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Methods for Improving Accuracy of Paper-Sheet Moisture Control
METHODS FOR IMPROVING ACCURACY OF PAPER-SHEET MOISTURE CONTROL Yu. S. Zhukova, and V. Z. Ponizovski, VNIIB, Leningrad, U.S.S .R.
AUTOMATION
PAPER 12.3 loss ~c and cl. i ), the consumptions of dP the steam and condensate are the same and equal to the mean for the group.
(d
SUMMARY A mathematical model of the drying part, as a controlled member for paper sheet moisture control is being constructed . I n constructing it the distribution of temperature along the thickness of the cylinder as well as the distribution of the moisture along the length of the drying part is taken into account. The optimization of the dynamic responses of the controlled member is achieved with using of model through the choice of the proper structure of the steam pipeline, of the proper design of the drying cylinder and of the proper value of some technological parameters. I n construction the combined system of control the vacuum in the chamber of the couch-roll is used as an indirect indicator of the basic disturbance. A further improvement of the efficiency of the moisture control is achieved by the statistical prediction of the indirect indicator of disturbance.
(6)
The machine speed (v) is constant.
In this way the complicated combined contact-convective drying is replaced by the contact paper drying which is equivalent to it. In case of 3 steam groups it may be represented as it is shown in fig. 1. The above mentioned indexes which will occur further: s, c, d, p denote: steam, condensate, dryer, paper. The process of the change of steam pressure in cylinders of steam group" i" is described by the equation of steam mass balance:
e t. ddt
..(1
INTRODUCTION There is some difficulty in the moisture control of the paper sheet in those machines where the paper drying takes place in revolving drying cylinders. This difficulty is caused by the fact that disturbancies include highfrequency parts, whereas the only available and inexpensive control channel (where the changes in moisture are caused by the changes of the pressure of steam in the the drying cylinders) is characterized by a large lag. I n view of th is difficulty it is necesseray to make the control system operate more quickly, that is: to find some method of neutralizing or diminishing the lag and the dead time of the control-channel.
P,' _
- ~ A
G'
f1
(1)
C - hydraulic capacity G s 1 and G s 2 - the consumptions of the steam introduced into the group through the control valve and of through steam. Gc 1 - the consumption of the condensate wh ich is formed in process. In accordance with assumption 3 the process of heat transfer from the condensing steam through the layer of the condensate and the shell of the cyl inder to the paper is described for group" i" by the problem of heat conduction for two-layer shell with boundary conditions of type 3 :
ch({(y,t) _ Jt - Cle
MATHEMATICAL MODEL OF THE DRYING PART AS A CONTROLLED MEMBER FOR PAPER SHEET MOISTURE CONTROL
G:52'- - L\ Get l
Sf -
tJf (~IO) :
t,; "', (H 'j~ J,
C'!{
(2 )
0
( 3)
The model of the drying part was constructed with the following basic assumptions: 1)
Deviations of the parameters from their nom inal values are small.
2)
The steam is saturated.
3)
Condensate in the cylinder is distributed evenly and its cross section has the shape of a ring whose thickness c) is constant.
4)
The process of heat and mass transfer is inertialess.
[ dt.f{'(/j,t)] d
6
(b
5)
L
i
a Bd ('1, t )=a 6
Jt
the coefficients of the heat
467
~ [cJl~ t) c 0,
6 ()~ (
be , t ) ==
(
t)j. [ 3()~;(~1 J'j ';j=d
In all the cylinders of group "i" the steam pressure (pt.), the thickness of the layer of the condensate, the temperature of the cylinder surface
(Bd ((( t )
~
=-
I;/=O.Ae
cl.
8B/ pi] (4) JP,' t. ~ J
Se ,t)
( s)
:;.~ ['d68~(f-/1 t)] c
/le
~:i6
Jl.j
V~ ~
e; (lj. t)
o~ t~oO
d,/2
~~!{lj{S
({,) p.)
11
P, '.I C f
2 G.J2
f
GSf
.3
G.s 1
Fig. 1. The diagram of the equivalent process of contact paper drying
[ ;h Bd'f;;, t)] Jy
Here
f:,
the initial condition : u(x, 0) = f (x) the boundary conditions are: u(O, t) = uo(t) u(13 ' t)= uf(t)
re )
De; flj, O)=0 -=
Sk
,-!=S)' d
(}(.I, t )
= arYl t)
(J (}p _1\ 6 (}~ (f"t) (,) J (}d
Here :
- )
- Bf?!, 0).
Water removal of the moving paper in the drying pa rt is described by the following system of equations :
d-~{j (:r t)
+
()t
v d {j (:r,t)= - R Jx
ire ' 2 ) 0 ~ t ~ dl
u -
moisture contents in paper;
Ri -
rate of drying in the 1st period in group" i" ;
li -
the length of paper in groups 1 to" i" ;
l
=
l (0, 0=
f.t - I !!x",t(
So'
(}t
3
-+
l(X,
t)
t) - the basis weight of oven dry paper;
= lO
(t);
U cr = U cr (Zo' So) - the critical moisture of paper when the period of falling rate of drying sets in;
(10)
As a concrete example we shall take the drying part consisting of 3 steam groups. In this case i = 1,2.
" t) dU(2',
(13) (14) (15)
beaten stuff condition in paper in front of the drying part :
lcr - co-ordinate along the length of the paper corresponding to critical moisture. The equations (10), (11), (12), (13) , (14), (15) can be substituted by the ordinary differential equations, When solving these equations one can get the expression for the final moisture of the paper. Passing over to deviations and to Laplace transform we obtain :
r"'eP
Ll Ii d LI ( x,t ) '" _ "03( Dd , l ) i..Ilx,i) 0'"- (/2) (J x U(/l. t(/l~X~~
468
(il P,t)8c i
l (t.P) f mIC..:}: t.P,
+,<,)/
can be reduced by means of the reduction of the thickness of the condensate layer and of the steam pressure in the drying cylinder. The curves of relative sensitivity on fig. 4 can serve as an illustration. The reduction of the th ickness of the condensate layer can be achieved by applying the proper types of the condensate removal units. When applying these units it is possible to diminish the steam pressure by means of creating sufficiently profound vacuum in group l. One can also reduce the lag of control channel by choosing the optimal structure of the pipeline and the methods of stabilization of the steam groups. The ratio of the first moment of the proper weighting function to zero moment is used as the measure of the process lag. I n accordance with equation (16) the dynam ics of control channel is greatly influenced by the distribution of the drying process along the length of the drying part. Besides this it is also dependent, on the lag of automatic control systems of pressure differences. That is why from the viewpoint of reduction of the lag of the control channel of paper moisture it would be best to control moisture by changing pressure in only one or several groups, nearest the reel. For instance moisture control ach ieved by changing pressure only in group 3 makes it possible to make the lag 20 % lower when compared to the case, where the pressure changes in all the groups (according to the data of machine NO 5 in Kondopoga paper mill). In order to diminish the lag of the control channel it is also necessary that the system of pressure stabilization in group 3 should be adjusted to the maximum of operation rate.
l
If I (S) .
l
l
1(2
t.B C (5) L
f(~ (5)
~8d
Fig. 2. Block diagram of heat transfer process where T 1, T 2, T 3 - the time of paper transfer through the 1st, 2nd, and 3rd groups respectively, and T = T 1 +T2+ T 3 ' The transfer function which we got in equation (16) can be easily simulated using the analogue computer (AC) . Fig. 2 shows the block diagram of the process of heat transfer. The transfer functions K2(S) u K 3 (S) are transcendental and cannot be simulated with the help of AC. That is why we performed the approximation of these transfer functions by transfer functions of fractionally /-/ rational type. When performing it we took the equality of the definite number of such integral characteristics as Simoju areas or the moments of weighting function for a criterion of the proximity of the real and approximating function /3/, /4/, /5/, /2/. It appears that the equality of the 1st two areas or moments guarantees sufficient accuracy. That is, the order of approximating fractionally-rational functions does not exceed 2. OPTIMISATION OF DYNAMIC RESPONSES OF CONTROLLED MEMBER The main source of the lag of controlled member is the process of heat transfer from the steam to the paper. This is clearly shown on fig. 3. The analysis of the sensitivity of transient response of cylinder surface temperature to the parameters showed, that in reality the lag of the process of heat transfer
469
THE SYSTEM OF PAPER MOISTURE FEED-FORWARD CONTROL WHEN APPLYING STATISTICAL PREDICTION OF THE BASIC DISTURBANCE The comparison of the autocorrelation function of paper moisture at the back of the drying part to cross-correlation function of moisture and vacuum in the couchroll chamber showed, that with frequencies above W = 0,01 the following transfer function of the channel (where the changes in moisture at the back of the drying part are caused by the changes of vacuum in couchroll) takes place: Kv(S)=kve-TvS
(17)
where Tv - the time of paper transfer from the couch-roll to the reel. Thus with the above mentioned frequencies the vacuum in the couch-roll can be taken as an indirect indicator of the basic disturbance in the feed-forward control system. The efficiency of th is control system can be improved by applying statistical prediction of vacuum. If a digital computer is used for paper machine control, the Box and Jenkins method of prediction /6/ can be recommended. The digital simulation of the feed-forward moisture control system with the prediction of vacuum in the couchroll chamber showed, that when the prediction step h = 45 sec the moisture mean-root-square error can be reduced by more than 30 % in comparison with the process without control (according to the data of machine NO 5 in Kondopoga paper mill) .
LJtJ/{t} . ~ Bd (t) .1l1l{.o) ) .18d (00) f,O
0,9
0,8
0,7
0.6 0.5
Pape7mac!tine tY!!.9
0,4-
Papep mitt in
So!ikamsk
0.3
0.2. O,f
o
zoo
100
300
500
'000
Fig. 3. Transient responses of the cylinder surface temperature and of the final paper moisture in case of perturbation by the set point of the steam pressure regulator.
9zad C f2
fO
9
8 7
Pape7mach,'ne t/Eg
6
Pape? mitt in SotiKaFnsk
S
4 3 2 f
o
t(se~)
L - - i__~__L - - i__~__L--L__~__L - - i__~~L--i__-1~
~-L--~_ _
(00
2U0
300
400
500
600
700
!lOo
- f
-z -3
-4 -5
Fig . 4. Transient response of the cylinder surface temperature in case of perturbation by the set point of the steam pressure regulator and the functions of relative sensitivity of this response to parameters Sc and ~). .
470
LIST OF REFERENCES
1.
D . P. Kempbell. Dynamics of Chemical Technology Processes. Goskhimizdat, Moscow, 1962
2.
V . Z . Ponizovski. Mathematical Description of paper drying process. VNIIB. Proceedings of the Institute. NO 54 . Moscow, 1969, p. 50-61,
3.
M. P. Simoyu. Determination of Transfer Function Coefficients of Linearized U nits and of Control Systems. "Avtomatika i Telemekhanika", vol. XVIII, NO 6,1957, p. 514-528.
4.
N. S. Khorkov. Calculation and Approximation of the heataccumulating Capacity Dynamic Responses. CN II KA. Proceedings of the Institute. NO 9, Moscow, 1963, p. 242-260.
5.
B. N. Devjatov . Theory of the Transient Responses in Technological Apparatus from the ViewPoint of Control Problems. Ac . of Se. USSR . Siberia branch. Novosibirsk. 1964.
6.
G. G. E. P. Box, G. M. Jenkins. Some Statistical Aspects of Adaptial Optim ization and Control. .. Journal of the Royal Statistical Society" , 1962, B, vol. 24 , p. 297-343.
471
AUTOMATION
PAPER 12.3 loss ~c and cl. i ), the consumptions of dP the steam and condensate are the same and equal to the mean for the group.
(d
SUMMARY A mathematical model of the drying part, as a controlled member for paper sheet moisture control is being constructed . I n constructing it the distribution of temperature along the thickness of the cylinder as well as the distribution of the moisture along the length of the drying part is taken into account. The optimization of the dynamic responses of the controlled member is achieved with using of model through the choice of the proper structure of the steam pipeline, of the proper design of the drying cylinder and of the proper value of some technological parameters. I n construction the combined system of control the vacuum in the chamber of the couch-roll is used as an indirect indicator of the basic disturbance. A further improvement of the efficiency of the moisture control is achieved by the statistical prediction of the indirect indicator of disturbance.
(6)
The machine speed (v) is constant.
In this way the complicated combined contact-convective drying is replaced by the contact paper drying which is equivalent to it. In case of 3 steam groups it may be represented as it is shown in fig. 1. The above mentioned indexes which will occur further: s, c, d, p denote: steam, condensate, dryer, paper. The process of the change of steam pressure in cylinders of steam group" i" is described by the equation of steam mass balance:
e t. ddt
..(1
INTRODUCTION There is some difficulty in the moisture control of the paper sheet in those machines where the paper drying takes place in revolving drying cylinders. This difficulty is caused by the fact that disturbancies include highfrequency parts, whereas the only available and inexpensive control channel (where the changes in moisture are caused by the changes of the pressure of steam in the the drying cylinders) is characterized by a large lag. I n view of th is difficulty it is necesseray to make the control system operate more quickly, that is: to find some method of neutralizing or diminishing the lag and the dead time of the control-channel.
P,' _
- ~ A
G'
f1
(1)
C - hydraulic capacity G s 1 and G s 2 - the consumptions of the steam introduced into the group through the control valve and of through steam. Gc 1 - the consumption of the condensate wh ich is formed in process. In accordance with assumption 3 the process of heat transfer from the condensing steam through the layer of the condensate and the shell of the cyl inder to the paper is described for group" i" by the problem of heat conduction for two-layer shell with boundary conditions of type 3 :
ch({(y,t) _ Jt - Cle
MATHEMATICAL MODEL OF THE DRYING PART AS A CONTROLLED MEMBER FOR PAPER SHEET MOISTURE CONTROL
G:52'- - L\ Get l
Sf -
tJf (~IO) :
t,; "', (H 'j~ J,
C'!{
(2 )
0
( 3)
The model of the drying part was constructed with the following basic assumptions: 1)
Deviations of the parameters from their nom inal values are small.
2)
The steam is saturated.
3)
Condensate in the cylinder is distributed evenly and its cross section has the shape of a ring whose thickness c) is constant.
4)
The process of heat and mass transfer is inertialess.
[ dt.f{'(/j,t)] d
6
(b
5)
L
i
a Bd ('1, t )=a 6
Jt
the coefficients of the heat
467
~ [cJl~ t) c 0,
6 ()~ (
be , t ) ==
(
t)j. [ 3()~;(~1 J'j ';j=d
In all the cylinders of group "i" the steam pressure (pt.), the thickness of the layer of the condensate, the temperature of the cylinder surface
(Bd ((( t )
~
=-
I;/=O.Ae
cl.
8B/ pi] (4) JP,' t. ~ J
Se ,t)
( s)
:;.~ ['d68~(f-/1 t)] c
/le
~:i6
Jl.j
V~ ~
e; (lj. t)
o~ t~oO
d,/2
~~!{lj{S
({,) p.)
11
P, '.I C f
2 G.J2
f
GSf
.3
G.s 1
Fig. 1. The diagram of the equivalent process of contact paper drying
[ ;h Bd'f;;, t)] Jy
Here
f:,
the initial condition : u(x, 0) = f (x) the boundary conditions are: u(O, t) = uo(t) u(13 ' t)= uf(t)
re )
De; flj, O)=0 -=
Sk
,-!=S)' d
(}(.I, t )
= arYl t)
(J (}p _1\ 6 (}~ (f"t) (,) J (}d
Here :
- )
- Bf?!, 0).
Water removal of the moving paper in the drying pa rt is described by the following system of equations :
d-~{j (:r t)
+
()t
v d {j (:r,t)= - R Jx
ire ' 2 ) 0 ~ t ~ dl
u -
moisture contents in paper;
Ri -
rate of drying in the 1st period in group" i" ;
li -
the length of paper in groups 1 to" i" ;
l
=
l (0, 0=
f.t - I !!x",t(
So'
(}t
3
-+
l(X,
t)
t) - the basis weight of oven dry paper;
= lO
(t);
U cr = U cr (Zo' So) - the critical moisture of paper when the period of falling rate of drying sets in;
(10)
As a concrete example we shall take the drying part consisting of 3 steam groups. In this case i = 1,2.
" t) dU(2',
(13) (14) (15)
beaten stuff condition in paper in front of the drying part :
lcr - co-ordinate along the length of the paper corresponding to critical moisture. The equations (10), (11), (12), (13) , (14), (15) can be substituted by the ordinary differential equations, When solving these equations one can get the expression for the final moisture of the paper. Passing over to deviations and to Laplace transform we obtain :
r"'eP
Ll Ii d LI ( x,t ) '" _ "03( Dd , l ) i..Ilx,i) 0'"- (/2) (J x U(/l. t(/l~X~~
468
(il P,t)8c i
l (t.P) f mIC..:}: t.P,
+,<,)/
can be reduced by means of the reduction of the thickness of the condensate layer and of the steam pressure in the drying cylinder. The curves of relative sensitivity on fig. 4 can serve as an illustration. The reduction of the th ickness of the condensate layer can be achieved by applying the proper types of the condensate removal units. When applying these units it is possible to diminish the steam pressure by means of creating sufficiently profound vacuum in group l. One can also reduce the lag of control channel by choosing the optimal structure of the pipeline and the methods of stabilization of the steam groups. The ratio of the first moment of the proper weighting function to zero moment is used as the measure of the process lag. I n accordance with equation (16) the dynam ics of control channel is greatly influenced by the distribution of the drying process along the length of the drying part. Besides this it is also dependent, on the lag of automatic control systems of pressure differences. That is why from the viewpoint of reduction of the lag of the control channel of paper moisture it would be best to control moisture by changing pressure in only one or several groups, nearest the reel. For instance moisture control ach ieved by changing pressure only in group 3 makes it possible to make the lag 20 % lower when compared to the case, where the pressure changes in all the groups (according to the data of machine NO 5 in Kondopoga paper mill). In order to diminish the lag of the control channel it is also necessary that the system of pressure stabilization in group 3 should be adjusted to the maximum of operation rate.
l
If I (S) .
l
l
1(2
t.B C (5) L
f(~ (5)
~8d
Fig. 2. Block diagram of heat transfer process where T 1, T 2, T 3 - the time of paper transfer through the 1st, 2nd, and 3rd groups respectively, and T = T 1 +T2+ T 3 ' The transfer function which we got in equation (16) can be easily simulated using the analogue computer (AC) . Fig. 2 shows the block diagram of the process of heat transfer. The transfer functions K2(S) u K 3 (S) are transcendental and cannot be simulated with the help of AC. That is why we performed the approximation of these transfer functions by transfer functions of fractionally /-/ rational type. When performing it we took the equality of the definite number of such integral characteristics as Simoju areas or the moments of weighting function for a criterion of the proximity of the real and approximating function /3/, /4/, /5/, /2/. It appears that the equality of the 1st two areas or moments guarantees sufficient accuracy. That is, the order of approximating fractionally-rational functions does not exceed 2. OPTIMISATION OF DYNAMIC RESPONSES OF CONTROLLED MEMBER The main source of the lag of controlled member is the process of heat transfer from the steam to the paper. This is clearly shown on fig. 3. The analysis of the sensitivity of transient response of cylinder surface temperature to the parameters showed, that in reality the lag of the process of heat transfer
469
THE SYSTEM OF PAPER MOISTURE FEED-FORWARD CONTROL WHEN APPLYING STATISTICAL PREDICTION OF THE BASIC DISTURBANCE The comparison of the autocorrelation function of paper moisture at the back of the drying part to cross-correlation function of moisture and vacuum in the couchroll chamber showed, that with frequencies above W = 0,01 the following transfer function of the channel (where the changes in moisture at the back of the drying part are caused by the changes of vacuum in couchroll) takes place: Kv(S)=kve-TvS
(17)
where Tv - the time of paper transfer from the couch-roll to the reel. Thus with the above mentioned frequencies the vacuum in the couch-roll can be taken as an indirect indicator of the basic disturbance in the feed-forward control system. The efficiency of th is control system can be improved by applying statistical prediction of vacuum. If a digital computer is used for paper machine control, the Box and Jenkins method of prediction /6/ can be recommended. The digital simulation of the feed-forward moisture control system with the prediction of vacuum in the couchroll chamber showed, that when the prediction step h = 45 sec the moisture mean-root-square error can be reduced by more than 30 % in comparison with the process without control (according to the data of machine NO 5 in Kondopoga paper mill) .
LJtJ/{t} . ~ Bd (t) .1l1l{.o) ) .18d (00) f,O
0,9
0,8
0,7
0.6 0.5
Pape7mac!tine tY!!.9
0,4-
Papep mitt in
So!ikamsk
0.3
0.2. O,f
o
zoo
100
300
500
'000
Fig. 3. Transient responses of the cylinder surface temperature and of the final paper moisture in case of perturbation by the set point of the steam pressure regulator.
9zad C f2
fO
9
8 7
Pape7mach,'ne t/Eg
6
Pape? mitt in SotiKaFnsk
S
4 3 2 f
o
t(se~)
L - - i__~__L - - i__~__L--L__~__L - - i__~~L--i__-1~
~-L--~_ _
(00
2U0
300
400
500
600
700
!lOo
- f
-z -3
-4 -5
Fig . 4. Transient response of the cylinder surface temperature in case of perturbation by the set point of the steam pressure regulator and the functions of relative sensitivity of this response to parameters Sc and ~). .
470
LIST OF REFERENCES
1.
D . P. Kempbell. Dynamics of Chemical Technology Processes. Goskhimizdat, Moscow, 1962
2.
V . Z . Ponizovski. Mathematical Description of paper drying process. VNIIB. Proceedings of the Institute. NO 54 . Moscow, 1969, p. 50-61,
3.
M. P. Simoyu. Determination of Transfer Function Coefficients of Linearized U nits and of Control Systems. "Avtomatika i Telemekhanika", vol. XVIII, NO 6,1957, p. 514-528.
4.
N. S. Khorkov. Calculation and Approximation of the heataccumulating Capacity Dynamic Responses. CN II KA. Proceedings of the Institute. NO 9, Moscow, 1963, p. 242-260.
5.
B. N. Devjatov . Theory of the Transient Responses in Technological Apparatus from the ViewPoint of Control Problems. Ac . of Se. USSR . Siberia branch. Novosibirsk. 1964.
6.
G. G. E. P. Box, G. M. Jenkins. Some Statistical Aspects of Adaptial Optim ization and Control. .. Journal of the Royal Statistical Society" , 1962, B, vol. 24 , p. 297-343.
471