International Sugar Journal

# Crystal growth and crystallization control tactics in industrial sugar crystallizers Part 1. Crystal growth

## Abstract

The problems of crystal growth in industrial scale crystallization, where there are billions of crystals in the massecuite are rarely studied, and validation of the findings is rare, or is completely missing. Production of good quality sugar, however, should be based on an in-depth knowledge of the processes which characterize mass production. The major difficulty in the study of crystal growth under industrial scale production is the lack of suitable instruments to monitor online its governing parameters. SeedMaster Instruments developed by K-Patents Oy, have facilitated online measurement of crystallization parameters, namely supersaturation, crystal content and mother liquor purity. The acquired data can be used to check, refine and validate crystal growth rate models in real industrial environments. This can help us not only to have a better understanding of the process, but also to find better ways to its advanced control as well. Some studies on sugar crystallization have sought to formulate an equation pertaining to a single parameter, like supersaturation, temperature or mother liquor purity on crystal growth. In this paper a multi-component crystal growth rate model is formulated. It contains all of the governing parameters which can be measured by dedicated instruments or calculated by a SeedMaster in industrial crystallizers. The model is based on single-parameter components (correlated to a single parameter) selected from the literature. The components have been combined in a single equation. It is only natural that the model must be validated with data acquired during the normal operation of industrial crystallizers. Validation of the growth rate model relies on the comparison of the important parameters

1. directly calculated online by a SeedMaster Instrument (reference data) and
2. the same ones calculated by making use of the growth rate model (model data).

In our case the validation was completed in two batch vacuum pans at sugar refineries in Europe and in West Africa.The validated growth rate model provides a better understanding of the process of crystallization and can be used to design and implement more effective control strategies.

## Introduction

Crystallization process is really complex: there are a large number of parameters affecting the process. Besides sugar manufacturing, in other sectors such as pharmaceutical, crystallization is also a major process step. Understanding and researching sugar crystallization will help maximize crystal product yield. Though there are a lot of similarities with non-sugar products supporting crystallization, this paper focuses on sugar crystallization.

There is a large body of research on crystal growth. Its study can be characterized as

• experimental (confined mostly to experiments in the laboratories), or
• a mixture of the above two methods.

These studies have underscored the complexity of the problem and the lack of instruments able to provide reliable online data on the complete set of parameters governing crystal growth. However, results usually provide information on the effect of only one parameter on crystal growth. The often un-specified or completely neglected parameters and the sometimes inconsistent use of terms make the comparison and use of the results difficult.

Cost-effective and high-quality mass production of any product, sugar included, needs advanced instrumentation and automatic control. When confronted with the task to implement an effective automatic control strategy of sugar crystallization by up-to-date PLC-s or DCS-s, control engineers generally need an approach different from the purely academic or small-scale laboratory type one. They are less interested in the intricacies of single-crystal growth on the molecular and microscopic level, when the task is to produce good quality crystals by billions in a single strike. To be successful, it is vital to gain insight that is as accurate as possible into the inner workings of the process of industrial-scale sugar crystallization. Selection of a good control strategy needs

• good knowledge of the parameters governing the process, and
• knowledge of the relative and combined effect of these parameters on crystal growth,

that is a control engineering approach.

In this paper, it is assumed that only crystal growth takes place in the process, where full seeding with slurry or footing magma is being practiced.

Studies on the subject in most cases are directed at the effect of a single parameter on crystal growth. While these attempts are certainly useful, they alone are not able to provide a clear, proven and reliable picture on the relative and combined effects of all of the parameters governing the process of crystallization. However, development and implementation of good crystallization control strategy and practice is very much in need of this kind of information.

It is very well known that supersaturation is one of the parameters (often referred to as the “driving force”) having considerable effect on the rate of crystal growth. Contrary to false claims, there is no single instrument capable of providing reliable data on supersaturation in real time. With the introduction of the SeedMaster Instruments by K Patents (Rozsa (1996), Rozsa (2006), Rozsa (2011), (Rozsa, Arriaza, Romero (2011) online calculation of supersaturation and quite a few other important massecuite parameters became reality and actual practice in an increasing number of mills and refineries.

Aided by the online data acquired while monitoring and controlling hundreds of strikes in batch pans (pans from 30 to 120 m3 volume) at many sugar factories, an attempt is made here to:

• test in practice previous results documented in the literature on the effect of particular parameters on the rate of crystal growth and to
• formulate and validate in real industrial environments a single equation of crystal growth.

The main aim of this study is to help to develop an effective control strategy subject of another forthcoming publication.

## Crystal growth

### Crystal growth: terminology

In the studies related to crystal growth

• linear crystal growth (growth along the different axes of a crystal) and
• volumetric crystal growth

are investigated. However, it is not always clearly defined which one is being studied. This may lead to inconsistent or even contradictory claims regarding crystal growth. In this paper, our main objective is to determine and demonstrate the effect of the most important parameters on linear crystal growth. The results can be easily converted to its volumetric equivalent, if needed. In what follows, the term “crystal growth” is clearly related to linear crystal growth specified as da/dat, where

• a is the characteristic length of a crystal and
•  da/dt is the rate of crystal growth

### Crystal size (mean aperture MA), size distribution (CSD) and coefficient of variation (CV)

The most important parameters of product sugar crystals are mean crystal size (MA), size distribution (CSD characterized by the coefficient of variation (CV)) and color. Only the first two of these parameters will be briefly discussed here.

The MA of the product crystals should be as close to the target product size as possible. It depends very much on the seeding and control practice in use.

A large number of studies resulting in an equally large number of papers have been devoted to the subject of crystal growth.  There are three major theories regarding growth rate with more or less supporters and opponents for each one.

#### The common growth rate (CGR) theory

According to McCabe’s ΔL law (McCabe 2004): “The total linear growth of all crystals in the massecuite over the same time interval and along similar axes is the same, if they have similar and invariant geometric shapes and supersaturation is the same for all crystals in all locations in the massecuite. The linear growth rate is independent of crystal size.”

As a direct consequence of this theory a crystal crop having a wide size distribution like the ones used in seed slurries (for example: CV=80) will grow by the same growth rate (da/dt) resulting in a continuously improving (decreasing) CV if the requirements stated above are met.

#### Size dependent growth rate (SDG) theory

One form of the growth rate G   $(G=\frac { da }{ dt } )$   (Pot 1983) based on the SDG theory is expressed by Eq. 1

G(L) =G(0)*(1+α*L)  Eq. 1

According to the equation large crystals grow faster (constant α is positive). This results in a widening size distribution.

#### Growth rate dispersion (GRD) theory

Instead of differences in size, differences in the growth rates are due to random variations over time or crystal population. There are supporters and opponents of this theory as well. “GRD is a real phenomenon, whereas SDG is largely an artifact” (Srisanga 2014).

#### A practical approach

It is evident that with billions of crystals in a limited massecuite volume the strict requirements posted by McCabe’s law cannot be met; only approximated to some extent. The approximation is the best when the crystals are small and separated far from each other due to turbulent flow: that begins right after seeding has been completed. This is the period during which a fairly common growth rate prevails and results in an improving (decreasing) CV value. Later on, as crystal content increases the distances among them become less and less, and meeting McCabe’s requirements becomes more and more impossible. It can be proved that at about 30 to 34 % (by volume) crystal content (or 86 to 87 % massecuite solids content) the crystals begin to touch each other even when they are evenly distributed, resulting in increasing differences in the local parameters. On the other hand, with increasing crystal content, vacuum pans can less and less be termed as “perfectly mixed vessels”, the assumption so often taken for granted in theoretical studies. This means that as the crystals keep growing, there will be more and more locations in the massecuite where the major parameters (mother liquor purity, concentration, temperature, flow velocity and crystal content) will not be exactly the same and the chances of providing fresh feed syrup to supply growth are different.

Figure 1. Variation of CV and MA versus pan level during 3 strikes

In Figure 1. the variations of CV and MA data versus pan level (60, 75 and 90 %) in three strikes are shown (the strikes were ended at 100 % level). The data were determined by the use of a laser diffraction size analyzer in a sugar mill in Brazil. Seeding of the pan was completed at 32 % level, but no data were available from that time. Assuming seed crystal CV~ 80…90 in the slurry, there was already a considerable improvement in the CV value when level reached 60 %. This is the period of crystal growth where the fairly common growth rate prevails. The improvement continued, but slowed down after about 75 % level, the time when crystal content probably reached about 30 to 34 % by volume.

Lessons to learn: it is important

• to have high quality (small CV) seed crystals in the slurry when practicing full seeding, because

right after seeding, the improvement will start from the original CV value, and

• to have good, supersaturation-based control of the crystallization all over the strike in order to prevent nucleation or dissolution of the crystals.

## The major massecuite parameters governing crystal growth

It is a well known fact that crystal growth depends on many parameters of the massecuite, such as

1. overall crystallization growth rate,
2. supersaturation,
3. temperature,
4. non-sugar concentration,
5. crystal content and
6. massecuite circulation.

Based on literature data, the effect of these parameters on crystal growth will be discussed in a condensed form one by one.

### Supersaturation (SS)

It is a well-known and accepted fact that supersaturation (SS for short) is the most important parameter of crystallization. The multitude of its different definitions does not help to compare the results of research on its effect on crystal growth. The one which is most often used is as follows:

$SS=\frac { sugar\quad in\quad solution\quad (\frac { g }{ 100\quad g\quad water } ) }{ sugar\quad needed\quad for\quad saturation\quad (\frac { g }{ 100\quad g\quad water } ) } (both\quad at\quad the\quad same\quad temperature)$

It can be shown (Rozsa 1996, Rozsa 2006) that SS is a function of several variables, namely:

• liquid concentration (C),
• mother liquor purity (P),
• temperature (T),
• liquid (feed syrup) quality parameters (m, b, c), which are different for feed syrups of beet or cane origin and should be determined locally.

It is clear that SS is a multiparameter function of several variables of the feed syrup or mother liquor (that is parameters of the technical sugar solution):

SS = f(C, P, T, m, b, c)  Eq. 2

SS = 1.00 characterizes saturation. If SS is less than 1.00 fast dissolution of the existing crystals takes place, while above some limit (for example above SS = 1.15 depending on purity) spontaneous nucleation will be started. The normal area of control is in the range in between the two, close but safely below the nucleation limit.

As a rough approximation + / – 0.07 change in SS can be caused by:

+ / – 1 % change in liquid concentration (C), or

+ / – 5  change in temperature (T), or

+ / – 16 % change in feed syrup / mother liquor purity (P)

It is clear from Eq. 2 that SS cannot be measured by any single instrument. Online data on SS can only be provided by calculation, taking into account all of its governing parameters. In order to have reliable data on SS, liquid concentration has to be measured selectively with a least +/-0.1 % accuracy. In addition, the measurement should be undisturbed by crystal content, color and vapor bubbles. The only instrument capable to meet these requirements is the digital process refractometer pioneered by K-PATENTS Oy, Finland. It is to be noted that purity (P) of the solution is not constant during the process of crystallization; it has to be calculated online as well.

### Overall crystallization growth rate (K; f0)

Crystal growth depends on two processes:

• mass transport of the sucrose molecules to the reaction layer at the surface of the crystals and
• integration of the molecules into the crystal lattice.

The process depends on quite many parameters, like:

liquid velocity (v) and width (δ) of the stagnant layer on the crystal surface, liquid density (ρ) and viscosity (η), coefficient of diffusion (D), temperature (T) and purity (P). Non-sugar composition (Nc) has also a strong effect on crystal growth and shape as well (Belhami & Mathlouti 2004). Dextran gives the most important decrease in growth.

K = f0(ν, δ, a, ρ, η, D, T, P, Nc)    Eq. 3

As in many cases, when we do not have reliable and quantified information on different parameters governing a complex process, like in this case, all the variables (knowns and unknowns as well) are lumped in a multivariable parameter K. Unfortunately though, K is quite often regarded and treated in the simplified equations on growth rate as a constant, but it is anything, but constant.

### Crystal growth versus supersaturation (f1(SS))

The general and most often used, simplified equation of crystal growth is:

$\frac { da }{ dt } =K*(SS-b)^{ n }$  Eq. 4

da/dt:  growth rate (linear)

K:        overall crystallization growth rate

b:         constant

n:         constant

Opinion on the value of “b” and “n” is fairly common: in most cases both are equal to 1,0, so

f1(SS) = SS – 1        Eq. 5

Based on experiments the authors in their paper (Heffels et al. 1987) report on a somewhat different form:

$f1(SS)=\frac { da }{ dt } =41.22*(SS-1.038)(\frac { \mu m }{ min } )$     Eq.6

According to Eq. 6 if SS = 1.138 the rate of crystal growth is 4.122  $\frac { \mu m }{ min }$ . By using K=41.22 as a constant, this simplified equation hides the multitude of massecuite parameters which have considerable effect on the rate of crystal growth. This is certainly a gross simplification.

### Temperature (f2(T))

Keeping other parameters constant, increasing process temperature results in faster crystal growth (neglecting here the effect of temperature on the parameters listed in f0 and f1). It is generally accepted that the rate of growth increases exponentially with temperature and the general form of the equation is:

$f2(T)=\frac { da }{ dt } =A*EXP(B*T)$ Eq. 7

T: temperature ⁰C

The equation with actual data (Zagrodzki and Krysicki 1979) and SS = 1,06 (constant) with flow velocity

$v=0\frac { m }{ sec }$:

$[f2(T)=\frac { da }{ dt } =0.067*EXP(0.0586*T)/latex] Eq. 8

[latex]v=0\frac { m }{ sec }$

:

$f2(T)=\frac { da }{ dt } =0.0502*EXP(0.0495*T)$   Eq. 9

The effect of temperature on growth rate used here:

$f2(T)=\frac { da }{ dt } =0.277*EXP(0.0186*T)$   Eq. 10

### Non-sugar concentration (f3(NS))

Increasing non-sugar concentration decreases the rate of crystal growth. The equation to be used is (Rodrigez 1980):

$f3(NS)=\frac { da }{ dt } =EXP(-1.75*\frac { NS }{ 100 } )$ Eq.11

NS: non-sugar concentration (g / 100 g water)

### Crystal content by volume (f4(y))

It is interesting to note that the effect of crystal content on crystal growth is rarely studied, even more so because it is evident that with increasing crystal content considerable changes in the process of crystallization take place, such as:

• decreasing heat transfer coefficient between the calandria and the massecuite,
• increasing difficulties in supplying fresh feed syrup to the crystals,
• increasing differences in the important parameters (supersaturation, temperature, non-sugar concentration etc.) among different parts of the massecuite.

In experiments in the laboratories, crystal to crystal interactions are in most cases carefully avoided. Crystal growth is mostly studied by growing a single, or only a very few (far away from each other) crystals in small crystallizers mostly in ideal conditions (constant temperature, syrup flow etc.). The rather simplified equations on crystal growth provide higher rate of growth than the one that can be really produced in an industrial crystallizer. “The rate of crystallization in supersaturated sugar solutions with a large number of crystals was less than that found with a single or only a few crystals under identical conditions. There is a strong correlation between the rate of crystal growth in supersaturated solutions and the volume of solution among the crystals” (Zagrodzki 1968). In a large number of laboratory experiments a coefficient h was determined defined as

$h=(\frac { { v }_{ m }-{ v }_{ c } }{ { v }_{ m } } )^{ 2 }$ Eq. 12

Vm: massecuite volume

VC: crystal volume

It was found that depending on crystal content the rate of crystallization was decreased by the value of h compared to the rate valid for a single crystal under identical conditions. The reduction is quite considerable: the value of h changes in the 1 (no crystals) to 0.16 (crystal content = 60 % by volume) range. It does not depend on crystal size, temperature and supersaturation. Eq. 12 can be changed to

$f4(y)=h={ (1-y) }^{ 2 }$ Eq. 13

y: crystal content by volume (abs.)

### Massecuite circulation

Opinions regarding the effect of stirring on crystal growth are varied.

“Opinions differ concerning the influence of stirring and agitation upon the crystallization velocity of sucrose. Experiments on this question have led different investigators to conclude that there is either a pronounced effect or relatively none”. This is attributed to the fact that “conditions and circumstances are not uniform and comparable” (VanHook 1959). The effect of flow velocity on crystal growth has already been demonstrated by Eq. 8 and Eq. 9.

Some studies regarding circulation in vacuum pans quite often rely on surprising simplifications:

• stirrers and their mode of operation (single speed, double speed, variable speed) are neglected,
• in most cases only the flow of liquid phase process medium (no crystals) is studied,
• syrup feed and location of feed entry into the pan are neglected.

Good circulation, however, has definite advantages like “thorough and unified distribution of feed, minimum variations of temperature, improved crystal formation, … however the stirrer does not show significant results in crystal size distribution or massecuite exhaustion” (Chen & Chou 1993). It is also well known that high supersaturation and bad massecuite circulation are considered responsible for conglomerate formation.

## Modeling crystal growth in industrial sugar crystallizers

### The simple model of crystal growth

The most often used model describing crystal growth is:

$\frac { da }{ dt } =K*(SS-1)$ Eq. 14

It is evident from the above discussions that this model is a very simplified one. It neglects the effect of quite a few important parameters, or implicitly assumes that they are included in K, the overall crystallization growth rate, which is certainly not constant.

### Complex models

No.1.

An often quoted complex model of crystal growth rate is given by the expression (Wright and White 1968):

$\frac { da }{ dt } =G=P1*(OS-P2)*EXP(P3*(\frac { 1 }{ w } ))*EXP(-FT)$ Eq. 15

where:

P1, P2, P3:                        constants

OS:             SS-1

FT:             $(\frac { Eact }{ R } )*(\frac { 1 }{ 273.16+T } -\frac { 1 }{ 333.16 } )$

Eact:          15.0-0.2*(T-60)

R:               1.9872*E-0.3

This model relies on supersaturation (SS), non-sugar concentration (1/w) and temperature (T) data. All other missing parameters should be taken into account by the proper determination of constants P1, P2 and P3.

No. 2.

Another complex model based on mass, energy and population balances with constant K determined on the basis of data from eight strikes in an industrial crystallizer (Georgieva et al. 2003):

$\frac { da }{ dt } =Ka*EXP[-\frac { 57000 }{ R*(T+273) } ]*(SS-1)*EXP[-Kb*(1-P)]*(1+2\frac { v }{ V } )(\frac { m }{ sec } )$  Eq. 16

Ka                       265

Kb                       13.863

R                         8.314 (gas constant)

T                          temperature (C)

P                         mother liquor purity (abs)

v                          volume of crystals (m3)

V                         massecuite volume (m3)

v/V                   crystal content by volume (abs)

According to Azevedo et al (1994) the crystal growth rate:

$*(SS-1)*EXP[-Kb*(1-P)]*(1+2\frac { v }{ V } )(\frac { m }{ sec } )]$ Eq. 17

Ka:                      100

Kb:                      8

Notes:

• Eq. 16 and Eq. 17 have the same structure, but their two constants (Ka and Kb) and the results are different.
• This model contains all of the most important parameters; however, its last term $(1+2*\frac { v }{ V } )$ is in direct contradiction with Eq. 13. Increasing crystal content definitely will not increase the rate of crystal growth.

## Formulation and validation of a crystal growth model

There are not many attempts to be found in the relevant literature resulting in the formulation and validation of a model describing crystal growth: it is not easy even in a laboratory environment. One basic requirement, online monitoring of supersaturation, the most important parameter of crystallization in industrial crystallizers became available only with the introduction and use of the SeedMaster Instruments.

By making use of the model components discussed above, a complex model will be formulated and validated using all of the required data, including supersaturation acquired online in industrial batch vacuum pans in operation in two different countries.

### Model formulation

The crystal growth model is based on the individual components discussed in Point 2. and has the following form:

$\frac { da }{ dt } =K*f1(SS)8f2(T)*f3(NS)*f4(y)$

or

$\frac { da }{ dt } =K*(SS-b)*0.277*EXP[0.0186*T]*EXP[-1.75*\frac { NS }{ 100 } ]{ (1-y) }^{ A }$

Eq. 18

da/dt:                 linear crystal growth rate (μm/min)

K:                  overall crystallization growth rate

SS:               supersaturation (–)

b:                  constant

T:                  temperature (⁰C)

NS:               non-sugar concentration (g/100 g water)

y:                  crystal content (by volume, abs.)

A:                  constant

Crystal content by volume (abs.) depends on the number N and size “a” of crystals in a unit volume of massecuite. The volume of a single crystal depends on its size and shape. There are different shape and volume form factors in the literature for sugar crystals regarded as “normal”. In an experiment data acquired by measuring the length along the 3 axes of a large number of crystals in a beet sugar factory L.b : L.c : L.a = 1,33 : 1,0 : 0,73 was found (Pot 1983). The volume of a similar crystal with L.c = 1 mm is V1 = 0,971 mmᶾ, while the volume of one of cubic shape with a = 1 mm is V2 = 1 mmᶾ. For sake of simplicity cubic shape with crystal size “a” will be used here.

### Model validation

#### Methodology

It is natural that the model of crystal growth as described by Eq.18 must be validated by using industrial-size sugar crystallizer(s) where mass production of sugar takes place. In our case this meant monitoring crystallization online in a large number of batch pans already in use in different countries.

Our method of validation is based on the active online monitoring of (among others) the most important parameter of crystallization: supersaturation. This was made possible due to the development and active use of the novel SeedMaster Instruments.

#### Instruments

It is clear from Eq.18 that supersaturation (SS) plays a very important role. There is no single instrument to measure it: it can only be calculated by taking into account all its governing parameters (see Eq.1). The SeedMaster 2 (SM-2) and the new SeedMaster 3 (SM-3,) can provide all of the required data online every 10 second intervals.

Inputs:

1. syrup / mother liquor concentration (provided by a K-Patents process refractometer),
2. temperature (provided by a K-Patents process refractometer),
3. massecuite solids content (microwave transmitter), OR
4. massecuite density (transmitter), plus
5. massecuite level (transmitter).

Based on the above process data it is possible to calculate online the parameters which are really important in crystallization control:

Outputs:

• supersaturation SS
• crystal content y (by volume, % and tons)
• mother liquor purity P (%)
• mean crystal size (mm), (if no nucleation or dissolution of crystals takes place during the strike)
• calculated massecuite solids content (%), or density (kg/m3), depending on the type of the selected instrument.

#### Method of model validation

The basic process of model validation consisted of the following steps:

1. Selection of strikes characterized by:
• full seeding with slurry
• no nucleation all over the strike
• no dissolution of crystals all over the strike
• crystal size (MA) close to target (Case No.1: 0,65 mm; Case No.2: 0,25 mm)

Monitoring of the strikes from seeding till its end (time of crystallization).

1. Storing and trending data. All of the input and calculated data can be trended on the LCD of the instruments. The same data can be downloaded and trended in Excel sheets as well. These are the “reference” or “SM-2” / “SM-3” data.
2. Calculation of the crystal growth rate da/dt according to Eq. 18 and its integral “a(n)” (crystal size) in Excel by using the SM-2 / SM-3 input data stored at 10 second intervals.

$a(n)=a.s+\sum _{ 1 }^{ B }{ (\frac { da(n) }{ dt } } )(mm)$ Eq. 19

a.s:               seed crystal size (mm)

B:                 number of 10 sec intervals from seeding till the end of the strike

Time of crystallization:                             T.cryst =  B/6 minutes

With final crystal size (MA) and stored data on level, the number N of crystals in a unit volume (1 cm3) of massecuite can be calculated for every 10 second interval. Finally: using data on N and crystal size “a(n)”, crystal content, solids content and purity of the mother liquor can be calculated, too. The calculated data are based on the simulation (modeling) of crystal growth represented by Eq. 18. and will be referred to as “SIMUL”.

1. Trending the data of the same selected reference (“SM-2” / “SM-3”) and model (“SIMUL”) parameter in a common Excel figure for comparison.
2. Eventual differences between the reference and calculated parameters are used to fine-tune the model equation (adjustable parameters K, b and A) accordingly.

## Case studies

### A well-known refinery in Western Europe

Instruments used: K-Patents PR-23-GP refractometer and SM-2

Strike parameters.

• Feed syrup purity : 98 %
• Supersaturation when seeding : 1.09
• Time of crystallization (from seeding till the end) : 141 min
• Massecuite density (at the end of strike) : 1499 kg / mᶾ
• Product crystal size (MA) : 0.65 mm
• Massecuite volume ratio (V.max/V.seed): 2.51
• Number of product crystals in 1 cm of massecuite : 1815
• max : 1.145
• min : 1.061

Measured data (data inputs) trended on the instrument display (figures 2-4)

Figure 2. Mother liquor concentration

Figure 3. Temperature

Figure 4. Massecuite density

Notes:

1. Massecuite level was controlled as a function of massecuite solids content.
2. All trends were started when seeding was carried out and lasted till the end of the strike.

#### Comparing data monitored by the SM-2 with those calculated by making use of the model equation Eq.18

Data are compared by showing the trends of the selected parameter. The trend window with white characters on black background shows the original trend displayed on the SM-2 display. The one beside it shows the same parameter displayed for comparison in an Excel diagram

• with solid line marked: SM-2 (exactly the same as the one on the instrument display) and
• with dotted line marked: SIMUL: calculated after fine-tuning the K, b, A parameters of Eq. 18.
• The horizontal axis is time expressed as a number (B). Actual Time = B/6 minutes

Supersaturation (figure 5):

Figure 5. Monitored and simulated supersaturation data

Note: There is no noticeable difference between SM-2 and SIMUL data.

Massecuite solids content (figure 6):

Figure 6. Monitored and simulated massecuite solids content

Crystal content (7):

Figure 7. Crystal content

Mother liquor purity (figure 8):

Figure 8. Mother liquor purity

Crystal size (figure 9):

Figure 9. Crystal size

Note: Considerable difference of crystal size data in the first 40 % or so of crystallization time. One of the possible causes: density probe calibration problem. The difference is fast reduced later on.

Massecuite density (figure 10):

Figure 10. Massecuite density

#### The validated crystal growth equation

Based on the process of validation it was found that the best fit was reached with parameters K, b and A as follows:

K:        0.0182

b:         1.035

A:        2

As already discussed, the overall crystallization growth rate K is a multi-parameter function of several variables (see Eq. 3).

Using these data, the (linear) crystal growth rate was established:                                                                                                                                               $\frac { da }{ dt } =0.0182*(SS-1.035)*0.277*EXP(0.0186*T)*EXP(-1.75*\frac { NS }{ 100 } )*{ (1-y) }^{ 2 }(\frac { \mu m }{ 10\quad sec) }$  Eq. 20

Conversion of the equation in order to get the rate of crystal growth in the more accustomed engineering unit  μm/min can be done by multiplying K by 6. However, by doing so eventual changes in the values of the governing parameters during one minute will be neglected.

$\frac { da }{ dt } =0.1092*(SS-1.035)*0.277*EXP(0.0186*T)*EXP(-1.75*\frac { NS }{ 100 } )*{ (1-y) }^{ 2 }(\frac { \mu m }{ min } )$  Eq. 21

where for completeness:

SS:                      supersaturation (-)

T:                         temperature (C)

NS:                     non-sugar concentration (g/100 g water)

y:                         crystal content by volume (abs)

NOTE: When using online data of supersaturation for crystallization control, it is highly advisable to refresh all measured and calculated data at least every 10 seconds (it is a built-in feature of the SeedMaster Instruments). Doing so only at 1 minute intervals will result in difficulties with control.

#### Relative strength of the individual components (Eq.20)

Component: K

K is a function of quite many parameters of the process and the equipment used (Eq. 2). The value of K determined validating the growth rate equation:

$K=0.0182(sampling\quad rate:\frac { 1 }{ 10\quad sec } )$

$K=0.1092(sampling\quad rate:\frac { 1 }{ min } )$

Component: Supersaturation (SS) (Figure 11.)

f1(SS) =(SS-1.035)         Eq. 22

Figure 11. The most important component: supersaturation

Eq.22 shows similarity to Eq.4 and it means that there is no crystal growth if SS =< 1.035. The effect of temperature T (see Figure 5.) on supersaturation is demonstrated.

Component: Temperature (T), (Figure12.)

f2(T) = 0.277*EXP(0.0186*T)      Eq. 10

It was proved that Eq. 10 represents correctly the direct effect of temperature on crystal growth. However, while increasing the rate of crystal growth (Figure12) it can only slightly compensate the large drop in supersaturation when temperature is increased (see Figure13. and Figure15.). Temperature has a complex effect on the overall rate of crystallization (K) as well.

Figure 12. The effect of temperature on crystal growth

Figure 13. The combined effect of supersaturation and temperature

Component: Non-sugar concentration (NS), (Figure13.)

$f3(NS)=EXP(-1.75*\frac { NS }{ 100 } )$ Eq. 11

Component: Crystal content by volume (y (abs.)) (Figure14.)

f4(y) =(1-y)2    Eq. 13

Figure 14. The effect of non-sugar concentration on crystal growth

Figure 15. The effect of crystal content on crystal growth

The effect of non-sugar concentration increases fast with low purity feed syrup, where the purity drop can be quite considerable. It was proved that Eq.13 represents correctly the very strong effect of crystal content on the rate of crystal growth.

The combined effect of supersaturation, temperature, non-sugar concentration and crystal content on the rate of crystal growth is shown in Figure 16.

Figure 16. The combined effect of all of the important parameters on crystal growth

Finally: when the data (Figure16., vertical axis) are multiplied by 6000*K we get the rate of crystal growth in μm/min. The result is shown in Figure 17.

Figure 17. Rate of crystal growth (Case study No.1)

### Refinery in West Africa

Instruments used: K-Patents PR-23-GP refractometer and SM-3

Strike parameters

• Feed syrup purity : 98 %
• Supersaturation when seeding : 1.12
• Time of crystallization (from seeding till the end) : 99 min
• Massecuite density (at the end of strike) : 1495 kg / m
• Product crystal size (MA) : 0.28 mm
• Massecuite volume ratio (V.max/V.seed): 2.50
• Number of product (footing magma) crystals in 1 cm of massecuite : 22400
• max : 1.14
• min : 1.09
• Full seeding with slurry

In this case only the major results of the model validation will be presented.

Figure 18. Supersaturation

Figure 19. Crystal content (% by vol.)

Supersaturation (Figure 18.): supersaturation was almost constant at about SS=1.14 for a while after seeding, but due to increasing shortage of steam supply, supersaturation began to drop considerably. No noticeable difference between the instrument and simulated data.

Crystal content (% by vol.) (Figure 19.): having fine-tuned the overall crystallization growth rate K, simulated crystal content tracks the one calculated by the SM-3 device.

Based on the process of validation in Case No. 2 it was found that the best fit was reached with parameters K, b and A as follows (see table 1):

K:        0.0090

b:         1.035

A:        2

Table 1. Comparing parameters of Eq. 18:

 K b A Case No. 1 0.0182 1.035 2.0 Case No. 2 0.0090 1.035 2.0

## Summary

As can be seen in Table 1, there is a considerable difference in the values of K (ratio: 2:1), while the other 2 parameters (b and A) are the same. The large difference in the overall crystallization growth rates (Eq. 3) could be expected: it lumps together the effect of all of the parameters which we are unable to measure or calculate. K is certainly not constant during the strike but depends on quite many changing parameters.

There are two considerable differences between the two cases:

1. final (product) crystal size (0,65 and 0,282 mm, respectively), and
2. the fact that there was not enough steam during most of the strike in Case No.2.

Remembering the effect of liquid velocity v and the width of the stagnant layer δ at the crystal surface it is logical to recall that the larger settling velocity of the larger size crystals (Case study No. 1) has a positive (increasing) effect on K. Also, lack of steam supply (Case study No. 2) to keep the rate of evaporation at the required (by the control system) level results in poor massecuite circulation and in a degrading effect on liquid velocity (v) and width of the stagnant layer (δ) on the crystal surface. These are only two of the many possible sources of the difference in the values of K in the two cases.

The crystal growth rate model was validated in batch vacuum pans in two different locations using 98 % purity feed syrups in both cases. What if the feed syrup purity is quite lower? Is the growth rate model still valid?

1. Supersaturation is being calculated by the SeedMaster Instruments on-line, taking into account all of its important parameters, including feed syrup / mother liquor purity, quality and purity drop during crystallization (see Eq. 2). Being the most important parameter of crystallization, crystal growth mainly depends on the correctly calculated supersaturation (see McCabe’ law, McCabe 2004).
2. Changing (during the strike) non-sugar concentration (NS) is involved in the growth rate model (see Eq. 11).
3. The overall crystallization growth rate K (see Eq. 3) is being used to reflect the effect of other non-measurable parameters, like viscosity (η), width of the stagnant layer (δ), flow velocity (v) etc. It depends on local parameters and should be determined locally.

## Conclusion

Crystal growth is a very complex process that depends on a large number of massecuite parameters, crystallizer designs and control practice. The process is highly nonlinear.

In this paper a multiparameter growth rate model and its validation was presented. The major findings are summarized below.

Supersaturation: its importance is widely acknowledged. Unwanted nucleation should be avoided all over the process of crystallization. The zone of control is very narrow: there is no crystal growth below SS <= 1,035 (see Eq. 20) and unwanted nucleation begins if SS > = 1,15 … 1,20, depending on mother liquor purity.

Fluctuations, like in the Case study No.1 (SS.max = 1,145, SS.min = 1,061) result in almost 3:1 change in crystal growth rate (Figure 13). This certainly has a very strong effect on the time of crystallization (time from seeding to the end of the strike). Supersaturation should be kept constant and close, but safely below the nucleation limit all over the strike. This needs:

• good, undisturbed steam supply,
• well controlled (preferably: individual) vacuum system,
• reliable online data on supersaturation and
• advanced crystallization control based on supersaturation (Rozsa et al. 2011, 2013).

Temperature:  temperature has multiple effects on crystal growth: via supersaturation, directly and via the parameters affecting the overall crystallization growth rate (K).

Syrup / mother liquor purity: the effect of non-sugar concentration on crystal growth is considerable (Figure 16) even with high purity syrups like in the two cases discussed here.

Crystal content:  crystal content has a very strong effect on crystal growth (Figure 17). Well designed level control is needed to find an optimal solution.

Overall crystallization growth rate (K): it is a multivariable parameter depending – among others – on equipment designs and control practice as well. It is a parameter to be determined locally.

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