positions from semibatch precipitation. (CaOx, Rushton turbine, 40 min feed time, total concentration 0:008 M.) (After Zauner & Jones, 2000b.)

accounts for local inhomogenities in the reactor in terms of micro- and mesomixing times within compartments. Both micro- and mesomixing strongly limit precipitation processes, as the incoming unreacted feed is not completely mixed with the bulk when precipitation starts. Therefore, inhomogenities in the level of supersaturation occur throughout the reactor and the kinetic values di7er locally. As the nucleation rate is extremely sensitive to the level of supersaturation, it shows a very high dependency on the mixing conditions. The interplay of the kinetics with non-ideal mixing can lead to bimodal particle size distributions under non-steady-state conditions, which are most likely due to the high non-linearity between the level of supersaturation and the nucleation kinetics. The number mean size exhibits oscillating behaviour before it achieves steady state, while the second moment reaches its 9nal level far more quickly. The SFM also provides other explanations for e7ects that are usually observed in precipitation processes. It can account for the in uence on the particle size distribution of the • • • • •

feed point position, feed rate, feed tube diameter, average and local energy dissipation and impeller type.

830

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

1 l reactor, exp. 5 l reactor, exp. 25 l reactor, exp. 1 l reactor, model 5 l reactor, model 25 l reactor, model

30

L43 [µm]

25 20 15 10 5 1E-3

0.01

0.1

1

10

Specific power input [W/kg] Fig. 10. Mean particle size from semibatch precipitation. (CaOx, Rushton turbine, 40 min feed time, feed point position close to the impeller, total concentration 0:008 M.) (After Zauner & Jones, 2000a.)

The di7erences in the 9nal product size and properties are due to di7erent intensities of micro- and mesomixing, giving rise to di7erent kinetic rates. The volume of the feed zones covers a very small proportion of the total volume, and the reaction and fast nucleation processes take place in a very limited volume around the feed zones. Thus, the conditions in the feed zone (reaction zone) become crucial for the 9nal product. The model gives qualitative predictions about the consequences when changing geometric parameters of the reactor, mixing conditions or process parameters. Since it is based on physical parameters, it is potentially suitable for predicting scale-up e7ects, and does not require the long computation times currently needed for direct solution of coupled population and momentum balance equations. Notation ag Ahom b(v; xk ) bg B B0 Bi0 0 Bhom c cg cj

growth parameter, dimensionless homogeneous nucleation rate constant, m−3 s−1 breakage function, dimensionless growth parameter, dimensionless birth rate, m−4 s−1 nucleation rate (general), m−3 s−1 nucleation rate in compartment i, m−3 s−1 homogeneous nucleation rate, m−3 s−1 concentration, mol m−3 growth parameter, dimensionless concentration of reactant j, mol m−3

cj0 D g g G ka kg kn kv Ksp L L0 L10 Lu Lv ULj mj MWj n n ni; k nk nP n0 Ni Qk r rj; k

feed concentration of reactant j, mol m−3 death rate, m−4 s−1 growth rate order, dimensionless acceleration of free fall, m s−2 growth rate, m s−1 surface area shape factor, dimensionless growth rate coeIcient, m s−1 nucleation rate coeIcient, m−4 s−1 volumetric shape factor, dimensionless solubility product, mol2 m−6 particle size, m particle size of lower bound of 9rst size class, m number mean size, m particle size of class u, m particle size of class v, m width of particle size class j, m jth moment of distribution, m−3 molecular weight of component j, kg mol−1 nucleation rate order, dimensionless population density (general), m−4 weighting term (i,k) in weighting matrix, dimensionless population density of stream k, m−4 population density of product P, m−4 population density of nuclei, m−3 number of particles in size class i, m−3 ow rate of stream k, m3 s−1 volumetric discretisation ratio, dimensionless reaction rate of component j in compartment k, dimensionless

R. Zauner, A. G. Jones / Chemical Engineering Science 57 (2002) 821–831

S tmeso; j tmicro; j uj; ik U Vj Vtot xk

supersaturation (general), dimensionless time constant for mesomixing in compartment j, s time constant for micromixing in compartment j, s exchange ow of component j between compartments i; k, mol s−1 velocity gradient, s−1 subvolume of compartment j, m3 reactor volume, m3 discretised particle volume, m3

Greek letters &aggl &disr ' , i s $j -

agglomeration kernel disruption kernel speci9c power input, W kg−1 volume correction function, dimensionless stoichiometric coeIcient of component i, dimensionless solids density, kg m−3 supersaturation in compartment j, dimensionless residence time, s

References Austin, L., Shoji, V., Jindal, V., Savage, K., & Kimpel, R. (1976). Some results on the description of size reduction as a rate process in various mills. Industrial Engineering Chemistry Process Design Development, 15, 187–196. Baldyga, J., & Bourne, J. R. (1984). A uid mechanical approach to turbulent mixing and chemical reaction. Chemical Engineering Communication, 28, 231–281. Baldyga, J., & Bourne, J. R. (1989). Simpli9cation of micromixing calculations. Chemical Engineering Journal, 42, 83–101. Baldyga, J., & Bourne, J. R. (1992). Interactions between mixing on various scales in stirred tank reactors. Chemical Engineering Science, 47, 1839–1848. Baldyga, J., Bourne, J. R., & Hearn, S. J. (1997). Interaction between chemical reactions and mixing on di7erent scales. Chemical Engineering Science, 52, 457–466. Baldyga, J., Podgorska, W., & Pohorecki, R. (1995). Mixing-precipitation model with application to double feed semibatch precipitation. Chemical Engineering Science, 50, 1281–1300. Baldyga, J., & Pohorecki, R. (1995). Turbulent micromixing in chemical reactors—a review. Chemical Engineering Journal, 58, 183–195. Chang, L.-J., Mehta, R. V., & Tarbell, J. M. (1986). An evaluation of models of mixing and chemical reaction with a turbulence analogy. Chemical Engineering Communication, 42, 139–155. Curl, R. L. (1963). Dispersed phase mixing-theory and e7ects in simple reactors. A.I.Ch.E. Journal, 9, 175. David, R., & Marcant, B. (1994). Prediction of micromixing e7ects in precipitation: Case of double-jet precipitators. AIChE Journal, 40, 424–432. Franck, R., David, R., Villermaux, J., & Klein, J. P. (1988). Crystallization and precipitation engineering—II. A chemical reaction engineering

831

approach to salicylic acid precipitation: Modelling of batch kinetics and application to continuous operation. Chemical Engineering Science, 43, 69–77. Garside, J., & Tavare, N. S. (1985). Mixing, reaction and precipitation: limits of micromixing in an MSMPR crystallizer. Chemical Engineering Science, 40, 1485–1493. Harada, M., Arima, K., Eguchi, W., & Nagata, S. (1962). Micromixing in a continuous ow reactor. Memoirs of the Faculty of Engineering. Kyoto University Japan, 24, 431. Hill, P. J., & Ng, K. M. (1995). New discretization procedure for the breakage equation. A.I.Ch.E. Journal, 41, 1204–1217. Klimpel, R. R., & Austin, L. G. (1984). The back-calculation of speci9c rates of breakage from continuous mill data. Powder Technology, 38, 77–91. Kumar, S., & Ramkrishna, D. (1996a). On the solution of population balance by discretization I. A 9xed pivot technique. Chemical Engineering Science, 51, 1311–1332. Kumar, S., & Ramkrishna, D. (1996b). On the solution of population balance by discretization II. A moving pivot technique. Chemical Engineering Science, 51, 1333–1342. Marcant, B. (1996). Prediction of mixing e7ects in precipitation from laser sheet visualisation. Proceedings of Industrial Crystallization 1996, Progep, Toulouse, France, pp. 531–538. Mehta, R. V., & Tarbell, J. M. (1983). A four environment model of mixing and chemical reaction. Part I—model development. A.I.Ch.E. Journal, 29, 320. Mersmann, A. (1994). Crystallization technology handbook. New York: Dekker, p. 612. Pohorecki, R., & Baldyga, J. (1983). The use of new model of micromixing for determination of crystal size in precipitation. Chemical Engineering Science, 38, 79–83. Pohorecki, R., & Baldyga, J. (1988). The e7ects of micromixing and the manner of reactor feeding on precipitation in stirred tank reactors. Chemical Engineering Science, 43, 1949–1954. Ramkrishna, D. (1985). The status of population balances. Reviews in Chemical Engineering, 3, 49–95. Randolph, A. D., & Larson, M. A. (1998). Theory of particulate processes. Second edition, Academic Press, New York. Ritchie, B. W., & Togby, A. H. (1979). A three-environment micromixing model for chemical reactors with arbitrary separate feedstreams. Chemical Engineering Journal, 17, 173. Schuler, H. (1996). ProzeAsimulation. Weinheim: VCH-Verlag. Tosun, G. (1992). A mathematical model of mixing and polymerization in a semibatch stirred tank reactor. A.I.Ch.E. Journal, 38, 425–437. van Leeuwen, M.L.J. (1998). Precipitation and mixing. Ph.D. thesis, Univ. of Delft, The Netherlands. Villermaux, J. (1989). A simple model for partial segregation in a semibatch reactor A.I.Ch.E. Annual Meeting, San Francisco, Paper 114a. Villermaux, J., & Falk, L. (1994). A generalised mixing model for initial contacting of reactive uids. Chemical Engineering Science, 49, 5127– 5140. Wei, H. Y., & Garside, J. (1997). Application of CFD modelling to precipitation systems. Transition Institute of Chemical Engineers, 75, 219–227. Zauner, R. (1999). Scale-up of precipitation processes. Ph.D. thesis, University College London, UK. Zauner, R., & Jones, A. G. (2000a). Scale-up of continuous and semi-batch precipitation processes. Industrial Engineering Chemistry, 39, 2392– 2403. Zauner, R., & Jones, A. G. (2000b). Mixing e7ects on product characteristics from semi-batch precipitation. Transition Institute of Chemical Engineers, 78(A), 894–901.