towards predictive simulation of single feed semibatch reaction crystallization

Chemical Engineering Science 64 (2009) 1559 -- 1576

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Chemical Engineering Science

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Towards predictive simulation of single feed semibatch reaction crystallization

Marie Ståhl, Åke C. Rasmuson∗

Department of Chemical Engineering and Technology, Royal Institute of Technology, 100 44 Stockholm, Sweden

A R T I C L E I N F O A B S T R A C T

Article history:Received 18 October 2007Received in revised form 26 November 2008Accepted 2 December 2008Available online 9 December 2008

Keywords:PrecipitationMixingPopulation balanceScale-upKineticsBenzoic acidBatch

A population balance model is developed over single-feed semi-batch reaction crystallization of benzoicacid. The model is evaluated by comparison with experimental data, and simulations are carried out toadvance the understanding of the process. The model accounts for chemical reaction, micro and me-somixing, primary nucleation, crystal growth and growth rate dispersion (GRD). Two mechanistic mixingmodels are evaluated: the segregated feed model and the engulfment model (E-model) with mesomixing.When the mixing is described by the E-model (engulfment model) and GRD is accounted for, the modelquite well captures the influence of reactant concentrations, agitation rate, feed point location, feed pipediameter, total feeding time and crystallizer volume, on the product weight mean size. When using theSF-model (segregated feed model) the results are less satisfactory. The kinetics of nucleation and crystalgrowth have a great impact on the results of the simulations, influencing the product weight mean sizeas well as the response to changes in the processing conditions. A new set of kinetic data for benzoicacid derived from semi-batch experimental results are presented.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Reaction crystallization or precipitation is used in the productionof both inorganic and organic compounds, for example organic finechemicals and photographic materials. A common procedure is tofeed one reactant into an agitated solution of the other reactant ina semi-batch (or fed-batch) process. In reaction crystallization pro-cesses, the solubility of the formed compound is normally low orvery low, and a region with high supersaturation and subsequentrapid nucleation and growth will form around the feed point. Nucle-ation starts before the solution has been completely homogenized,and the crystallization proceeds in a partially segregated solution.The mixing conditions in the crystallizer will have a significant in-fluence on the final product characteristics.

The energy provided by the agitator is transported through a cas-cade of eddies of decreasing size and is finally dissipated as heat, andthe active motions along this process results in mixing. The processis often divided into three levels of mixing: macromixing, mesomix-ing and micromixing. Micromixing describes mixing in the smallestvortices and includes the final molecular diffusion by which the fluidis mixed in the true, molecular sense. Mesomixing is of particularrelevance in processes where feed pipes are used, and accounts for

∗ Corresponding author. Present address: Department of Chemical and Environ-mental Science, Material and Surface Science Institute, University of Limerick, Lim-erick, Ireland. Tel.: +4687908227, +35361234617; fax: +468105228.

E-mail addresses: [email protected], [email protected] (Å.C. Rasmuson).

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2008.12.001

the initial mixing of the feed with the bulk. Two main mesomix-ing mechanisms have been described: turbulent dispersion and mix-ing by inertial-convective disintegration (Baldyga and Bourne, 1999).Macromixing describes the distribution of the fluid on a macroscopiclevel in the tank. When the circulation time is much shorter thanthe total feeding time, the vessel can be considered well-mixed onthe macroscopic level, except for the region around the feed point(Baldyga and Bourne, 1999). In some mixing models, the circulationin the tank is accounted for even if the vessel is well-mixed on themacroscale. The macroscopic circulation in the tank may still havean influence by the transport of the feed through regions in thetank of different mixing conditions and hence may influence on themeso and micromixing that is experienced by a feed lump travelingthrough different regions of the tank.

There are more studies on the influence of mixing on chemicalreactions than on precipitations. However, precipitation and sys-tems of fast reactions are influenced by mixing in a similar way.The selectivity in the case of fast reactions, and the mean size in thecase of precipitation, often shows a similar dependency on the mix-ing conditions. Still, it is easier to apply a complex mixing model tocompeting chemical reactions, since the reaction terminates quickly,making it possible to identify a chemically active region around thefeed point in an otherwise chemically passive bulk (Baldyga et al.,2005). Crystal growth, on the other hand, operates on a longertime scale and is active in the whole bulk as long as the solution issupersaturated, and this makes the calculations more complicated.Tosun (1988) used competing reactions to characterize the mixingintensity in a vessel, which was later used in a precipitation study.

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1560 M. Ståhl, Å.C. Rasmuson / Chemical Engineering Science 64 (2009) 1559 -- 1576

He found that even though two feed points gave the same selectivitywith competing reactions, they resulted in different mean sizes inprecipitation.

Studies ofmixing effects on precipitation or fast reactions have of-ten focused on the micromixing-controlled regime. The correspond-ing experiments are specifically designed to eliminate the influenceof mesomixing by using a low feeding rate (Phillips et al., 1999;Uehara-Nagamine, 2001; Akiti and Armenante, 2004). Micromix-ing can be described by phenomenological models with parameterswhich must be either determined from experimental data or inter-preted from underlying physics, e.g. the “interaction by exchangewith the mean” model and the “segregated feed model (SF-model)”(Villermaux and Falk, 1994). A different approach is to describe themost relevant flow and diffusion processes in a physical model. Theparameters can then be determined from fluid dynamics theory ormeasurements. Examples are the EDD-model and the E-model (en-gulfment model) (Baldyga and Bourne, 1999), as well as the diffusionmodel by Lindberg and Rasmuson (1999).

Mesomixing has been included in the modeling in just a few stud-ies. Baldyga et al. (1997) and Verschuren et al. (2001) combined theE-model with mesomixing and applied it to fast chemical reactions.Verschuren et al. (2001) used flow maps from laser-Doppler ve-locimetry (LDV) measurements to calculate changing hydrodynamicenvironment during the reaction, and obtained a relatively good de-scription of the variation in selectivity with feed time. The SF-model(Villermaux and Falk, 1994) includes both micro and mesomixing.Zauner and Jones (2000a,b, 2002) claimed that the model gave goodresults for precipitation of calcium oxalate, but the model was notcompared with experimental data on the influence of the feed timeand feed pipe diameter.

Nowadays, computational fluid dynamics (CFD) is often used tocalculate the flow field. The population balance equations can ei-ther be solved together with the CFD equations (Baldyga et al., 2005;Piton et al., 2000; Vicum et al., 2004; Marchisio and Barresi, 2003;Baldyga and Orciuch, 2001), or the CFD simulation can be done sep-arately to calculate energy dissipation rate values, which are thenused in a mechanistic mixing model (Uehara-Nagamine, 2001; Akitiand Armenante, 2004; Zauner and Jones, 2000a). Another, more sim-ple approach was used by Phillips et al. (1999), in which the reactorwas divided into zones with different hydrodynamic properties andthe flow map was used to calculate the corresponding variation inthe mixing parameters as mixing proceeded.

Multiscale Eulerian mixing models that account for all levels ofmixing are usually combined with a CFD code to calculate the flowfield. Baldyga et al. (2005) used the turbulent mixer model (Baldygaand Bourne, 1999) and a beta-distribution to describe micromixingfor a single feed semi-batch process. The variation of selectivity withvarying mean energy dissipation rate and feed concentration waswell described, but the influence of feed time on the selectivity wasless pronounced than in the experiments. The authors also appliedthe model to precipitation, but it is difficult to judge the model's ca-pability from the results. Vicum et al. (2004) used the same modelfor fast reactions in a double feed semi-batch arrangement. Accord-ing to the authors, mesomixing had a comparably small influence atthe particular experimental conditions. Marchisio and Barresi (2003)used CFD together with a finite-mode PDF approach for micromixingto model a semi-batch system for fast reactions in a Taylor–Couettereactor. A relatively good description of the decrease in selectivitywith increasing feed time was obtained. Wei et al. (2001) modeledthe precipitation of BaSO4 in a semi-batch reactor and comparedtheir results with the experimental results of Chen et al. (1996) con-cerning the variation of mean size with agitation rate. A correct de-scription of the trend in mean size was obtained even though themodel simply ignored possible fluctuations in concentrations on ascale smaller than the grid in the CFD simulation. However, the

product size predicted by the model was about four times largerthan the experimental.

The semi-batch reaction crystallization of benzoic acid have beeninvestigated in several experimental studies of our research group(Åslund and Rasmuson, 1992; Torbacke and Rasmuson, 2001, 2004),and the kinetics and aging of benzoic acid has also been studied,based on experiments and population balance modeling (Ståhl et al.,2001, 2004). The semi-batch experiments showed that mesomixinghas a decisive influence on the experimental results. The productweight mean size was influenced by the total feeding time and bythe feed pipe diameter. In the present work a model is developedof single feed semi-batch reaction crystallization, having the aimof capturing the influence of various processing conditions on theproduct size distribution. The model retrieves the full crystal sizedistribution, an advantage compared to most studies of mixing andprecipitation where moment transformation is used in the numericalsolution. Benzoic acid is used as the model compound. The ability ofthe model to reproduce the variation in weight mean size with vary-ing agitation rate, feed point location, reactant concentration, totalfeed time, feed pipe diameter and crystallizer volume is evaluatedby comparison with experimental data.

2. Modeling

Benzoic acid is produced in a semi-batch reaction crystallizationby adding hydrochloric acid through a feed pipe to an agitated so-lution of sodium benzoate. Benzoic acid and sodium chloride areproduced:

HCl + C6H5COONa → C6H5COOH (aq) + NaCl

A B P D

C6H5COOH (aq) → C6H5COOH (s)

It is assumed that benzoic acid crystallizes as a molecularcompound, similarly to salicylic acid (Mersmann, 1995). The pop-ulation balance for a perfectly mixed semi-batch reactor withsize-independent growth rate and a crystal-free feed is

�n�t

+ G�n�L

+ nVdVdt

= 0 (1)

where n is the population density, G is the growth rate and L isthe characteristic crystal size. In the region around the feed point,the induction time for nucleation is of the same order of magnitudeas the time needed for homogenization, and the solution cannotbe considered perfectly mixed. Mixing effects are accounted for byintroducing a mixing description in the population balance model.Two different mechanistic mixing models are used and compared:the SF-model (Villermaux and Falk, 1994) and the E-model withmesomixing (Baldyga et al., 1997). The semi-batch crystallizationmodel accounts for chemical reaction, mixing, nucleation, growth,and, in some cases, growth rate dispersion (GRD).

The models are based on the following assumptions:

• The reaction is very fast in comparison with nucleation.• All benzoic acid is in undissociated form.• No significant secondary nucleation or aging is taking place.• Nuclei are formed at zero size. The mass consumption due to

nucleation is negligible.

2.1. The E-model with mesomixing

The E-model for micromixing is a simplification of theengulfment-deformation-diffusion model. It is valid for fluids wherethe Schmidt number is lower than 4000 (Baldyga and Bourne, 1999).The assumption is that the viscous-convective incorporation of bulk

M. Ståhl, Å.C. Rasmuson / Chemical Engineering Science 64 (2009) 1559 -- 1576 1561

fluid into the smallest concentration spots controls the rate of mi-cromixing. To apply the E-model to crystallization, the continuousfeed is discretized into a number of small feed volumes which areadded in sequence. Each feed drop is followed in a Lagrangian man-ner; the mixing equations are applied to calculate both the expand-ing volume and the variation in concentration as mixing proceeds.Benzoic acid forms immediately when bulk reactant is mixed intothe drop, and the local supersaturation in the drop increases. Thiscontinues until a stoichiometric amount of bulk reactant has beenmixed in and all acid has reacted. In the drop, rapid nucleation andgrowth may occur. After all acid has reacted the supersaturation inthe drop starts to decrease. When one feed drop is completely mi-cromixed with the bulk, a new drop is added and the calculation isrepeated. Engulfment of another feed-rich eddy, instead of the bulksolution, does not advance mixing. This self-engulfment will becomeincreasingly frequent when the volume fraction of the feed increases.

Mesomixing is included in the model by envisioning the solutionas mesomixed “islands” where feed reactant is present, surroundedby a “sea” where only bulk reactant is present. Micromixing canonly proceed in the islands. The bulk solution is assumed to be well-mixed. Taking both mesomixing and self-engulfment into account,the volume of a feed drop increases according to

dVdt

= E(1 − Xmicro

Xmeso

)V (2)

with E being the engulfment rate constant, Xmicro the micromixedvolume fraction and Xmeso the mesomixed volume fraction (Baldygaand Bourne, 1999). The micromixed and mesomixed volume frac-tions change according to

dXmicro

dt= E

(1 − Xmicro

Xmeso

)Xmicro (3)

dXmeso

dt= 1

tmeso(1 − Xmeso)Xmeso (4)

tmeso is the mesomixing time constant. When Eq. (2) is combinedwith a mass balance on each substance, equations describing theevolution of the concentrations in the drop are obtained (Baldygaand Bourne, 1999):

dcidt

= E(1 − Xmicro

Xmeso

)(〈ci〉 − ci) + ri (5)

The consumption or production of each species is denoted ri. The bulkconcentrations are denoted 〈ci〉. Benzoate is the limiting reactant inthe feed region, giving cB = 0 and dcB/dt = 0 until all acid has beenconsumed.

Therefore, the consumption rate of acid and the production rateof benzoic acid and sodium chloride is limited by the rate at whichbenzoate is transported from the bulk into the feed drop

ri = ±E(1 − Xmicro

Xmeso

)〈cB〉 (6)

Benzoic acid is also consumed by growth of crystals. The followingdifferential equations apply until all acid in the drop has reacted:

dcAdt

= E(1 − Xmicro

Xmeso

)(〈cA〉 − cA) − E

(1 − Xmicro

Xmeso

)〈cB〉 (7)

dcBdt

= 0 (8)

dcPdt

= E(1 − Xmicro

Xmeso

)(〈cP〉 − cP) + E

(1 − Xmicro

Xmeso

)〈cB〉

− 3kv�Mc

Gm2 (9)

dcDdt

= E(1 − Xmicro

Xmeso

)(〈cD〉 − cD) + E

(1 − Xmicro

Xmeso

)〈cB〉 (10)

with m2 denoting the second moment of the crystal size distribu-tion. After all acid has been consumed, the supersaturation in thefeed drop decreases due to dilution and consumption by growth,described by Eq. (11). The concentration of sodium chloride also de-creases until mixing is complete (Xmicro = 1)

dcPdt

= E(1 − Xmicro

Xmeso

)(〈cP〉 − cP) − 3kv

�Mc

Gm2 (11)

dcDdt

= E(1 − Xmicro

Xmeso

)(〈cD〉 − cD) (12)

The concentration of benzoic acid in the well-mixed bulk decreasescontinuously due to crystal growth:

d(cPVbulk)dt

= −3kv�Mc

Gbulk〈m2〉 (13)

When the feed drop has been completely micromixed, the bulk con-centration of benzoic acid is updated by setting it equal to the con-centration in the feed drop (which now has a volume fraction of 1).Growth in the bulk continues for the remainder of the time interval.At the end of each time interval, new bulk concentrations of ben-zoate and sodium chloride are calculated from the mass balances

〈cB〉new = 〈cB〉old − �tfeedQf /Vtot(c0A + 〈cB〉) (14)

〈cD〉new = 〈cD〉old + �tfeedQf /Vtot(c0A − 〈cD〉) (15)

where c0A is the acid concentration in the feed and Qf is the volumetricfeed flow rate.

The population balance for the feed drop is derived in a similarway as the equations for the concentrations. The population densitychanges due to nucleation, growth and mixing with the bulk giving

�n�t

+ G�n�L

= E(1 − Xmicro

Xmeso

)(〈n〉 − n)

n(0, t) = BfG

n(L, 0) = 0 (16)

Bf is the nucleation rate and G is the growth rate. The populationbalance in the bulk is written:

dndt

+ GbulkdndL

= 0

〈n〉(0, t) = BbulkVbulk

Gbulk

〈n〉(L, 0) = 0 (17)

When GRD is excluded, the population balance is solved via mo-ment transformation (Randolph and Larson, 1988). The follow-ing equations for the moments in the feed drop are obtained(Uehara-Nagamine, 2001):

dmj

dt= 0jBf + jGmj−1 + E

(1 − Xmicro

Xmeso

)(〈mj〉 − mj)

j = 0, 1, 2, . . . (18)

The moments in the bulk change as

d〈mj〉dt

= 0jBbulkVbulk + jGbulk〈mj−1〉 j = 0, 1, 2, . . . (19)


2.2. Growth rate dispersion

To include GRD in the model, we use the moment analysis devel-oped by Klug and Pigford (1989). Each crystal is born with a growthrate activity kg, which remains constant during the crystal's lifetime.A distribution of growth rate activities within the crystal popula-tion results in GRD. As the crystals grow, an initially monosized sizedistribution attains the same functional form as the distribution ofgrowth rate activities. This allows us to retrieve the whole size dis-tribution even though the simulation program only calculates thechanges in moments of subpopulations.

The growth rate of each individual crystal is expressed by

G = kgD(S) (20)

where D(S) is the driving force function for growth (Klug andPigford, 1989). The distribution of growth rate activities is given bythe distribution function

Pg = Pg(kg ,�g , kg) (21)

The population balance is solved by the method of characteristics.The formed crystals are divided into subpopulations that consist ofcrystals that were born in the same feed time interval.

The population balance for each subpopulation is written

�ni�t

+ kgD(S)�ni�L

= 0 (22)

where ñi is the total population density (unit: [#/m]) of the sub-population i. The total population density is not influenced by thevolume changes.

The population density for the whole population is obtained byintegration over all growth rate activities

n(L, t) =∫ ∞

0n(L, t, kg) dkg (23)

The evolution of the moments of each subpopulation is described bythe following equations (Klug and Pigford, 1989; Ståhl et al., 2001):

dm(i)L1

d�i= m(i)

L0m(i)g1 (24)

dm(i)L2

d�i= 2m(i)

L1(0)m(i)g1 + 2�im

(i)L0m

(i)g2 (25)

dm(i)L3

d�i= 3m(i)

L2(0)m(i)g1 + 6�im

(i)L1(0)m

(i)g2 + 3�2

i m(i)L0m

(i)g3 (26)

wheremLr represents the rth moment of the crystal size distribution,mgr the rth moment of the growth rate activity distribution, and �is the transformed time which is defined as

d�dt

= D(S) (27)

Unfortunately, with the software available, GRD growth cannot beapplied to the nuclei directly as they form. As an approximation, itis assumed that the nuclei grow without GRD in the feed droplet.

The consumption term must account for growth of both nucleiand earlier formed crystals that are mixed in from the bulk. Weobtain

rgrowth = −3kv�Mc

GmnucL2 − kv

�Mc

dmL3

d�

(d�dt

)f

(28)

The benzoic acid concentration in the bulk is given by

d〈cP〉dt

= −kv�Mc

dmL3

d�

(d�dt

)bulk

(29)

The evolution of the crystals can be traced provided that the initialmoments and the value of � are known for each subpopulation. Thenuclei born in the feed drop during a time step forms a subpopula-tion, and the initial moments of the subpopulation are the momentsof the nuclei at tXmicro = 1,

miLr(0) = mnuc

Lr (tXmicro=1) r = 0, 1, 2, . . . (30)

Since the bulk is well-mixed, all crystals are equally probable to befound in the feed region at a certain time. The probability is givenby the volume fraction of the feed drop at that time. The increase of�i for each subpopulation depends on how much time is spent athigh supersaturation compared to the time in the bulk:(d�dt

)i= Xmicro

(d�dt

)f+ (1 − Xmicro)

(d�dt

)bulk

(31)

The moments of each crystal subpopulation at a certain time is givenby

m(i)Lk(�i) =

r=k∑r=0

(kr

)m(i)

L(k−r)(0)m(i)gr (�i)

r k = 0, 1, 2, . . . (32)

2.3. The SF-model

The SF-model is a Eulerian model which was originally presentedby Villermaux (1989). It divides the crystallizer into two zones: thefeed zone around the feed point and the bulk zone. The exchangebetween the two zones is determined by two time constants, whereone constant describes dilution by convection of fluid from the feedzone into the bulk and the other describes further exchange betweenthe two zones by an “interaction by exchange with the mean” mech-anism. In the original model, the time constants are phenomenologi-cal constants without a concrete physical basis, but Zauner and Jones(2000a,b) used the mesomixing and micromixing time constants, intheir model of calcium oxalate precipitation.

The SF-model is described by the following equations for thevolume of the zones (Villermaux and Falk, 1994):

dVf

dt= Qf − Vf

tmeso(33)

dVtot

dt= Qf (34)

The volume of the feed zone reaches a constant value after a fewseconds. The volume is then:

Vf = Qf tmeso (35)

At the hydrodynamic conditions in this work, the feed zone is gen-erally very small (in the order of �l) with a residence time of tmeso.The limiting reactant in the feed zone is benzoate, which means thatthe supersaturation generation in the feed zone is determined bythe transfer rate of benzoate from the bulk to the feed zone.

The flow of component j between the feed zone and the bulkzone is given by

uj =Vf cjftmeso

+ Vf (cjf − cjb)tmicro

(36)

Using Eq. (36), a mass balance for each component is written in thefeed and the bulk zone, respectively. In total, there are eight massbalance equations. For example, for hydrochloric acid (componentA) in the feed zone:

d(Vf cAf )dt

= Qf c0A − rA − Vf cAf

tmeso− Vf (cAf − cAb)

tmicro(37)

where rA is the rate of consumption due to reaction.


Since the residence time in the feed zone is very short, it is as-sumed that growth in the feed zone does not significantly influencethe final crystal size. The nuclei that are formed are moved to thebulk zone immediately, and the population balance is only solvedfor the bulk zone. The population balance is written in terms of thetotal population density in the crystallizer (n = nV , unit: [#/m])

�n�t

= −G�n�L

n(0, L) = 0

n(t, 0) = (Bf Vf + BbVb)G

(38)

2.4. Crystallization and mixing kinetics

Crystallization kinetics for nucleation and growth, as well as ex-pressions for the mesomixing and micromixing constants must beinserted into the model. The supersaturation is expressed by the su-persaturation ratio

S = cc∗ (39)

The solubility of benzoic acid varies with temperature and concen-tration of sodium chloride (Åslund, 1994):

log10 c∗ = −5.6239 + 0.0137T − 0.1771cNaCl (40)

The volume shape factor of a spherical particle has been used(kv = �/6) since the experimental size distributions were deter-mined by an electro sensing zone instrument.

The main crystallization processes are primary nucleation andcrystal growth, and are described by

B = Kp1 exp(

− Kp2

ln2 S

)(41)

G = kg(S − 1)g (42)

If GRD is included, the driving force for growth is

D(S) = (S − 1)g (43)

The micromixing time constant in the model is

tmicro = 1E

= 12.7√

��

(44)

and the mesomixing time constant is

tmeso = 2

(d2

�

)1/3

(45)

as was proposed by Torbacke and Rasmuson (2004). The mixing con-stants are estimated from the energy dissipation rate at the feedpoint location, calculated using the mean energy dissipation ratecombined with literature data of energy dissipation rate distribu-tions. The used hydrodynamic constants and energy dissipation ratesfor different impellers and feed point locations are summarized inTable 1.

The weight mean size and the corresponding coefficient of vari-ation are calculated from the moments

L43 = mL4

mL3(46)

and

CV43 =(mL3mL5

m2L4

− 1

)0.5

(47)

Table 1Hydrodynamic constants for the different impellers.

Np NQ �/� (impeller) �/� (bulk) �/� (surface)

Rushton turbine 5 0.8 15 1.2 0.075Pitched blade turbine 1.3 0.8 6

Energy dissipation rates are taken from Jaworski and Fort (1991) for the pitchedblade turbine and from Bourne and Yu (1994) for the Rushton turbine.

2.5. Retrieving the crystal size distribution from the moments

For the E-model with GRD, the entire crystal size distributionmaybe retrieved from the moments. The functional form of a subpop-ulation is determined by the growth rate activity distribution, andthe simulated size distribution can be constructed from the distri-bution function Pg. The parameters kg and �g are substituted by themean size Li and the standard deviation �L

(i). Since the distributionis normalized, it is multiplied with the total number of crystals inthe population to determine the population density at size L as

ni(L) = NiPL(Li,�(i)L , L) (48)

The total population density is the sum of the population densitiesof all subpopulations:

n(L) =imax∑i=1

ni(L) (49)

The growth rate activity distribution is assumed to be log-normal(Ståhl et al., 2001), and is described by

Pg = 1√2� ln �′

gexp

⎡⎣− (ln kg − ln k

′g)

2

2 ln2 �′g

⎤⎦ (50)

where k′g is the geometric mean, and �′

g is the standard deviation of

kg/k′g (Randolph and Larson, 1988):

k′g = kg

exp(0.5 ln2 �′g)

(51)

and

ln �′g =

√ln(CV2

g + 1) (52)

The rth moment of the growth activity distribution is calculated as(Randolph and Larson, 1988)

mgr = k′rg exp(0.5r2 ln2 �′

g) r = 0, 1, 2, . . . (53)

When the log-normal distribution is inserted into Eq. (48), the fol-lowing expression for the population density distribution is obtained

ni(L) = Ni1√

2� ln �′(i)L

1(L − L0)

exp

(− (ln(L − L0) − ln L′

i)2

2 ln2 �′(i)L

)(54)

where

L′i =

Li − L0

exp(0.5 ln2 �′(i)L )

(55)

ln �′(i)L =

√√√√√ln

⎛⎝(

�(i)L

Li − L0

)2

+ 1

⎞⎠ (56)


2.6. Initial values in the simulations

In the beginning of the experiments in the 10- and 200-L scale,the bulk is undersaturated (Torbacke and Rasmuson, 2004). Nucleithat are formed in the region of high supersaturation will dissolvewhen they enter the undersaturated bulk. This part of the experi-ment is, therefore, simulated by assuming a perfectly mixed vessel towhich benzoic acid and sodium chloride are added until the bulk issaturated. The time to reach saturation and the corresponding bulkconcentrations are then used as initial values in the simulation. Forthe 1-L experiments, the bulk solution was saturated with benzoicacid and a stoichiometric amount of sodium chloride was added be-fore the experiment started (Åslund and Rasmuson, 1992).

The feed drop initially consists of pure acid-rich feed solution.The initial value of the micromixed and mesomixed volume fractionsboth equal X0 (Baldyga and Bourne, 1999). Two different approachesto determine X0 (Baldyga and Bourne, 1999) have been evaluated.When

X0 = Qf /(Qf + qc) (57)

where qc is the bulk volume flow calculated as

qc = 2NQNsD3s (58)

X0 becomes constant independent of the initial volume of the feeddrop (i.e. of the feed discretization). However, using this approach,the number of particles formed during a time step, and the totalmass crystallized depends on the initial volume of the drop. Thetotal mass of benzoic acid may deviate by more than 50%, and achange in �tfeed changes the simulated weight mean size and thetotal number concentration significantly. Eq. (57) should only beused in a circulation-loop model (Baldyga et al., 2001). Instead, thefollowing initial value (also used by Uehara-Nagamine, 2001) is usedin the present work:

X0 = Qf�tfeed/Vtot (59)

With this equation, the mass balance is satisfied and the predictedCSD is reasonably constant even if �tfeed is changed.

2.7. Numerical solution and parameter estimation

When the E-model without GRD is used, the population bal-ance is solved using the method of moments (Randolph andLarson, 1988), while the E-model with GRD is solved using themethod of characteristics. When the SF-model is used to describethe mixing process, the population balance is solved using themethod of lines with a first-order, upwind differentiation scheme.A grid size of 0.05�m was deemed sufficient to reduce the nu-merical diffusion to an acceptable level. In all cases, the resultingsystem of ODEs is solved using software from the Matlab ODE-suite(Mathworks Inc., www.mathworks.com).

For the E-model with mesomixing and GRD, parameter estima-tions are carried out to determine nucleation and growth rate con-stants that correctly reproduces the variation in weight mean sizein the experiments. The objective function is defined as

F =∑

R2 (60)

R = (Lsim43 − Lexp43 ) × 106 (61)

A subset of four experiments is included in the optimization forthe 1- and 10-L scale. Several optimizations are run for differentcombinations of experiments, and the kinetics that gives the bestoverall description of the variation in weight mean size is manuallyselected. At the 200-L scale, the three experiments are all included

Table 2The kinetic parameters that were used in the simulations with the E-model withmesomixing and GRD.

Kp1 Kp2 kg g CVg

1-L 1.78×1016 17.9 9.9×10−9 1.35 0.2910-L 3.8×1015 17.9 6.9×10−9 1.35 0.29200-L 8.1×1014 17.9 4.45×10−9 1.35 0.29

in the optimization. The Levenberg–Marquardt optimization methodin the implementation in the Matlab optimization toolbox is used.The parameters and the residuals are scaled to be in the order ofunity by dividing each constant by its typical order of magnitude.

The confidence range for an estimate obtained by optimizationwith a least-squares method can be estimated from the covariancematrix of the estimate (Fletcher, 1987). The variance of estimate i iscalculated as

�2i = 2F

m − nHii (62)

where F is the sum of squares at the optimum, m is the number ofdata points, n is the number of estimated parameters and Hii is theith diagonal element of the Hessian at the optimum. In least-squaresoptimizations, the Hessian can be approximated by

H = 2JJT (63)

where J is the Jacobian at the optimum (Fletcher, 1987). The Jacobianis calculated by the optimization routine.

3. Results

The simulation results are compared with experimental data pre-viously published by our research group. The 1-L semi-batch reactioncrystallization data was published by Åslund and Rasmuson (1992),the 10- and 200-L data was published by Torbacke and Rasmuson(2004). These studies include experiments at different agitation rate,feed point location, reactant concentration, feed pipe diameter, totalfeed time, and crystallizer volume.

The E-model with GRD can give a qualitatively correct descriptionof the semi-batch reaction crystallization of benzoic acid, and theresults of this approach are presented in this chapter. The model'sability to describe the influence of mixing phenomena on the prod-uct crystal size distribution is sensitive to the kinetics used to de-scribe the rates of nucleation and growth. In addition, it is foundthat the optimum kinetic representation is not independent of scale.The results reported in this section are obtained using the kineticsin Table 2, for which the E-model correctly captures the variation inweight mean size for most processing conditions.

The results for the SF-model are unsatisfactory as will be pre-sented and analyzed in the discussion section, where also the im-portance of GRD will be examined. In the discussion we will alsoexamine the influence of the crystallization kinetics.

3.1. General behavior of the E-model with GRD

An example of a simulated crystal size distribution and the cor-responding relative mass distribution is given in Fig. 1. The generalshape of the simulated CSD and how it compares to the experimentalone is typical for all experiments. There is some deviation at smallersizes while a good description is obtained for larger sizes. The pre-dicted evolution of the supersaturation is shown in Fig. 2. The to-tal rate of nucleation in the crystallizer, is strongly governed by themaximum supersaturation in the feed region. The figure shows howthe maximum supersaturation decreases with time since the con-centration of the bulk reactant (benzoate) decreases with time. The

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0 50 100 1501012

1014

1016

1018

Crystal size (μm)

Pop

ulat

ion

dens

ity, (

#/m

3 , m

)

ModelExp.

0 50 100 1500

0.5

1

1.5

2

2.5

Crystal size (μm)

Rel

ativ

e M

ass

dens

ity (%

/μm

)

ModelExp.

Fig. 1. Experimental and simulated population density distributions and relative mass distributions for a 10-L experiment at Ns = 500 rpm, d = 1mm, and tf = 180min.

0 50 100 150 2000

2

4

6

8

10

Time (min)

Sup

ersa

tura

tion

ratio

(−)

BulkSmax

0 50 100 150 2000

0.5

1

1.5

2

2.5x 108

Time (min)

Nuc

leat

ion

rate

in b

ulk,

(#/m

3 , s

)

Fig. 2. The predicted bulk supersaturation and maximum feed region supersaturation. The second plot shows the rate of nucleation rate in the bulk. A 10-L experiment atNs = 500 rpm, d = 1mm, and tf = 180min

supersaturation in the bulk increases at first, reaches a maximum,and then decreases again. The supersaturation in the bulk dependson the rate of supersaturation consumption by crystal growth andhence on the mass of crystals, and depends on the transfer of su-persaturation from the feed to the bulk and hence on the conditionsin the feed region and the mixing. When the bulk supersaturationpeaks, nucleation occurs in the bulk as shown in the diagram to theright. However, in this particular experiment only 0.1% of the totalnumber of crystals is formed in the bulk.

The behavior of the model when the total feed time, agitationrate and feed pipe diameter are varied is shown in Fig. 3. As ex-pected, the weight mean size increases with increasing total feed-ing time, (i.e. reduced feeding rate) approaching a constant value atlonger feeding times, where micromixing controls. The weight meansize decreases with increasing feed pipe diameter, because the initialscale of segregation increases which leads to that the time constantof mesomixing increases, Eq. (45). Increased agitation rate leads tolarger crystals, but at high agitation rates this dependence levelsoff. Increased agitation increases the rate of dilution of the super-saturation generated in the feed region, and hence the feed regionnucleation decreases. However, at very fast mixing, the increasedsupersaturation in the bulk increases the importance of the bulknucleation, which counteracts the reduced feed region nucleation.The fraction of the crystals that are formed in the bulk increaseswith increasing agitation rate as is shown in the fourth diagram ofFig. 3.

When the concentration of the feed reactant is increased, theproduct weight mean size decreases (see Fig. 4) as expected. At theend of each experiment, stoichiometric amounts are always added,and hence at increasing feed concentration either the volumetricfeed rate is constant and the total time is reduced, or the totaltime of feeding is constant and the volumetric feed rate is reduced.The decrease in the product weight mean size is stronger when thevolumetric feed rate is kept constant, i.e. when the feed time is short-ened. The reason is that in this case the rate of supersaturation gen-eration increases in the feed volume as well as in the bulk, as a resultof the higher molar feeding rate. In the case where the feed time isconstant the molar rate of addition of the reactant is constant and itis only the local conditions in the feed volume that are influenced.

3.2. Comparison with experimental data

As shown in Fig. 5, in general the agreement between the simu-lated and experimental weight mean size is quite satisfactory whenthe kinetics of Table 2 are used. The 1- and the 200-L scale are betterdescribed than the 10-L scale. If nothing else is stated in the presen-tation to follow, the conditions for the 1- and 10-L scale experimentsare those given in Table 3. At the 200-L scale, three experimentswere reported where agitation rate, feed pipe diameter and feedtime were varied simultaneously (Torbacke and Rasmuson, 2004).

In Figs. 6 is shown the ability of the model to capture the influ-ence of reactant concentration at the 1-L scale. In all experiments


0 200 400 60035

40

45

50

55

Total feed time (min)

Wei

ght m

ean

size

(μm

)

0 2 4 6 830

35

40

45

50

Feed pipe diameter (mm)

Wei

ght m

ean

size

(μm

)

0 500 1000 150025

30

35

40

45

Stirrer speed (rpm)

Wei

ght m

ean

size

(μm

)

0 500 1000 15000

10

20

30

40

Stirrer speed (rpm)

Cry

stal

s fo

rmed

in b

ulk

(%)

Fig. 3. Model prediction of influence on product weight mean size of the feed time, stirrer speed and feed pipe diameter. Base case conditions: tf = 90min, d = 1.8mm, andNs = 500 rpm.

0 1 2 3 420

30

40

50

60

70

HCl concentration (M)

Wei

ght m

ean

size

(μm

)

Wei

ght m

ean

size

(μm

)

Constant feed time

0 1 2 3 420

30

40

50

60

70


Constant feed rate

Fig. 4. Model prediction of influence of feed acid concentration.

10 20 30 40 50 60 70

10

20

30

40

50

60

70

Simulated weight mean size (μm)

Exp

erim

enta

l wei

ght m

ean

size

(μm

)

1 litre

10 litre200 litre

Fig. 5. Comparison between predicted and experimental product weight mean sizes.

the total amount of reactant added is equal. When the concentra-tion is decreased, final stoichiometry is maintained by increasing thefeed volume. The decrease in weight mean size at increasing feedconcentration is correctly reproduced by the model, both when thevolumetric feed rate is kept constant, and when the total feed timeis kept constant. In Fig. 7 is shown that the model is able to describethe influence of the feed point location when the local energy dis-sipation rate corresponds to the data of Table 1. Fig. 8 shows thatthe model description is satisfactory with respect to the influence ofthe total feeding time at different reactant concentrations in the 1-Lscale.

The influence of the feeding time in the 10-L scale experimentsis well described by the model, Fig. 9. The only exception is forthe 3.8mm feed pipe, where backmixing is believed to significantlyinfluence the experimental results (Torbacke and Rasmuson, 2004).Regarding the feed pipe diameter, the simulations correctly describethe overall trend of the experimental results, Fig. 10. However, at60min feed time, the experimental weight mean size decreases atfirst, and then increases. This is not captured by the model, andis difficult to explain within the inertial-convective mesomixing


Table 3The default processing conditions in the experiments (Åslund and Rasmuson, 1992; Torbacke and Rasmuson, 2004).

Agitation rate (rpm) Feed point location Feed pipe diameter (mm) Feed time (min) Bulk concentration (M) Feed concentration (M) Impeller type

1-L 400 Impeller 1.5 90 0.35 1.4 RT10-L 500 Impeller 1.8 90 0.35 1.4 PBT

RT—Rushton turbine and PBT—pitched blade turbine.

0 1 2 3 420

30

40

50

60


Wei

ght m

ean

size

(μm

)

Qf ≈ constant

0 1 2 3 430

35

40

45

50

HCl concentration (M)W

eigh

t mea

n si

ze (

μm)

tf = 90 min

Exp.

Model

Exp.

Model

Fig. 6. Comparison between simulations and experiments regarding the influence of reactant concentration in the 1-L experiments.

Impeller Bulk Surface15

20

25

30

35

Feed point

Wei

ght

mea

n si

ze (

μm)

N = 200 rpm

Exp

Model

ExpModel

Impeller Bulk28

30

32

34

36

38

Feed point

Wei

ght

mea

n si

ze (

μm)

N = 400 rpm

Fig. 7. Comparison between model and experiment regarding the influence of the feed point location in the 1-L experiments.

mechanism. Perhaps, this behavior stems from a transition in thegoverning mesomixing mechanism in the experiments. Fig. 11shows that the description of the variation in mean size with vary-ing agitation rate is satisfactory. The model predicts an increasein mean size with increasing agitation rate at all investigatedconditions. In the experiments, the weight mean size increaseswith increased agitation, except for the experiments with 1mmfeed pipe and 90min feed time, where the mean size decreasessomewhat.

In the three 200-L scale experiments (Torbacke and Rasmuson,2004), the feed pipe diameter, agitation rate, and total feed timewere varied simultaneously. As shown in Fig. 5 the model gives agood prediction of the results of these experiments.

4. Discussion

4.1. The E-model with inertial-convective mesomixing

The E-model with GRD is able to predict the influence of process-ing conditions on the product weight mean size, provided that ap-propriate crystallization kinetics are used. The model can predict the

general influence of reactant concentration, agitation rate, feed pipediameter, feed point position and feeding rate. However, concerningthe scale of operation some systematic discrepancy is indicated. Itis found in the comparison of model predictions and experimentalresults that the optimum set of crystallization kinetics is somewhatscale dependent, Table 2. At increasing scale there is a systematic de-crease in both the nucleation rate constant and the growth rate con-stant. If the crystallization kinetics are assumed constant the modelwill predict a decrease in the product weight mean size with in-creasing scale of operation, as is shown in Fig. 12 (left). However, inthe experimental results, Fig. 12 (right), there is no clear influence ofscale. Other processing conditions appear to have a stronger influ-ence on the product than the scale of processing. Please note that thecrystallizers used in the experimental studies are not geometricallysimilar (Torbacke and Rasmuson, 2004) and that the required localenergy dissipation rates used for the simulations are just estimatedvalues. Torbacke and Rasmuson (2004) found that the experimen-tal product weight mean size was reasonably well correlated to thedimensionless number TR,

TR = ubulktfd

(64)


20 40 60 80 10025

30

35


Wei

ght m

ean

size

(μm

)

cHCl = 3.51 M

Exp.

Model

Exp.

Model

50 100 150 20025

30

35

40


Wei

ght m

ean

size

(μm

)

cHCl = 1.4 M

Exp.

Model

0 50 100 150 20020

30

40

50

60


Wei

ght m

ean

size

(μm

)cHCl = 0.56 M

Fig. 8. Comparison between simulations and experiments regarding the influence of the total feed time at the 1-L scale. Ns = 200 rpm except for one experiment at 400 rpm.

50 100 150 20035

40

45

50

55


Wei

ght m

ean

size

(μm

)W

eigh

t mea

n si

ze (

μm)

Wei

ght m

ean

size

(μm

)W

eigh

t mea

n si

ze (

μm)

d = 1.0 mm

Exp.

Model

Exp.

Model

Exp.

Model

Exp.

Model

50 100 150 20030

35

40

45

50


d = 1.8 mm

50 100 150 20030

35

40

45

50


d = 2.7 mm

50 60 70 80 90 10030

32

34

36

38

40


d = 3.8 mm

Fig. 9. Comparison between simulations and experiments regarding the influence of the total feeding time at the 10-L scale.

where ubulk is the resultant bulk flow velocity past the feed point, tfis the total feeding time and d is the feed pipe diameter. Thus, theexperimental data correlate to the total feeding time rather than tothe volumetric feed rate, which is the parameter entering into themodel.

Of course, the mixing time constants in the model are approx-imate since they are calculated from somewhat rough estimatesof the local energy dissipation rate at the feed points, and themesomixing time constant, according to Eq. (45) assumes that theintegral scale of concentration fluctuations depends on the feed pipe


0 1 2 3 425

30

35

40


Wei

ght m

ean

size

(μm

)W

eigh

t mea

n si

ze (

μm)

Wei

ght m

ean

size

(μm

)W

eigh

t mea

n si

ze (

μm)

tf = 60 min

Exp.

Model

Exp.

Model

Exp.

Model

Exp.

Model

0 1 2 3 435

40

45


0 1 2 3 4


0 1 2 3 4


tf = 90 min

40

42

44

46

48

50tf = 120 min

40

45

50

55tf = 180 min

Fig. 10. Comparison between simulations and experiments regarding the influence of the feed pipe diameter at the 10-L scale.

200 400 600 800 100030

35

40

45

Agitation rate (rpm)

Wei

ght m

ean

size

(μm

)

Wei

ght m

ean

size

(μm

)

d = 2.7 mm, tf = 90 min

400 500 600 700 80040

42

44

46

48


d = 1.0 mm, tf = 90 min

Exp.

Model

Exp.

Model

Wei

ght m

ean

size

(μm

)

Exp.

Model

400 600 800 100050

55

60

65


d = 1.0 mm, tf = 180 min

Fig. 11. Comparison between simulations and experiments regarding the influence of agitation rate at the 10-L scale.

diameter only. Furthermore, the model disregards effects causedby varying energy dissipation rates during the course of mixing.The feed point is generally located in the discharge stream of the

impeller, where both the turbulence and the linear velocity of thefluid are high. However, the feed will be convected away from theimpeller and into zones of lower energy dissipation rate before the


1 10 100

10

20

30

40

50

Exp

erim

enta

l wei

ght m

ean

size

(μm

)

Exp

erim

enta

l wei

ght m

ean

size

(μm

)

Model

1 litre10 litre200 litre

1 10 100

20

30

40

50

60

70

80

Volume (dm3)Volume (dm3)

Experimental


Fig. 12. Comparison between simulations and experiments regarding the influence of the crystallizer volume, using the kinetics from the second row of Table 4.

mixing and nucleation are completed. For fast reactions, Bourne(2003) analyzed the effect of inhomogeneous turbulence whilescaling-up at geometric similarity. He concluded that the reactionzone occupies a smaller relative volume at larger scales. Baldygaet al. (2001) accounted for effects of macromixing and distributionof the mixing conditions in the tank by a one-dimensional single-circulation-loop plug flow model. If such a model is included, theinfluence of scale may be different since the circulation time, theintegral scale of turbulent flow, and X0 will depend on scale.

The E-model with mesomixing can without too much difficultybe adapted to include a variation in energy dissipation rate along thetrajectory of each feed volume, even though it would lengthen thesimulation time. It could be done by dividing the vessel into regionswith different energy dissipation rates, and assume that the feeddrop spends a certain part of a circulation in each region. An alter-native would be to use CFD simulations of the vessel together witha prediction of how the feed drop spreads out to calculate an aver-age time-dependant energy dissipation rate for the feed drop. On theother hand, we may note that the TR-number was successfully usedto correlate the experimental results and is also based on the hydro-dynamic conditions at the feed point only (Torbacke and Rasmuson,2004). Furthermore, it turns out that the simulation results are notthat sensitive to changes in the local energy dissipation rate. Fig. 13shows how the weight mean size varies in the 200-L experimentswhen the feed point energy dissipation rate is increased/decreasedone order of magnitude. If the 10-L scale kinetics from Table 2 areused in the 200-L simulations, the energy dissipation rate must beincreased by a factor of 40 to reach a reasonable agreement with theexperimental results.

Ostwald ripening can influence the size distribution in a semi-batch crystallization. When the supersaturation in the feed region isconsiderably higher than in the bulk, nuclei may dissolve if they areswept from the feed region into the bulk before they have had timeto grow to a size that is stable at the supersaturation prevailing inthe bulk solution. This may explain some of the deviation betweenthe simulated and experimental size distributions. However, someapproximate simulations reveal that under the influence of Ostwaldripening, the mean size only increases to a small extent, and theshape of the CSD is essentially unchanged.

Åslund and Rasmuson (1992), found in the experimental results amaximum in the product weight mean size versus feed point energydissipation rate. Even though the crystals were small the decrease athigher mixing intensity is perhaps partly due to breakage. However,at least in one experiment a decrease in size was found when theincrease in local mixing intensity was achieved by moving the feedpoint to a more intensively mixed region at constant rate of agita-tion. It was speculated that the rate of mixing may influence on the

2000 4000 6000 8000 10000 1200010

20

30

40

50

60

TR number

Wei

ght m

ean

size

(μm

)

0.1*ε

1*ε

10*ε

Fig. 13. Model prediction of the influence of feed point energy dissipation rate onthe product weight mean size in the 200-L experiments.

rate of supersaturation generation in the bulk of the tank. Increasedmixing leads to reduced time for nucleation and growth in the feedregion and to faster dilution of the supersaturation generated at thefeed point. This supersaturated liquid is diluted into the bulk, whichmay lead to increased levels of supersaturation in the bulk and mayperhaps induce primary nucleation in the bulk in a way that is com-parable to what is happening early in unseeded well-mixed batchcooling crystallizers. This mechanism is actually indicated already inFigs. 2 and 3. When this happens, the product weight mean size willdecrease with increasing agitation rate, and result in a maximum inthe dependence of the weight mean size on the rate of agitation. InFig. 14 is presented simulations for a 10-L crystallizer at 60min feedtime and 1mm feed pipe. At increasing agitation rate the weightmean size increases at first, reaches a maximum and then decreasesagain. Generally speaking the product weight mean size is deter-mined by the number of crystals generated and hence by the nucle-ation. Accordingly, minimum nucleation overall is found at approxi-mately 500 rpm. In the diagram to the right in Fig. 14, it is shown thatat increasing agitation a decreasing fraction of the crystals are nu-cleated in the feed drops before they have been mixed into the bulk,and hence the bulk nucleation gradually increase in importance.


24

26

28

30

32

34

36

Wei

ght m

ean

size

(μm

)

0 500 1000 150050

60

70

80

90

100

Stirrer speed (rpm)

0 500 1000 1500

Stirrer speed (rpm)

Per

cent

age

form

ed a

t fee

dpo

int (

%)

Fig. 14. Model prediction of the influence of agitation rate on the product weight mean size at 10 L volume, 60min feed time and 1mm feed pipe. The right-hand graphshows the percentage of crystals that are nucleated in the feed region.

Table 4Estimated kinetics using the T-mixer experimental data (Ståhl et al., 2001).

Kp1 Kp2 kg g CVg

Ståhl et al. (2001) (2.0 ± 0.4)×1019 17.9 ± 0.4 (5.6 ± 0.8)×10−6 1.66 ± 0.1 0.25 ± 0.004This study (9.1 ± 4)×1016 17.9 ± 0.7 (3.6 ± 0.6)×10−8 1.35 ± 0.3 0.29 ± 0.004

0 2 4 6 8

1018

1020

Crystal size (μm)

Pop

ulat

ion

dens

ity (#

/m3 ,

m)

S0=4.1S0=4.4S0=4.6S0=4.9

0 0.5 1 1.5 21

2

3

4

5

Time (s)

Sup

ersa

tura

tion,

c/c

*, (−

)

Fig. 15. Comparisons of the simulated and experimental size distributions using the kinetics reported by Ståhl et al. (2001), row 1 in Table 4. The figure also shows thecalculated supersaturation profile for the experiment at S0 = 4.6.

0 2 4 6 8

1018

1020

Crystal size (μm)

Pop

ulat

ion

dens

ity (#

/m3 ,

m)

S0=4.1

S0=4.4

S0=4.6S0=4.9

0 20 40 601

2

3

4

5

Time (s)

Sup

ersa

tura

tion,

c/c

*, (−

)

Fig. 16. Comparisons of the simulated and experimental size distributions using the T-mixer kinetics in this study, row 2 in Table 4. The figure also shows the calculatedsupersaturation profile for the experiment at S0 = 4.6.

4.2. Fitting the model with appropriate kinetics

The product weight mean size predicted by the model, and itsdependence on processing conditions is significantly influenced bythe kinetics of crystallization. Unfortunately, an optimization includ-ing all experiments and all parameters would have taken weeks to

perform with the computer resources available to us. In addition,it would perhaps not even converge to a satisfactory optimum, e.g.considering the difference in the kinetics depending on scale shownin Table 2.

In Table 4, first row, are given the previous benzoic acid crystal-lization kinetics from T-mixer experiments (Ståhl et al., 2001) here


reevaluated using the driving force: S-1, for growth. Obviously theseparameters differ significantly from those given in Table 2. In theoriginal work, it is assumed that mixing effects in the T-mixer can beneglected and that nucleation has terminated before the suspensionleaves the T-mixer pipe. If the T-mixer experiments are simulatedusing the kinetics of Table 2, nucleation and growth continue for amuch longer time. In the second row of Table 4, a new set of kineticsare presented where the T-mixer model is extended to account fornucleation also in the sampling vessel (well mixed) after the T-mixer,and the kinetics are estimated from the T-mixer experiments by

10 20 30 40 50 60 70

10

20

30

40

50

60

70

Simulated weight mean size (μm)

Exp

erim

enta

l wei

ght m

ean

size

(μm

)


Fig. 17. Comparison of predicted and experimental weight mean sizes for semi-batchcrystallization using the kinetics of row 2 in Table 4.

50 100 150 20020

30

40

50

60


Wei

ght m

ean

size

(μm

)

d = 1.0 mm

Exp.Model

50 100 150 200

20

30

40

50

60


Wei

ght m

ean

size

(μm

)

d = 1.8 mm

Exp.Model

Fig. 18. Comparison between simulations and experiments regarding the influence of total feed time at the 10-L scale, using the kinetics of row 2 in Table 4.

50 100 150 20020

30

40

50kg

Feed time (min)

Wei

ght m

ean

size

(μm

)

50 100 150 2000

20

40

60

80

100Kp1

Feed time (min)

Wei

ght m

ean

size

(μm

)

3.8.1015

1.0.1014

8.0.1016

1.0.10−8

6.9.10−9

4.0.10−9

Fig. 19. Influence of rate constants on the predicted product weight mean size.

minimizing the objective function

F =∑

R2

R = nsim − nexp

nsim + nexp + 1(65)

using the Levenberg–Marquardt non-linear least squares optimiza-tion method in the Matlab Optimization Toolbox (Mathworks Inc.,www.mathworks.com). In Figs. 15 and 16 is shown that the first setof kinetics (row 1 Table 4), do give a better description of the ex-perimental size distributions from the T-mixer experiments, and amuch faster decay of the supersaturation. However, the second setof kinetics (row 2 Table 4) gives an acceptable fit to the experimen-tal T-mixer data, and is much closer to those required (Table 2) togive a reasonable description of the semi-batch experiments. As canbe seen in Table 4, the values of the parameters Kp2, g, and CVg aresimilar for the two sets, while the rate constants of nucleation andgrowth differ. If the first set of T-mixer kinetics (row 1 Table 4), isused to simulate the semi-batch experiments, the calculated crystalmean size is about 5 times smaller than that found experimentally,and the variation in mean size when the feed pipe diameter or feedtime are changed is quite weak. The T-mixer kinetics from the ex-tended model (row 2 Table 4), allow for a better prediction of thesemi-batch experiments, but as shown in Fig. 17, the predicted meansize is always lower than the experimental, and the deviations in-crease when the crystallizer volume increases. Fig. 18 illustrates thatthe influence of total feeding time is considerably more pronouncedin the experiments than in the simulations for these kinetics.

In Fig. 19, is shown the sensitivity of the performance of thesemi-batch crystallization model to the rate constants of nucleationand growth (Kp1 and kg). The parameters of row 2 in Table 4 areused except for that the value of Kp1 and kg, respectively, are at first

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determined from the 10-L semibatch experiments by optimization,and then allowed to vary. Obviously, within the range of differencein the data between Table 2 and row 2 of Table 4, the predictions aresensitive to the values of Kp1, and kg, and wemay note that in Table 4,the standard deviations for these two parameters is relatively largerthan for the other parameters.

Obviously, in this case we have yet not reached the point wherecrystallization kinetics can be determined in dedicated experimentsin the laboratory, and then be combined with an adequate descrip-tion of the mixing conditions in the crystallizer, to predict the prod-uct weight mean size. It appears as if further work is required bothon the determination of kinetics as well as on the modeling of themixing in the crystallizer. Wemay note that if the kinetics are slowerthan originally assumed (Ståhl et al., 2001), the assumption of per-fect mixing in the evaluation of the T-mixer experiments is evenbetter fulfilled, while the design of the T-mixer apparatus shouldbe altered. Even though overall stoichiometric amounts are addedin the semi-batch experiments, the local conditions at each specificmoments are not stoichiometric. For ionic crystals there is evidencethat non-stoichiometric conditions may influence the nucleation andgrowth kinetics (Aoun et al., 1996; Vicum et al., 2003). However, ben-zoic acid crystallizes as a molecular compound, and in calculationsusing the simulated concentration of the different reactants duringan experiment and the dissociation constant for benzoic acid, it ispredicted that less than 1% of the benzoic acid molecules would bedissociated.

Another aspect is the fact that benzoic acid crystals formed athigh supersaturation have an irregular shape initially (Åslund andRasmuson, 1992), which rapidly heals and grows more regular ifthe crystals are left in the mother liquid (Ståhl et al., 2004), andthis may influence on the determination of kinetics. There is someevidence that this healing applies to the semi-batch experiments aswell, even though the product crystals in the semi-batch experimentsare generally more regularly shaped and the product crystal sizedistribution is quite stable, which was not the case for the productcrystals in the T-mixer experiments (Åslund and Rasmuson, 1992).It is not unreasonable to assume that newly formed crystals in thesemi-batch experiments are similar to the T-mixer crystals, sincethey form at high supersaturation in the feed point region.

4.3. The E-model without GRD

Without allowing for GRD, the E-model gives a good descriptionof how the product weight mean size depends on the different pro-cessing conditions, but the predicted size distributions are alwaysmuch more narrow than those found in the experiments. In com-plete absence of GRD, the method of Klug and Pigford (1989) cannotbe used. In this case we use moment transformation for the numeri-cal solution, and hence the full size distribution cannot be retrieved.However, using the model with GRD (for which the entire distribu-tion is retrieved), at low values of CVg, reveals that the simulateddistribution is too narrow and lacks the tailing towards larger sizeswhich is characteristic for the experimental distribution. In addition,the simulated CSD receives a bimodal shape, which is not found inthe experiments. This bimodal character is likely a result from thatmost nuclei are formed in the feed region, while the crystal growthoccurs mainly in the bulk. Since the nucleation rate in the feed re-gion is always high, and the bulk supersaturation first increases andthen decreases, a bimodal size distribution may develop.

4.4. The SF-model

To evaluate the SF-model, the nucleation and growth rate con-stants (Kp1 and kg) are estimated using different subsets of 10-Lscale experiments, in a manner similar to that used for the E-model.

Table 5Estimated kinetics for the SF-model.

Kp1 Kp2 kg g

10-L 2.93×1016 17.9 1.8×10−8 1.35

20 30 40 50 60 7020

30

40

50

60

70

Simulated weight mean size (μm)E

xper

imen

tal w

eigh

t mea

n si

ze (μ

m)

10 litre

Fig. 20. Comparison of predicted and experimental weight mean sizes for semi-batchcrystallization using the SF-model.

The estimated kinetics are given in Table 5. As shown in Fig. 20, theagreement between simulated and experimental mean sizes is lesssatisfactory than for the E-model. In Fig. 21 is shown the variationof the weight mean size when the feed pipe diameter and total feedtime are changed. The mean size increases with increasing total feedtime, and, as expected, this increase levels off at longer feed times.Versus the feed pipe diameter, the weight mean size exhibits a max-imum, and an increase in the feed pipe diameter changes the shapeof the simulated size distribution, as shown in Fig. 22. In the cor-responding experiments (Torbacke and Rasmuson, 2004), the meansize versus feed pipe diameter rather exhibits a minimum. The pre-dictions regarding the influence of the agitation rate depends on thefeed pipe diameter, as is shown in Fig. 23. For the larger feed pipe,the weight mean size increases steadily with increasing agitationrate, while the weight mean size shows a maximum at intermediateagitation rates when the smaller feed pipe is used. For low agita-tion rates, an increased feed pipe diameter increases the mean size,while the opposite is predicted at higher agitation rates. Comparedto the actual experimental data (Torbacke and Rasmuson, 2004), themodel description is quite good.

The SF-model does not behave in a consistent manner, whenthe mesomixing and micromixing time constants are varied, as isshown in Fig. 24. In essentially all the experimental results reducedmixing leads to reduced product weight mean size. When the me-somixing time constant increases in the model (i.e. mesomixing isreduced), the weight mean size increases at first, passes a maximumand then decreases, passes a minimum and then increases. When themicromixing time constant increases (i.e. micromixing is reduced)the product crystal weight mean size at first increases, but passesa maximum and then decreases again. In the experiments, the me-somixing time constant varies between 0.012 and 0.069 s, and themicromixing time constant varies between 0.006 and 0.030 s.

The SF-model, is attractive in that it is simple, easy to implement,and computationally inexpensive, but the model do not predict theexperimental results with the same confidence as the E-model. Evenwith optimized kinetics, the overall accuracy in the predictions islower, the influence of some processing conditions appears to be


0 1 2 3 410

20

30

40

50


Wei

ght m

ean

size

(μm

)

60 min90 min

0 100 200 30010

20

30

40

50


Wei

ght m

ean

size

(μm

)

Fig. 21. The variation in weight mean size with feed pipe diameter and feed time for the SF-model. Base case conditions: Ns = 500 rpm, tf = 90min, and d = 2.7mm.

0 20 40 60 801012

1014

1016

1018

Crystal size (μm)

Pop

ulat

ion

dens

ity (#

/m3 ,

m)

ModelExp.

0 20 40 60 80Crystal size (μm)

ModelExp.

1014

1013

1015

1016

1017

Pop

ulat

ion

dens

ity (#

/m3 ,

m)

Fig. 22. Comparison of simulated and experimental product size distributions using the SF-model. tf = 60min, Ns = 500 rpm: (a) d = 1mm and (b) d = 2.7mm for the 10-Lexperiments.

0 200 400 600 800 100025

30

35

40

45


Wei

ght m

ean

size

(μm

)

d=2.7 mmd=1 mm

Fig. 23. Predicted influence on the product weight mean size of agitation rate usingthe SF-model.

erratic, and the response to direct changes in the mixing time con-stants, is not in accordance with expectation. Another weakness isthat the mesomixing convection term in Eq. (37) does not dependon the bulk concentration. At the end of the feeding time, the feedzone can become undersaturated, even though the bulk zone is stillsupersaturated. It is likely that a major source of error in the modelis the assumption that there is a small well-mixed feed region ofhigh supersaturation in a large bulk volume of low supersaturation.

CFD simulations of a semi-batch precipitation by Wei et al. (2001),revealed the existence of two well-defined regions with differentsupersaturation levels, comparable to the zones in the SF-model.However, the reported supersaturation in the region of high su-persaturation varied by two or more orders of magnitude. Nucle-ation has a highly non-linear dependence on supersaturation, whichmeans that the nucleation rate would show an even stronger vari-ation inside the feed region. Another simplification in the model(however shared with the standard E-model used here) is that theenergy dissipation rate is assumed to be constant during the mixingof the feed, and is set to the value at the feed point. In reality, thefeed is convected away from the feed point and into regions withother hydrodynamic conditions, before nucleation and growth hasfinished.

5. Conclusions

A population balance model combined with a mechanistic modelof meso and micromixing can correctly capture the influence ofimportant processing conditions in single feed semi-batch reactioncrystallization of benzoic acid. When the E-model with inertial-convective mesomixing is used, a correct representation is deliveredof the influence on the product weight mean size of the agitationrate, the feed point location, the feed pipe diameter, the reactantconcentration, and the total feeding time. Of course, of decisive im-portance is the adoption of appropriate crystallization kinetics. Thework shows that when inadequate crystallization kinetics are in-serted, not only the magnitude of the weight mean size becomesincorrect but also the influence of the processing conditions can es-sentially vanish. In its present version themodel cannot fully account


0 0.05 0.120

25

30

35

40

45

Mesomixing time constant (s)

Wei

ght m

ean

size

(μm

)

0 0.02 0.04 0.0620

30

40

Micromixing time constant (s)

Wei

ght m

ean

size

(50μ

m)

Fig. 24. Predicted influence on the product weight mean size of the mesomixing and micromixing time constants using the SF-model.

for the influence (or lack of influence) of crystallizer size. This can bedue to that the bulk is assumed to be well-mixed and then introduc-tion of a circulation loop concept may lead to improved predictions.However, this may also be due to deficiencies in the estimation ofthe mixing intensity in the various crystallizers.

When the SF-model is used to describe mixing, the model pre-dictions are less satisfactory. The influence of processing conditionson the simulated weight mean size is sometimes inconsistent andcontradictory to experimental data. For a wide range of values, themean size decreases when the mixing intensity increases, while inthe experiments, increased mixing generally leads to a larger crystalweight mean size.

Previously determined kinetics of nucleation and growth esti-mated from T-mixer experiments give unsatisfactory results whenused in simulation of the semi-batch agitated tank crystallization.The predicted weight mean sizes are significantly smaller than theexperimental, and the influence of processing conditions is less pro-nounced. It is suggested that the kinetics are not fully appropriate,because of unrealistic assumptions in the evaluation of the T-mixerexperiments. A reevaluation of the T-mixer experiments is givenhere. Even though the new parameters do not give the best represen-tation of the semi-batch experiments, the result suggest that furtherwork on determination of reaction crystallization kinetics as well ason describing the mixing in agitated tank crystallizers will allow forsuccessful model based predictions of the product size distributionof full scale crystallizers.

Notations

B nucleation rate, #/m3 sc solute concentration, mol/m3

〈c〉 bulk concentration, mol/m3

c* solubility, mol/m3

c0A acid concentration in feed, mol/m3

CV43 coefficient of variation for the size distributionCVg coefficient of variation for the growth rate activity

distributiond feed pipe diameter, mDs impeller diameter, mD(S) driving force functionE engulfment rate constant, 1/sF objective functiong exponent in growth rate expressionsG crystal growth rate = dL/dt, m/sk Boltzmann constant (1.3805×10−23), J/Kkd mass transfer coefficient, m/skg growth rate activity

kg growth rate constant in power lawskv volume shape factorKp1, Kp2 primary nucleation rate parametersL characteristic crystal size, mL43 weight mean size, mmgr rth moment of growth rate activity distributionmLj, mj jth moment of size distribution, mj/m3

Mc molar weight, kg/moln population density, #/mm3

N number of crystals, #/m3

NP power number of impellerNQ pumping number of impellerNs agitation rate, 1/sPg growth rate activity distribution functionPL size distribution functionqc bulk pumping flow, m3/sQf volumetric feed rate, m3/sri production rate of substance i, mol/m3,sR residualS supersaturation ratio, c/c*

t time, stf total feed time, s�tfeed time step in the feed discretization, stmicro, tmeso micro and mesomixing time constants, sT temperature, KTR dimensionless number = ubulktf /�ubulk resultant velocity past the feed point, m/suj flow between zones in SF-modelV volume, m3

X volume fraction

Greek letters

� energy dissipation rate, W/kg� transformed time = ∫

D(S) dt, s� kinematic viscosity, m2/s� crystal density, kg/m3

�g standard deviation of growth rate activity distribu-tion

�L standard deviation of size distribution

Superscripts

〈〉 value in bulk– mean value∼ total value with respect to the whole crystallizer

volume (as opposed to per unit volume)′ variable in log-normal distribution


i for the ith subpopulationnuc crystals born in the particular droplet

Subscripts

b bulkf feed regioni for the ith subpopulation0 initial

Acknowledgment

The financial support of the Swedish Industrial Association forCrystallization Research and Development (IKF) is gratefully ac-knowledged.

References

Akiti, O., Armenante, P.M., 2004. Experimentally-validated micromixing-based CFDmodel for fed-batch stirred-tank reactors. A.I.Ch.E. Journal 50, 566–577.

Aoun, M., Plasari, E., David, R., Villermaux, J., 1996. Are barium sulphatekinetics sufficiently known for testing precipitation reactor models? ChemicalEngineering Science 51, 2449–2458.

Åslund, B., 1994. Kinetics and processing in reaction crystallization of benzoicacid. Ph. D. Thesis, Department of Chemical Engineering and Technology, RoyalInstitute of Technology, Stockholm.

Åslund, B., Rasmuson, Å.C., 1992. Semibatch reaction crystallization of benzoic acid.A.I.Ch.E. Journal 38, 328–342.

Baldyga, J., Bourne, J.R., 1999. Turbulent Mixing and Chemical Reactions. Wiley,Chichester.

Baldyga, J., Orciuch, W., 2001. Barium sulphate precipitation in a pipe—anexperimental study and CFD modelling. Chemical Engineering Science 56,2435–2444.

Baldyga, J., Bourne, J.R., Hearn, S.J., 1997. Interaction between chemical reactionsand mixing on various scales. Chemical Engineering Science 52, 457–466.

Baldyga, J., Henczka, M., Makowski, L., 2001. Effects of mixing on parallel chemicalreactions in a continuous-flow stirred-tank reactor. Transactions of the IChemE,Part A, Chemical Engineering Research and Design 79, 895–900.

Baldyga, J., Makowski, L., Orciuch, W., 2005. Interaction between mixing, chemicalreactions and precipitation. Industrial Engineering Chemistry and Research 44,5342–5352.

Bourne, J.R., 2003. Mixing and the selectivity of chemical reactions. Organic ProcessResearch and Development 7, 471–508.

Bourne, J.R., Yu, S., 1994. Investigation of micromixing in stirred tank reactors usingparallel reactions. Industrial Engineering Chemistry and Research 33, 41–55.

Chen, J., Zheng, C., Chen, G., 1996. Interactions of macro- and micromixing onparticle size distribution in reactive precipitation. Chemical Engineering Science51, 1957–1966.

Fletcher, R., 1987. Practical Methods of Optimization. second ed. Wiley, Chichester.

Jaworski, Z., Fort, I., 1991. Energy dissipation rate in a baffled vessel with pitchedblade turbine impeller. Collection of Czechoslovak Chemical Communications56, 1856–1867.

Klug, D.L., Pigford, R.L., 1989. The probability distribution of growth rates ofanhydrous sodium sulfate crystals. Industrial Engineering Chemistry andResearch 28, 1718–1725.

Lindberg, M., Rasmuson, Å.C., 1999. Product concentration profile in strained reactingfluid films. Chemical Engineering Science 54, 483–494.

Marchisio, D.L., Barresi, A.A., 2003. CFD simulation of mixing and reaction:the relevance of the micro-mixing model. Chemical Engineering Science 58,3579–3587.

Mersmann, A., 1995. Crystallization Technology Handbook. Marcel Dekker, Inc., NewYork.

Phillips, R., Rohani, S., Baldyga, J., 1999. Micromixing in a single-feed semibatchprecipitation process. A.I.Ch.E. Journal 45, 82–92.

Piton, D., Fox, R.O., Marcant, B., 2000. Simulation of fine particle formation byprecipitation using computational fluid dynamics. Canadian Journal of ChemicalEngineering 78, 983–993.

Randolph, A.D., Larson, M.A., 1988. Theory of Particulate Processes. second ed.Academic Press, San Diego.

Ståhl, M., Åslund, B.L., Rasmuson, Å.C., 2001. Reaction crystallization kinetics ofbenzoic acid. A.I.Ch.E. Journal 47, 1544–1560.

Ståhl, M., Åslund, B.L., Rasmuson, Å.C., 2004. Aging of reaction-crystallized benzoicacid. Industrial Engineering Chemistry Research 43, 6694–6702.

Torbacke, M., Rasmuson, Å.C., 2001. Influence of different scales of mixing in reactioncrystallization. Chemical Engineering Science 56, 2459–2473.

Torbacke, M., Rasmuson, Å.C., 2004. Mesomixing in semi-batch reactioncrystallization and influence of reactor size. A.I.Ch.E. Journal 50, 3107–3119.

Tosun, G., 1988. An experimental study of the effect of mixing on the particlesize distribution in BaSO4 precipitation reaction. In: Proceedings from the SixthEuropean Conference on Mixing, Pavia, Italy, pp. 161–170.

Uehara-Nagamine, E., 2001. Modeling and experimental validation of a single-feed semi-batch precipitation process. Ph.D. Thesis, Department of ChemicalEngineering, New Jersey Institute of Technology, Newark.

Verschuren, I.L.M., Wijers, J.G., Keurentjes, J.T.F., 2001. Effect of mixing on productquality in semibatch stirred tank reactors. A.I.Ch.E. Journal 47, 1731–1739.

Vicum, L., Mazotti, M., Baldyga, J., 2003. Applying a thermodynamic model to the non-stoichiometric precipitation of barium sulfate. Chemical Engineering Technology26, 325–333.

Vicum, L., Ottiger, S., Mazotti, M., Makowski, L., Baldyga, J., 2004. Multi-scalemodelling of a reactive mixing process in a semibatch stirred tank. ChemicalEngineering Science 59, 1767–1781.

Villermaux, J., 1989. A simple model for partial segregation in a semi-batch reactor.AIChE Meeting San Francisco, Paper 114a.

Villermaux, J., Falk, L., 1994. A generalized mixing model for initial contacting ofreactive fluids. Chemical Engineering Science 49, 5127–5140.

Wei, H., Zhou, W., Garside, J., 2001. Computational fluid dynamics modelling ofthe precipitation process in a semibatch crystallizer. Industrial EngineeringChemistry and Research 40, 5255–5261.

Zauner, R., Jones, A.G., 2000a. Mixing effects on product particle characteristics fromsemi-batch crystal precipitation. Transactions of the IChemE 78A, 894–902.

Zauner, R., Jones, A.G., 2000b. Scale-up of continuous and semibatch precipitationprocesses. Industrial Engineering Chemistry and Research 39, 2392–2403.

Zauner, R., Jones, A.G., 2002. On the influence of mixing on crystal precipitationprocesses—application of the segregated feed model. Chemical EngineeringScience 57, 821–831.

towards predictive simulation of single feed semibatch reaction crystallization

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