Modelling the Moisture Content of Multi-Ply Paperboard in the Paper Machine Drying Section

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Modelling the Moisture Content of Multi-Ply Paperboard in the Paper Machine Drying Section

CHRISTELLE GAILLEMARD

Licentiate Thesis Stockholm, Sweden 2006

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TRITA-MAT-06-OS-01 ISSN 1401-2294 ISBN 91-7178-302-4

KTH SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan fram- lägges till offentlig granskning för avläggande av Licentiatexamen fredagen den 7 april 2006 klockan 10.00 i rum 3721, plan 7, Lindstedsvägen 25, Kungl Tekniska Högskolan, Stockholm.

c

° Christelle Gaillemard, April 2006 Tryck: Universitetsservice US AB

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To Pär-Anders

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Abstract

This thesis presents a grey-box model of the temperature and moisture content for each layer of the multi-ply paperboard inside the drying section of a paper mill. The distribution of the moisture inside the board is an important variable for the board quality, but is un- fortunately not measured on-line. The main goal of this work is a model that predicts the moisture evolution during the drying, to be used by operators and process engineers as an estimation of the unmeasurable variables inside the drying section.

Drying of carton board is a complex and nonlinear process. The physical phenomena are not entirely understood and the drying depends on a number of unknown parameters and unmodelled or unmeasurable features. The grey-box modelling approach, which consists in using the available measurements to estimate the unknown disturbances, is therefore a suitable approach for modelling the drying section.

A major problem encountered with the modelling of the drying section is the lack of measurements to validate the model. Consequently, the correctness and uniqueness of the estimated variables and parameters are not guaranteed. We therefore carry out observability and identifiability analyses and the results suggest that the selected model structure is observable and identifiable under the assumption that specific measurements are available.

Based on this analysis, static measurements in the drying section are carried out to identify the parameters of the model. The parameters are identified using one data set and the results are validated with other data sets.

We finally simulate the model dynamics to investigate if predicting the final board properties on-line is feasible. Since only the final board temperature and moisture content are measured on-line, the variables and parameters are neither observable nor identifiable. We therefore regard the predictions as an approximation of the estimated variables. The semi- physical model is complemented with a nonlinear Kalman filter to estimate the unmeasured inputs and the unmodelled disturbances. Data simulations show a good prediction of the final board temperature and moisture content at the end of the drying section. The model could therefore possibly be used by operators and process engineers as an indicator of the board temperature and moisture inside the drying section.

Keywords: Drying section modelling, multi-ply paperboard, moisture content, identifica- tion, grey-box modelling

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Acknowledgements

I have been fortunate to interact with remarkable people that have contributed in many ways to the completion of this thesis.

First of all, I would like to thank my advisor Per-Olof Gutman, for introducing me to the project after my master thesis. His insightful suggestions and his support despite the geographical distance were a source of inspiration and motivation.

These years have been challenging but I now feel it was worth going through all the struggles.

I am very grateful to my advisor Anders Lindquist for the opportunity to join the Optimization and System Theory group. His enthusiasm for research provides a creative and friendly working environment.

I also wish to acknowledge AssiDomän Frövi for financing the project. I am grateful to Bengt Nilsson for the opportunity to join the welcoming Process Con- trol group. I also thank my master-thesis advisors, Stefan Ericsson and Lars Jon- hed, for sharing their knowledge about the drying section, the paper machine and the simulation tool Dymola. Lars and his family deserve special thanks for their kindness and sincere concern.

My free time in Frövi would have been lonelier without my friend Dorothée Millon whom I thank for all the time spent together and for her hospitality.

In AssiDomän Frövi, I would like to thank Antero Jauhiainen for helping me with the measurements, and Kent Åkerberg, Gunnar Pålsson and Anders Hen- riksson for sharing their knowledge about the drying section. I am also thankful to Magnus Karlsson for answering my questions about the physical model, Jens Pettersson for helping me with the IPOPT interface and Jenny Ekvall for the discussions about paper machine drying sections.

The faculty members and graduate students at the division of Optimization and System Theory at KTH have contributed to making these years of study stim- ulating and enjoyable. I am very grateful to my colleague Gianantonio Bortolin for sharing his experience of grey-box modelling. His coaching and friendship were a great source of motivation. I can not thank enough my roommate Vanna Fanizza for her support and all those discussions, sometimes work related. Her spontane- ity and our true friendship always made me happy to go to the office. I also want to thank Ryozo Nagamune for his kindness and precious advises.

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My love and gratitude go to my parents, Gérard and Bernadette, and my brother Flavien. They have indirectly contributed to this work with their constant support and encouragement. My Swedish family also deserves to be mentioned for their warm welcome that made me feel home in a new country. I deeply thank all my relatives and friends, for providing me escapes from work. All the time spent together gave me kicks of energy that I needed to continue.

Finally, I thank my future husband Pär-Anders for his love, patience and support during the completion of the thesis. He helped me believe I could do it!

Train Stockholm - Nyköping, February 2006 Christelle Gaillemard

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Nomenclature

Some notations may have a different meaning locally.

Symbol Description Dimension

vx Speed of the machine m/s

u Moisture content kgw/kgdry

t Time index s

T Temperature ^◦C or K

G Basis weight kg/m²

W Flow of paper kg/s

l Paper width m

ρ Density kg/m³

d Thickness m

h Heat transfer coefficient W/m^{2 ◦}C

Dab Diffusion coefficient of m²/s

element a into element b

FRF Fabric heat reduction factor %

Cp Specific capacity J/kg^◦C

k Thermal conductivity W/m^◦C

P Absolute pressure Pa

p Partial pressure of water Pa

˙

m Evaporation rate kg/m²s

λ Heat of evaporation J/kg

Kg Mass transfer coefficient m/s

R Gas constant J/mol K

M Molar mass kg/mol

L Length in the machine direction m

x Space coordinate in the machine direction m z Space coordinate in the thickness direction m

φ Relative humidity −

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Dimensionless numbers Description

Nu Nusselt number

Sc Schmidt number

Pr Prandtl number

Re Reynolds number

Subscripts Description

p Paper

c Cylinder

dry Dry material, fibers

a Air

w Vapor

f Fabric

s Steam

f d Free draw

cz Contact zone

lam Laminar flow

turb Turbulent flow

sorp Sorption

sat Saturation

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Chapter 1

Introduction

Paperboard manufacturing is a challenging enterprise requiring advanced technology and great financial investments. In this competitive field of business, con- siderable resources are put into process optimization and control systems to increase productivity, reduce manufacturing costs and improve quality of the board.

During such research and development, simulation tools provide a cost-effective approach for verifying and validating new ideas without requiring risky and costly experiments on the operational machines.

The aim of this thesis is to contribute to the efforts on modelling the complete manufacturing process of a paper machine. More specifically, the thesis deals with the modelling of the multi-cylinder drying section of a paper mill. Drying is a critical part of the paperboard manufacturing process since it consumes a great amount of energy and affects the quality variables of the paperboard considerably.

The idea behind the present work was initiated by two modelling approaches used in AssiDomän Frövi: the grey-box modelling approach for implementing on-line predictors as decision tools for the process engineers and operators, and object-oriented modelling to obtain a model of the paper plant. Inspired by these two modelling approaches, this thesis presents a model that predicts the moisture content for each layer of the board in the drying section.

1.1 Paper-machine modelling in AssiDomän Frövi

The modelling interest in AssiDomän Frövi started in 1991, with the modelling of the bending stiffness¹. Gutman and Nilsson [18] made a first attempt with a quasi-linear ARMA-model. Bohlin [6] reported a grey-box model that Petters- son [36] improved by ameliorating the physical description in the sub-models.

With a similar approach as Pettersson [36], Bortolin [9, 10] developed a model of the curl and twist². The semi-physical models of bending stiffness [36] and curl

1The bending stiffness represents the force needed to bend the board.

2Curl is defined as the departure from flat form.

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and twist [9, 10] were complemented with a nonlinear Kalman filter to estimate the unmodelled disturbances, and implemented in the mill information system as quality predictors for the operators and process engineers.

Another modelling project in AssiDomän Frövi is based on the object-oriented language Modelica together with the simulation tool Dymola. The project aims for a model of the paper machine, by creating and reusing libraries adapted to the pulp and paper manufacturing process. To this end, the following parts have been modelled in Dymola: the bleach plant [30], the wet end [24, 8], the press [11] and the drying section [16].

1.2 Drying-section modelling

The drying of paper is an essential part for paper manufacturing. Firstly, it requires a great amount of energy, and secondly it is an important parameter for the quality variables of the board such as curl and twist, bending stiffness, shrinkage, wrinkle and delamination³.

Some models of the drying section of AssiDomän Frövi are available from previous work [16, 28]. The first model is a one-layer model [16] based on the work of Persson [35]. The model is implemented in Dymola, and fitted with static measurements. The second model, developed by Karlsson [28], is a considerably more complex physical model that includes both internal mass and heat transfer inside the board.

The main objective of the present work is a three-layer model of the board, since we are interested in estimating the moisture content for each layer. The model should therefore be detailed enough to reproduce the important physical behaviour inside the board. Furthermore, we want to apply a similar approach as Pettersson [36] and Bortolin [9, 10] to obtain a predictor of the moisture inside the board in the drying section. This approach implies the need to design a model that is simple enough to allow the simulation to be run on-line. Thus, the chosen model will result from a compromise between simplicity and completeness. Since a simple one-layer model was already implemented in Dymola [16], the original idea of this work was to extend it to a three-layer model and investigate the grey-box modelling approach with Dymola.

A major problem encountered in the modelling of the drying section is the lack of measurement data to validate the model. Few on-line sensors are present in the drying section for two main reasons: the drying hood is very hot and the board is difficult to reach. Moreover, experiments are not allowed due to high cost of non-sellable products. In this work, we attempt to answer the questions of observability and identifiability that arise when modelling the drying section:

With the few available measurements, can we estimate the board moistures and temperatures in the drying section?

3Delamination is the separation of the board layers.

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1.3 Objectives and contributions 3

1.3 Objectives and contributions

Based on a similar approach as the predictors for the bending stiffness [36] and the curl and twist [9], this thesis aims for a grey-box model that predicts the board moistures and temperatures inside the drying section of a paper machine. The objective of the thesis is not to derive a complex model of the drying section, since this can be found, for example, in the works of Baggerud [3] and Karlsson [28]. The model should instead be simple enough to be usable for simulation and running tests and as a control tool for the operators. Additionally, we aim to describe the board moisture content in each layer, since it affects several quality parameters.

Moreover, we want to investigate the possibility of using grey-box modelling with the simulation tool Dymola, to apply the technique on the other modelled parts of the paper machine.

In short, the contribution of the thesis is as follows:

• An observability and identifiability analysis of the drying section model is performed. Based on this analysis, we specify a set of static measurements that ensures observability and identifiability of the estimated variables and parameters during the identification. We also show, however, that the model is not observable for on-line conditions, and the predictions are therefore regarded as an approximation of the estimated variables.

• The grey-box modelling approach for a multi-cylinder drying section is ap- plied. Since on-line measurements for the board moisture content and temperature inside the drying section are difficult, we implement an extended Kalman filter that uses the few available on-line measurements to compensate for the unmodelled or not measured features. The resulting stochastic model gives an approximation of the board properties in the drying section.

• We investigate the grey-box modelling approach on a model implemented in the simulation tool Dymola. The parameter identification method and the nonlinear Kalman filtering technique are performed for the model. The resulting algorithms can be used to apply the approach on the other modelled parts of the board machine [30, 24, 8, 11].

1.4 Outline

The structure of the thesis is as follows:

Chapter 2 provides a brief literature review and a description of the drying pro- cess.

Chapter 3 describes the semi-physical model of the drying section. The model derives the equations for the temperature of the cylinders, the temperature and the moisture content of a three-layer board. The parameters and inputs of the model are introduced.

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Chapter 4 presents an analysis of observability and identifiability of the model.

A sensitivity analysis of the parameters to identify is carried out to estimate their impact on the model and to select the dominant ones for the identification.

Chapter 5 first describes the measurements carried out to evaluate the model with static process data. The process of identification of the unknown parameters is then introduced and the identification results are presented.

Chapter 6 investigates the behaviour of the model under on-line conditions. The deterministic model is first studied and then complemented by an extended Kalman filter to add disturbances and uncertainties in the model.

Chapter 7 concludes the thesis and discusses possible directions for future work.

Appendix A describes the model structure in the simulation tool Dymola and the algorithm developed for parameter identification within the Dymola environment.

Appendix B further analyses the observability conditions for one cylinder.

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Chapter 2

Background

2.1 Brief literature review

Drying of paper

The paper manufacturing process is described in e.g. [43] and the drying of paper is detailed in [29]. The drying process is an important part in the manufacturing of paper since it requires a lot of energy and affects the quality variables of the paper, such as bending stiffness and curl and twist. Various models of the drying of paper can be found in the literature, depending on which goal one has with the model; some are detailed and complex to get an insight into the physical phenomena and others are simplified for control purposes. The research group at Chemical engineering, Lund Institute of Technology provides physical modelling of the drying of paper [54, 35, 28, 3], the condensate flow inside the dryer [47, 48], infrared drying [37] and internal transport of water inside the paper [33, 53].

Slätteke [41] modelled the dynamics from the steam valve to the steam pressure with a black-box IPZ-model (one Integrator, one Pole and one Zero) for control tuning and derived a grey-box model to get an insight into the physical laws behind the black-box model. He then expanded the steam pressure model with a model for the paper to test several moisture controls [42]. Ekvall [14] examined a control strategy to improve the restart of the machine after a web break.

Wilhelmsson [54] developed a dynamic model of the multi-cylinder drying section by using the heat transport in the cylinder and paper, which was extended by Persson [35]. These two previous models do not include the internal mass transport of water in the thickness direction and assume that all the evaporation occurs at the surface of the paper. Baggerud [3] developed a detailed model for general drying of paper that includes both internal transfer of water and heat inside the paper, and Karlsson [28] a model of the drying section of the Frövi board. None of these models include the cross direction (CD) profile.

Wingren [55] simplified Persson’s model [35] by considering one steam group (a group of cylinders with the same supplied steam) as one cylinder with modified

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dimensions. This approach is not applied in this work, since the resulting model was not considered satisfactory.

The distribution of the moisture inside the paper is an important factor for quality parameters of the board, for example, shrinkage, curl and twist, wrinkle and delamination. Research has therefore been performed to understand and evaluate the distribution of the moisture in the thickness direction. Bernada et al. [4, 5]

carried out experiments to observe the internal moisture during drying by using the magnetic resonance imaging (MRI) technique. Harding et al. [19] studied the water profile and diffusion inside the board by using nuclear magnetic resonance (NMR) imaging. Wessman [53] investigated the transport of water in the thickness direction when watering or drying the board. Baggerud [2] developed a model of the moisture gradient to fit the data of Bernada et al. [4]. These works focussed on convective drying, i.e. drying by hot air. The moisture gradient was measured on a small sample of board, which is hardly feasible on-line on the hot cylinder drying systems because of the configuration (access point difficulty) and the speed of the machine.

Modelling in AssiDomän Frövi

The interest in modelling in AssiDomän Frövi started in 1991. A first attempt of modelling the bending stiffness was made by Gutman and Nilsson [18] with a quasi-linear ARMA-model with slow adaptation of the model parameters and fast adaptation of a bias compensation term. Bohlin [6] used grey-box modelling, where the parameters of the model were first identified on one set of data and bias was then compensated on-line with an Extended Kalman Filter. Pettersson [36] improved the physical behaviour in the sub-models and achieved a satisfactory model usable for the operators. With a similar approach as Pettersson [36], Bortolin [9, 10] developed a model of the curl and twist.

The grey-box modelling approach in AssiDomän Frövi has also been considered by Funkquist [15] for the continuous pulp digester, a nonlinear distributed parameter process.

The semi-physical models of bending stiffness [36] and curl and twist [9] are complemented with a nonlinear Kalman filter to estimate the unmodelled disturbances, and implemented in the mill information system as quality predictors for the operators and process engineers. The two grey-box models require the amount of water per layer as input. It is therefore of interest to develop a model predicting the moisture content per layer.

Several master theses and reports provide models of various parts of the pulp and paper plant in order to achieve a complete model of the paper manufacturing process. To this end, the following parts have been modelled: the bleach plant [30], the wet end [24, 8], the press [11] and the drying section [16]. The model of the drying section [16] is a simplified model of Persson [35], where the cartonboard is considered to be one layer. The present work is an extension of the simple model [16] to a three-layer grey-box model with the addition of the moisture content

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2.2 Process description 7

profile in the thickness direction and an extended Kalman filter to compensate for unmodelled features.

Observability and Identifiability of nonlinear systems

Since different structures of models can be chosen, the goal of this work is to find a compromise between a complete and simple model, to preserve the important physical phenomena while keeping a simple-enough model to be run on- line. Since the task of this work is to implement the model on-line and correct the bias with an observer, it is important to check if the model is observable and identifiable. In other words, we want to know if it is possible to reconstruct all the interesting states given the few measurements available. Observability and identifiability are related subjects, since identifiability can be considered as the observability of the parameters [1]. Analysis of linear observability and identifiability is a well-studied subject. However, for nonlinear systems, the complexity is increased and the subject is still under investigation. Anguelova [1] offers a literature review on the subject. The main tools for the study of observability and identifiability are differential geometry and differential algebra. The differential geometric approach can be found in the work of [21, 23, 45, 49] and consists in computing the Lie derivatives of the output up to rank n where n is the number of states in the system. The idea behind the differential algebraic approach is to express the Lie derivatives of the inputs and outputs as polynomial expressions.

This approach is easier for rational or polynomial functions. These two tools are of high complexity that increases with the number of states. An alternative is to investigate observability and identifiability of the system linearized around some operating point [44] which yields local properties only. In this thesis, we follow the latter approach.

2.2 Process description

Carton board manufacturing is a complex industrial process. The board at As- sidomän Frövi is composed of four fiber layers (or plies) and two coating layers.

The two middle fiber layers are composed of fibers of low density to get light weight and high bending stiffness. The bottom fiber layer is made of unbleached pulp while the top layer is composed of bleached pulp to achieve good printing properties. The two middle layers are considered identical as they use the same fiber mixture.

The board machine in Frövi, depicted in figure 2.1, is divided into five main processes: The wet end, the press, the drying section and the calendering and coating.

1. The wet end: This is the first step of the paper manufacturing. For each of four layers, the paper stock is spread by a headbox onto a fabric drained by water.

The thickness of the stock jet is determined by the opening of the headbox

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Figure 2.1: The board machine at Frövi. The felts in the drying section are not depicted.

slice while the velocity is provided by the headbox pressure. These two parameters will determine the spatial distribution of the fibers in the paper and the basis weight. Each layer is formed independently and then added to the previous layer in the order bottom ply, middle ply and top ply.

2. The press section: The main purposes of the press are to remove the water from the paper, consolidate the web and provide a surface smoothness. Since the water removal is more economical by mechanical means in the press section than by drying, as much water as possible is removed in the press section.

The water removal should be uniform across the machine to obtain a level moisture profile for the pressed sheet entering the drying section. After the press, the paper contains between 56 % and 64 % of water.

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3. The drying section: In the drying section, the board passes over and under steam-heated cylinders and the water inside the board is removed by evaporation. The concentration of water is around 8 % at the end of the drying section. The drying process is detailed further in this section.

4. Calendering: After the drying section, the paper is processed in a heated press nip to provide a smooth surface of the paper before the coating.

5. The coating: The coating is applied on the top layer of the sheet in two layers to improve the paper printing properties.

The drying section

In the drying section, the paper is passed over a series of 93 rotating steam-heated cylinders where water is evaporated and carried away by ventilation air. The wet web is held tightly against the cylinders by a synthetic permeable fabric called drying felt. Between two cylinders the paper is only in contact with the air; this part is called free draw.

The evaporation of the water in the paper inside the drying section is divided into four zones [43]: the warming up, the constant evaporation rate, the falling evaporation rate and the bound water zone. In the first zone, the paper is warmed up. During the constant evaporation rate zone, the water is situated on the fiber surfaces or within the large capillaries. When the free moisture is concentrated in the smaller capillaries, the evaporation rate decreases and reaches the falling rate zone. In the bound water zone, the residual water is more tightly held by physiochemical phenomena.

The rest of this section further describes the main parts of the drying section.

Steam and condensate system: The steam inside the cylinders provides the heat energy referred to latent heat when it condenses inside the cylinder shell.

The temperature of the saturated steam depends of the pressure. The steam is the main variable used to control the drying. At high machine speed, a layer of condensate film is formed inside the cylinder shell because of the centrifugal force. Even a thin layer of condensate is undesirable, since it affects the heat transfer considerably. To improve the heat transfer, spoiler bars are placed inside the shell to create turbulent flow (this increases the heat transfer) and rotative siphons are used to remove the condensate. The force controlling the flow of condensate outside the cylinder is the differential pressure between the incoming and outgoing steam. Together with the condensate, approximately 15 to 20 % of the incoming steam, called blow through steam, is also removed [43]. The outgoing steam and the condensate are conducted to a separator tank, where the steam is reused for the other steam groups in a cascade system configuration.

Hood ventilation: The surrounding air is an important parameter for the drying.

It must be drier than the paper to ensure evaporation and the temperature

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should be higher than the dew point¹. The main task of the ventilation system is to remove the evaporated water, to prevent condensation. The incoming flow should be the same as the outgoing flow to avoid disturbances and to get the total pressure equal to the atmospheric pressure.

Drying felts: The main purpose of the drying felt (also called fabric) is to keep the paper tight against the cylinders to get a better contact surface, and hence a better heat transfer, and to control the shrinkage in the cross direction. The speed of the felt is often higher than the speed of the machine to prevent shrinkage in the machine direction. The felts are run by the help of rolls whose speed determines the tension of the felts.

VIB device: At cylinder 53, water is sprayed over the board bottom layer by a steam actuator called VIB since the cross direction (CD) profile after the drying section shows that the middle of the paper is drier than the edges. The control of the VIB gives a more uniform profile and releases the risk of web breaks in the stack dryers. The sprayed water is taken from the condensate tanks and is around 80^◦C. A full opening of the actuator corresponds to an increase of 2 % units in the final moisture concentration.

Stack dryers: In most of the steam groups, the drying felts are situated below and under the cylinders; this is called a two-tier configuration. After the fifth steam group, the board enters a critical zone and can break easily in the free draw. Therefore, a single-tier configuration is adopted. The felt is holding the board even in the free draw. Between cylinders, vacuum rolls are leading the paper web to prevent folding when the board comes in contact with the cylinders. The first stack group is composed of lower cylinders which warms the bottom layer whereas the second group warms the top layer. The effect of the vacuum rolls is not well understood but their presence increases the drying rate. One possible explanation is that the vacuum rolls create a turbulent flow of the air in contact with the paper that increases the heat transfer coefficient. Another assumption is that the air is drier in the stacks because the vacuum cylinders suck it up. The pressure of the vacuum rolls is about 3000 P a.

Infrared dryers: At the end of the drying section, two infrared dryers are used, together with the VIB, to control the CD moisture profile. A full effect of the infrared dryers corresponds to a decrease of 2 % units in the final moisture concentration.

Measuring frame: A measuring frame is situated at the end of the drying section, between the two infrared dryers. The frame measures the average moisture content (in the thickness direction) and the temperature of the top layer. The

1the temperature at which water vapour begins to condense.

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frame is not used continuously, because it is sensitive to the heat of the infrared dryers. For the control of the CD moisture profile, measurements from the measurement frame located before the coating section are used.

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Chapter 3

Model Description

The physical model is essentially based on the model derived by Persson [35] with the addition of the diffusion of water in the thickness direction. The model computes the temperature of the cylinders and the temperature and moisture content of the carton board, using the heat balance of the cylinder and the heat and mass balance of the board. This chapter first explains the choice of discretization for the temperature and moisture in the paper. The three following sections describe the derivation of the temperature of the cylinder and the temperature and moisture inside the paper. The physical properties are then described: the properties of the cylinder, the paper, the surrounding air and at the interface between the paper and the air. A summary of the model is given in section 3.6. Finally, we present the semi-physical adjustments and the parameters, inputs and outputs of the model.

3.1 Discretization of the paper moisture and temperature

We consider that the carton board contains three layers by gathering the two middle layers, since they have the same properties. The present work discretizes the moisture content and temperature of the board in the machine only in the machine direction x (MD) and thickness direction z. For the cross direction y, only the prop- erties in the middle of the sheet are considered, i.e. the edges are not modelled.

In the machine direction, each cylinder is divided into two blocks: the contact zone (we use the same notation as Karlsson [28]), where the board is in contact with the cylinder, and the free draw, where the board is in the free draw. For each block, only one node is computed in the machine direction. The calculated temperatures Tp and moisture contents u are situated at the end of the contact zone or the free draw and the computed states of the previous block are used as incoming bound- ary condition (Tp,inand uin). The discretization of the temperature and moisture of the board in the machine direction is illustrated in figure 3.1.

The discretization of the moisture content and the temperature of the paper in the thickness direction is displayed in figure 3.2. The moisture content is computed

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Figure 3.1: Discretization of the temperature and moisture of the board in the ma- chine direction, where Lcpand Lf dare the lengths of contact zone and free draw.

Tpand u, the computed temperature and moisture content of the paper, are used as incoming boundary conditions Tpinand uinfor the next block.

for each layer since we want to know the distribution of the moisture per layer.

The temperature of the paper is considered for seven locations along the thickness axis. The temperatures for each layer, Tp,2, Tp,4and Tp,6are needed to compute the moisture in each layer. The temperatures at each side of the paper Tp,1and Tp,7

are required since the amount of evaporated water depends on the temperatures at each surface of the paper. The temperatures between two layers, Tp,3and Tp,5

are used to compute the other temperatures (see further in section 3.3). For ease of notation, the indices of the layers and moisture content are the same as the ones for the temperature. The bottom layer (BS) is layer 2, the two middle layers (MS) are grouped in layer 4 and the top layer (TS) is layer 6. An analysis of the observability and identifiability of the model is presented in chapter 4 in order to investigate the appropriateness of this choice of discretization.

In the following sections 3.2, 3.3, 3.4, the equations are derived for the case where the paper is in contact with an upper cylinder (i.e. the bottom layer is in contact with the cylinder). If the paper is in contact with a lower cylinder, the equations are obtained by switching the indices. The equations for the paper in the free draw are identical; one just needs to apply the equations for the paper in contact with the air for both sides of the paper and replace the length of contact between the paper and the cylinder Lcpwith the length of free draw Lf d.

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3.2 Heat balance of the cylinder 15

Figure 3.2: Discretization of the temperature and moisture content of the board in the thickness direction, when the board is in contact with an upper cylinder. T_p and u are the temperature and moisture content of the paper, Tcis the temperature of the cylinder shell, Tsthe steam temperature and Tathe air temperature.

3.2 Heat balance of the cylinder

The heat balance in the cylinder shell is given from Persson [35]:

∂Tc

∂t = kc

ρcCpc

∂²Tc

∂²zc − vx∂Tc

∂x (3.1)

where the term vx∂T_c

∂x [^◦C/s] is the convection transport of energy in the machine direction x, and _ρ^k^c

cCpc

∂²Tc

∂²z [^◦C/s] is the conductive heat transfer in the thickness direction of the cylinder. Tc [^◦C] is the temperature of the cylinder shell, t [s]

is the time index, zc [m] is the space coordinate in the thickness direction of the cylinder, and x [m] is the space coordinate in the machine direction. The properties of the cylinder, kc [W/m ^◦C] , Cpc [J/kg ^◦C] and ρc [kg/m³], are the thermal

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conductivity, the specific capacity, and the density of the cylinder, respectively.

They are described in section 3.5.

Persson [35] computes one cylinder temperature in the machine direction and three in the thickness direction for each cylinder. Videau and Lemaitre [50] observe a variation below one degree in the machine direction in their simulation.

Therefore, we assume, as [54, 35, 50], that the temperature of the outer surface of the shell is constant during a turn of the cylinder and consequently the convec- tive term vx∂Tc

∂x is removed. For the thickness direction, since we are only able to measure the temperature at the surface of the cylinder in contact with the air, we compute only one point, at the outer surface of the cylinder shell. The modifica- tions from Persson’s model [35] are derived in this section.

The differential equation (3.1) for the point at the surface of the cylinder becomes:

∂Tc

∂t

¯¯

dc

= kc

ρcCpcd²_c(T_c,d_c− T_c,0) (3.2) where dc[m] is the thickness of the cylinder shell, Tc,dc[^◦C] the temperature of the cylinder shell at the surface in contact with the air or the paper, and Tc,0[^◦C] the temperature of the cylinder shell at the surface in contact with the steam.

In order to compute (Tc,dc− Tc,0), we use the boundary conditions in the thickness direction, which are defined in the next section.

Boundary conditions for the temperature of the cylinder

The boundary conditions (3.3), (3.6) and (3.7), displayed in figure 3.3, are based from Persson [35].

Surface of the cylinder shell in contact with the steam

The heat transferred from the steam is conducted through the cylinder shell [35]:

hsc(Ts− Tc,0) = −kc

∂Tc

∂zc

¯¯

zc=0

= −kc

dc(Tc,0− Tc,dc) (3.3) where hsc [W/m^{2 ◦}C], the heat transfer coefficient between the steam and the cylinder is a parameter to identify (see section 3.8). The temperature of the steam Ts [^◦C] inside the cylinder is considered as saturated and is calculated directly from the steam pressure [35].

Since we want to compute the temperature of the cylinder only at the outside surface, we need to remove T_c,0in the left side of equation (3.3) and replace it by an expression of Tc,dc. Consequently, we modify the heat transfer coefficient between steam and cylinder hsc, in order to include the conduction of heat from the inside surface of the cylinder shell to the outside surface. The heat transferred from the steam to the inside shell and the heat conducted through the shell are connected

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3.2 Heat balance of the cylinder 17

Figure 3.3: Boundary conditions for the temperature of the cylinder. Lcpand Lca

are the length of contact zone and free draw, respectively, dcis the thickness of the shell, Tc,0and Tc,dcare the temperatures of the inside and outside of the shell and Tsis the temperature of the steam.

in series. Thus, the modified heat transfer coefficient from the steam to the outside of the shell ¯hscis calculated as follows:

1

¯hsc

= 1 hsc+ 1

kc

dc

⇒ ¯hsc= hsckc

dc

h_sc+^k_d^c

c

(3.4)

Therefore, the boundary condition (3.3) becomes:

¯hsc(Ts− Tc,dc) = −^k_d^c

c(Tc,0− Tc,dc) (3.5) Surface of the cylinder shell in contact with the paper

The conductive heat inside the cylinder is transferred to the paper [35]:

hcp(Tc,dc− Tp,1) = −kc∂Tc

∂z_c

¯¯

dc

= −kc

d_c(Tc,dc− Tc,0) (3.6)

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where hcp [W/m^{2 ◦}C], the heat transfer coefficient between the cylinder and the paper, is a parameter to identify (see section 3.8) and Tp,1[^◦C] is the temperature of the paper surface in contact with the cylinder.

Surface of the cylinder shell in contact with the air

The conductive heat inside the cylinder is transferred to the air:

h_ca(T_c,d_c− T_a) = −k_c∂Tc

∂zc

¯¯

dc

= −kc

dc

(T_c,d_c− T_c,0) (3.7)

where T_a[^◦C] is the temperature of the surrounding air. h_ca[W/m^{2 ◦}C] is the heat transfer coefficient between the cylinder and the air and is calculated in the same manner as the heat transfer coefficient between paper and air hpa(see section 3.5).

The temperature of the outer side of the cylinder

Since we assume that there is only one node per cylinder, we gather equations (3.5), (3.6) and (3.7) to get the boundary condition for the outer side temperature of the cylinder. If we call L_cp [m] the length of contact between the cylinder and the paper and Lca[m] the length of contact cylinder–air, the boundary condition becomes:

(Tc,dc−Tc,0) = −dc

kc

µLcphcp(Tc,dc− Tp,1) + Lcahca(Tc,dc− Ta)

Lcp+ Lca − ¯hsc(Ts− Tc,dc)

¶

(3.8) To simplify the notation, we remove the index d_csince there is only one node for the cylinder and define Tc:= Tc,d_c. Inserting (3.8) in (3.2), the differential equation for the temperature at the surface of the cylinder is defined as follows:

∂Tc

∂t = 1

ρcCpcdc

µLcphcp(Tp,1− Tc) + Lcahca(Ta− Tc)

Lcp+ Lca + ¯hsc(Ts− Tc)

¶

(3.9)

3.3 Heat balance of the paperboard

The heat balance of the paper web is described by the following equation [35]:

∂T_p

∂t = k_p ρpCpp

∂²T_p

∂²zp − vx

∂T_p

∂x (3.10)

where the term vx∂Tp

∂x [^◦C/s] is the convection transport of energy in the machine direction x, and _ρ^k^p

pCpp

∂²Tp

∂²z [^◦C/s] is the conductive heat transfer in the thickness direction of the cylinder. Tp[^◦C] is the temperature of the paper, t [s] is the time index, zp [m] is the space coordinate in the thickness direction of the paper, and x [m] is the space coordinate in the machine direction. The properties of the paper,

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3.3 Heat balance of the paperboard 19

kp [W/m^◦C], Cpp [J/kg^◦C] and ρp [kg/m³], are the thermal conductivity, the specific capacity and the density of the paper, respectively. They are described in section 3.5.

Numerical solution

In the machine direction (x), since there is only one node, we use the same method as Persson [35]: the backwards differentiation method of the first order.

∂Tp,i,j

∂x = 1

∆xj(Tp,i,j− Tp,i,j−1) (3.11)

where i and j are the indices in the thickness direction and in the machine direc- tion, respectively.

The temperature of the paper at the point Tp,i,j−1 is given by the boundary condition in the machine direction (see figure 3.1). For ease of notation, we skip the index j:

∂Tp,i

∂x = 1

Lcp(Tp,i− Tpin,i) (3.12)

where Lcpis the length of the contact between the paper and the cylinder.

For the thickness direction (z), we use the same method as in [35]: the centre differentiation method.

For the first order, the discretization is written:

∂Tp

∂z

¯¯

i

= 1

2∆zi(Tp,i+1− Tp,i−1) (3.13)

where ∆z_iis the discretization step at the point z_i. The centre differentiation method of second order is:

∂²Tp

∂²z

¯¯

i

= 1

(∆zi)²(Tp,i+1− 2Tp,i+ Tp,i−1) (3.14) Since the thickness of each layer is different, the step is varying. Therefore, equations (3.13) and (3.14) are modified into (3.15) and (3.16).

∂T_p

∂z

¯¯

i

= 1

∆zi−1,i+1(Tp,i+1− Tp,i−1) (3.15)

∂²T

∂²z

¯¯

i

= 1

(∆zi,i+1)²(Tp,i+1− Tp,i) + 1

(∆zi−1,i)²(Tp,i−1− Tp,i) (3.16) where ∆zl,kis the distance between the points situated at zland zk.

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Paper temperature in the middle of each layer

For the nodes in the middle of each layer i = 2, 4, 6, the step sizes ∆zi−1,i and

∆zi,i+1 are equal, ∆zi−1,i = ∆zi,i+1 = dp,i/2, where dp,i is the thickness of the layer i. Inserting (3.12) and (3.16) into (3.10) the differential equations of the nodes in the middle of each layer Tp,iare described as follows:

∂Tp,i

∂t = 4kp,i

ρp,iCpp,id²_p,i(Tp,i+1− 2Tp,i+ Tp,i−1) − vx

Lcp(Tp,i− Tpin,i) (3.17) where the properties of the layer i, kp,iρp,iCpp,idp,iare described in section 3.5.

Paper temperature between two layers

For the nodes between two layers i = 3, 5, the differential equations for the tem- perature Tp,iare derived by inserting (3.12) and (3.16) into (3.10):

∂Tp,i

∂t = _ρ ^4k^p,i−1

p,i−1Cpp,i−1d²_p,i−1(Tp,i−1− Tp,i) +_ρ ^4k^p,i+1

p,i+1Cpp,i+1d²_p,i+1(−Tp,i+ Tp,i+1) −_L^v^x

cp(Tp,i− Tpin,i) (3.18) Paper temperature at the surface in contact with the fabric/air

To compute the heat differential equation for the point situated at the surface of the paper in contact with the fabric or the air, Tp,7, we use the same method as Persson [35] and introduce a virtual point Tp,8, situated at the distance dp,6/2, on the opposite side of Tp,6. The equation (3.16) is then written:

∂²Tp

∂²z

¯¯

7

= 4

d²_p,6(Tp,8− 2Tp,7+ Tp,6) (3.19) To remove Tp,8in the equation, we use the boundary conditions between the surface of the paper and the surrounding air, represented in figure 3.4.

Boundary condition between the surface of the paper and the fabric/air

At the surface of the paper in contact with the air, the heat of conduction inside the paper and the heat of evaporation of water in the air are transferred to the air.

The boundary condition is described by the following equation [35]:

˙

mλ = hpa(Ta− Tp,7) − kp,6 ∂Tp

∂z

¯¯

7

(3.20)

where hpa[W/m^{2 ◦}C] is the heat transfer coefficient between the paper and the air,

˙

m [kg/m²s] is the evaporation rate of water and λ [J/kg] is the heat of evaporation.

These properties are described in section 3.5.

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3.3 Heat balance of the paperboard 21

Figure 3.4: Boundary condition at the surface of the paper in contact with the air, where Tp,iis the board temperature at position i, Ta is the temperature of the surrounding air and dp,6the thickness of layer 6.

Using equation (3.15), the previous equation can be written:

˙

mλ = hpa(Ta− Tp,7) − kp,6Tp,8− Tp,6

dp,6

(3.21) We can now extract Tp,8:

→ Tp,8=dp,6

kp,6(− ˙mλ + hpa(Ta− Tp,7)) + Tp,6 (3.22) and insert it in (3.19):

∂²Tp

∂²z

¯¯

7

= 4 d²_p,6

µ

2Tp,6− 2Tp,7+dp,6

kp,6(− ˙mλ + hpa(Ta− Tp,7))

¶

(3.23) Finally, the differential equation (3.10) for Tp,7 is given by inserting (3.12) and (3.23):

∂Tp,7

∂t = _ρ ^4k^p,6

p,6Cpp,6d²_p,6

³

2Tp,6− 2Tp,7+^d_k^p,6

p,6(− ˙mλ + hpa(Ta− Tp,7))

´

−_L^v^x

cp(Tp,7− Tpin,7) (3.24)

Paper temperature at the surface in contact with the cylinder

To compute the heat differential equation for the point situated at the surface of the paper in contact with the cylinder, Tp,1, we use the same method as previously and introduce a virtual point Tp,0, situated at the distance dp,2/2, on the opposite side of Tp,2[35]. The equation (3.16) is then written:

∂²Tp

∂²z

¯¯

1

= 4

d²_p,2(Tp,2− 2Tp,1+ Tp,0) (3.25)

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To remove Tp,0in the equation, we use the boundary conditions between the surface of the paper and the cylinder, represented in figure 3.5.

Figure 3.5: Boundary condition at the surface of the paper in contact with the cylinder, where Tp,iis the board temperature at position i, Tcis the temperature of the cylinder and dp,2the thickness of layer 2.

Boundary condition at the surface of the paper in contact with the cylinder At the surface of the paper in contact with the cylinder shell, the heat given from the cylinder is conducted inside the paper [35]:

hcp(Tc− Tp,1) = −kp,2 ∂Tp

∂z

¯¯

i=1

(3.26)

Using equation (3.15), the previous equation can be written:

hcp(Tc− Tp,1) = −kp,2Tp,2− Tp,0

dp,2

(3.27)

We can now extract T_p,0:

Tp,0= dp,2

kp,2

(hcp(Tc− Tp,1)) + Tp,2 (3.28)

and insert it in (3.25):

∂²Tp

∂²z

¯¯

1

= 4

d²_p,2(2Tp,2− 2Tp,1+dp,2

kp,2

(hcp(Tc− Tp,1)) (3.29)

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3.4 Mass balances within the paper web 23

Finally, the differential equation (3.10) for Tp,1is derived by combining (3.12) and (3.29):

∂Tp,1

∂t = 4kp,2

ρp,2Cpp,2d²_p,2 µ

2Tp,2− 2Tp,1+dp,2

kp,2

(hcp(Tc− Tp,1)

¶

− vx

Lcp

(Tp,1− Tpin,1) (3.30)

3.4 Mass balances within the paper web

The present model derives the mass balances of dry material and water. The paper is assumed to be composed of two components: the dry component, consisting of fibers and fillers (subscriptdry) and the water (subscriptw). To not increase the complexity of the model, the distinction between the two phases of water, liquid or vapor, is not included in this work, and the presence of air inside the paper, formed when the water leaves the pores, is not modelled. Descriptions of mass balances including the air, liquid and vapor can be found in the works of Baggerud [2] and Karlsson [28].

Mass balance of dry material

The mass of dry material per layer Gdry,i[kgdry/m²] remains constant during the drying. In the numerical computations in the thesis, the mass of dry matter in each layer is obtained from the bending stiffness predictor estimates of Pettersson [36].

The mass balance is expressed as in Persson [35]:

∂Gdry,i

∂t = −vx∂Gdry,i

∂x (3.31)

Mass balance of water

The moisture content u [kgw/kgdry] represents the amount of water in the paper.

To compute the mass balance of water in the paper, we consider the following two types of water transport:

1. Evaporation of water into the air ˙mevap[kgw/m²s] occurring at the surface [35], derived in section 3.5.

2. Diffusion of water in the thickness direction ˙mdif f[kgw/m²s] given by Fick’s law [3]. The driving force for the diffusive transport of water is the gradient in the thickness direction of the moisture content (we consider that the diffusion of water only occurs in the thickness direction):

˙

mdif f = −ρdryDwp∂u

∂z =Gdry

∆z Dwp∂u

∂z (3.32)

where Dwp[m²/s] is the diffusion coefficient of water in the paper, described in section 3.8, and ∆z is the thickness where we consider the diffusion.

Modelling the Moisture Content of Multi-Ply Paperboard in the Paper Machine Drying Section