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Modeling of Fourdrinier and Cylinder Machines.

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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.
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5.
6.
Received: 15 June 1999; Accepted aper revision: 15 May 2000.
76
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