# Modelling the Moisture Content of Multi-Ply CHRISTELLE GAILLEMARD Licentiate Thesis

Modelling the Moisture Content of Multi-Ply Paperboard in the Paper Machine Drying Section CHRISTELLE GAILLEMARD Licentiate Thesis Stockholm, Sweden 2006 TRITA-MAT-06-OS-01 ISSN 1401-2294 ISBN 91-7178-302-4 Departement of Mathematics KTH SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av Licentiatexamen fredagen den 7 april 2006 klockan 10.00 i rum 3721, plan 7, Lindstedsvägen 25, Kungl Tekniska Högskolan, Stockholm. c Christelle Gaillemard, April 2006 ° Tryck: Universitetsservice US AB To Pär-Anders iii Abstract This thesis presents a grey-box model of the temperature and moisture content for each layer of the multi-ply paperboard inside the drying section of a paper mill. The distribution of the moisture inside the board is an important variable for the board quality, but is unfortunately not measured on-line. The main goal of this work is a model that predicts the moisture evolution during the drying, to be used by operators and process engineers as an estimation of the unmeasurable variables inside the drying section. Drying of carton board is a complex and nonlinear process. The physical phenomena are not entirely understood and the drying depends on a number of unknown parameters and unmodelled or unmeasurable features. The grey-box modelling approach, which consists in using the available measurements to estimate the unknown disturbances, is therefore a suitable approach for modelling the drying section. A major problem encountered with the modelling of the drying section is the lack of measurements to validate the model. Consequently, the correctness and uniqueness of the estimated variables and parameters are not guaranteed. We therefore carry out observability and identifiability analyses and the results suggest that the selected model structure is observable and identifiable under the assumption that specific measurements are available. Based on this analysis, static measurements in the drying section are carried out to identify the parameters of the model. The parameters are identified using one data set and the results are validated with other data sets. We finally simulate the model dynamics to investigate if predicting the final board properties on-line is feasible. Since only the final board temperature and moisture content are measured on-line, the variables and parameters are neither observable nor identifiable. We therefore regard the predictions as an approximation of the estimated variables. The semiphysical model is complemented with a nonlinear Kalman filter to estimate the unmeasured inputs and the unmodelled disturbances. Data simulations show a good prediction of the final board temperature and moisture content at the end of the drying section. The model could therefore possibly be used by operators and process engineers as an indicator of the board temperature and moisture inside the drying section. Keywords: Drying section modelling, multi-ply paperboard, moisture content, identification, grey-box modelling Acknowledgements I have been fortunate to interact with remarkable people that have contributed in many ways to the completion of this thesis. First of all, I would like to thank my advisor Per-Olof Gutman, for introducing me to the project after my master thesis. His insightful suggestions and his support despite the geographical distance were a source of inspiration and motivation. These years have been challenging but I now feel it was worth going through all the struggles. I am very grateful to my advisor Anders Lindquist for the opportunity to join the Optimization and System Theory group. His enthusiasm for research provides a creative and friendly working environment. I also wish to acknowledge AssiDomän Frövi for financing the project. I am grateful to Bengt Nilsson for the opportunity to join the welcoming Process Control group. I also thank my master-thesis advisors, Stefan Ericsson and Lars Jonhed, for sharing their knowledge about the drying section, the paper machine and the simulation tool Dymola. Lars and his family deserve special thanks for their kindness and sincere concern. My free time in Frövi would have been lonelier without my friend Dorothée Millon whom I thank for all the time spent together and for her hospitality. In AssiDomän Frövi, I would like to thank Antero Jauhiainen for helping me with the measurements, and Kent Åkerberg, Gunnar Pålsson and Anders Henriksson for sharing their knowledge about the drying section. I am also thankful to Magnus Karlsson for answering my questions about the physical model, Jens Pettersson for helping me with the IPOPT interface and Jenny Ekvall for the discussions about paper machine drying sections. The faculty members and graduate students at the division of Optimization and System Theory at KTH have contributed to making these years of study stimulating and enjoyable. I am very grateful to my colleague Gianantonio Bortolin for sharing his experience of grey-box modelling. His coaching and friendship were a great source of motivation. I can not thank enough my roommate Vanna Fanizza for her support and all those discussions, sometimes work related. Her spontaneity and our true friendship always made me happy to go to the office. I also want to thank Ryozo Nagamune for his kindness and precious advises. v vi My love and gratitude go to my parents, Gérard and Bernadette, and my brother Flavien. They have indirectly contributed to this work with their constant support and encouragement. My Swedish family also deserves to be mentioned for their warm welcome that made me feel home in a new country. I deeply thank all my relatives and friends, for providing me escapes from work. All the time spent together gave me kicks of energy that I needed to continue. Finally, I thank my future husband Pär-Anders for his love, patience and support during the completion of the thesis. He helped me believe I could do it! Train Stockholm - Nyköping, February 2006 Christelle Gaillemard Contents Nomenclature 1 ix . . . . 1 1 2 3 3 2 Background 2.1 Brief literature review . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Process description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 3 Model Description 3.1 Discretization of the paper moisture and temperature 3.2 Heat balance of the cylinder . . . . . . . . . . . . . . . 3.3 Heat balance of the paperboard . . . . . . . . . . . . . 3.4 Mass balances within the paper web . . . . . . . . . . 3.5 Physical properties . . . . . . . . . . . . . . . . . . . . 3.6 Summary of the physical model equations . . . . . . . 3.7 Semi-physical adjustments . . . . . . . . . . . . . . . . 3.8 Parameters, inputs and outputs of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 15 18 23 25 29 31 32 Observability, Identifiability and Sensitivity Analyses 4.1 Observability analysis of the physical model . . . . 4.2 Identifiability analysis of the physical model . . . . 4.3 Sensitivity analysis of the semi-physical model . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 45 48 65 Identification of the Parameters 5.1 Description of the static measurements . 5.2 Identification procedure . . . . . . . . . 5.3 Parameter selection . . . . . . . . . . . . 5.4 Identification results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 67 72 73 73 4 5 Introduction 1.1 Paper-machine modelling in AssiDomän Frövi 1.2 Drying-section modelling . . . . . . . . . . . . 1.3 Objectives and contributions . . . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 5.5 CONTENTS Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6 Dynamic Simulations 6.1 Deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Grey-box modelling of the disturbances . . . . . . . . . . . . . . . . 6.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 80 88 7 Conclusions and Future Work 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Directions for future work . . . . . . . . . . . . . . . . . . . . . . . . 91 91 92 A Implementation A.1 Simulation program . . . . . . . . . . . . . . . . . . A.2 Model structure . . . . . . . . . . . . . . . . . . . . A.3 Algorithm for identification with a Dymola model A.4 Optimization routine . . . . . . . . . . . . . . . . . B Observability Analysis for one Cylinder Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 95 96 97 98 99 103 Nomenclature Some notations may have a different meaning locally. Symbol vx u t T G W l ρ d h Dab FRF Cp k P p ṁ λ Kg R M L x z φ Description Speed of the machine Moisture content Time index Temperature Basis weight Flow of paper Paper width Density Thickness Heat transfer coefficient Diffusion coefficient of element a into element b Fabric heat reduction factor Specific capacity Thermal conductivity Absolute pressure Partial pressure of water Evaporation rate Heat of evaporation Mass transfer coefficient Gas constant Molar mass Length in the machine direction Space coordinate in the machine direction Space coordinate in the thickness direction Relative humidity ix Dimension m/s kgw /kgdry s ◦ C or K kg/m2 kg/s m kg/m3 m W/m2 ◦ C m2 /s % J/kg ◦ C W/m ◦ C Pa Pa kg/m2 s J/kg m/s J/mol K kg/mol m m m − x Nomenclature Dimensionless numbers Nu Sc Pr Re Description Nusselt number Schmidt number Prandtl number Reynolds number Subscripts Description Paper Cylinder Dry material, fibers Air Vapor Fabric Steam Free draw Contact zone Laminar flow Turbulent flow Sorption Saturation p c dry a w f s fd cz lam turb sorp sat Chapter 1 Introduction Paperboard manufacturing is a challenging enterprise requiring advanced technology and great financial investments. In this competitive field of business, considerable resources are put into process optimization and control systems to increase productivity, reduce manufacturing costs and improve quality of the board. During such research and development, simulation tools provide a cost-effective approach for verifying and validating new ideas without requiring risky and costly experiments on the operational machines. The aim of this thesis is to contribute to the efforts on modelling the complete manufacturing process of a paper machine. More specifically, the thesis deals with the modelling of the multi-cylinder drying section of a paper mill. Drying is a critical part of the paperboard manufacturing process since it consumes a great amount of energy and affects the quality variables of the paperboard considerably. The idea behind the present work was initiated by two modelling approaches used in AssiDomän Frövi: the grey-box modelling approach for implementing on-line predictors as decision tools for the process engineers and operators, and object-oriented modelling to obtain a model of the paper plant. Inspired by these two modelling approaches, this thesis presents a model that predicts the moisture content for each layer of the board in the drying section. 1.1 Paper-machine modelling in AssiDomän Frövi The modelling interest in AssiDomän Frövi started in 1991, with the modelling of the bending stiffness1 . Gutman and Nilsson [18] made a first attempt with a quasi-linear ARMA-model. Bohlin [6] reported a grey-box model that Pettersson [36] improved by ameliorating the physical description in the sub-models. With a similar approach as Pettersson [36], Bortolin [9, 10] developed a model of the curl and twist2 . The semi-physical models of bending stiffness [36] and curl 1 The bending stiffness represents the force needed to bend the board. is defined as the departure from flat form. 2 Curl 1 2 Introduction and twist [9, 10] were complemented with a nonlinear Kalman filter to estimate the unmodelled disturbances, and implemented in the mill information system as quality predictors for the operators and process engineers. Another modelling project in AssiDomän Frövi is based on the object-oriented language Modelica together with the simulation tool Dymola. The project aims for a model of the paper machine, by creating and reusing libraries adapted to the pulp and paper manufacturing process. To this end, the following parts have been modelled in Dymola: the bleach plant [30], the wet end [24, 8], the press [11] and the drying section [16]. 1.2 Drying-section modelling The drying of paper is an essential part for paper manufacturing. Firstly, it requires a great amount of energy, and secondly it is an important parameter for the quality variables of the board such as curl and twist, bending stiffness, shrinkage, wrinkle and delamination3 . Some models of the drying section of AssiDomän Frövi are available from previous work [16, 28]. The first model is a one-layer model [16] based on the work of Persson [35]. The model is implemented in Dymola, and fitted with static measurements. The second model, developed by Karlsson [28], is a considerably more complex physical model that includes both internal mass and heat transfer inside the board. The main objective of the present work is a three-layer model of the board, since we are interested in estimating the moisture content for each layer. The model should therefore be detailed enough to reproduce the important physical behaviour inside the board. Furthermore, we want to apply a similar approach as Pettersson [36] and Bortolin [9, 10] to obtain a predictor of the moisture inside the board in the drying section. This approach implies the need to design a model that is simple enough to allow the simulation to be run on-line. Thus, the chosen model will result from a compromise between simplicity and completeness. Since a simple one-layer model was already implemented in Dymola [16], the original idea of this work was to extend it to a three-layer model and investigate the grey-box modelling approach with Dymola. A major problem encountered in the modelling of the drying section is the lack of measurement data to validate the model. Few on-line sensors are present in the drying section for two main reasons: the drying hood is very hot and the board is difficult to reach. Moreover, experiments are not allowed due to high cost of non-sellable products. In this work, we attempt to answer the questions of observability and identifiability that arise when modelling the drying section: With the few available measurements, can we estimate the board moistures and temperatures in the drying section? 3 Delamination is the separation of the board layers. 1.3 Objectives and contributions 1.3 3 Objectives and contributions Based on a similar approach as the predictors for the bending stiffness [36] and the curl and twist [9], this thesis aims for a grey-box model that predicts the board moistures and temperatures inside the drying section of a paper machine. The objective of the thesis is not to derive a complex model of the drying section, since this can be found, for example, in the works of Baggerud [3] and Karlsson [28]. The model should instead be simple enough to be usable for simulation and running tests and as a control tool for the operators. Additionally, we aim to describe the board moisture content in each layer, since it affects several quality parameters. Moreover, we want to investigate the possibility of using grey-box modelling with the simulation tool Dymola, to apply the technique on the other modelled parts of the paper machine. In short, the contribution of the thesis is as follows: • An observability and identifiability analysis of the drying section model is performed. Based on this analysis, we specify a set of static measurements that ensures observability and identifiability of the estimated variables and parameters during the identification. We also show, however, that the model is not observable for on-line conditions, and the predictions are therefore regarded as an approximation of the estimated variables. • The grey-box modelling approach for a multi-cylinder drying section is applied. Since on-line measurements for the board moisture content and temperature inside the drying section are difficult, we implement an extended Kalman filter that uses the few available on-line measurements to compensate for the unmodelled or not measured features. The resulting stochastic model gives an approximation of the board properties in the drying section. • We investigate the grey-box modelling approach on a model implemented in the simulation tool Dymola. The parameter identification method and the nonlinear Kalman filtering technique are performed for the model. The resulting algorithms can be used to apply the approach on the other modelled parts of the board machine [30, 24, 8, 11]. 1.4 Outline The structure of the thesis is as follows: Chapter 2 provides a brief literature review and a description of the drying process. Chapter 3 describes the semi-physical model of the drying section. The model derives the equations for the temperature of the cylinders, the temperature and the moisture content of a three-layer board. The parameters and inputs of the model are introduced. 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 estimate their impact on the model and to select the dominant ones for the identification. Chapter 5 first describes the measurements carried out to evaluate the model with static process data. The process of identification of the unknown parameters is then introduced and the identification results are presented. Chapter 6 investigates the behaviour of the model under on-line conditions. The deterministic model is first studied and then complemented by an extended Kalman filter to add disturbances and uncertainties in the model. Chapter 7 concludes the thesis and discusses possible directions for future work. Appendix A describes the model structure in the simulation tool Dymola and the algorithm developed for parameter identification within the Dymola environment. Appendix B further analyses the observability conditions for one cylinder. Chapter 2 Background 2.1 Brief literature review Drying of paper The paper manufacturing process is described in e.g. [43] and the drying of paper is detailed in [29]. The drying process is an important part in the manufacturing of paper since it requires a lot of energy and affects the quality variables of the paper, such as bending stiffness and curl and twist. Various models of the drying of paper can be found in the literature, depending on which goal one has with the model; some are detailed and complex to get an insight into the physical phenomena and others are simplified for control purposes. The research group at Chemical engineering, Lund Institute of Technology provides physical modelling of the drying of paper [54, 35, 28, 3], the condensate flow inside the dryer [47, 48], infrared drying [37] and internal transport of water inside the paper [33, 53]. Slätteke [41] modelled the dynamics from the steam valve to the steam pressure with a black-box IPZ-model (one Integrator, one Pole and one Zero) for control tuning and derived a grey-box model to get an insight into the physical laws behind the black-box model. He then expanded the steam pressure model with a model for the paper to test several moisture controls [42]. Ekvall [14] examined a control strategy to improve the restart of the machine after a web break. Wilhelmsson [54] developed a dynamic model of the multi-cylinder drying section by using the heat transport in the cylinder and paper, which was extended by Persson [35]. These two previous models do not include the internal mass transport of water in the thickness direction and assume that all the evaporation occurs at the surface of the paper. Baggerud [3] developed a detailed model for general drying of paper that includes both internal transfer of water and heat inside the paper, and Karlsson [28] a model of the drying section of the Frövi board. None of these models include the cross direction (CD) profile. Wingren [55] simplified Persson’s model [35] by considering one steam group (a group of cylinders with the same supplied steam) as one cylinder with modified 5 6 Background dimensions. This approach is not applied in this work, since the resulting model was not considered satisfactory. The distribution of the moisture inside the paper is an important factor for quality parameters of the board, for example, shrinkage, curl and twist, wrinkle and delamination. Research has therefore been performed to understand and evaluate the distribution of the moisture in the thickness direction. Bernada et al. [4, 5] carried out experiments to observe the internal moisture during drying by using the magnetic resonance imaging (MRI) technique. Harding et al. [19] studied the water profile and diffusion inside the board by using nuclear magnetic resonance (NMR) imaging. Wessman [53] investigated the transport of water in the thickness direction when watering or drying the board. Baggerud [2] developed a model of the moisture gradient to fit the data of Bernada et al. [4]. These works focussed on convective drying, i.e. drying by hot air. The moisture gradient was measured on a small sample of board, which is hardly feasible on-line on the hot cylinder drying systems because of the configuration (access point difficulty) and the speed of the machine. Modelling in AssiDomän Frövi The interest in modelling in AssiDomän Frövi started in 1991. A first attempt of modelling the bending stiffness was made by Gutman and Nilsson [18] with a quasi-linear ARMA-model with slow adaptation of the model parameters and fast adaptation of a bias compensation term. Bohlin [6] used grey-box modelling, where the parameters of the model were first identified on one set of data and bias was then compensated on-line with an Extended Kalman Filter. Pettersson [36] improved the physical behaviour in the sub-models and achieved a satisfactory model usable for the operators. With a similar approach as Pettersson [36], Bortolin [9, 10] developed a model of the curl and twist. The grey-box modelling approach in AssiDomän Frövi has also been considered by Funkquist [15] for the continuous pulp digester, a nonlinear distributed parameter process. The semi-physical models of bending stiffness [36] and curl and twist [9] are complemented with a nonlinear Kalman filter to estimate the unmodelled disturbances, and implemented in the mill information system as quality predictors for the operators and process engineers. The two grey-box models require the amount of water per layer as input. It is therefore of interest to develop a model predicting the moisture content per layer. Several master theses and reports provide models of various parts of the pulp and paper plant in order to achieve a complete model of the paper manufacturing process. To this end, the following parts have been modelled: the bleach plant [30], the wet end [24, 8], the press [11] and the drying section [16]. The model of the drying section [16] is a simplified model of Persson [35], where the cartonboard is considered to be one layer. The present work is an extension of the simple model [16] to a three-layer grey-box model with the addition of the moisture content 2.2 Process description 7 profile in the thickness direction and an extended Kalman filter to compensate for unmodelled features. Observability and Identifiability of nonlinear systems Since different structures of models can be chosen, the goal of this work is to find a compromise between a complete and simple model, to preserve the important physical phenomena while keeping a simple-enough model to be run online. Since the task of this work is to implement the model on-line and correct the bias with an observer, it is important to check if the model is observable and identifiable. In other words, we want to know if it is possible to reconstruct all the interesting states given the few measurements available. Observability and identifiability are related subjects, since identifiability can be considered as the observability of the parameters [1]. Analysis of linear observability and identifiability is a well-studied subject. However, for nonlinear systems, the complexity is increased and the subject is still under investigation. Anguelova [1] offers a literature review on the subject. The main tools for the study of observability and identifiability are differential geometry and differential algebra. The differential geometric approach can be found in the work of [21, 23, 45, 49] and consists in computing the Lie derivatives of the output up to rank n where n is the number of states in the system. The idea behind the differential algebraic approach is to express the Lie derivatives of the inputs and outputs as polynomial expressions. This approach is easier for rational or polynomial functions. These two tools are of high complexity that increases with the number of states. An alternative is to investigate observability and identifiability of the system linearized around some operating point [44] which yields local properties only. In this thesis, we follow the latter approach. 2.2 Process description Carton board manufacturing is a complex industrial process. The board at Assidomän Frövi is composed of four fiber layers (or plies) and two coating layers. The two middle fiber layers are composed of fibers of low density to get light weight and high bending stiffness. The bottom fiber layer is made of unbleached pulp while the top layer is composed of bleached pulp to achieve good printing properties. The two middle layers are considered identical as they use the same fiber mixture. The board machine in Frövi, depicted in figure 2.1, is divided into five main processes: The wet end, the press, the drying section and the calendering and coating. 1. The wet end: This is the first step of the paper manufacturing. For each of four layers, the paper stock is spread by a headbox onto a fabric drained by water. The thickness of the stock jet is determined by the opening of the headbox 8 Background Figure 2.1: The board machine at Frövi. The felts in the drying section are not depicted. slice while the velocity is provided by the headbox pressure. These two parameters will determine the spatial distribution of the fibers in the paper and the basis weight. Each layer is formed independently and then added to the previous layer in the order bottom ply, middle ply and top ply. 2. The press section: The main purposes of the press are to remove the water from the paper, consolidate the web and provide a surface smoothness. Since the water removal is more economical by mechanical means in the press section than by drying, as much water as possible is removed in the press section. The water removal should be uniform across the machine to obtain a level moisture profile for the pressed sheet entering the drying section. After the press, the paper contains between 56 % and 64 % of water. 2.2 Process description 9 3. The drying section: In the drying section, the board passes over and under steam-heated cylinders and the water inside the board is removed by evaporation. The concentration of water is around 8 % at the end of the drying section. The drying process is detailed further in this section. 4. Calendering: After the drying section, the paper is processed in a heated press nip to provide a smooth surface of the paper before the coating. 5. The coating: The coating is applied on the top layer of the sheet in two layers to improve the paper printing properties. The drying section In the drying section, the paper is passed over a series of 93 rotating steam-heated cylinders where water is evaporated and carried away by ventilation air. The wet web is held tightly against the cylinders by a synthetic permeable fabric called drying felt. Between two cylinders the paper is only in contact with the air; this part is called free draw. The evaporation of the water in the paper inside the drying section is divided into four zones [43]: the warming up, the constant evaporation rate, the falling evaporation rate and the bound water zone. In the first zone, the paper is warmed up. During the constant evaporation rate zone, the water is situated on the fiber surfaces or within the large capillaries. When the free moisture is concentrated in the smaller capillaries, the evaporation rate decreases and reaches the falling rate zone. In the bound water zone, the residual water is more tightly held by physiochemical phenomena. The rest of this section further describes the main parts of the drying section. Steam and condensate system: The steam inside the cylinders provides the heat energy referred to latent heat when it condenses inside the cylinder shell. The temperature of the saturated steam depends of the pressure. The steam is the main variable used to control the drying. At high machine speed, a layer of condensate film is formed inside the cylinder shell because of the centrifugal force. Even a thin layer of condensate is undesirable, since it affects the heat transfer considerably. To improve the heat transfer, spoiler bars are placed inside the shell to create turbulent flow (this increases the heat transfer) and rotative siphons are used to remove the condensate. The force controlling the flow of condensate outside the cylinder is the differential pressure between the incoming and outgoing steam. Together with the condensate, approximately 15 to 20 % of the incoming steam, called blow through steam, is also removed [43]. The outgoing steam and the condensate are conducted to a separator tank, where the steam is reused for the other steam groups in a cascade system configuration. Hood ventilation: The surrounding air is an important parameter for the drying. It must be drier than the paper to ensure evaporation and the temperature 10 Background should be higher than the dew point1 . The main task of the ventilation system is to remove the evaporated water, to prevent condensation. The incoming flow should be the same as the outgoing flow to avoid disturbances and to get the total pressure equal to the atmospheric pressure. Drying felts: The main purpose of the drying felt (also called fabric) is to keep the paper tight against the cylinders to get a better contact surface, and hence a better heat transfer, and to control the shrinkage in the cross direction. The speed of the felt is often higher than the speed of the machine to prevent shrinkage in the machine direction. The felts are run by the help of rolls whose speed determines the tension of the felts. VIB device: At cylinder 53, water is sprayed over the board bottom layer by a steam actuator called VIB since the cross direction (CD) profile after the drying section shows that the middle of the paper is drier than the edges. The control of the VIB gives a more uniform profile and releases the risk of web breaks in the stack dryers. The sprayed water is taken from the condensate tanks and is around 80◦ C. A full opening of the actuator corresponds to an increase of 2 % units in the final moisture concentration. Stack dryers: In most of the steam groups, the drying felts are situated below and under the cylinders; this is called a two-tier configuration. After the fifth steam group, the board enters a critical zone and can break easily in the free draw. Therefore, a single-tier configuration is adopted. The felt is holding the board even in the free draw. Between cylinders, vacuum rolls are leading the paper web to prevent folding when the board comes in contact with the cylinders. The first stack group is composed of lower cylinders which warms the bottom layer whereas the second group warms the top layer. The effect of the vacuum rolls is not well understood but their presence increases the drying rate. One possible explanation is that the vacuum rolls create a turbulent flow of the air in contact with the paper that increases the heat transfer coefficient. Another assumption is that the air is drier in the stacks because the vacuum cylinders suck it up. The pressure of the vacuum rolls is about 3000 P a. Infrared dryers: At the end of the drying section, two infrared dryers are used, together with the VIB, to control the CD moisture profile. A full effect of the infrared dryers corresponds to a decrease of 2 % units in the final moisture concentration. Measuring frame: A measuring frame is situated at the end of the drying section, between the two infrared dryers. The frame measures the average moisture content (in the thickness direction) and the temperature of the top layer. The 1 the temperature at which water vapour begins to condense. 2.2 Process description 11 frame is not used continuously, because it is sensitive to the heat of the infrared dryers. For the control of the CD moisture profile, measurements from the measurement frame located before the coating section are used. Chapter 3 Model Description The physical model is essentially based on the model derived by Persson [35] with the addition of the diffusion of water in the thickness direction. The model computes the temperature of the cylinders and the temperature and moisture content of the carton board, using the heat balance of the cylinder and the heat and mass balance of the board. This chapter first explains the choice of discretization for the temperature and moisture in the paper. The three following sections describe the derivation of the temperature of the cylinder and the temperature and moisture inside the paper. The physical properties are then described: the properties of the cylinder, the paper, the surrounding air and at the interface between the paper and the air. A summary of the model is given in section 3.6. Finally, we present the semi-physical adjustments and the parameters, inputs and outputs of the model. 3.1 Discretization of the paper moisture and temperature We consider that the carton board contains three layers by gathering the two middle layers, since they have the same properties. The present work discretizes the moisture content and temperature of the board in the machine only in the machine direction x (MD) and thickness direction z. For the cross direction y, only the properties in the middle of the sheet are considered, i.e. the edges are not modelled. In the machine direction, each cylinder is divided into two blocks: the contact zone (we use the same notation as Karlsson [28]), where the board is in contact with the cylinder, and the free draw, where the board is in the free draw. For each block, only one node is computed in the machine direction. The calculated temperatures Tp and moisture contents u are situated at the end of the contact zone or the free draw and the computed states of the previous block are used as incoming boundary condition (Tp,in and uin ). The discretization of the temperature and moisture of the board in the machine direction is illustrated in figure 3.1. The discretization of the moisture content and the temperature of the paper in the thickness direction is displayed in figure 3.2. The moisture content is computed 13 14 Model Description Figure 3.1: Discretization of the temperature and moisture of the board in the machine direction, where Lcp and Lf d are the lengths of contact zone and free draw. Tp and u, the computed temperature and moisture content of the paper, are used as 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 thickness axis. The temperatures for each layer, Tp,2 , Tp,4 and Tp,6 are needed to compute the 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 temperatures at 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 of notation, the indices of the layers and moisture content are the same as the ones for the temperature. The bottom layer (BS) is layer 2, the two middle layers (MS) are grouped in layer 4 and the top layer (TS) is layer 6. An analysis of the observability and identifiability of the model is presented in chapter 4 in order to investigate the appropriateness of this choice of discretization. In the following sections 3.2, 3.3, 3.4, the equations are derived for the case where the paper is in contact with an upper cylinder (i.e. the bottom layer is in contact with the cylinder). If the paper is in contact with a lower cylinder, the equations are obtained by switching the indices. The equations for the paper in the free draw are identical; one just needs to apply the equations for the paper in contact with the air for both sides of the paper and replace the length of contact between the paper and the cylinder Lcp with the length of free draw Lf d . 3.2 Heat balance of the cylinder 15 Figure 3.2: Discretization of the temperature and moisture content of the board in the thickness direction, when the board is in contact with an upper cylinder. Tp and u are the temperature and moisture content of the paper, Tc is the temperature of 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 kc ∂ 2 Tc ∂Tc = − vx ∂t ρc Cpc ∂ 2 zc ∂x (3.1) ◦ c where the term vx ∂T ∂x [ C/s] is the convection transport of energy in the machine ∂ 2 Tc ◦ c direction x, and ρckCp [ C/s] is the conductive heat transfer in the thickness 2 c ∂ z direction of the cylinder. Tc [◦ C] is the temperature of the cylinder shell, t [s] is the time index, zc [m] is the space coordinate in the thickness direction of the cylinder, and x [m] is the space coordinate in the machine direction. The properties of the cylinder, kc [W/m ◦ C] , Cpc [J/kg ◦ C] and ρc [kg/m3 ], are the thermal 16 Model Description conductivity, the specific capacity, and the density of the cylinder, respectively. They are described in section 3.5. Persson [35] computes one cylinder temperature in the machine direction and three in the thickness direction for each cylinder. Videau and Lemaitre [50] observe a variation below one degree in the machine direction in their simulation. Therefore, we assume, as [54, 35, 50], that the temperature of the outer surface of the shell is constant during a turn of the cylinder and consequently the convecc tive term vx ∂T ∂x is removed. For the thickness direction, since we are only able to measure the temperature at the surface of the cylinder in contact with the air, we compute only one point, at the outer surface of the cylinder shell. The modifications from Persson’s model [35] are derived in this section. The differential equation (3.1) for the point at the surface of the cylinder becomes: ¯ kc ∂Tc ¯¯ = (Tc,dc − Tc,0 ) (3.2) ∂t ¯dc ρc Cpc d2c where dc [m] is the thickness of the cylinder shell, Tc,dc [◦ C] the temperature of the cylinder shell at the surface in contact with the air or the paper, and Tc,0 [◦ C] the temperature of the cylinder shell at the surface in contact with the steam. In order to compute (Tc,dc − Tc,0 ), we use the boundary conditions in the thickness direction, which are defined in the next section. Boundary conditions for the temperature of the cylinder The boundary conditions (3.3), (3.6) and (3.7), displayed in figure 3.3, are based from Persson [35]. Surface of the cylinder shell in contact with the steam The heat transferred from the steam is conducted through the cylinder shell [35]: ¯ ∂Tc ¯¯ kc hsc (Ts − Tc,0 ) = −kc (3.3) = − (Tc,0 − Tc,dc ) ¯ ∂zc zc =0 dc where hsc [W/m2 ◦ C], the heat transfer coefficient between the steam and the cylinder is a parameter to identify (see section 3.8). The temperature of the steam Ts [◦ C] inside the cylinder is considered as saturated and is calculated directly from the steam pressure [35]. Since we want to compute the temperature of the cylinder only at the outside surface, we need to remove Tc,0 in the left side of equation (3.3) and replace it by an expression of Tc,dc . Consequently, we modify the heat transfer coefficient between steam and cylinder hsc , in order to include the conduction of heat from the inside surface of the cylinder shell to the outside surface. The heat transferred from the steam to the inside shell and the heat conducted through the shell are connected 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 the shell, Tc,0 and Tc,dc are the temperatures of the inside and outside of the shell and Ts is the temperature of the steam. in series. Thus, the modified heat transfer coefficient from the steam to the outside of the shell h̄sc is calculated as follows: k hsc dcc 1 1 1 = + kc ⇒ h̄sc = hsc h̄sc hsc + kdcc dc (3.4) Therefore, the boundary condition (3.3) becomes: h̄sc (Ts − Tc,dc ) = − kdcc (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 ¯¯ kc = − (Tc,dc − Tc,0 ) ∂zc ¯dc dc (3.6) 18 Model Description where hcp [W/m2 ◦ C], the heat transfer coefficient between the cylinder and the paper, is a parameter to identify (see section 3.8) and Tp,1 [◦ C] is the temperature of the paper surface in contact with the cylinder. Surface of the cylinder shell in contact with the air The conductive heat inside the cylinder is transferred to the air: ¯ ∂Tc ¯¯ kc hca (Tc,dc − Ta ) = −kc = − (Tc,dc − Tc,0 ) ∂zc ¯dc dc (3.7) where Ta [◦ C] is the temperature of the surrounding air. hca [W/m2 ◦ C] is the heat transfer coefficient between the cylinder and the air and is calculated in the same manner as the heat transfer coefficient between paper and air hpa (see section 3.5). The temperature of the outer side of the cylinder Since we assume that there is only one node per cylinder, we gather equations (3.5), (3.6) and (3.7) to get the boundary condition for the outer side temperature of the cylinder. If we call Lcp [m] the length of contact between the cylinder and the paper and Lca [m] the length of contact cylinder–air, the boundary condition becomes: µ ¶ dc Lcp hcp (Tc,dc − Tp,1 ) + Lca hca (Tc,dc − Ta ) (Tc,dc −Tc,0 ) = − − h̄sc (Ts − Tc,dc ) kc Lcp + Lca (3.8) To simplify the notation, we remove the index dc since there is only one node for the cylinder and define Tc := Tc,dc . Inserting (3.8) in (3.2), the differential equation for the temperature at the surface of the cylinder is defined as follows: µ ¶ ∂Tc 1 Lcp hcp (Tp,1 − Tc ) + Lca hca (Ta − Tc ) = + h̄sc (Ts − Tc ) (3.9) ∂t ρc Cpc dc Lcp + Lca 3.3 Heat balance of the paperboard The heat balance of the paper web is described by the following equation [35]: ∂Tp kp ∂ 2 Tp ∂Tp = − vx 2 ∂t ρp Cpp ∂ zp ∂x (3.10) ∂Tp ◦ ∂x [ C/s] is the convection transport of energy in the machine kp ∂ 2 Tp ◦ and ρp Cp [ C/s] is the conductive heat transfer in the thickness 2 p ∂ z the cylinder. Tp [◦ C] is the temperature of the paper, t [s] is the time where the term vx direction x, direction of index, zp [m] is the space coordinate in the thickness direction of the paper, and x [m] is the space coordinate in the machine direction. The properties of the paper, 3.3 Heat balance of the paperboard 19 kp [W/m ◦ C], Cpp [J/kg ◦ C] and ρp [kg/m3 ], are the thermal conductivity, the specific capacity and the density of the paper, respectively. They are described in section 3.5. Numerical solution In the machine direction (x), since there is only one node, we use the same method as Persson [35]: the backwards differentiation method of the first order. ∂Tp,i,j 1 = (Tp,i,j − Tp,i,j−1 ) ∂x ∆xj (3.11) where i and j are the indices in the thickness direction and in the machine direction, respectively. The temperature of the paper at the point Tp,i,j−1 is given by the boundary condition in the machine direction (see figure 3.1). For ease of notation, we skip the index j: ∂Tp,i 1 = (Tp,i − Tpin,i ) (3.12) ∂x Lcp 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 ¯¯ 1 (Tp,i+1 − Tp,i−1 ) (3.13) = ∂z ¯i 2∆zi where ∆zi is the discretization step at the point zi . The centre differentiation method of second order is: ¯ ∂ 2 Tp ¯¯ 1 (Tp,i+1 − 2Tp,i + Tp,i−1 ) = ¯ 2 ∂ z i (∆zi )2 (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 ¯¯ 1 (Tp,i+1 − Tp,i−1 ) = ∂z ¯i ∆zi−1,i+1 ¯ 1 ∂ 2 T ¯¯ 1 (Tp,i+1 − Tp,i ) + (Tp,i−1 − Tp,i ) = ¯ 2 2 ∂ z i (∆zi,i+1 ) (∆zi−1,i )2 where ∆zl,k is the distance between the points situated at zl and zk . (3.15) (3.16) 20 Model Description Paper temperature in the middle of each layer For the nodes in the middle of each layer i = 2, 4, 6, the step sizes ∆zi−1,i and ∆zi,i+1 are equal, ∆zi−1,i = ∆zi,i+1 = dp,i /2, where dp,i is the thickness of the layer i. Inserting (3.12) and (3.16) into (3.10) the differential equations of the nodes in the middle of each layer Tp,i are described as follows: ∂Tp,i 4kp,i vx (Tp,i+1 − 2Tp,i + Tp,i−1 ) − (Tp,i − Tpin,i ) = ∂t ρp,i Cpp,i d2p,i Lcp (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 temperature Tp,i are derived by inserting (3.12) and (3.16) into (3.10): ∂Tp,i ∂t 4kp,i−1 (Tp,i−1 − Tp,i ) ρp,i−1 Cpp,i−1 d2p,i−1 4kp,i+1 x + ρp,i+1 Cpp,i+1 d2 (−Tp,i + Tp,i+1 ) − Lvcp (Tp,i p,i+1 = − Tpin,i ) (3.18) Paper temperature at the surface in contact with the fabric/air To compute the heat differential equation for the point situated at the surface of the paper in contact with the fabric or the air, Tp,7 , we use the same method as Persson [35] and introduce a virtual point Tp,8 , situated at the distance dp,6 /2, on the opposite side of Tp,6 . The equation (3.16) is then written: ¯ ∂ 2 Tp ¯¯ 4 (3.19) = 2 (Tp,8 − 2Tp,7 + Tp,6 ) ¯ 2 ∂ z 7 dp,6 To remove Tp,8 in the equation, we use the boundary conditions between the surface of the paper and the surrounding air, represented in figure 3.4. Boundary condition between the surface of the paper and the fabric/air At the surface of the paper in contact with the air, the heat of conduction inside the paper and the heat of evaporation of water in the air are transferred to the air. The boundary condition is described by the following equation [35]: ¯ ∂Tp ¯¯ ṁλ = hpa (Ta − Tp,7 ) − kp,6 (3.20) ∂z ¯7 where hpa [W/m2 ◦ C] is the heat transfer coefficient between the paper and the air, ṁ [kg/m2 s] 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 the air, where Tp,i is the board temperature at position i, Ta is the temperature of the surrounding air and dp,6 the thickness of layer 6. Using equation (3.15), the previous equation can be written: ṁλ = hpa (Ta − Tp,7 ) − kp,6 Tp,8 − Tp,6 dp,6 (3.21) We can now extract Tp,8 : → Tp,8 = dp,6 (−ṁλ + hpa (Ta − Tp,7 )) + Tp,6 kp,6 and insert it in (3.19): ¯ µ ¶ ∂ 2 Tp ¯¯ dp,6 4 2T − 2T + (− ṁλ + h (T − T )) = p,6 p,7 pa a p,7 ∂ 2 z ¯7 d2p,6 kp,6 (3.22) (3.23) Finally, the differential equation (3.10) for Tp,7 is given by inserting (3.12) and (3.23): ´ ³ ∂Tp,7 4k dp,6 = ρp,6 Cpp,6 (− ṁλ + h (T − T )) 2T − 2T + 2 pa a p,7 p,6 p,7 ∂t k d p,6 p,6 p,6 (3.24) x − Lvcp (Tp,7 − Tpin,7 ) Paper temperature at the surface in contact with the cylinder To compute the heat differential equation for the point situated at the surface of the paper in contact with the cylinder, Tp,1 , we use the same method as previously and introduce a virtual point Tp,0 , situated at the distance dp,2 /2, on the opposite side of Tp,2 [35]. The equation (3.16) is then written: ¯ ∂ 2 Tp ¯¯ 4 = 2 (Tp,2 − 2Tp,1 + Tp,0 ) (3.25) ∂ 2 z ¯1 dp,2 22 Model Description To remove Tp,0 in the equation, we use the boundary conditions between the surface of the paper and the cylinder, represented in figure 3.5. Figure 3.5: Boundary condition at the surface of the paper in contact with the cylinder, where Tp,i is the board temperature at position i, Tc is the temperature of the 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 from the cylinder is conducted inside the paper [35]: hcp (Tc − Tp,1 ) = −kp,2 ¯ ∂Tp ¯¯ ∂z ¯i=1 (3.26) Using equation (3.15), the previous equation can be written: Tp,2 − Tp,0 dp,2 (3.27) dp,2 (hcp (Tc − Tp,1 )) + Tp,2 kp,2 (3.28) hcp (Tc − Tp,1 ) = −kp,2 We can now extract Tp,0 : Tp,0 = and insert it in (3.25): ¯ ∂ 2 Tp ¯¯ 4 dp,2 (hcp (Tc − Tp,1 )) = 2 (2Tp,2 − 2Tp,1 + ∂ 2 z ¯1 dp,2 kp,2 (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): µ ¶ ∂Tp,1 4kp,2 dp,2 vx 2T − 2T + − = (h (T − T ) (Tp,1 − Tpin,1 ) p,2 p,1 cp c p,1 ∂t ρp,2 Cpp,2 d2p,2 kp,2 Lcp (3.30) 3.4 Mass balances within the paper web The present model derives the mass balances of dry material and water. The paper is assumed to be composed of two components: the dry component, consisting of fibers and fillers (subscript dry ) and the water (subscript w ). To not increase the complexity of the model, the distinction between the two phases of water, liquid or vapor, is not included in this work, and the presence of air inside the paper, formed when the water leaves the pores, is not modelled. Descriptions of mass balances including the air, liquid and vapor can be found in the works of Baggerud [2] and Karlsson [28]. Mass balance of dry material The mass of dry material per layer Gdry,i [kgdry /m2 ] remains constant during the drying. In the numerical computations in the thesis, the mass of dry matter in each layer is obtained from the bending stiffness predictor estimates of Pettersson [36]. The mass balance is expressed as in Persson [35]: ∂Gdry,i ∂Gdry,i = −vx ∂t ∂x (3.31) Mass balance of water The moisture content u [kgw /kgdry ] represents the amount of water in the paper. To compute the mass balance of water in the paper, we consider the following two types of water transport: 1. Evaporation of water into the air ṁevap [kgw /m2 s] occurring at the surface [35], derived in section 3.5. 2. Diffusion of water in the thickness direction ṁdif f [kgw /m2 s] given by Fick’s law [3]. The driving force for the diffusive transport of water is the gradient in the thickness direction of the moisture content (we consider that the diffusion of water only occurs in the thickness direction): ṁdif f = −ρdry Dwp ∂u Gdry ∂u = Dwp ∂z ∆z ∂z (3.32) where Dwp [m2 /s] is the diffusion coefficient of water in the paper, described in section 3.8, and ∆z is the thickness where we consider the diffusion. 24 Model Description Mass balance for the layer in contact with the air For the layer in contact with the air, we take into account the water evaporated into the air and the diffusion from the middle layer: ¯ ∂u ¯¯ 4Dwp ṁ6 vx =¡ − (u6 − uin,6 ) (3.33) ¢2 (u4 − u6 ) − ¯ ∂t 6 Gdry,6 Lcp dp(6) + dp(4) where the first term of the right hand side represents the diffusion of water from layer 4 to layer 6, with Dwp [m2 /s] the diffusion coefficient of water into the paper, dp (i) [m] the thickness of layer i and ui [kgw /kgdry ] the moisture content of layer i. The second term represents the evaporation from the paper surface to the surrounding air, where ṁ6 is the evaporation rate [kgw /m2 s] from the surface of layer 6 and Gdry,6 [kgdry /m2 ] is the dry basis weight of layer 6. The last term represents the convection transport of water in the machine direction, where vx [m/s] is the speed of the machine, Lcp [m] the length of the contact zone and uin,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 have to take into account the diffusion to the middle layer: ¯ ∂u ¯¯ vx 4Dwp (u2 − uin,2 ) (3.34) =¡ ¢2 (u4 − u2 ) − ¯ ∂t 2 Lcp dp(2) + dp(4) where the first term of the right hand side represents the diffusion of water from layer 2 to layer 4, with Dwp [m2 /s] the diffusion coefficient of water into the paper, dp (i) [m] the thickness of layer i and ui [kgw /kgdry ] the moisture content of layer i. The last term represents the convection transport of water in the machine direction, where vx [m/s] is the speed of the machine, Lcp [m] the length of the contact 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 ¯¯ 4Dwp vx 4Dwp (u4 −uin,4 ) (3.35) =¡ ¢2 (u2 −u4 )+ ¡ ¢2 (u6 −u4 )− ¯ ∂t 4 Lcp dp(2) + dp(4) dp(6) + dp(4) where the first and second term of the right hand side represent the diffusion of water from layer 2 to layer 4 and layer 6 to layer 4, with Dwp [m2 /s] the diffusion coefficient of water into the paper, dp (i) [m] the thickness of layer i and ui [kgw /kgdry ] the moisture content of layer i. The last term represents the convection transport of water in the machine direction, where vx [m/s] is the speed of the machine, Lcp [m] the length of the contact zone and uin,4 [kgw /kgdry ] the incoming moisture content of layer 4. 3.5 Physical properties 3.5 25 Physical properties Properties of the cylinder All the cylinders in the drying section are made of cast iron and have the same properties: • 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 computed for each layer i. Board heat capacity The heat capacity depends on the moisture content and the heat capacity of the fibers: Cpdry,i + ui · Cp,w (Tp,i ) Cpp,i (ui , Tp,i ) = (3.36) 1 + ui where the heat capacity of the fibers Cpdry,i is a parameter that we identify (see section 3.8) and the heat capacity of the water Cpw (Tp,i ) is obtained from a heat capacity table ([51] p. 772). Board thermal conductivity The thermal conductivity of the layer i, kp,i [W/m ◦ C] is depending on the moisture 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 identify and is described in section 3.8. The thermal conductivity of the water inside the web is obtained from a thermal conductivity table ([51] p. 772). 26 Model Description Density of the paper We compute the density ρp,i [kg/m3 ] of layer i as follows: ρp,i (ui , Tp,i ) = 1 + ui 1 + ρdry,i ui ρw (Tp,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 is taken from the bending stiffness predictor estimates computed by Pettersson [36]. However, since the densities in Pettersson [36] are computed after the calendering that increases the densities, we need to compute the dry densities in the drying section. Two approaches are possible: • The first alternative is to compute the densities using the estimated potential densities (equations (4.39) and (4.40) in [36]). • The other possibility is to take the estimated densities after the calendering and calculate backwards the density in the drying section, by considering the 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 the pulp properties of the layers. However, the second method should be more accurate 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 for the middle layer; therefore only the density of the middle layer differs when taking one or the other methods. In this work, we choose to use the density computed with the Kalman filter. Thickness of the paper Once we have the layer density ρp,i , the thickness dp,i [m] of layer i is computed as 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 numerical computations, the dry basis weight is obtained from the bending stiffness predictor estimates computed by Pettersson [36]. Paper width The paper width is around 7 m at beginning of the drying section and around 6.7 m at the end of the drying. A fitted coefficient is calculated to take into account the 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 dimensionless numbers: N u(Ta ) · ka (Ta ) hpa (Ta ) = (3.40) Lcp where N u [-] is the Nusselt number, ka [W/m2 ◦ C] is the conductivity of the air and 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 transfer coefficient from the paper to the air: hpf a = hpa (3.41) Persson [35] and Wilhelmsson [54] suggest adjusting this coefficient to take into account both the heat and mass transfer with the following: h?pa = p,v ṁC ṁC if Tp < Ta p,v −1) hpa ṁCp,v ṁC p,v − exp − −1 hpa exp 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 also depending on dimensionless numbers: µ Kgpa (Ta ) = N u(Ta ) · Sc(Ta ) P r(Ta ) ¶ 13 · Dwa(Ta ) Lcp (3.43) where the dimensionless numbers Sc and P r are the Schmidt Number and the Prandtl Number respectively. Dwa [m2 /s] is the diffusion coefficient of water into the air, computed by using calculations and tables in [51]. The presence of the fabric is assumed to reduce the mass transfer of water between the paper and the air. This reduction is modelled by a coefficient F RF (Fabric reduction factor), described in section 3.8: µ Kgpf a = 100 − F RF 100 ¶ · Kgpa (3.44) 28 Model Description Vapor partial pressure of the paper The vapor partial pressure for free water in the paper psat [Pa] is calculated from Antoine equation (3.45) [35]: psat (Tp ) = 133.322 · e a 1 18.3036− Tp +227.03 (3.45) where the coeficient a1 [K−1 ] is usually set to the value 3816. In this work, however, 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 inside the paper remains inside the pores, the amount of free water is limited. The vapor partial 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 of the air is assumed to be equal to 1 when the water inside the paper is free water and decreases when the moisture in the paper is lower than a critical content (around 0.4 kgw /kgdry according to [35] and [3]). We further discuss the choice of sorption isotherm in the semi-physical adjustments in section 3.7. If the partial pressure of water in the paper is higher than the atmospheric pressure, flash evaporation occurs (this phenomena is modelled, for example, in the work of Wilhelmsson (see eq 3.12 in [54]) and Karlsson [28]). In this work, we do not consider the flash evaporation, since when the board reaches high temperatures, the moisture content at the surface is low, and consequently the board vapor partial pressure at the surface is not higher than the partial pressure of water in the air. Evaporation rate The evaporation rate is calculated by the Stefan equation [35]: µ ¶ Kgpa (Ta ) · Mw · Pa Pa − pa ṁ(Tp , Ta , u) = ln R · (Tp + 273.15) 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 in the air, described in section 3.5 and pp [Pa] is the partial pressure of water in the paper, 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 taken into account [35, 28, 29]. The heat of evaporation becomes: λ0 (Tp ) = λ(Tp ) + λsorp (Tp , u) (3.49) where λsorp [J/kg], the heat of sorption, is discussed in the semi-physical adjustments 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, we model the temperature and humidity of the air as a linear interpolation between a constant and the properties of the paper. The measured temperature of the air is displayed in figure 5.1. In the beginning of the drying, where the paper is cooler, the temperature of the air is lower. Therefore, since we believe that the temperature of the paper influences the temperature of 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 from the paper to the air due to the low partial pressure of water in the paper (since the temperature of the paper is low). Therefore, we model the partial pressure of water in the air as a linear interpolation between a constant pa,ave and the partial pressure 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 interpolation 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 constant, since the influence of the temperature is very small on these values. We also remove the dependency of the air temperature in equations (3.40) as well as for the dimensionless numbers N u, Sc and P r. The temperature dependencies are kept in the simulation of the model, but they are omitted in the observability and identifiability analysis in chapter 4 for ease of computation. To simplify the notation, 30 Model Description we define the following functions: f1 (u) = f2 (u) = f3 (Tp , u, Ta ) = f4 (u) = f5 (uj , uk ) = f6 (Tp , Ta , u) = 4kp (u) ρp (u)Cpp (u)dp (u)2 4hcp (u) ρp (u)Cpp (u)dp (u) 4ṁ(Tp ,Ta ,u)λ(Tp ) − ρp (u)Cp p (u)dp (u) 4hpa ρp (u)Cpp (u)dp (u) 4Dwp (uj (dp (uj )+dp (uk ))2 ṁ(Tp ,Ta ,u) − Gdry,6 (3.52) − uk ) 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 ∂Tp,2 ∂t ∂Tp,3 ∂t ∂Tp,4 ∂t ∂Tp,5 ∂t ∂Tp,6 ∂t ∂Tp,7 ∂t = = = = = = = ∂u2 ∂t ∂u4 ∂t ∂u6 ∂t = = = f1 (u2 )(2Tp,2 − 2Tp,1 ) + f2 (u2 )(Tc − Tp,1 ) − vLx (Tp,1 − Tpin,1 ) f1 (u2 )(Tp,3 − 2Tp,2 + Tp,1 ) − vLx (Tp,2 − Tpin,2 ) f1 (u2 )(Tp,2 − Tp,3 ) + f1 (u4 )(−Tp,3 + Tp,4 ) − vLx (Tp,3 − Tpin,3 ) f1 (u4 )(Tp,5 − 2Tp,4 + Tp,3 ) − vLx (Tp,4 − Tpin,4 ) f1 (u4 )(Tp,4 − Tp,5 ) + f1 (u6 )(−Tp,5 + Tp,6 ) − vLx (Tp,5 − Tpin,5 ) f1 (u6 )(Tp,7 − 2Tp,6 + Tp,5 ) − vLx (Tp,6 − Tpin,6 ) f1 (u6 )(2Tp,6 − 2Tp,7 ) + f3 (Tp,7 , u6 , Ta ) + f4 (u6 )(Ta − Tp,7 ) − vLx (Tp,7 − Tpin,7 ) f5 (u4 , u2 ) − vLx (u2 − uin,2 ) f5 (u2 , u4 ) + f5 (u6 , u4 ) − vLx (u4 − uin,4 ) f5 (u4 , u6 ) + f6 (Tp,7 , Ta , u6 ) − vLx (u6 − uin,6 ) (3.53) Paper in the free draw ∂Tp,1 ∂t = ∂Tp,2 ∂t ∂Tp,3 ∂t ∂Tp,4 ∂t ∂Tp,5 ∂t ∂Tp,6 ∂t ∂Tp,7 ∂t = = = = = = ∂u2 ∂t ∂u4 ∂t ∂u6 ∂t = = = f1 (u2 )(2Tp,2 − 2Tp,1 ) + f3 (Tp,1 , u2 , Ta ) + f4 (u2 )(Ta − Tp,1 ) − vLx (Tp,1 − Tpin,1 ) f1 (u2 )(Tp,3 − 2Tp,2 + Tp,1 ) − vLx (Tp,2 − Tpin,2 ) f1 (u2 )(Tp,2 − Tp,3 ) + f1 (u4 )(−Tp,3 + Tp,4 ) − vLx (Tp,3 − Tpin,3 ) f1 (u4 )(Tp,5 − 2Tp,4 + Tp,3 ) − vLx (Tp,4 − Tpin,4 ) f1 (u4 )(Tp,4 − Tp,5 ) + f1 (u6 )(−Tp,5 + Tp,6 ) − vLx (Tp,5 − Tpin,5 ) f1 (u6 )(Tp,7 − 2Tp,6 + Tp,5 ) − vLx (Tp,6 − Tpin,6 ) f1 (u6 )(2Tp,6 − 2Tp,7 ) + f3 (Tp,7 , u6 , Ta ) + f4 (u6 )(Ta − Tp,7 ) − vLx (Tp,7 − Tpin,7 ) f5 (u4 , u2 ) + f6 (Tp,1 , Ta , u2 ) − vLx (u2 − uin,2 ) f5 (u2 , u4 ) + f5 (u6 , u4 ) − vLx (u4 − uin,4 ) f5 (u4 , u6 ) + f6 (Tp,7 , Ta , u6 ) − vLx (u6 − uin,6 ) (3.54) 3.7 Semi-physical adjustments 3.7 31 Semi-physical adjustments Vapor partial pressure for free water When simulating the physical model, the temperature of the paper is decreasing too much in the free draw compared to the measurements and seems to reach an equilibrium point (see, for example, figure 4.3). Tests with the physical parameters could not remedy this problem. A possible explanation, suggested by Wilhelmsson [54], is that the drying section is a fast process, while the time required for a porous material to reach its equilibrium state is very long. Therefore, we modify the 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 in equation (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 capacity is decreased. This phenomenon is called sorption. The partial pressure of vapor inside the paper is equal to the partial pressure of vapor for free water multiplied by a sorption isotherm φ(T p, u). The sorption isotherm is defined as the relative humidity of air [-] as a function of the equilibrium moisture content of the paper [kgw /kgdry ] for a given temperature [29], p. 67. The sorption phenomenon is described, for example, in [29]. Several formulas of the sorption isotherm can be found in the literature, but according to Karlsson [28], the choice of the sorption isotherm does not influence the drying significantly. We therefore consider the sorption isotherm in Heikkilä [20], which is suitable for mechanical pulp (see figure 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 surface layer, is assumed to be considerably higher than the one at the surface. Therefore, the sorption isotherm is modified to take the sorption phenomenon into account also for higher moisture contents (the relative humidity of air increases until 0.8 kgw /kgdry ). Figure 3.6 shows the modified sorption isotherm. In the new sorption isotherm, the temperature effect has been removed since the influence of the temperature is already taken into account by the coefficient a1 . A similar approach is applied to the heat of sorption. The chosen expressions for the relative humidity of air and the heat of sorption are: ¡ ¢ ¡ ¢ (−5·u−0.3) φ(u, Tp ) = φ(u) = 1 − · 1 − e(−50·u) ³e ´ 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 humidity of air (heat of sorption) similar to the sorption isotherm (heat of sorption) 32 Model Description Sorption isotherm: Relative humidity of air as a function of the moisture content u 1 0.9 900 Heikkilä (T=90 C) Modified sorption isotherm 0.8 0.7 0.6 0.5 0.4 0.3 700 600 500 400 300 0.2 200 0.1 100 0 0 0.2 0.4 0.6 0.8 1 1.2 moisture content u in kgw/kgdry (a) Sorption isotherm 1.4 1.6 1.8 Heikkilä (T=90 C) Modified heat of sorption 800 Heat of sorption [kJ/kg] Relative humidity of air [−] Heat of Sorption as a function of the moisture content 1000 0 0 0.2 0.4 0.6 0.8 1 1.2 moisture content u in kgw/kgdry 1.4 1.6 1.8 (b) Heat of sorption Figure 3.6: Comparison between the sorption isotherm and the heat of sorption in Heikkilä [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 not identify the coefficients of equation (3.55) in order to not increase the complexity of 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 inputs are not measured on-line; we therefore also need to estimate them. We make a distinction between the parameters, assumed to be the constant, and the unmeasured inputs that are time-varying. An identifiability analysis and a sensitivity analysis of the parameters are presented in sections 4.2 and 4.3. The identification procedure 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 steam to the cylinder is difficult to estimate since the amount of condensate inside the cylinder is not known. It is a common assumption that the condensate is totally removed by the siphons and that the heat transfer coefficient between steam and cylinder is a constant to identify [35, 54, 28]. According to Wilhelmsson [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 increases when decreasing the diffusion coefficient, to compensate the total heat transferred. Heat transfer coefficient between cylinder and paper hcp : The heat transfer coefficient between the cylinder and the paper is supposed to depend on the contact area between cylinder and paper [3, 35, 54]. Wilhelmsson [54] and Persson [35] assume that the contact area is linearly dependent on the moisture 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. The heat transfer should be independent of the fabric tension if the tension is in the interval of 2–6 kN/m, which is in the range of the typical values applied in the machine [35, 54]. Baggerud [3] and Karlsson [28] consider that hcp can be modelled as a constant when the internal mass transfer is included. However, since this model only describes the diffusion of water in the thickness direction and the contact area is difficult to model, we use equation (3.56). Karlsson and Stenström [27] found a fitted value of 2200 W/m2 ◦ C but point out that the fitted value is also dependent on the diffusion coefficient of the water inside the paper. 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 result of the smoother surface. This improvement can be seen by observing the measured temperature of the first seven cylinders (for example in figure 5.4). Even though the supplied steam is the same for all cylinders, the temperature of the upper cylinders is higher than the lower cylinders since the lower cylinders release more heat to the board. We therefore multiply the heat transfer coefficient hcp by two for the cylinders 2, 4 and 6, since this value gives the best fit for the temperature of the cylinders. Diffusion coefficient of water/vapor into the paper Dwp : The coefficient for the diffusion through the paper should be in the range of 1 ·10−10 –2 ·10−6 m2 /s for the water (liquid) and 2.1 ·10−8 –5.4 ·10−6 m2 /s for the vapor [3]. Baggerud [3] and Karlsson [28] model the liquid and vapor diffusion coefficients as functions of the moisture content, but since in this thesis, we do not differentiate the two phases of the water, we model the water diffusion coefficient as a constant. Fabric reduction factor F RF : This coefficient is used to take into account the effect of the fabric on the mass transfer between the paper and the air. The reduction of mass transfer due to the felt is around 30–50 % [35] and is assumed to be the same for all the cylinders in the drying section. 34 Model Description Dry heat capacity Cpdry : Since there are few static measurements available for the identification, we assume in this work that the dry heat capacity does not depend 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 the dry thermal conductivity does not depend on the quality of the pulp. Baggerud [3] and Karlsson [28] suggest a dry thermal conductivity of the fibers of 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 evaporation on the decrease of temperature. Since it is a dummy parameter, the minimal and maximal values are set manually. The minimal value is set to the physical value and the maximal value is set to 3880. Unmeasured inputs The unmeasured inputs are not measured on-line. However, they are measured occasionally, for example in the static measurements described in chapter 5. Incoming moisture content uin : The water concentration of the paperboard leaving the press section is known to be between 0.56 % and 0.64 %, which corresponds to a moisture content of 1.27–1.78 kgw /kgdry . Incoming temperature of the paper Tp,in : The temperature of the board entering the drying section is in the range of 45–55 ◦ C. Properties of the surrounding air: The parameters for the properties of the surrounding 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 the information 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 calculations of the bending stiffness predictor [36] and are available in the on-line information system. To compute the density in the drying section, we use the load of 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 two following outputs: • The total moisture content of the board uout • The top layer board temperature Tp7,out Chapter 4 Observability, Identifiability and Sensitivity 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 measurable outputs, a nonlinear system can be expressed as: ½ Σnl Ẋ(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 estimate them using the observations of the outputs Y (t) and the inputs U (t) for a certain set of parameters θ. To know if this is possible, we need to study the observability of the system. It is also of interest to know if the parameters are identifiable, i.e. the set of parameters giving a specific input–output trajectory is unique. Moreover, in order to evaluate the impact of the parameters on the model and select the most influent ones that remain for the identification, we carry out a sensitivity analysis of the parameters. We should point out that in the general case, 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 observable 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 determined only for the set of X in the neighbourhood of a given point X0 . A brief literature review on the analysis of nonlinear observability and identifiability is presented in section 2.1. The sections in this chapter describe an analysis of the three concepts — observability, identifiability and sensitivity — for the model described in chapter 3. 37 38 4.1 Observability, Identifiability and Sensitivity Analyses Observability analysis of the physical model A system is said to be observable, if we can determine the states X(t) given the outputs Y (t), the inputs U (t), and the set of parameters θ. Since the tools of differential geometry and differential algebra mentioned in section 2.1 are of high complexity increasing with the number of states, we use instead the linearization principle for observability [44] and determine the observability of the linearization of the system at a constant operational point. If the linearized system is observable, we can conclude that the nonlinear system is locally observable. The theory used 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 to apply. It is therefore easier to study the linearized system. In this section, we first state the linearization principle for observability and then show how to linearize a system. Finally, we describe how to analyse the observability of a linear system by transforming it into the canonical form. Linearization Principle for Observability To analyse the observability of a nonlinear system, one can study the observability of the linearized system at a constant operational point by using the linearization principle for observability that states that if the linearization around x̃ is observable, then the nonlinear system is locally observable around x̃. Theorem 1 (Linearization Principle for Observability [44], p. 209) Assume that Σnl is a continuous-time system over R of class C 1 , 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 about the operational point x̃ is that the linearized system Σl is observable on I = [σ, τ ]. Linearization of a system The linearized system at the constant operational point (X̃(t), Ũ (t)), given the constant output Ỹ (t) is obtained by the following expression: ½ Σl where δ Ẋ(t) = AδX(t) + BδU (t) δY (t) = CδX(t) δX(t) = X(t) − X̃(t) δU (t) = U (t) − Ũ (t) δY (t) = Y (t) − Ỹ (t) A= B= C= (4.2) ¯ ∂F ¯ ∂X ¯ X̃,Ũ ∂F ¯ ∂U ¯X̃,Ũ ∂H ¯ ∂X X̃ 4.1 Observability analysis of the physical model 39 Observability analysis of a linear system Consider the following linear system. ½ Ẋ(t) = AX(t) + BU (t) Σl Y (t) = CX(t) (4.3) To study the observability of a linear system, one needs to check the rank of the 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 linear system is not observable. If A has n distinct real eigenvalues, with a coordinate change X̂ = T X(t), we can transform the system into the modal canonical form (also called Jordan canonical form): ( ˙ X̂(t) = ÂX̂(t) + B̂U (t) Σ̂ (4.4) Y (t) = Ĉ X̂(t) where Â = T AT −1 is a diagonal matrix containing the eigenvalues of A, T −1 is a matrix containing the eigenvectors of A, B̂ = T B and Ĉ = CT −1 . Note that this transformation 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 Ω̂i = (Ĉi , Ĉi Â, · · · , Ĉi Ân−1 )T is the number of non zero items in the row Ĉi , since Â is diagonal and has distinct eigenvalues. A zero in the j th column of the row Ĉi corresponds to an unobservable state X̂j by the output Yi . Application to the physical model We are mainly interested in the feasibility of building an observer. The observability analysis is therefore derived for the physical model, i.e. to simplify the study the semi-physical adjustments in section 3.8 are not considered. Since the model computes 21 states for each cylinder (11 nodes for the contact zone and 10 nodes for the free draw), a complete observability analysis for the whole drying section seems 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 of a complete cylinder (concatenation of a contact-zone sub-model and a free-draw sub-model). In this section, we derive the observability analysis for the contactzone sub-model. The observability analysis for the free-draw sub-model is similar and is mentioned at the end of the section. In appendix B, the observability analysis 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 moisture dependency of ṁ in the functions f3 and f6 in equation (3.52) is removed. 40 Observability, Identifiability and Sensitivity Analyses The chosen states of the model are the temperature and moisture inside the paper. We assume that we know the properties of the incoming sheet, Tp,in and uin , as well as the temperature Tc of the cylinder surface and the temperature of the surrounding air Ta , and use them as inputs. Tc Tpin,1 Tp,1 Tp,2 Tpin,2 Tp,3 Tpin,3 Tp,4 Tpin,4 Tp,5 Tpin,5 X = U = , Tp,6 Tpin,6 Tp,7 Tpin,7 u2 Ta uin,2 u4 uin,4 u6 uin,6 For the output, we assume that we can measure the board temperature at the two surfaces, 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 the cylinder is difficult to measure. The observability with only the temperature as output is discussed at the end of this section. The obtained output is linear: 1 0 0 0 0 0 0 0 0 0 X 0 0 0 Y = 0 0 0 0 0 0 1 (4.5) Gdry,2 Gdry,4 Gdry,6 0 0 0 0 0 0 0 Gdry,tot Gdry,tot Gdry,tot 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 δTc δ Ṫp,1 δTp,2 δTpin,1 µ ¶ δ Ṫp,2 = A1 δTp,3 + A2 δu2 + B 1 δTpin,2 δ Ṫp,3 δu4 δTp,4 δTpin,3 δ Ṫp,4 δTp,5 δTpin,4 δTp,4 δTpin,5 ¶ µ δ Ṫp,5 δTp,5 δu δTpin,6 4 δ Ṫp,6 = A3 + B2 δTp,6 + A4 δu6 δTpin,7 δ Ṫp,7 δTa δTp,7 δTp,7 δTa δ u̇2 δu2 δuin,2 δ u̇4 = A5 δu4 + B3 δuin,4 δ u̇6 δu6 δuin,6 4.1 Observability analysis of the physical model 41 where A1 A1 A2 = = = 0 . A1 − vLx [I4 ] .. with 0 −2f1 (ũ2 ) − f2 (ũ2 ) 2f1 (ũ2 ) 0 f (ũ ) −2f (ũ ) f (ũ 1 2 1 2 1 2) 0 f1 (ũ2 ) −f1 (ũ2 ) − f1 (ũ4 ) 0 0 f1 (ũ4 ) ∂f2 (u2 ) ∂f1 (u2 ) (2T̃p,2 − 2T̃p,1 ) + (T̃c ∂u2 ∂u2 ũ ũ ∂f1 (u2 ) ( T̃ − 2 T̃ + T̃ p,1 p,2 p,3 ) ∂u2 ũ ∂f1 (u2 ) (T̃p,2 − T̃p,3 ) ∂u2 ũ B1 A3 = = A3 − vx L = = f1 (ũ4 ) 0 −f1 (ũ4 ) − f1 (ũ6 ) f1 (ũ6 ) 0 0 ∂f1 (u4 ) (T̃p,4 − T̃p,5 ) ∂u 4 = ũ 0 £ ¤ vx L · I3 A5 = A5 = B3 = 0 0 0 0 f1 (ũ4 ) 0 −2f1 (ũ4 ) f1 (ũ 4) − T̃p,1 ) 0 0 ∂f1 (u4 ) (−T̃p,3 + T̃p,4 ) ∂u4 ũ ∂f1 (u4 ) ( T̃p,5 − 2T̃p,4 + T̃p,3 ) ∂u4 ũ f1 (ũ6 ) −2f1 (ũ6 ) ! 0 f1 (ũ6 ) ∂f3 (Tp,7 ,u6 ,Ta ) − f4 (ũ6 ) 2f1 (ũ6 ) −2f1 (ũ6 ) + ∂Tp,7 T̃p ,ũ,T̃a ∂f1 (u6 ) (−T̃p,5 + T̃p,6 ) ∂u6 ũ ∂f1 (u6 ) ( T̃p,7 − 2T̃p,6 + T̃p,5 ) ∂u6 ũ ∂f3 (u6 ,Tp,7 ,Ta ) ∂f4 (u6 ) ∂f1 (u6 ) + 2(T̃p,6 − T̃p,7 ) + (T̃a − T̃p,7 ) ∂u6 ∂u6 ∂u6 ũ ũ 0 ¯0 ∂f3 (u6 ,Tp,7 ,Ta ) ¯ + f4 (ũ6 ) ¯ ∂Ta ũ,T̃p ,T̃a T̃p ,ũ,T̃a 0 .. . [I3 ] with 0 ¯ ¯ ∂f5 (u4 ,u2 ) ¯ ∂f5 (u4 ,u2 ) ¯ 0 ¯ ¯ ∂u2 ∂u4 ¯ũ ¯ ũ ¯ ∂f5 (u2 ,u4 ) ¯ ∂f5 (u2 ,u4 ) ¯ ∂f5 (u6 ,u4 ) ¯ 0 + ¯ ¯ ¯ ∂u2 ∂u4 ¯ ũ ũ ¯ ∂u4 ũ ¯ ∂f6 (Tp,7 ,Ta ) ¯ ∂f5 (u4 ,u6 ) 0 ¯ ¯ ∂Tp,7 ∂u4 T̃p ,T̃a ũ 0 ¤ £ vx 0 · I3 ∂f (T ,T ) ¯¯ L 6 p,7 a ¯ ∂Ta A5 − 0 .. . [I3 ] with 0 0 B2 £ ¤ vx · I4 L 0 A4 f2 (ũ2 ) 0 .. . 0 Ã A3 vx L T̃p ,T̃a 0 ¯ ∂f5 (u6 ,u4 ) ¯ ¯ ∂u6 ¯ũ ∂f5 (u4 ,u6 ) ¯ ¯ ∂u6 ũ 42 Observability, Identifiability and Sensitivity Analyses where In is the n × n identity matrix. To summarize, the linearized system is given by the following expression. δ Ṫp,1 δ Ṫp,2 δ Ṫp,3 δ Ṫp,4 δ Ṫp,5 δ Ṫp,6 δ Ṫp,7 δ u̇2 δ u̇4 δ u̇6 0 = 0 0 0 0 0 + B1 0 0 0 0 0 0 0 0 0 0 A1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A2 0 0 0 A3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A4 A5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B3 δTp,1 δTp,2 δTp,3 δTp,4 δTp,5 δTp,6 δTp,7 δu2 δu4 δu6 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 derived for an upper cylinder. To strengthen the results, the observability analysis is also carried 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 4.1 Observability analysis of the physical model 43 The operational point is obtained from the simulation: 103.7 100.3 98.0 88.7 81.6 80.3 79.0 0.5375 0.6877 0.5627 X̃(t) = Ũ (t) = 124.6 74.1 75.6 76.6 81.48 83.2 82.4 80.2 88 0.5290 0.7033 0.5786 We use the symbolic computer program Maple to derive the matrices A and B, and get: A 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 −0.009 0 0 0 0.12 −2.3 0 19.6 2.2 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 −0.0003 ! 0 0 2.2 = Ã C 98.5 = B −120.3 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 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0.23 0.52 0.25 0 0 0 44 Observability, Identifiability and Sensitivity Analyses Observability analysis of the linearized system We transform the linearized system Σl into the modal canonical form Σ̂l , using the command canon.m in the program Matlab, and get the following: Â = B̂ = −189 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 −2.4 0 0 0 0 0 −12 0 −1.3 0 0 1.8 0 −0.7 0 0 0.03 0 0 0 0 0 0 −0.007 −0.002 0 0 −2.2 −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 −4.2 2.6 0.29 0.59 0.37 −1.9 0.42 0.64 0.31 −0.045 0.64 −1.6 4.2 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 0.063 750 −200 −730 −0.14 −0.015 −0.035 −0.042 −0.079 −0.8 0.0013 −0.76 −0.059 Ĉ = −6.7e−4 −9 −8.2e −0.82 −5 −1.1e −0.049 −7 −1.4e 0.75 −5 2.4e 0.44 0.46 0.24 −0.12 0.25 −0.13 0.38 −0.066 −0.33 0.26 −5 3.4e −0.38 −4 −1.2e −0.28 −4 −2.2e −0.018 400 330 200 −0.0014 −0.0094 0.0045 0.0082 −4 −2e −0.016 −4 −3.9e Â has distinct eigenvalues and Ĉ has no zeros. We therefore conclude that the observability 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: µ ¶ 1 0 0 0 0 0 0 0 0 0 Y = X (4.7) 0 0 0 0 0 0 1 0 0 0 Since only the average moisture content and the top side temperature are measured at the end of the drying section, the following output is also of interest: Ã ! 0 0 0 0 0 0 1 0 0 0 Y = X (4.8) Gdry,2 Gdry,4 Gdry,6 0 0 0 0 0 0 0 Gdry,tot Gdry,tot Gdry,tot 0.0085 0.0025 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 submodel becomes not observable. The observability analysis for the free-draw sub-model is similar and leads to the same conclusion: the free-draw sub-model is observable if we measure at least two of the board variables. In appendix B, we concatenate a contact-zone sub-model and a free-draw submodel to further investigate the observability for one cylinder. The analysis shows that we need to measure at least the temperatures of both surfaces of the board in the contact zone and in the free draw to ensure observability. Conclusion The observability analysis shows that the contact-zone and free-draw sub-models are observable if we measure at least two of the following board variables: the board temperature at the top surface or the bottom surface and the average board moisture content in the thickness direction. Consequently, since the drying section is a concatenation of observable models, we consider that the model described in chapter 3 is observable under the assumption that we measure the output (4.7) for the contact zone and free draw of all cylinders. In appendix B, we show that these are 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 determined given an input–output trajectory. In this work, determine means that the parameters given a special input–output trajectory are unique. Ljung [31] distinguishes bewteen two kinds of identifiability: the structure identifiability, that considers the model structure, and the persistence of excitation, that examines if the inputs are informative enough to enable the identification of parameters. For the identification, described in chapter 5, we are dealing with static data and a machine with constant settings. We can therefore investigate the structural identifiability of the model, but not the persistence of the input. The identifiability analysis is carried out on the physical model, with the same simplifications as for the observability analysis. Since the primary goal of this identifiability analysis is to validate our choice of model structure, the study is carried out before identification (and the sensitivity analysis) has been tested. To simplify the computations, the parameters Cpdry , kdry and a1 are set to physical values from literature (see section 3.8). The set of parameters that we chose for the identifiability analysis is Θ = {hsc , hcp (0), F RF , Dwp }. We follow the approach suggested in [1, 56, 49] that views the identifiability problem as a special case of the observability problem where the constant parameters θ are considered as states with zero derivative. Since we want to estimate the 46 Observability, Identifiability and Sensitivity Analyses identifiability of the heat transfer coefficient between steam and cylinder hsc , we consider the cylinder temperature as a state instead of an input. The identifiability system is thus obtained by extending the observability system in previous section with the vector of new states Θ = {hsc , hcp (0), F RF , Dwp }. δ Ṫc δ Ṫp,1 δ Ṫp,2 δ Ṫp,3 δ Ṫp,4 δ Ṫp,5 δ Ṫp,6 δ Ṫp,7 δ u̇2 δ u̇4 δ u̇6 = Ax δTc δTp,1 δTp,2 δTp,3 δTp,4 δTp,5 δTp,6 δTp,7 δu2 δu4 δu6 + Aθ δhsc δhcp0 δhcpinc + B δF RF δDwp δTs δTpin,1 δTpin,2 δTpin,3 δTpin,4 δTpin,5 δTpin,6 δTpin,7 δTa δuin,2 δuin,4 δuin,6 where, Ax = Aθ = A01 A02 f2 (ũ2 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ā11 Ā12 0 Ā22 0 0 0 0 0 0 0 0 A1 0 0 0 0 0 0 0 0 0 Ā13 Ā23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ā84 0 0 Ā114 A4 A5 0 0 O5×5 0 0 0 0 0 0 0 0 A2 0 0 0 A3 0 Ā95 Ā105 Ā115 A09 (4.9) 4.2 Identifiability analysis of the physical model B= B01 0 0 0 0 0 0 B08 0 0 0 0 0 vx 0 I 0 0 0 0 4 L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B3 A1 , A2 , A3 , A4 , A5 , B2 , B3 are defined in the observability analysis O is the zero matrix ³ ´ A01 = A02 = A09 = Ā11 = Ā12 = Ā13 = Ā22 = 1 ρc Cpc dc 1 ρc Cpc dc 1 ρc Cpc dc −Lcp hcp −Lca hca Lcp +Lca Lcp hcp Lcp +Lca ³ Lcp hcpi nc T̃p,1 − Lcp +Lca ³ ∂ h̄sc 1 ρc Cpc dc ∂hsc T̃s³− T̃c Lcp 1 ρc Cpc dc Lcp +Lca ³T̃p,1 Lcp ũ 1 ρc Cpc dc Lcp +Lca ³T̃p,1 Lcp ũ 4 ρc Cpc dc Lcp +Lca T̃p,1 ³ ´ ´ T̃c ´ − T̃c − T̃c − T̃c Lcp ũ 4ũ2 ρc Cpc dc Lcp +Lca T̃p,1 − ∂f (Tp,7 ,u6 ,Ta ) Ā84 = 3 ∂F RF (u4 ,u2 ) Ā95 = ∂f5∂D wp (u2 ,u4 ) (u6 ,u4 ) Ā105 = ∂f5∂D + ∂f5∂D wp wp ∂f (T ,Ta ) Ā114 = 6∂Fp,7 RF (u4 ,u6 ) Ā115 = ∂f5∂D wp 1 B01 = ρc Cpc dc h̄sc Lca hca 1 B08 = ρc Cp c dc Lcp +Lca Ā23 = − h̃sc ´ ´ ´ T̃c We assume that we can measure the temperature of the cylinder and the output becomes: 1 0 Y = 0 0 0 1 0 0 0 0 0 0 0 0 0 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 Gdry,2 Gdry,tot Gdry,4 Gdry,tot Gdry,6 Gdry,tot X + (O4×5 ) Θ 48 Observability, Identifiability and Sensitivity Analyses To resume, the linearized extended system is µ δ Ẋ δ Θ̇ δY ¶ µ = ¶µ ¶ µ ¶ Aθ δX B + δU Oµ5×5 ¶ δΘ O5×12 δX O4×5 ) δΘ Ax O5×11 = (Cx The matrices are computed in Maple, similarly to the observability analysis. Checking the rank of the observability matrix Ωident , we note that it only has rank 15 (for 16 states), implying that one of the parameters is not identifiable. To find out which, we fix one parameter to a constant value assuming it is known and run an identifiability test. If the new system is identifiable, we have found the unidentifiable parameter. Removing only hsc , F RF or Dwp , we get a rank of observability of 14 (for 15 states), which means that these parameters are identifiable. But when removing hcp,0 or hcp,inc , the observability rank is 15, and the system is identifiable. This result means that the combination of hcp,0 and hcp,inc is not observable, which is reasonable since we consider only one position, where the moisture content u is constant. However, in the drying section, the moisture content is varying and the coefficients hcp,0 and hcp,inc could therefore be identifiable. As for the observability, we also consider other outputs. Measuring the temperature 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 only one board variable and the temperature of the cylinder, the sub-model is no longer identifiable. The identifiability results for the free-draw sub-model are identical. Conclusion The linearized system is identifiable (with Θ = {hsc , hcp (0), F RF , Dwp }) when measuring the cylinder and board surface temperatures. Therefore, since the whole system is a concatenation of identifiable systems, we assume that it is identifiable if the board temperature at both surfaces and the cylinder temperature are measured for the contact zone and free draw of every cylinder. In appendix B, we show that these are the minimal measurements required to ensure observability. If the model is not observable, this implies that it is not identifiable either. We therefore conclude that we need to measure at least the temperature of the cylinders and the temperatures of both surfaces of the board for each contact zone and each free 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 output, 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), on the semi-physical model described in chapter 3. To understand and estimate the effect of the parameters on the outputs, we first carry out a qualitative sensitivity analysis. We then apply the Dominant Parameter Selection (DPS) [22] to select the most 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 and maximal value, while the others are fixed to their nominal values and we observe the resulting outputs. Table 4.1 shows the respective values of each parameter. Some of the minimal and maximal values are exaggerated in order to visualize the effect of the parameter. The outputs are the temperature of the cylinder, the temperature of the paper at the bottom side (BS) and the top side (TS), and the total moisture content of the paper along the drying section. The slope of the moisture content curve represents the amount of evaporated water. Table 4.1: Minimal, nominal and maximal values of the parameters for the qualitative sensitivity analysis. parameter hsc hcp (0) hcp,inc Dwp F RF Cpdry kdry a1 uin Tp,in pa Ta min. value 500 100 200 10−9 10 400 0.02 3750 1.27 40 8000 50 nom. value 1900 450 955 2 · 10−8 40 1550 0.157 3816 1.38 50 18000 80 max. value 3000 1000 2000 0.001 70 5000 2 3870 1.94 65 28000 100 Heat transfer between steam and cylinder hsc Figure 4.3 displays the temperature of the cylinder, the temperature of the paper and the moisture content for the minimal and maximal values of hsc . We note that the parameter hsc affects all the variables considerably. A change in the heat transfer between the steam and cylinders creates a bias for the temperatures of the cylinder and the paper along the drying section. The moisture content is also affected by the parameter, an expected result since hsc represents how much heat 50 Observability, Identifiability and Sensitivity Analyses Temperature at the top side of the paper for different values of hsc 100 95 95 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] Temperature at the bottom side of the paper for different values of hsc 100 90 85 80 75 70 65 hsc=500 hsc=3000 60 55 0 10 20 30 Cylinders 40 50 90 85 80 75 70 65 hsc=500 60 hsc=3000l 55 50 60 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of hsc Temperature of the cylinders for different values of hsc 130 1.6 hsc=500 Moisture content [kgw/kgs] Temperature of the cylinders [°C] 110 100 hsc=500 hsc=3000 90 80 70 hsc=3000 1.4 120 1.2 1 0.8 0.6 0.4 0 10 20 30 Cylinders 40 50 (c) Temperature of the cylinders 60 0.2 0 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.1: Effect of the heat transfer coefficient hsc on the simulated outputs for the first 53 cylinders of the drying section. is used to dry the paper. If the coefficient increases, more heat is transferred to the cylinder and consequently to the paper (the temperatures increase), which results in 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 moisture content for the minimal and maximal values of hcp (0) and hcp,inc . These two parameters have approximately the same effect on the outputs, except at the beginning of the drying section where hcp,inc is more influent, since it is linearly dependent of the moisture content. An increase in hcp increases the heat transfer between the cylinder and paper, the temperature of the cylinder thus decreases, while the temperature of the paper increases, resulting in an increase of evaporation. 4.3 Sensitivity analysis of the semi-physical model Temperature at the bottom side of the paper for different values of hcp(0) 51 Temperature at the top side of the paper for different values of hcp(0) 105 110 100 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] 100 95 90 85 80 75 70 hcp(0)=100 65 60 90 80 70 hcp(0)=100 hcp(0)=1000 60 hcp(0)=1000 0 10 20 30 Cylinders 40 50 50 60 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of hcp(0) Temperature of the cylinders for different values of hcp(0) 130 1.6 hcp(0)=100 hcp(0)=1000 1.4 120 Moisture content [kgw/kgs] Temperature of the cylinders [°C] 125 115 110 hcp(0)=100 hcp(0)=1000 105 100 1.2 1 0.8 0.6 95 0.4 90 85 0 10 20 30 Cylinders 40 50 (c) Temperature of the cylinders 60 0.2 0 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.2: Effect of the heat transfer coefficient hcp (0) on the simulated outputs for 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 paper (see figure 4.4). When the diffusion coefficient decreases, the water moves slowly through the paper. Therefore, the water remains in the middle layer and the moisture at the surface layers decreases faster since the loss of water from the evaporation is stronger than the gain of water coming from the middle layer. The total 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 moisture 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 when the moisture content decreases (since it also depends on the heat conductivity, heat capacity, density and thickness of the paper) and the temperature of the paper also increases with a lower diffusion coefficient. 52 Observability, Identifiability and Sensitivity Analyses Temperature at the top side of the paper for different values of hcp,inc 100 100 95 95 90 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] Temperature at the bottom side of the paper for different values of hcp,inc 105 90 85 80 75 70 85 80 75 70 65 hcp,inc=200 65 hcp,inc=200 60 hcp,inc=2000 60 hcp,inc=2000 55 55 0 10 20 30 Cylinders 40 50 50 60 0 (a) Board temperature (BS) 30 Cylinders 40 50 60 Moisture content for different values of hcp,inc Temperature of the cylinders for different values of hcp,inc 1.6 120 1.4 hcp,inc=200 hcp,inc=2000 115 Moisture content [kgw/kgs] Temperature of the cylinders [°C] 20 (b) Board temperature (TS) 125 110 105 hcp,inc=200 hcp,inc=2000 100 95 1.2 1 0.8 0.6 0.4 90 85 10 0 10 20 30 Cylinders 40 50 (c) Temperature of the cylinders 60 0.2 0 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.3: Effect of the heat transfer coefficient hcp,inc on the simulated outputs for the first 53 cylinders of the drying section. Fabric reduction factor F RF Figure 4.5 shows the influence of F RF on the outputs. A small value of F RF represents a small decrease of the mass transfer coefficient due to the fabric. Consequently the evaporation increases, which results in a decrease of the board temperature, 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 the dry 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 the warming phase of the paper, the temperature of the paper is lower, while it gets higher at the end of the drying. Therefore, the evaporation of water is lower in 4.3 Sensitivity analysis of the semi-physical model Temperature at the top side of the paper for different values of Dwp 105 100 100 95 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] Temperature at the bottom side of the paper for different values of Dwp 95 90 85 80 75 70 65 60 0 10 20 30 Cylinders 90 85 80 75 70 65 Dwp=10−9 60 Dwp=0.001 55 40 50 50 60 Dwp=10−9 Dwp=0.001 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of Dwp Temperature of the cylinders for different values of Dwp 130 1.6 Dwp=10−9 125 Dwp=0.001 1.4 120 Moisture content [kgw/kgs] Temperature of the cylinders [°C] 53 115 110 105 Dwp=10−9 100 Dwp=0.001 1.2 1 0.8 0.6 95 0.4 90 85 0 10 20 30 Cylinders 40 50 (c) Temperature of the cylinders 60 0.2 0 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.4: Effect of the diffusion coefficient Dwp on the simulated outputs for the first 53 cylinders of the drying section. the warming phase and higher at the end of the drying. Since the derivative of the paper temperature is lower when the paper is in contact with the cylinder, more heat 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 board temperature in the free draw. When kdry increases, the heat transport inside the paper increases. Therefore, in the free draw, the increase of temperature due to the heat coming from the cylinder dominates the decrease of temperature due to the 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 important relatively to the heat coming from the cylinder. The board temperature is therefore colder, which results in a lower cylinder temperature. Since the board temperature 54 Observability, Identifiability and Sensitivity Analyses Temperature at the bottom side of the paper for different values of FRF Temperature at the top side of the paper for different values of FRF 100 100 95 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] 95 90 85 80 75 70 FRF=10 FRF=70 65 60 90 85 80 75 70 FRF=10 FRF=70 65 60 55 0 10 20 30 Cylinders 40 50 50 60 0 (a) Board temperature (BS) 10 40 50 60 Moisture content for different values of FRF Temperature of the cylinders for different values of FRF 1.6 120 1.4 FRF=10 FRF=70 115 110 Moisture content [kgw/kgs] Temperature of the cylinders [°C] 30 Cylinders (b) Board temperature (TS) 125 FRF=10 FRF=70 105 100 95 1.2 1 0.8 0.6 0.4 90 85 20 0 10 20 30 Cylinders 40 50 (c) Temperature of the cylinders 60 0.2 0 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.5: Effect of the Fabric Reduction Factor F RF on the simulated outputs for 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. An increase in a1 lowers the evaporation and its effect on the decrease of temperature of the paper in the free draw. Therefore, the temperature of the paper increases, as well as the cylinder temperature, but the evaporation decreases. The incoming moisture content uin The incoming moisture content mainly affects the moisture content along the drying section, but also significantly the temperature of the cylinder and slightly the temperature of the paper in the free draw (see figure 4.9). An increase of the mois- 4.3 Sensitivity analysis of the semi-physical model Temperature at the top side of the paper for different values of Cpdry 105 100 100 95 95 90 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] Temperature at the bottom side of the paper for different values of Cpdry 90 85 80 75 70 Cpdry=400 65 Cpdry=5000 60 55 55 85 80 75 70 Cpdry=400 65 Cpdry=5000 60 55 0 10 20 30 Cylinders 40 50 50 60 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of Cpdry Temperature of the cylinders for different values of Cpdry 125 1.6 120 1.4 115 110 Moisture content [kgw/kgs] Temperature of the cylinders [°C] Cpdry=400 Cpdry=400 Cpdry=5000 105 100 95 1.2 1 0.8 0.6 0.4 90 85 Cpdry=5000 0 10 20 30 Cylinders 40 50 60 0.2 0 (c) Temperature of the cylinders 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.6: Effect of the dry heat capacity Cpdry on the simulated outputs for the first 53 cylinders of the drying section. ture increases the heat of the cylinder given to the paper, which explains the decrease of the temperature of the cylinder, but the temperature of the paper remains almost the same since it depends of a combination between the heat from the cylinder, 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. The incoming board temperature affects the temperature of the board and the cylinders only for the first seven cylinders. For the rest of the drying section, the temperatures are almost equal. For a higher incoming board temperature, the initial evaporation is higher only for the first cylinders, which explains the bias in the moisture content for the rest of the drying section. 56 Observability, Identifiability and Sensitivity Analyses Temperature at the top side of the paper for different values of kdry 100 100 95 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] Temperature at the bottom side of the paper for different values of kdry 105 95 90 85 80 75 70 kdry=0.02 kdry=2 65 60 0 10 20 30 Cylinders 40 50 90 85 80 75 70 kdry=0.02 65 kdry=2 60 55 50 60 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of kdry Temperature of the cylinders for different values of kdry 125 1.6 120 1.4 115 110 Moisture content [kgw/kgs] Temperature of the cylinders [°C] kdry=0.02 kdry=0.02 kdry=2 105 100 95 1.2 1 0.8 0.6 0.4 90 85 kdry=2 0 10 20 30 Cylinders 40 50 60 (c) Temperature of the cylinders 0.2 0 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.7: Effect of the dry thermal conductivity kdry on the simulated outputs for 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 moisture content and temperature of the cylinders. If the surrounding air is cold, it cools down the cylinders. Therefore less heat is transferred to the paper, which results in 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 (figure 4.12). A low humidity in the air increases the drying capacity. The increase of evaporation results in a higher decrease of the board temperature. The cylinders temperature decreases since they release more heat to the paper. The board temperature in the free draw seems to reach an equilibrium point. 4.3 Sensitivity analysis of the semi-physical model Temperature at the top side of the paper for different values of a1 105 100 100 95 95 90 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] Temperature at the bottom side of the paper for different values of a1 90 85 80 75 70 65 a1=3750 60 55 10 20 30 Cylinders 40 50 85 80 75 70 65 a1=3750 60 a1=3870 55 a1=3870 0 50 60 0 (a) Board temperature (BS) 10 120 1.4 115 Moisture content [kgw/kgs] Temperature of the cylinders [°C] 1.6 a1=3750 a1=3870 100 95 40 50 60 a1=3750 a1=3870 1.2 1 0.8 0.6 0.4 90 85 30 Cylinders Moisture content for different values of a1 Temperature of the cylinders for different values of a1 105 20 (b) Board temperature (TS) 125 110 57 0 10 20 30 Cylinders 40 50 (c) Temperature of the cylinders 60 0.2 0 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.8: Effect of the dummy coefficient a1 on the simulated outputs for the first 53 cylinders of the drying section. Quantitative sensitivity analysis of the parameters In this section we want to select the most influent parameters for the identification described in chapter 5. Since only few measurements are available, we use the Dominant Parameter Selection (DPS) method suggested by Ioslovich et al. [22]. For this study, we assume that we have found an optimal point; the analysis is therefore carried out after some identification tests. The theory behind the DPS method 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 as follows: δYi S(i, j) = (4.10) δθj 58 Observability, Identifiability and Sensitivity Analyses Temperature at the bottom side of the paper for different values of uin Temperature at the top side of the paper for different values of uin 100 100 95 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] 95 90 85 80 75 70 uin=1.27 uin=1.94 65 60 0 10 20 30 Cylinders 40 50 90 85 80 75 70 uin=1.27 65 uin=1.94 60 55 50 60 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Temperature of the cylinders for different values of uin Moisture content for different values of uin 125 2 120 1.8 115 110 105 uin=1.27 uin=1.94 100 95 1.4 1.2 1 0.8 0.6 90 85 uin=1.94 1.6 Moisture content [kgw/kgs] Temperature of the cylinders [°C] uin=1.27 0.4 0 10 20 30 Cylinders 40 50 60 0.2 0 (c) Temperature of the cylinders 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.9: Effect of the incoming moisture content uin on the simulated outputs for the first 53 cylinders of the drying section. Since the parameters and outputs differ in magnitude, we use the relative sensitivity Sr (i, j), which is obtained by multiplying the sensivity by θj /Yi : Sr (i, j) = δYi θj δθj Yi (4.11) The relative sensitivity is convenient because it is dimensionless. With the relative sensitivity matrix, we compute the modified Fisher matrix Fm : Fm = SrT Λ−1 Sr (4.12) where Λ is a diagonal matrix with the square roots of the weights λ in the loss function, see equation (5.3). We notice that the Fisher matrix (which is obtained in the same way, using the sensitivity instead of the relative sensitivity) corresponds to the approximation of the Hessian of the loss function (5.3). The rank 4.3 Sensitivity analysis of the semi-physical model Temperature at the bottom side of the paper for different values of Tp,in 59 Temperature at the top side of the paper for different values of Tp,in 100 100 95 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] 90 90 85 80 75 70 Tp,in=40 65 Tp,in=65 80 70 Tp,in=40 Tp,in=65 60 50 60 55 0 10 20 30 Cylinders 40 50 40 0 60 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of Tp,in Temperature of the cylinders for different values of Tp,in 125 1.6 Tp,in=40 Tp,in=65 1.4 115 110 Moisture content [kgw/kgs] Temperature of the cylinders [°C] 120 Tp,in=40 Tp,in=65 105 100 95 1.2 1 0.8 0.6 90 0.4 85 80 0 10 20 30 Cylinders 40 50 0.2 0 60 (c) Temperature of the cylinders 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.10: Effect of incoming board temperature Tp,in on the simulated outputs for the first 53 cylinders of the drying section. of the matrix Fm indicates whether the matrix is singular. If the matrix is nonsingular, the parameters are theoretically identifiable. However, this matrix can be ill-conditioned, i.e. almost singular. In practice, this means that we can only estimate some parameters. To measure the importance of the parameters, we compute the relative sensitivity vector my obtained by computing the square root of the diagonal elements of Fm . my = p diag(Fm ) (4.13) A large value at the k th position indicates a big influence of the parameter θk for the 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 60 Observability, Identifiability and Sensitivity Analyses Temperature at the bottom side of the paper for different values of Ta Temperature at the top side of the paper for different values of Ta 100 100 95 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] 95 90 85 80 75 70 Ta=50 Ta=100 65 60 0 10 20 30 Cylinders 40 50 90 85 80 75 70 Ta=50 65 Ta=100 60 55 50 60 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of Ta Temperature of the cylinders for different values of Ta 125 1.6 120 1.4 115 Moisture content [kgw/kgs] Temperature of the cylinders [°C] Ta=50 110 105 Ta=50 Ta=100 100 95 1.2 1 0.8 0.6 0.4 90 85 Ta=100 0 10 20 30 Cylinders 40 50 60 0.2 0 (c) Temperature of the cylinders 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.11: Effect of the temperature of the air Ta on the simulated outputs for the first 53 cylinders of the drying section. the smallest eigenvalue. If the condition number is below the threshold, we consider the matrix well-conditioned. If this is not the case, we take only the first n parameters corresponding to eigenvalues with a ratio below the threshold (starting from the largest eigenvalue); we then check the condition number of the Fisher submatrix corresponding to the n most important parameters. If the submatrix is well-conditioned, we select the first n parameters for the identification. The DPS method [22] permits to select the most influent parameters, but also provides an examination of the correlation of those parameters, by analysing the normalized modified Fisher matrix Fm,n : T Fm,n = Sr,n Sr,n (4.14) where Sr,n is the normalized sensitivity matrix with normalized sensitivity vectors 4.3 Sensitivity analysis of the semi-physical model Temperature at the top side of the paper for different values of pa 105 100 100 95 95 90 Temperature of the paper (TS) [°C] Temperature of the paper (BS) [°C] Temperature at the bottom side of the paper for different values of pa 90 85 80 75 70 65 pa=8000 60 55 10 20 30 Cylinders 40 50 85 80 75 70 65 pa=8000 60 pa=28000 55 pa=28000 0 61 50 60 0 (a) Board temperature (BS) 10 20 30 Cylinders 40 50 60 (b) Board temperature (TS) Moisture content for different values of pa Temperature of the cylinders for different values of pa 125 1.6 120 1.4 115 110 Moisture content [kgw/kgs] Temperature of the cylinders [°C] pa=8000 pa=8000 pa=28000 105 100 95 1.2 1 0.8 0.6 0.4 90 85 pa=28000 0 10 20 30 Cylinders 40 50 0.2 60 0 (c) Temperature of the cylinders 10 20 30 Cylinders 40 50 60 (d) Board moisture content Figure 4.12: Effect of the partial pressure of water in the air pa on the simulated outputs for the first 53 cylinders of the drying section. j Sr,n of parameter j such that: Srj j Sr,n =q T Srj Srj (4.15) All the diagonal elements of Fm,n are equal to 1, and the elements Fm,n (i, j), i 6= i j j, correspond to the cosine between the sensitivity vectors Sr,n and Sr,n . If the sensitivity vectors are correlated, i.e. we can not identify both in the same time, the absolute value of their scalar product is close to 1. To determine if two parameters are correlated, we put a threshold α2 [-] on the off-diagonal elements of Fm,n . If the first n selected parameters are uncorrelated, they are identifiable. Ioslovich et al. [22] point out that the thresholds α1 and α2 are related. They consider that the condition number of the matrix Fm should be under 40 for Fm to be well- 62 Observability, Identifiability and Sensitivity Analyses 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 only on the nominal values of the parameters, but also on the inputs, initial conditions and structure of the system [39, 22, 31]. Therefore, we should be aware that the sensitivity results may differ if we choose to run the tests at different operational points, 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 do not include the unmeasured inputs of the model, since they change according to which set of measurements we use (only their relative sensitivity is displayed in table 4.3, to evaluate their importance). The nominal parameter vector for this analysis, displayed in table 4.2, is assumed to be close to the identification optimal point θ̂ (see chapter 5), thus some identification tests have been carried out before performing 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 used for the parameter selection. parameter hsc hcp (0) hcp,inc Dwp F RF Cpdry kdry a1 value 1800 1100 500 10−9 40 5000 1.5 3860 input uin Tp,in pa Ta value 1.38 48 15000 48 The minimal and maximal values are respectively set to -1 % and +1 % of the nominal value of table 4.2, so the relative sensitivity is computed as: Sr (i, j) = Yi (θmax ) − Yi (θmin ) θ̂ θmax − θmin Yi (θ̂) (4.16) where θmax = 1.01 × θ̂, θmin = 0.99 × θ̂ and the vector Y regroups the measured outputs, described in section 5.1. The previous equation can be rewritten as: Sr (i, j) = Yi (1.01 × θ̂) − Yi (0.99 × θ̂) 0.02 × Yi (θ̂) (4.17) 4.3 Sensitivity analysis of the semi-physical model 63 The matrix Sr is a N × r matrix, where N is the number of available measurements and r = 8 is the number of parameters. We compute the modified Fisher matrix Fm , in equation (4.12), with the weights λ given in section 5.2. The vector of relative sensitivities my is displayed in table 4.3. The unmeasured inputs, defined in section 3.8, are not considered for the parameter selection, but table 4.3 shows that the incoming board moisture content and temperature strongly affect the simulated outputs, since their relative sensitivity is high. The partial pressure of water in the air is considerable while the temperature of the surrounding air is neglectible (low relative sensitivity). Table 4.3: Relative sensitivity of the parameters and unmeasured inputs. High relative sensitivity represents a strong effect of the item on the model. The relative sensitivity of the inputs is displayed as an indication of their importance. parameter a1 hcp (0) hsc Dwp Cpdry hcp,inc F RF kdry rel. sensitivity 10.6 0.39 0.32 0.18 0.14 0.12 0.11 0.08 input uin Tp,in pa Ta rel. sensitivity 5.48 0.56 0.25 0.0272 The resulting matrix Fm is found to be ill-conditioned and we do not display the eigenvalues. We decide to remove the two less influent parameters (kdry and F RF ) and redo the analysis. The eigenvalues of the corresponding modified Fisher submatrix are reported in table 4.4. Table 4.4: Eigenvalues of the modified Fisher submatrix of the 6 more dominant parameters. parameter a1 hcp (0) hsc Dwp Cpdry hcp,inc eigenvalue 112 0.15 0.10 0.034 0.019 0.015 Table 4.4 shows that the parameter a1 is very dominant, since its eigenvalue is large compared to the eigenvalues of the other parameters. The ratio of the 64 Observability, Identifiability and Sensitivity Analyses eigenvalues λa1 /λhcp (0) ' 750 is already above the threshold, which implies that we can only identify the parameter a1 . But if we consider the eigenvalues of the other 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 identifiable. Therefore, we assume that the identification has found the optimal a1 , we fix it to its optimal value and analyse the sensitivity of the remaining parameters. The eigenvalues of the corresponding modified Fisher submatrix are reported in table 4.5. Table 4.5: Eigenvalues of the modified Fisher submatrix, after identification of a1 . parameter hcp (0) hsc Dwp Cpdry hcp,inc eigenvalue 0.23 0.045 0.032 0.013 0.0011 From table 4.5, only 4 parameters (hcp (0), hsc , Dwp and Cpdry ) are selected since λhcp (0) /λCpdry ' 18 < 40 but λhcp (0) /λhcp,inc ' 200 > 40. Before concluding about the identifiability of the selected parameters, we need to verify that they are not correlated. The normalized modified submatrix Fm,n corresponding to those parameters is: 1 0.70 −0.57 −0.36 0.70 1 −0.65 −0.06 Fm,n = (4.18) −0.57 −0.65 1 −0.07 −0.36 −0.06 −0.07 1 The absolute values of the off-diagonal elements of the matrix (4.18) are well under the threshold of 0.95, thus the parameters hcp (0), hsc , Dwp and Cpdry are assumed to be uncorrelated. Conclusion The sensitivity analysis provides a practical parameter selection for the identification. It shows that the dummy parameter a1 is strongly influent, relatively to the other parameters, and therefore suggests to first identify it to its optimal value with respect to the measurements and afterwards identify the other influent parameters — hcp (0), hsc , Dwp and Cpdry — since they are uncorrelated. The parameters — kdry , F RF and hcp,inc — are considered not identifiable and should be fixed manually to a constant. The unmeasured inputs are not included in the identification since they are time-varying, but we show that they affect the outputs of the model considerably. 4.4 Summary 4.4 65 Summary The observability and identifiability analyses show that the minimal measurements required to ensure observability and identifiability are the cylinder temperature and the board temperatures Tp,1 and Tp,7 for the contact zone and the free draw of each cylinder. Based on those results, special measurements were ordered to get the necessary data for the identification of the unknown parameters, described in the next chapter. Since only the final board moisture content and top surface temperature are measured on-line, the model is not observable under on-line conditions. We can therefore not guarantee that the model estimates the true temperatures and moistures inside the board in the drying section. However, since the model is considered observable and identifiable during the parameter identification, we can view it as an indication of those variables. The sensitivity analysis gives an understanding of the effect of the unknown parameters on the outputs and an estimation of which parameters can be identified in practice. It also suggests that the unmeasured inputs are influent for the model. Chapter 5 Identification of the Parameters The goodness of a model is evaluated by comparing the simulated outputs with measurements. The parameter identification consists in estimating the optimal values of the parameters giving the best fit to reality, represented by the measurements. In chapter 4, we show that the model of the drying section is not observable under on-line conditions and present the minimal measurements required to enable the identification of the parameters. Those measurements would be too expensive to perform under dynamic conditions and are therefore carried out under static conditions on four special occasions. We first select one measurement set to identify the parameters, and validate the results with the three other measurement sets. After a description of the static measurements, the process of identification is introduced. We then describe the selection of the parameters to identify and the optimization routine. In the last section, the identification and validation results are presented. 5.1 Description of the static measurements With only the final moisture content and the temperature of the board measured on-line in the drying section, the model described in chapter 3 is neither observable nor identifiable. The presence of on-line sensors is, however, difficult to achieve for two main reasons: firstly, the drying hood is very hot and secondly, the board is not easy to reach because of the lack of space. The observability and identifiability analyses in chapter 4 conclude that the model is observable and identifiable if we measure at least two variables per contact zone and per free draw. Static measurements were therefore carried out to measure the temperatures and the moisture content at different positions in the drying section, to enable the identification of the unknown parameters. Those measurements were performed on four occasions and 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. Since the model does not consider the cross direction of the machine, all the measur67 68 Identification of the Parameters ing positions are approximately 50 cm from the front side of the machine towards the middle of the sheet. The rest of this section further describes the measured variables. Properties of the surrounding air The properties of the air in the ventilation pockets are measured with an hygrometer. The humidity of the air and the partial pressure of water in the air are calculated from the wet and dry measured temperatures. Figures 5.1 and 5.2 show the measured temperature and partial pressure of water for the four sets of static measurements. Measured Temperature of the air for different sets of measurements 110 16/03/04 28/09/04 05/04/05 14/06/05 100 Temperature [C] 90 80 70 60 50 Groups 1−5 40 0 20 Stack 40 60 Groups 6−7 80 100 Cylinders Figure 5.1: Measured temperature [◦ C] of the surrounding air for different sets of measurements. The temperature between cylinders 53 and cylinder 70 (stack dryers) is not considered reliable since it is difficult to measure. The temperatures are 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 evaporation occurs, the properties of the air (temperature and partial pressure of water) 5.1 Description of the static measurements 69 Partial pressure of water [kPa] are lower in the beginning of the drying section and increase with the properties of the board. In the stack dryers, the properties of the air are difficult to measure and less reliable. We therefore assume that the temperature of the air is constant and that the air is completely dry (as an effect of the vacuum rolls). In the last two groups, 6 and 7, the board is almost dry, and the properties of the air are considered as constant. The fitted values of equations (3.50) and (3.51) for the three parts of 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 unreasonably too high, we replace it by the values of the measurement set 28/09/04. Measured Partial pressure of water in the air for different sets of measurements 70 16/03/04 28/09/04 60 05/04/05 14/06/05 50 40 30 20 10 Groups 1−5 0 0 20 Stack 40 60 Groups 6−7 80 100 Cylinders Figure 5.2: Measured partial pressure of water [kPa] of the surrounding air for different 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 water in 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 with the air (see figure 5.3), using a contact thermometer. The standard deviation of the measurement error is roughly around 1 ◦ C. 70 Identification of the Parameters Figure 5.3: Position of the measured (Y ) and computed (Ŷ ) variables for the cylinder k, where Tc is the temperature of the cylinder, Tp the board temperature and u the average moisture content of the board. p1 and p2 are the positions after the contact 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 thermometer on two positions per cylinder, at the end of the contact zone (position p1 ) and at the end of the following free draw (position p2 ), see figure 5.3. It is not possible to measure exactly at the point where the paper leaves or reaches the cylinder. Consequently, the measurement points are situated approximately 5 cm from the contact between the cylinder and the paper. Since a decrease of approximately 10 ◦ C is measured in the free draw, and the simulation points are situated before (position p1 ) and after (position p2 ) the measurements, the simulation point in position p1 /p2 should be approximately 1 ◦ C more/less than the measurement point. Adding the instrument uncertainty (around 1 ◦ C) to the uncertainty due to the position, the uncertainty in the measurements of the temperature is approximately 2 ◦ C. 5.1 Description of the static measurements 71 The measurements show a strange behaviour for the temperature of the surface of the board that was not in contact with the cylinder (in figure 5.3, this corresponds to the board top layer from position p1 to position p2 ). According to the measurements, the temperature increases in the free draw, while it should decrease due to the evaporation. This behaviour is not well understood and has to be further investigated. This could be due to some reflection from the cylinders at position p2 . We decide nevertheless to omit the points of the surface layer that was not in contact with the previous cylinders at position p2 since we believe that those measures are biased. Average moisture content inside the paper To validate the computation of the moisture content for each layer, the ideal situation would be to use sensors that are able to measure the distribution of the moisture in the thickness direction. Such instruments are starting to appear, but they require cutting a sample of the edge of paper during the drying. This is not desired, since there is a big risk to provoke a web break. The sample would moreover represent the moisture at the edges of the paper, while we are interested in modelling the middle of the sheet (in the CD direction). The average moisture in the thickness direction is instead measured with a γ-radiation device. Since the instrument is voluminous, there are only few spots where it can safely reach the paper during the drying. The moisture content is consequently measured only in the free draw (see figure 5.3) between the different steam groups. Table 5.1: Fitted parameter values for the properties of the surrounding air, in equations (3.50) and (3.51). The values of the data set 05/04/05 are modified to get more realistic values. date Ta,ave [◦ C] groups 1–5 αT [-] pa,ave [Pa] αp [-] Ta,ave [◦ C] stacks αT [-] pa,ave [Pa] αp [-] Ta,ave [◦ C] groups 6–7 αT [-] pa,ave [Pa] αp [-] 16/03/04 60 0.3 10000 0.9 80 1 0 1 90 1 10000 1 28/09/04 60 0.3 15000 0.9 85 1 0 1 95 1 18000 1 05/04/05 60 0.2 15000 0.9 80 1 0 1 85 1 18000 1 14/06/05 60 0.2 15000 0.9 80 1 0 1 90 1 15000 1 72 5.2 Identification of the Parameters Identification procedure To estimate the parameters, we use the same approach as Pettersson [36] and Bortolin [9]: the predictor error method (see [31]). The model can be written as an output 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 the nonlinear function describing the model, θ is the vector of constant parameters to identify, and e(t) is the error that we assume to be white noise. The outputs available from the measurements are the following: the temperature 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 ² as the difference between the measurements Y and the predicted output Ŷ . ²T c (k) ²T p,1 (k) ²T p,7 (k) ²u (k) = = = = Tc (k) − T̂c (k, θ), Tp,1 (k) − T̂p,1 (k, θ), Tp,7 (k) − T̂p,7 (k, θ), u(k) − û(k, θ), k k k k = 1 . . . NT c = 1 . . . NT p,1 = 1 . . . NT p,7 = 1 . . . Nu (5.2) where NT c , NT p,1 , NT p,7 and Nu are the numbers of available measurements for each output. Since we are using static measurements, the index k represents the position of the measure instead of the time index. Similarly to Pettersson [36] and Bortolin [9], we define the loss function VN : ³P PNT p,1 ²T p,1 (i)2 PNT p,7 ²T p,7 (i)2 PNu ²u (i)2 ´ NT c ²T c (i)2 + VN (θ, Z N ) = N1 i=1 i=1 i=1 i=1 λu λT c λT p,1 + λT p,7 + (5.3) where N = NT c + NT p,1 + NT p,7 + Nu is the total number of measurements, Z N is the input–output data and the weights λ are used to normalise the squared prediction errors: λT c = maxi |²T c (i)| λT p,1 = maxi |²T p,1 (i)| (5.4) λT p,7 = maxi |²T p,7 (i)| λu = 0.02 The weight λu is set to a low value to give more importance to the moisture content in the loss function, to conpensate for the few measures available. The process of identification of the parameters consists in finding the set of optimal parameters θ̂ that minimizes the loss function. θ̂ = arg min VN (θ, Z N ) θ (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 outliers. However in this work, the possible outliers are removed manually, since we are dealing with a small amount of data. 5.3 Parameter selection Solving the optimization problem (5.5) requires a large number of iterations. The simulation time of the model of the whole drying section, after a change of parameters, is considerable because of the high complexity of the model. The drying section has therefore been divided into two parts for the identification of the parameters. The first part contains the first 53 cylinders (groups 1–5) and the second part regroups the stack dryers and the groups 6 and 7. This division is also justified by the fact that we believe that the board has a different behaviour for high or low moisture content. The properties in the stack dryers are also assumed to be different because of the presence of vacuum rolls, but since the measurements in the stack dryers are poor and not very reliable, we decide to fix the parameters in the 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 are estimated 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 F RF and kdry can be set to a constant. The sensitivity analysis also suggests to first identify a1 and afterwards {hsc , hcp (0), Dwp , Cpdry }, setting hcp,inc to a constant. For the identification of the second part of the drying section, we fix more parameters to a constant value, since there are fewer measurements available. The parameters a1 , kdry , and Cpdry are fixed to their physical values. We also assume that the heat transfer coefficient between the cylinder and the paper does not depend on the moisture content (hcp,inc = 0). The parameters that we choose to identify are therefore hsc , hcp (0) and Dwp . Table 5.2 summarizes which parameters 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 because it provides the most available measurements and the resulting outputs are displayed in figure 5.4. To validate the model, the model is simulated for three other sets of measurements, with the estimated parameters during the identification. The validation results are displayed in figures 5.5, 5.6 and 5.7. The simulated outputs for the identification are acceptable, except in the stack dryers where the simulated temperature of the cylinders is 15 ◦ C higher than the measurements. 74 Identification of the Parameters Table 5.2: Estimated values of the parameters during the identification, for the three parts of the drying section. The values in parentheses are fixed manually. parameter hsc hcp (0) hcp,inc Dwp F RF Cpdry a1 kdry groups 1–5 1500 1000 (500) 1 · 10−9 (40) 6000 3860 2 stacks (1900) (742) (0) (2 · 10−9 ) (40) (1550) (3816) (0.157) groups 6–7 300 400 (0) 2 · 10−9 (40) (1550) (3816) (0.157) The final simulated board temperature and moisture content are in accordance with the measurements. To quantitatively evaluate the results, the standard deviations of the residuals are compared with the standard deviations of the measurement errors, see table 5.3. The standard deviations of the residuals are close to the standard deviation of the measurements for the first part of the drying section, but generally higher for the second part. The validation results are less good than the identification. For both data sets 28/09/04 and 16/03/04, the simulated temperatures of the cylinders are too high and the temperature of the board is too low. The simulated evaporation is slightly higher than the measured output for the 16/03/04 set, probably because the air is drier. For the data set 14/06/05, the temperature of the cylinders is reasonable but 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 difficult because of the lack of data. The measurements are poor and uncertain (for example the board temperature is showing a strange behaviour), and important inputs, such as the incoming moisture content, are not measured. Moreover, the identification is carried out under static conditions and some input–output relations may be misrepresented. The resulting model can be considered as satisfactory for the identification set but not for the validation sets. Besides, some identified values of the parameters are outside the typical boundaries found in literature (see section 3.8). To improve the model, more measurements on a regular basis are needed, to be able to judge which measures are relevant, or distinguish which features are not well modelled. The final moisture content seems, however, to be in accordance with the measurements. We therefore assume that the parameter 5.5 Conclusion 75 identification gives satisfactory results, considering the limitations, and we further investigate the behaviour of the model under on-line conditions in the next chapter. Identification set 05/04/05: Temperature of the board (TS) in the drying section 120 110 110 Temperature of the paper in °C Temperature of the paper in °C Identification set 05/04/05: Temperature of the board (BS) in the drying section 120 100 90 80 70 simulation measurements 60 50 100 90 80 70 simulation measurements 60 0 10 20 30 40 50 Cylinders 60 70 80 90 50 100 0 10 (a) Board temperature (BS) 20 30 40 50 Cylinders 60 70 80 90 100 (b) Board temperature (TS) Identification set 05/04/05: Temperature of the cylinders in the drying section Identification set 05/04/05: Moisture content in the drying section 130 1.4 simulation measurements 1.2 Moisture content in kgw/kgdry Temperature of the cylinders in °C 120 110 100 simulation measurements 90 80 70 1 0.8 0.6 0.4 0.2 0 10 20 30 40 50 Cylinders 60 70 80 (c) Temperature of the cylinders 90 100 0 0 10 20 30 40 50 Cylinders 60 70 80 90 100 (d) Board moisture content Figure 5.4: Measured and simulated outputs in the drying section for the identification set 05/04/05. 76 Identification of the Parameters Validation set 16/03/04: Temperature of the board (TS) in the drying section 130 120 120 110 110 Temperature of the paper in °C Temperature of the paper in °C Validation set 16/03/04: Temperature of the board (BS) in the drying section 130 100 90 80 70 simulation measurements 90 80 70 simulation measurements 60 60 50 100 50 0 10 20 30 40 50 Cylinders 60 70 80 90 40 100 0 10 (a) Board temperature (BS) 20 30 40 50 Cylinders 60 70 80 90 100 (b) Board temperature (TS) Validation set 16/03/04: Temperature of the cylinders in the drying section Validation set 16/03/04: Moisture content in the drying section 140 1.4 130 1.2 Moisture content in kgw/kgdry Temperature of the cylinders in °C simulation measurements 120 110 100 simulation measurements 90 80 70 1 0.8 0.6 0.4 0.2 0 10 20 30 40 50 Cylinders 60 70 80 (c) Temperature of the cylinders 90 100 0 0 10 20 30 40 50 Cylinders 60 70 80 90 100 (d) Board moisture content Figure 5.5: Measured and simulated outputs in the drying section for the validation set 16/03/04. 5.5 Conclusion 77 Validation set 28/09/04: Temperature of the board (TS) in the drying section 120 120 110 110 Temperature of the paper in °C Temperature of the paper in °C Validation set 28/09/04: Temperature of the board (BS) in the drying section 130 100 90 80 simulation measurements 70 90 80 70 simulation measurements 60 60 50 100 0 10 20 30 40 50 Cylinders 60 70 80 90 50 100 0 10 (a) Board temperature (BS) 20 30 40 50 Cylinders 60 70 80 90 100 (b) Board temperature (TS) Validation set 28/09/04: Temperature of the cylinders in the drying section Validation set 28/09/04: Moisture content in the drying section 140 1.4 130 1.2 Moisture content in kgw/kgdry Temperature of the cylinders in °C simulation measurements 120 110 100 simulation measurements 90 80 70 1 0.8 0.6 0.4 0.2 0 10 20 30 40 50 Cylinders 60 70 80 (c) Temperature of the cylinders 90 100 0 0 10 20 30 40 50 Cylinders 60 70 80 90 100 (d) Board moisture content Figure 5.6: Measured and simulated outputs in the drying section for the validation set 28/09/04:. 78 Identification of the Parameters Validation set 14/06/05: Temperature of the board (TS) in the drying section 120 110 110 Temperature of the paper in °C Temperature of the paper in °C Validation set 14/06/05: Temperature of the board (BS) in the drying section 120 100 90 80 70 simulation measurements 60 50 100 90 80 70 60 simulation measurements 50 0 10 20 30 40 50 Cylinders 60 70 80 90 40 100 0 10 (a) Board temperature (BS) 30 40 50 Cylinders 60 70 80 90 100 90 100 (b) Board temperature (TS) Validation set 14/06/05: Temperature of the cylinders in the drying section Validation set 14/06/05: Moisture content in the drying section 125 1.6 120 1.4 simulation measurements 115 Moisture content in kgw/kgdry Temperature of the cylinders in °C 20 110 105 100 simulation measurements 95 90 1.2 1 0.8 0.6 0.4 85 0.2 80 75 0 10 20 30 40 50 Cylinders 60 70 80 90 (c) Temperature of the cylinders 100 0 0 10 20 30 40 50 Cylinders 60 70 80 (d) Board moisture content Figure 5.7: Measured and simulated outputs in the drying section for the validation set 14/06/05. Table 5.3: Standard deviations of the measurements error and the residuals for the identification and validation sets. Part 1 contains the first 5 groups and part 2 the stacks dryers together with groups 6 and 7. The standard deviation of the measurements error is assumed to be larger in the stack dryers. meas. Tcyl [◦ C] Tp,1 [◦ C] Tp,7 [◦ C] u [kg/kg] ≈1 ≈2 ≈2 ≈ 0.04 ident. 05/04/05 part 1 part 2 1.83 3.20 3.19 2.64 2.92 3.29 0.035 0.022 valid. 16/03/04 part 1 part 2 3.04 3.36 4.12 3.77 5.07 1.73 0.073 0.025 valid. 28/09/04 part 1 part 2 1.89 2.26 3.85 4.53 3.63 2.80 - valid. 14/06/05 part 1 part 2 2.73 3.75 5.83 3.74 4.42 3.15 0.097 0.075 Chapter 6 Dynamic Simulations In the previous chapter, the model is validated under static conditions, i.e. the inputs of the machine are constant. In this chapter, we investigate the dynamics of the model, since we want to simulate the on-line conditions. We consider only the available on-line measurements: the board properties at the end of the drying section. Those measurements are not sufficient to ensure observability for the whole model (see chapter 4), but the simulations can be used to give an approximation of the board temperature and moisture content for different positions in the drying section. We first examine the simulations of the deterministic model, then include disturbances 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 machines are varying. We consider only the two available measurements at the end of the drying section: the board temperature at the top side and the average board moisture content in the thickness direction. In these conditions, we do not know the following model inputs described in section 3.8: the board moisture content and temperature after the press and the air properties. The board moisture content is computed with the press model [11] and the board temperature and air properties are considered constant. We use a smoothing filter to reduce high-frequency disturbances that could introduce numerical problems. Since we are dealing with a small amount of data, the outliers are removed manually. The measurement frame situated at the end of the drying section is not always active; we thus need to find a good data set for the simulations. We consider a period between 18/08/05 and 29/08/05, where a reasonable amount of measurements is available. In the measurements set, lack of data is detected for some time periods. Either some measured inputs or outputs are missing or the bending stiffness predictor [36] computations fail. The missing data are replaced by a linear interpolation between the available data. 79 80 Dynamic Simulations The simulation results are displayed in figures 6.1 and 6.2. The model is first simulated with the parameters identified in chapter 5, displayed in table 5.2. The model is not satisfactory since it is obviously missing some dynamics and showing some large biases. Since some identified parameters in chapter 5 are outside the range of the physical values found in the literature (see section 3.8), we also simulate the model with more physical values of the parameters. The physical values, chosen after some trial and error with the deterministic model, are displayed in table 6.1. The simulation results are different from the model with identified values, but we can not really determine which model is the best. To compare the performance of the two models, the mean and the standard deviation of the error vectors are displayed in table 6.2, p. 87. The error vector is defined as the difference between the predicted (ŷ) and the measured (y) output: ²u = ûout − uout ²Tp = T̂p7,out − Tp7,out (6.1) where uout and Tp7,out are, respectively, the average moisture content and the top side temperature of the board, at the end of the drying section. The standard deviations of the error vectors are similar for the two models, but the model with identified parameters have a larger error mean for the temperature (this corresponds to a larger bias). For the moment, we can not conclude which model alternative is the best. In the next section, we attempt to improve the model by including disturbances to compensate for unmodelled features. Table 6.1: Physical values of the parameters for the simulation of the deterministic model described in chapter 3. parameter hsc hcp (0) hcp,inc Dwp F RF Cp,dry a1 kdry 6.2 groups 1–5 1700 200 850 5 · 10−9 40 1550 3816 0.157 stacks 1700 200 850 5 · 10−9 40 1550 3816 0.157 groups 6–7 500 200 850 5 · 10−9 40 1550 3816 0.157 Grey-box modelling of the disturbances In the previous section, the only uncertainty considered is the process noise, which is 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 Moisture content at the end of the drying section 0.2 measured computed using identified parameters computed using physical parameters 0.18 Moisture content in kgw/kgdry 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 50 100 150 Time in hours 200 250 Figure 6.1: Dynamic simulation of the average moisture content at the end of the drying section of the deterministic model. The short time periods between vertical lines represent the data replaced by a linear interpolation. Unmeasured inputs: We showed in section 4.3 that the properties of the incoming 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 dynamics of the steam and air systems are not considered. The presence of air and the distinction between the liquid or vapor phases of water inside the board are neglected. We also believe that the pulp properties influence the drying parameters, (such as Cpdry , kdry , Dwp , see section 3.8), but we did not include them in the model, due to identifiability problems. Parameter uncertainty: The model with the parameters identified in chapter 5 shows a good agreement for the identification set, but a poor agreement for the static validations sets and the dynamic simulations (in section 6.1). Since the identification is carried out for static conditions, the inputs are apparently not informative enough. Indeed, the identified values of the parameters 82 Dynamic Simulations Top side board temperature at the end of the drying section 120 measured computed using identified parameters computed using physical parameters Temperature of the paper in °C 115 110 105 100 95 90 85 80 0 50 100 150 Time in hours 200 250 Figure 6.2: Dynamic simulation of the top side board temperature at the end of the drying section of the deterministic model. The short time periods between vertical lines represent the data replaced by a linear interpolation. are outside the range of typical values found in literature and dynamic simulation with more physical values achieves a similar performance. Input uncertainty: The inputs in the model are given by measurements and can be subject to measurements errors. In this section, we apply a grey-box modelling method, the nonlinear Kalman filter to use the available measurements to estimate the uncertainties that the deterministic model can not catch. After a short description of the Extended Kalman filter, 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 local linearization about the current mean and covariance of the random variable. The EKF may not perform well in some applications where the linearization is too crude. Variations of the EKF, such as the unscented Kalman filter [25], preserve the 6.2 Grey-box modelling of the disturbances 83 normal distributions of the random variables under the nonlinear transformation, but require several function evaluations for each time step. In our case, a function evaluation represents a simulation of the model for the whole drying section, which is very time consuming. We therefore choose to test the Extended Kalman filter since, for each time step, it requires only one extra simulation per disturbance variable. Extended Kalman Filter We consider the following system: ½ Σnl xk+1 yk+1 = = f (xk , uk , wk ) 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 f and l are nonlinear functions. The noises wk and vk are supposed to be mutually independent and have normal probability distributions with respective covariance matrices Q and R. In practice, we do not know the process noise wk and the measurement noise vk at time k. So we compute an a priori estimate of xk+1 (denoted x̂− k+1 ), using an a posteriori estimate of xk (denoted x̂k ), computed at previous time k: ½ x̂− k+1 − ŷk+1 = = f (x̂k , uk , 0) l(x̂k , 0) (6.3) By introducing the following quantities, Ak[i,j] = Wk,[i,j] = Lk,[i,j] = Vk,[i,j] = ∂f[i] ∂x[j] (x̂k , uk , 0) ∂f[i] ∂w[j] (x̂k , uk , 0) ∂l[i] ∂x[j] (x̂k , 0) ∂l[i] ∂v[j] (x̂k , 0) Jacobian of f with respect to x Jacobian of f with respect to w Jacobian of l with respect to x (6.4) Jacobian of l with respect to v we can estimate the actual state xk and output yk by xk yk ≈ x̂− k + Ak ¡(xk−1 − x̂ ¢k−1 ) + Wk wk−1 ≈ ŷk− + Lk xk − x̂− k + Vk v k (6.5) The Extented Kalman Filter equations are given by: Time update equations: ½ x̂− k Π− k = f (x̂k−1 , uk−1 , 0) = Ak Πk−1 ATk + Wk Qk−1 WkT (6.6) 84 Dynamic Simulations Measurement update equations: ¡ ¢ − T T T −1 Kk = Π − k Πk Lk + Vk R k Vk k Lk L¡ ¢ − x̂ = x̂− k + Kk yk − l(x̂k , 0) k − Πk = (I − Kk Lk ) Πk (6.7) Modelling the disturbances To choose which disturbances to include in the model is a trial-and-error procedure. One possibility is to add noise to all uncertain variables in the model, but this includes redundancies and is therefore inefficient. Since we need to compute the derivative of the function evaluation with respect to each Kalman state, the computation time increases with the number of disturbance variables. We therefore include as few uncertainties as possible. In addition, the derivative is calculated using the backwards differentiation method that requires only one extra function evaluation per derivative. ¢ ∂f[i] 1 ¡ = f[i] (x[j] ) − f[i] (x[j] − δx[j] ) ∂x[j] δx[j] (6.8) The qualitative sensitivity analysis shows that the incoming board moisture content and the partial pressure of water in the air affect the computed variables considerably, (see figures 4.9 and 4.12). The quantitative sensitivity analysis suggests that the incoming board temperature has also a strong effect on the variables in the drying section, but we do not consider it since it affects mostly the warming-up zone of the board (first cylinders) and not the board properties at the end of the drying section. Moreover, the partial pressure of water in the air represents the drying potential since the evaporation is driven by the difference of partial pressure between the air and the paper. Consequently, we choose to add disturbances to the incoming moisture content and partial pressure of water in the air. We model the disturbances in the same manner as Pettersson [36] and Bortolin [10], by multiplying the uncertain variables with an exponential factor. This approach is convenient since the multiplying factor is always positive which prevents the allocation of unphysical values to the following disturbed variables (denoted with a superscript d ): udin (k) = pda (k) = uin (k)e−x1 (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 mean and covariance rξ i . This is a typical way to model slow varying process. The covariance rξ i describes how fast the disturbance xi changes. 6.2 Grey-box modelling of the disturbances 85 We transform the model into the standard Kalman filter form (6.2) by including the measured output and the disturbances in the Kalman state X: ûout (k − 1) T̂p7,out (k − 1)/100 X(k) = (6.11) x1 (k) x2 (k) where ûout (k) = l1 (Xk , Uk , 0) and T̂p7,out = l2 (Xk , Uk , 0) are respectively the computed average moisture content and the computed top side board temperature at the end of the drying section at time k (given by the simulation of the model described in chapter 3). The temperature is divided by 100, to normalize the error vector. Note that U (k) is the input vector and not the moisture content. The governing equations for the overall model can then be written in the standard Kalman filter form (6.2): l1 (X(k), U (k), 0) 0 l2 (X(k), U (k), 0) + 0 X(k + 1) = x1 (k) ξ1 (k) (6.12) x2 (k) · ξ2 (k) ¸ £ ¤ v1 (k) I 0 X(k) = + Y (k) v2 (k) With this transformation, the Jacobian matrices are easy to compute: ¸ ¸ · · O2×2 O2×2 ∂l(k) ∂x W = A(k) = I2×2 £ O2×2 I2×2 ¤ I2×2 O2×2 L = V = I2×2 where O is the null matrix and I the identity matrix. Only the first matrix A(k) is dependent 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: · ¸ O2×2 O2×2 Q= R=R O2×2 Q where Q = diag(rξ 1 , rξ 2 ) is the covariance matrix of the states x1 and x2 and R is the covariance matrix of the measurements uout , Tp7,out . Simulations results The stochastic model is simulated with the same data set used for the deterministic model, with both the identified values and physical values of the parameters. In order to apply the Kalman filter algorithm, we need to set initial values to the estimated Kalman state x̂(0) and the covariance matrix Π(0). x̂(0) = 0, Π(0) = 0.02 · I2×2 (6.13) 86 Dynamic Simulations The initial conditions do not matter in our case, since the model is first simulated with constant inputs. We also need to estimate the process and measurements covariance matrices Q and R. Finding the optimal values is a trial-and-error procedure, changing the values of Q and R determines if we give more weight to the process or the measurements. For example, a larger value of Q increases the Kalman gain K and allows larger changes in the process disturbances; the algorithm therefore tracks the measurements more. After some tests, the following values are chosen. ¸ · 0.002 0 Q = 0.02 · I2×2 , R = (6.14) 0 0.03 The simulation results of the stochastic model are depicted in figures 6.3 and 6.4. The Kalman filter improves the model considerably for both simulations with identified and physical parameters values. The identified model performs less well than the physical model for some times periods. This discrepancy is not well understood but could be due to numerical problems. Moisture content at the end of the drying section 0.16 measured with KF and identified parameters with KF and physical parameters Moisture content in kgw/kgdry 0.14 0.12 0.1 0.08 0.06 0.04 0 50 100 150 Time in hours 200 250 Figure 6.3: Dynamic simulation of the average moisture content at the end of the drying section, of the stochastic model. The short time periods between vertical lines represent the data replaced by a linear interpolation. The mean and standard deviations of the prediction error, displayed in table 6.2, show a great improvement for both models with the Kalman filter compared to the deterministic models. 6.2 Grey-box modelling of the disturbances 87 Top side board temperature at the end of the drying section 115 measured with KF and identified parameters with KF and physical parameters Temperature of the paper in °C 110 105 100 95 90 85 80 75 0 50 100 150 Time in hours 200 250 Figure 6.4: Dynamic simulation of the top side board temperature at the end of the drying section of the stochastic model. The short time periods between vertical lines represent the data replaced by a linear interpolation. Since the predictions of the moisture and the temperature at the end of the drying section are satisfactory, we examine the values of the disturbances estimated by the Kalman filter. To verify that the Kalman filter computes reasonable values, the disturbed variables udin and pda are depicted in figure 6.5. The variables have similar trends for the identified and physical models and are within physical limits. Table 6.2: Mean and standard deviations of the prediction error ², for the deterministic and stochastic model. prediction errors det. model stoch. model ident. parameters phys. parameters ident. parameters phys. parameters ²Tp [◦ C] mean Std 9.50 4.14 1.57 4.02 -0.74 2.54 -0.025 1.25 ²u [kgw /kgdry ] mean Std -0.005 0.0155 0.0006 0.0229 -0.0009 0.0048 -0.0001 0.0031 88 Dynamic Simulations d uin estimated by the Kalman filter 1.9 model with identified parameters model with physical parameters Moisture content in kgw/kgdry 1.8 1.7 1.6 1.5 1.4 1.3 0 50 100 150 Time in hours 200 250 (a) Incoming moisture content d pa estimated by the Kalman filter 4 Partial pressure of water in the air in Pa 3 x 10 model with identified parameters model with physical parameters 2.5 2 1.5 1 0.5 0 0 50 100 150 Time in hours 200 250 (b) Partial pressure of water in the air Figure 6.5: Values of the disturbed variables estimated by the Kalman filter for the physical and identified models. For the incoming moisture content, the lower and upper 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, but the stochastic model improves the fit considerably and shows satisfactory results. Since the identified values in chapter 5 are outside the range of physical values found in literature, we also simulate the model with more physical values. The results 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 physical values shows a good agreement of the final board temperature and average moisture content and we believe it could be used by the operators as an indicator of the board 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 difference between the performances of the two models is not well understood and should be further investigated. The remaining biases for the identified values can be due to numerical problems, since the simulated outputs show strong disturbances for those time periods. Another possible explanation is the fact that the identification is carried out under static conditions. The inputs are therefore not excited and some input-output relations are not well represented. • Does the Kalman filter give a correct estimation of the unmeasured input? We have checked that the Kalman filter assigns reasonable values to the disturbed variables. However, we should be aware that if we measure the inputs estimated by the Kalman filter on-line, we may not find the values obtained by the stochastic model. The Kalman filter compensates for the unmodelled features. For example, the biases in the physical model seem to be related to the basis weight: for low (respectively high) basis weight, the simulated moisture content seems too high (respectively low). The relation drying capacity–basis weight is therefore not well represented and the Kalman filter algorithm uses the partial pressure of water in the air to adjust the drying capacity. We believe that if we measure the incoming moisture and partial pressure of water in the air on-line, other disturbances (for example in the parameters hsc , Dwp , Cpdry or kdry ) have to be included in the Kalman filter to 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 inside the drying section of a paper mill. The distribution of the moisture inside the board is an important variable for the board quality, but is unfortunately not measured on-line. The main goal of this work was a model that predicts the moisture evolution during the drying, to be used by operators and process engineers as an estimation of the unmeasurable variables inside the drying section. A major limitation in the drying-section modelling field is the lack of measurements to validate the model. If the model is not observable or identifiable, we can not guarantee that our estimation of the variables or parameters is correct. We therefore carried out observability and identifiability analyses to verify the relevance 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 free draw, the model presented in chapter 3 is observable and identifiable1 . Moreover, we showed in appendix B that the model is not observable or identifiable under on-line conditions, i.e. when only the final board moisture and temperature are measured. A sensitivity analysis was also performed to select the most influent parameters for the identification. Based on the observability and identifiability analyses, four special measurements sessions were carried out for the identification of the unknown parameters under static conditions, i.e. the inputs of the machine were constant. The parameters were identified using one data set and the resulting model was validated using the three other data sets. The results were considered satisfactory for the identification set but poor for the validation sets. The poor agreement is believed to 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 examine the grey-box modelling technique for a drying section to investigate if we could 1 with 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 simulated under on-line conditions and the results were far from satisfactory. However, the addition of disturbances in the process by a Kalman filter showed a good prediction of the final board temperature and moisture content at the end of the drying section. We therefore believe that the model could be used by operators and process engineers as an indicator of the board properties inside the drying section. 7.2 Directions for future work The modelling of the board properties inside the drying section is a challenging field. The complexity of the process and the limitations of measurements suggest several 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? Where should we drop constraints: on the physical description of the model or on the observability? The model purpose provides an indication for answering those questions. If the model is used to predict quality variables, we believe the physical description should be retained. But if the model is used for moisture control, a one-layer model is probably more appropriate. • The dynamic behaviour of the deterministic model in section 6.1 shows poor agreement with the measurements. The discrepancy between identified and physical parameters suggests that the values of the parameters, identified under static conditions, are not optimal. A careful study of the correlations 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 also enhance the dynamic behaviour of the model. • The numerical solution of the model is another field to investigate. The simulations were sometimes stopped by numerical errors for some parameter values. Modifying the sorption equations (3.55) would probably remedy this numerical problem. The simulation time of the model is also an important limitation for trial-and-error procedures. For example, on a standard high-end PC, the simulation time for the Kalman filter algorithm, with two Kalman states, and a sampling time of 12 minutes, is equal to the simulated time period. Improving the computation time would allow more freedom to test new ideas. • Any addition of on-line sensors in the drying section would contribute to improve the model. We showed that the incoming moisture and the partial pressure 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 measure on-line inside the drying section, but temperature sensors could be used to improve the parameters. Two sensors, one before and one after the stack dryers can contribute to improve the stack model. • The research on improving sensors is obviously of great interest. The development of sensors able to measure the moisture distribution in the sheet would be a huge contribution to this project. Appendix A Implementation The first objective of this work was to apply a grey-box modeling approach, similar to Pettersson [36] and Bortolin [9], to the model of the drying section. Since a simple model, as well as other parts of the paper mill, was already implemented in Dymola the idea was to investigate grey-box modeling with Dymola. Furthermore, to the knowledge of the author, no work has been reported on this particular topic. However, since Dymola does not provide useful tools to perform parameters studies or optimization runs, we combine it with Matlab to perform the identification of the parameters. After an introduction to the simulation program Dymola, this appendix describes the structure of the model, the algorithm of identification of parameters with Dymola and the optimization routine used in this work. A.1 Simulation program The model is implemented in the simulation tool Dymola [12] based on the objectoriented language Modelica [32]. The object-oriented approach is suitable for physical modelling, because it supports hierarchical structuring. Models are defined in classes which can be changed and reused in a modular manner. Dymola generates an executable called Dymosim that simulates the model. Dymosim can be called either from Dymola or from other environments, for example Matlab. More details can be found in the Dymola user manual [13]. Among the possible implemented integration methods in Dymola, we choose to solve the equations system with the method DASSL [38]. DASSL, a variable step size algorithm, is a suitable solver for stiff ODE (Ordinary Differential Equation) or DAE (Differential Algebraic Equation). In the steady state, the ODEs are equal to zero. So, the closer the computation is to the steady state, the closer are the ODEs to zero, and hence the faster is the simulation. Therefore, if small changes are applied 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 simulations. A.2 Model structure The structure of the model, displayed in figure A.1, follows the structure of the drying section. An overall class, Dry Section (figure A.2), is composed of objects of the class Air Group that contain the disposition of the cylinders. A class Steam Group, containing the steam temperature is linked to the class Air Group. Finally, a class 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 Free Draw where the paper is in the free draw. Figure A.1: Model structure, where Ps represents the steam pressure, Ts the steam temperature, Tc the cylinder temperature, Tp and u the temperature and the moisture 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 class Air Group (AirGr) linked to instances of the class Steam Group (Steam). The paper sheet, passing trough the air groups, is the input (white square) and output (black square) 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, and other environment than Matlab. It could be used to optimize the other implemented parts of the paper machine. This algorithm is described for steady-state computation, but can easily be modified for identification of dynamic models. Algorithm 1 Identification algorithm for a dymola model Step 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 previous simulation or the default values assigned in the model and Xf (0) is the final states of 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: Ŷ (k) = Xf (k) - Compute the vector error: ²(k) = Y − Ŷ (k) PN - Compute the loss function: VN (k) = N1 i=1 ²i (k) λi - Given θ(k) and VN (k), the optimization routine either - stops (θ̂ = 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. The parameters are bounded in order to avoid that the optimization routine sets them to unrealistic values. Different approaches were tried to solve the optimization problem. Gradients methods were tested: the nonlinear least square solver function lsqnonlin in the Matlab optimization toolbox [34] and the software package IPOPT (interior point line search filter method) [52], but both routines did not converge to an optimal solution. The routines were staying around the starting point without converging to a solution. The reason for the failure of these two routines is unknown and should be further investigated. The problem is probably due to the computation of the gradient. The methods using random search were more successful. Both the elliptical random search (see [9], Appendix D) or the pattern search algorithm in the Matlab optimization toolbox [34] converged to the same solution for identical problems. Appendix B Observability Analysis for one Cylinder In section 4.1, we carry out an observability analysis for the linearized system of a contact-zone sub-model. In this appendix, we investigate the observability for a whole cylinder, i.e. we concatenate the contact zone and free draw models into one cylinder model. The numerical values for this analysis are the same as the one used in section 4.1, for the cylinders 1 and 53. They are not displayed for the sake of clarity, and only the results are mentioned. One cylinder is made of the following state vector: Tp,cz ucz Tp,f d uf d where the subscripts cz and f d are for the contact zone and the free draw respectively. Tp is a vector of the seven temperatures in the thickness direction and u the vector of the three moisture contents in the thickness direction. With these notations, we rewrite equation (4.6) as: µ δ Ṫp,cz δ u̇cz ¶ where Bcz is divided into tion (4.5) is written: ¡ µ = Acz δTp,cz δucz Bcz,Tc Bcz,Tp µ δy = Ccz 99 ¶ δTc δTp,cz,in + Bcz δTa δucz,in Bcz,Ta δTp,cz δucz ¶ Bcz,u ¢ . The output in equa- 100 Observability Analysis for one Cylinder The equations in the free draw are similar, except the temperature of the cylinder that is not considered: µ ¶ µ ¶ δTp,f d,in δTp,f d δ Ṫp,f d = Af d + Bf d δTa δuf d δ u̇f d δu f d,in µ ¶ δTp,f d δy = Cf d δuf d ¡ ¢ where Bf d is divided into Bf d,Tp Bf d,Ta Bf d,u . We now want to concatenate the two systems to consider the observability of a whole cylinder. The incoming temperature and moisture content of the free draw is then equal to the temperature and moisture content of the contact zone (see figure 3.1). uf d,in = ucz Tp,f d,in = Tp,cz The system becomes: δ Ṫp,cz µ δ u̇cz = δ Ṫp,f d δ u̇f d µ δy = Acz Bf d,Tp Bf d,u O Af d ¶ Ccz O2×10 ¶ O2×10 Cf d δTp,cz δucz δTp,f d + Bcz δuf d δTc δTp,cz,in δTa δucz,in δTp,cz δucz δTp,f d δuf d (B.1) The resulting system is linear with 20 states (temperatures and moisture contents in the contact zone and free draw) and 12 inputs (the temperatures of the cylinder and the air and the incoming temperatures and moisture contents). In section 4.1, we conclude that if we measure at least the temperature at both surfaces of the paper for every contact zone and every free draw, the model is observable. 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