control model for compressible cake ﬁltration of green ...576310/fulltext01.pdfcompressible filter...

Control model forcompressible cake filtration

of green liquor in cassette filter

KAJSA BORNEFELT

Masters’ Degree ProjectStockholm, Sweden 2006

XR-EE-RT 2006:013

AbstractIn the closed chemical recovery cycle in the sulphate pulp mill it is important to remove non-process elements. This is done by clarification of the green liquor, either in clarifiers or in filters. This project focuses on a cassette filter developed by Kvaerner Pulping AB. The cassette filter is semi-continuous and the aim of the project was to model the filter in order to be able to control cycle time and feed towards optimization of the capacity. The green liquor sludge forms a compressible filter cake when filtered.

The model was built on the filter equation for compressible cake filtration and parameters such as filter cloth resistance, compressibility index and specific resistance in the cake were to be determined. The parameters were calculated by minimizing the difference between the calculated model and the measured data. Some simulation experiments were done to examine if optimization was possible.

It turned out that the two parameters describing the green liquor (specific resistance in the cake and compressibility) were not identifiable from each other and the third parameter (resistance in the filter cloth) was also sometimes unidentifiable. The simulation experiments showed that the capacity of the cassette filter is hard to optimize during unfavourable conditions controlling only cycle time and feed. Proper actions might be to add lime mud or aluminium to increase the filterability of the green liquor sludge or to wash the filter socks to decrease the resistance in the filter cloths.

Control model for compressible cake filtration of green liquor in cassette filter 2

Table of contents ABSTRACT.....................................................................................................................................2

TABLE OF CONTENTS................................................................................................................3

ACKNOWLEDGEMENTS............................................................................................................4

1 INTRODUCTION........................................................................................................................5

2 THE SULPHATE PULP PROCESS ..........................................................................................6

2.1 DESCRIPTION OF THE SULPHATE PULP PROCESS.......................................................................62.2 GREEN LIQUOR ........................................................................................................................8

2.2.1 Green liquor clarification............................................................................................... 82.2.2 Problems with non-process elements.............................................................................. 8

3 THE CASSETTE FILTER........................................................................................................10

3.1 DESCRIPTION OF THE CASSETTE FILTER.................................................................................103.2 DESCRIPTION OF THE PROBLEM .............................................................................................12

3.2.1 The project .................................................................................................................... 123.3 SOME TYPICAL FILTRATION CYCLES ......................................................................................13

4 THEORY ....................................................................................................................................15

4.1 FILTRATION THEORY .............................................................................................................154.1.1 Derivation of the filter equation ................................................................................... 154.1.2 The filter equation for compressible cake filtration...................................................... 174.2.1 Estimation of parameters from the model..................................................................... 194.2.2 Identifiability ................................................................................................................ 19

5 METHOD ...................................................................................................................................21

5.1 MEASUREMENTS ...................................................................................................................215.2 DESCRIPTION OF THE IDENTIFICATION ALGORITHM ...............................................................22

6 MODEL.......................................................................................................................................24

6.1 DATA RELIABILITY - IDENTIFIABILITY ...................................................................................246.3 SOURCES OF ERROR ...............................................................................................................30

6.3.1 Measurement errors...................................................................................................... 306.3.2 Model errors ................................................................................................................. 30

7 OPTIMIZATION.......................................................................................................................31

7.1 OPTIMIZATION EXPERIMENT..................................................................................................31

8 RESULTS AND DISCUSSION.................................................................................................33

9 CONCLUSIONS ........................................................................................................................34

REFERENCES..............................................................................................................................35

APPENDIX


Acknowledgements This master thesis was done at Kvaerner Pulping AB in cooperation with Peterson AS, Moss mill. I would like to thank Claes Lysén on Kvaerner Pulping AB and Elling Jacobsen in the lab for Automatic Control at KTH for supervising.

I also want to thank Alexander Vedeler and Lars-Roar Karlsen from Moss mill, Michael Berggren from Aspa mill and Emil Karlsson from Mönsterås mill for helping me in this work.

Finally I want to thank Patrik Löwnertz and the rest of the recausticizing group for their support and good company during these months.


Introduction

1 Introduction In a sulphate pulp mill, boiling wood chips in a solution of chemicals makes pulp. For both economical and environmental reasons, the chemicals in the process are recycled. As the recovery cycle gets more and more closed it is important to remove non-process elements. Otherwise they will accumulate in the cycle and cause problems like corrosion and form incrusts that lead to bad heat transfer. The non-process substances are traditionally removed by sedimentation in the green liquor but relatively new filtration techniques are also in use. This work focuses on one such filter, the cassette filter developed by Kvaerner Pulping AB and especially the cassette filter in Moss mill in Norway.

The cassette filter is semi-continuous and works in cycles. There are several problems with the filter capacity; it is very sensitive to changes in the green liquor content and the green liquor sludge easily blocks the filter cloths. The green liquor sludge builds a filter cake during filtration which is compressible and the behaviour of the cake affects the pressure in the filter vessel as well as the flow and hence the capacity of the filter.

When the green liquor is difficult to filtrate or there is high resistance towards flow in the filter cloth, the filter capacity decreases. Because of problems with capacity it would be desirable to optimize the filter by controlling the running conditions such as cycle time and feed. The aim of this project was to create a model for such an optimization.


The sulphate pulp process

2 The sulphate pulp process

2.1 Description of the sulphate pulp process To make pulp, wood chips are boiled in a solution of chemicals, called white liquor. The white liquor contains the active chemicals sodium hydroxide (NaOH) and sodium sulphide (Na2S). The boiled pulp is washed with water and the chemicals are sent into the recovery cycle. The washed pulp (the cellulose from the wood) is treated in several steps; washing, bleaching and drying.

The water from the washing has high concentrations of used chemicals, different organic and inorganic substances, lignin and other polymers from the wood. This is called black liquor and contains a lot of chemical energy. To be able to use this energy, the black liquor is going through several evaporating steps. It starts off with approximately 15 % solids of which two thirds are organic and one third inorganic substances. After the evaporation it has a dry content of approximately 70 % and has very high viscosity. The thick black liquor is burned in the soda recovery boiler and the released energy is often enough to supply the whole mill with steam and electricity.

The chemicals in the black liquor form a melt in the bottom of the soda recovery boiler. The melt is dissolved in water to form green liquor. The green liquor contains sodium carbonate (Na2CO3), sodium sulphide (Na2S) and some sodium sulphate (Na2SO4).

For both economical and environmental reasons, the chemicals in the green liquor are recycled. To prevent that the non-process elements accumulate in the system, the green liquor is clarified. After the clarification it is important to turn as much as possible of the inactive sodium carbonate into the active sodium hydroxide.

The green liquor is mixed with lime (calcium oxide) in the causticizing reaction.

CaO + H2O Ca(OH)2 + heat

Na2CO3 + Ca(OH)2 2NaOH + CaCO3(s)

The solid calcium carbonate (CaCO3) is burned in the lime kiln to be able to use it again.

CaCO3 + heat CaO + CO2(g)

The liquor now contains active chemicals again and can be used to boil the wood chips. [1], [5]



Washing

The rest of the fibre line; washing, bleaching, drying etc.

Digester

Wood chips

White liquor

Black liquor (~15 % solids)

Sodarecovery boiler Thick black liquor

(~70 % solids)

Green liquor clarification

Causticizing

Lime mud clarifier

Lime kiln

Green liquor sludge

Clear green liquor

Evaporation steps

Water

Lime mud

Lime

Pulp

Figure 2.1. Flow sheet of the causticizing cycle. The green liquor filter is in the box for green liquor clarification. The clarification can also be done by sedimentation. [1], [9]



2.2 Green liquorThe melt from the soda recovery boiler consists of 60-65 % sodium carbonate (Na2CO3) and 25-30 % sodium sulphide (Na2S). The rest is non-process substances and 10-15 % sodium sulphate (Na2SO4). [5]

The sodium sulphide is not stable in aqueous solution and is therefore hydrated to sodium hydroxide (NaOH) and sodium hydrosulphide (NaHS) when the melt is solved in water to form green liquor.

Most of the non-process substances in the green liquor have low solubility in alkali solutions and are therefore solids in the green liquor. There are also some elements that are soluble in the green liquor, most important are potassium and chloride. Potassium form salts in the same way as sodium, K2S, K2SO4 etc. but also with chloride, KCl and NaCl.

In Swedish mills there is about 3-8 g/l of NaCl in the green liquor while the amount of solid non-process particles is about 0,5-2 g/l. [5]

Mg Al Si P Cl Mn Fe K Na Ca

Dry wood chips mg/kg 100-250 10 10-40 40-80 >100 50-150 50-150 200-600 10 400-800

Bark mg/kg 600 100-700 300-800 400-600 100-200 300-700 100-250 1400-2200 10-40 4000

Table 2.1. Examples of origin of non-process elements. [5]

2.2.1 Green liquor clarification Traditionally, the green liquor is clarified in so called clarifiers. That is basically a big sedimentation tank where the dregs sink to the bottom and the clear green liquor spills over at the top. This system is still used in most mills around the world. The advantages with clarifiers are that they are durable and dependable and require minimum of maintenance. In the last 10-15 years however, the use of green liquor filters have increased. The filters have higher production rate in smaller area and they produce cleaner green liquor than the clarifiers. On the other hand, they are more expensive, both in operation and investment and they require better control and operator attention.

The cassette filter leaves dregs with 50-60 % liquor, i.e. 40-50 % solids. This is about twice the solid content than from a normal clarifier, leading to lower pH (less basic) in the landfill and less soda loss for the mill. [4]

The green liquor sludge forms a compressible filter cake in green liquor filtration. This cake is relatively compressible and behaves like other compressible materials. The specific resistance to filtrate flow in the cake can be reduced significantly by adding lime mud to the green liquor. The filtration can also be improved by having green liquor with a lot of aluminium or calcium ions and it has been shown that the ratio between aluminium and magnesium plays an important role in the filtration properties of the green liquor. [3], [8]

2.2.2 Problems with non-process elements As the mills try to have a more closed system with both economical and environmental benefits, the problems with non-process elements grow more important. The green liquor clarification is the “kidney” of the pulp mill.



The problems with non-process elements are many and sometimes severe; corrosion and plugging in the recovery boiler and formation of scales and other incrusts on heat transfer surfaces. The latter diminish the effect of the heat transfer. In the causticizing and lime kiln plants there are problems with settling and filtration disturbances. On the fibre line, in the bleaching, the non-process elements can cause decomposing of the bleaching agents. A problem that can be troubling over time is the accumulation of inerts in the lime cycle that reduce the efficiency of the causticizing. [7]


The cassette filter

3 The cassette filter

3.1 Description of the cassette filter A cassette filter is made of a number of tubes, all placed inside a tank, from now on referred to as the filter vessel (fig.3.1). The tubes have a length of app. 2 m and a radius of app. 5 cm. The tubes are made of steel and have perforated mantel area. Around the mantel area is a filter cloth; the filter socks. The top of the tubes is sealed and the bottom continues as a pipe leading downwards, so that all liquid inside the tubes flows down to a tank for the green liquor filtrate (fig.3.2).

When a filtration cycle starts, the filter vessel is filled from the bottom with unfiltrated green liquor from an equalization tank. As soon as the level reaches the perforated mantel area and filter cloth of the tubes, the filtration starts. The filtrate flows from the outside to the inside of the tubes, leaving the solid particles on the outside of the filter socks, forming a so called filter cake around the tubes. After a few minutes the vessel is filled with green liquor and the pressure inside the filter vessel increases. If the pressure difference reaches a maximum level the flow is decreased so that the pressure difference stays at that level. In Moss the maximum level is set at 2,7 bar.

When the filtration time is over, the pump stops and the unfiltrated green liquor flows back. At the same time air is blown into the filter vessel from the top to keep the filter cakes from falling down from the socks and to dry out as much liquor as possible. When the vessel is drained, the valve in the bottom of the vessel closes and the back flushing starts. Heated water is pumped into the tubes in the opposite direction of the filtrate, cleaning the filter cloth and taking away the cakes. When the back flushing is finished, the valve in the bottom opens to let the sludge out to a sludge filter. Now the filter vessel and the tubes are cleaned and a new filtration cycle can start. Normally, the filtration lasts for about 30-40 minutes and the draining and back flushing takes app. 5 minutes.

In most mills, a system for acid wash of the filter socks is installed. The socks are then washed with acid regularly, for example every second month, to remove all the small particles that accumulates in the filter cloth. In Moss however, this system is not yet installed, and therefore the cassettes have to be changed every 6 months.

The feed is set by the operators and is normally decided by the level in the equalization tank before the filter vessel and the level in the tank for filtrated green liquor. The cycle time does normally not vary so much but can be changed by the operators for different reasons.


The cassette filter

Figure 3.1. Schematic picture of the green liquor flow during the filtration. The unfiltrated green liquor is green and the filtrate is yellow.

Figure 3.2. Schematic picture of the filter tubes. The filtrate is collected in the bottom of the tubes and lead out of the filter vessel.


The cassette filter

3.2 Description of the problem In many mills, the cassette filter causes problems from time to time. If there are problems in the soda recovery boiler, this will result in incomplete combustion and more solid particles in the green liquor. Different types of wood, which contains significant amounts of non-process elements, raise the solid concentration of the green liquor and change the filtration properties.

Over time the amount of small particles in the filter socks will increase and so will the resistance in the filter cloth. Hence the filter capacity will decrease with time from the latest change of cassette or acid wash. But it is not only time that describe the change in filter resistance, the efficiency of the back flush will also influence the filter cloth resistance. If the cycle time or feed is increased, there will be more particles in the cloth and it will be harder to wash it away with the back flush. If the filtration time is reduced the washing will be more efficient and the resistance will decrease even without an acid wash.

If the cycle time is too long or the feed too high, the filter cakes will grow together since the space between the tubes can be as small as 2 cm. This phenomenon is called bridging and reduces the filter area significantly which also reduces the capacity of the filter. With smaller filter area the filter cakes will grow faster and cause more bridging. Having this problem, it is essential to open the vessel and manually clean the tubes or change the whole cassette.

During the back flushing, the seams in the filter socks are put to great stress. This is why it is not a good idea to increase the pressure to remove the bridges. In Moss, where the back-flush pressure was very low, it was possible to remove the bridges by increasing the pressure. This also led to an overall better capacity of the filter since the filter socks were cleaned more thoroughly.

The porosity of the cake decreases with increased pressure difference. The smaller the porosity, the harder for the liquor to flow through the filter and the greater the pressure drop. Eventually the flow will stop totally because the filter has become tight. This happens when the cake collapses at a specific pressure drop. That is why it is very important to make sure that the pressure drop stays below the critical level.

There are suggestions to how to reduce the compressibility and thus improve the filterability. In Moss mill there is a possibility to add a small amount of lime mud to the feed to increase the porosity of the filter cake. This method is used when the capacity of the filter is low but the results are not fully evaluated

3.2.1 The project The project was planned to include several steps. First to collect the necessary data from the cassette filter, then to use the data to get the parameters needed for the model. These parameters would then be used to create a model of the cassette filter. Having the model it would be possible to calculate the optimal running conditions for different values of the parameters and then create an algorithm that predicts the behaviour of the filter and calculates the optimal feed and cycle time for the cassette filter in order to optimize the capacity.


The cassette filter

3.3 Some typical filtration cycles This first example of a filtration cycle is from a period where the filter was working with sufficient capacity. The diagrams in figures 3.3-3.5 show how the measured level, feed and pressure difference changes with time. The feed pump recycles the unfiltrated green liquor during the back flushing period. That is why it looks like the feed continues even during that period. In this example the cycle starts at time 2 and approximately at time 33 the filtration stops and when the level is back to zero, at time 34, the back flush starts.

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Time [min]

Leve

l [%

]

Figure 3.3. Example of a filtration cycle during good conditions. Level vs. time

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Time [min]

Feed

[m3/

h]

Figure 3.4. Example of a filtration cycle during good conditions. Feed vs. time

-0,5

0

0,5

1

1,5

2

0 5 10 15 20 25 30 35

Time [min]

Pres

sure

diff

eren

ce [b

ar]

Figure 3.5. Example of a filtration cycle during good conditions. Pressure difference vs. time.

The next example is from period of larger resistance against filtration quicker raising pressure drop. When the pressure drop in figure 3.8 reaches the critical level of 2,7 bars, the feed in figure 3.7 is reduced so that the pressure stays at 2,7 bars. As soon as the pressure drop reaches the maximum level, the feed cannot be controlled by the operators anymore and is automatically controlled.


The cassette filter

0

20

40

60

80

100

0 10 20 30 40

Time [min]

Leve

l [%

]

Figure 3.6. Example of a filtration cycle during bad conditions. Level vs. time.

0

20

40

60

80

100

0 10 20 30 40

Time [min]

Feed

[m3/

h]

Figure 3.7. Example of a filtration cycle during bad conditions. Feed vs. time.

0

0,5

1

1,5

2

2,5

3

0 10 20 30 40

Time [min]

Pres

sure

diff

eren

ce [b

ar]

Figure 3.8. Example of a filtration cycle during bad conditions. Pressure difference vs. time.


Theory

4 Theory

4.1 Filtration theory

4.1.1 Derivation of the filter equation Filtration theory starts with Darcy’s law, an equation based on experimental data that describes the flow of water through a vertical sand bed. Darcy discovered in 1855 that the volume flow rate was proportional to the pressure difference, indicating that the flow trough the bed was laminar.

dzdpku (1)

Darcy himself did not include the viscosity, μ, but the equation is normally written as in (1) where k is the permeability of the porous medium, dp is the dynamic pressure across the thickness of the medium, dz. This relation is of course only valid for laminar flow but even if the flow increases it will in most cases never get turbulent. Darcy’s law only breaks down at really low Reynolds numbers, in the order of 1-10, because of the big inertial forces in the laminar

flow. Reynolds number (Re) in packed beds is defined as uxRe where x is

the particle size and is the density of the liquid.

In filtration the permeability is often replaced by the specific resistance, ,depending of the density ( s) of the solids and the porosity ( ) of the filtration cake.

)1(1

s

k (2)

The pressure gradient over the thickness of the medium is replaced by a pressure loss per unit mass of solid deposited on the medium.

dzdw s )1( (3)

Combining (1), (2) and (3) gives Darcy’s law in filtration.

dwdpu 1 (4)

When a liquid flows through a filter cloth, there will be a pressure gradient over the filter; the magnitude will depend on the superficial velocity, the viscosity and the resistance in the filter. When a cake of deposited solids starts to form on one side of the filter, the resistance in the cake will also contribute to the pressure drop.


Theory

Figure 4.1. Pressure drop over the filter cloth alone to the left and over both filter cake and cloth to the right. [10]

Now introducing R as the resistance to fluid flow through the filter cloth, defined as the ratio between the thickness and the permeability gives Darcy’s law in a new shape.

Rp

Rpp

u m01 (5)

Assuming that the cake has a constant resistance across the cake, this relation can be used for the cake as well.

c

c

c Rp

Rppu 1 (6)

The total pressure drop is then given by

)(0 RRuppppp cmc (7)

The flow rate is often measured as the flow rate divided by the filter area.

dtdV

Au 1 (8)

Now the total filter equation is given by equations (7) and (8).

)(1

RRp

dtdV

A c

(9)

When the resistance in the cake is proportional to the mass of dry solids deposited per unit area (w), the specific resistance, is reintroduced.

wRc (10)

The mass of solids on the filter cloth (ms) is related to the filtrate volume (V) and the concentration (c) of solids in the fluid.

(11) cVwAms

Equation (10) and (11) gives the expression for the resistance to fluid flow through the filter cake (Rc).

AcVRc (12)

Combining equation (9) and (12) gives the total filter equation for non compressible cake filtration.

)(1

RAcVpA

dtdV

A (13)


Theory

[10]

4.1.2 The filter equation for compressible cake filtration filter cakes are more or less compressible; increased pressure drop over the

w. In the s constant

t

Figure 4.2. The variation he hy lic pressure and the solids compressive pressure through the filter cake. The total pr ure is always constant and the change in hydraulic pressure equals

e change in the solids compressive pressure. [101]

Manycake decreases the porosity and increases the resistance towards fluid flofilter equation (13) for compressible cake filtration, it is assumed that iacross the cake. To be able to use (13), it is essential to find an expression for the harmonic mean of the specific resistance, av. The drag that the fluid exerts on each particle leads to a mechanical pressure between the particles in the cake. The larger the pressure drop from the fluid, the stronger the pressure between particles and the more compressed the cake. The magnitude of change in the two differenpressures is always the same through the cake.

0ls dpdp (14)

of t drauess

th

Combining equations (6), (10) and (14) gives

AdtdwdwdVdpdp sl 1 (15)

is a function of the pressure difference so (15) is rearranged to

pp

dwu0 1

w

sdp0

(16)

tegrating (16) In1

0

ppsdpuw (17)

Filter cake

Filter medium

essure ps

d pressure p1

dp

Pp-p1

0

x=0 x x+ x x=L

dp

dp1

Filtrate s

Liquidp1

Solids compressive prp1 s


Theory

Defining the local resistance by sp0 where n is a compressibility index, th

n

en av is

0 00 )1(1

sav np

Combining equation (5), (7) and (19) gives

111

ppn

cns pdppp

(18)

(19) ncav pn)1(0

nRav AdtdV )1 (20) pn)(1(0

0 is a parameter that describes the resistance in the cake independent of both cation and pressure drop. Finally, (20) can b

= av, and this is the filter equation for compressible cake filtration. lo e put into equation (13) by setting

))1)(1((

1

0 ARcVRAdt

dVpn

pAdtdV

Au

n (21)

A Filter cloth area [m2]p Pressure drop over the filter and the filter cake [N/m2]

u Volume flow per area unit, superficial velocity [m/s]

]

th [m ]

μ Viscosity [Ns/m2]0 Specific resistance at unit applied pressure [m/(kg, Pan)]

c Concentration of solids [kg solids/m3]3V Filtrate volume [m

n Compressibility index (dimensionless) -1R Resistance to fluid through the filter clo

[10]1

1 A similar derivation can be found in Coulson & Richardson’s Chemical Engineering, volume 2, Particle Technology and Separation Processes, 4 th edition, 1996.


Theory

4.2 Modelling theoryThe model in this project had two important purposes. The first was to predict what would happen to the filter if any of the controllable inputs changed; cycle time, feed of green liquor or maximum pressure drop. The second was to calculate the parameters of the filter cloth and of the green liquor; specific resistance, compressibility index and filter cloth resistance.

4.2.1 Estimation of parameters from the model To estimate parameters in a model, there are two different methods. The first uses known physical relations that describe the phenomenon modelled and the parameters have known physical meaning. The second one is often referred to as “black-box” model where different general models are used calculating parameters that describe the relation between inputs and outputs but have no physical meaning.

Typical is to predict a value of the output y(t), depending on the parameters , the predicted value being )|(ˆ ty . The predicted value is then compared with the real value from collected data.

)|(ˆ)(),( tytyt (22)

Over a period of time, the errors can be summoned up to see how good the model predicts reality.

N

tN t

NV

1

2 ),(1)( (23)

The parameters that best describe reality are those that give the smallest value of VN.

In this model, two error functions have been added, one for the filling of the filter vessel (t = [1:k]) and one for the filtration (t = [k+1:N]).

(24) N

kt

k

tN twtwV

1

22

1

21 ),(),()(

[6]

4.2.2 Identifiability It is important to know if the parameters can be fully identified from the model. If two parameters depend too much of each other they are not identifiable. That is, if one of the parameters is fixed and the prediction still is successful by compensating with the other parameter.

A prediction is identifiable if

** )|(ˆ)|(ˆ tyty (25)

This relation is not valid in two cases, one is that there simply are two different values of that gives the same prediction and one is that the two different values of gives different models but because there are problems with the input y(t), the predictions are the same anyway. [6]


Theory

If no minimum is found, it is necessary to either find a relationship between the two dependant parameters or fix one of them to a value. If the two parameters represent real physical states, then the new parameter can be hard to interpret.

Problem occurs when the parameters are slightly identifiable; that there is a minimum to be found but other values of the parameters gives almost as good a model. It is important to be aware of the problems with identifiability.


Method

5 Method Data was collected from three different mills but only the data from Moss mill was used. The cassette filter in Aspa mill (Munksjö AB) is very big in comparison to the needed capacity so there is no use for an optimization. The data from Mönsterås mill (Södra) was received too late to be used in this model.

Since the data comes from a running production and not from any pilot plant, there was no possibility to for example change the level for maximum pressure to see what happens. Therefore there is no information about when the filter cake might collapse or what happens with a pressure drop over 2,7 bars.

5.1 Measurements The pressure difference is measured as the difference between a point inside the filter vessel 55 cm from the bottom and a point in the horizontal pipe that leads the filtrate to the tank for filtrated green liquor. Assuming that the liquid never forms a pillar inside the tubes throughout the filtration cycle, this pressure difference can be considered the same as the one over the filter at the measuring point. To find the pressure difference for all other points along the filter tube, the pressure from the liquid pillar is either added or removed.

The measurement of the level is also a measurement of pressure difference. One measuring point is located 67 cm from the top of the vessel and the other 40 cm from the bottom. Due to calibration, the measured level never exceeds 97,5 %. It is assumed that at this level, the filter vessel is completely filled with liquor. The operators in Moss mill have confirmed this assumption.

The flow of green liquor into the vessel is measured after the pump from the equalization tank. During back flushing as well as when the filter is drained, the pump continues to work. There is a system for recirculation so that the green liquor is pumped back to the tank instead of to the filter vessel.

In the period 24-30 of April, the data was collected with a time interval of 15 s. and in the period 1-23 of February, the time interval was 30 s.

0,55 m 0,40 m

0,67 m

PDI 98

LI 132

Figure 5.1. The meters’ placing in the cassette filter


Method

5.2 Description of the identification algorithm The whole calculation was made to estimate the values of the specific resistance in the cake, the compressibility index and the resistance in the filter cloth. Rewriting the filter equation (21), it is possible to pick out three parameters to be calculated.

RARAdt

dVpncV

pAdtdV

A n)1)(1(

1

0

That is μ 0c, n and Rμ.The algorithm begins with guessing values of the parameters. During the first part of the filtration cycle, the filling of the filter vessel, the level is calculated for each time unit. The calculation is based on the filter equation without the cake (5).

Rp

dtdV

A1

The flow of filtrate at a level L corresponding to a height h can be written as

RghNCdh

Rp

AdtdV dh

dhdh

Where C is the circumference of the tubes, N is the number of tubes and dh is the height of the small element around h.Summing up all these filtrate flows gives the total flow at the time k.

RgNCdhh

RghNCdh

dtdVF

i

nn

ni

nkkout

00, )()(

Now the level can be calculated at each time k by comparing the flow in with the flow out.

tottioncross

k

nnout

k

nnink hA

tFFLsec

int

0,

0,

100)(

Where tint is the time interval, htot is the total height of the tubes, Across section is the cross section area of the filter vessel minus the cross section areas of the filter tubes and it is multiplied with 100 to get the level in percent. This procedure is repeated until the measured value of the level has reached a set value (ex. 90 %). Because of the calibration error of the level measurement, it is difficult to know when the filter vessel is full. About 1-2 minutes after reaching the set value, the filling is considered finished and it is assumed that the filter vessel is totally filled with green liquor. Now the pressure drop is calculated for each time unit by using the filter equation for compressible cake filtration. This time it is no longer possible to assume that the filtrate flow depends only on the pressure. Both the pressure and the thickness of the filter cake will vary over the length of the filter tubes and will influence the filtrate flow. First the pressure difference is calculated for every centimetre of the tube length, it is larger below the measuring point and smaller above the measuring point.


Method

mpimpi pghhp )(

pi is the pressure drop at a point i along the tube, hi is the height at point i, hmp is the height at measuring point and pmp is the pressure drop at the measuring point. The superficial velocity is first calculated to be put into the filter equation. The pressure drop from the filter cloth is neglected in this first step.

RApncVApu

dhniidh

dhii )1(,0

Then this velocity is used to estimate the flow of filtrate at point i.

RARupncVAp

dtdV

dhn

iiidh

dhi

idh ))(1(,0

2

,

The flow of filtrate is summoned up to get the total flow. Since the filter vessel is full, the same amount of liquor going in must go out.

kinidhi

kout FdtdVF ,

,,

From this relation, the pressure difference at the time k can be calculated. This procedure is repeated for each time unit throughout the cycle. Finally, the calculated level from the beginning of the cycle is being compared with the measured data and the calculated pressure drop is compared with the measured pressure drop. The squared errors from each time unit are summoned up. Then the starting guess of the parameters is changed and the whole calculation repeated to minimize the size of the sum of errors.


Model

6 Model

6.1 Data reliability - identifiability When trying to calculate the three parameters Rμ, 0μc and n according to the calculation algorithm described in the previous chapter, the MATLAB function fminsearch did not find the parameters within the set tolerance frame that minimizes the error function. To be able to get any values at all, a limit for the number of iterations was set. The prediction seemed to be quite accurate even without finding the minimum. It was first when the number of iterations was increased to 500 that it became clear that 0μc and n was depending on each other. Rμ did not change when the number of iterations was increased. Knowing that the compressibility index and the specific resistance were not identifiable from each other, the identifiability between the filter resistance (Rμ)and the new parameter ( ) had to be examined. being the value of 0μc when nis fixed at 0,8. Since the viscosity is a part of both expressions, the two parameters would have to be somehow related. First, was fixed (at 3,0*105) to see if Rμ alone could make as good a prediction as the two parameters together. It turned out that the value of Rμ could compensate for very well in some cycles but the prediction got much worse in others. The average sum of errors was about three times as big when only Rμ was varied. Plots of Rμ with both μ 0c and n fixed and of the errors can be found in appendix 4.1. There are also some examples of different cycles and how the prediction differs if both Rμ and or only Rμ is varied in that appendix. The next way to check for identifiability is to see if the function VN(Rμ, )(equation 24) is strictly convex. If it is, then there must be only one minimum and also only one solution to the minimization problem. Figure 6.1 and 6.2 show the sum of errors as a function of Rμ and for one filtration cycle. It is likely that this function is convex but there is a large area with very small values of the sum of errors that indicates that the values of Rμ and could be changed and still give a good prediction of the error.

Figure 6.1. The graph shows the area function VN(Rμ, ).


Model

Figure 6.2. The figure shows the same function as in figure 6.1 but the sum of errors are projected in the x-y plane represented in different colours. The dark blue area shows the minimum.

To examine this further the value of Rμ and was calculated for one cycle. Then was fixed at 75 % of its value and Rμ was varied to find the new minimum. The

process was then repeated for plus 25 % and for the case where Rμ was fixed and varied. Figures 6.3-6.5 shows some of the results. For the first cycle (nr 1, 1st Feb), the errors was between 5-30 times bigger for the predictions where one parameter was fixed. For the second one (nr 1, 5th Feb), the error was between 4-120 times bigger. When was fixed at 1,25 times the calculated value, the prediction was good without variation in , see figures 6.6-6.8. Important to notice when comparing these two cycles is that the total pressure is much lower in the second cycle. All results from this test are found in appendix 4.2. For the whole test, the prediction started after 10 minutes to be sure that the cake had been built up over the whole tube. A corresponding test, with earlier start time was also done showing similar results. The data from this test can also be found in appendix 4.2.

Figure 6.3. The green line shows the measured data and the blue line shows the calculated prediction. Here both parameters are varied.


Model

Figure 6.4. The green line shows the measured data and the blue line shows the calculated prediction. Here is fixed at 0,75 times the calculated value (from the prediction in figure 6.3). First cycle 1st February.

Figure 6.5. The green line shows the measured data and the blue line shows the calculated prediction. Here is fixed at 1,25 times the calculated value (from the prediction in figure 6.3). First cycle 1st February.

Figure 6.6. The green line shows the measured data and the blue line shows the calculated prediction. Here both parameters are varied. First cycle 5th February. The step at 35 minutes is due to a change in the feed.


Model

Figure 6.7. The green line shows the measured data and the blue line shows the calculated prediction. Here is fixed at 0,75 times the calculated value (from the prediction in figure 6.6). First cycle 5th February. The step at 35 minutes is due to a change in the feed.

Figure 6.8. The green line shows the measured data and the blue line shows the calculated prediction. Here is fixed at 1,25 times the calculated value (from the prediction in figure 6.6). First cycle 5th February. The step at 35 minutes is due to a change in the feed.

To try to see if Rμ and depend with some unknown relation, they were plotted against each other in figure 6.9.

400000000

500000000

600000000

700000000

800000000

900000000

1000000000

1100000000

1200000000

1300000000

1400000000

0 100000 200000 300000 400000 500000 600000 700000 800000 900000

R

Figure 6.9. Rμ plotted against . Model data from 1-23 Feb.

If the correct calculation of Rμ and was possible, this would provide a useful tool in analysing the filtration problem. A high value of Rμ would indicate a problem with the filter cloth or the back flushing. This could then be solved by cleaning the filter socks or improving the back flush system. A high value of would indicate a problem with the green liquor and addition of lime mud would then help solving the problem. The back flush system was not working well in


Model

February in Moss mill and high values of Rμ were expected. Unfortunately the calculated data from February showed that even if Rμ had increased slightly, had increased more. This also indicates that the data is not very reliable. Appendix 3.1 shows plots of the calculated data. The plots show that the parameters, especially Rμ, did not vary as much in April than it did in February. The errors are evenly distributed over both periods. In February the pressure drop reaches the maximum level more often than in April but it does not seem to be any other period where low pressure drop gives more even parameters. On the contrary, the test on the previous page shows that it is likely that the problems with identifiability are larger when the pressure drop is low. It is interesting to see how much the viscosity contributes to the problems with identifiability. To do that, the viscosity (μ) was picked out as a parameter as well as the specific filter resistance together with the concentration ( 0c) and the resistance in the filter cloth (R) from the filter equation, still having the fourth parameter, the compressibility index n, fixed. Separating the concentration would not be necessary since it is obvious that c and 0 not are identifiable. It is not possible to plot the sum of errors against three parameters but looking at the distribution of data it might still be possible to guess whether the parameters are really identifiable or not. The parameters are plotted in appendix 4.2 together with the errors. In this calculation, n was fixed at a value of 0,8. None of the plotted trends are very consistent and this indicates that the values are not very trustworthy.To see if the values of the calculated parameters were even close to any real physical meaning, a rough estimate was done. A normal value of 0 is about 108

[3] the approximate size of the viscosity of water is 10-3 and the concentration of solids in the green liquor is about 101. This means that 0μc should be about 106.In this model, the values of 0μc are about 105.The conclusion is that Rμ and are partly identifiable, i.e. they are identifiable in some cycles and in other they are not. The reason for this is probably that the model is too different from reality to detect the differences between the parameters. The model seem to be more inaccurate in some cycles and better in other, this might be because some of the errors, for example the unreliable filter area, probably is bigger in some cycles than in others.


Model

6.2 Importance of the parametersA simulation experiment was done to evaluate the importance of the two different parameters. The efficiency was tested and then compared to the normal values, also the time when the pressure drop reached the maximum value (2,7 bar) was noted.

Normal: : 3*105 Feed: 80 m3/h R : 7,4*108 Cycle time: 30 min

EfficiencyR Efficiency Efficiency normal

Reaches max pressure drop

Normal Normal 1,0515 100,0% 29 min Normal*1,25 Normal 1,0393 98,8% 24 min Normal normal*1,25 1,0144 96,5% 17,5 min Normal*1,25 normal*1,25 0,9866 93,8% 15 min Normal*0,75 Normal 1,0518 100,0% -Normal normal*0,75 1,0563 100,5% -Normal*0,75 normal*0,75 1,0563 100,5% -

Table 6.1. Importance of R compared to .

It seems like both R and have an impact on the efficiency of the filter even though R is more important and changes the efficiency more. To see what the two parameters contribute with in the prediction, a simulation was done. The feed was set to 80 m3/h and the cycle time was 30 minutes. With the average parameter value (R = 8*108, = 2,9*105) this gives a pressure difference that reaches the maximum value after 19 min. R was then changed to 6,4*108 and was changed to try to compensate for the change in R . The result is show in figure 6.10.

Figure 6.10. The dark blue line shows the attempt to fit the light blue line. The horizontal dotted line is the maximum pressure drop.

R is the parameter that decides where the curve should start bending off and sets the slope of the curve.


Model

6.3 Sources of error There are two different types of errors that will affect this model, first the errors that comes from insufficient measurements and then are the noise in the process that the model do not account for.

6.3.1 Measurement errors The measurement of the level in the filter vessel is incorrect; when the vessel is full it measures about 97 %. This is probably due to bad calibration and the result is that the used data gets inaccurate, especially in the last phase of filling.

The data received from the feed pump during the start of each cycle may be incorrect since there is no information as to when the feed stops the recirculation and when it starts to fill the filter vessel.

The measurement of the pressure difference might be incorrect since it compares the pressure in the filter vessel to the pressure in the tubes for outgoing filtrate. If this tube is filled with fluid, the measurement will not show the pressure difference over the filter cake.

There is no information whether the measurements of the feed and the pressure difference are well calibrated or not.

6.3.2 Model errors Cake bridges might have blocked the filter tubes during all of or parts of the measuring period so the real filter area is unknown. There is also a risk that the bridges appeared during one cycle and then was washed away in the back flushing. This is probably one of the biggest reasons of the inaccuracy of the model. The filter area is a very important parameter in the filter equation.

In the model the building of a filter cake during the filling of the filter vessel is neglected.

In the model it is presumed that the cake is equally distributed over the tube length after the filling is finished.

There is no information about possible unevenness in the filter cake that can cause model prediction difficulties.


Optimization

7 Optimization The aim for this project was to develop a control model to optimize the cassette filter towards increased capacity, in particular when the filter has very low capacity. To be able to do this, there are two demands that must be fulfilled. The first one is that the relation between the pressure drop and the filter parameters must be know, i.e. the model must be reliable, and the second is that optimization should be a significant positive change to the current situation. Since the model parameters turned out to be very unreliable because of identifiability problems, the optimization based on a model of physical relations seems to fail the first demand. If the model was made like a “black-box” model however, the same problems would probably occur because it would be hard to distinguish the difference between problems with the filter cloth and problems with the green liquor. The proper actions for optimizing the capacity will depend on which type of problem there is. There are suggestions to how to reduce the compressibility and thus improve the filterability. In Moss mill there is a possibility to add a small amount of lime mud to the feed to increase the porosity of the filter cake. This method is used when the capacity of the filter is low but the results are not fully evaluated.

7.1 Optimization experiment If the parameters had been truly identifiable, then the optimization probably would have worked. The question is then: is it possible to get a significant increase in capacity and make it worth the effort? The possible change in capacity was examined by changing the values of Rμ and 0μc, calculating the optimal flow or cycle time and comparing the efficiency with the one using the recommended cycle time and normal flow. The outcome is shown in table 7.1.


Optimization

Constant feed+ 0 -

8,1 2,5 0,85 *105 Feed: 80 m3/hRμ 12,6 7,8 1,6 *108 Max pressure drop: 2,7 bar

Optimal time Efficiency

Maxfiltrate

Efficiencywith 30 min

Percent gained capacity

Rμ# 1 + + 33 0,7196 0,1005 0,7184 0,17 % # 2 + 0 28 0,8822 0,1066 0,8813 0,10 % # 3 + - 31 1,0151 0,1354 1,0140 0,11 % # 4 0 + 53 0,8802 0,1888 0,8444 4,24 % # 5 0 0 51 1,0973 0,1988 1,0317 6,36 % Limited # 6 0 - 39 1,1025 0,1972 1,0432 5,68 % Limited # 7 - + 50 0,9690 0,1986 0,8967 8,06 % Limited # 8 - 0 42 1,1087 0,1969 1,0317 7,46 % Limited # 9 - - 35 1,0794 0,1964 1,0432 3,47 % Limited

Constant cycle time Cycle time: 30 min Max pressure drop: 2,7 bar

Optimal feed Efficiency

Maxfiltrate

Efficiencywith 30 min

Percent gained capacity

Rμ# 1 + + 95 0,7239 0,0932 0,7184 0,77 %

# 2 + 0 120 0,9539 0,1182 0,8813 8,24 % # 3 + - 150 1,2390 0,1521 1,0140 22,19 % Limited*# 4 0 + 75 0,8446 0,1098 0,8444 0,02 % # 5 0 0 119 1,2535 0,1582 1,0317 21,50 % # 6 0 - 113 1,4838 0,1989 1,0432 42,24 % Limited # 7 - + 76 0,8971 0,1172 0,8967 0,04 % # 8 - 0 119 1,4280 0,1822 1,0317 38,41 % # 9 - - 102 1,3346 0,1996 1,0432 27,93 % Limitation: Maximal size of element of filtrate volume (Max filtrate): 0,20 * maximal feed: 150 m3/h

Table 7.1. Optimization experiment with and Rμ. First the feed was varied and then the cycle time.

It is known that the cakes will be harder to wash off if the flow is too high or the time is too long (see more in chapter 3.2). It was therefore decided that for the element of the tube where the largest amount of filtrate passes must not be greater than 0,20. This value is from a normal feed (80 m3/h) with a cycle time of 40 min and normal values of and Rμ. Of course this limitation puts hard constrains on the optimization. It is important to consider that the feed in table 7.1 automatically decreases as soon as the maximal pressure drop is reached. The feed is then controlled by the pressure, keeping it at the maximum level. From table 7.1 the conclusion can be drawn that it seems to be best to control the feed instead of the cycle time but also that the better conditions, the larger possibility to optimize. Unfortunately, that kind of optimization is not useful since the cassette filter is only one process step in the pulp mill.


Results and discussion

8 Results and discussion Because of the problems with identifiability, the results from this project are very uncertain and must be treated with much scepticism. The goal of this project was not reached. This was mainly because it turned out that the filtration data was not sufficient to calculate the parameters wanted. I.e. there is too much noise and too many unknown parameters to create a good enough model. The two parameters that describe the green liquor, 0 and n, are not identifiable from each other in the model based on the filter equation for compressible cake filtration. There are also identification problems between the combination of 0and n ( ) and the parameter describing the resistance in the filter cloth (Rμ). The problems are greater in some cycles and smaller in other. Tests show that Rμ has a larger influence on the behaviour of the filter than .Large values of both parameters decreases the efficiency and maximum pressure drop is reached earlier with large values of Rμ and . Small values of both parameters increases the efficiency and maximum pressure drop is reached later with small values of Rμ and . Simulation tests show that there are differences in the influence of Rμ and . This means that the parameters cannot be totally unidentifiable.When trying to minimize VN(R, 0c,μ) a minimum was found but the same problems with partly unidentifiable parameters remained. A rough estimate indicates that the specific resistance in the cake has approximately the right magnitude. This means that the model probably is built on the correct principles and the problem is lack of accuracy and noise. When there are problems with identifiability the solution is to cut down on the number of variables. In this case however, this causes two problems; first the prediction is not as good any more and second the physical meaning of the parameters is lost. This also means that the use of the parameter in the control of the filter is lost since it does no longer tell anything else than that the filter is running good or bad. To be able to successfully calculate the parameters in this model, other types of measurement could have been helpful. If the pressure drop was measured over the cake and filter cloth, preferably in several places along the tubes, this would have given a better value of the pressure drop. If there also was a possibility to know whether the green liquor was easily filtrated or not and whether the filter cloth had high or low resistance towards flow, then it would have been easier to evaluate the reliability of the parameters. If the process had been available for changes, then it would have been easier to analyze the cycles because they would have been more consistent, for example same feed and cycle time for longer periods. Another way of dealing with the problem with identifiability would have been to create a black-box model. In this model the physical meaning of the parameters would have been lost. If it at the same time was possible to get information about the green liquor and the resistance in the filter cloth, then the parameters probably could have been related to these conditions.


Conclusions

9 Conclusions The parameters that describe the pressure difference and the level in the filter vessel cannot be calculated simply using production data. It is not even possible to find two reliable variables that describe the green liquor and the filter cloth resistance respectively.

There is more optimization potential when the cassette filter is running without problems, i.e. when the filtration properties are favourable.

Because of the low optimization potential, other optimization tools should be considered, for example adding lime mud or aluminium.

The feed is a more efficient control tool for optimization than the cycle time. To improve the over all capacity, the feed should be increased so that the maximum pressure difference is reached faster.

The efficiency of the filter depends more of the filter cloth conditions than of the filterability of the green liquor.

Future work in this area should focus on improvements of the filter cloth. A filter cloth that is thinner and has smaller resistance towards flow would be desirable, although the cloth must be strong enough to bear the back-flush.


References

References[1] Biermann, Christopher J. Handbook of PULPING AND PAPERMAKING, 2nd edition

Academic Press, 1996

[2] Coulson, J. M. and Richardson, J.F. with Backhurst J.R. and Harker, J.H. Coulson & Richardson’s Chemical Engineering Volume 2, Particle Technology and Separation Processes, 4th edition

Butterworth-Heinemann, 1996

[3] Dusic, Sandra Some aspects of the filtration properties of green liquor sludge Chalmers University of Technology, Master Thesis, 1997 Department of Chemical Engineering Design

[4] Headley, Richard L. New Liquor Filtration Options Provide Production and Quality Improvements

Pulp & Paper, p. 69-71, nr 13, Dec. 1996

[5] Kassberg, Mats and Pettersson, Bertil VITLUTBEREDNING Yrkesbok Y-214 Skogsindustrins Utbildning i Markaryd, 1997

[6] Ljung, Lennart and Glad Torkel Modellbygge och simulering Studentlitteratur, 2004

[7] Montanhese, M., Vilar de Carvalho, M. M. and Puig, F. P. Green liquor filtration at Aracruz Celulose Mill: Causticizing and effluent close-up, pilot plant and practical results Pulp & Paper Canada, p. 39-41, vol. 102:1, 2001

[8] Sedin, Peter and Theliander, Hans Filtration properties of green liquor sludge Nordic Pulp and Paper Research Journal, v. 19, nr 1, p. 67-74, 2004

[9] Vennerberg, Karl and Vennerberg, Nils-Gösta Skogens kemi, om massa, trä och fibrer

Liber AB, 2004

[10] Wakeman, R. J. and Tarleton, E. S. FILTRATION Equipment selection, modelling and process simulation, 1st

editionElsevier Science Ltd., 1999


APPENDIX

List of appendices APPENDIX 1 OPERATION DATA FROM MOSS MILL, 24-30TH APRIL......................... A.3

A.1.1 Level............................................................................................................................... A.3A.1.2 Pressure drop .................................................................................................................. A.5A.1.3 Feed ................................................................................................................................ A.7

APPENDIX 2 OPERATION DATA FROM MOSS MILL, 1-23RD FEBRUARY................. A.9

A.2.1 Level............................................................................................................................... A.9A.2.2 Pressure drop ................................................................................................................ A.14A.2.3 Feed .............................................................................................................................. A.19

APPENDIX 3 CALCULATED PARAMETER DATA ......................................................... A.24

A.3.1 Errors, highest pressure drop, Rμ and 0μc with n fixed to 0,8.................................... A.24A.3.2 Errors, R, 0c and μ with n fixed to 0,8 ........................................................................ A.25

APPENDIX 4 IDENTIFIABILITY TESTS............................................................................ A.26

A.4.1 Rμ with 0c and n fixed. ............................................................................................... A.26A.4.1.1 Examples of different cycles ...................................................................................A.26A.4.1.2 Data plots, Rμ and errors. .....................................................................................A.28

A.4.2 Identifiability tests ........................................................................................................ A.29A.4.2.1 Test with prediction started after ~10 minutes, no 1, 1st February ........................A.29A.4.2.2 Test with prediction started after ~10 minutes, no 1, 5th February ........................A.30A.4.2.3 Test with prediction started after ~5 minutes, no 1, 1st February ..........................A.31A.4.2.4 Test with prediction started after ~5 minutes, no 1, 5th February ..........................A.32

Control model for compressible cake filtration of green liquor in cassette filter A.2

Appendix 1

Appendix 1 Operation data from Moss mill, 24-30th April

A.1.1 Level

0

20

40

60

80

100

120

1 1001 2001 3001 4001 5001

Time units (15 s)

Leve

l (%)

Figure A.1.1.1. Level 24th April

0

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1 1001 2001 3001 4001 5001

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1 1001 2001 3001 4001 5001

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Appendix 1

0

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1 1001 2001 3001 4001 5001

Time units (15 s)

Leve

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1 1001 2001 3001 4001

Time units (15 s)

Leve

l (%)


0

20

40

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1 1001 2001

Time units (15 s)

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Appendix 1

A.1.2 Pressure drop

0

0,5

1

1,5

2

2,5

3

1 1001 2001 3001 4001 5001

Time units (15 s)

Pres

sure drop (bar)

Figure A.1.2.1. Pressure drop 24th April

0

0,5

1

1,5

2

2,5

3

1 1001 2001 3001 4001 5001

Time units (15 s)

Press

ure drop

(bar)


0

0,5

1

1,5

2

2,5

1 1001 2001 3001 4001 5001

Time units (15 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

1 1001 2001 3001 4001 5001

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Pres

sure drop (bar)



Appendix 1

0

0,5

1

1,5

2

2,5

1 1001 2001 3001 4001 5001

Time units (15 s)

Press

ure drop

(bar)


0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

1 1001 2001 3001 4001

Time units (15 s)

Pres

sure drop (bar)


0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

1 1001 2001

Time units (15 s)

Pres

sure drop (bar)



Appendix 1

A.1.3 Feed

0

20

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1 1001 2001 3001 4001 5001

Ti me uni t s ( 15 s)

Figure A.1.3.1. Feed 24th April

0

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1 1001 2001 3001 4001 5001

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(m3/h)


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(m3/h)


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Feed

(m3/h)



Appendix 1

0

10

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1 1001 2001 3001 4001 5001

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Feed

(m3/h)


0

10

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1 1001 2001 3001 4001

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Feed

(m3/h)


0

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1 1001 2001

Time units (15 s)

Feed

(m3/h)



Appendix 2

Appendix 2 Operation data from Moss mill, 1-23rd February

A.2.1 Level

0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Leve

l (%)

Figure A.2.1.1. Level 1st February

0

20

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1 501 1001 1501 2001 2501

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Leve

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Figure A.2.1.2. Level 2nd February

0

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1 501 1001 1501 2001 2501

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Figure A.2.1.3. Level 3rd February

0

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1 501 1001 1501 2001 2501

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Figure A.2.1.4. Level 4th February


Appendix 2

0

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1 501 1001 1501 2001 2501

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Leve

l (%)


0

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1 501 1001 1501 2001

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Leve

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1 501 1001 1501 2001

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Leve

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Appendix 2

0

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120

1 501 1001 1501 2001 2501

Time units (30 s)

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0

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l (%)


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Appendix 2

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Appendix 2

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Figure A.2.1.21. Level 21st February

0

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120

1 501 1001 1501 2001 2501

Time units (30 s)

Leve

l (%)

Figure A.2.1.22. Level 22nd February

0

20

40

60

80

100

120

1 501 1001 1501

Time units (30 s)

Leve

l (%)

Figure A.2.1.23. Level 23rd February


Appendix 2

A.2.2 Pressure drop

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)

Figure A.2.2.1. Pressure drop 1st February

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)

Figure A.2.2.2. Pressure drop 2nd February

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)

Figure A.2.2.3. Pressure drop 3rd February

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)

Figure A.2.2.4. Pressure drop 4th February


Appendix 2

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001

Time units (30 s)

Pres

sure drop (bar)



Appendix 2

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)



Appendix 2

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001

Time units (30 s)

Pres

sure drop (bar)



Appendix 2

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)


0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)

Figure A.2.2.21. Pressure drop 21st February

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501 2001 2501

Time units (30 s)

Pres

sure drop (bar)

Figure A.2.2.22. Pressure drop 22nd February

0

0,5

1

1,5

2

2,5

3

1 501 1001 1501

Time units (30 s)

Pressu

re drop (bar)

Figure A.2.2.23. Pressure drop 23rd February


Appendix 2

A.2.3 Feed

0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)

Figure A.2.3.1. Feed 1st February

0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)

Figure A.2.3.2. Feed 2nd February

0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)

Figure A.2.3.3. Feed 3rd February

0

20

40

60

80

100

120

140

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)

Figure A.2.3.4. Feed 4th February


Appendix 2

0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

160

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

1 501 1001 1501 2001

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

160

1 501 1001 1501 2001

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

1 501 1001 1501 2001

Time units (30 s)

Feed

(m3/h)



Appendix 2

0

20

40

60

80

100

120

140

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

160

1 501 1001 1501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

160

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)



Appendix 2

0

10

20

30

40

50

60

70

80

90

100

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

10

20

30

40

50

60

70

80

90

100

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

1 501 1001 1501 2001

Time units (30 s)

Feed

(m3/h)



Appendix 2


0

20

40

60

80

100

120

140

160

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)


0

20

40

60

80

100

120

140

160

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)

Figure A.2.3.21. Feed 21st February

0

20

40

60

80

100

120

1 501 1001 1501 2001 2501

Time units (30 s)

Feed

(m3/h)

Figure A.2.3.22. Feed 22nd February

0

20

40

60

80

100

120

140

1 501 1001 1501

Time units (30 s)

Feed

(m3/h)

Figure A.2.3.23. Feed 23rd February

Appendix 3


Appendix 3 Calculated parameter data

A.3.1 Errors, highest pressure drop, Rµ and 0µc with n fixed to 0,8

0

1000

00

2000

00

3000

00

4000

00

5000

00

6000

00

7000

00

8000

00

9000

00

R

0

0000

00

0000

00

0000

00

0000

00

0000

00

0000

00

0000

00

0000

00

200

400

600

800

1000

1200

1400

1600

Erro

r

%%%%%%%%%%%

0,00

0

0,10

0

0,20

0

0,30

0

0,40

0

0,50

0

0,60

0

0,70

0

0,80

0

0,90

0

1,00

0

High

est p

ress

ure

drop

per

cyc

le

0,00

0

0,50

0

1,00

0

1,50

0

2,00

0

2,50

0

3,00

0

Fig

ure

A.3

.1.1

. To

the

left

of th

e lin

e is

dat

a fr

om F

ebru

ary

and

to th

e rig

ht is

dat

a fr

om A

pril.

Appendix 3

A.3.2 Errors, R, 0c and µ with n fixed to 0,8

R

0

+11

+11

+11

+11

+12

+12

+12

2E4E6E8E1E

1,2E

1,4E

*103

0

0,2

0,4

0,6

0,81

1,2

1,4

1,6

1,82

Erro

r (%

)

00%

00%

00%

00%

00%

00%

00%

00%

00%

00%

00%

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0c

0

1000

0000

0

2000

0000

0

3000

0000

0

4000

0000

0

5000

0000

0

6000

0000

0

7000

0000

0

8000

0000

0

9000

0000

0

Fig

ure

A.3

.2.1

. Onl

y fo

r the

per

iod

1-23

Feb

ruar

y


Appendix 4

Appendix 4 Identifiability tests

A.4.1 Rµ with 0c and n fixed.

A.4.1.1 Examples of different cycles

Figure A.4.1.1. The blue line shows the predicted data and the green line shows the calculated data. The left graph shows the prediction where is fixed and the right graph shows the prediction where both Rμ and are varied. In this case it is obvious that Rμ and are identifiable since the prediction can not be made with Rμ alone.


Appendix 4

Figure A.4.1.2. The blue line shows the predicted data and the green line shows the calculated data. The left graph shows the prediction where is fixed and the right graph shows the prediction where both Rμ and are varied. In this case the prediction seems to be quite successful even without which means that the parameters are not fully identifiable or the fixed value of is very close to the calculated one.


Appendix 4

A.4.1.2 Data plots, Rμ and errors.

R

0

2000

0000

0

4000

0000

0

6000

0000

0

8000

0000

0

1000

0000

00

1200

0000

00

1400

0000

00

Erro

rs w

ith o

nly

R v

arie

d

%%%%%%%%

0,00

0

2,00

0

4,00

0

6,00

0

8,00

0

10,0

00

12,0

00

14,0

00

Erro

rs w

ith b

oth

R a

nd

var

ied

0%0%0%0%0%0%0%0%0%0%

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

Fig

ure

A.4

.1.3

. Onl

y fo

r the

per

iod

1-23

Feb

ruar

y. T

he e

rror

s are

big

ger w

hen

is fi

xed,

indi

catin

g th

at th

e pa

ram

eter

s are

iden

tifia

ble.


Appendix 4

A.4.2 Identifiability tests

A.4.2.1 Test with prediction started after ~10 minutes, no 1, 1st

February

(*10-5) Rμ (*10-8) error (%) Nocalculated 4,31 9,75 0,056% 1

*0,75 3,23 10,60 0,255% 2*1,25 5,39 8,62 0,470% 3

Rμ*0,75 5,89 7,31 1,590% 4Rμ*1,25 1,36 12,18 1,160% 5

Table A.4.2.1. A small value of is compensated with a large value of Rμ and vice versa. The bold figures are fixed and the numbers refer to the graphs in figure A.4.2.1.

1

2 3

54

Figure A.4.2.1. The blue line shows the predicted data and the green line shows the calculated data. 25 % change in Rμ makes the prediction worse than 25 % change in .


Appendix 4

A.4.2.2 Test with prediction started after ~10 minutes, no 1, 5th

February

(*10-5) Rμ (*10-8) error (%) No calculated 2,14 5,84 0,014% 1

*0,75 1,61 6,06 0,054% 2*1,25 2,68 5,58 0,078% 3

Rμ*0,75 3,85 4,38 1,661% 4Rμ*1,25 0 7,30 2,150% 5


1

2 3

4 5



Appendix 4

A.4.2.3 Test with prediction started after ~5 minutes, no 1, 1st

February


*0,75 3,60 10,02 0,785% 2*1,25 6,00 7,84 1,298% 3

Rμ*0,75 6,13 6,83 2,305% 4Rμ*1,25 2,60 11,39 1,908% 5


1

2 3

54



Appendix 4

A.4.2.4 Test with prediction started after ~5 minutes, no 1, 5th

February


*0,75 1,74 5,94 0,101% 2*1,25 2,90 5,45 0,143% 3

Rμ*0,75 3,93 4,30 2,098% 4Rμ*1,25 - 7,16 -

Table A.4.2.4. A small value of is compensated with a large value of Rμ and vice versa. The bold figures are fixed and the numbers refer to the graphs below. With Rμ = 7,16, no real value of a was found.

21

3 4



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