Azucar- Vacum Panes

Department of Chemical EngineeringSt. Lucia, Queensland.

Individual Inquiry 2000

By

Terry V Tahal

Supervisor: Emeritus Prof. E T WhiteIndustrial Supervisor: Dr R. Broadfoot, SRI MackaySubmission date: 27th October 2000

SteamFlow

Feed

Water BatchDischarge

Condensate

EvaporationCE1

CC1

Vacuum

CE1

New Models For Sugar Vacuum Pans i

Individual Inquiry-2000 Terry Vickram Tahal

ABSTRACT

One of the key steps in the sugar processing is the crystallisation of sugar. This is the

step that produces the final product and as such must be monitored very closely as the

quality of the final product determines profit. Crystallisation occurs by boiling a sugar

solution until it becomes oversaturated and the crystals grow. Care must be taken not to

exceed a critical oversaturation (OScrit) to prevent nucleation of new crystals, as these

greatly affect the quality of the final product. The process can either be batch or

continuous with the batch process being more common in the industry at present.

The crystallisation process is complex and hence a model is required to understand the

effect of operating conditions on parameters such as growth rate and crystal size. Models

have been previously developed, however they included a term for growth rate dispersion

(GRD) that is now known not to be correct. As a result they over-estimated the spread of

sizes (CV) in the product.

In the new interpretation of GRD crystals should be characterised not only in terms of

their size, but also by their relative growth rates. This interpretation does not allow the

simple moments relation modelling to be used. In this report the new modelling

methodology, which allows GRD to be included will be outlined and implemented in

Matlab codes and the results of the new model discussed

Within the simulation, the pan operation is controlled through the feedback control. A

PID algorithm was implemented. The measured variables were fractional oversaturation

(OSfrac) and the crystal content (cc). The fractional oversaturation measures how close

the oversaturation is to the critical value. Faster pan operation results if this value is

maximised but the oversaturation must never exceed the critical oversaturation, as new

crystals will be nucleated. Crystal content affects the pan circulation and a loss of

circulation is catastrophic. Crystal content is used to terminate a strike when it reaches

the desired value.

A working model for the batch is provided and discussed. The dynamic continuous pan

model is currently being worked on and the general mass balance equations for the

continuous model are included in this report however the model is not fully developed to

date.

New Models For Sugar Vacuum Pans ii


ACKNOWLEDGMENT

The completion of this project would not have been possible without the help and

guidance from people whom I have associated with. I would therefore like to thank all

the people that have helped and assisted me during this project.

Firstly I would to thank Emeritus Professor E .T White. I have enjoyed working with you

during this project and I would like to thank you for patience and time to explain to the

details behind this project.

I would also like to thank my family and friends for their support. Also I would like to

thank my classmates for the friendly atmosphere that they created throughout the course

of my degree.

I would also like to say thank to the lord for keeping me healthy throughout the project.

New Models For Sugar Vacuum Pans iii


TABLE OF CONTENTS

LIST OF FIGURES............................................................................................................................... V

LIST OF TABLES................................................................................................................................. V

1.0 INTRODUCTION............................................................................................................................. 1

1.1 SUGAR CRYSTALLISATION ............................................................................................................... 1

1.2 OBJECTIVES .................................................................................................................................... 1

1.3 RELEVANCE OF TOPIC ...................................................................................................................... 1

1.4 LITERATURE OVERVIEW: ................................................................................................................. 1

1.4.1. Doucet and Giddey (1966)...................................................................................................... 2

1.4.2 Evans et al. (1970) .................................................................................................................. 2

1.4.3 Wright (1971).......................................................................................................................... 2

1.4.4 Broadfoot (1980)..................................................................................................................... 3

1.4.5 Wilson (1990).......................................................................................................................... 3

1.4.6 Schneider (1996)..................................................................................................................... 3

1.4.7 Summary................................................................................................................................. 4

2.0 DYNAMIC MODEL OF A CRYSTALLISER................................................................................. 5

2.1 ASSUMPTIONS ................................................................................................................................. 5

2.2 MATERIAL BALANCE ....................................................................................................................... 5

2.3 ENERGY BALANCE .......................................................................................................................... 6

2.4 KINETIC EXPRESSIONS .................................................................................................................... 6

2.5 CRYSTAL POPULATION BALANCES AND MOMENTS RELATIONSHIPS .................................................... 6

3.0 DERIVATION OF MODEL RELATIONS FOR A SUGAR VACUUM PAN................................ 7

3.1 MATERIAL BALANCES: .................................................................................................................... 7

3.1.1 Mass balance of Water in molasses ......................................................................................... 7

3.1.2 Mass balance of Impurity in molasses...................................................................................... 7

3.1.3 Mass balance of Sucrose in molasses....................................................................................... 7

3.1.4 Mass balance of Crystals......................................................................................................... 7

3.1.5 Mass balance of massecuite..................................................................................................... 8

3.2 PHASE EQUILIBRIUM........................................................................................................................ 8

3.3 CRYSTAL GROWTH RATE RELATIONS: ........................................................................................... 11

3.4 MOMENTS RELATIONS FOR CRYSTALLISATION WITH GRD............................................................... 13

3.4.1 Number size distribution........................................................................................................ 13

3.4.2 A Batch crystalliser with GRD............................................................................................... 14

3.5 ENERGY BALANCES ....................................................................................................................... 15

3.5.1 Heat transfer relationships .................................................................................................... 16

3.6 DENSITY RELATIONSHIPS .............................................................................................................. 16

New Models For Sugar Vacuum Pans iv


4.0 STRUCTURE OF THE MODEL ................................................................................................... 17

4.1 INPUT DATA .................................................................................................................................. 17

4.2 OUTPUT DATA ............................................................................................................................... 18

4.3 THE DRIVER FILE .......................................................................................................................... 19

4.4 THE FUNCTION FILE ...................................................................................................................... 19

5.0 CONTROL OF THE PAN. ............................................................................................................. 20

5.1 FEEDBACK CONTROL ..................................................................................................................... 21

5.1.1 The three basic feedback control mode actions are; ............................................................... 22

5.1.1 1 Proportional control .......................................................................................................................22

5.1.1.2 Integral control ..............................................................................................................................22

5.1.1.3 Derivative control ..........................................................................................................................23

5.2 TUNING THE CONTROLLER ............................................................................................................. 23

6.0 RESULTS AND DISCUSSION....................................................................................................... 24

6.1 MODEL VERIFICATION................................................................................................................... 24

6.2 TYPICAL RESULTS ......................................................................................................................... 25

6.3 USE OF MODEL .............................................................................................................................. 26

6.4 SUMMARY..................................................................................................................................... 30

7.0 RECOMMENDATIONS AND CONCLUSIONS .......................................................................... 31

8.0 NOMENCLATURE (INCLUDES VARIABLES USED IN MATLAB CODE) ............................ 32

9.0 REFERENCES................................................................................................................................ 35

10.0 GLOSSARY OF TERMS.............................................................................................................. 37

APPENDIX A – MORE RESULTS..................................................................................................... A1

APPENDIX B – MATLAB CODES .................................................................................................... B1

(1)DRIVER FILE:..................................................................................................................................B1

(2) FUNCTION FILE: .............................................................................................................................B9

(3)INITIAL CODE FOR CONTINUOUS MODEL .........................................................................................B12

APPENDIX C – TUNING THE MODEL CONTROL ....................................................................... C1

New Models For Sugar Vacuum Pans v


LIST OF FIGURES

FIGURE 2.1 – DIAGRAM OF THE INPUTS AND OUTPUTS FOR A SUGAR CRYSTALLISER ....................5FIGURE 3.1 – THE SECONDARY NUCLEATION LIMIT AS IT VARIES WITH PURITY [WHITE, 2000] ..11FIGURE 3.2 – DEMONSTRATING OF GROWTH RATE DISPERSION ..................................................13FIGURE 4.1- STRUCTURE OF MODEL USED.................................................................................17FIGURE 5.1 - PID CONTROLLER ON A BATCH PAN ......................................................................21FIGURE 5.2 –MAIN FEATURES OF A FEEDBACK CONTROL ...........................................................21FIGURE 6.1 – PURITY, BRIX AND CRYSTAL CONTENT FOR SCHNEIDER AND FOR MODEL

DEVELOPED IN THIS STUDY................................................................................................24FIGURE 6.2 – OVERSATURATION AND CRITICAL OVERSATURATION ..........................................25FIGURE 6.3 – MASS OF IMPURITIES, WATER, SUCROSE, CRYSTAL AND TOTAL MASS IN PAN. .......26FIGURE 6.4 – CHARACTERISTIC CRYSTAL SIZE VS. TIME ............................................................27FIGURE 6.5 – CHARACTERISTIC CRYSTAL SIZE FOR A, B AND C STRIKES ...................................28FIGURE 6.6 – MOLASSES PURITY AND CRYSTAL CONTENT FOR A, B AND C STRIKES..................29FIGURE 6.7 – BRIX FOR A, B AND C STRIKES.............................................................................29FIGURE A1- MASS OF SUCROSE, WATER, IMPURITIES, CRYSTALS AND TOTAL PAN MASS ......... A1FIGURE A2 – SYRUP FEED RATE AND EVAPORATION RATE ........................................................ A1FIGURE A3 – GROWTH RATE AGAINST TIME ............................................................................. A2FIGURE A4 – BRIX, PURITY AND CRYSTAL CONTENT ................................................................ A2FIGURE A5 – OVERSATURATION FRACTION VS. TIME................................................................ A2FIGURE A6 – COEFFICIENT OF VARIATION AND SKEWNESS ....................................................... A3FIGURE A7 – THE RATIOS OF SUCROSE AND IMPURITIES TO WATER........................................... A3FIGURE A8 – PLOT OF CHARACTERISTIC SIZE VS. TIME ............................................................. A3

LIST OF TABLES

TABLE 6.1- CONDITIONS FOR A TYPICAL A STRIKE PAN..............................................................24TABLE 6.2 – CHANGES FOR THE TESTS 1-4.................................................................................26TABLE 6.3 – TYPICAL PAN CONDITIONS FOR B AND C STRIKES ..................................................28TABLE A1 - OPERATING CONDITIONS FOR A STRIKE ................................................................. A1TABLE C1 – INITIAL CONDITIONS FOR STRIKES ........................................................................ C1

New Models For Sugar Vacuum Pans 1


1.0 INTRODUCTION

1.1 Sugar Crystallisation

Sugar is crystallised under vacuum in a sugar pan, because sucrose decomposes

enormously at high temperatures. By operating under vacuum the boiling temperature is

reduced. This process can either operate as a batch or continuous system. Now pan

operation is complex and hence for us to freely understand it, a model is required.

There have been models but these are out of date and do not include the latest growth

dispersion theory, Growth Rate Dispersion (GRD). Equations will be developed in this

report that will apply to both batch and continuous operations. The simple moments

method used in the previous models does not apply with the new theory and hence a new

methodology which describes GRD will be discussed.

1.2 Objectives

The aim of the project is to develop dynamic batch and continuous models that account

for GRD. The models will be controlled using feedback control technique and tests will

be done to verify and validate the models.

1.3 Relevance of topic

The main purpose of the thesis is to develop a dynamic model that will assist the SRI

(Sugar Research Institutes, Mackay) with its research on vacuum pans. The model will

show the effect of the changes to certain parameters such crystal size and feed

composition on pan behaviour in particular the crystal size distribution.

From a personal point of view I have a particular interest in this area of work as I have

worked, and will be working in the sugar industry in Guyana after I have completed my

studies. By developing reliable controls for a continuous pan it further encourages more

mills to implement them.

1.4 Literature overview:

Considerable earlier work has been done on the modelling and control of sugar

crystallisers, in particular by Wright (1971), Broadfoot (1982), Wilson (1990) and

Schneider (1996). They all developed models effectively to monitor the oversaturation



and the crystal content of the batch and continuous crystallisers. In practice the crystal

content and oversaturation as yet cannot be measured directly. There are several ways of

inferring the crystal content and the oversaturation with the most predominant method for

oversaturation in the Australian sugar industry being conductivity measurements.

1.4.1. Doucet and Giddey (1966)

They were the first to publish a model for a batch sugar pan. The model they developed

was a dynamic model for high and medium purity systems. The model used integral

equations that were solved by analogue computation. The concept in their model is that

the water evaporated from the pan must be balanced by two effects

(1) the sucrose removed from the pan molasses to maintain the oversaturation at

high enough value for crystallisation.

(2) the water added from the feed required to replace the sucrose loss due to

deposition

They considered impure solution, but did not model the crystal size distribution nor

considered controlling the crystal content.

1.4.2 Evans et al. (1970)

They suggested a model for refinery pan operation. Their model included mass and

energy balances and also intensive variables such as solution concentrations and crystal

mass fraction. This work was done to analyse the sensitivity of sugar production to

process variables and also to fit industrial data to the model.

1.4.3 Wright (1971)

Prior to the release of the model by Wright (1971), Wright and White (1968) published a

paper on a mathematical simulation of the fed-batch impure sucrose crystallisation

system. The model included the following;

Ø Mass and energy balances

Ø Equilibrium phase relationships

Ø Crystal and nucleation rate expressions

Ø Dynamic crystal inventory, as done by Ciolan (1966)



It was made clear that most of the equations except the mass and energy balance may not

be exact expressions.

The work done by Wright (1971) resulted in the development of the model for the batch

pan that included the population balance dynamics as moment relations, crystal size

distribution and the growth rate parameters for an industrial batch pan. He also did some

industrial testing of the batch model.

1.4.4 Broadfoot (1980)

Broadfoot carried out designs on the continuous pan and then optimised the continuous

pan for various operating conditions. His designs were based on a steady state model and

contained moments relations.

1.4.5 Wilson (1990)

Wilson dealt with the control of a fed-batch impure sucrose crystallisation in a vacuum

pan. Wilson’s task was the application of a non-linear control algorithm (Generic Model

Control) to the vacuum pan. Wilson showed that given sufficient plant model mismatch,

algorithm performance deteriorates to the point where the GMC was equivalent to the

PID controller or in the worst case became unstable.

Wilson also attempted to implement a state estimator that would deliver the on-line

predictions of the process states and their derived variables, such as the sucrose OS and

the crystal content (CC). The estimator used by Wilson was based on the Extended

Kalman filter (EFK), (Kalman, 1960) presented by Hamilton et al. (1973).

1.4.6 Schneider (1996)

Schneider developed further the work by Wilson and others by improving the controls for

the fed-batch pans. He studied the addition of the state estimator (Schmidt-Kalman Filter

or SEKF). This estimator was capable of accounting for the impact of uncertain process

parameters and/or unobservable model states in such a way that they kept the other model

states open to the process measurements employed for feedback to the model.



1.4.7 Summary

I will continue from the previous work done by Wright (1971), Broadfoot (1980), Wilson

(1990), Schneider (1996) and others. The main change that will be made to the model is

the replacement of the population balance equations to account for growth rate effects on

the crystal size distribution, known as Growth Rate Dispersion (GRD). The model will

be written in Matlab, as it is widely available and powerful. A control system will be

placed on the model to mimic the actual operation of the pan. The model will then be

tested and compared to previous models.

A dynamic batch model will be developed to completion and then work will then begin

on developing a dynamic model for the continuous pan. The continuous model will then

be tested and controlled similarly to the batch model.



2.0 DYNAMIC MODEL OF A CRYSTALLISER

To model a system requires nomination of the system states which are important and

those that are considered less important. The state space is that set of process variables

that are chosen to describe the system so that a meaningful outcome can be accomplished.

The following sections presents the general model (batch or continuous) for a sugar

crystalliser. Figure 2.1 illustrates the inputs and outputs for a crystalliser.

Figure 2.1 – Diagram of the inputs and outputs for a sugar crystalliser

2.1 Assumptions

Ø Isothermal ie. it operates under a constant vacuum

Ø seeded (with common history seed)

Ø no nucleation, breakage or agglomeration ie. no Birth or Death

Ø well mixed

2.2 Material balance

The important balances for the model are listed below. They are the ones used in

industry.

Ø Water balance - W

Ø Sucrose balance - S

Ø Impurity balance - I

Sucrose SWater WImpurities ICrystal mass XTotal Mass M

Feed Syrup, Fsyrup

Water, Fwater

Steam, St

Evaporation rate, E

Product, PMassecuite, F Mass



Ø Crystal mass balance - X

Ø Total material balance - M

Note: The impurities contain all materials that are not water, sucrose or crystals. The

total material balance is a sum of the water, sucrose, impurities and crystals in the pan

and so it is not separate balance. In addition to the above balances there are other

important characteristics that are useful in operating a pan efficiently. All are related

(except CV) to the above variables. These include

Ø The purity of the molasses during the strike of the pan - Pmol = IS

S+

Ø The dry substance of the molasses or brix - Bmol = WS

I++

+

Ø The oversaturation of the pan molasses - OSmol = 1)/(

/*

−WS

WS

Ø Crystal content of the pan - CCpan = MX

Ø Coefficient of variation during the strike - CV

2.3 Energy Balance

The energy balance was not included in this project, as the main area of concern was the

material balances and the moments. The energy balance is used only to evaluate the

steam rates and can be calculated easily after the simulation is completed. The

temperature of the pan is usually constant so the major heat requirements will be the

evaporation rates and raising the temperature of the feed to the pan. There are also heat

losses from the walls of the pan.

2.4 Kinetic Expressions – The kinetic relationships needed in this study are,

Ø Nucleation kinetics (mainly as the nucleation threshold)

Ø Growth rate (including growth rate dispersion)

2.5 Crystal population balances and moments relationships

This follows the size of the crystals in the strike and thus the size distribution. The

distribution shape is expressed as the coefficient of variation (CV) and the skewness (Sk)

of the crystal size distribution throughout the strike. These involve the distribution

moments related to number, area, volume and mass of crystals.

The mass balance of the water describes the change in the water for the pan, the water

comes in via the syrup and movement water (Wmove) and is taken away by evaporation

and the product.

dt

dWt = Ft (1 – BF) + H (1-BH) + Wmove - Et - P (1- BM) (1)

3.1.2 Mass balance of Impurity in molasses

The impurity comes from the syrup being fed to the pan and is taken away in the product.

dt

dI t = Ft BF (1 – PF) + BH (1-PH) – P BM (1-PM) (2)

3.1.3 Mass balance of Sucrose in molasses

The sucrose in the pan changes by the addition of sucrose from the syrup and removal by

the product and the formation of the crystals.

dt

dS t = Ft BF PF – P BM PM + H BH PH - dt

dX t (3)

3.1.4 Mass balance of Crystals

It is assumed that the size is measured as the volume equivalent size. The mass of the

crystals in the pan will be described in terms of third moment, which can also be written

in terms of CV (coefficient of variation) and Sk (skewness).



Mass of crystal in pan, X= N ρc 36µπ

= N ρc )33(6

32

123 LLmLmm +++π

= N ρc 3

6L

π[1+ 3 CV2 +Sk3 CV3]

= N ρc Φ3

6L

π(4)

where Φ = [1+ 3 CV2 +Sk3CV3]

Then

GLNdtdX

c

2

63 Φ=

πρ (5)

3.1.5 Mass balance of massecuite

The total massecuite in the pan is the sum of the water, impurities, sucrose and crystals in

the pan.

dt

dTt = dt

dWt + dt

dI t + dt

dS t + dt

dX t (6)

Note: For batch pans the product stream (P) does not exist

3.2 Phase equilibrium

The phase equilibrium property considered is the oversaturation (OS). Also considered is

the nucleation limit defined by the critical oversaturation (OScrit). The following section

defines and develops the oversaturation and critical oversaturation relationships.

The solubility of sucrose is defined as the concentration of sucrose in a saturated solution,

which is in equilibrium with sucrose in the solid state - Bubnik and Kaldec (1995). The

solubility of a pure sucrose solution is temperature dependent and the following equation

was given by Charles (1960).

100

10004.9

10057.20

100251.7407.64

32

*

−

+

+

=

+

TTT

WSS

(7)



When linearized to 65oC the equation becomes,

)65(00225.07533.0*

−+=

+T

WSS

(8)

This equation can now be converted to the mass ratio of sucrose to water which varies

with temperature is

))65(00225.07533.0(1

)65(00225.07533.0*

−+−−+

=

TT

WS

(9)

Supersaturation is defined as,

SS = etemperatursametheatsolutionsaturatedaofratioWaterSucrose

solutioninratioWaterSucrose/

/

(10)

=( )( )*/

/

WS

WS(11)

If the solution is saturated then the value of SS will be 1, while if it is over-saturated the

value is greater than 1 and if under-saturated the value is less than 1.

In practice sucrose solutions contain impurities and hence SS is redefined as;

SS = *)/(

)/(

i

i

WS

WS =

*)/(

)/(

WSSC

WS i (12)

Where SC (saturation coefficient) is the ratio of the impure sucrose equilibrium

concentration to that of the pure sucrose equilibrium concentration.

SC = *)/(

)/(

WS

WS i (13)

Various authors have given correlations for the saturation coefficient. Some of those are,

Ø Wright (1971) suggested

−=WI

SC ψ1 (14)

where Ψ = 0.088



Ø Wright and White (1968) gave another relationship, which was temperaturedependent;

−

−−=WIT

SC200

601.01 (15)

Ø Broadfoot and Steindl (1980) later presented a correlation for Queensland conditions

( )ashRS

WI

TSC

/28.01.075.5

1+

−= (16)

A more recent correlation presented by Schneider (1996) that was proposed by SRI

(Sugar Research Institute) is;

−−++

=

WI

PPPWI

PSC 7665 exp)1( (17)

where P5 = 0.011 +0.00046T (18)

P6 = 0.67+0.0021T –0.07 RS/ash (19)

P7 = 0.54 +0.0049T (20)

Oversaturation is defined as

OS = SS –1 (21)

An equation for the critical oversaturation is given by White (2000) as,

3

6.311.0

++=

ISI

OScrit (22)

The driving force for crystallisation is oversaturation, which is a measure of the sucrose

content above equilibrium in the molasses. As shown in equation 22 when the impurities

increase, the critical oversaturation increases. One of the major objectives with efficient

pan boiling is to avoid nucleation in the pan so the oversaturation must be controlled so

that it remains below the limits of the critical oversaturation-ie below the secondary

nucleation limit shown in Figure 3.1.



Figure 3.1 – The secondary nucleation limit as it varies with purity [White, 2000]

3.3 Crystal Growth Rate Relations:

The growth of crystals is defined in terms of the normalised first moment as follows;

dt

dG 1µ

= (23)

Several different experimental growth rate expressions have been used in past work.

However more recently, the BCF equation (White, 2000) was found to approximate the

growth rate equally as well as other expressions. Some of the previous approximations

used are now discussed.

Wright (1971) considered the effect of temperature and impurities on oversaturation. The

oversaturation gives a measure of the excess sucrose to water in the pan at any time. The

effect of OS/G is non-linear. As shown in equation 24 the results may be fitted by a

displaced linear relationship for high oversaturation while at low oversaturation (equation

25) the linear relationship may be used. Further equation 26 shows when at negative

oversaturations dissolution occurs.

OScrit > OS > 1.5 P2 G = kT (OS –P2) (24)

1.5P2 > OS > 0 G = kT/3 OS (25)

0 > OS G = 5 kT OS (26)

P2 is the oversaturation to change from the low oversaturation (OS) equation to the higher

oversaturation. This value of kT applies at a given temperature and impurity level.



The effect of temperature is correlated by an Arrhenius relation with variable activationenergy,

=

−

+−

OShmm

ekk TR

E

CT

act

.16.333

1

16.273

1

600 (27)

where

Eact = 62.76 – 0.8368(T – 60)

molgkJ

(28)

T is the temperature( oC)

The C

k 060 term includes the effect of impurities, which is taken as varying exponentially

with the impurities to water ratio.

=

OShmm

ePk W

IP

C .

3

0 160(29)

Wright (1971) estimated the three growth related parameters to be;

P1 = 7.418

⋅OShmm

(30)

P2 = 0.04 [OS] (31)

P3 = -1.75 unitless (32)

Therefore the relation for growth rate as given by Wright (1971) is;

−=

−

+−

hmm

POSeePG TR

E

W

IP act

)( 216.333

1

16.273

1

1

3

for OS> P2 (33)

The following growth rate relation by White (2000) is based on the BCF screw

dissolution theory given for pure solutions,

)/tanh(2 OSOSOSAG bom = (34)

))60(00031.0)60(022.031.3( 2

10 −−−+= TTA (35)

043.0=bOS (36)

Using the impurity correlation given by Wright (1971) the general correlation for the

growth rate is,

)/tanh(2)/75.1( OSOSOSAeG bWI−= (37)



3.4 Moments relations for crystallisation with GRD

To account for the number of crystals in a pan with a particular size the population

balance is used. Using the moments of the population balance is a simple way of

carrying out the simulation. The previous models considered growth rate dispersion but

did not include recent development in the area. White et al. (1998) describes the effect of

Growth Rate Dispersion for sugar crystallisation. With GRD some crystals are fast

growers while others are slow growers. Hence they grow at different rates even though

they may have been of the same size initially. This is illustrated in Figure 3.2.

crystal 1

crystal 2

Intially

Crystallisation

After crystallization

crystal 2

crystal 1Slow grower

Faster grower

Figure 3.2 – Demonstrating of growth rate dispersion

3.4.1 Number size distribution

The moments are always defined in terms of )(Lf N the number size distribution, with

terms defined as;

L Size, taken as volume equivalent diameter)(Lf N Number density function

dLLf N )( Fraction of total number in size range L è L + dL

N Total number of crystals system

Conversion to a volume distribution is given by,

33 /)()( µLfLLf Nv = (38)



The kth moment of the number distribution is defined by

dLLfL Nk

k )(0∫∞

=µ (39)

kµ is the normalised k th moment about origin (L =0). Now km is normalised k th moment

about mean, LL = is given by

dLLfLLm Nk

k )()(0∫∞

−= (40)

Some specified results for the moments are;

10 == omµ ; L=1µ , 01 =m ; iancem var2 =

Inter-conversion relationships between µ and m are;

( ) i

ik

k

iik

k Lm −=∑=

0

µ (41)

eg )1(2 222

2

2

122 CVLLmLLmm +=+=++=µ (42)

).31(33 32332

233 CVSkCVLLLmLmm ++=+++=µ (43)

where

CV is coefficient of variation = 2

2 / Lm - a dimensionless measure of spread

Sk is skewness = 2/323 / mm - a dimensionless measure of distribution shape

Not all size distribution are based on the mean size L , so consider some other

characteristic size, Lc. An example is the volume median size,

Let cLL /=α (44)

Hence in terms of Lc

cLαµ =1 (45)

2222 )1( cLCV+= αµ (46)

33233 )31( cLCVskCV ++= αµ (47)

3.4.2 A Batch crystalliser with GRD

Some of the assumptions for a batch crystalliser are,

Ø Common history seed

Ø No nucleation, agglomeration or breakage



With these assumptions it follows that the shape of size distribution is the same, just

scaled (expanded by a factor) as growth occurs.

Therefore α=cLL / is a constant during growth. Also CV and Sk are constants. The

moment relationships now become;

Zeroth Moment

00 ==dtdN

dt

dNµi.e. no nucleation (48)

First Moment

01 GN

dt

dN=

µ(49)

cGNdt

dNα

µ=1 (50)

cc GN

dt

dNL= (51)

Second Moment

NGLCVdt

dNcc)1(2 222 += α

µ(52)

Third Moment

NGLCVSkCVdt

dNcc

23233 )31(3 ++= αµ(53)

3.5 Energy balances In this model the temperature is assumed to be constant and thus the energy balances do

not affect the model directly if the evaporation is taken as a mass rate. The energy

relationships for the pan is,

losses

QxTCpmTCpmTCpmdt

mTCpdsteamevapevapevapevapwaterwaterwatersyrupsyrupsyrup

massmassmass

+

++−+= )()( λ&&&

(54)



There are others thermal effects, which could be included in the above equations, but

these will not be taken into consideration as they probably negligible. If the above

equation is considered at steady state, then the left hand side of the equation is zero and

the relationship can be rearranged to give the following;

)(

)(

evapevapevap

waterwaterwatersteamsyrupsyrupsyrup

TCp

TCpmQTCpmm

λ+

++=

&&& (55)

3.5.1 Heat transfer relationships

The relationships below give the heat transfer relationships which can be calculated once

the simulation is done.

)( masssteamcalancalsteam TTAUQ −= (56)

)( condensatecondensateoncondensatisteamsteam TCpmQ += λ& (57)

3.6 Density Relationships

The density of massecuite is,

+

+=

ccrystal

crystal

mol

mol

molmass

mm

Xm

ρρ

ρ )((58)

where ρmol is defined by Wilson (1990) to be;

3160

20036.01)8.0(6827.1441

m

kg

T

TDSmol

−−

−−−=ρ (59)

The density of crystal, ρcrystal = 1585 kg/m3



4.0 STRUCTURE OF THE MODEL

Many of the previous models were written in Fortran with only a few in Matlab. The

model developed in this study is written in Matlab as it is widely available and is capable

of handling the modelling and controller. The structure of the model is shown in Figure

4.1 and a listing of the program is given in Appendix B.

Figure 4.1- Structure of model used

4.1 Input data

The model developed applies to A, B and C batch pan strikes. The results of testing the

model will be displayed in section 6. The following data are required inputs for the

computer model. Typical values for an A strike are shown.

Driver file

Output data

Algebraic Equations - Oversaturation , Growth

rate,

Differential Equations - Sucrose, Water,Impurities, Crystal mass, Total mass, First moment, SecondMoment

Properties of footing - PFo, BF

o ' CV, Sk,

cc, rs/ash, Li, alpha

Properties of feed syrup - PF, BF

Operation conditions of pan - T, Pansize,Footf

Initial conditions - footing, total crystal number

Differential solver - ODE15s

Control loops - PID feedback control

Batch Model Structure

Input Data File

Function file

All results are display as graphs(1) mass of sucrose water impurities crystal Pan total(2) feed syrup and evaporation rates(3) oversaturation(4) growth rate(5) crystal content, pan purity, brix(6) Oversaturation vs growth rate(7) Critical oversaturation(8) Charateristic size(9) Coefficient of variation(10) Sucrose to water and Impurities towater

Calculate and globalisevariables

'



T operating temperature of the pan - typical value = 65 oC

RS/ash ratio of reducing sugar to ash of seed, as a fraction - typical value = 0.1

Pfeed purity of syrup, as a fraction - typical value - 0.9

Bfeed Brix of syrup, as a fraction - typical value - 0.65

Sk skewness of the distribution - typical value - 0.001

CV CV of seed crystal - typical value = 0.25

Footf fraction of per volume which is footing typical value = 0.25

Pansize size of pan in kg

Bfoot brix of footing molasses, as a fraction - typical value = 0.79

Li average crystal size, in micrometers - typical value = 350 um

Pfoot purity of footing molasses, as a fraction - typical value = 0.835

cci crystal content of footing - typical value = 0.20

alpha ratio of L/Lc - typical value = 0.7

All sizes used by the model are volume equivalent sizes. To convert from other sizes

such as the size of the crystal obtained from a sieve, Dalziel and White (1999) gave,

vL15.075.0 +=β (60)

Where Lv is the volume equivalent size

β is the sieve size / volume equivalent size

Note: The relationship applies for the range (0.3mm < Lv < 1.4 mm)

4.2 Output data

All the data from the model can be displayed as graphs. Output from the model include

1. Masses of crystals, sucrose, impurities, water and the total

2. The syrup flowrate and evaporation rates

3. Purity and brix of the pan molasses

4. The crystal content of the pan

5. Oversaturation of pan molasses

6. Fraction of oversaturation to critical oversaturation

7. Growth rate of crystals during strike

8. Characteristic crystal size

9. Coefficient of variation and skewness



Examples of some plots for an A strike are shown in Appendix A.

Note: In using the program the user can select which plots are required and only those

will be output.

4.3 The Driver File

The driver file calls the function file containing the differential equations and solves

them. It contains the following;

Ø Initial conditions

Ø the solver (ODE 15s in Matlab)

Ø The control algorithm

Ø The time constants and gains to tune the controller.

4.4 The Function File

The function contains the differential equations and is called by the driver file during the

simulation. It contains the following;

Ø Differential equations for material and population balances

Ø Supersaturation equations

Ø The growth rate equations



5.0 CONTROL OF THE PAN.

The optimal control of a batch pan has been worked on by many but the scheme

suggested by Frew (1973) would be the most relevant for this study. Frew suggested that

the primary objective is to minimise the time required for seed crystals to grow to their

required size without the formation of false grains. The two important crystallisation

parameters that were included in Frew’s proposal were;

Ø Oversaturation

Ø Crystal content

Both are critical in pan boiling. The oversaturation is the driving force for crystal growth.

In practice this cannot be measured directly, however for simulation purposes its value

will be taken as known. Hence the oversaturation was chosen as the measured variable

for pan boiling in this simulation. Crystal content should also be considered as this can

seriously affect the pan circulation. So in the simulation, the oversaturation is controlled

during the pan strike. When the pan contents reaches a particular value, feed to the pan is

stopped and “heavying up” started.

The actual variable that is controlled is OSfrac, the ratio of the oversaturation (OS) to the

critical oversaturation (OScrit). This was thought to be the better control variable as the

critical oversaturation increases during a strike, since it changes with impurities (Figure

2). Therefore to maximize crystallization and avoid slowing the growth of crystals OSfrac

needs to be maximised. Since in practice there will be fluctuations in the control

variable, the setpoint should be below the critical value. OSfrac was taken to be 0.5 by

Schneider (1996). The new model has OSfrac as an input and it can be tested for higher

ratio.

Previous research work by Schneider (1996), Wilson (1990) and others used control

methods such as SEFK (Schmidt Extended Kalman Filter). In this study feedback control

is the most suitable and also the one industry uses. In the Australian industry the

oversaturation is estimated from conductivity and then control actions are made. In the

model a full PID controller is used. The control action can be applied to either the steam

(evaporation rates) or the feed syrup rates but in the simulation the feed rate is the

manipulated variable. The reason for this choice is it fits the previous model by

Schneider (1996). Figure 5.1 shows a batch pan with the control loop.



Figure 5.1 - PID controller on a batch pan

5.1 Feedback control

Feedback control is when the information about the control variable is fed back to adjust

the manipulated variable. In industrial control of a batch pan a conductivity meter detects

the solution concentration and this is sent to the controller where it changes the feed flow

rate. The main theoretical features in the PID feedback control loop are shown in Figure

5.2.

Figure 5.2 –Main features of a feedback control

The transfer function for the process in terms of the Laplace transform variables isdefined as;

)(

)()(

sUsY

sG p = (61)

SteamFlow

Feed

Water BatchDischarge

Condensate

Evaporation

Et

CE1

CC1

Vacuum

CE1

U(s) Y(s)

Gs(s)

ε(s)R(s)Gc(s) Gs(s)

X(s)



For the controller it is defined as,

)(

)()(

ssU

sGc ε= (62)

while the sensor is given by,

Gs(s) = )(

)(

sYsX

(63)

whereØ U(s) refers to the manipulated input variable, the feed rate

Ø Y(s) is the measured output, the conductivity (oversaturation)

Ø R(s) is the setpoint, OSfrac

Ø ε(s) = R(s) –Y(s) is the error

5.1.1 The three basic feedback control mode actions are;

(1) Proportional (P)

(2) Integral(I)

(3) Derivative (D)

5.1.1 1 Proportional control

In feedback control the objective is to reduce the error signal to zero. The error signal is

the difference between the desired and the actual operating point. Proportional control

action is represented by,

)()( tKutu co ε+= (64)

where

Ø u (t) is the controller output

Ø uo is the previous controller output

Ø Kc is the proportional gain

Ø ε(t) is the error

5.1.1.2 Integral control

Integral control action is also referred to as the reset or floating controls. For this type of

action the output depends on the integral of the error signal over time scaled by the



integral time constant or reset time, τ1, which has units of time. Proportional plus integral

control is described as,

++= ∫

1

0

)(1

)()( dtttKutui

co εε (65)

5.1.1.3 Derivative control

Derivative control is used to speed up the response of the system. It works by

anticipating where a process is going and apply corrections early. It anticipates the future

behavior of the error signal by measuring its rate of change. Derivative control is used in

conjunction with the proportional and integral action controllers. The algorithm for the

proportional plus integral and derivative (PID) is,

+++= ∫ dt

tddtttKutu d

ico

)()(

1)()(

1

0

ετετ

ε (66)

where τD is the derivative time constant

For small time intervals the terms ∫1

0)( dttε and

dttd )(ε

can be written in the form as;

s

n

kkTedtt∫ ∑

=

≈1

01

)(ε (67)

s

nn

T

ee

dttd 1)( −−

≈ε

(68)

Rewriting equation 64 for small time intervals gives,

−+++= ∑

=−

n

knn

s

Dk

i

sncon ee

Te

TeKuu

11)(

ττ

(69)

This algorithm is known as the positional formula and this is the one used for the pan

model in this thesis.

5.2 Tuning the controller

When initial conditions are changed the program will need to be re tuned to get the

desired results. To tune the model the time constants and the gain will need to be

changed. The values of the time constants and gain for the tuned simulations that were

reported are given in Appendix C.



6.0 RESULTS AND DISCUSSION

6.1 Model Verification

This section compares the new batch model against published information. The

continuous model is currently being worked on and as such the results are not available.

The completed codes for the batch model are listed in Appendix B. Simulation from the

new batch model used similar conditions to those used by Schneider (1996) as shown in

Table 6.1. This simulation is for a typical A strike.

Table 6.1- conditions for a typical A strike pan

ConditionsInitial Boiling Terminal

Mass (fraction of mass) 0.45 0.98Size, mm 0.55 0.85Crystal content(fraction of mass) 0.365 0.47 (heavy up starts) 0.55Molasses dry substance 0.79 0.81Molasses Purity 0.835 0.73OSfrac 0.5Liquor Dry Substance 0.675Liquor Purity 0.915

Figure 6.1 – Purity, brix and crystal content for Schneider and for model developed inthis study

0 0.5 1 1.5 2 2.5 30.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85Plot of crystal content, pan purity and brix vs time

Time-hr

crys

tal c

onte

nt,p

an p

urity

and

brix

- fr

actio

n

crystal content

molasses purity

molasses dry substance

Improved ModelSchnieder



From Figure 6.1 it can be seen that the model compares well with Schneider’s model.

The difference is probably due to the use of a new growth rate equation. This model

predicts the time for the strike is 2.25 hrs where as Schneider’s model predicted the time

for strike to 2.75 hrs.

6.2 Typical Results

This section illustrates the type of output the model can produce. As mentioned in

section 4 the model is capable of reproducing most of the operation features of a batch

pan. Typical outputs such as oversaturation, critical oversaturation and species masses in

the are shown below. The conditions for the simulation are those given in Table 6.1.

Other rest of the outputs from the model are given in Appendix A.

0 0.5 1 1.5 2 2.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Plot of oversaturation vs time

Time -hr

OVERSATURATION

OversaturationCritical oversaturation

Figure 6.2 – Oversaturation and Critical Oversaturation

Oversaturation and critical oversaturation is a most important parameter in pan boiling

and can be predicted by the model as shown in Figure 6.2. From the graph it can be seen

that the operating oversaturation was kept below the critical oversaturation thus avoiding

nucleation.

The model also predicts changes for the masses of water, crystal, impurities, sucrose and

total pan over a strike (Figure 6.3).



Figure 6.3 – Mass of impurities, water, sucrose, crystal and total mass in pan.

6.3 Use of model

In this section the use of the model will be demonstrated for some changes in operation

conditions. Some of the testing done on the model are,

1. The purity was set at constant value and low OSfrac

2. The purity of the syrup was ramped from a low purity to high purity

3. The OSfrac was increased

4. OSfrac increased even further

5. Simulation of a typical B and C strikes

Table 6.2 – Changes for the tests 1-4.

Parameters Test1 2 3 4

Pfeed (%) 92 85-92 92 92OSfrac 0.5 0.5 0.6 0.7

The result of tests 1-4 for an A strike will be displayed by showing how the changes

affect the characteristic crystal size and pan time. These are shown in Figure 6.4. The

changes for the tests are quantified in Table 6.2.

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

9

10x 10

4 Plot of pan mass vs time

Time -hr

Mas

s, k

g)

Total Mass

Mass of crystals

Mass of water Mass of Impurities

Mass of Sucrose

Total Mass



Figure 6.4 – Characteristic crystal size vs. time

Test 2 took longer to acquire the crystal size of ~840µm than test 1 because test 1 had a

feed of constant high purity where as the feed for test 2 had a low purity that was ramped

to a higher purity (from 88% to 91.5%).

As the OSfrac was increased from 0.5 to 0.7 (tests 3 and 4) the time for the strike

decreased because oversaturation is the driving force and the higher it is the faster the

crystals will grow.

A test was done to simulate B and C strikes. Table 6.3 gives the conditions for the

boiling. The results are displayed compared to the A strike as characteristic crystal size,

molasses purity, crystal content and molasses dry substances (Brix) during a strike.

Typical values for control tuning of the three strikes are given in Appendix C.

0 0.5 1 1.5 2 2.5 3550

600

650

700

750

800

850Plot of Characteristic Size vs time

Time-hr

Characteris

tic

SIze,um

Test 1

Test 2

Test 3

Test 4



Table 6.3 – Typical pan conditions for B and C strikes

B strike conditions C strike conditionsInitial Boiling Terminal Initial Boiling Terminal

Mass (fraction of mass) 0.45 0.98 0.35 0.98Size, mm 0.55 0.85 0.180 0.35Crystal content(fraction ofmass)

0.365 0.47 0.55 0.2 0.55

Footing Molasses drysubstance

0.79 0.81 0.79 0.89

Footing Molasses Purity 0.835 0.73 0.7 0.5OSfrac 0.5 0.5Liquor Dry Substance 0.675 -Liquor Purity 0.915 -

A molasses purity 0.73 -

A molasses dry substance 0.81 -

B molasses purity - 0.65

B molasses dry substance - 0.835

Figure 6.5 – Characteristic crystal size for A, B and C strikes

Figure 6.5 shows that the model works for all strikes and this is confirmed by the desired

crystal sizes being achieved in the usual times of 2 to 2.5 hrs. It should be noted that the

C strikes takes longer than the other strikes since the feed for the C strike is B molasses

which has a lower purity.

0 0.5 1 1.5 2 2.5 3100

200

300

400

500

600

700

800

900Plot of Characterist ic Size vs t ime

Time-hr

Cha

ract

eris

tic S

ize,

um

B str ikeA str ike

C str ike



Figure 6.6 – Molasses purity and crystal content for A, B and C strikes

The main thing to observe from Figure 6.6 is the purity of the various strikes. The purity

of the strike decreased more for the B strike than the A strike and even more for the C

strike. The brix of the strikes (Figure 6.7) follows a similar though increasing trend to

the purity in terms of the changes for the respective strikes.

Figure 6.7 – Brix for A, B and C strikes

0 0.5 1 1.5 2 2.5 30.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Plot of molasses puri ty and crystal vs t ime

Time-hr

mol

asse

s pu

rity

and

crys

tal -

frac

tion

Molasses Pur i ty

crystal content

A str ike

B str ike

C str ike

0 0.5 1 1.5 2 2.50.75

0.8

0.85

0.9

0.95Plot of Brix(Molasses Dry Substance)

Time-hr

Brix

(Mol

asse

s D

ry S

ubst

ance

) -

frac

tion

C str ike

B str ike

A str ike



6.4 Summary

The batch model for common history (CH) seed was fully developed and tested as shown

above. The model was only tested against Schneider’s model and showed similar trends,

but due to the short time frame of the project the model was not tested against industrial

data or for other disturbances such as noises.

The dynamic continuous model was only partially completed. The general equations

(section 3) for the crystalliser is applicable to the continuous models but other

relationships that fully describe the continuous model were not detailed. The partially

completed code for the continuous model to date is provided in Appendix B and is given

as a starting point for further work.



7.0 RECOMMENDATIONS AND CONCLUSIONS

A fully developed model is available for the batch and work is currently being done on

the continuous model. The data from the batch model was only compared against

Schneider’s model but it recommended that the model compared against industrial data

before it can be fully validated. It also recommended other testing such as adding

disturbances be done.

This report also presents a partial model for the continuous pan. The mass balance and

population balance equations for a common history seed scenario are present but other

equations that fully describes the continuous models needs to be obtained. The initial

coding of the continuous model is given in Appendix B as a starting point for further

work.

The achievements of the study are,

(1) A new model for the batch was developed - The previous model was revised and new

relations, which allow the correct interpretation of growth rate dispersion were

implemented. It is expected this will give better results than the previous model and

be able to predict the coefficient of variation for any strike.

(2) The Matlab coding for the working dynamic model of batch vacuum pan has been

completed and tested. - The model works for batch scenarios A B and C strikes.

Thus it is expected it will predict batch behaviour for all similar scenarios. Tests to

show that the model respond to changes made such as to the feed properties.

(3) The model compares well with industrial and previous models – The model gave

results that were similar to those of Schneider (1996). Differences are probably due

to the new growth rate equation.

(4) Feedback control was used and appears to work well. This mode of control is

compatible with industry practice.

(5) And finally it is hoped that the model will be used by SRI to predict pan behaviour.



8.0 NOMENCLATURE (Includes variables used in Matlabcode)

B Birth rate [#/m3 hr]

BFo Brix of molasses in footing

solutiong

solidsg100

BF Dissolved solids in feed syrup

solutiong

solidsg100

BH Dissolved solids concentration of seed material fed to the

pan

solutiong

solidsg100

BUM Boil back brix eg A molasses brix

solutiong

solidsg100

BM Dissolved solids concentration of material in the pan

solutiong

solidsg100

Brix Dry substance in molasses

solutiong

solidsg100

CC Crystal content

Cpvariable Specific heat capacity of the subscripted variable [kJ/kg oC]

CV Coefficient of variation

D Death rate [#/m3 hr]

e Error (derivative from set point)

ecc End crystal content

Et Water Evaporation rate [kg/hr]

Ft Syrup feed rate [kg/hr]

G Linear crystal growth rate [um/hr]

Gc Characteristic growth rate [um/hr]

hbcc Boil back crystal content

hcc Heavying up crystal content

H Massecuite feed rate to pan [kg/hr]

I Impurities in the molasses in pan [kg]

Kc Gain for control



L Mean crystal size [um]

Lc Characteristic crystal size (Volume mean size)[um]

M Total mass of massecuite in the pan [kg]

mk Normalized kth moment about mean, L = L [µmk]

N Total number of crystals in pan #

OS Oversaturation of molasses

OScrit Critical oversaturation

OSset Set point for oversaturation

OSfrac Fractional oversaturation

PF Purity of feed syrup

solutiong

sucroseg100

PM Purity of molasses in pan

solutiong

sucroseg100

PH Purity of molasses fed to material

solutiong

sucroseg100

PFo Purity of molasses in footing

solutiong

sucroseg100

P Product (Massecuite) [kg/hr]

P1 Growth rate expression constant [mm/h OS]

P2 Null oversaturation in growth rate expression

P3 Impurity parameter in growth rate expression

P5,6,7 Solubility parameters

PUM Boil-back purity eg A molasses purity

R Set point

SC Solubility coefficient

SS Supersaturation of sucrose in molasses

Sk Skewness

S Sucrose content in the molasses in pan [kg]

T Temperature [oC]

Ts Sampling time or control interval for control[hr]

TI Integral time for control [hr]

Td Derivative time for control [hr]

t Time [hr]



u (t) The controller output

uo The derived controller output

U Manipulated input variable for control

V Total volume of pan material [m3]

Wt Water feed rate to pan [kg/hr]

W Water in the molasses in pan [kg]

Wmove Movement water [kg/hr]

X Crystal content of the pan [kg]

Y Measured output for control

Greek Variablesµk Normalised kth

moment about origin, L = 0 [µmk]

α Ratio of L /Lc

)(Lf N Number density function

ε Error (derivative from set point)

ρcrystal Density of crystal

ρmol Density of molasses

ρmass Density of massecuite

Subscripted variables

Mass Massecuite

mol Molasses

Pansize Capacity of pan [kg]

rho Density of crystals [kg/m3]

Rhomol Density of molasses [kg/m3]



9.0 REFERENCES

Broadfoot, R., (1980) Modelling and Optimum Design of Continuous Sugar Pans.

Ph.D. Thesis, The University of Queensland, Brisbane 4072, Australia.

Broadfoot R. and Steindl R. J: (1980), Solubility – crystallization characteristics of

Queensland molasses, Proceedings of the International Society of Sugar Cane

Technologists: 2557-2581.

Butler B. K., (1998) Lactose Crystallisation, Ph.D. Thesis, The University of

Queensland, Brisbane 4072, Australia.

Butler B. K, Zhang H., Johns M. R, Mackintosh D. L and White E. T. (1997) The

Influence of Growth Rate Dispersion in Crystallisation, Proc. Chemeca 97, Rotorua

(NZ). Paper PD 4a on CDROM, published by IPENZ, New Zealand.

Bubnik Z. and Kadlec P., (1995) in Mathouthi M. and Reiser P. (Eds) Sucrose:

Properties and Application. Blackie Academic and Professional, Glasgow.

Charles D.F., (1960) The solubility of pure sucrose in water. The International Sugar

Journal. 62, 126.

Chen J. C.P. and Chou C .C (1993) Cane Sugar Handbook- A Manual for Sugar

Manufacturers and Their Chemists, John Wiley and Sons, New York 12th Edition

Ciolan, I., (1966) Distribution des dimension des cristaux dans les appeals de

cristallisation. Genie Chimique, 95(6): 1381-1387.

Dalziel S. M., Yan S. Y., White E.T., and Broadfoot R., (1999) An image analysis

system for sugar crystal sizing, Proc. Aust. Soc. Sugar Cane Technol., 21: 366-372

Doucet, J and Giddey, C.: (1966) Automatic Control of Sucrose Crystallization from

High – Medium-purity Syrups. International Sugar Journal. 68: 131 – 136.

Evans, L. B., Trearches , G. P and Jones, C.: (1970) Simulation Study of Vacuum Pan

Sugar Crystallizer, Sugar y Azucar. October: 19-37.



Evans, L. B., Trearches, G. P and Jones, C.: (1970) Simulation Study of Vacuum Pan

Sugar Crystallizer, Sugar y Azucar. December: 19-25.

Frew, J.A. (1973) Direct Digital Conductivity Controller for Vacuum Pan Crystalliser

Software Design, Report Number CE/M33, Division of Chemical Engineering,

C.S.I.R.O., Australia.

Hamilton, J. C., Seborg, D.E. and Fisher, D.G. (1973) An Experimental Evaluation of

Kalman Filtering, American Institute of Chemical Engineers Journal. 19(5): 901-909

Kalman, R.E., (1960) A new approach to linear filtering and prediction problems,

Journal of Basic Engineering, 82, 35-45.

Schneider P. A., (1996) Advanced Control of an Industrial Crystalliser, Ph.D. Thesis,

The University of Queensland, St. Lucia 4072, Australia.

Smith C. A. and Corripio A.B., (1997), Principles And Practice Of Automatic Process

Control. John Wiley and Sons, Inc, New York, 2nd Edition.

Wilson D.I., (1990) Advanced Control of a Batch Raw Sugar Crystalliser. Ph.D. Thesis,

The University of Queensland, Brisbane 4072, Australia.

White E.T., (1998) A Review of the Crystallisation of Sugar, Proc. Intl. Conf.- Mixing

and Crystallisation, Malaysia.

White E.T, (2000) Sugar Crystallization- E1460 Sugar Technology, The University of

Queensland, Brisbane 4072, Australia.

White E.T., Butler B. K., Zhang H., Johns M. R. and Mackintosh D. L., (1998)

Modelling growth rate dispersion (GRD) in sugar crystallization, Proc. Aust. Soc. Sugar

Cane Technol., 20:524-531

Wright P.G., (1971) A Model of Industrial Sugar Crystallisation. Ph.D. Thesis, The

University of Queensland, Brisbane 4072, Australia.

Wright, P.G., and White, E.T., (1968) Digital simulation of the vacuum pan

crystallisation process, Proceedings of the International Society of Sugar Cane

Technologists: 1697-1710.



10.0 GLOSSARY OF TERMS

Dry Substance Total solids dissolved in molasses, syrup or juice usually expressed

as a percentage of the total mass of solution.

False grains These are small crystals which result through undesirable

nucleation.

Footing The initial amount of massecuite in a batch pan

Heavy up The stage when the pan becomes full and some of the remaining

sucrose is being exhausted from the solution

Juice The impure sugar solution extracted from sugar cane.

Massecuite A mixture of crystals suspended in molasses

Molasses A less pure sugar solution with a higher brix than syrup or juice.

Pan Vacuum pan

Purity The amount of sucrose as a fraction of the dry substance of

molasses, syrup or juice

Strike A complete cycle of a batch crystallization.

Syrup Concentrated clarified juice from the final vessel of the

evaporation train

New Models for Sugar Vacuum Pans A1


APPENDIX A – MORE RESULTS

This Appendix contains results for a typical A strike with conditions as shown in theTable A1.

Table A1 - Operating conditions for A strike

ConditionsInitial Boiling Terminal

Mass (fraction of mass) 0.45 0.98Size, mm 0.55 0.85Crystal content(fraction of mass) 0.365 0.47 (heavy up starts) 0.55Molasses dry substance 0.79 0.81Molasses Purity 0.835 0.73OSfrac 0.5Liquor Dry Substance 0.675Liquor Purity 0.915

Figure A1-Mass of sucrose, water, impurities, crystals and total pan mass

Figure A2 – Syrup feed rate and evaporation rate

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

9

10x 10

4 Plot of pan mass vs time

Time -hr

Mas

s, k

g)

Mass of crystals Mass of water Mass of ImpuritiesMass of Sucrose Total Mass

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4x 10

4 Syrup and Evaporation rates vs time

Time-hr

Mas

s, k

g/hr

)

Syrup rate Evaporation rate

Dynamic Models of Sugar Vacuum Pans A2


Figure A3 – Growth rate against time

Figure A4 – Brix, purity and crystal content

Figure A5 – Oversaturation fraction vs. time

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25Plot of growth rate vs time

Time-hr

Gro

wth

rat

e -

um/h

r

0 0.5 1 1.5 2 2.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Plot of OSsetpoint vs time

Time-hr

OS

setp

oint

0 0.5 1 1.5 2 2.50.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85Plot of crystal content, pan purity and brix vstime

Time-hr

crystalcontent,panpurityandbrix -fracti

Brix

Purity

Crystal content

Dynamic Models of Sugar Vacuum Pans A3


Figure A6 – Coefficient of variation and skewness

Figure A7 – The ratios of sucrose and impurities to water

Figure A8 – Plot of characteristic size vs. time

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35Plot of Coefficient of variation and skewness vs time

Time-hr

Act

ual C

oeffi

cien

t of v

aria

tion

and

skew

ness

CVsk

0 0.5 1 1.5 2 2.5550

600

650

700

750

800

850Plot of Characteristic Size vs time

Time-hr

Cha

ract

eris

tic S

ize,

um

0 0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

3

3.5ratio of sucrose to water and impurity to water vstime

Time-hr

S/WandI/Wrat

S/W

I/W

Dynamic Models of Sugar Vacuum Pans B1


APPENDIX B – MATLAB CODES

This Appendix has the completed code the batch model and initial coding for thecontinuos model.

Ø Batch model codes;(1) Driver file(2) Function files.

Ø Continuous model codes

(3) Initial code for continuous model

Details about the files are contained in their respective sections.

(1)Driver file:

This file contains the following,

(1) Initial conditions for strikes(2) Control loops(3) Plot of various pan conditions(4) ODE solvers

% Individual enquiry 2000% Written by: Terry Tahal% Supervisor: Prof. T White% This script file runs the mass balance equations and GRD equationsfor a crystallization process of a sugar mill using ode15s% Plots the results.% Initialise system parameters and variables% Driver for bat.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This file contains the driver file that can plot all the parametersfor a % pan strike.% The model can be adopted for any of the strike A, B and C% The file is divided in to four sections

% (1) The input tables% (2) testing for feed changes such as purity and brix.% (3) The control loop% (4) The ODE solver - ODE 15s% (5) The parameters displayed as plots%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%This section contains the variables that can be changed%In the Input table - replace the number below with your operatingconditions%Any variable in the table can be changed except the brix and purity ofthe footing



% Input table%%%%%%%%%%%%%%%%%%%%%% These parameters should be changed for every strike - The currentparameters are preset for% An A strike.T = 65; %operating temperature of the pan - typical value = 65oC')RS = 0.1;%reducing sugar in seed, as a fraction - typical value = 0.1ash = 0.1;%ash content of seed used, as a fraction- typical value = 0.1PF = 0.915%purity of syrup, as a fraction - typical value - 0.915BF =0.675%Brix of syrup, as a fraction - typical value - 0.675Sk = 0.001;%skewness of the distribution - typical value - 0.001CV = 0.25;%give initial CV of seed material - typical value = 0.25Footf = 0.45;%fraction of seed for pan') - typical value =0.45Pansize = 100000;%Size of pan in kg')Li = 550;%average crystal size, in micrometers - typical value = 550umcci = 0.365;%crystal content of seed used - typical value =0.365alpha = 0.7;%ratio of Lbar/Lc') - typical value = 0.7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Other Changeable variables - These parameters should only be changed ifthe type of strike% is to be changed. This section is preset for an A strike.

OSset = 0.6; %Set point for oversaturationhcc = 0.47; %heavying up crystal contenthbcc = 0.4; %boil back crystal contentecc = 0.55; %end crystal contentPUM = 0.915; %boilback purity eg A molasses purityBUM = 0.675; %boil back brix eg A molasses brixBFo = 0.79; %brix of footing molasses, as a fraction - typicalvalue = 0.79PFo = 0.835; %purity of footing molasses, as a fraction -typical value = 0.83%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Tuning the program% The following parameters will be used to tune the program when anychanges are made to% The inputs.% Adjust the gains for the evaporation rate and the syrup.

%Time constants and gain

TD = 0.01; % Differential time for syrup flowrate -hrTD1 = 0.038; % Differential time for evaporation rate -hrTI = 0.030; % Integration time for syrup flowrate -hrTI1 = 0.047; % Integration time for evaporation rate -hrK1 =-Pansize/13; % Gain for evaporation rate K = Pansize/3.9; % Gain for syrupK2 = Pansize/70; % factor for reducing evaporation rate duringheaving upt0 = 0; % start time -hrh = 0.001; % Time to initiate the differential equations -hrtf = 4; % Finish time for the pan -hr% If the sampling time is need to be changed, it can be found on the

% first line of section 4%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% If information about the ramping of the batch pan is required theparameters min_B and min_P can be changed. Give the minimum purity andbrix of the feed and run simulation.% The pan will be ramped be changing both the purity and brix with timeduring the strike% The following are the minimum conditions for the A% A strikemin_B = 0.675; % Minimum Brix of syrupmin_P = 0.915; % Minimum purity

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Section 3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$%Note: !!!Do not change anything in section 3!!!!!!!!!!!!!!!!!!!!!!!!!!!%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

bat2 =[T ;RS ;ash ;PF ;BF ;SK ;CV ;Footf ;Pansize ;BFo ;Li/1000000 ;PFo;cci ;alpha];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Globalises the variables used in the modelglobal T Ts Tw Tst rsash Ft1 PF BF PH PMo Et OScrit PU Brix G sk CVaalpha ska Lc SE PFoglobal Li cc OS rho N Wt BFo tf min_P min_B BUM PUM hbcc hcc ecc%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%T = bat2(1); % Operating temperature of pan (C)Ts = 35; % Temperature of feed syrup (C)Tw = 25; % Temperature of water (C)Tst = 70; % Temperature of evaporating vapour (C)rs = bat2(2); % reducing sugar in the massecuiteash = bat2(3); % ash content of feed to the panrsash = RS/ash; % assume ratio of reducing sugar to ash is 1Pansize = bat2(9); % Capacity of pan (kg)Ft1 = 4000; % Initial feed syrup rate (kg/hr)Sk = bat2(6); % SkewnessCV = bat2(7); % Initial coefficient of variationPF = bat2(4); % Purity of feed syrup (g suc/gsolid)BF = bat2(5); % solids conc. in feed syrup (g suc/ gsolution)PH = 0.5; % Purity of fed seed material (g suc/gsolid)PMo = 0; % Purity of molasses to pan (g suc/gsolid)Wt = 0; %Movement waterEt = 0; % Initial evaporation rate (kg/hr)alpha = bat2(14); % ratio of Lbar/Lc

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Footf = bat2(8); % Fraction of footing in panSE = Footf*Pansize; % Amount of footing in the pan (kg)BFo = bat2(10); % Brix of footing (g suc/ gsolutionPFo = bat2(12); % Purity of footing (g suc/gsolid)cci = bat2(13); % crystal content in footing (fraction)



Li = bat2(11); % Mean crystal size (m)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Creating tablesOScrit1 = []; % Critical OversaturationPUa = []; % Purity of pan material during strike (Fraction)Brixa = []; % Brix of pan during strike (Fraction)G2 = []; % Creates tables for growth rateCV1 =[]; % Creates table for CVCVb = []; % CV through the strikesk1 =[]; % Creates table for skewnessLC = []; % Characteristic size of crystals in the panI1 = []; % creates table for integrationcc1 =[]; % calculates crystal content of pan during strikej1 = [];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%rho = 1585; % density of crystals (kg/m^3)rhomol = 1000/(0.76925 -0.00055*T + 0.00767*BFo +0.00228*cci);

% Density of molasses in footing (kg/m^3)rhos = SE/((cci*SE/rho)+((1-cci)*SE/rhomol));

% Density of footing (kg/m^3)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Calculating the number of crystals in the pan

N = cci*SE/((1+3*CV^2+sk^3*CV^3)*rhos*(pi/6)*Li^3);% Number of crystals in footing

u1 = alpha*N*Li*1000000; % Calculates the initial values for first momentu2 = (alpha^2)*N*(1+CV^2)*(Li*1000000)^2; % Calculates the initial value for the secondmomentu3 = (alpha^3)*N*(1+3*CV^2 + Sk*CV^3)*(Li*1000000)^3; % Initial value for third momentS1 = (1-cci)*SE*PFo*BFo; % Initial sucrose in the pan (kg)X = cci*SE; % Initial crystal in pan (kg)W1 = (1-cci)*SE*(1-BFo);

% Initial water in the pan (kg)IM = ((1-cci)*SE)-S1-W1; % Initial water in the pan (kg)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%M0 = [X W1 IM S1 SE u1 u2 u3];

%set initial conditiontspan1 = [t0 h]; % time span%This will integrate your equation between the times specified[t,M] = ode15s('bat',tspan1,M0);

%**************************************************************************Ftact =[]; % Creates table for syrup feed rateEtact =[]; % Creates table for evaporation rateta2 =[t]; % creates table for timelast1 =[M]; % Creates table for mass balancesOSact = [];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

L = length(M);las = M(L,:);OScritm = OS/OScrit;%OScrita = [OScritm];



% Generates the initial values for errors, integration of errors anderror differentialE = OScritm - OSset;Etab = [E];I=(E + OSset)*h/2;Err=(E - OSset)/(h);En =E;last =las;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Section 4%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Control loopTs = 0.01; % Sampling time -hr

for j = h:Ts:tf % Gives the time interval for the for the second ode jlas=last;[ta,Ma] = ode15s('bat',[j j+Ts],las); % calculates the mass balances attime intervalsm = length(Ma);last = Ma(m,:);ta1 = ta(m,:);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calculates the increment of for the flow rateEtact = [Etact;Et];steam = 1000*((Etact/(1000*0.95))-0.6298)/0.9946;glow = length(Etact);if cc < hccEFt = K1*(En+ (I/TI1) + TD1*Err);else EFt = K2*(ecc-cc)end

if ta < 0.1Et = Et;elseif cc < hcc ; Et = Etact(glow) + (EFt);else Et = Etact(glow) - EFt;end

if Et < 0 Et = 0;end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Calculates the increment for syrup flowrateDFt = K*(En + (I/TI) + TD*Err);if DFt < 0 DFt = 0; end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



% Generates the initial values for errors, integration of errors anderror differentialOScritm1 = OS/OScrit;En = OScritm1 - OSset; % Calculates the error for the oversaturation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Stores values in the tables created aboveFtact = [Ftact;Ft1];OSact = [OSact;OS];Etab = [Etab;En];last1 = [last1;last];ta2 = [ta2;ta1];G2 =[G2;G];cc1 = [cc1;cc];CV1 = [CV1;CV];CVb = [CVb ;CVa];LC = [LC;Lc];OScrita = [OScrita;OScritm1];OScrit1 = [OScrit1;OScrit];PUa = [PUa;PU];Brixa = [Brixa;Brix];sk1 =[ska;sk1];j1 = [j1;j];fg = length(j1);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculates the errors at two timesa = length(Etab);E1=Etab(a);E2=Etab(a-1);b =length(ta2);t1 = ta2(b);t2 = ta2(b-1);Time = t1-t2;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Integrates the error for the PI controllerTotal =(E1+E2)*Time/(2);I1=[I1;Total];I=(sum(I1));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Differentiates the errors for the PD controllerErr=(E1-E2)/(Time);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Turns off control for heaving up purposesif cc > hcc DFt = 0;end

flow = length(Ftact);

if cc < hcc ; Ft1 = Ftact(flow) + DFt;else Ft1 = 0;



endif cc > ecc break end end tp = j1(fg);%End of control loop

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% There are 12 plots that can be obtained from the following section .To plots the required% graph remove semi-colons from them.

%Schneider's resultccs =[0.365,0.375,0.385,0.395,0.405,0.41,0.42,0.43,0.44,0.455,0.465,0.47,0.485,0.535,0.55];pus =[0.835,0.83,0.827,0.825,0.822,0.82,0.818,0.815,0.810,0.807,0.803,0.785,0.765,0.745,0.73];brs =[0.79,0.785,0.783,0.785,0.79,0.791,0.793,0.795,0.798,0.80,0.805,0.81,0.812,0.8125,0.810];times =[0,0.2,0.4,0.65,0.8,1.05,1.25,1.45,1.65,1.85,2.1,2.3,2.5,2.7,2.75];

%Plotting of results% plot mass in pan vs. timefigure(1);plot(ta2,last1(:,1),'m-.',ta2,last1(:,2),'r-',ta2,last1(:,3),'g--',ta2,last1(:,4),'k:',ta2,last1(:,5),'b-'); %plot the resultstitle('Plot of pan mass vs. time')xlabel('Time -hr')ylabel('Mass, kg)')legend('Mass of crystals','Mass of water','Mass of Impurities','Mass ofSucrose','Total Mass')hold

% plots flowrate of syrup and evaporation ratefigure(2);plot(0:Ts:tp,Ftact,'--',0:Ts:tp,Etact,'-')title('Syrup and Evaporation rates vs. time')xlabel('Time-hr')ylabel('Mass, kg/hr)')legend('Syrup rate','Evaporation rate')hold

% plots the third moment against timefigure(3);plot(ta2,(1585*(pi/6))*last1(:,8),'b-')title('Plot of Moments vs. time')xlabel('Time -hr')ylabel('Mass, kg)')hold



% plots oversaturation during the pan strikefigure(4);plot(0:Ts:tp,OSact,'-',0:Ts:tp,OScrit1,'--')title('Plot of oversaturation vs. time')xlabel('Time -hr')ylabel('Oversaturation)')legend('Oversaturation','Critical oversaturation')hold

% plots the growth rate of crystals during the strikefigure(5);plot(0:Ts:tp,G2)title('Plot of growth rate vs. time')xlabel('Time-hr')ylabel('Growth rate - um/hr')hold

% plots the purity, crystal content and brix during the strikefigure(6);%plot(0:Ts:tp,PUa,'--', times,pus,'-',0:Ts:tp,cc1,'--',times,ccs,'-',0:Ts:tp,Brixa,'--',times,brs,'-')plot(0:Ts:tp,Brixa,':')title('Plot of Brix(Molasses Dry Substance)')xlabel('Time-hr')ylabel('Brix(Molasses Dry Substance) - fraction')%legend('Improved Model','Schneider')%,'crystal content','panpurity','brix')axis([0 tp 0.75 0.95])hold on

% plots oversaturation against growth ratefigure(7);plot(OSact,G2*1000/60)title('Plot of Oversaturation vs Growth rate')xlabel('Oversaturation')ylabel('Growth rate- um/min')hold

%plots critical oversaturationfigure(8);plot(0:Ts:tp+Ts,OScrita)title('Plot of OSsetpoint vs time')xlabel('Time-hr')ylabel('OSsetpoint')hold

%plots characteristic size of crystalsfigure(9);plot(0:Ts:tp,LC)title('Plot of Characteristic Size vs time')xlabel('Time-hr')ylabel('Characteristic Size,um')hold on

% Plots the coefficient of variationfigure(10);plot(0:Ts:tp,CVb,'-',0:Ts:tp,sk1,'--')title('Plot of Coefficient of variation and skewness vs time')xlabel('Time-hr')ylabel('Actual Coefficient of variation and skewness')legend('CV','sk')hold



%Plots the ratio of sucrose to water and impurity to waterfigure(11);plot(ta2,(last1(:,4)./last1(:,2)),'-',ta2,(last1(:,3)./last1(:,2)),'--')title('ratio of sucrose to water and impurity to water vs time')xlabel('Time-hr')ylabel('S/W and I/W ratio')hold

% plots the purity and crystal content of molassesfigure(12);%plot(0:Ts:tp,PUa,'--', times,pus,'-',0:Ts:tp,cc1,'--',times,ccs,'-',0:Ts:tp,Brixa,'--',times,brs,'-')plot(0:Ts:tp,PUa,'-', 0:Ts:tp,cc1,'--',0:Ts:tp,Brixa,':')title('Plot of molasses purity and crystal vs time')xlabel('Time-hr')ylabel('molasses purity and crystal - fraction')legend('A strike','B strike', 'C strike')hold on

(2) Function file:

The file contains the following:

(1) Differential equation for mass and moments(2) Growth rate equations(3) Supersaturation and oversaturation relationships

% Individual enquiry 2000% Model for batch pan% Differential equations% Prepared by Terry Tahal% 2000-07-11% Supervisor: Prof. T White%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This file has three sections% (1) oversaturation and saturation equations% (2) growth rate equation% (3) Differential equationsfunction dM = bat(ta,Ma)

%ConstantsdX = Ma(1); % Takes calculated values of mass of crystals to beused in the equationdW = Ma(2); % Takes calculated values of water to be used inthe equationdI = Ma(3); % Takes calculated values of impurities to be usedin the equationdS = Ma(4); % Takes calculated values of sucrose to be used inthe equationdT = Ma(5); % Takes calculated values of total mass to be usedin the equationdNu1 = Ma(6); % Takes calculated values of characteristic sizedNu2 = Ma(7); % Takes calculated values of massdNu3 = Ma(8); % Takes calculated values of mass of crystals%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Inputs



global Wt % Balance water feed rate (kg/hr)H = 0; % Massecuite feed rate (kg/hr)P = 0; % Production rate (kg/hr)global SE % Mass of seed added to pan (kg) - 1/3mass of full panglobal rho % Density of crystals (kg/m^3)global PF % Purity of syrup global BF % Solid concentration of syrupglobal PH % Purity of feed to panglobal PMo % Purity of molasses to panglobal rsash % Assumes ratio of reducing sugar and ash stays thesameglobal tf%Temperaturesglobal T % Operating temperature of pan (oC)global Ts % Temperature of syrup (oC)global Tw % Temperature of water used (oC)global Tst % Temperature of steam (oC)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Section 1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Saturation and oversaturation%Calculates oversaturation and supersaturation from the initialdifferential equationP5 = 0.011 +0.00046*T;P6 = 0.67 + 0.0021*T -0.07*rsash;P7 = 0.54 + 0.0049*T;SC = P5*(dI/dW) + P6 + (1-P6)*exp(-P7*(dI/dW));global SWr %Ratio of sucrose to water of saturated solutionSWr = (0.75328365 + 0.00225338*(T-65))/(1-(0.7532865+0.002225338*(T-65)));SS = (dS/dW)/(SC*(SWr)); %Supersaturationglobal OSOS =SS-1; %Oversaturation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Section 2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Growth rateglobal G%Growth rateglobal PU BUM PUM hbcc hcc eccPU = dS/(dS+dI); % Purity if molasses in panglobal Brix % Brix of molasses in panBrix = (dS+dI)/(dS+dI+dW);global OScrit % Critical oversaturation during the strikeOScrit =0.11 + 3.6*(dI/(dS+dI))^3;% Growth rate correlation - (White, 2000)OSb = 0.043;A = 10^(3.31 + 0.022*(T-60)-0.00031*(T-60)^2);G = (60/1000)*A*(OS^2)*tanh(OSb/OS)*exp(-1.75*(dI/dW));%

(um/hr)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Calculating the number of crystalsglobal LiIO = 1; % For monosizeglobal cccc = dX/dT; % Crystal contentglobal sk % Skewnesss



global N % number of crystals in panLb = Li*1000000; % umglobal alpha % Ratio of mean number crystal size tocharacteristic mean size(50%)global LcLc = dNu1/(alpha*N) ;% Characteristic crystal size (um)global CVaCVa = sqrt(((dNu2/N)/(alpha*Lc)^2)-1);CM = dX/dW; % solids conc. of material in the pan (g suc/ gsolution)CH = 0.1; % solids conc. of seed material (g suc/ gsolution)PM = dS/(dI+dS); % purity of material in pan (g suc/gsolid)global ska % calculates skewnessska = (((dNu3/N)/((alpha^3)*((Lc)^3)))-(1+3*CVa^2))/(CVa^3);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if cc < hbcc PF = PF;BF = BF;elseif cc < hcc PF = PUM;BF = BUM;else PF = PF;BF = BF;end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Evaporation from the pan%Globalize flowrate and evaporation rateglobal Ft1 % Flowrate of syrup (kg/hr)global Et % Evaporation rate (kg/hr)global BFo % Brix of footing (g suc/ gsolution)global PFo % Purity of footing (g suc/gsolid)global min_Bglobal min_P%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%PFi = min_P + ((PF-min_P)*ta/tf);BFi = min_B + ((BF - min_B)*ta/tf);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Section 3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Mass balance equations%Mass balance for the amount of crystals in the pandX = 3*rho*(pi/6)*IO*N*((Lc/1000000)^2)*G/1000;%Mass balance of Water insolution (kg/hr)%Mass balance of water in the pandW = Ft1*(1 - BFi) + Wt + H*(1-CH) - Et - P*(1-CM);%Mass balance of Impurity in solution (kg/hr)dI = Ft1*BFi*(1 - PFi) + H*CH*(1-PH) - P*CM*(1-PM);%Mass balance ofSucrose in solution (kg/hr)%Mass balance of sucrose in pandS = Ft1*BFi*PFi + H*CH*PH - dX - P*CM*PM; %Mass balance of Massecuite (kg/hr)dT = dW + dI + dX + dS;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%Growth rate dispersion expressions%First momentsdNu1 = alpha*N*G*1000;%Second momentsdNu2 = 2*(alpha^2)*(1+CVa^2)*Lc*(G*1000)*N;%Third momentsdNu3 = 3*(alpha^3)*(1+ 3*CVa^2+ ska*CVa^3)*(Lc^2)*(G*1000)*N;dM = [dX ; dW ; dI ; dS ; dT; dNu1; dNu2; dNu3];

(3)Initial code for continuous model

T = 65;rsash = 1;Li = 350;SE =25000;PFo = 0.9;BFo = 0.70;cci = 0.15;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%S1 = (1-cci)*SE*PFo*BFo; % Initial sucrose in the pan (kg)X = cci*SE; % Initial crystal in pan (kg)W1 = (1-cci)*SE*(1-BFo);

% Initial water in the pan (kg)IM = ((1-cci)*SE)-S1-W1; % Initial water in the pan (kg)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

P5 = 0.011 +0.00046*T;P6 = 0.67 + 0.0021*T -0.07*rsash;P7 = 0.54 + 0.0049*T;SC = P5*(IM/W1) + P6 + (1-P6)*exp(-P7*(IM/W1));

%Ratio of sucrose to water of saturated solutionSWr = (0.75328365 + 0.00225338*(T-65))/(1-(0.7532865+0.002225338*(T-65)));SS = (S1/W1)/(SC*(SWr)); %SupersaturationOS =SS-1; %Oversaturation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Growth rateglobal G%Growth rateglobal PUPU = dS/(dS+dI); % Purity if molasses in panglobal Brix % Brix of molasses in panBrix = (dS+dI)/(dS+dI+dW);global OScrit % Critical oversaturation during the strikeOScrit =0.11 + 3.6*(dI/(dS+dI))^3;% Growth rate correlation - (White, 2000)OSb = 0.043;A = 10^(3.31 + 0.022*(T-60)-0.00031*(T-60)^2);G = (60/1000)*A*(OS^2)*tanh(OSb/OS)*exp(-1.75*(dI/dW));% (um/hr%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%n1 = 1/20;m1 = 1/20;



Gtrue = [0:m1:1];Ltrue = 300 + ([0:n1:1]*(Li*1000000-300));N1= [N];N3 = [0];N2 = [N];

for i = 1:1/n1 delL = Ltrue(i+1) - Ltrue(i); for jj= 1:1/m1 tou1 = 2; % hours tou2 = 2;dL = G*Gtrue(jj)*dt; %cell1Nf = N2(jj)*(1-(dL/delL)-(dt/tou1)) + N2(i)*(dL/delL) +N2(jj)*(m1*n1)*(dt/tou2);dX = 3*rho*(pi/6)*IO*Nf*((Ltrue(i)/1000000)^2)*G/1000 -P*cc;%Massbalance of Water in solution (kg/hr)N2 = [N2;Nf];dX = 3*rho*(pi/6)*IO*Nf*((Lc/1000000)^2)*G/1000 -P*cc;%Mass balance ofWater in solution (kg/hr)%Mass balance of water in the pandW = Ft1*(1 - BFi) + Wt + H*(1-CH) - Et - P*(1-CM);%Mass balance of Impurity in solution (kg/hr)dI = Ft1*BFi*(1 - PFi) + H*CH*(1-PH) - P*CM*(1-PM);%Mass balance ofSucrose in solution (kg/hr)%Mass balance of sucrose in pandS = Ft1*BFi*PFi + H*CH*PH - dX - P*CM*PM; %Mass balance of Massecuite (kg/hr)dT = dW + dI + dX + dS;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Growth rate dispersion expressions%First momentsdNu1 = alpha*Nf*G*1000;%Second monemtsdNu2 = 2*(alpha^2)*(1+CVa^2)*Lc*(G*1000)*Nf;%Third momentsdNu3 = 3*(alpha^3)*(1+ 3*CVa^2+ ska*CVa^3)*(Lc^2)*(G*1000)*Nf;dM = [dX ; dW ; dI ; dS ; dT; dNu1; dNu2; dNu3];N1 = [N1;N2];N3 = [N3;N2];

endend

Dynamic Models of Sugar Vacuum Pans C1


APPENDIX C – TUNING THE MODEL CONTROL

The table below gives some typical initial conditions for A, B and C strikes and it alsogives the values of the tuning parameters for those conditions.

Table C1 – Initial Conditions for Strikes

Parameter A strike B strike C strike

K1 Pansize/13 Pansize/13 Pansize/5.5

K Pansize/3.9 Pansize/4.9 Pansize/140.5

K2 Pansize/70 Pansize/70 Pansize/300

PUM 0.915 0.73 0.65

BUM 0.675 0.81 0.835

BFo 0.79 0.79 0.79

PFo 0.835 0.835 0.70

hcc 0.47 0.47 0.35

hbcc 0.4 0.44 0.2

ecc 0.55 0.55 0.55

Footf 0.45 0.45 0.35

cci 0.365 0.365 0.2

Azucar- Vacum Panes

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