Dev. Chem Eng. Mineral Process., 9(1/2},pp.69-76, 2001. Modeling of Fourdrinier and Cylinder Machines W.D. Zhang*, Y.X. Sun‘ and X.M. Xu Department of Automation, Shanghai Jiaotong University, Shanghai 200030, P R. China # National Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 31 0027, P. R. China In this paper an “equivalent paper machine” is presented which possesses the features of both the fourdrinier machines and cylinder machines, and a universal model is developed by mechanism analysis. The model is not onlj an external description as ofien obtained by system identification, but an internal one in which internal process variables are included. Experiments show that the model agrees closely with the plant data. Introduction Papermaking process is a complex process with heat and mass transfer. Many mathematical models have been reported [1-4]. However, most of these works were only concerned with its sub-processes, such as a head-box. The model of the whole process was usually built by identification [5,6]. The m e c h s m model can characterize not only the input-output behavior, but also the internal mechanism. Therefore, it is useful for analysis, design and control of paper machmes. A discrete model for fourdrinier machines by mechamsm analysis has been proposed by [5,6]. However, it was limited to the special case. The universal model has not been obtained yet for fourdrinier and cylinder machines, which are widely used in paper mills of many developing countries. In this paper, an “equivalent paper machine” possessing the features of both the fourdnnier machines and cylinder machines is presented, and the system dynamic model is developed by m e c h s m analysis. It is found that the previous model presented by [5,6] is only a special case of the new model. “Equivalent paper machine” and control strategy There are many kinds of paper machines in modern paper mills, of which the fourdnnier machine and cylinder machine are widely used in many developing countries. In some ways the basic operating principles of the two paper machines are the same except for the sheet forming device (see Figure 1). * Author for correspondence (mail: wdzhang@mail.sjtu.edu.cn). 69 W.D.Zhang,Y.X. SunandX.M.Xu Finished paper Thick pulp I , . 1 ; whitewater I Figure 1. The structure of paper machine. Since there are less than five cylinder molds and there are less than sixty dryers in the fourdrinier and cylinder machines, the proposed “equivalent paper machine” consists of a mixing tank,a head-box, a fourdrinier table, five cylinder molds, a press and sixty dryers. Its operating principles are similar to those of fourdrher and cyhder machines. Then the mechanism model based on “equivalent paper machme” is suitable for fourdrinier machines, cylinder machines and combined fourdrinier and cylinder machines. When the model is applied to a special plant, some part of it can simply be cancelled. For example, when the plant is a fourdrinier m a c b e , the cylinder mold and some of dryers can be set to zero; When the plant is a cylinder machine, the fourdrhier table and some of dryers can be set to zero; When the plant is a combined fourdrinier and cylinder machine, some cylinder mold and some of dryers will be zero. Although we can obtain a model as discussed above, it is not equal to the simple combination of the fourdrinier machine model and cylinder machine model. The modeling of a paper machine is closely related to the control strategy of the system. In light of the technological process of production, the following control strategy about the paper machine basis weight and moisture content control is considered. First, retain the pulp consistencies, pulp flow rates and stream pressures of dryers constant by the three basic feedback loops, then the basis weight and the moisture content are constant over all. Second, measure the basis weight and moisture content continuously on-line, the measured value of the process output and the setpoint are compared within the computer, and the difference between them is used to adjust the process. The computer sends a signal based upon the difference to a final control element. The basis weight is adjusted by the pulp flow rate and the moisture content stxeam pressure (see Figure 2). Dynamic model The dynamic model of the papermaking process is developed here. Miring tank and headbox In the “equivalent paper machine”, there are six flows of pulp. As their modeling procedures are similar to each other, only one flow of pulp is considered. In this section, the thick pulp is mixed with the white water in a mixing tank,and the mixture flows through a pipe into head-box. Regard the mixing tank and head-box as one container. Assume that the mixture is well-proportioned, of which the density does not vary with the consistency of pulp. The dynamic model equations can be established by a non-stationary fluid flow balance. 70 Modeling of Fourdrinier and Cylinder Machines ,-, -z Thick pulp White water Stem Reel -8 I- I I 1 Table and -_ ,Lpress section - _ _ t .c Head section ~ Dryer section Caiendei stack Figure 2. Control strategy of the system. where H, A, are the depth of pulp, the cross section of the container, and the density of pulp, respectively; G, , G, , G, are the flow rates of thick pulp, the white water and the pulp from the head-box, respectively; C, Cw , Cm are the consistencies of thick pulp, the white water and the pulp from the head-box, + G,, = G,, , respectively. Consider the conditions of steady state: GPO Gp,Cpo + G,C, = GnoCno , and the relationshrp between the variation of pulp flow rate and that of the pulp depth: AGn = AH I R , where R is flow resistance, we obtain the following models by Laplace transform: where 5, is the central time delay. TJg and Tk are the time constants, a, and a, are coefficients used to correct error. b, b, are constants which relate to the physical plant. If we want to cancel a fourdrinier table or a cylinder mold from the system, we can let G, and Gwsimply be zero. - Forming devices and presses De-watering process of fourdrinier table is similar to that of cylinder mold. Suppose that the basis weight and the moisture content of the whole mat are well-proportioned. By mass balance we have: 71 W.D. Zhang, Y.X. Sun andX.M. Xu (3) where Bw and M , are the basis weight and moisture content of the mat, respectively. g is the amount of de-watered white water from per unit area of mat per unit time. If the rate of de-watering is a constant, then B,M,, = B , M s , - W ( l - C w ) , B , = B , - W ,where Bw, and B , are the basis weight before de-watering and after de-watering, respectively. M I ,and M , , are the moisture content before de-watering and after de-watering, respectively. W = jgdt is the amount of de-watered white water from per unit area. The initial conditions B,, and M,, are determined by the state of pulp. Bw, = G, / DY , M,, = 1- CA) , where D is the width of the mat and V is the velocity of paper machme. It is very important to calculate the amount of de-watered whte water. Careful study shows that the de-watering process is a complex process influenced by many factors. Thus,it is difficult to build its model by mechanism analysis and we will use empirical formula instead. For cylinder machine, the de-watering amount is mainly affected by the pulp flow rate G, , the pulp consistency C , , the difference h between the pulp inside the mold and outside the mold, and the velocity V, .Then the de-watering amount W,can be described by W,= K , A h ‘ C : ‘ G ~ ~ * . For fourdrinier machine, the main affecting factors are the pulp flow rate G,,, the pulp consistency C,,, and the velocity V, . Then the de-watering amount W, can be described by W, = K , C ~ ~ G. ~ ~ ~ ” We find that the above two formulas are similar. If we stipulate c, = 0 for fourdrinier machhe, the latter is a special case of the former. The effect caused by velocity is relatively small and can be omitted. The introduced error will be corrected in the last section. Consider the time delays of the table section, as a result of transform and linearization we obtain the simplified model: The section consists of many forming devices, then the basis weight and moisture content that get into presses are given by the equations: 72 Modeling of Fourdrinier and Cylinder Machines Suppose that there is no fiber loss in press process. Let A h 4 ~and AM,, be the variations of moisture content that get into presses and get out of presses, respectively. Then there exists the following relationship between them: AM- = KwAMs,, , where K w is the de-watering rate of presses. The model of forming devices and presses can be obtained from it. Dryers Drying process is a relatively large, complex, and poorly understood process. In order to get a model of low order but with high accuracy, an “equivalent dryer” is presented, which consists of a group of dryers with slmilar functions. Then the dryers can be divided into two sections: drymg section (one equivalent dryer consists of fifty dryers) and sizing and coating section (one equivalent dryer consists of ten dryers). Investigations of the technological process prove that the drymg process is mainly related to the temperature of “equivalent dryer” T, , the initial basis weight M s 2 ,the initial moisture content B,, , and velocity V. According to the empirical formula obtained by [ 6 ] ,the model of the section can be written as 1 Bw3(s) e-rhs = T,S+l(ATh(’) + f 2 B w 2 ( S ) + f 3 M s 2 ( s ) +f4V(s)1 (6) e-W M,, 0 )= -k , T , (s) + g2Bw,(s) + &M,,(s) + g,V(s)l T,s + 1 The surface temperature of equivalent dryer, T,, can be obtained by the surface temperature of every single dryer. However this will result in a very hlgh order model. So it is considered to let the parameter be a linear combination of the surface so h,Th,(s). Take M c be the thermal temperature of every single dryer, i.e. T,(s) = ,=I capacity, Q,, and Q, be the amount of heat that get into and get out of the dryer. Then, we have M e(dT&,/ dt) = Q,. - Q,, . Thus where P, is the stream pressure, H is the enthalpy of stream. It is very dfficult to measure the enthalpy, and it is best considered as a constant. Sizing and coating have similar effects on the basis weight and moisture content of mat. Thus only the coating process is discussed here. Let B, be the the amount of coating per unit area, and its moisture content W, . We have the linearized equations: Bw4= B , + B , M,, = k,M,, + k , M , + k,Bw, + k,B . Thus, the final model of the section is 73 W.D. Zhang, Y.X. SunandX.M. Xu Universal model of the whole paper machine Based on the above discussions, together with other mathematical treatments, the model of the equivalent paper machine can be established: where GPis the flow rate of thick pulp which is used to control the basis weight, P is the stream pressure which is used to control the moisture content, A is the transfer function matrix of control variables, and B is the transfer function matrix of dsturbance variables. Because of the d ~ e r e n c eof the time constants and time delays of each section, the elements of A or B are somewhat complicated. After being simplified, they will be in the form of first order or second order plus time delay transfer functions. Applications Due to limited space only the application to fourdrinier machines is given here. The validty of the model has been tested and verified in normal operating conditions. The fourdnnier machine consists of one fourdrinier table and seven dryers. The thick pulp flow rate G,,, and the drymg section stream pressure 4 are selected as control variables. The paper machine is old and the velocity changes frequently. Because of the high quality requirement, the disturbance from the consistency variations of the thick pulp and white water are relatively large. Let all the flow rate of cylinder mold be zero and select the thick pulp consistency C,, and the white water consistency CM,,as disturbance variables, we establish the model of the paper machine, 5.158e-2.8' - 0.2e-I" 2.23S+l 1.8S+1 A= 0 , ~ e - 2 . 8 ' - 1.2&-'.2s - 1.8S+1 - - 2.238+1 - B= Q.4e-2"' 1.94S+1 4.5 x e-l." - 1.94S+1 - 11.6e-2.ss 1.94s + 1 5.3 x lo-' e-"" 1.94S+1 The actual outputs of the paper machine and those calculated from the model are given in Figures 3 and 4, which show that the model has high accuracy. 74 Modeling of Fourdrinier and Cylinder Machines Figure 3. Responses to a step variation of thick p u b flow rate. (Solid linefor the measured and dashed linefor the calculated) ao! "I Figure 4. Responses to a step variation of stream pressure. (Solid line for the measured and dashed line for the calculated) On the basis of the model, a robust controller was designed to control the basis weight and moisture content. After the system was put into operation, the rate of first class production increased by 15%. The heat system saving is 12.5%. The variation of basis weight decreased to 1.5 g/m . Conclusions A universal dynamic model of fourdrinier and cylinder machines has been established based mainly on the mechanism analysis of the papermaking processes. The test data from a paper mill agree closely with the model, which shows that it is accurate enough to be used in control system design. Since the mechanism analysis emphasizes the common characteristics of papennaking process, the models of different paper machmes have the same structure. The only difference is that they take different parameters. These parameters are determined by the production data in the steady state. The proposed model has been used successfully for the basis weight and moisture content control of two kinds of paper machines. Acknowledgment This project was supported by the Natural Science Foundation of Chma (69804007) and the Science and Technology Phosphor Program of Shanghai (99QD14012). 75 W.D. Bang, Y.X. Sun andX.M. Xu References , Valtchev V., and L. Guntchev. 1983. Optimal Paper Machine Control with Adaptive Properties. 5th IFAC PRP Conf.,Antwerp., 129-133 Astrom,K. J. ,Borisson U., Ljung L.,and B. Wittenmark. 1977. Theory and Applications of Self-Tuning Regulators. Automatica, 13 ,457-465 Codieux,S. M. 1983. A Mathematical Drainage Method for Fourdrinier paper machine. J. Pulp&Paper Scie., TKI 11 Gentil,S. 1975. difference Methods for Dynamic Identification of an Experimental Paper Machine. Roc. 3rd IFAC Sym. on Identificationand System Parameter Estimation, The Hague, 478-481 Wang,Q.G. 1984. Modeling and Control of Papermaking Process(1n Chinese). M.S. thesis, Zhejiang University Sun,Y.X. 1993. Modeling and Control of Papermaking Processes (In Chinese). Zhejiang University Publishing House 1. Angel0v.V. 2. 3. 4. 5. 6. Received: 15 June 1999; Accepted aper revision: 15 May 2000. 76

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