modelling the moisture content of multi-ply paperboard in ...9961/fulltext01.pdf · 1.2...

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

CHRISTELLE GAILLEMARD

Licentiate ThesisStockholm, Sweden 2006

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

Departement of MathematicsKTH

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan fram-lägges till offentlig granskning för avläggande av Licentiatexamen fredagen den7 april 2006 klockan 10.00 i rum 3721, plan 7, Lindstedsvägen 25, Kungl TekniskaHögskolan, Stockholm.

c© Christelle Gaillemard, April 2006

Tryck: Universitetsservice US AB

To Pär-Anders

iii

Abstract

This thesis presents a grey-box model of the temperature and moisture content for eachlayer of the multi-ply paperboard inside the drying section of a paper mill. The distributionof 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 themoisture evolution during the drying, to be used by operators and process engineers as anestimation of the unmeasurable variables inside the drying section.

Drying of carton board is a complex and nonlinear process. The physical phenomenaare not entirely understood and the drying depends on a number of unknown parametersand unmodelled or unmeasurable features. The grey-box modelling approach, which con-sists in using the available measurements to estimate the unknown disturbances, is there-fore a suitable approach for modelling the drying section.

A major problem encountered with the modelling of the drying section is the lack ofmeasurements to validate the model. Consequently, the correctness and uniqueness of theestimated variables and parameters are not guaranteed. We therefore carry out observabil-ity and identifiability analyses and the results suggest that the selected model structure isobservable and identifiable under the assumption that specific measurements are available.Based on this analysis, static measurements in the drying section are carried out to iden-tify the parameters of the model. The parameters are identified using one data set and theresults are validated with other data sets.

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

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

Acknowledgements

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

First of all, I would like to thank my advisor Per-Olof Gutman, for introducingme to the project after my master thesis. His insightful suggestions and his supportdespite the geographical distance were a source of inspiration and motivation.These years have been challenging but I now feel it was worth going through allthe struggles.

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

I also wish to acknowledge AssiDomän Frövi for financing the project. I amgrateful 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 andthe simulation tool Dymola. Lars and his family deserve special thanks for theirkindness and sincere concern.

My free time in Frövi would have been lonelier without my friend DorothéeMillon 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 mewith the measurements, and Kent Åkerberg, Gunnar Pålsson and Anders Hen-riksson for sharing their knowledge about the drying section. I am also thankfulto Magnus Karlsson for answering my questions about the physical model, JensPettersson for helping me with the IPOPT interface and Jenny Ekvall for the dis-cussions about paper machine drying sections.

The faculty members and graduate students at the division of Optimizationand 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 forsharing his experience of grey-box modelling. His coaching and friendship were agreat source of motivation. I can not thank enough my roommate Vanna Fanizzafor 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 wantto thank Ryozo Nagamune for his kindness and precious advises.

v

vi

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

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

Train Stockholm - Nyköping, February 2006

Christelle Gaillemard

Contents

Nomenclature ix

1 Introduction 11.1 Paper-machine modelling in AssiDomän Frövi . . . . . . . . . . . . 11.2 Drying-section modelling . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Objectives and contributions . . . . . . . . . . . . . . . . . . . . . . . 31.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 52.1 Brief literature review . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Process description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Model Description 133.1 Discretization of the paper moisture and temperature . . . . . . . . 133.2 Heat balance of the cylinder . . . . . . . . . . . . . . . . . . . . . . . 153.3 Heat balance of the paperboard . . . . . . . . . . . . . . . . . . . . . 183.4 Mass balances within the paper web . . . . . . . . . . . . . . . . . . 233.5 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 Summary of the physical model equations . . . . . . . . . . . . . . . 293.7 Semi-physical adjustments . . . . . . . . . . . . . . . . . . . . . . . . 313.8 Parameters, inputs and outputs of the model . . . . . . . . . . . . . 32

4 Observability, Identifiability and Sensitivity Analyses 374.1 Observability analysis of the physical model . . . . . . . . . . . . . 384.2 Identifiability analysis of the physical model . . . . . . . . . . . . . 454.3 Sensitivity analysis of the semi-physical model . . . . . . . . . . . . 484.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Identification of the Parameters 675.1 Description of the static measurements . . . . . . . . . . . . . . . . . 675.2 Identification procedure . . . . . . . . . . . . . . . . . . . . . . . . . 725.3 Parameter selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4 Identification results . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

vii

viii CONTENTS

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Dynamic Simulations 796.1 Deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Grey-box modelling of the disturbances . . . . . . . . . . . . . . . . 806.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 88

7 Conclusions and Future Work 917.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2 Directions for future work . . . . . . . . . . . . . . . . . . . . . . . . 92

A Implementation 95A.1 Simulation program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.2 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96A.3 Algorithm for identification with a Dymola model . . . . . . . . . . 97A.4 Optimization routine . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B Observability Analysis for one Cylinder 99

Bibliography 103

Nomenclature

Some notations may have a different meaning locally.

Symbol Description Dimensionvx Speed of the machine m/su Moisture content kgw/kgdry

t Time index sT Temperature C or KG Basis weight kg/m2

W Flow of paper kg/sl Paper width mρ Density kg/m3

d Thickness mh Heat transfer coefficient W/m2 CDab Diffusion coefficient of m2/s

element a into element bFRF Fabric heat reduction factor %Cp Specific capacity J/kg Ck Thermal conductivity W/m CP Absolute pressure Pap Partial pressure of water Pam Evaporation rate kg/m2sλ Heat of evaporation J/kgKg Mass transfer coefficient m/sR Gas constant J/mol KM Molar mass kg/molL Length in the machine direction mx Space coordinate in the machine direction mz Space coordinate in the thickness direction mφ Relative humidity −

ix

x Nomenclature

Dimensionless numbers DescriptionNu Nusselt numberSc Schmidt numberPr Prandtl numberRe Reynolds number

Subscripts Descriptionp Paperc Cylinderdry Dry material, fibersa Airw Vaporf Fabrics Steamfd Free drawcz Contact zonelam Laminar flowturb Turbulent flowsorp Sorptionsat Saturation

Chapter 1

Introduction

Paperboard manufacturing is a challenging enterprise requiring advanced tech-nology and great financial investments. In this competitive field of business, con-siderable resources are put into process optimization and control systems to in-crease productivity, reduce manufacturing costs and improve quality of the board.During such research and development, simulation tools provide a cost-effectiveapproach for verifying and validating new ideas without requiring risky and costlyexperiments on the operational machines.

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

The idea behind the present work was initiated by two modelling approachesused in AssiDomän Frövi: the grey-box modelling approach for implementingon-line predictors as decision tools for the process engineers and operators, andobject-oriented modelling to obtain a model of the paper plant. Inspired by thesetwo modelling approaches, this thesis presents a model that predicts the moisturecontent 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 modellingof the bending stiffness1. Gutman and Nilsson [18] made a first attempt with aquasi-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 ofthe 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.

1

2 Introduction

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

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

1.2 Drying-section modelling

The drying of paper is an essential part for paper manufacturing. Firstly, it requiresa great amount of energy, and secondly it is an important parameter for the qualityvariables of the board such as curl and twist, bending stiffness, shrinkage, wrinkleand delamination3.

Some models of the drying section of AssiDomän Frövi are available from pre-vious work [16, 28]. The first model is a one-layer model [16] based on the workof Persson [35]. The model is implemented in Dymola, and fitted with static mea-surements. The second model, developed by Karlsson [28], is a considerably morecomplex physical model that includes both internal mass and heat transfer insidethe 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. Themodel should therefore be detailed enough to reproduce the important physicalbehaviour inside the board. Furthermore, we want to apply a similar approach asPettersson [36] and Bortolin [9, 10] to obtain a predictor of the moisture inside theboard in the drying section. This approach implies the need to design a model thatis simple enough to allow the simulation to be run on-line. Thus, the chosen modelwill result from a compromise between simplicity and completeness. Since a sim-ple one-layer model was already implemented in Dymola [16], the original ideaof this work was to extend it to a three-layer model and investigate the grey-boxmodelling approach with Dymola.

A major problem encountered in the modelling of the drying section is thelack of measurement data to validate the model. Few on-line sensors are presentin the drying section for two main reasons: the drying hood is very hot and theboard is difficult to reach. Moreover, experiments are not allowed due to highcost of non-sellable products. In this work, we attempt to answer the questionsof observability and identifiability that arise when modelling the drying section:With the few available measurements, can we estimate the board moistures andtemperatures in the drying section?

3Delamination is the separation of the board layers.

1.3 Objectives and contributions 3

1.3 Objectives and contributions

Based on a similar approach as the predictors for the bending stiffness [36] andthe curl and twist [9], this thesis aims for a grey-box model that predicts the boardmoistures and temperatures inside the drying section of a paper machine. Theobjective of the thesis is not to derive a complex model of the drying section, sincethis can be found, for example, in the works of Baggerud [3] and Karlsson [28]. Themodel should instead be simple enough to be usable for simulation and runningtests and as a control tool for the operators. Additionally, we aim to describe theboard moisture content in each layer, since it affects several quality parameters.Moreover, we want to investigate the possibility of using grey-box modelling withthe simulation tool Dymola, to apply the technique on the other modelled parts ofthe paper machine.

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

• An observability and identifiability analysis of the drying section model isperformed. Based on this analysis, we specify a set of static measurementsthat ensures observability and identifiability of the estimated variables andparameters during the identification. We also show, however, that the modelis not observable for on-line conditions, and the predictions are thereforeregarded 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 tem-perature inside the drying section are difficult, we implement an extendedKalman filter that uses the few available on-line measurements to compen-sate for the unmodelled or not measured features. The resulting stochasticmodel gives an approximation of the board properties in the drying section.

• We investigate the grey-box modelling approach on a model implementedin the simulation tool Dymola. The parameter identification method and thenonlinear Kalman filtering technique are performed for the model. The re-sulting algorithms can be used to apply the approach on the other modelledparts 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 modelderives the equations for the temperature of the cylinders, the temperatureand the moisture content of a three-layer board. The parameters and inputsof the model are introduced.

4 Introduction

Chapter 4 presents an analysis of observability and identifiability of the model.A sensitivity analysis of the parameters to identify is carried out to estimatetheir impact on the model and to select the dominant ones for the identifica-tion.

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

Chapter 6 investigates the behaviour of the model under on-line conditions. Thedeterministic model is first studied and then complemented by an extendedKalman 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 thealgorithm developed for parameter identification within the Dymola envi-ronment.

Appendix B further analyses the observability conditions for one cylinder.

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 paperis detailed in [29]. The drying process is an important part in the manufactur-ing of paper since it requires a lot of energy and affects the quality variables ofthe paper, such as bending stiffness and curl and twist. Various models of thedrying of paper can be found in the literature, depending on which goal one haswith the model; some are detailed and complex to get an insight into the physicalphenomena and others are simplified for control purposes. The research group atChemical engineering, Lund Institute of Technology provides physical modellingof 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 pres-sure with a black-box IPZ-model (one Integrator, one Pole and one Zero) for con-trol tuning and derived a grey-box model to get an insight into the physical lawsbehind the black-box model. He then expanded the steam pressure model with amodel for the paper to test several moisture controls [42]. Ekvall [14] examined acontrol strategy to improve the restart of the machine after a web break.

Wilhelmsson [54] developed a dynamic model of the multi-cylinder drying sec-tion by using the heat transport in the cylinder and paper, which was extended byPersson [35]. These two previous models do not include the internal mass trans-port of water in the thickness direction and assume that all the evaporation occursat the surface of the paper. Baggerud [3] developed a detailed model for generaldrying of paper that includes both internal transfer of water and heat inside thepaper, and Karlsson [28] a model of the drying section of the Frövi board. None ofthese 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

5

6 Background

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

The distribution of the moisture inside the paper is an important factor forquality parameters of the board, for example, shrinkage, curl and twist, wrinkleand delamination. Research has therefore been performed to understand and eval-uate 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 usingthe magnetic resonance imaging (MRI) technique. Harding et al. [19] studied thewater profile and diffusion inside the board by using nuclear magnetic resonance(NMR) imaging. Wessman [53] investigated the transport of water in the thicknessdirection when watering or drying the board. Baggerud [2] developed a model ofthe moisture gradient to fit the data of Bernada et al. [4]. These works focussed onconvective drying, i.e. drying by hot air. The moisture gradient was measured ona small sample of board, which is hardly feasible on-line on the hot cylinder dry-ing systems because of the configuration (access point difficulty) and the speed ofthe machine.

Modelling in AssiDomän Frövi

The interest in modelling in AssiDomän Frövi started in 1991. A first attemptof modelling the bending stiffness was made by Gutman and Nilsson [18] witha quasi-linear ARMA-model with slow adaptation of the model parameters andfast 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 andbias was then compensated on-line with an Extended Kalman Filter. Pettersson[36] improved the physical behaviour in the sub-models and achieved a satisfac-tory 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 consid-ered by Funkquist [15] for the continuous pulp digester, a nonlinear distributedparameter process.

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

Several master theses and reports provide models of various parts of the pulpand paper plant in order to achieve a complete model of the paper manufacturingprocess. 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 thedrying section [16] is a simplified model of Persson [35], where the cartonboard isconsidered 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

2.2 Process description 7

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

Observability and Identifiability of nonlinear systems

Since different structures of models can be chosen, the goal of this work is tofind a compromise between a complete and simple model, to preserve the im-portant 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 correctthe bias with an observer, it is important to check if the model is observable andidentifiable. In other words, we want to know if it is possible to reconstruct allthe interesting states given the few measurements available. Observability andidentifiability are related subjects, since identifiability can be considered as theobservability of the parameters [1]. Analysis of linear observability and identifi-ability is a well-studied subject. However, for nonlinear systems, the complexityis increased and the subject is still under investigation. Anguelova [1] offers a lit-erature review on the subject. The main tools for the study of observability andidentifiability are differential geometry and differential algebra. The differentialgeometric approach can be found in the work of [21, 23, 45, 49] and consists incomputing the Lie derivatives of the output up to rank n where n is the numberof states in the system. The idea behind the differential algebraic approach is toexpress the Lie derivatives of the inputs and outputs as polynomial expressions.This approach is easier for rational or polynomial functions. These two tools areof high complexity that increases with the number of states. An alternative is toinvestigate observability and identifiability of the system linearized around someoperating point [44] which yields local properties only. In this thesis, we followthe 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 lightweight and high bending stiffness. The bottom fiber layer is made of unbleachedpulp while the top layer is composed of bleached pulp to achieve good printingproperties. The two middle layers are considered identical as they use the samefiber mixture.

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

1. The wet end: This is the first step of the paper manufacturing. For each of fourlayers, 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

8 Background

Figure 2.1: The board machine at Frövi. The felts in the drying section are notdepicted.

slice while the velocity is provided by the headbox pressure. These two pa-rameters will determine the spatial distribution of the fibers in the paper andthe basis weight. Each layer is formed independently and then added to theprevious 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 fromthe paper, consolidate the web and provide a surface smoothness. Since thewater removal is more economical by mechanical means in the press sectionthan 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 levelmoisture profile for the pressed sheet entering the drying section. After thepress, the paper contains between 56 % and 64 % of water.


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

4. Calendering: After the drying section, the paper is processed in a heated pressnip 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 layersto improve the paper printing properties.

The drying section

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

The evaporation of the water in the paper inside the drying section is dividedinto four zones [43]: the warming up, the constant evaporation rate, the fallingevaporation rate and the bound water zone. In the first zone, the paper is warmedup. During the constant evaporation rate zone, the water is situated on the fibersurfaces or within the large capillaries. When the free moisture is concentratedin the smaller capillaries, the evaporation rate decreases and reaches the fallingrate zone. In the bound water zone, the residual water is more tightly held byphysiochemical 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 heatenergy referred to latent heat when it condenses inside the cylinder shell.The temperature of the saturated steam depends of the pressure. The steamis the main variable used to control the drying. At high machine speed, alayer of condensate film is formed inside the cylinder shell because of thecentrifugal force. Even a thin layer of condensate is undesirable, since itaffects the heat transfer considerably. To improve the heat transfer, spoilerbars are placed inside the shell to create turbulent flow (this increases theheat transfer) and rotative siphons are used to remove the condensate. Theforce controlling the flow of condensate outside the cylinder is the differen-tial pressure between the incoming and outgoing steam. Together with thecondensate, approximately 15 to 20 % of the incoming steam, called blowthrough steam, is also removed [43]. The outgoing steam and the conden-sate are conducted to a separator tank, where the steam is reused for theother 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

10 Background

should be higher than the dew point1. The main task of the ventilation sys-tem is to remove the evaporated water, to prevent condensation. The incom-ing flow should be the same as the outgoing flow to avoid disturbances andto get the total pressure equal to the atmospheric pressure.

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

VIB device: At cylinder 53, water is sprayed over the board bottom layer by asteam actuator called VIB since the cross direction (CD) profile after the dry-ing section shows that the middle of the paper is drier than the edges. Thecontrol of the VIB gives a more uniform profile and releases the risk of webbreaks in the stack dryers. The sprayed water is taken from the condensatetanks and is around 80C. A full opening of the actuator corresponds to anincrease of 2 % units in the final moisture concentration.

Stack dryers: In most of the steam groups, the drying felts are situated belowand under the cylinders; this is called a two-tier configuration. After thefifth steam group, the board enters a critical zone and can break easily inthe free draw. Therefore, a single-tier configuration is adopted. The felt isholding the board even in the free draw. Between cylinders, vacuum rolls areleading the paper web to prevent folding when the board comes in contactwith the cylinders. The first stack group is composed of lower cylinderswhich warms the bottom layer whereas the second group warms the toplayer. The effect of the vacuum rolls is not well understood but their presenceincreases the drying rate. One possible explanation is that the vacuum rollscreate a turbulent flow of the air in contact with the paper that increases theheat transfer coefficient. Another assumption is that the air is drier in thestacks because the vacuum cylinders suck it up. The pressure of the vacuumrolls is about 3000 Pa.

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 theinfrared dryers corresponds to a decrease of 2 % units in the final moistureconcentration.

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

1the temperature at which water vapour begins to condense.


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

Chapter 3

Model Description

The physical model is essentially based on the model derived by Persson [35] withthe addition of the diffusion of water in the thickness direction. The model com-putes the temperature of the cylinders and the temperature and moisture contentof the carton board, using the heat balance of the cylinder and the heat and massbalance of the board. This chapter first explains the choice of discretization forthe temperature and moisture in the paper. The three following sections describethe derivation of the temperature of the cylinder and the temperature and mois-ture inside the paper. The physical properties are then described: the properties ofthe cylinder, the paper, the surrounding air and at the interface between the paperand the air. A summary of the model is given in section 3.6. Finally, we present thesemi-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 mid-dle layers, since they have the same properties. The present work discretizes themoisture content and temperature of the board in the machine only in the machinedirection 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 contactzone (we use the same notation as Karlsson [28]), where the board is in contact withthe 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 temperaturesTp and moisture contents u are situated at the end of the contact zone or the freedraw and the computed states of the previous block are used as incoming bound-ary condition (Tp,in and uin). The discretization of the temperature and moistureof the board in the machine direction is illustrated in figure 3.1.

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

13

14 Model Description

Figure 3.1: Discretization of the temperature and moisture of the board in the ma-chine direction, where Lcp and Lfd are the lengths of contact zone and free draw.Tp and u, the computed temperature and moisture content of the paper, are usedas incoming boundary conditions Tpin and uin for 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 thicknessaxis. The temperatures for each layer, Tp,2, Tp,4 and Tp,6 are needed to computethe moisture in each layer. The temperatures at each side of the paper Tp,1 and Tp,7

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

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

In the following sections 3.2, 3.3, 3.4, the equations are derived for the casewhere the paper is in contact with an upper cylinder (i.e. the bottom layer is incontact with the cylinder). If the paper is in contact with a lower cylinder, theequations are obtained by switching the indices. The equations for the paper inthe free draw are identical; one just needs to apply the equations for the paper incontact with the air for both sides of the paper and replace the length of contactbetween the paper and the cylinder Lcp with the length of free draw Lfd.

3.2 Heat balance of the cylinder 15

Figure 3.2: Discretization of the temperature and moisture content of the board inthe thickness direction, when the board is in contact with an upper cylinder. Tp

and u are the temperature and moisture content of the paper, Tc is the temperatureof the cylinder shell, Ts the steam temperature and Ta the 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∂Tc

∂x [C/s] is the convection transport of energy in the machinedirection x, and kc

ρcCpc

∂2Tc

∂2z [C/s] is the conductive heat transfer in the thicknessdirection 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 thecylinder, and x [m] is the space coordinate in the machine direction. The propertiesof the cylinder, kc [W/m C] , Cpc [J/kg C] and ρc [kg/m3], are the thermal


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 andthree in the thickness direction for each cylinder. Videau and Lemaitre [50] ob-serve 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 ofthe 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 tomeasure the temperature at the surface of the cylinder in contact with the air, wecompute 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 be-comes:

∂Tc

∂t

∣∣∣∣dc

=kc

ρcCpcd2c

(Tc,dc − Tc,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] thetemperature 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 thick-

ness 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 basedfrom 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/m2 C], the heat transfer coefficient between the steam and thecylinder is a parameter to identify (see section 3.8). The temperature of the steamTs [C] inside the cylinder is considered as saturated and is calculated directlyfrom the steam pressure [35].

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

3.2 Heat balance of the cylinder 17

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

are the length of contact zone and free draw, respectively, dc is the thickness of theshell, Tc,0 and Tc,dc are the temperatures of the inside and outside of the shell andTs is the temperature of the steam.

in series. Thus, the modified heat transfer coefficient from the steam to the outsideof the shell hsc is calculated as follows:

1hsc

=1

hsc+

1kc

dc

⇒ hsc =hsc

kc

dc

hsc + kc

dc

(3.4)

Therefore, the boundary condition (3.3) becomes:

hsc(Ts − Tc,dc) = −kc

dc(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

∂zc

∣∣∣∣dc

= −kc

dc(Tc,dc − Tc,0) (3.6)


where hcp [W/m2 C], the heat transfer coefficient between the cylinder and thepaper, is a parameter to identify (see section 3.8) and Tp,1 [C] is the temperatureof 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:

hca(Tc,dc − Ta) = −kc∂Tc

∂zc

∣∣∣∣dc

= −kc

dc(Tc,dc − Tc,0) (3.7)

where Ta [C] is the temperature of the surrounding air. hca [W/m2 C] is the heattransfer coefficient between the cylinder and the air and is calculated in the samemanner 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 temperatureof the cylinder. If we call Lcp [m] the length of contact between the cylinder andthe paper and Lca [m] the length of contact cylinder–air, the boundary conditionbecomes:

(Tc,dc−Tc,0) = −dc

kc

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

Lcp + Lca− hsc(Ts − Tc,dc)

)

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

∂Tc

∂t=

1ρcCpcdc

(Lcphcp(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]:

∂Tp

∂t=

kp

ρpCpp

∂2Tp

∂2zp− vx

∂Tp

∂x(3.10)

where the term vx∂Tp

∂x [C/s] is the convection transport of energy in the machine

direction x, and kp

ρpCpp

∂2Tp

∂2z [C/s] is the conductive heat transfer in the thicknessdirection of the cylinder. Tp [C] is the temperature of the paper, t [s] is the timeindex, zp [m] is the space coordinate in the thickness direction of the paper, andx [m] is the space coordinate in the machine direction. The properties of the paper,

3.3 Heat balance of the paperboard 19

kp [W/m C], Cpp [J/kg C] and ρp [kg/m3], are the thermal conductivity, thespecific capacity and the density of the paper, respectively. They are described insection 3.5.

Numerical solution

In the machine direction (x), since there is only one node, we use the same methodas 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 boundarycondition in the machine direction (see figure 3.1). For ease of notation, we skipthe index j:

∂Tp,i

∂x=

1Lcp

(Tp,i − Tpin,i) (3.12)

where Lcp is 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 ∆zi is the discretization step at the point zi.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).

∂Tp

∂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,k is the distance between the points situated at zl and zk.


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 thelayer i. Inserting (3.12) and (3.16) into (3.10) the differential equations of the nodesin the middle of each layer Tp,i are described as follows:

∂Tp,i

∂t=

4kp,i

ρp,iCpp,id2p,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,i ρp,i Cpp,i dp,i are 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,i are derived by inserting (3.12) and (3.16) into (3.10):

∂Tp,i

∂t = 4kp,i−1

ρp,i−1Cpp,i−1d2p,i−1

(Tp,i−1 − Tp,i)

+ 4kp,i+1

ρp,i+1Cpp,i+1d2p,i+1

(−Tp,i + Tp,i+1)− vx

Lcp(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 ofthe paper in contact with the fabric or the air, Tp,7, we use the same method asPersson [35] and introduce a virtual point Tp,8, situated at the distance dp,6/2, onthe opposite side of Tp,6. The equation (3.16) is then written:

∂2Tp

∂2z

∣∣∣∣7

=4

d2p,6

(Tp,8 − 2Tp,7 + Tp,6) (3.19)

To remove Tp,8 in the equation, we use the boundary conditions between the sur-face 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 insidethe 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/m2 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.

3.3 Heat balance of the paperboard 21

Figure 3.4: Boundary condition at the surface of the paper in contact with theair, where Tp,i is the board temperature at position i, Ta is the temperature of thesurrounding air and dp,6 the 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

d2p,6

(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 = 4kp,6

ρp,6Cpp,6d2p,6

(2Tp,6 − 2Tp,7 + dp,6

kp,6(−mλ + hpa(Ta − Tp,7))

)

− vx

Lcp(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 ofthe paper in contact with the cylinder, Tp,1, we use the same method as previouslyand introduce a virtual point Tp,0, situated at the distance dp,2/2, on the oppositeside of Tp,2 [35]. The equation (3.16) is then written:

∂2Tp

∂2z

∣∣∣∣1

=4

d2p,2

(Tp,2 − 2Tp,1 + Tp,0) (3.25)


To remove Tp,0 in the equation, we use the boundary conditions between the sur-face 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 thecylinder, where Tp,i is the board temperature at position i, Tc is the temperature ofthe cylinder and dp,2 the 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 fromthe 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 Tp,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

d2p,2

(2Tp,2 − 2Tp,1 +dp,2

kp,2(hcp(Tc − Tp,1)) (3.29)

3.4 Mass balances within the paper web 23

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

∂Tp,1

∂t=

4kp,2

ρp,2Cpp,2d2p,2

(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 paperis assumed to be composed of two components: the dry component, consisting offibers and fillers (subscript dry) and the water (subscript w). To not increase thecomplexity of the model, the distinction between the two phases of water, liquid orvapor, is not included in this work, and the presence of air inside the paper, formedwhen the water leaves the pores, is not modelled. Descriptions of mass balancesincluding the air, liquid and vapor can be found in the works of Baggerud [2] andKarlsson [28].

Mass balance of dry material

The mass of dry material per layer Gdry,i [kgdry/m2] remains constant during thedrying. In the numerical computations in the thesis, the mass of dry matter in eachlayer 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 twotypes 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 mdiff [kgw/m2s] given by Fick’slaw [3]. The driving force for the diffusive transport of water is the gradientin the thickness direction of the moisture content (we consider that the dif-fusion of water only occurs in the thickness direction):

mdiff = −ρdryDwp∂u

∂z=

Gdry

∆zDwp

∂u

∂z(3.32)

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


Mass balance for the layer in contact with the air

For the layer in contact with the air, we take into account the water evaporatedinto the air and the diffusion from the middle layer:

∂u

∂t

∣∣∣∣6

=4Dwp(

dp(6) + dp(4)

)2 (u4 − u6)− m6

Gdry,6− vx

Lcp(u6 − uin,6) (3.33)

where the first term of the right hand side represents the diffusion of water fromlayer 4 to layer 6, with Dwp [m2/s] the diffusion coefficient of water into the pa-per, dp(i) [m] the thickness of layer i and ui [kgw/kgdry] the moisture content oflayer i. The second term represents the evaporation from the paper surface tothe surrounding air, where m6 is the evaporation rate [kgw/m2s] from the sur-face of layer 6 and Gdry,6 [kgdry/m2] is the dry basis weight of layer 6. The lastterm represents the convection transport of water in the machine direction, wherevx [m/s] is the speed of the machine, Lcp [m] the length of the contact zone anduin,6 [kgw/kgdry] the incoming moisture content of layer 6.

Mass balance for the layer in contact with the cylinder

For the surface in contact with the cylinder, there is no evaporation; we just haveto take into account the diffusion to the middle layer:

∂u

∂t

∣∣∣∣2

=4Dwp(

dp(2) + dp(4)

)2 (u4 − u2)− vx

Lcp(u2 − uin,2) (3.34)

where the first term of the right hand side represents the diffusion of water fromlayer 2 to layer 4, with Dwp [m2/s] the diffusion coefficient of water into the pa-per, dp(i) [m] the thickness of layer i and ui [kgw/kgdry] the moisture content oflayer i. The last term represents the convection transport of water in the machinedirection, where vx [m/s] is the speed of the machine, Lcp [m] the length of thecontact zone and uin,2 [kgw/kgdry] the incoming moisture content of layer 2.

Mass balance for the middle layer

For the node in the middle layer, we consider the diffusion to the two other layers:

∂u

∂t

∣∣∣∣4

=4Dwp(

dp(2) + dp(4)

)2 (u2−u4)+4Dwp(

dp(6) + dp(4)

)2 (u6−u4)− vx

Lcp(u4−uin,4) (3.35)

where the first and second term of the right hand side represent the diffusion ofwater from layer 2 to layer 4 and layer 6 to layer 4, with Dwp [m2/s] the dif-fusion coefficient of water into the paper, dp(i) [m] the thickness of layer i andui [kgw/kgdry] the moisture content of layer i. The last term represents the con-vection transport of water in the machine direction, where vx [m/s] is the speedof the machine, Lcp [m] the length of the contact zone and uin,4 [kgw/kgdry] theincoming moisture content of layer 4.

3.5 Physical properties 25

3.5 Physical properties

Properties of the cylinder

All the cylinders in the drying section are made of cast iron and have the sameproperties:

• Thickness of the shell: dc = 0.034 m

• Width: 7.15 m

• Diameter: 1.8 m

• Thermal conductivity: kc = 45 W/m C

• Specific heat capacity: Cpc = 500 J/kg C

• Density: ρc = 7300 kg/m3

Properties of the board

All the equations in this section, derived from the thesis of Persson [35], are com-puted for each layer i.

Board heat capacity

The heat capacity depends on the moisture content and the heat capacity of thefibers:

Cpp,i(ui, Tp,i) =Cpdry,i + ui · Cp,w(Tp,i)

1 + ui(3.36)

where the heat capacity of the fibers Cpdry,i is a parameter that we identify (seesection 3.8) and the heat capacity of the water Cpw(Tp,i) is obtained from a heatcapacity table ([51] p. 772).

Board thermal conductivity

The thermal conductivity of the layer i, kp,i [W/m C] is depending on the mois-ture content and the thermal conductivity of the fibers kdry,i [W/m C]:

kp,i(ui, Tp,i) =kdry,i + ui · kw(Tp,i)

1 + ui(3.37)

The thermal conductivity of the dry material kdry,i is a parameter that we identifyand is described in section 3.8. The thermal conductivity of the water inside theweb is obtained from a thermal conductivity table ([51] p. 772).


Density of the paper

We compute the density ρp,i [kg/m3] of layer i as follows:

ρp,i(ui, Tp,i) =1 + ui

ui

ρw(Tp,i)+ 1

ρdry,i

(3.38)

where the density of water ρw [kg/m3] is given from a density table ([51] p772).In the numerical computations, the dry density ρdry,i [kg/m3] of each layer i istaken from the bending stiffness predictor estimates computed by Pettersson [36].However, since the densities in Pettersson [36] are computed after the calenderingthat increases the densities, we need to compute the dry densities in the dryingsection. Two approaches are possible:

• The first alternative is to compute the densities using the estimated potentialdensities (equations (4.39) and (4.40) in [36]).

• The other possibility is to take the estimated densities after the calenderingand calculate backwards the density in the drying section, by consideringthe line loads in the calenders (equations (4.41) to (4.43) in [36]).

The first method seems more natural since we compute the densities using thepulp properties of the layers. However, the second method should be more accu-rate since the estimates of the densities are corrected by the Kalman filter in [36].Since only the total density is measured, the Kalman filter corrects the density forthe middle layer; therefore only the density of the middle layer differs when tak-ing one or the other methods. In this work, we choose to use the density computedwith the Kalman filter.

Thickness of the paper

Once we have the layer density ρp,i, the thickness dp,i [m] of layer i is computedas follows from (3.38):

dp,i(ui, Tp,i) =Gdry,i · (1 + ui)

ρp,i(ui, Tp,i)(3.39)

where Gdry,i [kg/m2] is the basis weight of dry material for the layer i. In the nu-merical computations, the dry basis weight is obtained from the bending stiffnesspredictor estimates computed by Pettersson [36].

Paper width

The paper width is around 7 m at beginning of the drying section and around6.7 m at the end of the drying. A fitted coefficient is calculated to take into accountthe shrinkage in the width direction. This coefficient is applied in the free draw,since the shrinkage happens there, but not in the stack dryers where the fabric,tightly holding the paper, prevents shrinkage.

3.5 Physical properties 27

Properties of the paper surface in contact with the fabric or the air

Heat transfer coefficient between paper and air hpa

The heat transfer coefficient between paper and air is calculated using the dimen-sionless numbers:

hpa(Ta) =Nu(Ta) · ka(Ta)

Lcp(3.40)

where Nu [-] is the Nusselt number, ka [W/m2 C] is the conductivity of the airand Lcp [m] is the length of contact between the cylinder and the paper.

We assume, like Persson [35], that the fabric does not affect the heat transfercoefficient from the paper to the air:

hpfa = hpa (3.41)

Persson [35] and Wilhelmsson [54] suggest adjusting this coefficient to take intoaccount both the heat and mass transfer with the following:

h?pa =

mCp,v

exp

mCp,vhpa

−1)

if Tp < Ta

mCp,v

− exp−

mCp,vhpa

−1 if Tp > Ta

(3.42)

where Cp,v [J/kg C] is the heat capacity of the vapor in the paper.

Mass transfer coefficient

The mass transfer coefficient between the paper and the air Kgpa [m/s] is alsodepending on dimensionless numbers:

Kgpa(Ta) = Nu(Ta) ·(

Sc(Ta)Pr(Ta)

) 13

· Dwa(Ta)Lcp

(3.43)

where the dimensionless numbers Sc and Pr are the Schmidt Number and thePrandtl Number respectively. Dwa [m2/s] is the diffusion coefficient of water intothe air, computed by using calculations and tables in [51].

The presence of the fabric is assumed to reduce the mass transfer of water be-tween the paper and the air. This reduction is modelled by a coefficient FRF(Fabric reduction factor), described in section 3.8:

Kgpfa =(

100− FRF

100

)·Kgpa (3.44)


Vapor partial pressure of the paper

The vapor partial pressure for free water in the paper psat [Pa] is calculated fromAntoine equation (3.45) [35]:

psat(Tp) = 133.322 · e18.3036− a1

Tp+227.03

(3.45)

where the coeficient a1 [K−1] is usually set to the value 3816. In this work, how-ever, we choose to identify it (see further in section 3.7).

If the paper is wet, the vapor partial pressure of the water at the surface pp [Pa]is equal to the vapor partial pressure for free water. But when the water insidethe paper remains inside the pores, the amount of free water is limited. The vaporpartial pressure of the paper at the surface is therefore:

pp(Tp, u) = φ(u, Tp) · psat(Tp) (3.46)

where φ [-], the relative humidity of the air, is expressed as a sorption isotherm.This phenomena is described in [29], p. 67 and [35], p. 37. The relative humidity ofthe air is assumed to be equal to 1 when the water inside the paper is free water anddecreases when the moisture in the paper is lower than a critical content (around0.4 kgw/kgdry according to [35] and [3]). We further discuss the choice of sorptionisotherm in the semi-physical adjustments in section 3.7.

If the partial pressure of water in the paper is higher than the atmosphericpressure, flash evaporation occurs (this phenomena is modelled, for example, inthe work of Wilhelmsson (see eq 3.12 in [54]) and Karlsson [28]). In this work, wedo not consider the flash evaporation, since when the board reaches high tempera-tures, the moisture content at the surface is low, and consequently the board vaporpartial pressure at the surface is not higher than the partial pressure of water in theair.

Evaporation rate

The evaporation rate is calculated by the Stefan equation [35]:

m(Tp, Ta, u) =Kgpa(Ta) ·Mw · Pa

R · (Tp + 273.15)ln

(Pa − pa

Pa − pp(Tp, u)

)(3.47)

where Mw [kg/mol] is the molar mass of water, R [J/mol K] is the gas constant,Pa [Pa] is the absolute pressure of the air, pa [Pa] is the partial pressure of water inthe air, described in section 3.5 and pp [Pa] is the partial pressure of water in thepaper, computed in equation (3.46).

Heat of vaporization

The heat of evaporation λ [J/kg] is a linear function of the temperature of the paper[35]:

λ(Tp) = (2503.28− 2.43665 · Tp) · 1000 (3.48)

3.6 Summary of the physical model equations 29

For moisture below the critical content, the heat of sorption should also be takeninto account [35, 28, 29]. The heat of evaporation becomes:

λ′(Tp) = λ(Tp) + λsorp(Tp, u) (3.49)

where λsorp [J/kg], the heat of sorption, is discussed in the semi-physical adjust-ments in section 3.7.

Properties of the surrounding air

The properties of the surrounding air are measured during static measurements,but since they are neither constant in the drying section nor measured on-line, wemodel the temperature and humidity of the air as a linear interpolation between aconstant and the properties of the paper.

The measured temperature of the air is displayed in figure 5.1. In the beginningof the drying, where the paper is cooler, the temperature of the air is lower. There-fore, since we believe that the temperature of the paper influences the temperatureof the air, we model the temperature of the air as follows:

Ta = αT · Ta,ave + (1− αT ) · Tp with 0 < αT < 1 (3.50)

where the average air temperature, Ta,ave [C] and the interpolation coefficient αa

[-] are fitted to the measurements (see chapter 5, table 5.1).The measured partial pressure of water in the air, displayed in figure 5.2, is

lower at the beginning of the drying section, because there is less evaporation fromthe paper to the air due to the low partial pressure of water in the paper (sincethe temperature of the paper is low). Therefore, we model the partial pressure ofwater in the air as a linear interpolation between a constant pa,ave and the partialpressure of water in the paper:

pa = αp · pa,ave + (1− αp) · pp, with 0 < αp < 1 (3.51)

where the average partial pressure of water in the air, pa,ave [Pa] and the interpo-lation coefficient αp [-] are fitted to the measurements (see chapter 5, table 5.1).

3.6 Summary of the physical model equations

For ease of notation, we remove the temperature dependencies in equations (3.36)(3.37), (3.39) and the values of ρw(Tp), Cpw(Tp) and kw(Tp) are considered con-stant, since the influence of the temperature is very small on these values. We alsoremove the dependency of the air temperature in equations (3.40) as well as for thedimensionless numbers Nu, Sc and Pr. The temperature dependencies are keptin the simulation of the model, but they are omitted in the observability and iden-tifiability analysis in chapter 4 for ease of computation. To simplify the notation,


we define the following functions:

f1(u) = 4kp(u)ρp(u)Cpp(u)dp(u)2

f2(u) = 4hcp(u)ρp(u)Cpp(u)dp(u)

f3(Tp, u, Ta) = − 4m(Tp,Ta,u)λ(Tp)ρp(u)Cpp(u)dp(u)

f4(u) = 4hpa

ρp(u)Cpp(u)dp(u)

f5(uj , uk) = 4Dwp

(dp(uj)+dp(uk))2(uj − uk)

f6(Tp, Ta, u) = − m(Tp,Ta,u)Gdry,6

(3.52)

The nonlinear differential equations for the temperature of the paper (3.17),(3.18), (3.24), (3.30) and for the moisture content inside the paper (3.33), (3.34),(3.35) are summarized as follows:

Paper in contact with an upper cylinder∂Tp,1

∂t = f1(u2)(2Tp,2 − 2Tp,1) + f2(u2)(Tc − Tp,1)− vx

L (Tp,1 − Tpin,1)∂Tp,2

∂t = f1(u2)(Tp,3 − 2Tp,2 + Tp,1)− vx


∂t = f1(u2)(Tp,2 − Tp,3) + f1(u4)(−Tp,3 + Tp,4)− vx


∂t = f1(u4)(Tp,5 − 2Tp,4 + Tp,3)− vx




∂t = f1(u6)(Tp,7 − 2Tp,6 + Tp,5)− vx


∂t = f1(u6)(2Tp,6 − 2Tp,7) + f3(Tp,7, u6, Ta) + f4(u6)(Ta − Tp,7)−vx

L (Tp,7 − Tpin,7)∂u2∂t = f5(u4, u2)− vx

L (u2 − uin,2)∂u4∂t = f5(u2, u4) + f5(u6, u4)− vx

L (u4 − uin,4)∂u6∂t = f5(u4, u6) + f6(Tp,7, Ta, u6)− vx

L (u6 − uin,6)(3.53)

Paper in the free draw∂Tp,1



∂t = f1(u2)(Tp,3 − 2Tp,2 + Tp,1)− vx




∂t = f1(u4)(Tp,5 − 2Tp,4 + Tp,3)− vx




∂t = f1(u6)(Tp,7 − 2Tp,6 + Tp,5)− vx



L (Tp,7 − Tpin,7)∂u2∂t = f5(u4, u2) + f6(Tp,1, Ta, u2)− vx

L (u2 − uin,2)∂u4∂t = f5(u2, u4) + f5(u6, u4)− vx

L (u4 − uin,4)∂u6∂t = f5(u4, u6) + f6(Tp,7, Ta, u6)− vx

L (u6 − uin,6)(3.54)

3.7 Semi-physical adjustments 31

3.7 Semi-physical adjustments

Vapor partial pressure for free water

When simulating the physical model, the temperature of the paper is decreasingtoo much in the free draw compared to the measurements and seems to reach anequilibrium point (see, for example, figure 4.3). Tests with the physical parameterscould not remedy this problem. A possible explanation, suggested by Wilhelms-son [54], is that the drying section is a fast process, while the time required for aporous material to reach its equilibrium state is very long. Therefore, we modifythe coefficient a1 in the equation for the vapor partial pressure for free water (3.45)in order to lower the effect of the evaporation on the decrease of temperature inequation (3.24).

Sorption phenomena

When the moisture inside the paper is below a certain rate (around 0.4 kgw/kgdry),the amount of free water at the surface becomes limited and the evaporation ca-pacity is decreased. This phenomenon is called sorption. The partial pressure ofvapor inside the paper is equal to the partial pressure of vapor for free water mul-tiplied by a sorption isotherm φ(Tp, u). The sorption isotherm is defined as therelative humidity of air [-] as a function of the equilibrium moisture content of thepaper [kgw/kgdry] for a given temperature [29], p. 67. The sorption phenomenonis described, for example, in [29]. Several formulas of the sorption isotherm canbe found in the literature, but according to Karlsson [28], the choice of the sorp-tion isotherm does not influence the drying significantly. We therefore considerthe sorption isotherm in Heikkilä [20], which is suitable for mechanical pulp (seefigure 3.6).

Since we believe that the diffusion of water inside the paper is a slow process,the moisture content in the computation, situated in the middle of the surfacelayer, is assumed to be considerably higher than the one at the surface. There-fore, the sorption isotherm is modified to take the sorption phenomenon into ac-count also for higher moisture contents (the relative humidity of air increases until0.8 kgw/kgdry). Figure 3.6 shows the modified sorption isotherm. In the new sorp-tion isotherm, the temperature effect has been removed since the influence of thetemperature is already taken into account by the coefficient a1. A similar approachis applied to the heat of sorption. The chosen expressions for the relative humidityof air and the heat of sorption are:

φ(u, Tp) = φ(u) =(1− e(−5·u−0.3)

) · (1− e(−50·u))

λsorp(u) = 200 · u0.4 ·(

1−φ(u)φ(u)

) (3.55)

where the coefficients of this expression are fitted manually to get a relative hu-midity of air (heat of sorption) similar to the sorption isotherm (heat of sorption)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Sorption isotherm: Relative humidity of air as a function of the moisture content u

Rel

ativ

e hu

mid

ity o

f air

[−]

moisture content u in kgw

/kgdry

Heikkilä (T=90 C)Modified sorption isotherm

(a) Sorption isotherm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

100

200

300

400

500

600

700

800

900

1000Heat of Sorption as a function of the moisture content

Hea

t of s

orpt

ion

[kJ/

kg]

moisture content u in kgw

/kgdry

Heikkilä (T=90 C)Modified heat of sorption

(b) Heat of sorption

Figure 3.6: Comparison between the sorption isotherm and the heat of sorption inHeikkilä [20] and in this work.

of Heiikilä for lower moistures (around 0.3 kgw/kgdry) but reaches the maximum(minimum) at higher moisture content (around 0.8 kgw/kgdry). We choose to notidentify the coefficients of equation (3.55) in order to not increase the complexityof the process of identification of the parameters.

3.8 Parameters, inputs and outputs of the model

This section describes the parameters, the inputs and the outputs of the model.The parameters are unknown constants to be identified. Some of the model inputsare not measured on-line; we therefore also need to estimate them. We make a dis-tinction between the parameters, assumed to be the constant, and the unmeasuredinputs that are time-varying. An identifiability analysis and a sensitivity analysisof the parameters are presented in sections 4.2 and 4.3. The identification proce-dure is described in chapter 5.

Parameters to identify

The following parameters are candidates for the identification.

Heat transfer between steam and cylinder hsc: The heat transferred from the steamto the cylinder is difficult to estimate since the amount of condensate insidethe cylinder is not known. It is a common assumption that the condensateis totally removed by the siphons and that the heat transfer coefficient be-tween steam and cylinder is a constant to identify [35, 54, 28]. According toWilhelmsson [54], the constant should be in the interval 800–5000 W/m2 C.Karlsson and Stenström [27] obtain a value of 1900 W/m2 C for the board

3.8 Parameters, inputs and outputs of the model 33

machine in Frövi and notice that the fitted value of this coefficient increaseswhen decreasing the diffusion coefficient, to compensate the total heat trans-ferred.

Heat transfer coefficient between cylinder and paper hcp: The heat transfer coef-ficient between the cylinder and the paper is supposed to depend on thecontact area between cylinder and paper [3, 35, 54]. Wilhelmsson [54] andPersson [35] assume that the contact area is linearly dependent on the mois-ture content:

hcp(u) = hcp(0) + hcp,inc · u (3.56)

where hcp(0) is in the interval 200–500 W/m2 C and hcp,inc is set to 955. Theheat transfer should be independent of the fabric tension if the tension is inthe interval of 2–6 kN/m, which is in the range of the typical values appliedin the machine [35, 54].

Baggerud [3] and Karlsson [28] consider that hcp can be modelled as a con-stant when the internal mass transfer is included. However, since this modelonly describes the diffusion of water in the thickness direction and the con-tact area is difficult to model, we use equation (3.56). Karlsson and Sten-ström [27] found a fitted value of 2200 W/m2 C but point out that the fittedvalue is also dependent on the diffusion coefficient of the water inside thepaper.

In the first steam group, the lower cylinders (2, 4, 6) are covered with teflon.The heat transfer from the cylinder to the board therefore increases as a re-sult of the smoother surface. This improvement can be seen by observingthe measured temperature of the first seven cylinders (for example in fig-ure 5.4). Even though the supplied steam is the same for all cylinders, thetemperature of the upper cylinders is higher than the lower cylinders sincethe lower cylinders release more heat to the board. We therefore multiplythe heat transfer coefficient hcp by two for the cylinders 2, 4 and 6, since thisvalue gives the best fit for the temperature of the cylinders.

Diffusion coefficient of water/vapor into the paper Dwp: The coefficient for thediffusion through the paper should be in the range of 1 ·10−10–2 ·10−6 m2/sfor the water (liquid) and 2.1 ·10−8–5.4 ·10−6 m2/s for the vapor [3]. Bag-gerud [3] and Karlsson [28] model the liquid and vapor diffusion coefficientsas functions of the moisture content, but since in this thesis, we do not differ-entiate the two phases of the water, we model the water diffusion coefficientas a constant.

Fabric reduction factor FRF : This coefficient is used to take into account the ef-fect of the fabric on the mass transfer between the paper and the air. Thereduction of mass transfer due to the felt is around 30–50 % [35] and is as-sumed to be the same for all the cylinders in the drying section.


Dry heat capacity Cpdry : Since there are few static measurements available for theidentification, we assume in this work that the dry heat capacity does notdepend on the quality of the pulp. Several values are found in the literature:1256 J/kg C [35], 1300 J/kg C [3], 1550 J/kg C [28].

Dry thermal conductivity kdry : We assume, as for the dry heat capacity, that thedry thermal conductivity does not depend on the quality of the pulp. Bag-gerud [3] and Karlsson [28] suggest a dry thermal conductivity of the fibersof 0.157 W/m C and Persson [35] a value of 0.08 W/m C.

Dummy parameter a1: This parameter is applied to reduce the effect of the evap-oration on the decrease of temperature. Since it is a dummy parameter, theminimal and maximal values are set manually. The minimal value is set tothe physical value and the maximal value is set to 3880.

Unmeasured inputs

The unmeasured inputs are not measured on-line. However, they are measuredoccasionally, for example in the static measurements described in chapter 5.

Incoming moisture content uin: The water concentration of the paperboard leav-ing the press section is known to be between 0.56 % and 0.64 %, which cor-responds to a moisture content of 1.27–1.78 kgw/kgdry .

Incoming temperature of the paper Tp,in: The temperature of the board enteringthe drying section is in the range of 45–55 C.

Properties of the surrounding air: The parameters for the properties of the sur-rounding air (Ta,ave, αT , pa,ave, αp ), introduced in equations (3.50) and(3.51), section 3.5, are fitted to each set of measurements (see chapter 5).

On-line measured inputs

The following variables are measured or calculated on-line and are available in theinformation system of the machine:

• The steam pressure for each steam group Ps (measured)

• The speed of the machine vx (measured)

• The dry basis weight of each layer Gdry (calculated)

• The dry board density of each layer ρdry (calculated)

The dry basis weight and density of the finished board are obtained from the cal-culations of the bending stiffness predictor [36] and are available in the on-lineinformation system. To compute the density in the drying section, we use the loadof the calendering and the estimated load coefficients (see equations (4.41) to (4.43)in [36]).

3.8 Parameters, inputs and outputs of the model 35

On-line measured outputs

The measuring frame situated at the end of the drying section measures the twofollowing outputs:

• The total moisture content of the board uout

• The top layer board temperature Tp7,out

Chapter 4

Observability, Identifiability andSensitivity Analyses

Define X(t) the vector of n states, θ the vector of r constant parameters to identify,U(t) the vector of the m measurable inputs, and Y (t) the vector of p measurableoutputs, a nonlinear system can be expressed as:

Σnl

X(t) = F (X(t), θ, U(t))Y (t) = H(X(t))

(4.1)

Since all the computed state variables X(t) can not be measured, we want to es-timate them using the observations of the outputs Y (t) and the inputs U(t) fora certain set of parameters θ. To know if this is possible, we need to study theobservability of the system. It is also of interest to know if the parameters areidentifiable, i.e. the set of parameters giving a specific input–output trajectory isunique. Moreover, in order to evaluate the impact of the parameters on the modeland select the most influent ones that remain for the identification, we carry outa sensitivity analysis of the parameters. We should point out that in the generalcase, the concepts of observability and identifiability are global for linear systems,but only local for nonlinear systems. In other words, if a linear system is observ-able or identifiable at some point X0, it is also observable or identifiable for all X .But for nonlinear systems, the observability and identifiability can be determinedonly for the set of X in the neighbourhood of a given point X0. A brief literaturereview on the analysis of nonlinear observability and identifiability is presentedin section 2.1. The sections in this chapter describe an analysis of the three con-cepts — observability, identifiability and sensitivity — for the model described inchapter 3.

37

38 Observability, Identifiability and Sensitivity Analyses

4.1 Observability analysis of the physical model

A system is said to be observable, if we can determine the states X(t) given theoutputs Y (t), the inputs U(t), and the set of parameters θ. Since the tools of dif-ferential geometry and differential algebra mentioned in section 2.1 are of highcomplexity increasing with the number of states, we use instead the linearizationprinciple for observability [44] and determine the observability of the linearizationof the system at a constant operational point. If the linearized system is observ-able, we can conclude that the nonlinear system is locally observable. The theoryused in this analysis is first introduced and then applied to the physical model.

Theoretical background

The theory on observability for linear system is well established and simple toapply. It is therefore easier to study the linearized system. In this section, we firststate the linearization principle for observability and then show how to linearizea system. Finally, we describe how to analyse the observability of a linear systemby transforming it into the canonical form.

Linearization Principle for Observability

To analyse the observability of a nonlinear system, one can study the observabilityof the linearized system at a constant operational point by using the linearizationprinciple for observability that states that if the linearization around x is observ-able, then the nonlinear system is locally observable around x.

Theorem 1 (Linearization Principle for Observability [44], p. 209) Assume that Σnl

is a continuous-time system over R of class C1, and let Γ = (ξ, ω) be a trajectory for Σnl

on a interval I = [σ, τ ]. Then a sufficient condition for Σnl to be locally observable aboutthe operational point x is that the linearized system Σl is observable on I = [σ, τ ].

Linearization of a system

The linearized system at the constant operational point (X(t), U(t)), given theconstant output Y (t) is obtained by the following expression:

Σl

δX(t) = AδX(t) + BδU(t)δY (t) = CδX(t)

(4.2)

whereδX(t) = X(t)− X(t)δU(t) = U(t)− U(t)δY (t) = Y (t)− Y (t)

A = ∂F∂X

∣∣X,U

B = ∂F∂U

∣∣X,U

C = ∂H∂X

∣∣X

4.1 Observability analysis of the physical model 39

Observability analysis of a linear system

Consider the following linear system.

Σl

X(t) = AX(t) + BU(t)Y (t) = CX(t)

(4.3)

To study the observability of a linear system, one needs to check the rank ofthe observability matrix Ω = [C; CA; CA2; · · · ; CAn−1]T . If Ω has full rank, i.e.rank(Ω) = n, the linear system is observable. Otherwise (rank(Ω) < n), the linearsystem is not observable.

If A has n distinct real eigenvalues, with a coordinate change X = TX(t),we can transform the system into the modal canonical form (also called Jordancanonical form):

Σ

˙

X(t) = AX(t) + BU(t)Y (t) = CX(t)

(4.4)

where A = TAT−1 is a diagonal matrix containing the eigenvalues of A, T−1 is amatrix containing the eigenvectors of A, B = TB and C = CT−1. Note that thistransformation is possible since A has n distinct real eigenvalues, and thus T−1

and T are non singular.The modal canonical form provides an easy way to check observability. For

each output i, the rank of the observability matrix Ωi = (Ci, CiA, · · · , CiAn−1)T is

the number of non zero items in the row Ci, since A is diagonal and has distincteigenvalues. A zero in the jth column of the row Ci corresponds to an unobserv-able state Xj by the output Yi.

Application to the physical model

We are mainly interested in the feasibility of building an observer. The observabil-ity analysis is therefore derived for the physical model, i.e. to simplify the studythe semi-physical adjustments in section 3.8 are not considered. Since the modelcomputes 21 states for each cylinder (11 nodes for the contact zone and 10 nodesfor the free draw), a complete observability analysis for the whole drying sectionseems hardly feasible. We therefore consider the observability for the sub-models(i.e. the contact-zone sub-model and the free-draw sub-model) and for the model ofa complete cylinder (concatenation of a contact-zone sub-model and a free-drawsub-model). In this section, we derive the observability analysis for the contact-zone sub-model. The observability analysis for the free-draw sub-model is similarand is mentioned at the end of the section. In appendix B, the observability analy-sis is further examined for a complete cylinder.

For ease of computation, the sorption phenomenon is neglected in this study,since it only affects the computation for low moisture content. Thereby, the mois-ture dependency of m in the functions f3 and f6 in equation (3.52) is removed.


The chosen states of the model are the temperature and moisture inside thepaper. We assume that we know the properties of the incoming sheet, Tp,in anduin, as well as the temperature Tc of the cylinder surface and the temperature ofthe surrounding air Ta, and use them as inputs.

X =

Tp,1

Tp,2

Tp,3

Tp,4

Tp,5

Tp,6

Tp,7

u2

u4

u6

, U =

Tc

Tpin,1

Tpin,2

Tpin,3

Tpin,4

Tpin,5

Tpin,6

Tpin,7

Ta

uin,2

uin,4

uin,6

For the output, we assume that we can measure the board temperature at the twosurfaces, and the average moisture content of the board in the thickness direction.In practice, however, the moisture content where the paper is in contact with thecylinder is difficult to measure. The observability with only the temperature asoutput is discussed at the end of this section. The obtained output is linear:

Y =

1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 Gdry,2

Gdry,tot

Gdry,4Gdry,tot

Gdry,6Gdry,tot

X (4.5)

where Gdry,tot = Gdry,2 + Gdry,4 + Gdry,6 is the total dry basis weight.

Linearization of the equations (3.53)

We linearize the system at the constant operational point, using equation (4.2).

δTp,1

δTp,2

δTp,3

δTp,4

= A1

δTp,1

δTp,2

δTp,3

δTp,4

δTp,5

+ A2

(δu2

δu4

)+ B1

δTc

δTpin,1

δTpin,2

δTpin,3

δTpin,4

δTp,5

δTp,6

δTp,7

= A3

δTp,4

δTp,5

δTp,6

δTp,7

+ A4

(δu4

δu6

)+ B2

δTpin,5

δTpin,6

δTpin,7

δTa

δu2

δu4

δu6

= A5

δTp,7

δu2

δu4

δu6

+ B3

δTa

δuin,2

δuin,4

δuin,6


where

A1 =

A1 − vx

L

[I4]

0...0

with

A1 =

−2f1(u2)− f2(u2) 2f1(u2) 0 0 0f1(u2) −2f1(u2) f1(u2) 0 0

0 f1(u2) −f1(u2)− f1(u4) f1(u4) 00 0 f1(u4) −2f1(u4) f1(u4)

A2 =

∂f1(u2)∂u2

u

(2Tp,2 − 2Tp,1) +∂f2(u2)

∂u2

u

(Tc − Tp,1) 0

∂f1(u2)∂u2

u

(Tp,1 − 2Tp,2 + Tp,3) 0

∂f1(u2)∂u2

u

(Tp,2 − Tp,3)∂f1(u4)

∂u4

u

(−Tp,3 + Tp,4)

0∂f1(u4)

∂u4

u

(Tp,5 − 2Tp,4 + Tp,3)

B1 =

f2(u2)0...0

[vx

L · I4

]

A3 =

A3 − vx

L

0...0

[I3]

with

A3 =

(f1(u4) −f1(u4)− f1(u6) f1(u6) 0

0 f1(u6) −2f1(u6) f1(u6)

0 0 2f1(u6) −2f1(u6) +∂f3(Tp,7,u6,Ta)

∂Tp,7

Tp,u,Ta

− f4(u6)

)

A4 =

∂f1(u4)∂u4

u

(Tp,4 − Tp,5)∂f1(u6)

∂u6

u

(−Tp,5 + Tp,6)

0∂f1(u6)

∂u6

u

(Tp,7 − 2Tp,6 + Tp,5)

0∂f1(u6)

∂u6

u

2(Tp,6 − Tp,7) +∂f3(u6,Tp,7,Ta)

∂u6

Tp,u,Ta

+∂f4(u6)

∂u6

u

(Ta − Tp,7)

B2 =

[vx

L · I3

]

00

∂f3(u6,Tp,7,Ta)∂Ta

∣∣∣u,Tp,Ta

+ f4(u6)

A5 =

A5 − vx

L

0...0

[I3]

with

A5 =

0 ∂f5(u4,u2)∂u2

∣∣∣u

∂f5(u4,u2)∂u4

∣∣∣u

0

0 ∂f5(u2,u4)∂u2

∣∣∣u

∂f5(u2,u4)∂u4

∣∣∣u

+ ∂f5(u6,u4)∂u4

∣∣∣u

∂f5(u6,u4)∂u6

∣∣∣u

∂f6(Tp,7,Ta)∂Tp,7

∣∣∣Tp,Ta

0 ∂f5(u4,u6)∂u4

∣∣∣u

∂f5(u4,u6)∂u6

∣∣∣u

B3 =

00

∂f6(Tp,7,Ta)∂Ta

∣∣∣Tp,Ta

[vx

L · I3

]


where In is the n× n identity matrix.To summarize, the linearized system is given by the following expression.

δTp,1

δTp,2

δTp,3

δTp,4

δTp,5

δTp,6

δTp,7

δu2

δu4

δu6

=

0 0 00 0 0

A1 0 0 A2 00 0 0

0 0 0 00 0 0 A3 0 A4

0 0 0 00 0 0 0 0 00 0 0 0 0 0 A5

0 0 0 0 0 0

δTp,1

δTp,2

δTp,3

δTp,4

δTp,5

δTp,6

δTp,7

δu2

δu4

δu6

+

+

0 0 0 0 0 0 00 0 0 0 0 0 0

B1 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 B2 0 0 00 0 0 0 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 B3

0 0 0 0 0 0 0 0

δTc

δTpin,1

δTpin,2

δTpin,3

δTpin,4

δTpin,5

δTpin,6

δTpin,7

δTa

δuin,2

δuin,4

δuin,6

(4.6)

with the output, given by (4.5), already linear.We now insert the numerical values. We consider the contact zone of the upper

cylinder 53. We choose this position, since the displayed equations are derivedfor an upper cylinder. To strengthen the results, the observability analysis is alsocarried out for the cylinder 1 and leads to the same conclusions.

The parameters required for the simulation are:

• The length of contact of paper with the cylinder 53: L = 3.67 m

• The speed of the machine: vx = 8.1 m2/s

• The partial pressure of water in the air: pa = 24451 Pa

• The dry basis weight per layer: Gdry,2 = 0.056 kg/m2, Gdry,4 = 0.129 kg/m2,Gdry,6 = 0.062 kg/m2

• The dry density per layer: ρdry,2 = 654 kg/m3, ρdry,4 = 561 kg/m3, ρdry,6 =755 kg/m3


The operational point is obtained from the simulation:

X(t) =

103.7100.398.088.781.680.379.0

0.53750.68770.5627

U(t) =

124.674.175.676.681.4883.282.480.288

0.52900.70330.5786

We use the symbolic computer program Maple to derive the matrices A and B, andget:

A =

−120.3 98.5 0 0 0 0 0 188.7 0 0

49.2 −100.7 49.2 0 0 0 0 −43.2 0 0

0 49.2 −58.8 7.4 0 0 0 −92.6 52.8 0

0 0 7.4 −16.9 7.4 0 0 0 −12.1 0

0 0 0 7.4 −53.2 43.6 0 0 −40.7 48.7

0 0 0 0 43.6 −89.4 43.6 0 0 3.8

0 0 0 0 0 87.2 −113.1 0 0 159.6

0 0 0 0 0 0 0 −2.3 0.12 0

0 0 0 0 0 0 0 0.13 −2.4 0.13

0 0 0 0 0 0 −0.009 0 0.12 −2.3

B =

19.6 2.2 0 0 0 0 0 0 0 0 0 0

0 0 2.2 0 0 0 0 0 0 0 0 0

0 0 0 2.2 0 0 0 0 0 0 0 0

0 0 0 0 2.2 0 0 0 0 0 0 0

0 0 0 0 0 2.2 0 0 0 0 0 0

0 0 0 0 0 0 2.2 0 0 0 0 0

0 0 0 0 0 0 0 2.2 −0.3 0 0 0

0 0 0 0 0 0 0 0 0 2.2 0 0

0 0 0 0 0 0 0 0 0 0 2.2 0

0 0 0 0 0 0 0 0 −0.0003 0 0 2.2

C =

( 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0.23 0.52 0.25

)


Observability analysis of the linearized system

We transform the linearized system Σl into the modal canonical form Σl, using thecommand canon.m in the program Matlab, and get the following:

A =

−189 0 0 0 0 0 0 0 0 0

0 −171 0 0 0 0 0 0 0 0

0 0 −82 0 0 0 0 0 0 0

0 0 0 −76 0 0 0 0 0 0

0 0 0 0 −20 0 0 0 0 0

0 0 0 0 0 −8.9 0 0 0 0

0 0 0 0 0 0 −4.9 0 0 0

0 0 0 0 0 0 0 −2.6 0 0

0 0 0 0 0 0 0 0 −2.4 0

0 0 0 0 0 0 0 0 0 −2.2

B=

−12 −1.3 1.8 −0.7 0.03 −0.007 −0.002 −0.002 0.0004 1.4 0.21 −0.03

0.02 0.002 −0.002 −0.002 0.03 −0.68 1.8 −1.3 0.2 −0.16 −0.22 1.8

−9.4 −1 −0.81 1.8 −0.21 0.13 −0.046 −0.066 0.0099 4.2 −1.1 −0.05

−0.74 −0.081 −0.077 0.12 0.21 −1.8 0.88 1 −0.15 0.46 −1.1 −1.2

2.6 0.29 0.59 0.37 −1.9 0.42 0.64 0.31 −0.045 0.64 −1.6 −4.2

4.2 0.46 1 0.99 −0.088 −1.1 −1.1 −0.46 0.066 8.1 −16 19

−3.6 −0.4 −0.92 −0.99 −1.2 −0.92 −0.84 −0.33 0.045 −22 −0.79 37

−0.88 −0.1 −0.22 −0.26 −0.48 −0.68 −0.7 −0.37 −0.084 640 −1400 1000

0.54 0.059 0.14 0.16 0.31 0.44 0.46 0.24 0.063 750 −200 −730

−0.14 −0.015 −0.035 −0.042 −0.079 −0.12 −0.13 −0.066 −0.018 400 330 200

C =

−0.8 0.0013 −0.76 −0.059 0.25 0.38 −0.33 −0.0014 −0.0094 0.0045

−6.7e−4 −0.82 −0.049 0.75 0.26 −0.38 −0.28 0.0082 −0.016 0.0085

−8.2e−9 −1.1e−5 −1.4e−7 2.4e−5 3.4e−5 −1.2e−4 −2.2e−4 −2e−4 −3.9e−4 0.0025

A has distinct eigenvalues and C has no zeros. We therefore conclude that theobservability matrix has full rank and the linearized system is observable.

The observability analysis can easily be carried out for other outputs than (4.5).The board moisture content is difficult to measure for all cylinders (see section 5.1);we therefore consider the case when only the surface temperatures Tp,1 and Tp,7

are measured:

Y =(

1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0

)X (4.7)

Since only the average moisture content and the top side temperature are mea-sured at the end of the drying section, the following output is also of interest:

Y =

(0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 Gdry,2

Gdry,tot

Gdry,4Gdry,tot

Gdry,6Gdry,tot

)X (4.8)

4.2 Identifiability analysis of the physical model 45

The contact-zone sub-model is also found observable with the outputs (4.7) and(4.8). However, if we measure only one board variable (Tp,1, Tp,7 or u), the sub-model becomes not observable.

The observability analysis for the free-draw sub-model is similar and leads tothe same conclusion: the free-draw sub-model is observable if we measure at leasttwo of the board variables.

In appendix B, we concatenate a contact-zone sub-model and a free-draw sub-model to further investigate the observability for one cylinder. The analysis showsthat we need to measure at least the temperatures of both surfaces of the board inthe contact zone and in the free draw to ensure observability.

Conclusion

The observability analysis shows that the contact-zone and free-draw sub-modelsare observable if we measure at least two of the following board variables: theboard temperature at the top surface or the bottom surface and the average boardmoisture content in the thickness direction. Consequently, since the drying sectionis a concatenation of observable models, we consider that the model described inchapter 3 is observable under the assumption that we measure the output (4.7) forthe contact zone and free draw of all cylinders. In appendix B, we show that theseare indeed the minimal measurements required to ensure observability.

4.2 Identifiability analysis of the physical model

The unknown parameters of a model are said to be identifiable if they can be de-termined given an input–output trajectory. In this work, determine means that theparameters given a special input–output trajectory are unique. Ljung [31] distin-guishes bewteen two kinds of identifiability: the structure identifiability, that con-siders the model structure, and the persistence of excitation, that examines if theinputs are informative enough to enable the identification of parameters. For theidentification, described in chapter 5, we are dealing with static data and a ma-chine with constant settings. We can therefore investigate the structural identifia-bility of the model, but not the persistence of the input.

The identifiability analysis is carried out on the physical model, with the samesimplifications as for the observability analysis. Since the primary goal of thisidentifiability analysis is to validate our choice of model structure, the study iscarried out before identification (and the sensitivity analysis) has been tested. Tosimplify the computations, the parameters Cpdry , kdry and a1 are set to physicalvalues from literature (see section 3.8). The set of parameters that we chose for theidentifiability analysis is Θ = hsc, hcp(0), FRF , Dwp.

We follow the approach suggested in [1, 56, 49] that views the identifiabilityproblem as a special case of the observability problem where the constant param-eters θ are considered as states with zero derivative. Since we want to estimate the


identifiability of the heat transfer coefficient between steam and cylinder hsc, weconsider the cylinder temperature as a state instead of an input.

The identifiability system is thus obtained by extending the observability sys-tem in previous section with the vector of new states Θ = hsc, hcp(0), FRF , Dwp.

δTc

δTp,1

δTp,2

δTp,3

δTp,4

δTp,5

δTp,6

δTp,7

δu2

δu4

δu6

= Ax

δTc

δTp,1

δTp,2

δTp,3

δTp,4

δTp,5

δTp,6

δTp,7

δu2

δu4

δu6

+ Aθ

δhsc

δhcp0

δhcpinc

δFRFδDwp

+ B

δTs

δTpin,1

δTpin,2

δTpin,3

δTpin,4

δTpin,5

δTpin,6

δTpin,7

δTa

δuin,2

δuin,4

δuin,6

(4.9)

where,

Ax =

A01 A02 0 0 0 0 0 0 A09

f2(u2) 0 0 00 0 0 00 A1 0 0 A2 00 0 0 0

0 0 0 0 00 0 0 0 A3 0 A4

0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0 A5

0 0 0 0 0 0 0

Aθ =

A11 A12 A13 0 00 A22 A23 0 0

O5×5

0 0 0 A84 00 0 0 0 A95

0 0 0 0 A105

0 0 0 A114 A115

4.2 Identifiability analysis of the physical model 47

B =

B01 0 0 0 0 0 0 B08 0 0 00 0 0 0 0 0 0 00 vx

L I4 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 B2 0 0 00 0 0 0 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 B3

0 0 0 0 0 0 0 0

A1, A2, A3, A4, A5, B2, B3 are defined in the observability analysisO is the zero matrixA01 = 1

ρcCpcdc

(−Lcphcp−Lcahca

Lcp+Lca− hsc

)

A02 = 1ρcCpcdc

Lcphcp

Lcp+Lca

A09 = 1ρcCpcdc

Lcphcpinc

Lcp+Lca

(Tp,1 − Tc

)

A11 = 1ρcCpcdc

∂hsc

∂hsc

(Ts − Tc

)

A12 = 1ρcCpcdc

Lcp

Lcp+Lca

(Tp,1 − Tc

)

A13 = 1ρcCpcdc

LcpuLcp+Lca

(Tp,1 − Tc

)

A22 = 4ρcCpcdc

LcpuLcp+Lca

(Tp,1 − Tc

)

A23 = 4u2ρcCpcdc

LcpuLcp+Lca

(Tp,1 − Tc

)

A84 = ∂f3(Tp,7,u6,Ta)∂FRF

A95 = ∂f5(u4,u2)∂Dwp

A105 = ∂f5(u2,u4)∂Dwp

+ ∂f5(u6,u4)∂Dwp

A114 = ∂f6(Tp,7,Ta)∂FRF

A115 = ∂f5(u4,u6)∂Dwp

B01 = 1ρcCpcdc

hsc

B08 = 1ρcCpcdc

Lcahca

Lcp+Lca

We assume that we can measure the temperature of the cylinder and the outputbecomes:

Y =

1 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 Gdry,2

Gdry,tot

Gdry,4Gdry,tot

Gdry,6Gdry,tot

X + (O4×5)Θ


To resume, the linearized extended system is(

δX

δΘ

)=

(Ax Aθ

O5×11 O5×5

)(δXδΘ

)+

(B

O5×12

)δU

δY = (Cx O4×5)(

δXδΘ

)

The matrices are computed in Maple, similarly to the observability analysis. Check-ing the rank of the observability matrix Ωident, we note that it only has rank 15 (for16 states), implying that one of the parameters is not identifiable. To find outwhich, we fix one parameter to a constant value assuming it is known and run anidentifiability test. If the new system is identifiable, we have found the unidentifi-able parameter.

Removing only hsc, FRF or Dwp, we get a rank of observability of 14 (for 15states), which means that these parameters are identifiable. But when removinghcp,0 or hcp,inc, the observability rank is 15, and the system is identifiable. Thisresult means that the combination of hcp,0 and hcp,inc is not observable, which isreasonable since we consider only one position, where the moisture content u isconstant. However, in the drying section, the moisture content is varying and thecoefficients hcp,0 and hcp,inc could therefore be identifiable.

As for the observability, we also consider other outputs. Measuring the tem-perature of the cylinder and at least two of the following board variables (Tp,1, Tp,7

or u), the contact-zone sub-model is still identifiable. However, if we measure onlyone board variable and the temperature of the cylinder, the sub-model is no longeridentifiable. The identifiability results for the free-draw sub-model are identical.

Conclusion

The linearized system is identifiable (with Θ = hsc, hcp(0), FRF , Dwp) whenmeasuring the cylinder and board surface temperatures. Therefore, since the wholesystem is a concatenation of identifiable systems, we assume that it is identifiableif the board temperature at both surfaces and the cylinder temperature are mea-sured for the contact zone and free draw of every cylinder. In appendix B, weshow that these are the minimal measurements required to ensure observability. Ifthe model is not observable, this implies that it is not identifiable either. We there-fore conclude that we need to measure at least the temperature of the cylindersand the temperatures of both surfaces of the board for each contact zone and eachfree draw to ensure identifiability of the chosen parameters.

4.3 Sensitivity analysis of the semi-physical model

The parameters and unmeasured inputs of the model are introduced in section 3.8.In the present section, we are considering the effect of these parameters on the out-put, in order to select which ones are relevant for the identification. The sensitivity

4.3 Sensitivity analysis of the semi-physical model 49

analysis is carried out for the first part of the drying section (cylinders 1 to 53), onthe semi-physical model described in chapter 3. To understand and estimate theeffect of the parameters on the outputs, we first carry out a qualitative sensitivityanalysis. We then apply the Dominant Parameter Selection (DPS) [22] to select themost influent parameters to keep for the identification described in chapter 5.

Qualitative sensitivity analysis

For the qualitative sensitivity analysis, we fix each parameter to its minimal andmaximal value, while the others are fixed to their nominal values and we observethe resulting outputs. Table 4.1 shows the respective values of each parameter.Some of the minimal and maximal values are exaggerated in order to visualizethe effect of the parameter. The outputs are the temperature of the cylinder, thetemperature of the paper at the bottom side (BS) and the top side (TS), and thetotal moisture content of the paper along the drying section. The slope of themoisture content curve represents the amount of evaporated water.

Table 4.1: Minimal, nominal and maximal values of the parameters for the quali-tative sensitivity analysis.

parameter min. value nom. value max. valuehsc 500 1900 3000

hcp(0) 100 450 1000hcp,inc 200 955 2000Dwp 10−9 2 · 10−8 0.001FRF 10 40 70Cpdry 400 1550 5000kdry 0.02 0.157 2a1 3750 3816 3870uin 1.27 1.38 1.94Tp,in 40 50 65pa 8000 18000 28000Ta 50 80 100

Heat transfer between steam and cylinder hsc

Figure 4.3 displays the temperature of the cylinder, the temperature of the paperand the moisture content for the minimal and maximal values of hsc. We notethat the parameter hsc affects all the variables considerably. A change in the heattransfer between the steam and cylinders creates a bias for the temperatures ofthe cylinder and the paper along the drying section. The moisture content is alsoaffected by the parameter, an expected result since hsc represents how much heat


0 10 20 30 40 50 6055

60

65

70

75

80

85

90

95

100

Temperature at the bottom side of the paper for different values of hsc

Tem

pera

ture

of t

he p

aper

(B

S)

[°C

]

Cylinders

hsc

=500

hsc

=3000

(a) Board temperature (BS)

0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of hsc

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

hsc

=500

hsc

=3000l

(b) Board temperature (TS)

0 10 20 30 40 50 6070

80

90

100

110

120

130

Temperature of the cylinders for different values of hsc

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

hsc

=500

hsc

=3000

(c) Temperature of the cylinders

0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of hsc

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

hsc

=500

hsc

=3000

(d) Board moisture content

Figure 4.1: Effect of the heat transfer coefficient hsc on the simulated outputs forthe first 53 cylinders of the drying section.

is used to dry the paper. If the coefficient increases, more heat is transferred to thecylinder and consequently to the paper (the temperatures increase), which resultsin an increase of evaporation (the moisture content decreases faster).

Heat transfer coefficient between cylinder and paper hcp(0) and hcp,inc

Figures 4.2 and 4.3 show the cylinder and board temperatures and the moisturecontent for the minimal and maximal values of hcp(0) and hcp,inc. These two pa-rameters have approximately the same effect on the outputs, except at the begin-ning of the drying section where hcp,inc is more influent, since it is linearly depen-dent of the moisture content. An increase in hcp increases the heat transfer betweenthe cylinder and paper, the temperature of the cylinder thus decreases, while thetemperature of the paper increases, resulting in an increase of evaporation.


0 10 20 30 40 50 6060

65

70

75

80

85

90

95

100

105

Temperature at the bottom side of the paper for different values of hcp

(0) T

empe

ratu

re o

f the

pap

er (

BS

) [°

C]

Cylinders

hcp

(0)=100

hcp

(0)=1000


0 10 20 30 40 50 6050

60

70

80

90

100

110

Temperature at the top side of the paper for different values of hcp

(0)

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

hcp

(0)=100

hcp

(0)=1000


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

130

Temperature of the cylinders for different values of hcp

(0)

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

hcp

(0)=100

hcp

(0)=1000


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of hcp

(0)

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

hcp

(0)=100

hcp

(0)=1000


Figure 4.2: Effect of the heat transfer coefficient hcp(0) on the simulated outputsfor the first 53 cylinders of the drying section.

Diffusion coefficient of water in the thickness direction Dwp

The diffusion coefficient mainly influences the total moisture content in the pa-per (see figure 4.4). When the diffusion coefficient decreases, the water movesslowly through the paper. Therefore, the water remains in the middle layer andthe moisture at the surface layers decreases faster since the loss of water from theevaporation is stronger than the gain of water coming from the middle layer. Thetotal moisture appears to be higher, since more water remains in the middle layer.Since the heat transferred from the cylinder to the paper decreases with the mois-ture content at the surface of the paper, the temperature of the cylinder increases.However, the heat transferred from the cylinder to the paper is increasing whenthe moisture content decreases (since it also depends on the heat conductivity, heatcapacity, density and thickness of the paper) and the temperature of the paper alsoincreases with a lower diffusion coefficient.


0 10 20 30 40 50 6055

60

65

70

75

80

85

90

95

100

105

Temperature at the bottom side of the paper for different values of hcp,inc

T

empe

ratu

re o

f the

pap

er (

BS

) [°

C]

Cylinders

hcp,inc

=200

hcp,inc

=2000


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of hcp,inc

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

hcp,inc

=200

hcp,inc

=2000


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of hcp,inc

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

hcp,inc

=200

hcp,inc

=2000


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of hcp,inc

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

hcp,inc

=200

hcp,inc

=2000


Figure 4.3: Effect of the heat transfer coefficient hcp,inc on the simulated outputsfor the first 53 cylinders of the drying section.

Fabric reduction factor FRF

Figure 4.5 shows the influence of FRF on the outputs. A small value of FRFrepresents a small decrease of the mass transfer coefficient due to the fabric. Con-sequently the evaporation increases, which results in a decrease of the board tem-perature, and therefore a decrease of the temperature of the cylinders.

Dry heat capacity Cpdry

Figure 4.6 displays the effect of the dry heat capacity on the output. When thedry heat capacity increases, the derivative of the paper temperature decreases,both in the free draw and when the paper is in contact with the cylinder. In thewarming phase of the paper, the temperature of the paper is lower, while it getshigher at the end of the drying. Therefore, the evaporation of water is lower in


0 10 20 30 40 50 6060

65

70

75

80

85

90

95

100

105

Temperature at the bottom side of the paper for different values of Dwp

T

empe

ratu

re o

f the

pap

er (

BS

) [°

C]

Cylinders

Dwp

=10−9

Dwp

=0.001


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of Dwp

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

Dwp

=10−9

Dwp

=0.001


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

130

Temperature of the cylinders for different values of Dwp

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

Dwp

=10−9

Dwp

=0.001


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of Dwp

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

Dwp

=10−9

Dwp

=0.001


Figure 4.4: Effect of the diffusion coefficient Dwp on the simulated outputs for thefirst 53 cylinders of the drying section.

the warming phase and higher at the end of the drying. Since the derivative of thepaper temperature is lower when the paper is in contact with the cylinder, moreheat is needed to warm the paper, and the temperature of the cylinder decreases.

Dry thermal conductivity kdry

In figure 4.7, the dry conductivity affects the cylinder temperature and the boardtemperature in the free draw. When kdry increases, the heat transport inside thepaper increases. Therefore, in the free draw, the increase of temperature due tothe heat coming from the cylinder dominates the decrease of temperature due tothe evaporation of water in the air. When the paper is in contact with the cylinder,the heat coming from the cold side of the paper becomes also more importantrelatively to the heat coming from the cylinder. The board temperature is thereforecolder, which results in a lower cylinder temperature. Since the board temperature


0 10 20 30 40 50 6060

65

70

75

80

85

90

95

100Temperature at the bottom side of the paper for different values of FRF

Tem

pera

ture

of t

he p

aper

(B

S)

[°C

]

Cylinders

FRF=10FRF=70


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100Temperature at the top side of the paper for different values of FRF

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

FRF=10FRF=70


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125Temperature of the cylinders for different values of FRF

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

FRF=10FRF=70


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6Moisture content for different values of FRF

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

FRF=10FRF=70


Figure 4.5: Effect of the Fabric Reduction Factor FRF on the simulated outputsfor the first 53 cylinders of the drying section.

is higher in general, the evaporation is also higher.

Dummy coefficient a1

Figure 4.8 shows that the parameter a1 considerably influences all the outputs. Anincrease in a1 lowers the evaporation and its effect on the decrease of temperatureof the paper in the free draw. Therefore, the temperature of the paper increases, aswell as the cylinder temperature, but the evaporation decreases.

The incoming moisture content uin

The incoming moisture content mainly affects the moisture content along the dry-ing section, but also significantly the temperature of the cylinder and slightly thetemperature of the paper in the free draw (see figure 4.9). An increase of the mois-


0 10 20 30 40 50 6055

60

65

70

75

80

85

90

95

100

105

Temperature at the bottom side of the paper for different values of Cpdry

T

empe

ratu

re o

f the

pap

er (

BS

) [°

C]

Cylinders

Cpdry

=400

Cpdry

=5000


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of Cpdry

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

Cpdry

=400

Cpdry

=5000


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of Cpdry

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

Cpdry

=400

Cpdry

=5000


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of Cpdry

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

Cpdry

=400

Cpdry

=5000


Figure 4.6: Effect of the dry heat capacity Cpdry on the simulated outputs for thefirst 53 cylinders of the drying section.

ture increases the heat of the cylinder given to the paper, which explains the de-crease of the temperature of the cylinder, but the temperature of the paper remainsalmost the same since it depends of a combination between the heat from the cylin-der, the heat transferred to the air and the heat of evaporation.

The incoming temperature of the paper Tp,in

The effect of the incoming paper temperature is represented in figure 4.10. Theincoming board temperature affects the temperature of the board and the cylin-ders only for the first seven cylinders. For the rest of the drying section, the tem-peratures are almost equal. For a higher incoming board temperature, the initialevaporation is higher only for the first cylinders, which explains the bias in themoisture content for the rest of the drying section.


0 10 20 30 40 50 6060

65

70

75

80

85

90

95

100

105

Temperature at the bottom side of the paper for different values of kdry

T

empe

ratu

re o

f the

pap

er (

BS

) [°

C]

Cylinders

kdry

=0.02

kdry

=2


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of kdry

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

kdry

=0.02

kdry

=2


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of kdry

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

kdry

=0.02

kdry

=2


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of kdry

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

kdry

=0.02

kdry

=2


Figure 4.7: Effect of the dry thermal conductivity kdry on the simulated outputsfor the first 53 cylinders of the drying section.

The temperature of the air Ta

Figure 4.11 shows that the temperature of the air mainly influences the moisturecontent and temperature of the cylinders. If the surrounding air is cold, it coolsdown the cylinders. Therefore less heat is transferred to the paper, which resultsin a lower paper temperature and a decrease of evaporation.

The partial pressure of water in the air pa

The partial pressure of water in the air has a strong influence on the outputs (fig-ure 4.12). A low humidity in the air increases the drying capacity. The increaseof evaporation results in a higher decrease of the board temperature. The cylin-ders temperature decreases since they release more heat to the paper. The boardtemperature in the free draw seems to reach an equilibrium point.


0 10 20 30 40 50 6055

60

65

70

75

80

85

90

95

100

105

Temperature at the bottom side of the paper for different values of a1

Tem

pera

ture

of t

he p

aper

(B

S)

[°C

]

Cylinders

a1=3750

a1=3870


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of a1

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

a1=3750

a1=3870


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of a1

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

a1=3750

a1=3870


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of a1

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

a1=3750

a1=3870


Figure 4.8: Effect of the dummy coefficient a1 on the simulated outputs for the first53 cylinders of the drying section.

Quantitative sensitivity analysis of the parameters

In this section we want to select the most influent parameters for the identificationdescribed in chapter 5. Since only few measurements are available, we use theDominant Parameter Selection (DPS) method suggested by Ioslovich et al. [22].For this study, we assume that we have found an optimal point; the analysis istherefore carried out after some identification tests. The theory behind the DPSmethod in [22] is first shortly described, and then applied to the identification set.

Methodology

The sensitivity function S(i, j) of the output Yi to the parameter θj is defined asfollows:

S(i, j) =δYi

δθj(4.10)


0 10 20 30 40 50 6060

65

70

75

80

85

90

95

100

Temperature at the bottom side of the paper for different values of uin

Tem

pera

ture

of t

he p

aper

(B

S)

[°C

]

Cylinders

uin

=1.27

uin

=1.94


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of uin

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

uin

=1.27

uin

=1.94


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of uin

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

uin

=1.27

uin

=1.94


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Moisture content for different values of uin

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

uin

=1.27

uin

=1.94


Figure 4.9: Effect of the incoming moisture content uin on the simulated outputsfor the first 53 cylinders of the drying section.

Since the parameters and outputs differ in magnitude, we use the relative sen-sitivity Sr(i, j), which is obtained by multiplying the sensivity by θj/Yi:

Sr(i, j) =δYi

δθj

θj

Yi(4.11)

The relative sensitivity is convenient because it is dimensionless. With the relativesensitivity matrix, we compute the modified Fisher matrix Fm:

Fm = STr Λ−1Sr (4.12)

where Λ is a diagonal matrix with the square roots of the weights λ in the lossfunction, see equation (5.3). We notice that the Fisher matrix (which is obtainedin the same way, using the sensitivity instead of the relative sensitivity) corre-sponds to the approximation of the Hessian of the loss function (5.3). The rank


0 10 20 30 40 50 6055

60

65

70

75

80

85

90

95

100

Temperature at the bottom side of the paper for different values of Tp,in

T

empe

ratu

re o

f the

pap

er (

BS

) [°

C]

Cylinders

Tp,in

=40

Tp,in

=65


0 10 20 30 40 50 6040

50

60

70

80

90

100

Temperature at the top side of the paper for different values of Tp,in

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

Tp,in

=40

Tp,in

=65


0 10 20 30 40 50 6080

85

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of Tp,in

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

Tp,in

=40

Tp,in

=65


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of Tp,in

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

Tp,in

=40

Tp,in

=65


Figure 4.10: Effect of incoming board temperature Tp,in on the simulated outputsfor the first 53 cylinders of the drying section.

of the matrix Fm indicates whether the matrix is singular. If the matrix is non-singular, the parameters are theoretically identifiable. However, this matrix canbe ill-conditioned, i.e. almost singular. In practice, this means that we can onlyestimate some parameters.

To measure the importance of the parameters, we compute the relative sensi-tivity vector my obtained by computing the square root of the diagonal elementsof Fm.

my =√

diag(Fm) (4.13)

A large value at the kth position indicates a big influence of the parameter θk forthe output.

To evaluate whether the Fm matrix is ill-conditioned, we put a threshold α1 [-]on the condition number, which is the ratio between the largest eigenvalue and


0 10 20 30 40 50 6060

65

70

75

80

85

90

95

100

Temperature at the bottom side of the paper for different values of Ta

Tem

pera

ture

of t

he p

aper

(B

S)

[°C

]

Cylinders

Ta=50

Ta=100


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of Ta

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

Ta=50

Ta=100


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of Ta

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

Ta=50

Ta=100


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of Ta

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

Ta=50

Ta=100


Figure 4.11: Effect of the temperature of the air Ta on the simulated outputs for thefirst 53 cylinders of the drying section.

the smallest eigenvalue. If the condition number is below the threshold, we con-sider the matrix well-conditioned. If this is not the case, we take only the first nparameters corresponding to eigenvalues with a ratio below the threshold (start-ing from the largest eigenvalue); we then check the condition number of the Fishersubmatrix corresponding to the n most important parameters. If the submatrix iswell-conditioned, we select the first n parameters for the identification.

The DPS method [22] permits to select the most influent parameters, but alsoprovides an examination of the correlation of those parameters, by analysing thenormalized modified Fisher matrix Fm,n:

Fm,n = STr,nSr,n (4.14)

where Sr,n is the normalized sensitivity matrix with normalized sensitivity vectors


0 10 20 30 40 50 6055

60

65

70

75

80

85

90

95

100

105

Temperature at the bottom side of the paper for different values of pa

Tem

pera

ture

of t

he p

aper

(B

S)

[°C

]

Cylinders

pa=8000

pa=28000


0 10 20 30 40 50 6050

55

60

65

70

75

80

85

90

95

100

Temperature at the top side of the paper for different values of pa

Tem

pera

ture

of t

he p

aper

(T

S)

[°C

]

Cylinders

pa=8000

pa=28000


0 10 20 30 40 50 6085

90

95

100

105

110

115

120

125

Temperature of the cylinders for different values of pa

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

[°C

]

pa=8000

pa=28000


0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Moisture content for different values of pa

Moi

stur

e co

nten

t [kg

w/k

g s]

Cylinders

pa=8000

pa=28000


Figure 4.12: Effect of the partial pressure of water in the air pa on the simulatedoutputs for the first 53 cylinders of the drying section.

Sjr,n of parameter j such that:

Sjr,n =

Sjr√

Sjr

TSj

r

(4.15)

All the diagonal elements of Fm,n are equal to 1, and the elements Fm,n(i, j), i 6=j, correspond to the cosine between the sensitivity vectors Si

r,n and Sjr,n. If the

sensitivity vectors are correlated, i.e. we can not identify both in the same time, theabsolute value of their scalar product is close to 1. To determine if two parametersare correlated, we put a threshold α2 [-] on the off-diagonal elements of Fm,n. Ifthe first n selected parameters are uncorrelated, they are identifiable. Ioslovich etal. [22] point out that the thresholds α1 and α2 are related. They consider thatthe condition number of the matrix Fm should be under 40 for Fm to be well-


conditioned, which corresponds to an absolute value of Fm,n(i, j) less than 0.95,for the parameters i and j to be uncorrelated.

It is important to keep in mind that the sensitivity analysis depends not onlyon the nominal values of the parameters, but also on the inputs, initial conditionsand structure of the system [39, 22, 31]. Therefore, we should be aware that thesensitivity results may differ if we choose to run the tests at different operationalpoints, or if we make changes in the model.

Application to the semi-physical model

For this study, we select only the parameters that we want to identify and donot include the unmeasured inputs of the model, since they change according towhich set of measurements we use (only their relative sensitivity is displayed intable 4.3, to evaluate their importance). The nominal parameter vector for thisanalysis, displayed in table 4.2, is assumed to be close to the identification optimalpoint θ (see chapter 5), thus some identification tests have been carried out beforeperforming this study. The data set 05/04/05 is used for the computations.

Table 4.2: Values of the nominal vector of parameters and unmeasured inputs usedfor the parameter selection.

parameter value input valuehsc 1800 uin 1.38

hcp(0) 1100 Tp,in 48hcp,inc 500 pa 15000Dwp 10−9 Ta 48FRF 40Cpdry 5000kdry 1.5a1 3860

The minimal and maximal values are respectively set to -1 % and +1 % of thenominal value of table 4.2, so the relative sensitivity is computed as:

Sr(i, j) =Yi(θmax)− Yi(θmin)

θmax − θmin

θ

Yi(θ)(4.16)

where θmax = 1.01 × θ, θmin = 0.99 × θ and the vector Y regroups the measuredoutputs, described in section 5.1. The previous equation can be rewritten as:

Sr(i, j) =Yi(1.01× θ)− Yi(0.99× θ)

0.02× Yi(θ)(4.17)


The matrix Sr is a N×r matrix, where N is the number of available measurementsand r = 8 is the number of parameters.

We compute the modified Fisher matrix Fm, in equation (4.12), with the weightsλ given in section 5.2. The vector of relative sensitivities my is displayed in ta-ble 4.3. The unmeasured inputs, defined in section 3.8, are not considered for theparameter selection, but table 4.3 shows that the incoming board moisture contentand temperature strongly affect the simulated outputs, since their relative sensi-tivity is high. The partial pressure of water in the air is considerable while thetemperature of the surrounding air is neglectible (low relative sensitivity).

Table 4.3: Relative sensitivity of the parameters and unmeasured inputs. Highrelative sensitivity represents a strong effect of the item on the model. The relativesensitivity of the inputs is displayed as an indication of their importance.

parameter rel. sensitivity input rel. sensitivitya1 10.6 uin 5.48

hcp(0) 0.39 Tp,in 0.56hsc 0.32 pa 0.25Dwp 0.18 Ta 0.0272

Cpdry 0.14hcp,inc 0.12FRF 0.11kdry 0.08

The resulting matrix Fm is found to be ill-conditioned and we do not dis-play the eigenvalues. We decide to remove the two less influent parameters (kdry

and FRF ) and redo the analysis. The eigenvalues of the corresponding modifiedFisher submatrix are reported in table 4.4.

Table 4.4: Eigenvalues of the modified Fisher submatrix of the 6 more dominantparameters.

parameter eigenvaluea1 112

hcp(0) 0.15hsc 0.10Dwp 0.034

Cpdry 0.019hcp,inc 0.015

Table 4.4 shows that the parameter a1 is very dominant, since its eigenvalueis large compared to the eigenvalues of the other parameters. The ratio of the


eigenvalues λa1/λhcp(0) ' 750 is already above the threshold, which implies thatwe can only identify the parameter a1. But if we consider the eigenvalues of theother parameters, the ratio λhcp(0)/λhcp,inc = 10 is under the threshold, so with a1

fixed to its optimal value with respect to the measurements, they could be identi-fiable. Therefore, we assume that the identification has found the optimal a1, wefix it to its optimal value and analyse the sensitivity of the remaining parameters.The eigenvalues of the corresponding modified Fisher submatrix are reported intable 4.5.

Table 4.5: Eigenvalues of the modified Fisher submatrix, after identification of a1.

parameter eigenvaluehcp(0) 0.23hsc 0.045Dwp 0.032

Cpdry 0.013hcp,inc 0.0011

From table 4.5, only 4 parameters (hcp(0), hsc, Dwp and Cpdry) are selectedsince λhcp(0)/λCpdry

' 18 < 40 but λhcp(0)/λhcp,inc ' 200 > 40. Before concludingabout the identifiability of the selected parameters, we need to verify that they arenot correlated. The normalized modified submatrix Fm,n corresponding to thoseparameters is:

Fm,n =

1 0.70 −0.57 −0.360.70 1 −0.65 −0.06−0.57 −0.65 1 −0.07−0.36 −0.06 −0.07 1

(4.18)

The absolute values of the off-diagonal elements of the matrix (4.18) are well underthe threshold of 0.95, thus the parameters hcp(0), hsc, Dwp and Cpdry are assumedto be uncorrelated.

Conclusion

The sensitivity analysis provides a practical parameter selection for the identifi-cation. It shows that the dummy parameter a1 is strongly influent, relatively tothe other parameters, and therefore suggests to first identify it to its optimal valuewith respect to the measurements and afterwards identify the other influent pa-rameters — hcp(0), hsc, Dwp and Cpdry — since they are uncorrelated. The pa-rameters — kdry , FRF and hcp,inc — are considered not identifiable and shouldbe fixed manually to a constant. The unmeasured inputs are not included in theidentification since they are time-varying, but we show that they affect the outputsof the model considerably.

4.4 Summary 65

4.4 Summary

The observability and identifiability analyses show that the minimal measure-ments required to ensure observability and identifiability are the cylinder tem-perature and the board temperatures Tp,1 and Tp,7 for the contact zone and thefree draw of each cylinder. Based on those results, special measurements wereordered to get the necessary data for the identification of the unknown parame-ters, described in the next chapter. Since only the final board moisture content andtop surface temperature are measured on-line, the model is not observable underon-line conditions. We can therefore not guarantee that the model estimates thetrue temperatures and moistures inside the board in the drying section. However,since the model is considered observable and identifiable during the parameteridentification, we can view it as an indication of those variables.

The sensitivity analysis gives an understanding of the effect of the unknownparameters on the outputs and an estimation of which parameters can be identi-fied in practice. It also suggests that the unmeasured inputs are influent for themodel.

Chapter 5

Identification of the Parameters

The goodness of a model is evaluated by comparing the simulated outputs withmeasurements. The parameter identification consists in estimating the optimalvalues of the parameters giving the best fit to reality, represented by the measure-ments. In chapter 4, we show that the model of the drying section is not observ-able under on-line conditions and present the minimal measurements required toenable the identification of the parameters. Those measurements would be too ex-pensive to perform under dynamic conditions and are therefore carried out understatic conditions on four special occasions. We first select one measurement set toidentify the parameters, and validate the results with the three other measurementsets. After a description of the static measurements, the process of identificationis introduced. We then describe the selection of the parameters to identify and theoptimization routine. In the last section, the identification and validation resultsare presented.

5.1 Description of the static measurements

With only the final moisture content and the temperature of the board measuredon-line in the drying section, the model described in chapter 3 is neither observablenor identifiable. The presence of on-line sensors is, however, difficult to achievefor two main reasons: firstly, the drying hood is very hot and secondly, the board isnot easy to reach because of the lack of space. The observability and identifiabilityanalyses in chapter 4 conclude that the model is observable and identifiable if wemeasure at least two variables per contact zone and per free draw. Static measure-ments were therefore carried out to measure the temperatures and the moisturecontent at different positions in the drying section, to enable the identification ofthe unknown parameters. Those measurements were performed on four occasionsand named after the date of their occurrence (with the format day/month/year).During a given measurement set, the inputs of the machine are constant. Sincethe model does not consider the cross direction of the machine, all the measur-

67

68 Identification of the Parameters

ing positions are approximately 50 cm from the front side of the machine towardsthe middle of the sheet. The rest of this section further describes the measuredvariables.

Properties of the surrounding air

The properties of the air in the ventilation pockets are measured with an hygrom-eter. The humidity of the air and the partial pressure of water in the air are cal-culated from the wet and dry measured temperatures. Figures 5.1 and 5.2 showthe measured temperature and partial pressure of water for the four sets of staticmeasurements.

0 20 40 60 80 10040

50

60

70

80

90

100

110Measured Temperature of the air for different sets of measurements

Cylinders

Tem

pera

ture

[C]

16/03/0428/09/0405/04/0514/06/05

Groups 1−5 Stack Groups 6−7

Figure 5.1: Measured temperature [C] of the surrounding air for different setsof measurements. The temperature between cylinders 53 and cylinder 70 (stackdryers) is not considered reliable since it is difficult to measure. The temperaturesare approximately the same for all sets of measurements, except for the 05/04/05(before the change of pockets ventilators), where it is below the average.

The properties of the air can be divided into three parts in the drying section.In the first five groups, where the board is still quite humid and thus more evapo-ration occurs, the properties of the air (temperature and partial pressure of water)

5.1 Description of the static measurements 69

are lower in the beginning of the drying section and increase with the propertiesof the board. In the stack dryers, the properties of the air are difficult to measureand less reliable. We therefore assume that the temperature of the air is constantand that the air is completely dry (as an effect of the vacuum rolls). In the last twogroups, 6 and 7, the board is almost dry, and the properties of the air are consid-ered as constant. The fitted values of equations (3.50) and (3.51) for the three partsof the drying section and the four sets of measurements are displayed in table 5.1.Since the humidity of the air in the measurement set 05/04/05 seem unreasonablytoo high, we replace it by the values of the measurement set 28/09/04.

0 20 40 60 80 1000

10

20

30

40

50

60

70Measured Partial pressure of water in the air for different sets of measurements

Cylinders

Par

tial p

ress

ure

of w

ater

[kP

a]

16/03/0428/09/0405/04/0514/06/05

Groups 1−5Stack Groups 6−7

Figure 5.2: Measured partial pressure of water [kPa] of the surrounding air for dif-ferent sets of measurements. The values for the 05/04/05 are unreasonably high,but after a change of pockets ventilators (14/06/05), the partial pressure of waterin the air, seem to be in accordance with the previous measurements data.

Temperature of the cylinders

The temperature of each cylinder is measured where the cylinder is in contact withthe air (see figure 5.3), using a contact thermometer. The standard deviation of themeasurement error is roughly around 1 C.


Figure 5.3: Position of the measured (Y ) and computed (Y ) variables for the cylin-der k, where Tc is the temperature of the cylinder, Tp the board temperature andu the average moisture content of the board. p1 and p2 are the positions after thecontact zone and at the end of the free draw respectively.

Board temperature

The temperature at both surfaces of the board is measured with an infrared ther-mometer on two positions per cylinder, at the end of the contact zone (positionp1) and at the end of the following free draw (position p2), see figure 5.3. It isnot possible to measure exactly at the point where the paper leaves or reaches thecylinder. Consequently, the measurement points are situated approximately 5 cmfrom the contact between the cylinder and the paper. Since a decrease of approxi-mately 10 C is measured in the free draw, and the simulation points are situatedbefore (position p1) and after (position p2) the measurements, the simulation pointin position p1/p2 should be approximately 1 C more/less than the measurementpoint. Adding the instrument uncertainty (around 1 C) to the uncertainty due tothe position, the uncertainty in the measurements of the temperature is approxi-mately 2 C.

5.1 Description of the static measurements 71

The measurements show a strange behaviour for the temperature of the sur-face of the board that was not in contact with the cylinder (in figure 5.3, this cor-responds to the board top layer from position p1 to position p2). According tothe measurements, the temperature increases in the free draw, while it should de-crease due to the evaporation. This behaviour is not well understood and has tobe further investigated. This could be due to some reflection from the cylindersat position p2. We decide nevertheless to omit the points of the surface layer thatwas not in contact with the previous cylinders at position p2 since we believe thatthose measures are biased.

Average moisture content inside the paper

To validate the computation of the moisture content for each layer, the ideal sit-uation would be to use sensors that are able to measure the distribution of themoisture in the thickness direction. Such instruments are starting to appear, butthey require cutting a sample of the edge of paper during the drying. This is notdesired, since there is a big risk to provoke a web break. The sample would more-over represent the moisture at the edges of the paper, while we are interested inmodelling the middle of the sheet (in the CD direction). The average moisture inthe thickness direction is instead measured with a γ-radiation device. Since theinstrument is voluminous, there are only few spots where it can safely reach thepaper during the drying. The moisture content is consequently measured only inthe free draw (see figure 5.3) between the different steam groups.

Table 5.1: Fitted parameter values for the properties of the surrounding air, inequations (3.50) and (3.51). The values of the data set 05/04/05 are modified toget more realistic values.

date 16/03/04 28/09/04 05/04/05 14/06/05Ta,ave [C] 60 60 60 60

groups 1–5 αT [-] 0.3 0.3 0.2 0.2pa,ave [Pa] 10000 15000 15000 15000

αp [-] 0.9 0.9 0.9 0.9Ta,ave [C] 80 85 80 80

stacks αT [-] 1 1 1 1pa,ave [Pa] 0 0 0 0

αp [-] 1 1 1 1Ta,ave [C] 90 95 85 90

groups 6–7 αT [-] 1 1 1 1pa,ave [Pa] 10000 18000 18000 15000

αp [-] 1 1 1 1


5.2 Identification procedure

To estimate the parameters, we use the same approach as Pettersson [36] and Bor-tolin [9]: the predictor error method (see [31]). The model can be written as anoutput error model:

Y (t) = G(U(t), θ) + e(t) (5.1)

where Y (t) is the vector of outputs, U(t) is the vector of inputs, G(U(t), θ) is thenonlinear function describing the model, θ is the vector of constant parameters toidentify, and e(t) is the error that we assume to be white noise.

The outputs available from the measurements are the following: the tempera-ture of the cylinders Tc, the temperature at the surfaces of the paper Tp,1 and Tp,7

and the board moisture content u. We define the predictor errors ε as the differencebetween the measurements Y and the predicted output Y .

εT c(k) = Tc(k)− Tc(k, θ), k = 1 . . . NT c

εT p,1(k) = Tp,1(k)− Tp,1(k, θ), k = 1 . . . NT p,1

εT p,7(k) = Tp,7(k)− Tp,7(k, θ), k = 1 . . . NT p,7

εu(k) = u(k)− u(k, θ), k = 1 . . . Nu

(5.2)

where NT c, NT p,1, NT p,7 and Nu are the numbers of available measurements foreach output. Since we are using static measurements, the index k represents theposition of the measure instead of the time index.

Similarly to Pettersson [36] and Bortolin [9], we define the loss function VN :

VN (θ, ZN ) = 1N

(∑NT c

i=1εT c(i)

2

λT c+

∑NT p,1i=1

εT p,1(i)2

λT p,1+

∑NT p,7i=1

εT p,7(i)2

λT p,7+

∑Nu

i=1εu(i)2

λu

)

(5.3)where N = NT c + NT p,1 + NT p,7 + Nu is the total number of measurements,ZN is the input–output data and the weights λ are used to normalise the squaredprediction errors:

λT c = maxi|εT c(i)|λT p,1 = maxi|εT p,1(i)|λT p,7 = maxi|εT p,7(i)|λu = 0.02

(5.4)

The weight λu is set to a low value to give more importance to the moisture contentin the loss function, to conpensate for the few measures available.

The process of identification of the parameters consists in finding the set ofoptimal parameters θ that minimizes the loss function.

θ = arg minθ

VN (θ, ZN ) (5.5)

The procedure to solve the optimization problem is described in appendix A.Sometimes, some observations can differ considerably from the rest of the data;they are called outliers. Pettersson [36] and Bortolin [9] suggest to remove the

5.3 Parameter selection 73

largest residuals (least trimmed square (LTS) method), to discard the eventual out-liers. However in this work, the possible outliers are removed manually, since weare dealing with a small amount of data.

5.3 Parameter selection

Solving the optimization problem (5.5) requires a large number of iterations. Thesimulation time of the model of the whole drying section, after a change of pa-rameters, is considerable because of the high complexity of the model. The dryingsection has therefore been divided into two parts for the identification of the pa-rameters. The first part contains the first 53 cylinders (groups 1–5) and the secondpart regroups the stack dryers and the groups 6 and 7. This division is also justi-fied by the fact that we believe that the board has a different behaviour for highor low moisture content. The properties in the stack dryers are also assumed to bedifferent because of the presence of vacuum rolls, but since the measurements inthe stack dryers are poor and not very reliable, we decide to fix the parameters inthe stacks manually.

The candidate parameters for the identification are described in section 3.8.The sorption phenomenon is fitted manually before performing the identification(see section 3.7). The properties of the surrounding air are fitted to each data set,see table 5.1. The values of the incoming temperature and moisture content areestimated manually to give a good agreement with the measurements.

In section 4.3, we suggest that for the first part of the drying section (groups 1–5), the parameters FRF and kdry can be set to a constant. The sensitivity analysisalso suggests to first identify a1 and afterwards hsc, hcp(0), Dwp, Cpdry, settinghcp,inc to a constant.

For the identification of the second part of the drying section, we fix more pa-rameters to a constant value, since there are fewer measurements available. Theparameters a1, kdry , and Cpdry are fixed to their physical values. We also assumethat the heat transfer coefficient between the cylinder and the paper does not de-pend on the moisture content (hcp,inc = 0). The parameters that we choose toidentify are therefore hsc, hcp(0) and Dwp. Table 5.2 summarizes which parame-ters are fixed (in parentheses) and which ones are identified.

5.4 Identification results

The identification of the parameters is carried out for the data set 05/04/05 be-cause it provides the most available measurements and the resulting outputs aredisplayed in figure 5.4. To validate the model, the model is simulated for threeother sets of measurements, with the estimated parameters during the identifica-tion. The validation results are displayed in figures 5.5, 5.6 and 5.7. The simulatedoutputs for the identification are acceptable, except in the stack dryers where thesimulated temperature of the cylinders is 15 C higher than the measurements.


Table 5.2: Estimated values of the parameters during the identification, for thethree parts of the drying section. The values in parentheses are fixed manually.

parameter groups 1–5 stacks groups 6–7hsc 1500 (1900) 300

hcp(0) 1000 (742) 400hcp,inc (500) (0) (0)Dwp 1 · 10−9 (2 · 10−9) 2 · 10−9

FRF (40) (40) (40)Cpdry 6000 (1550) (1550)

a1 3860 (3816) (3816)kdry 2 (0.157) (0.157)

The final simulated board temperature and moisture content are in accordancewith the measurements.

To quantitatively evaluate the results, the standard deviations of the residu-als are compared with the standard deviations of the measurement errors, see ta-ble 5.3. The standard deviations of the residuals are close to the standard deviationof the measurements for the first part of the drying section, but generally higherfor the second part.

The validation results are less good than the identification. For both data sets28/09/04 and 16/03/04, the simulated temperatures of the cylinders are too highand the temperature of the board is too low. The simulated evaporation is slightlyhigher than the measured output for the 16/03/04 set, probably because the airis drier. For the data set 14/06/05, the temperature of the cylinders is reasonablebut the simulated board temperature is too low. The moisture content is however,satisfactory.

5.5 Conclusion

The process of identification and validation of the drying section model is difficultbecause of the lack of data. The measurements are poor and uncertain (for exam-ple the board temperature is showing a strange behaviour), and important inputs,such as the incoming moisture content, are not measured. Moreover, the iden-tification is carried out under static conditions and some input–output relationsmay be misrepresented. The resulting model can be considered as satisfactory forthe identification set but not for the validation sets. Besides, some identified val-ues of the parameters are outside the typical boundaries found in literature (seesection 3.8). To improve the model, more measurements on a regular basis areneeded, to be able to judge which measures are relevant, or distinguish whichfeatures are not well modelled. The final moisture content seems, however, to bein accordance with the measurements. We therefore assume that the parameter

5.5 Conclusion 75

identification gives satisfactory results, considering the limitations, and we fur-ther investigate the behaviour of the model under on-line conditions in the nextchapter.

0 10 20 30 40 50 60 70 80 90 10050

60

70

80

90

100

110

120Identification set 05/04/05: Temperature of the board (BS) in the drying section

Cylinders

Tem

pera

ture

of t

he p

aper

in °

C

simulationmeasurements


0 10 20 30 40 50 60 70 80 90 10050

60

70

80

90

100

110

120Identification set 05/04/05: Temperature of the board (TS) in the drying section

Cylinders

Tem

pera

ture

of t

he p

aper

in °

C



0 10 20 30 40 50 60 70 80 90 10070

80

90

100

110

120

130Identification set 05/04/05: Temperature of the cylinders in the drying section

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

in °

C



0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4Identification set 05/04/05: Moisture content in the drying section

Cylinders

Moi

stur

e co

nten

t in

kgw

/kg dr

y



Figure 5.4: Measured and simulated outputs in the drying section for the identifi-cation set 05/04/05.


0 10 20 30 40 50 60 70 80 90 10050

60

70

80

90

100

110

120

130Validation set 16/03/04: Temperature of the board (BS) in the drying section

Cylinders

Tem

pera

ture

of t

he p

aper

in °

C



0 10 20 30 40 50 60 70 80 90 10040

50

60

70

80

90

100

110

120

130Validation set 16/03/04: Temperature of the board (TS) in the drying section

Cylinders

Tem

pera

ture

of t

he p

aper

in °

C



0 10 20 30 40 50 60 70 80 90 10070

80

90

100

110

120

130

140Validation set 16/03/04: Temperature of the cylinders in the drying section

Cylinders

Tem

pera

ture

of t

he c

ylin

ders

in °

C



0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4Validation set 16/03/04: Moisture content in the drying section

Cylinders

Moi

stur

e co

nten

t in

kgw

/kg dr

y



Figure 5.5: Measured and simulated outputs in the drying section for the valida-tion set 16/03/04.

5.5 Conclusion 77

0 10 20 30 40 50 60 70 80 90 10050

60

70

80

90

100

110

120


Cylinders

Tem

pera

ture

of t

he p

aper

in °

C



0 10 20 30 40 50 60 70 80 90 10050

60

70

80

90

100

110


Cylinders

Tem

pera

ture

of t

he p

aper

in °

C



0 10 20 30 40 50 60 70 80 90 10070

80

90

100

110

120

130


Cylinders

Tem

pera

ture

of t

he c

ylin

ders

in °

C



0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2


Cylinders

Moi

stur

e co

nten

t in

kgw

/kg dr

y



Figure 5.6: Measured and simulated outputs in the drying section for the valida-tion set 28/09/04:.


0 10 20 30 40 50 60 70 80 90 10050

60

70

80

90

100

110


Cylinders

Tem

pera

ture

of t

he p

aper

in °

C



0 10 20 30 40 50 60 70 80 90 10040

50

60

70

80

90

100

110


Cylinders

Tem

pera

ture

of t

he p

aper

in °

C



0 10 20 30 40 50 60 70 80 90 10075

80

85

90

95

100

105

110

115

120


Cylinders

Tem

pera

ture

of t

he c

ylin

ders

in °

C



0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4


Cylinders

Moi

stur

e co

nten

t in

kgw

/kg dr

y



Figure 5.7: Measured and simulated outputs in the drying section for the valida-tion set 14/06/05.

Table 5.3: Standard deviations of the measurements error and the residuals forthe identification and validation sets. Part 1 contains the first 5 groups and part2 the stacks dryers together with groups 6 and 7. The standard deviation of themeasurements error is assumed to be larger in the stack dryers.

ident. valid. valid. valid.meas. 05/04/05 16/03/04 28/09/04 14/06/05

part 1 part 2 part 1 part 2 part 1 part 2 part 1 part 2Tcyl [C] ≈ 1 1.83 3.20 3.04 3.36 1.89 2.26 2.73 3.75Tp,1 [C] ≈ 2 3.19 2.64 4.12 3.77 3.85 4.53 5.83 3.74Tp,7 [C] ≈ 2 2.92 3.29 5.07 1.73 3.63 2.80 4.42 3.15u [kg/kg] ≈ 0.04 0.035 0.022 0.073 0.025 - - 0.097 0.075

Chapter 6

Dynamic Simulations

In the previous chapter, the model is validated under static conditions, i.e. the in-puts of the machine are constant. In this chapter, we investigate the dynamics ofthe model, since we want to simulate the on-line conditions. We consider only theavailable on-line measurements: the board properties at the end of the drying sec-tion. Those measurements are not sufficient to ensure observability for the wholemodel (see chapter 4), but the simulations can be used to give an approximation ofthe board temperature and moisture content for different positions in the dryingsection. We first examine the simulations of the deterministic model, then includedisturbances in the model, apply a nonlinear Kalman filter and present the results.

6.1 Deterministic model

The model is simulated for dynamic conditions, i.e. the inputs of the machinesare varying. We consider only the two available measurements at the end of thedrying section: the board temperature at the top side and the average board mois-ture content in the thickness direction. In these conditions, we do not know thefollowing model inputs described in section 3.8: the board moisture content andtemperature after the press and the air properties. The board moisture content iscomputed with the press model [11] and the board temperature and air propertiesare considered constant. We use a smoothing filter to reduce high-frequency dis-turbances that could introduce numerical problems. Since we are dealing with asmall amount of data, the outliers are removed manually. The measurement framesituated at the end of the drying section is not always active; we thus need to finda good data set for the simulations. We consider a period between 18/08/05 and29/08/05, where a reasonable amount of measurements is available. In the mea-surements set, lack of data is detected for some time periods. Either some mea-sured inputs or outputs are missing or the bending stiffness predictor [36] compu-tations fail. The missing data are replaced by a linear interpolation between theavailable data.

79

80 Dynamic Simulations

The simulation results are displayed in figures 6.1 and 6.2. The model is firstsimulated with the parameters identified in chapter 5, displayed in table 5.2. Themodel is not satisfactory since it is obviously missing some dynamics and show-ing some large biases. Since some identified parameters in chapter 5 are outsidethe range of the physical values found in the literature (see section 3.8), we alsosimulate the model with more physical values of the parameters. The physical val-ues, chosen after some trial and error with the deterministic model, are displayedin table 6.1. The simulation results are different from the model with identifiedvalues, but we can not really determine which model is the best. To compare theperformance of the two models, the mean and the standard deviation of the er-ror vectors are displayed in table 6.2, p. 87. The error vector is defined as thedifference between the predicted (y) and the measured (y) output:

εu = uout − uout

εTp = Tp7,out − Tp7,out(6.1)

where uout and Tp7,out are, respectively, the average moisture content and the topside temperature of the board, at the end of the drying section. The standard devi-ations of the error vectors are similar for the two models, but the model with iden-tified parameters have a larger error mean for the temperature (this correspondsto a larger bias). For the moment, we can not conclude which model alternativeis the best. In the next section, we attempt to improve the model by includingdisturbances to compensate for unmodelled features.

Table 6.1: Physical values of the parameters for the simulation of the deterministicmodel described in chapter 3.

parameter groups 1–5 stacks groups 6–7hsc 1700 1700 500

hcp(0) 200 200 200hcp,inc 850 850 850Dwp 5 · 10−9 5 · 10−9 5 · 10−9

FRF 40 40 40Cp,dry 1550 1550 1550

a1 3816 3816 3816kdry 0.157 0.157 0.157

6.2 Grey-box modelling of the disturbances

In the previous section, the only uncertainty considered is the process noise, whichis supposed to be white noise. However, as Pettersson [36] and Bortolin [9, 10]point out, paper making is a complex process and other disturbances are present.

6.2 Grey-box modelling of the disturbances 81

0 50 100 150 200 2500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Moisture content at the end of the drying section

Moi

stur

e co

nten

t in

kgw

/kg dr

y

Time in hours

measuredcomputed using identified parameterscomputed using physical parameters

Figure 6.1: Dynamic simulation of the average moisture content at the end of thedrying section of the deterministic model. The short time periods between verticallines represent the data replaced by a linear interpolation.

Unmeasured inputs: We showed in section 4.3 that the properties of the incom-ing board and the surrounding air are important parameters for the drying.Unfortunately, they are not measured.

Unmodelled features: The model is based on simplifying assumptions. The dy-namics of the steam and air systems are not considered. The presence of airand the distinction between the liquid or vapor phases of water inside theboard are neglected. We also believe that the pulp properties influence thedrying parameters, (such as Cpdry, kdry, Dwp, see section 3.8), but we did notinclude them in the model, due to identifiability problems.

Parameter uncertainty: The model with the parameters identified in chapter 5shows a good agreement for the identification set, but a poor agreement forthe static validations sets and the dynamic simulations (in section 6.1). Sincethe identification is carried out for static conditions, the inputs are appar-ently not informative enough. Indeed, the identified values of the parameters


0 50 100 150 200 25080

85

90

95

100

105

110

115

120Top side board temperature at the end of the drying section

Time in hours

Tem

pera

ture

of t

he p

aper

in °

C

measuredcomputed using identified parameterscomputed using physical parameters

Figure 6.2: Dynamic simulation of the top side board temperature at the end of thedrying section of the deterministic model. The short time periods between verticallines represent the data replaced by a linear interpolation.

are outside the range of typical values found in literature and dynamic sim-ulation with more physical values achieves a similar performance.

Input uncertainty: The inputs in the model are given by measurements and canbe subject to measurements errors.

In this section, we apply a grey-box modelling method, the nonlinear Kalmanfilter to use the available measurements to estimate the uncertainties that the de-terministic model can not catch. After a short description of the Extended Kalmanfilter, we model the disturbances and show the results of the simulations.

Nonlinear Kalman filtering

We use the Extended Kalman Filter (EKF) algorithm, which is based on the lo-cal linearization about the current mean and covariance of the random variable.The EKF may not perform well in some applications where the linearization is toocrude. Variations of the EKF, such as the unscented Kalman filter [25], preserve the


normal distributions of the random variables under the nonlinear transformation,but require several function evaluations for each time step. In our case, a func-tion evaluation represents a simulation of the model for the whole drying section,which is very time consuming. We therefore choose to test the Extended Kalmanfilter since, for each time step, it requires only one extra simulation per disturbancevariable.

Extended Kalman Filter

We consider the following system:

Σnl

xk+1 = f(xk, uk, wk)yk+1 = l(xk, vk) (6.2)

where xk represents the unobserved state of the system, uk is the input signal, wk

is the process noise, yk is the observed signal, vk is the measurement noise and fand l are nonlinear functions. The noises wk and vk are supposed to be mutuallyindependent and have normal probability distributions with respective covariancematrices Q and R.

In practice, we do not know the process noise wk and the measurement noisevk at time k. So we compute an a priori estimate of xk+1 (denoted x−k+1), using ana posteriori estimate of xk (denoted xk), computed at previous time k:

x−k+1 = f(xk, uk, 0)y−k+1 = l(xk, 0)

(6.3)

By introducing the following quantities,

Ak[i,j] = ∂f[i]

∂x[j](xk, uk, 0) Jacobian of f with respect to x

Wk,[i,j] = ∂f[i]

∂w[j](xk, uk, 0) Jacobian of f with respect to w

Lk,[i,j] = ∂l[i]∂x[j]

(xk, 0) Jacobian of l with respect to x

Vk,[i,j] = ∂l[i]∂v[j]

(xk, 0) Jacobian of l with respect to v

(6.4)

we can estimate the actual state xk and output yk by

xk ≈ x−k + Ak (xk−1 − xk−1) + Wkwk−1

yk ≈ y−k + Lk

(xk − x−k

)+ Vkvk

(6.5)

The Extented Kalman Filter equations are given by:Time update equations:

x−k = f(xk−1, uk−1, 0)Π−k = AkΠk−1A

Tk + WkQk−1W

Tk

(6.6)


Measurement update equations:

Kk = Π−k LTk

(LkΠ−k LT

k + VkRkV Tk

)−1

xk = x−k + Kk

(yk − l(x−k , 0)

)Πk = (I −KkLk)Π−k

(6.7)

Modelling the disturbances

To choose which disturbances to include in the model is a trial-and-error proce-dure. One possibility is to add noise to all uncertain variables in the model, butthis includes redundancies and is therefore inefficient. Since we need to computethe derivative of the function evaluation with respect to each Kalman state, thecomputation time increases with the number of disturbance variables. We there-fore include as few uncertainties as possible. In addition, the derivative is cal-culated using the backwards differentiation method that requires only one extrafunction evaluation per derivative.

∂f[i]

∂x[j]=

1δx[j]

(f[i](x[j])− f[i](x[j] − δx[j])

)(6.8)

The qualitative sensitivity analysis shows that the incoming board moisture con-tent and the partial pressure of water in the air affect the computed variables con-siderably, (see figures 4.9 and 4.12). The quantitative sensitivity analysis suggeststhat the incoming board temperature has also a strong effect on the variables in thedrying section, but we do not consider it since it affects mostly the warming-upzone of the board (first cylinders) and not the board properties at the end of thedrying section. Moreover, the partial pressure of water in the air represents thedrying potential since the evaporation is driven by the difference of partial pres-sure between the air and the paper. Consequently, we choose to add disturbancesto the incoming moisture content and partial pressure of water in the air.

We model the disturbances in the same manner as Pettersson [36] and Bor-tolin [10], by multiplying the uncertain variables with an exponential factor. Thisapproach is convenient since the multiplying factor is always positive which pre-vents the allocation of unphysical values to the following disturbed variables (de-noted with a superscript d):

udin(k) = uin(k)e−x1(k)

pda(k) = pa(k)e−x2(k) (6.9)

The disturbance variables x are modelled as random-walk processes [31].

xi(k + 1) = xi(k) + ξi(k) i = 1, 2 (6.10)

where ξi is a sequence of normally distributed scalar variables, with zero meanand covariance rξi. This is a typical way to model slow varying process. Thecovariance rξi describes how fast the disturbance xi changes.


We transform the model into the standard Kalman filter form (6.2) by includingthe measured output and the disturbances in the Kalman state X :

X(k) =

uout(k − 1)Tp7,out(k − 1)/100

x1(k)x2(k)

(6.11)

where uout(k) = l1(Xk, Uk, 0) and Tp7,out = l2(Xk, Uk, 0) are respectively the com-puted average moisture content and the computed top side board temperature atthe end of the drying section at time k (given by the simulation of the model de-scribed in chapter 3). The temperature is divided by 100, to normalize the errorvector. Note that U(k) is the input vector and not the moisture content. The gov-erning equations for the overall model can then be written in the standard Kalmanfilter form (6.2):

X(k + 1) =

l1(X(k), U(k), 0)l2(X(k), U(k), 0)

x1(k)x2(k)

+

00

ξ1(k)ξ2(k)

Y (k) =[

I 0]X(k) +

[v1(k)v2(k)

](6.12)

With this transformation, the Jacobian matrices are easy to compute:

A(k) =[O2×2

∂l(k)∂x

O2×2 I2×2

]W =

[ O2×2

I2×2

]

L =[

I2×2 O2×2

]V = I2×2

where O is the null matrix and I the identity matrix. Only the first matrix A(k) isdependent of the step k, the other matrices are constant, we thus omit the index k.

The covariance matrixes of the process Q and measurements R are:

Q =[ O2×2 O2×2

O2×2 Q

]R = R

where Q = diag(rξ1, rξ2) is the covariance matrix of the states x1 and x2 and R isthe covariance matrix of the measurements uout, Tp7,out.

Simulations results

The stochastic model is simulated with the same data set used for the deterministicmodel, with both the identified values and physical values of the parameters. Inorder to apply the Kalman filter algorithm, we need to set initial values to theestimated Kalman state x(0) and the covariance matrix Π(0).

x(0) = 0, Π(0) = 0.02 · I2×2 (6.13)


The initial conditions do not matter in our case, since the model is first simulatedwith constant inputs. We also need to estimate the process and measurementscovariance matrices Q and R. Finding the optimal values is a trial-and-error pro-cedure, changing the values of Q and R determines if we give more weight tothe process or the measurements. For example, a larger value of Q increases theKalman gain K and allows larger changes in the process disturbances; the algo-rithm therefore tracks the measurements more. After some tests, the followingvalues are chosen.

Q = 0.02 · I2×2, R =[

0.002 00 0.03

](6.14)

The simulation results of the stochastic model are depicted in figures 6.3 and6.4. The Kalman filter improves the model considerably for both simulations withidentified and physical parameters values. The identified model performs less wellthan the physical model for some times periods. This discrepancy is not well un-derstood but could be due to numerical problems.

0 50 100 150 200 2500.04

0.06

0.08

0.1

0.12

0.14

0.16Moisture content at the end of the drying section

Moi

stur

e co

nten

t in

kgw

/kg dr

y

Time in hours

measuredwith KF and identified parameterswith KF and physical parameters

Figure 6.3: Dynamic simulation of the average moisture content at the end of thedrying section, of the stochastic model. The short time periods between verticallines represent the data replaced by a linear interpolation.

The mean and standard deviations of the prediction error, displayed in ta-ble 6.2, show a great improvement for both models with the Kalman filter com-pared to the deterministic models.


0 50 100 150 200 25075

80

85

90

95

100

105

110

115Top side board temperature at the end of the drying section

Time in hours

Tem

pera

ture

of t

he p

aper

in °

Cmeasuredwith KF and identified parameterswith KF and physical parameters

Figure 6.4: Dynamic simulation of the top side board temperature at the end of thedrying section of the stochastic model. The short time periods between verticallines represent the data replaced by a linear interpolation.

Since the predictions of the moisture and the temperature at the end of the dry-ing section are satisfactory, we examine the values of the disturbances estimatedby the Kalman filter. To verify that the Kalman filter computes reasonable val-ues, the disturbed variables ud

in and pda are depicted in figure 6.5. The variables

have similar trends for the identified and physical models and are within physicallimits.

Table 6.2: Mean and standard deviations of the prediction error ε, for the deter-ministic and stochastic model.

prediction errors εTp [C] εu[kgw/kgdry]mean Std mean Std

det. model ident. parameters 9.50 4.14 -0.005 0.0155phys. parameters 1.57 4.02 0.0006 0.0229

stoch. model ident. parameters -0.74 2.54 -0.0009 0.0048phys. parameters -0.025 1.25 -0.0001 0.0031


0 50 100 150 200 250

1.3

1.4

1.5

1.6

1.7

1.8

1.9

uind estimated by the Kalman filter

Time in hours

Moi

stur

e co

nten

t in

kgw

/kg dr

y

model with identified parametersmodel with physical parameters

(a) Incoming moisture content

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3x 10

4 pad estimated by the Kalman filter

Time in hours

Par

tial p

ress

ure

of w

ater

in th

e ai

r in

Pa

model with identified parametersmodel with physical parameters

(b) Partial pressure of water in the air

Figure 6.5: Values of the disturbed variables estimated by the Kalman filter for thephysical and identified models. For the incoming moisture content, the lower andupper bounds are displayed.

6.3 Summary and discussion

In this chapter, the dynamics of the model with on-line conditions are investigated.The deterministic model shows a poor agreement with the measured data, butthe stochastic model improves the fit considerably and shows satisfactory results.Since the identified values in chapter 5 are outside the range of physical valuesfound in literature, we also simulate the model with more physical values. Theresults are similar for the deterministic model, but the model with physical values

6.3 Summary and discussion 89

performs better for the stochastic model. The Kalman filter model with physicalvalues shows a good agreement of the final board temperature and average mois-ture content and we believe it could be used by the operators as an indicator of theboard temperature and moisture content in the drying section.

Two main issues are raised from these results:

• Why do the physical values perform better than the identified values? The differ-ence between the performances of the two models is not well understoodand should be further investigated. The remaining biases for the identifiedvalues can be due to numerical problems, since the simulated outputs showstrong disturbances for those time periods. Another possible explanation isthe fact that the identification is carried out under static conditions. The in-puts are therefore not excited and some input-output relations are not wellrepresented.

• Does the Kalman filter give a correct estimation of the unmeasured input? We havechecked that the Kalman filter assigns reasonable values to the disturbedvariables. However, we should be aware that if we measure the inputs es-timated by the Kalman filter on-line, we may not find the values obtainedby the stochastic model. The Kalman filter compensates for the unmodelledfeatures. For example, the biases in the physical model seem to be relatedto the basis weight: for low (respectively high) basis weight, the simulatedmoisture content seems too high (respectively low). The relation dryingcapacity–basis weight is therefore not well represented and the Kalman fil-ter algorithm uses the partial pressure of water in the air to adjust the dryingcapacity. We believe that if we measure the incoming moisture and partialpressure of water in the air on-line, other disturbances (for example in theparameters hsc, Dwp, Cpdry or kdry) have to be included in the Kalman filterto get a good fit.

Chapter 7

Conclusions and Future Work

7.1 Conclusions

The thesis presented a grey-box model of the board moisture and temperature in-side the drying section of a paper mill. The distribution of the moisture inside theboard is an important variable for the board quality, but is unfortunately not mea-sured on-line. The main goal of this work was a model that predicts the moistureevolution during the drying, to be used by operators and process engineers as anestimation of the unmeasurable variables inside the drying section.

A major limitation in the drying-section modelling field is the lack of measure-ments to validate the model. If the model is not observable or identifiable, wecan not guarantee that our estimation of the variables or parameters is correct.We therefore carried out observability and identifiability analyses to verify therelevance of the chosen model structure, which led to the following conclusions:If we measure at least two board variables for each contact zone and each freedraw, the model presented in chapter 3 is observable and identifiable1. Moreover,we showed in appendix B that the model is not observable or identifiable underon-line conditions, i.e. when only the final board moisture and temperature aremeasured. A sensitivity analysis was also performed to select the most influentparameters for the identification.

Based on the observability and identifiability analyses, four special measure-ments sessions were carried out for the identification of the unknown parametersunder static conditions, i.e. the inputs of the machine were constant. The param-eters were identified using one data set and the resulting model was validatedusing the three other data sets. The results were considered satisfactory for theidentification set but poor for the validation sets. The poor agreement is believedto be due to the lack of input excitation during the identification procedure.

Although we were aware that our model is not observable, we wanted to exam-ine the grey-box modelling technique for a drying section to investigate if we could

1with the set of parameters given in section 4.2

91

92 Conclusions and Future Work

predict the final board properties on-line. The deterministic model was first sim-ulated under on-line conditions and the results were far from satisfactory. How-ever, the addition of disturbances in the process by a Kalman filter showed a goodprediction of the final board temperature and moisture content at the end of thedrying section. We therefore believe that the model could be used by operatorsand process engineers as an indicator of the board properties inside the dryingsection.

7.2 Directions for future work

The modelling of the board properties inside the drying section is a challengingfield. The complexity of the process and the limitations of measurements suggestseveral directions for further studies:

• Observability is important for control purposes and raises more questions:Do we really need to estimate the moisture content for each layer? Whereshould we drop constraints: on the physical description of the model or onthe observability? The model purpose provides an indication for answeringthose questions. If the model is used to predict quality variables, we believethe physical description should be retained. But if the model is used formoisture control, a one-layer model is probably more appropriate.

• The dynamic behaviour of the deterministic model in section 6.1 shows pooragreement with the measurements. The discrepancy between identified andphysical parameters suggests that the values of the parameters, identifiedunder static conditions, are not optimal. A careful study of the correla-tions between the error vectors and the inputs could determine which input–output relations are not well modelled, as for example the basis weight–drying rate relation. The addition of the steam and air system would alsoenhance the dynamic behaviour of the model.

• The numerical solution of the model is another field to investigate. The sim-ulations were sometimes stopped by numerical errors for some parametervalues. Modifying the sorption equations (3.55) would probably remedythis numerical problem. The simulation time of the model is also an impor-tant limitation for trial-and-error procedures. For example, on a standardhigh-end PC, the simulation time for the Kalman filter algorithm, with twoKalman states, and a sampling time of 12 minutes, is equal to the simulatedtime period. Improving the computation time would allow more freedom totest new ideas.

• Any addition of on-line sensors in the drying section would contribute toimprove the model. We showed that the incoming moisture and the partialpressure of water in the air were important variables for the drying section.The identification and validation of the model parameters would also benefit

7.2 Directions for future work 93

from additional measured outputs. The moisture is still difficult to measureon-line inside the drying section, but temperature sensors could be used toimprove the parameters. Two sensors, one before and one after the stackdryers can contribute to improve the stack model.

• The research on improving sensors is obviously of great interest. The de-velopment of sensors able to measure the moisture distribution in the sheetwould be a huge contribution to this project.

Appendix A

Implementation

The first objective of this work was to apply a grey-box modeling approach, sim-ilar to Pettersson [36] and Bortolin [9], to the model of the drying section. Since asimple model, as well as other parts of the paper mill, was already implementedin Dymola the idea was to investigate grey-box modeling with Dymola. Further-more, to the knowledge of the author, no work has been reported on this par-ticular topic. However, since Dymola does not provide useful tools to performparameters studies or optimization runs, we combine it with Matlab to performthe identification of the parameters. After an introduction to the simulation pro-gram Dymola, this appendix describes the structure of the model, the algorithmof identification of parameters with Dymola and the optimization routine used inthis work.

A.1 Simulation program

The model is implemented in the simulation tool Dymola [12] based on the object-oriented language Modelica [32]. The object-oriented approach is suitable forphysical modelling, because it supports hierarchical structuring. Models are de-fined in classes which can be changed and reused in a modular manner.

Dymola generates an executable called Dymosim that simulates the model. Dy-mosim can be called either from Dymola or from other environments, for exampleMatlab. More details can be found in the Dymola user manual [13].

Among the possible implemented integration methods in Dymola, we chooseto solve the equations system with the method DASSL [38]. DASSL, a variable stepsize algorithm, is a suitable solver for stiff ODE (Ordinary Differential Equation) orDAE (Differential Algebraic Equation). In the steady state, the ODEs are equal tozero. So, the closer the computation is to the steady state, the closer are the ODEsto zero, and hence the faster is the simulation. Therefore, if small changes are ap-plied between two simulations, we usually load the final states of the previous

95

96 Implementation

simulation as initial conditions for the next simulation, to speed up the simula-tions.

A.2 Model structure

The structure of the model, displayed in figure A.1, follows the structure of thedrying section. An overall class, Dry Section (figure A.2), is composed of objectsof the class Air Group that contain the disposition of the cylinders. A class SteamGroup, containing the steam temperature is linked to the class Air Group. Finally, aclass Cylinder contains two subclasses where the physical equations are described:Shell Paper where the paper is in contact with the cylinder and the air and FreeDraw where the paper is in the free draw.

Figure A.1: Model structure, where Ps represents the steam pressure, Ts the steamtemperature, Tc the cylinder temperature, Tp and u the temperature and the mois-ture content of the paper.

A.3 Algorithm for identification with a Dymola model 97

Figure A.2: The drying section object in Dymola, containing elements of the classAir Group (AirGr) linked to instances of the class Steam Group (Steam). The papersheet, passing trough the air groups, is the input (white square) and output (blacksquare) of the drying section.

A.3 Algorithm for identification with a Dymola model

The algorithm presented here can be applied for other models in Dymola, andother environment than Matlab. It could be used to optimize the other imple-mented parts of the paper machine. This algorithm is described for steady-statecomputation, but can easily be modified for identification of dynamic models.

Algorithm 1 Identification algorithm for a dymola modelStep 0

- Compile the model in Dymola

- Choose the set of measurements for the identification

- Load the corresponding constant input U for the simulation

- Load the corresponding measured outputs Y

- Choose the vector θ of parameters to identify and set it to an initial value θ(0)

98 Implementation

- Run the simulation of the model: Xf (0) = dymosim(θ(0), U,Xi(0))where Xi(0) is the initial conditions of the states and is taken either from a previoussimulation or the default values assigned in the model and Xf (0) is the final statesof the simulation, and dymosim is a program that simulates the Dymola model.

- Set θ1 = θ0 and k = 1, go to step k

Step k

- Load initial values from the previous simulation: X0(k) = Xf (k − 1)

- Run a simulation: Xf (k) = dymosim(θ(k), U,X0(k))

- Load the results: Y (k) = Xf (k)

- Compute the vector error: ε(k) = Y − Y (k)

- Compute the loss function: VN (k) = 1N

∑Ni=1

εi(k)λi

- Given θ(k) and VN (k), the optimization routine either

- stops (θ = θ(k) or the optimization fails)

- or gives a new θ(k + 1), k = k + 1, go to step k.

A.4 Optimization routine

The optimization to solve is a nonlinear problem with linear constraints:

minθ VN (θ)s. t. θmin ≤ θ ≤ θmax

(A.1)

where θmin and θmax are the minimal and maximal values of the parameters. Theparameters are bounded in order to avoid that the optimization routine sets themto unrealistic values.

Different approaches were tried to solve the optimization problem. Gradientsmethods were tested: the nonlinear least square solver function lsqnonlin in theMatlab optimization toolbox [34] and the software package IPOPT (interior pointline search filter method) [52], but both routines did not converge to an optimalsolution. The routines were staying around the starting point without convergingto a solution. The reason for the failure of these two routines is unknown andshould be further investigated. The problem is probably due to the computationof the gradient. The methods using random search were more successful. Both theelliptical random search (see [9], Appendix D) or the pattern search algorithm inthe Matlab optimization toolbox [34] converged to the same solution for identicalproblems.

Appendix B

Observability Analysis for oneCylinder

In section 4.1, we carry out an observability analysis for the linearized system ofa contact-zone sub-model. In this appendix, we investigate the observability fora whole cylinder, i.e. we concatenate the contact zone and free draw models intoone cylinder model. The numerical values for this analysis are the same as theone used in section 4.1, for the cylinders 1 and 53. They are not displayed for thesake of clarity, and only the results are mentioned. One cylinder is made of thefollowing state vector:

Tp,cz

ucz

Tp,fd

ufd

where the subscripts cz and fd are for the contact zone and the free draw respec-tively. Tp is a vector of the seven temperatures in the thickness direction and uthe vector of the three moisture contents in the thickness direction. With thesenotations, we rewrite equation (4.6) as:

(δTp,cz

δucz

)= Acz

(δTp,cz

δucz

)+ Bcz

δTc

δTp,cz,in

δTa

δucz,in

where Bcz is divided into(

Bcz,Tc Bcz,Tp Bcz,Ta Bcz,u

). The output in equa-

tion (4.5) is written:

δy = Ccz

(δTp,cz

δucz

)

99

100 Observability Analysis for one Cylinder

The equations in the free draw are similar, except the temperature of the cylinderthat is not considered:

(δTp,fd

δufd

)= Afd

(δTp,fd

δufd

)+ Bfd

δTp,fd,in

δTa

δufd,in

δy = Cfd

(δTp,fd

δufd

)

where Bfd is divided into(

Bfd,Tp Bfd,Ta Bfd,u

).

We now want to concatenate the two systems to consider the observability of awhole cylinder. The incoming temperature and moisture content of the free drawis then equal to the temperature and moisture content of the contact zone (seefigure 3.1).

ufd,in = ucz

Tp,fd,in = Tp,cz

The system becomes:

δTp,cz

δucz

δTp,fd

δufd

=

(Acz O

Bfd,Tp Bfd,u Afd

)

δTp,cz

δucz

δTp,fd

δufd

+ Bcz

δTc

δTp,cz,in

δTa

δucz,in

δy =(

Ccz O2×10

O2×10 Cfd

)

δTp,cz

δucz

δTp,fd

δufd

(B.1)The resulting system is linear with 20 states (temperatures and moisture con-

tents in the contact zone and free draw) and 12 inputs (the temperatures of thecylinder and the air and the incoming temperatures and moisture contents). Insection 4.1, we conclude that if we measure at least the temperature at both sur-faces of the paper for every contact zone and every free draw, the model is observ-able. Indeed, we can check that the system (B.1) is observable with the followingoutput:

Ccz =(

1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0

)

Cfd =(

1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0

)

which is in accordance to the conclusion of section 4.1. If we measure only theboard temperature in the free draw,

Ccz = O2×10

Cfd =(

1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0

)

101

the rank of the observability matrix drops to 12 (<20), the system (B.1) becomestherefore not observable. If we add one measure of the board temperature in thecontact zone, for example,

Ccz =(

1 0 0 0 0 0 0 0 0 0)

Cfd =(

1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0

),

the rank of the observability matrix increases to 19 but the matrix is still not fullrank. We therefore conclude that, in order to ensure observability of the model, weneed to measure at least the temperature at both surfaces of the paper for everycontact zone and every free draw.

Bibliography

[1] Anguelova M. (2004) Nonlinear Observability and Identifiability: General Theoryand a Case Study of a Kinetic Model for S. cerevisiae, Licentiate Thesis, Depart-ment of Mathematics, Chalmers, Gothenburg, Sweden.

[2] Baggerud E., Stenström S. (2000) Modeling of Moisture Gradients for Convec-tive Drying of Industrial Pulp Sheets, Proceedings of the 12th International Dry-ing Symposium IDS2000, Noordwijkerhout, The Netherlands, Paper No. 308.

[3] Baggerud E. (2004) Modelling of Mass and Heat Transport in Paper, PhD Thesis,Department of Chemical Engineering, Lund University, Sweden.

[4] Bernada P., Stenström S., Månsson S. (1998) Experimental Study of the Mois-ture Distribution Inside a Pulp Sheet Using MRI. Part I: Principles of the MRITechnique, Journal of Pulp and Paper Science, Vol. 24 (12), pp 373–379.

[5] Bernada P., Stenström S., Månsson S. (1998) Experimental Study of the Mois-ture Distribution Inside a Pulp Sheet Using MRI. Part II: Drying Experiments,Journal of Pulp and Paper Science, Vol. 24 (12), pp 380–387.

[6] Bohlin T. (1991) Interactive System Identification: Prospects and Pitfalls, SpringerVerlag.

[7] Bohlin T. (1996) Modeling the bending stiffness of cartonboard at the Frövifors plantInternal report IR-S3-REG9608, Dep of Automatic Control, KTH, Stockholm,Sweden.

[8] Bortolin G., Gutman P. O., Borg S. (2001) Modeling of Wet-End Part of a PaperMill with Dymola, Proc. of IMACS/IFAC Symposium on Mathematical Modelingand Simulation in Agricultural and Bio-Industries, Haifa, Israel.

[9] Bortolin G. (2003) On Modelling and Estimation of Curl and Twist in Multi-plyPaperboard, Licentiate Thesis, Optimization and Systems Theory, Royal Insti-tute of Technology, Stockholm, Sweden.

[10] Bortolin G. (2005) Modelling and Grey-Box Identification of Curl and Twist in Pa-perboard Manufacturing, Doctoral Thesis in Mathematics, Royal Institute ofTechnology, Stockholm, Sweden.

103

104 BIBLIOGRAPHY

[11] Carlsson B. (2004) Dynamisk simulering av vira och presspartiet på en kar-tongmaskin, Master Thesis, Institutionen för kemiteknik och miljövetenskap,Chalmers Tekniska Högskola, Gothenburg, Sweden.

[12] Dymola, Dynamic Modeling Laboratory, Dynasim AB, Lund, Sweden.Homepage: www.dynasim.se

[13] Dymola User Manual (2004) Dymola, Multi-Engineering Modeling and Simu-lation, Dynasim AB.

[14] Ekvall J. (2004) Dryer Section Control in Paper Machines during Web Breaks, Li-centiate Dissertation, Department od Automatic Control, Lund Institute ofTechnology, Sweden.

[15] Funkquist J. (1995) Modelling and Identification of a Distributed Parameter Pro-cess: The Continuous Digester, PhD Thesis, Department of Automatic Control,KTH, Stockholm, Sweden.

[16] Gaillemard C. (2001) Modeling of the Drying Section in AssiDomän Frövi, MasterThesis, ESSTIN, France.

[17] Gaillemard C., Johansson A. (2004) Modelling of the Moisture Content of thePaper in the Drying Section of a Paper Mill, Proc. of Control Systems 2004,Quebec, Canada, pp 183–187.

[18] Gutman P.O., Nilsson B. (1996) Modelling and Predictor of Bending Stiffness forPaper Board Manufacturing IFAC’96.

[19] Harding S. G., Wessman D., Stenström S., Kenne L. (2001) Water transportduring the drying of cardboard studied by NMR imaging and diffusion tech-niques, Chemical Engineering Science, 56, pp 5269–5281.

[20] Heikkilä P. (1993) A Study on the Drying Process of Pigment Coated Paper Webs,PhD Thesis, Åbo Akademi, Finland.

[21] Hermann R., Krener A. J. (1977) Nonlinear Controllability and Observability,IEEE Trans. Automat. Control, vol. AC-22, No. 5, pp 728–740.

[22] Ioslovich I., Gutman P. O., Seginer I. (2004) Dominant parameter selection inthe marginally identifiable case, Mathematics and Computers in Simulation, Vol.65, pp 127–136.

[23] Isidori A. (2001) Nonlinear Control systems, third edition, Springer-Verlag.

[24] Jonhed L. (2000) Modellering, simulering och validering av en mälderimodell iModelica, Master Thesis, CTH, Gothenburg, Sweden.

[25] Julier S., Uhlmann J. (1997) A New Extension of the Kalman Filter to Nonlin-ear Systems, SPIE AeroSense Symposium.

105

[26] Karlsson M., Stenström S. (2005) Static and Dynamic Modelling of Carboarddrying, Part I: Theoretical model, Drying Technology 23 (1–2), pp 143–163.

[27] Karlsson M., Stenström S. (2005) Static and Dynamic Modelling of Carboarddrying, Part II: Simulations and Experimental results, Drying Technology 23(1–2), pp 165–186.

[28] Karlsson M. (2005) Static and Dynamic Modelling of the Drying Section of a PaperMachine, PhD Thesis, Department of Chemical Engineering, Lund University,Sweden.

[29] Karlsson M. (2000) Papermaking Science and technology, Papermaking, Part 2,Drying, Fapet Oy.

[30] Karlsson V. (2002) Dynamisk Modellering av ett Slutet Blekeri i Modelica, MasterThesis, KTH, Stockholm, Sweden.

[31] Ljung L. (1999) System identification, theory for the user, second edition, PrenticeHall.

[32] Modelica, Modelica Association, Homepage: www.modelica.org

[33] Nilsson L. (1996) Some Studies of the Transport Coefficients of Pulp and Paper,PhD Thesis, Lund University, Sweden.

[34] Optimization Toolbox- User’s guide (2000), The MathWorks Inc.

[35] Persson H. (1998) Dynamic Modelling and Simulation of Multi-Cylinder PaperDryers, Licentiate Dissertation, Lund University, Sweden.

[36] Pettersson J. (1998) On Model Based Estimation of Quality Variables for PaperManufacturing, Licentiate Thesis, Department of Automatic Control, KTHStockholm, Sweden.

[37] Pettersson M. (1999) Heat Transfer and Energy Efficiency in Infrared Paper Dryers,Doctoral Dissertation, Department of Chemical Engineering 1, Lund Univer-sity, Sweden.

[38] Petzold L. R. (1982) A description of DASSL: A differential/algebraic systemsolver, Proc. 10th IMACS World Congress, Montreal, Canada.

[39] Polak A. G. (2001) Indirect measurements: combining parameter selectionwith ridge regression, Measurement Science and Technology, vol. 12, pp 278–287.

[40] Pontremoli A. (2000) Modeling and Control of a Paper Dryer Section UsingModelicaTM , Master thesis, Department of Automatic Control, Lund Instituteof Technology, Sweden.

106 BIBLIOGRAPHY

[41] Slätteke O. (2003) Steam and Condensate System Control in Paper Making, Li-centiate Dissertation, Department of Automatic Control, Lund Institute ofTechnology, Sweden.

[42] Slätteke O. (2006) Modeling and Control of the Paper Machine Drying Section,Doctoral Dissertation, Department of Automatic Control, Lund Institute ofTechnology, Sweden.

[43] Smook G. A. (1992) Handbook for Pulp and Paper Technologists, Angus WildePublications.

[44] Sontag E. D. (1990) Mathematical Control Theory, Deterministic Finite Dimen-sional Systems, Springer-Verlag.

[45] Sontag E. D., Wang Y. (1991) I/O Equations for Nonlinear Systems and Obser-vation Spaces, Proceedings of the 30th conf. on Decision and Control, pp 720–725.

[46] Söderström T., Stoica P. (1989) System identification, Prentice Hall.

[47] Stenström S., Svanquist T. (1991) A general model for calculating pressuredrop in siphon and siphon riser tubes, Part 1: Principles and theoreticalmodel, Tappi Journal 74 (11), pp 154–162.

[48] Stenström S., Svanquist T. (1991) A general model for calculating pressuredrop in siphon and siphon riser tubes, Part 2: Experimental results, TappiJournal 74 (12), pp 157–162.

[49] Tunali E. T., Tarn T. J. (1987) New results for identifiability of nonlinear sys-tems, IEEE Trans. Automat. Control, Vol. AC-32, No. 2, pp 146–154.

[50] Videau J. L., Lemaitre A. (1982) An Improved Model of a Paper Machine Mul-ticylinder Drying Section, Drying 82, Section V, pp 129–138.

[51] Welty J., Wicks C. E., Wilson R. E. (1984) Fundamentals of Momentum, Heat, andMass Transfer, 3rd Edition, John Wiley & Sons.

[52] Wächter A., Biegler L. T. (2004) On the Implementation of a Primal-Dual Inte-rior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming,Research Report, IBM T. J. Watson Research Center, Yorktown, USA.

[53] Wessman D. (2001) Distribution and Redistribution of Moisture, Licentiate The-sis, Lund University, Sweden.

[54] Wilhelmsson B. (1995) An experimental and Theoretical Study of Multi-CylinderPaper Drying, PhD Thesis, Lund University, Sweden.

[55] Wingren A. (1999) Development of a simplified dynamic model for multi-cylinderpaper dryers, Master Thesis, Department of Chemical Engineering 1, LundInstitute of Technology, Lund, Sweden.

107

[56] Xia X., Moog C. H. (2003) Identifiability of Nonlinear Systems with Appli-cation to HIV/AIDS Models, IEEE Trans. Automat. Control, Vol. 48, No.2, pp330–336.

modelling the moisture content of multi-ply paperboard in ...9961/fulltext01.pdf · 1.2...

Documents