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A precision on-line model for the prediction of thermal crown in hot rolling processes
International Journal of Heat and Mass Transfer 78 (2014) 967–973
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A precision on-line model for the prediction of thermal crown in hot rolling processes Mian Jiang ⇑, Xuejun Li, Jigang Wu, Guangbin Wang Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment, Hunan University of Science and Technology, XiangTan 411201, China
a r t i c l e
i n f o
Article history: Received 15 September 2013 Received in revised form 17 March 2014 Accepted 19 July 2014
Keywords: Hot rolling processes Heat conduction Thermal crown Spectral method Neural networks Hybrid intelligent modeling
a b s t r a c t The tendency towards an increase in rolling speeds, which is characteristic of the development of modern sheet rolling, causes an increase requirement of accurate prediction on-line control models for the thermal crown of work rolls. In this paper, a precision on-line model is proposed for the prediction of thermal crown in hot-strip rolling processes. The heat conduction of the roll temperature can be described by a nonlinear partial differential equation (PDE) in the cylindrical coordinate. After selecting a set of proper basis functions, the spectral methods can be applied to time/space separation and model reduction, and the dynamics of the heat conduction can be described by a model of high-order nonlinear ordinary differential equations (ODE) with a few unknown nonlinearities. Using a technique for further reducing the dimensions of the ODE system, neural networks (NNs) can be trained to identify the unknown nonlinearities. The low-order predicted model of the thermal crown is given in state-space formulation and efficient in computation. The comparisons of prediction values for the thermal crown with the production data in an aluminum alloy hot rolling process show that the proposed method is effective and has high performance. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The tendency towards an increase in rolling speeds, which is characteristic of the development of modern sheet rolling, causes an increase in the instability of a number of production factors which influence the shape of the strip [1]. The flatness and profile of the strip are mainly dependent on the configuration of the roll gap across the width of the strip. The shape of the work rolls is one of the main factors affecting the shape of the roll gap, which has a significant effect on the roll gap contour. However, the shape of the work rolls changes dynamically due to the thermal deformation and wear of the rolls in continuous rolling of strip. With the exception of the work roll thermal crown expansion, the factors can be satisfactory compensated by a proper setup computation just before the rolling of each strip [2,3]. Therefore, real-time control of the uniformity of the thermal expansion of work rolls becomes the key factor in obtaining a good roll gap contour. Normally, an adequate design of the cooling system of work rolls can contribute to minimize the magnitude and shape irregularity of the thermal crown generated during the rolling process. It can keep the roll temperature and the thermal expansion within a ⇑ Corresponding author. Tel.: +86 0731 58290840; fax: +86 0731 58290848. E-mail address: [email protected] (M. Jiang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.061 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
proper range. In order to study the influence of cooling system for hot rolling processes, a good understanding of the thermal crown of rolls during hot rolling processes is critical. Due to the rapidity of the rolling process and the impossibility of measuring the thermal crown of work rolls online, an on-line model with precise computational efficiency is very important for the prediction of thermal crown. The problem of obtaining the thermal crown of work rolls is stated as a transient heat transfer problem. Heat transfer inside the work rolls is considered to occur by conduction and around the work rolls periphery by radiation and conduction (with the strip) and convection (with the water and air). And the fact that the influence of the rotation of work rolls in the heat transport phenomenon is negligible, when different methods of solution, such as analytical, experimental and numerical, have been used to predict the temperature and thermal crown profile of work rolls. An analytical solution was developed by Pawelski [4] for the heat transfer equation between work rolls and strip (roll bite region) to find the heat transfer coefficient in this region. This coefficient is a function of roll speed, scale thickness, physical properties of rolls and strip, and roll bite contact time. Tseng [5] also developed an analytical understanding of thermal expansion of work rolls to provide thermal displacements and expansion of the rolls. Guo [6] developed a semi-analytic solution of wok rolls
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thermal crown to establish a correlation between the thermal profile of the rolls and the strip shape. Because of the limitations of the analytical solution in which all the boundary conditions are not considered comprehensively, it cannot give the accurate predicted thermal crown of the work rolls. Most of the investigators preferred to use numerical and experimental solutions. A great many investigators have studied the thermal crown using the finite difference method (FDM) which is a common numerical method to solve the multidimensional heat transfer problems. Wilmotte and Mignon [7] used an axisymmetric finite difference method model to study the circumferential mean values of the roll thermal expansion. Zhang et al. [8] proposed a 2dimensional axisymmetric model developed by the finite-difference method to predict the transient temperature field and the thermal profile of the work rolls in hot strip rolling process. The calculation results were compared with the production data of a 1700 mm hot strip rolling mill, and good agreement was found. Nakagawa [9] studied the transient build-up of the thermal crown based on a three-dimensional Lagrangian finite difference model and concluded that the reduction, strip temperature, and cooling condition are three major influential parameters. Bennon [10] developed as Eulerian finite-difference scheme to predict the thermal expansion at different spray cooling patterns in cold rolling of aluminum. Zhang et al. [1] developed a finite difference model to simulate the thermal deformation of the continuously variable crown (CVC) work rolls in hot strip rolling. Generally speaking, the results of the calculation of the roll temperature using the finite difference method are in good agreement with the measured values. However, finite difference method for the calculation of the thermal crown in rolling processes will produce high-order models that are unsuitable for synthesizing implements and real-time control. There are also other published researches concerning mathematical modeling of the hot rolling processes, while numerical techniques particularly the finite element analysis have been utilized for determining the deformation behavior of work roll. Guo et al. [11] developed a simplified finite element method (FEM) to analyze the temperature field and thermal crown of the roll according to the practical boundary conditions. Park et al. [12] carried out the coupled analyses of heat transfer and deformation for casting rolling by using the finite element software MARC to examine the thermal crown. Benasciutti et al. [13] proposed a simplified numerical approach based on finite-elements to computer thermal stresses occurring in work roll of hot rolling mills, which are caused by a non-uniform temperature distribution over the work roll surface. The FEM can be used to analyze the temperature field and thermal deformation conveniently, and the results of the simulation have high precision. However, the FEM method is not efficient in computation and also produces high-order models that are unsuitable for synthesizing controller design and real-time prediction control. The present study derives an on-line model with high performance for the prediction of the thermal crown in hot rolling processes. The heat conduction of the roll temperature can be described by a nonlinear partial differential equation (PDE) in the cylindrical coordinate. After selecting a set of proper basis functions, the spectral methods can be applied to time/space separation and model reduction, and the dynamics of the heat conduction can be described by a model of high-order nonlinear ordinary differential equations (ODE) with a few unknown nonlinearities. Using a technique for further reducing the dimensions of the ODE system, neural networks (NNs) can be trained to identify the unknown nonlinearities using the production data of the rolling mill, and a lower-dimensional hybrid intelligent model of the thermal crown is given in state-space formulation, which is efficient in computation and suitable for the further application of the traditional
control techniques. The comparisons of prediction for the thermal crown with the production data in an aluminum alloy hot rolling process show that the proposed method is effective and has high performance. 2. Fundamental dynamics of the heat transfer of work rolls The thermal expansion is one of the important factors affecting the roll gap profile. The work roll periodically contacts with hot strips and cooling liquid and as a result the surface temperature changes drastically. Heat transfer in the work roll is considered to occur by conduction and around the work roll periphery by radiation and conduction with the strip and convection with the water and air [14]. The work roll, rotated at high speed, is considered as a cylinder and the partial differential equation governing the heat conduction in a cylindrical coordinate can be expressed as:
qc
@T @ 2 T 1 @T 1 @ 2 T @ 2 T ¼ kt þ þ þ @t @r 2 r @r r 2 @w2 @x2
! þ q þ lðTÞ þ gðTÞ
þ hðTÞ
ð1Þ
where r and w are the radial and circumferential direction, respectively. Because the influence of the work roll is second-order magnitude, the variation of temperature along the circumferential direction of the work roll can be ignored, i.e. o2T/ow2 = 0. Eq. (1) can then be transferred into
@T @ 2 T 1 @T @ 2 T qc ¼ kt þ þ @t @r 2 r @r @x2
! þ q þ lðTÞ þ gðTÞ þ hðTÞ
ð2Þ
where T = T(x, r, t) denotes the temperature of the work roll, t is the time variable. r e [0, R] and x e [0, l] are cylindrical coordinates in radial and axial direction. kt is the thermal conductivity coefficient, q, c are the density and specific heat, respectively. The rest of variables are introduced as follows: (1) q is the heat generation rate. (2) l(T) denotes the quantity of heat conduction between work rolls and the strip in unit time, which is a function of the surface temperature of the work rolls. (3) g(T) is the quantity of heat convection between work rolls and the coolant in unit time, which is also a function of the surface temperature of the work rolls. (4) h(T) is the quantity of heat convection between work rolls and the air in unit time. The surface around the work roll periphery by radiation and conduction with the strip and convection with the water and air, and the boundary conditions are not completely known. As it is difficult to obtain reasonable boundary conditions using only physical insights, for simplicity one can set the boundary conditions to be unknown nonlinear functions of the boundary temperature, the space coordinates (x, r) of the work roll and the ambient temperature TE as follows:
kt
@T ¼ fb1 ; @x x¼0
kt
@T ¼ fb2 @x x¼l
ð3:aÞ
kt
@T ¼ fb3 ; @r r¼0
kt
@T ¼ fb4 @r r¼R
ð3:bÞ
where
fb1 ¼ f1 ðx; r; T; T E Þjx¼0 ;
f b2 ¼ f2 ðx; r; T; T E Þjx¼l
fb3 ¼ f3 ðx; r; T; T E Þjr¼0 ;
f b4 ¼ f4 ðx; r; T; T E Þjr¼R
With fb1 fb4 being unknown nonlinear functions.
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Thus, a complete temperature field model of the work rolls is described by a parabolic partial differential (2) with the boundary conditions (3.a), (3.b). To our knowledge, it is the first work which presents a lowdimensional approximate system to control the thermal-crown of the work roll of a hot strip mill by varying coolant fluxes in real time. A parabolic PDE system typically involves spatial differential operators with eigenspectral that can be partitioned into a finitedimensional (slow) and an infinite-dimensional (fast) complements [15,16]. It is very convenient to use the eigenfunctions of a spatial differential operator to derive a low order ODE system for such a parabolic PDE system if the boundary conditions are homogeneous. However, the boundary conditions (3.a), (3.b) include unknown nonlinear functions, therefore a direct derivation of eigenfunctions for the operator is impossible. For the sake of derivation of eigenfunctions for the PDE model, the Eqs. (2) and (3) are transformed into the following equivalent model with homogeneous boundary conditions, while the proof is given in the Appendix A.
@T kt 2 ¼ r T þ FðTÞ þ Uðx; R; tÞ @t qc @T @T ¼ 0; ¼0 @x x¼0 @x x¼l @T @r
@T @r
¼ 0; r¼0
ð4Þ
¼0
ð5:bÞ
1
qc
ðq þ lðTÞ þ hðTÞÞ þ
kt 1 @T
qc r @r
dðx 0Þ
1
qc
fb1 þ dðx lÞ
1 1 1 f dðr 0Þ fb3 þ dðr RÞ fb4 qc b2 qc qc is the nonlinear term with some unknown nonlinearities; d() is the Dirac delta function; the U(x, R, t) denotes the spray input of the system at the surface of the work roll; x e [0, l] r e [0, R], t e [0, 1). Based on the axisymmetric temperature distribution, the thermal expansion of rolls is calculated by elastic analysis. The thermal expansion induced displacement y(x, t) at the roll surface, r = R, is
yðx; tÞ ¼ ð1 þ v Þ b R T AV ðx; tÞ T AV ðx; tÞ ¼
Z
R
2prTðx; r; tÞdr=pR2
ð6:aÞ ð6:bÞ
0
where m is Poisson’s ratio of the roll, b is linear expansion coefficient of roll; R is the radius of roll; and TAV(x, t) denotes the average temperature on the section of the work roll at x and t. Using the temperature field of the work roll, the thermal crown of the work roll can be calculated by numerical integration of the Eqs. (6.a) and (6.b).
The new boundary conditions (5.a) and (5.b) are homogeneous. To further reduce the PDE model (4) to a set of finitely many nonlinear ODE equations, the spectral methods [16,17] can be applied as follows. 3.1. Separation of time and spatial variables According to the theorem of separation of variables [17], the solution of system (4) can be expressed in the following time– space decouple form:
i¼1 j¼1
ð8:aÞ
j ¼ 0; 1; 2; . . .
ð8:bÞ
and the corresponding eigenvalues:
ni ¼
2 kt ip ; qc l
fj ¼
2 kt jp qc R
ð9Þ
It is interesting to note that the eigenspectrum of nonlinear PDEs (4) with boundary conditions (5.a), (5.b) can be separated into a finite-dimensional (slow) and an infinite-dimensional stable (fast) complements. This implies that the dynamical behavior of such a system can be approximately described by a finite-dimensional ODE system that captures the dynamics of the dominant (slow) modes of the PDE system. Let the number of the selected eigenfunctions (8.a), (8.b) is M, N respectively. The corresponding eigenvalues of the spatial basis functions /i(x)uj(r) set to be:
2 2 ! ip jp þ l R
k r ij ¼ t qc
ð10Þ
Thus the eigenvalues is rearranged in order of magnitude and then denoted as a new sequence of rn with sn(x, r) its corresponding eigenfunction, as follows:
sn ðx; rÞ ¼ f/i ðxÞuj ðrÞg; an ðtÞ ¼ faij ðtÞg where i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; N; n ¼ ðN 1Þi þ j 2 f1; 2; . . . ; MNg Then, the temperature T(x, r, t) of the work roll can be approximated by the time/space separation based on the sn(x, r) and the expansion coefficients an(t) as follows:
Tðx; r; tÞ
ð7Þ
M X N X
aij ðtÞ/i ðxÞuj ðrÞ ¼
MN X an ðtÞsn ðx; rÞ
ð11Þ
n¼1
i¼1 j¼1
3.2. Projection Due to the orthogonality of the basis functions sn(x, r), the insertion of the expansion (11) into (4) and the application of the Galerkin method[17] produces a system of MN first-order ordinary differential equations for the evolution of the expansion coefficients.
(
daðtÞ dt
¼ AaðtÞ þ BuðtÞ þ f ðaðtÞ; uðtÞÞ
yðtÞ ¼ CaðtÞ where
3. Spectral methods for the thermal crown
1 X 1 X Tðx; r; tÞ ¼ /i ðxÞuj ðrÞaij ðtÞ
i ¼ 0; 1; 2; . . .
rn ¼ fr ij g with r1 > r2 > > rn >
2
@ 1 where r2 ¼ @r@ 2 þ @x 2 ; Uðx; R; tÞ ¼ qc gðTÞ
FðTÞ ¼
rffiffiffi 2 ipx cos /i ðxÞ ¼ l l rffiffiffi 2 jpr cos uj ðrÞ ¼ R R
ð5:aÞ
r¼R
2
Thus, the eigenvalue problem for operator r2 in (4) can be solved analytically [18], as the eigenfunctions are selected as follows:
ð12Þ
aðtÞ ¼ ½a1 ðtÞ; a2 ðtÞ; . . . ; aMN ðtÞT ; yðtÞ ¼ ½yðx1 ; tÞ; yðx2 ; tÞ; . . . ;
yðxL ; tÞT ; uðtÞ ¼ ½u1 ðtÞ; u2 ðtÞ; . . . ; uK ðtÞT And {x1, x2, . . ., xL} denotes the axial locations of the work rolls, y(t) is the thermal crown at the axial location {x1, x2, . . ., xL}. u(t) = [u1(t), u2(t), . . ., uK(t)]T is the input of the spray considered in K intervals at the surface of work rolls. f(a(t), u(t)) is the nonlinear term with some unknown nonlinearities. The derivations of the matrix A, B are given:
A ¼ diagðr1 ; r2 ; ; rMN Þ 2 B1;1 B1;2 B1;K 6 B B2;2 B2;K 6 2;1 B¼6 4 BMN;1
BMN;2
BMN;K
3 7 7 7 5
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R 0:09þ0:08k
where Bn;k ¼ 0:09þ0:08ðk1Þ sn ðx; RÞdx, note that the we can find a (i, j) to satisfy n = (N 1)i + j and the [0.09 + 0.08(k 1), 0.09 + 0.08k] denotes the kth interval of the input of spray at the surface of the work roll, then
Bn;k
rffiffiffi l 1 ¼2 R ip
ip ip cosðjpÞ sin ð0:09 þ 0:08kÞ sin ð0:01 þ 0:08kÞ l l ð13Þ
The matrix C is calculated using the average temperature TAV(x, t) in the Eq. (6.b) and the approximation (11).
2
3
C 1;1
C 1;2
C 1;MN
6C 6 2;1 yðtÞ ¼ 6 4
C 2;2
C 2;MN 7 7 7 aðtÞ 5
C L;1
C L;2
Fig. 1. The aluminum alloy hot rolling process.
ð14Þ
C L;MN
RR
Table 1 Parameters used in the modeling methodology.
2
where C k;n ¼ ð1 þ v ÞbR 0 2pr sn ðxk ; rÞdr=pR , and note that we can find a (i,j) to satisfy n = (N 1)i + j, then C k;n ¼ C k;ðN1Þiþj ¼ ð1 þ v Þ qffiffi bR/i ðxk Þ 2R ðjp2Þ2 ðcos jp 1Þ i ¼ 1; . . . ; M; j ¼ 1; . . . ; N; k ¼ 1; . . . ; L.
Parameters
Value
l R
2.1 m 0.375 m 7800 kg/m3 586.15 J/(kg K) 34.88 w/(m K) 71 24
q c kt L K
4. Further model reduction and hybrid spectral/neural modeling
12
10
8
The coolant flux
In the above discussions, the truncation limits M, N are assumed to be sufficiently large such that the dynamical system of Eq. (12) captures the long-term behaviors of the PDE of Eq. (4) with sufficient accuracy. However, the order of the ODE system is still high, and its application for numerical implementation and practical control is difficult because of limited computation capacity and complex controller design. In the previous work [19], new spatial basis functions are derived to reduce the order of the dynamical system of the nonlinear PDEs, and model reduction performance of new spatial basis functions generated by balanced truncation has been proofed. Each new spatial basis function is a linear combination of eigenfunctions {sn(x, r), n = 1, 2, . . . , MN}.
0
ð15Þ
ðtÞ da dt
ðtÞ þ BuðtÞ þ f ða ðtÞ; uðtÞÞ ¼ Aa
ðtÞ ðtÞ ¼ C a y
-2
0
5
10
15
20
25
The spray operating locations
Fig. 2. The coolant flux of the spray at one moment.
0.3
Thermal Crown(mm)
where kb < MN. fx1 ðx; rÞ; x2 ðx; rÞ; . . . ; xkb ðx; rÞg and {s1(x, r), s2 (x, r), . . . , sMN(x, r)} denote new spatial basis functions and eigenfunctions, respectively. Basis function transform matrix Rb can be obtained from the linear part of PDEs. Let (A, B, C) be a MNth order stable state realization of corresponding linear time-invariant (LTI) system of (12). The LTI system has unique symmetric positive define controllability gramian P and observability gramian Q, which have full rank. Let P = GGT, Q = HHT be square root decompositions and defining GHT = WRVT as a singular value decomposition, then using the MATLAB style colon notation, the transform matrix Rb = [GWR1/2](:, 1:k). Because of the balanced truncation model reduction, the obtained matrix Rb is of column-orthogonality. Thus, RTb Rb ¼ Ikb . Based on the time/space separation and Galerkin method using the new spatial basis functions, a lower-dimensional dynamical system of (4) can be derived as follows:
(
4
2
fx1 ðx; rÞ; x2 ðx; rÞ; . . . ; xkb ðx; rÞg ¼ fs1 ðx; rÞ; s2 ðx; rÞ; . . . ; sMN ðx; rÞgRb
6
0.28 0.26 0.24 0.22 0.2 70 60 50
ð16Þ
ðtÞ ¼ ½a 1 ðtÞ; a 2 ðtÞ; . . . ; a kb ðtÞT ; y ðtÞ ¼ ½y ðx1 ; tÞ; y ðx2 ; tÞ; . . . ; where a T T T ðtÞ; uðtÞÞ ¼ ½f 1 ða ðtÞ; uðtÞÞ; ðxL ; tÞ ; A ¼ Rb ARb ; B ¼ Rb B; C ¼ CRb ; f ða y
40 30
Locations
20 10 0
0
5
10
15
20
25
30
35
Sampling(s)
Fig. 3. The data of thermal crown for testing.
40
45
50
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R R f 2 ða ðtÞ; uðtÞÞ; . . . ; f kb ða ðtÞ; uðtÞÞT and f i ða ðtÞ; uðtÞÞ ¼ 0R 0l FðTÞxi ðx; rÞ dxdr denotes the nominal nonlinearities. For practical implementation, a discrete-time model is often used. The previous reduced model (16) based on the new spatial basis functions is now discretized by Euler forward formula as follows:
ðk þ 1Þ ¼ ðI þ DtAÞa ðkÞ þ DtBuðkÞ þ Dtf ða ðkÞ; uðkÞÞ a ðkÞ ðkÞ ¼ DtC a y
ð17Þ
with Dt being the sampling time. For simplicity, one can replace ðkÞ; y ðkÞ with a(k), y(k) and the following can be derived as a
aðk þ 1Þ ¼ A0 aðkÞ þ B0 uðkÞ þ f ðaðkÞ; uðkÞÞ
ð18Þ
yðkÞ ¼ C 0 aðkÞ
with ðkÞ; uðkÞÞ. A0 ¼ I þ DtA; B0 ¼ DtB; C 0 ¼ DtC; f ðaðkÞ; uðkÞÞ ¼ Dtf ða A hybrid intelligent discrete system can be used to model the nonlinear dynamics (18). Using the production data from a 4-high mill of aluminum alloy hot rolling in China, a feedforward neural network is trained to identify the nominal nonlinear terms. The most often used neural networks [20] include the radial basis function (RBF) networks [21], back propagation (BP) neural networks [22], among others. The present study employs a feedforward BP neural network to construct a low-dimensional substitute model
The prediction of thermal crown(mm)
0.3 0.28 0.26 0.24 0.22 0.2 70 60
50 40
30 20
Locations
10
0
5
0
50 40 45 30 35 25 20 Sampling time(s) 10 15
Fig. 4. The approximate thermal crown of 5-order hybrid intelligent models.
Thermal crown
0.28 0.26 0.24 0.22
Thermal crown
35s thermal crown prediction Production data Predicted data
0.28 0.26 0.24 0.22
0
0.2
0.4
0.6
0.8 1 1.2 1.4 Length of work roll
1.6
1.8
2
0
25s thermal crown prediction
0.3
0.26 0.24
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Length of work roll
1.6
0.2
0.4
1.8
2
0.6
0.8 1 1.2 1.4 Length of work roll
1.6
1.8
2
45s thermal crown prediction
0.3 Production data Predicted data
0.28
0.22
The control of the thermal crown relies on varying the fluxes of coolant applied to work rolls in hot rolling processes. The predictive control technique uses the model to repeatedly foretell the effects of applying different amounts and distributions of fluxes in a multispray cooling system. To our knowledge, this is the first work which presents a system to control the thermal crown of the work rolls by varying coolant fluxes in real time in hot rolling processes. To evaluate the proposed modeling methodology for the prediction of thermal crown of the work rolls, the production data used for identification (19) were measured from an aluminum company in China, where the aluminum alloy hot rolling process is shown in Fig. 1, and its related parameters used in the modeling methodology are shown in Table 1. The number of spray operating locations for the control of the thermal expansion is 24, the coolant flux applied at each moment is measured using flow meters and set to be 0–10. The distribution of the coolant flux along the axial of the work roll at one moment is shown in Fig. 2. It is well known that the measurement of the thermal crown of the work rolls online is impossible. The paper belongs to the theoretical research for the low-dimensional prediction modeling, and the data of thermal crown used for training the (19) were collected from the monitoring system of a 4-high roll, which were calculated by the finite-difference method based on the measured temperature at the surface of the work roll. A total of 150 measurements were collected within 1 s sampling time in a hot rolling for a reel of aluminum alloy strip. And a set of 50 data is collected for testing to demonstrate the spatial–temporal performance of (19) and shown in Fig. 3. Based on the proposed method in Section 3, a 12-order spectralbased model is obtained, where the spatial basis functions at the
0.3 Production data Predicted data
ð19Þ
5. Model identification and verification
15s thermal crown prediction
0.3
^ðk þ 1Þ ¼ A0 a ^ðkÞ þ B0 uðkÞ þ NN½a ^ðkÞ; uðkÞ a ^ ^ðkÞ ¼ C 0 aðkÞ y
The hybrid neural network model can be trained offline. And it establishes the dynamical relationship between the spray of work ^ðkÞ, from which the roll u(k) and the expansion coefficients a thermal crown distribution of the work roll is developed using ^ðkÞ. ^ðkÞ ¼ C 0 a y
Thermal Crown
Thermal crown
(
for the nonlinear dynamics of spatially distributed processes. The hybrid intelligent model is expressed as follows:
Production data Predicted data
0.28 0.26 0.24 0.22 0
0.2
0.4
0.6
0.8 1 1.2 1.4 Length of work roll
Fig. 5. Thermal crown prediction at 15 s, 25 s, 35 s and 45 s.
1.6
1.8
2
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axial and radial direction of the work rolls are selected to be 4 and 3, respectively. Based on the model reduction method proposed in Section 4 and neural networks (NNs) can be trained to identify the unknown nonlinearities of (16), a 5-order hybrid intelligent models is obtained to predict the thermal crown of the work roll. A set of 100 data is used to train the hybrid neural network model, and the approximate thermal crown of the work roll based on 5-order hybrid intelligent models is shown in Fig. 4. As shown in Fig. 5, the prediction for thermal crown of work roll in compared with the production data obtained from the aluminum company at 15 s, 25 s, 35 s and 45 s, respectively. It is apparent that the proposed spectral-based intelligent model has good performance at the prediction for thermal crown of work roll in hot rolling processes. 6. Conclusions A precision on-line model is proposed for the prediction of thermal crown in hot-strip rolling processes. The heat conduction of the work rolls’ temperature is described by a nonlinear partial differential equation (PDE) in the cylindrical coordinate. Spectral methods are used to time/space separation and model reduction, and the dynamics of the heat conduction can be described by a model of high-order nonlinear ordinary differential equations (ODE) with a few unknown nonlinearities. The dimensions of the ODE system are further reduced and neural networks (NNs) are trained to identify the unknown nonlinearities. Finally, a lower-dimensional hybrid intelligent model given in state-space formulation is efficient in computation and suitable for the further application of the traditional control techniques. The predicted result for the thermal crown of work roll shows that the proposed method is indeed effective. This modeling methodology can be applied to the prediction for the thermal crown in hot rolling processes widely. Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of ‘‘A Precision On-line Model for the Prediction of Thermal Crown in Hot Rolling Processes’’. Acknowledgement This work is supported by National Natural Science Foundation of China (Grant No. 51305133). Open Fund of State Key Laboratory of Mechanical System and Vibration of China (Grant No. MSV201404), Financial Support from Natural Science Foundation of Hunan province (Grant No. 13JJB007), Project of The Education Department of Hunan Province (Grant No. 13C317), Open Fund of Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment (Grant No. E21214), Aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province are also gratefully acknowledged. Appendix A. Equivalence Proof of (4)–(5) and (2)–(3) When the solution of the system (4) with boundary conditions (5.a), (5.b) is expressed in the orthogonally decouple series (7), the residual
Z¼
@T kt 2 r T þ FðTÞ þ Uðx; R; tÞ @t qc
ðA:1Þ
is minimized with
ZZ X
Z/i ðxÞuj ðrÞdxdr ¼ 0
ðA:2Þ
where X is the domain ð0 6 x 6 l; 0 6 r 6 RÞ. The integration (A.2) becomes
Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z R Z l kt 2 ¼ r T þ FðTÞ þ Uðx; R; tÞ /i ðxÞuj ðrÞdxdr qc 0 0 0
ðA:3Þ
where FðTÞ ¼ F 0 ðTÞ þ F d ðTÞ 1 kt @T 1 dðx 0Þfb1 ¼ q þ lðTÞ þ hðTÞ þ qc r @r qc 1 1 1 þ dðx lÞfb2 dðr 0Þfb3 þ dðr RÞfb4 qc qc qc Note that (A.3) can be rewritten as
Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z R Z l Z RZ l kt 2 ¼ r T þ F d ðTÞ /i ðxÞuj ðrÞdxdr þ ðF 0 ðTÞ qc 0 0 0 0 0
þ Uðx; R; tÞÞ/i ðxÞuj ðrÞdxdr
ðA:4Þ
The integration of the first term of the right-hand side in the previous equation is
kt 2 r T þ F d ðTÞ /i ðxÞuj ðrÞdxdr qc 0 0 Z RZ l Z RZ l kt @T 2 kt @T 2 /i ðxÞuj ðrÞdxdr þ / ðxÞuj ðrÞdxdr ¼ 2 2 i 0 0 qc @x 0 0 qc @r Z RZ l F d ðTÞ/i ðxÞuj ðrÞdxdr þ
Z
R
Z l
0
0
Using the boundary conditions (5.a), (5.b), the previous equation becomes
kt 2 r T þ F d ðTÞ /i ðxÞuj ðrÞdxdr qc 0 0 Z RZ l Z RZ l duj ðrÞ kt @T d/i ðxÞ kt @T ¼ uj ðrÞdxdr /i ðxÞ dxdr dr dx 0 0 qc @x 0 0 qc @r Z RZ l F d ðTÞ/i ðxÞuj ðrÞdxdr þ 0 0 Z RZ l du ðrÞ kt @T d/i ðxÞ @T dxdr ¼ uj ðrÞ þ /i ðxÞ j dr dx @r 0 0 qc @x Z RZ l þ F d ðTÞ/i ðxÞuj ðrÞdxdr
Z
R
Z l
0
0
Substituting the previous equation into (A.3), one has
Z 0
R
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z RZ l du ðrÞ kt @T d/i ðxÞ @T dxdr ¼ uj ðrÞ þ /i ðxÞ j dr dx @r 0 0 qc @x Z RZ l Z RZ l þ F d ðTÞ/i ðxÞuj ðrÞdxdr þ ðF 0 ðTÞ 0
0
0
þ Uðx; R; tÞÞ/i ðxÞuj ðrÞdxdr
0
0
ðA:5Þ
On the other hand, using the solution (7) for (2) with the boundary conditions (3.a)-(3.b), the integration (A.2) gives
M. Jiang et al. / International Journal of Heat and Mass Transfer 78 (2014) 967–973
Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z R Z l kt 2 ¼ r T þ F 0 ðTÞ þ Uðx; R; tÞ /i ðxÞuj ðrÞdxdr qc 0 0
References
0
ðA:6Þ
Note also that (A.6) can be rewritten as Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr ¼ @t i
0
Z 0
þ
Z
R
Z
l
kt
q
c 0 R Z l
0
r2 T/i ðxÞuj ðrÞdxdr
ðF 0 ðTÞ þ Uðx; R; tÞÞ/i ðxÞuj ðrÞdxdr
0
The integration of the first term on the right-hand side of the previous equation is
Z
R
0
Z
l
0
kt 2 r T/i ðxÞuj ðrÞdxdr ¼ qc
Z 0
þ
R
Z
Z
R
l 0
0
Z
kt @ 2 T / ðxÞuj ðrÞdxdr qc @x2 i l 0
kt @ 2 T / ðxÞuj ðrÞdxdr qc @r2 i
Using the boundary condition (3.a), (3.b), the previous equation becomes
Z 0
R
Z 0
l
kt
qc
r2 T/i ðxÞuj ðrÞdxdr
! l Z l kt @T @T d/i ðxÞ dx uj ðrÞdr /i ðxÞ ¼ @x 0 dx 0 qc 0 @x ! R Z R Z l kt @T @T duj ðrÞ þ dr /j ðxÞdx uj ðrÞ dr @r 0 0 qc 0 @r Z RZ l du ðrÞ kt @T d/i ðxÞ @T dxdr uj ðrÞ þ /i ðxÞ j ¼ dx @r dr 0 0 qc @x Z RZ l þ F d ðTÞ/i ðxÞuj ðrÞdxdr Z
R
0
973
0
Substituting the previous equation into (A.6), one obtains the same results as (A.5). Thus, one can conclude that the (4) with boundary condition (5.a), (5.b) is equivalent to (2) with conditions (3.a), (3.b).
[1] X.M. Zhang, Z.Y. Jiang, A.K. Tieu, X.H. Liu, G.D. Wang, Numerical modeling of the thermal deformation of CVC roll in hot strip rolling, J. Mater. Process. Technol. 130–131 (2002) 219–223. [2] K.N. Shohet, N.A. Townsend, Roll bending methods of crown control in fourhigh plate mills, J. Iron Steel Inst. 11 (1968) 1088–1098. [3] W. Bald, G. Beisemann, H. Feldmann, T. Schultes, Continuously variable crown (CVC) rolling, Iron Steel Eng. 3 (1987) 32–41. [4] O. Pawelski, Arch. Eisenhuttenwes, 1968, pp. 821–827. [5] A.A. Tseng, S.X. Tong, T.C. Chen, Thermal expansion and crown evaluations in rolling processes, Mater. Des. 17 (1996) 193–204. [6] R.M. Guo, R. Kousha, J.H. Schunk, A semi-analytical solution of work roll thermal crown during hot rolling, in: Proceeding in 35th MWSP Conference, 1994, pp. 329–339. [7] S. Wilmotte, J. Mignon, Thermal variations of the camber of the working rolls during hot rolling, Metall. Rep. CRM 34 (1973) 17–34. [8] X.L. Zhang, J. Zhang, X.Y. Li, H.X. Li, G.C. Wei, Analysis of the thermal profile of work rolls in the hot strip rolling process, J. Univ. Sci. Technol. Beijing 11 (2004) 173–177. [9] K. Nakagawa, Heat crown of work rolls during aluminum hot rolling, Sum Light Metals Technol. Rep. 21 (1980) 45–51. [10] W.D. Bennon, Evaluation of selective coolant application for the control of work roll thermal expansion, J. Eng. Ind. Trans. ASME 107 (1985) 146–152. [11] Z.F. Guo, C.S. Li, J.Z. Xu, X.H. Liu, G.D. Wang, Analysis of temperature field and thermal crown of roll during hot rolling by simplified FEM, J. Iron. Steel Res. Int. 12 (2006) 27–30. [12] C.M. Park, J.T. Choia, H.K. Moona, G.J. Park, Thermal crown analysis of the roll in the strip casting process, J. Mater. Process. Technol. 209 (2009) 3714–3723. [13] D. Benasciutti, E. Brusa, G. Bazzaro, Finite element prediction of thermal stresses in work roll of hot rolling mills, Proc. Eng. 2 (2010) 707–716. [14] S. Serajzadeh, Effects of rolling parameters on work-roll temperature distribution in the hot rolling of steels, Int. J. Adv. Manuf. Technol. 35 (2008) 859–866. [15] B. Friedman, Partial Differential Equations, Holt, Rinehart, New York, 1976. [16] H. Deng, H.X. Li, G.R. Chen, Spectral-approximation-based intelligent modeling for distributed thermal processes, IEEE Trans. Control Syst. Technol. 13 (2005) 686–700. [17] P.D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes, Birkhauser, Boston, MA, 2001. [18] D.L. Powers, Boundary Value Problems, fourth ed., Harcourt, Orlando, FL, 1999. [19] H. Deng, M. Jiang, C.Q. Huang, New spatial basis function for model reduction of nonlinear distributed parameter systems, J. Process Control 22 (2012) 404– 411. [20] Sudipta Sikdar, Sabita Kumari, Neural network model of the profile of hotrolled strip, Int. J. Adv. Manuf. Technol. 42 (2009) 450–462. [21] H. Deng, H.X. Li, Y.H. Wu, Feekback-linearization-based neural adaptive control for unknown nonaffine nonlinear discrete-time systems, IEEE Trans. Neural Networks 9 (2008) 1615–1625. [22] H. Deng, Z. Xu, H.X. Li, A novel neural internal model control for multi-input multi-output nonlinear discrete-time processes, J. Process Control 19 (2009) 1392–1400.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A precision on-line model for the prediction of thermal crown in hot rolling processes Mian Jiang ⇑, Xuejun Li, Jigang Wu, Guangbin Wang Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment, Hunan University of Science and Technology, XiangTan 411201, China
a r t i c l e
i n f o
Article history: Received 15 September 2013 Received in revised form 17 March 2014 Accepted 19 July 2014
Keywords: Hot rolling processes Heat conduction Thermal crown Spectral method Neural networks Hybrid intelligent modeling
a b s t r a c t The tendency towards an increase in rolling speeds, which is characteristic of the development of modern sheet rolling, causes an increase requirement of accurate prediction on-line control models for the thermal crown of work rolls. In this paper, a precision on-line model is proposed for the prediction of thermal crown in hot-strip rolling processes. The heat conduction of the roll temperature can be described by a nonlinear partial differential equation (PDE) in the cylindrical coordinate. After selecting a set of proper basis functions, the spectral methods can be applied to time/space separation and model reduction, and the dynamics of the heat conduction can be described by a model of high-order nonlinear ordinary differential equations (ODE) with a few unknown nonlinearities. Using a technique for further reducing the dimensions of the ODE system, neural networks (NNs) can be trained to identify the unknown nonlinearities. The low-order predicted model of the thermal crown is given in state-space formulation and efficient in computation. The comparisons of prediction values for the thermal crown with the production data in an aluminum alloy hot rolling process show that the proposed method is effective and has high performance. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The tendency towards an increase in rolling speeds, which is characteristic of the development of modern sheet rolling, causes an increase in the instability of a number of production factors which influence the shape of the strip [1]. The flatness and profile of the strip are mainly dependent on the configuration of the roll gap across the width of the strip. The shape of the work rolls is one of the main factors affecting the shape of the roll gap, which has a significant effect on the roll gap contour. However, the shape of the work rolls changes dynamically due to the thermal deformation and wear of the rolls in continuous rolling of strip. With the exception of the work roll thermal crown expansion, the factors can be satisfactory compensated by a proper setup computation just before the rolling of each strip [2,3]. Therefore, real-time control of the uniformity of the thermal expansion of work rolls becomes the key factor in obtaining a good roll gap contour. Normally, an adequate design of the cooling system of work rolls can contribute to minimize the magnitude and shape irregularity of the thermal crown generated during the rolling process. It can keep the roll temperature and the thermal expansion within a ⇑ Corresponding author. Tel.: +86 0731 58290840; fax: +86 0731 58290848. E-mail address: [email protected] (M. Jiang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.061 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
proper range. In order to study the influence of cooling system for hot rolling processes, a good understanding of the thermal crown of rolls during hot rolling processes is critical. Due to the rapidity of the rolling process and the impossibility of measuring the thermal crown of work rolls online, an on-line model with precise computational efficiency is very important for the prediction of thermal crown. The problem of obtaining the thermal crown of work rolls is stated as a transient heat transfer problem. Heat transfer inside the work rolls is considered to occur by conduction and around the work rolls periphery by radiation and conduction (with the strip) and convection (with the water and air). And the fact that the influence of the rotation of work rolls in the heat transport phenomenon is negligible, when different methods of solution, such as analytical, experimental and numerical, have been used to predict the temperature and thermal crown profile of work rolls. An analytical solution was developed by Pawelski [4] for the heat transfer equation between work rolls and strip (roll bite region) to find the heat transfer coefficient in this region. This coefficient is a function of roll speed, scale thickness, physical properties of rolls and strip, and roll bite contact time. Tseng [5] also developed an analytical understanding of thermal expansion of work rolls to provide thermal displacements and expansion of the rolls. Guo [6] developed a semi-analytic solution of wok rolls
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thermal crown to establish a correlation between the thermal profile of the rolls and the strip shape. Because of the limitations of the analytical solution in which all the boundary conditions are not considered comprehensively, it cannot give the accurate predicted thermal crown of the work rolls. Most of the investigators preferred to use numerical and experimental solutions. A great many investigators have studied the thermal crown using the finite difference method (FDM) which is a common numerical method to solve the multidimensional heat transfer problems. Wilmotte and Mignon [7] used an axisymmetric finite difference method model to study the circumferential mean values of the roll thermal expansion. Zhang et al. [8] proposed a 2dimensional axisymmetric model developed by the finite-difference method to predict the transient temperature field and the thermal profile of the work rolls in hot strip rolling process. The calculation results were compared with the production data of a 1700 mm hot strip rolling mill, and good agreement was found. Nakagawa [9] studied the transient build-up of the thermal crown based on a three-dimensional Lagrangian finite difference model and concluded that the reduction, strip temperature, and cooling condition are three major influential parameters. Bennon [10] developed as Eulerian finite-difference scheme to predict the thermal expansion at different spray cooling patterns in cold rolling of aluminum. Zhang et al. [1] developed a finite difference model to simulate the thermal deformation of the continuously variable crown (CVC) work rolls in hot strip rolling. Generally speaking, the results of the calculation of the roll temperature using the finite difference method are in good agreement with the measured values. However, finite difference method for the calculation of the thermal crown in rolling processes will produce high-order models that are unsuitable for synthesizing implements and real-time control. There are also other published researches concerning mathematical modeling of the hot rolling processes, while numerical techniques particularly the finite element analysis have been utilized for determining the deformation behavior of work roll. Guo et al. [11] developed a simplified finite element method (FEM) to analyze the temperature field and thermal crown of the roll according to the practical boundary conditions. Park et al. [12] carried out the coupled analyses of heat transfer and deformation for casting rolling by using the finite element software MARC to examine the thermal crown. Benasciutti et al. [13] proposed a simplified numerical approach based on finite-elements to computer thermal stresses occurring in work roll of hot rolling mills, which are caused by a non-uniform temperature distribution over the work roll surface. The FEM can be used to analyze the temperature field and thermal deformation conveniently, and the results of the simulation have high precision. However, the FEM method is not efficient in computation and also produces high-order models that are unsuitable for synthesizing controller design and real-time prediction control. The present study derives an on-line model with high performance for the prediction of the thermal crown in hot rolling processes. The heat conduction of the roll temperature can be described by a nonlinear partial differential equation (PDE) in the cylindrical coordinate. After selecting a set of proper basis functions, the spectral methods can be applied to time/space separation and model reduction, and the dynamics of the heat conduction can be described by a model of high-order nonlinear ordinary differential equations (ODE) with a few unknown nonlinearities. Using a technique for further reducing the dimensions of the ODE system, neural networks (NNs) can be trained to identify the unknown nonlinearities using the production data of the rolling mill, and a lower-dimensional hybrid intelligent model of the thermal crown is given in state-space formulation, which is efficient in computation and suitable for the further application of the traditional
control techniques. The comparisons of prediction for the thermal crown with the production data in an aluminum alloy hot rolling process show that the proposed method is effective and has high performance. 2. Fundamental dynamics of the heat transfer of work rolls The thermal expansion is one of the important factors affecting the roll gap profile. The work roll periodically contacts with hot strips and cooling liquid and as a result the surface temperature changes drastically. Heat transfer in the work roll is considered to occur by conduction and around the work roll periphery by radiation and conduction with the strip and convection with the water and air [14]. The work roll, rotated at high speed, is considered as a cylinder and the partial differential equation governing the heat conduction in a cylindrical coordinate can be expressed as:
qc
@T @ 2 T 1 @T 1 @ 2 T @ 2 T ¼ kt þ þ þ @t @r 2 r @r r 2 @w2 @x2
! þ q þ lðTÞ þ gðTÞ
þ hðTÞ
ð1Þ
where r and w are the radial and circumferential direction, respectively. Because the influence of the work roll is second-order magnitude, the variation of temperature along the circumferential direction of the work roll can be ignored, i.e. o2T/ow2 = 0. Eq. (1) can then be transferred into
@T @ 2 T 1 @T @ 2 T qc ¼ kt þ þ @t @r 2 r @r @x2
! þ q þ lðTÞ þ gðTÞ þ hðTÞ
ð2Þ
where T = T(x, r, t) denotes the temperature of the work roll, t is the time variable. r e [0, R] and x e [0, l] are cylindrical coordinates in radial and axial direction. kt is the thermal conductivity coefficient, q, c are the density and specific heat, respectively. The rest of variables are introduced as follows: (1) q is the heat generation rate. (2) l(T) denotes the quantity of heat conduction between work rolls and the strip in unit time, which is a function of the surface temperature of the work rolls. (3) g(T) is the quantity of heat convection between work rolls and the coolant in unit time, which is also a function of the surface temperature of the work rolls. (4) h(T) is the quantity of heat convection between work rolls and the air in unit time. The surface around the work roll periphery by radiation and conduction with the strip and convection with the water and air, and the boundary conditions are not completely known. As it is difficult to obtain reasonable boundary conditions using only physical insights, for simplicity one can set the boundary conditions to be unknown nonlinear functions of the boundary temperature, the space coordinates (x, r) of the work roll and the ambient temperature TE as follows:
kt
@T ¼ fb1 ; @x x¼0
kt
@T ¼ fb2 @x x¼l
ð3:aÞ
kt
@T ¼ fb3 ; @r r¼0
kt
@T ¼ fb4 @r r¼R
ð3:bÞ
where
fb1 ¼ f1 ðx; r; T; T E Þjx¼0 ;
f b2 ¼ f2 ðx; r; T; T E Þjx¼l
fb3 ¼ f3 ðx; r; T; T E Þjr¼0 ;
f b4 ¼ f4 ðx; r; T; T E Þjr¼R
With fb1 fb4 being unknown nonlinear functions.
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Thus, a complete temperature field model of the work rolls is described by a parabolic partial differential (2) with the boundary conditions (3.a), (3.b). To our knowledge, it is the first work which presents a lowdimensional approximate system to control the thermal-crown of the work roll of a hot strip mill by varying coolant fluxes in real time. A parabolic PDE system typically involves spatial differential operators with eigenspectral that can be partitioned into a finitedimensional (slow) and an infinite-dimensional (fast) complements [15,16]. It is very convenient to use the eigenfunctions of a spatial differential operator to derive a low order ODE system for such a parabolic PDE system if the boundary conditions are homogeneous. However, the boundary conditions (3.a), (3.b) include unknown nonlinear functions, therefore a direct derivation of eigenfunctions for the operator is impossible. For the sake of derivation of eigenfunctions for the PDE model, the Eqs. (2) and (3) are transformed into the following equivalent model with homogeneous boundary conditions, while the proof is given in the Appendix A.
@T kt 2 ¼ r T þ FðTÞ þ Uðx; R; tÞ @t qc @T @T ¼ 0; ¼0 @x x¼0 @x x¼l @T @r
@T @r
¼ 0; r¼0
ð4Þ
¼0
ð5:bÞ
1
qc
ðq þ lðTÞ þ hðTÞÞ þ
kt 1 @T
qc r @r
dðx 0Þ
1
qc
fb1 þ dðx lÞ
1 1 1 f dðr 0Þ fb3 þ dðr RÞ fb4 qc b2 qc qc is the nonlinear term with some unknown nonlinearities; d() is the Dirac delta function; the U(x, R, t) denotes the spray input of the system at the surface of the work roll; x e [0, l] r e [0, R], t e [0, 1). Based on the axisymmetric temperature distribution, the thermal expansion of rolls is calculated by elastic analysis. The thermal expansion induced displacement y(x, t) at the roll surface, r = R, is
yðx; tÞ ¼ ð1 þ v Þ b R T AV ðx; tÞ T AV ðx; tÞ ¼
Z
R
2prTðx; r; tÞdr=pR2
ð6:aÞ ð6:bÞ
0
where m is Poisson’s ratio of the roll, b is linear expansion coefficient of roll; R is the radius of roll; and TAV(x, t) denotes the average temperature on the section of the work roll at x and t. Using the temperature field of the work roll, the thermal crown of the work roll can be calculated by numerical integration of the Eqs. (6.a) and (6.b).
The new boundary conditions (5.a) and (5.b) are homogeneous. To further reduce the PDE model (4) to a set of finitely many nonlinear ODE equations, the spectral methods [16,17] can be applied as follows. 3.1. Separation of time and spatial variables According to the theorem of separation of variables [17], the solution of system (4) can be expressed in the following time– space decouple form:
i¼1 j¼1
ð8:aÞ
j ¼ 0; 1; 2; . . .
ð8:bÞ
and the corresponding eigenvalues:
ni ¼
2 kt ip ; qc l
fj ¼
2 kt jp qc R
ð9Þ
It is interesting to note that the eigenspectrum of nonlinear PDEs (4) with boundary conditions (5.a), (5.b) can be separated into a finite-dimensional (slow) and an infinite-dimensional stable (fast) complements. This implies that the dynamical behavior of such a system can be approximately described by a finite-dimensional ODE system that captures the dynamics of the dominant (slow) modes of the PDE system. Let the number of the selected eigenfunctions (8.a), (8.b) is M, N respectively. The corresponding eigenvalues of the spatial basis functions /i(x)uj(r) set to be:
2 2 ! ip jp þ l R
k r ij ¼ t qc
ð10Þ
Thus the eigenvalues is rearranged in order of magnitude and then denoted as a new sequence of rn with sn(x, r) its corresponding eigenfunction, as follows:
sn ðx; rÞ ¼ f/i ðxÞuj ðrÞg; an ðtÞ ¼ faij ðtÞg where i ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; N; n ¼ ðN 1Þi þ j 2 f1; 2; . . . ; MNg Then, the temperature T(x, r, t) of the work roll can be approximated by the time/space separation based on the sn(x, r) and the expansion coefficients an(t) as follows:
Tðx; r; tÞ
ð7Þ
M X N X
aij ðtÞ/i ðxÞuj ðrÞ ¼
MN X an ðtÞsn ðx; rÞ
ð11Þ
n¼1
i¼1 j¼1
3.2. Projection Due to the orthogonality of the basis functions sn(x, r), the insertion of the expansion (11) into (4) and the application of the Galerkin method[17] produces a system of MN first-order ordinary differential equations for the evolution of the expansion coefficients.
(
daðtÞ dt
¼ AaðtÞ þ BuðtÞ þ f ðaðtÞ; uðtÞÞ
yðtÞ ¼ CaðtÞ where
3. Spectral methods for the thermal crown
1 X 1 X Tðx; r; tÞ ¼ /i ðxÞuj ðrÞaij ðtÞ
i ¼ 0; 1; 2; . . .
rn ¼ fr ij g with r1 > r2 > > rn >
2
@ 1 where r2 ¼ @r@ 2 þ @x 2 ; Uðx; R; tÞ ¼ qc gðTÞ
FðTÞ ¼
rffiffiffi 2 ipx cos /i ðxÞ ¼ l l rffiffiffi 2 jpr cos uj ðrÞ ¼ R R
ð5:aÞ
r¼R
2
Thus, the eigenvalue problem for operator r2 in (4) can be solved analytically [18], as the eigenfunctions are selected as follows:
ð12Þ
aðtÞ ¼ ½a1 ðtÞ; a2 ðtÞ; . . . ; aMN ðtÞT ; yðtÞ ¼ ½yðx1 ; tÞ; yðx2 ; tÞ; . . . ;
yðxL ; tÞT ; uðtÞ ¼ ½u1 ðtÞ; u2 ðtÞ; . . . ; uK ðtÞT And {x1, x2, . . ., xL} denotes the axial locations of the work rolls, y(t) is the thermal crown at the axial location {x1, x2, . . ., xL}. u(t) = [u1(t), u2(t), . . ., uK(t)]T is the input of the spray considered in K intervals at the surface of work rolls. f(a(t), u(t)) is the nonlinear term with some unknown nonlinearities. The derivations of the matrix A, B are given:
A ¼ diagðr1 ; r2 ; ; rMN Þ 2 B1;1 B1;2 B1;K 6 B B2;2 B2;K 6 2;1 B¼6 4 BMN;1
BMN;2
BMN;K
3 7 7 7 5
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M. Jiang et al. / International Journal of Heat and Mass Transfer 78 (2014) 967–973
R 0:09þ0:08k
where Bn;k ¼ 0:09þ0:08ðk1Þ sn ðx; RÞdx, note that the we can find a (i, j) to satisfy n = (N 1)i + j and the [0.09 + 0.08(k 1), 0.09 + 0.08k] denotes the kth interval of the input of spray at the surface of the work roll, then
Bn;k
rffiffiffi l 1 ¼2 R ip
ip ip cosðjpÞ sin ð0:09 þ 0:08kÞ sin ð0:01 þ 0:08kÞ l l ð13Þ
The matrix C is calculated using the average temperature TAV(x, t) in the Eq. (6.b) and the approximation (11).
2
3
C 1;1
C 1;2
C 1;MN
6C 6 2;1 yðtÞ ¼ 6 4
C 2;2
C 2;MN 7 7 7 aðtÞ 5
C L;1
C L;2
Fig. 1. The aluminum alloy hot rolling process.
ð14Þ
C L;MN
RR
Table 1 Parameters used in the modeling methodology.
2
where C k;n ¼ ð1 þ v ÞbR 0 2pr sn ðxk ; rÞdr=pR , and note that we can find a (i,j) to satisfy n = (N 1)i + j, then C k;n ¼ C k;ðN1Þiþj ¼ ð1 þ v Þ qffiffi bR/i ðxk Þ 2R ðjp2Þ2 ðcos jp 1Þ i ¼ 1; . . . ; M; j ¼ 1; . . . ; N; k ¼ 1; . . . ; L.
Parameters
Value
l R
2.1 m 0.375 m 7800 kg/m3 586.15 J/(kg K) 34.88 w/(m K) 71 24
q c kt L K
4. Further model reduction and hybrid spectral/neural modeling
12
10
8
The coolant flux
In the above discussions, the truncation limits M, N are assumed to be sufficiently large such that the dynamical system of Eq. (12) captures the long-term behaviors of the PDE of Eq. (4) with sufficient accuracy. However, the order of the ODE system is still high, and its application for numerical implementation and practical control is difficult because of limited computation capacity and complex controller design. In the previous work [19], new spatial basis functions are derived to reduce the order of the dynamical system of the nonlinear PDEs, and model reduction performance of new spatial basis functions generated by balanced truncation has been proofed. Each new spatial basis function is a linear combination of eigenfunctions {sn(x, r), n = 1, 2, . . . , MN}.
0
ð15Þ
ðtÞ da dt
ðtÞ þ BuðtÞ þ f ða ðtÞ; uðtÞÞ ¼ Aa
ðtÞ ðtÞ ¼ C a y
-2
0
5
10
15
20
25
The spray operating locations
Fig. 2. The coolant flux of the spray at one moment.
0.3
Thermal Crown(mm)
where kb < MN. fx1 ðx; rÞ; x2 ðx; rÞ; . . . ; xkb ðx; rÞg and {s1(x, r), s2 (x, r), . . . , sMN(x, r)} denote new spatial basis functions and eigenfunctions, respectively. Basis function transform matrix Rb can be obtained from the linear part of PDEs. Let (A, B, C) be a MNth order stable state realization of corresponding linear time-invariant (LTI) system of (12). The LTI system has unique symmetric positive define controllability gramian P and observability gramian Q, which have full rank. Let P = GGT, Q = HHT be square root decompositions and defining GHT = WRVT as a singular value decomposition, then using the MATLAB style colon notation, the transform matrix Rb = [GWR1/2](:, 1:k). Because of the balanced truncation model reduction, the obtained matrix Rb is of column-orthogonality. Thus, RTb Rb ¼ Ikb . Based on the time/space separation and Galerkin method using the new spatial basis functions, a lower-dimensional dynamical system of (4) can be derived as follows:
(
4
2
fx1 ðx; rÞ; x2 ðx; rÞ; . . . ; xkb ðx; rÞg ¼ fs1 ðx; rÞ; s2 ðx; rÞ; . . . ; sMN ðx; rÞgRb
6
0.28 0.26 0.24 0.22 0.2 70 60 50
ð16Þ
ðtÞ ¼ ½a 1 ðtÞ; a 2 ðtÞ; . . . ; a kb ðtÞT ; y ðtÞ ¼ ½y ðx1 ; tÞ; y ðx2 ; tÞ; . . . ; where a T T T ðtÞ; uðtÞÞ ¼ ½f 1 ða ðtÞ; uðtÞÞ; ðxL ; tÞ ; A ¼ Rb ARb ; B ¼ Rb B; C ¼ CRb ; f ða y
40 30
Locations
20 10 0
0
5
10
15
20
25
30
35
Sampling(s)
Fig. 3. The data of thermal crown for testing.
40
45
50
971
M. Jiang et al. / International Journal of Heat and Mass Transfer 78 (2014) 967–973
R R f 2 ða ðtÞ; uðtÞÞ; . . . ; f kb ða ðtÞ; uðtÞÞT and f i ða ðtÞ; uðtÞÞ ¼ 0R 0l FðTÞxi ðx; rÞ dxdr denotes the nominal nonlinearities. For practical implementation, a discrete-time model is often used. The previous reduced model (16) based on the new spatial basis functions is now discretized by Euler forward formula as follows:
ðk þ 1Þ ¼ ðI þ DtAÞa ðkÞ þ DtBuðkÞ þ Dtf ða ðkÞ; uðkÞÞ a ðkÞ ðkÞ ¼ DtC a y
ð17Þ
with Dt being the sampling time. For simplicity, one can replace ðkÞ; y ðkÞ with a(k), y(k) and the following can be derived as a
aðk þ 1Þ ¼ A0 aðkÞ þ B0 uðkÞ þ f ðaðkÞ; uðkÞÞ
ð18Þ
yðkÞ ¼ C 0 aðkÞ
with ðkÞ; uðkÞÞ. A0 ¼ I þ DtA; B0 ¼ DtB; C 0 ¼ DtC; f ðaðkÞ; uðkÞÞ ¼ Dtf ða A hybrid intelligent discrete system can be used to model the nonlinear dynamics (18). Using the production data from a 4-high mill of aluminum alloy hot rolling in China, a feedforward neural network is trained to identify the nominal nonlinear terms. The most often used neural networks [20] include the radial basis function (RBF) networks [21], back propagation (BP) neural networks [22], among others. The present study employs a feedforward BP neural network to construct a low-dimensional substitute model
The prediction of thermal crown(mm)
0.3 0.28 0.26 0.24 0.22 0.2 70 60
50 40
30 20
Locations
10
0
5
0
50 40 45 30 35 25 20 Sampling time(s) 10 15
Fig. 4. The approximate thermal crown of 5-order hybrid intelligent models.
Thermal crown
0.28 0.26 0.24 0.22
Thermal crown
35s thermal crown prediction Production data Predicted data
0.28 0.26 0.24 0.22
0
0.2
0.4
0.6
0.8 1 1.2 1.4 Length of work roll
1.6
1.8
2
0
25s thermal crown prediction
0.3
0.26 0.24
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Length of work roll
1.6
0.2
0.4
1.8
2
0.6
0.8 1 1.2 1.4 Length of work roll
1.6
1.8
2
45s thermal crown prediction
0.3 Production data Predicted data
0.28
0.22
The control of the thermal crown relies on varying the fluxes of coolant applied to work rolls in hot rolling processes. The predictive control technique uses the model to repeatedly foretell the effects of applying different amounts and distributions of fluxes in a multispray cooling system. To our knowledge, this is the first work which presents a system to control the thermal crown of the work rolls by varying coolant fluxes in real time in hot rolling processes. To evaluate the proposed modeling methodology for the prediction of thermal crown of the work rolls, the production data used for identification (19) were measured from an aluminum company in China, where the aluminum alloy hot rolling process is shown in Fig. 1, and its related parameters used in the modeling methodology are shown in Table 1. The number of spray operating locations for the control of the thermal expansion is 24, the coolant flux applied at each moment is measured using flow meters and set to be 0–10. The distribution of the coolant flux along the axial of the work roll at one moment is shown in Fig. 2. It is well known that the measurement of the thermal crown of the work rolls online is impossible. The paper belongs to the theoretical research for the low-dimensional prediction modeling, and the data of thermal crown used for training the (19) were collected from the monitoring system of a 4-high roll, which were calculated by the finite-difference method based on the measured temperature at the surface of the work roll. A total of 150 measurements were collected within 1 s sampling time in a hot rolling for a reel of aluminum alloy strip. And a set of 50 data is collected for testing to demonstrate the spatial–temporal performance of (19) and shown in Fig. 3. Based on the proposed method in Section 3, a 12-order spectralbased model is obtained, where the spatial basis functions at the
0.3 Production data Predicted data
ð19Þ
5. Model identification and verification
15s thermal crown prediction
0.3
^ðk þ 1Þ ¼ A0 a ^ðkÞ þ B0 uðkÞ þ NN½a ^ðkÞ; uðkÞ a ^ ^ðkÞ ¼ C 0 aðkÞ y
The hybrid neural network model can be trained offline. And it establishes the dynamical relationship between the spray of work ^ðkÞ, from which the roll u(k) and the expansion coefficients a thermal crown distribution of the work roll is developed using ^ðkÞ. ^ðkÞ ¼ C 0 a y
Thermal Crown
Thermal crown
(
for the nonlinear dynamics of spatially distributed processes. The hybrid intelligent model is expressed as follows:
Production data Predicted data
0.28 0.26 0.24 0.22 0
0.2
0.4
0.6
0.8 1 1.2 1.4 Length of work roll
Fig. 5. Thermal crown prediction at 15 s, 25 s, 35 s and 45 s.
1.6
1.8
2
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M. Jiang et al. / International Journal of Heat and Mass Transfer 78 (2014) 967–973
axial and radial direction of the work rolls are selected to be 4 and 3, respectively. Based on the model reduction method proposed in Section 4 and neural networks (NNs) can be trained to identify the unknown nonlinearities of (16), a 5-order hybrid intelligent models is obtained to predict the thermal crown of the work roll. A set of 100 data is used to train the hybrid neural network model, and the approximate thermal crown of the work roll based on 5-order hybrid intelligent models is shown in Fig. 4. As shown in Fig. 5, the prediction for thermal crown of work roll in compared with the production data obtained from the aluminum company at 15 s, 25 s, 35 s and 45 s, respectively. It is apparent that the proposed spectral-based intelligent model has good performance at the prediction for thermal crown of work roll in hot rolling processes. 6. Conclusions A precision on-line model is proposed for the prediction of thermal crown in hot-strip rolling processes. The heat conduction of the work rolls’ temperature is described by a nonlinear partial differential equation (PDE) in the cylindrical coordinate. Spectral methods are used to time/space separation and model reduction, and the dynamics of the heat conduction can be described by a model of high-order nonlinear ordinary differential equations (ODE) with a few unknown nonlinearities. The dimensions of the ODE system are further reduced and neural networks (NNs) are trained to identify the unknown nonlinearities. Finally, a lower-dimensional hybrid intelligent model given in state-space formulation is efficient in computation and suitable for the further application of the traditional control techniques. The predicted result for the thermal crown of work roll shows that the proposed method is indeed effective. This modeling methodology can be applied to the prediction for the thermal crown in hot rolling processes widely. Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of ‘‘A Precision On-line Model for the Prediction of Thermal Crown in Hot Rolling Processes’’. Acknowledgement This work is supported by National Natural Science Foundation of China (Grant No. 51305133). Open Fund of State Key Laboratory of Mechanical System and Vibration of China (Grant No. MSV201404), Financial Support from Natural Science Foundation of Hunan province (Grant No. 13JJB007), Project of The Education Department of Hunan Province (Grant No. 13C317), Open Fund of Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment (Grant No. E21214), Aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province are also gratefully acknowledged. Appendix A. Equivalence Proof of (4)–(5) and (2)–(3) When the solution of the system (4) with boundary conditions (5.a), (5.b) is expressed in the orthogonally decouple series (7), the residual
Z¼
@T kt 2 r T þ FðTÞ þ Uðx; R; tÞ @t qc
ðA:1Þ
is minimized with
ZZ X
Z/i ðxÞuj ðrÞdxdr ¼ 0
ðA:2Þ
where X is the domain ð0 6 x 6 l; 0 6 r 6 RÞ. The integration (A.2) becomes
Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z R Z l kt 2 ¼ r T þ FðTÞ þ Uðx; R; tÞ /i ðxÞuj ðrÞdxdr qc 0 0 0
ðA:3Þ
where FðTÞ ¼ F 0 ðTÞ þ F d ðTÞ 1 kt @T 1 dðx 0Þfb1 ¼ q þ lðTÞ þ hðTÞ þ qc r @r qc 1 1 1 þ dðx lÞfb2 dðr 0Þfb3 þ dðr RÞfb4 qc qc qc Note that (A.3) can be rewritten as
Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z R Z l Z RZ l kt 2 ¼ r T þ F d ðTÞ /i ðxÞuj ðrÞdxdr þ ðF 0 ðTÞ qc 0 0 0 0 0
þ Uðx; R; tÞÞ/i ðxÞuj ðrÞdxdr
ðA:4Þ
The integration of the first term of the right-hand side in the previous equation is
kt 2 r T þ F d ðTÞ /i ðxÞuj ðrÞdxdr qc 0 0 Z RZ l Z RZ l kt @T 2 kt @T 2 /i ðxÞuj ðrÞdxdr þ / ðxÞuj ðrÞdxdr ¼ 2 2 i 0 0 qc @x 0 0 qc @r Z RZ l F d ðTÞ/i ðxÞuj ðrÞdxdr þ
Z
R
Z l
0
0
Using the boundary conditions (5.a), (5.b), the previous equation becomes
kt 2 r T þ F d ðTÞ /i ðxÞuj ðrÞdxdr qc 0 0 Z RZ l Z RZ l duj ðrÞ kt @T d/i ðxÞ kt @T ¼ uj ðrÞdxdr /i ðxÞ dxdr dr dx 0 0 qc @x 0 0 qc @r Z RZ l F d ðTÞ/i ðxÞuj ðrÞdxdr þ 0 0 Z RZ l du ðrÞ kt @T d/i ðxÞ @T dxdr ¼ uj ðrÞ þ /i ðxÞ j dr dx @r 0 0 qc @x Z RZ l þ F d ðTÞ/i ðxÞuj ðrÞdxdr
Z
R
Z l
0
0
Substituting the previous equation into (A.3), one has
Z 0
R
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z RZ l du ðrÞ kt @T d/i ðxÞ @T dxdr ¼ uj ðrÞ þ /i ðxÞ j dr dx @r 0 0 qc @x Z RZ l Z RZ l þ F d ðTÞ/i ðxÞuj ðrÞdxdr þ ðF 0 ðTÞ 0
0
0
þ Uðx; R; tÞÞ/i ðxÞuj ðrÞdxdr
0
0
ðA:5Þ
On the other hand, using the solution (7) for (2) with the boundary conditions (3.a)-(3.b), the integration (A.2) gives
M. Jiang et al. / International Journal of Heat and Mass Transfer 78 (2014) 967–973
Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr @t i Z R Z l kt 2 ¼ r T þ F 0 ðTÞ þ Uðx; R; tÞ /i ðxÞuj ðrÞdxdr qc 0 0
References
0
ðA:6Þ
Note also that (A.6) can be rewritten as Z
R
0
Z
l
@T / ðxÞuj ðrÞdxdr ¼ @t i
0
Z 0
þ
Z
R
Z
l
kt
q
c 0 R Z l
0
r2 T/i ðxÞuj ðrÞdxdr
ðF 0 ðTÞ þ Uðx; R; tÞÞ/i ðxÞuj ðrÞdxdr
0
The integration of the first term on the right-hand side of the previous equation is
Z
R
0
Z
l
0
kt 2 r T/i ðxÞuj ðrÞdxdr ¼ qc
Z 0
þ
R
Z
Z
R
l 0
0
Z
kt @ 2 T / ðxÞuj ðrÞdxdr qc @x2 i l 0
kt @ 2 T / ðxÞuj ðrÞdxdr qc @r2 i
Using the boundary condition (3.a), (3.b), the previous equation becomes
Z 0
R
Z 0
l
kt
qc
r2 T/i ðxÞuj ðrÞdxdr
! l Z l kt @T @T d/i ðxÞ dx uj ðrÞdr /i ðxÞ ¼ @x 0 dx 0 qc 0 @x ! R Z R Z l kt @T @T duj ðrÞ þ dr /j ðxÞdx uj ðrÞ dr @r 0 0 qc 0 @r Z RZ l du ðrÞ kt @T d/i ðxÞ @T dxdr uj ðrÞ þ /i ðxÞ j ¼ dx @r dr 0 0 qc @x Z RZ l þ F d ðTÞ/i ðxÞuj ðrÞdxdr Z
R
0
973
0
Substituting the previous equation into (A.6), one obtains the same results as (A.5). Thus, one can conclude that the (4) with boundary condition (5.a), (5.b) is equivalent to (2) with conditions (3.a), (3.b).
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