A temperature predicting model for manufacturing processes requiring coiling

A Temperature Predicting Model for Manufacturing Processes Requiring Coiling

Nando Troyani

Luis Montano

Universidad de Oriente

Departamento de Mécanica

Puerto de La Cruz Venezuela

Abstract

A model for predicting temperature evolution for automatic controling systems in manufacturing processes requiring the coiling of bars in the transfer table is presented. Although the method is of a general nature, the presentation in this work refers to the manufacturing of steel plates in hot rolling mills. The predicting strategy is based on a mathematical model of the evolution of temperature in a coiling and uncoiling bar and is presented in the form of a parabolic partial differential equation for a shape changing domain. The mathematical model is solved numerically by a space discretization via geometrically adaptive finite elements which accomodate the change in shape of the domain, using a computationally novel treatment of the resulting thermal contact problem due to coiling. Time is discretized according to a Crank-Nicolson scheme. Since the actual physical process takes less time than the time required by the process controlling computer to solve the full mathematical model, a special predictive device was developed, in the form of a set of least squares polynomials, based on the off-line numerical solution of the mathematical model.

Keywords: Temperature Prediction, Thermal Contact, Finite Elements, Metal coiling

Work partially supported by Consejo de Investigación, Universidad de Oriente, Venezuela and CEGELEC Automation Inc., Pittsburgh, PA, USA

A streamlined version of this work appeared in the Proceedings of the 3d International Conference on Adaptive Computing in Design and Manufacturing. Plymouth, UK, 1998.

Introduction

In order to minimize heat loss to the environment in the transfer table (the stage between rougher rolling stands and finishing rolling stands in a hot mill) and for reasons of energy consumption, metallurgical uniformity and rollability, a variety of heat loss reducing strategies have been employed. One of the utilized strategies is provided by the Coilbox (Smith,1981) a Stelco Inc. patent, which has been succesfully used in some industrial instalations worldwide. By means of this device the hot bar coming out of the rougher is coiled and uncoiled, Fig. 1, prior to processing it in the finisher, resulting in a substantial heat loss rate reduction, due to the fact that once coiled the bar surfaces no longer "see the environmental temperature", except for the edges, and as a result the radiating heat to the environment is practically eliminated, and the convected heat is substantially reduced. To preset the parameters of the finishing rolling stands in order to make the finishing process more accurate and efficient it is necessary to have an estimate of the temperature distribution along the bar prior to rolling it.

Fig. 1 Finite element discretization and coordinate sytems in a partially coiled bar.

Here, a description of the mathematical model of the process is given. The reader is referred to Troyani (1996) for additional details, including the variational formulation of the problem and the corresponding temperature finite element representation and solution. Since the bar curves and straightens out during coiling and uncoiling elements with geometric adaptability were used, such as the 8-node isoparametric element (Becker, Carey and Oden, 1981). Based on the numerical solution of the mathematical model of the heat transfer process, an on-line fast temperature prediction system (OFTPS) was developed and is presented in this work. The OFTPS is a necessary device since the actual physical process of coiling and uncoiling takes less time than the numerical solution of the mathematical model for the particular computers used to run the software for automatic control of hot mills.

The model accounts for the different speeds of the Coilbox operation during both coiling and uncoiling, Fig. 2. This is considered a critical issue since this aspect of the operation directly affects the time of exposure of the different segments of hot bars to the environment, and as a result it affects the heat transfer process.

Fig. 2 Typical bar speed schedule according to the coiling device operating procedures

The mathematical model for the problem is stated in section 2. It consists of the non-linear parabolic partial differential equation of heat conduction together with the non-linear convective and non-linear radiative boundary conditions, initial conditions in the form of an initial distribution of temperature, as well as the equations which describe the motion and change of shape of the bar as it coils and uncoils. In section 3 the equations describing the motion of the bar are given together with a brief description of the operation of a coiling device in terms of its operational speeds.

In order to accomodate the shape changing domain of the bar the model is solved using geometrically adaptive finite elements for the space discretization. Time discretization was achieved via a Crank-Nicolson scheme (implicit parameter value of 0.5). The solution proceeds in a forward time stepping manner with a variable time step in order to comply with the coiling process variable speed schedule. The problem was treated as a first approach as two dimensional.

Special attention was given to the issue of continuosly changing boundary conditions due to thermal contact. The novel numerical treatment given to the thermal contact problem which arises as a result of adjacent coiling wraps coming into contact is described in section 4.

The numerical results for a 0.0254 m thick, 85.34 m long steel bar as well as a full decription of the OFTPS are given in sections 5 and 6 respectivelly.

The Mathematical Model

The equation governing the temperature evolution in the bar is the well known non-linear parabolic partial differential equation of heat transfer by conduction (Ozisik, 1993) given by:

(1)

Nomenclature

c ( T ) = temperature dependent heat capacity, [ J / kgº K]

C = capacitance matrix containing the heat capacity and density dependency

f = thermal load vector in the global system

h = film heat transfer coefficient, [W / m²º K]

k ( T ) = temperature dependent thermal conductivity, [W / mº K]

K = stiffness matrix for the problem

N = element number of nodes

R = radial coordinate

T = temperature, [ºK]

T = temperature nodal vector

T = polynomially calculated temperature

t = time, [s]

u = speed function, [m/s]

" = for all

x = space coordinate, [m]

Greek symbols

Î = belongs to

d = bar thickness

f( x ) = space dependent element shape function

¶W ( t ) = time dependent bar boundary

r( T )= temperature dependent material density, kg/m³

Ñ = vector differential operator

D = increment of

q = total arc of angle of bar

W ( t ) = time dependent bar domain

Subscripts

c = convective

cr = convective plus radiative

tccf = thermal contact conductance film coefficient

r = radiative

1,2 = coordinate directions

¥ = variable taken at a sufficient distance

Superscripts

* = initial

o = initial value of a given variable

The boundary condition is the combined convective and radiative boundary conditions given by

(2)

where , .The indicated derivative represents the outward normal derivative. The indicated film coefficients are computed in the actual calculations using standard established heat transfer procedures for convective and equivalent radiative film coefficients, (Ozisik, 1993).

With an initial distribution of temperature , an initial condition is assumed in the form:

(3)

The numerical solution of the parabolic mathematical model was achieved via a single time stepping procedure using a space discretization through finite elements. Due to time dependency of the problem the approximating element equations include time dependent nodal temperature values as well as space dependent shape functions as indicated below in Eq.(4)

(4)

when Eq.(4) is used to give approximate representation to the temperature field in each element and using Eq.1 the following form of the global system of ordinary differential equations results

(5)

Discretization of the time derivative in Eq.(5) was resolved using a standard Crank-Nicolson implicit scheme with an impliciticity parameter value of 0.5. The solution proceeds in a forward time stepping manner with a variable time step in order to comply with the coiling process and uncoiling process variable speed schedule.

In order to satisfy curving of the bar during coiling the 8-node isoparametric element was used (Becker, Carey and Oden, 1981).

Motion of the Bar

The bar beeing hot rolled is required to move according to specific mill operating speeds. If in addition coiling of the bar is required in the transfer table, then the bar will be complied to move according to the coiling device.

Figure 2 exhibits a typical speed schedule of a bar coiling device showing fifteen stages (each of the straight line segments). As an illustration it is indicated that the first stage corresponds to the speed coming out of the rougher set of stands. The second stage coresponds to deceleration of the bar to the coiling device threading speed. The third stage is a short constant speed stage. The fourth corresponds to acceleration of the bar to full coiling speed. And so on.

There are two distinct types of motion of the bar in the transfer table. Straight motion prior to coiling and after uncoiling, and curving and uncurving during coiling and uncoiling. Coiling in our case implies that all bar points (element nodes computationally) move so that the bar adopts a spiral shape.

These motions are fully described mathematically with the following equations based on the coordinate systems of Fig. 1.

1. Prior to any coiling the motion of the bar points is straight and is given by:

(6)

(7)

and represent initial coordinate value of bar points and speed function respectively.

2. Once coiling starts the motion is given by:

2.1 The c _l coordinate of points which have yet to be coiled change according to expression in Eq.(6) above

and the coordinate changes according to:

(8)

represents the total angle of arc of coiled bar up to time t, and d the thickness of the bar.

2.2 Points which have coiled ( crossed line c₁ = 0 at least once ), and here for simplicity the switch is made to cylindrical coordinates, move according to:

(9)

(10)

where r(t) is given by the following expressions:

for points on the inner surface of the wraps

(11)

for points on the middle plane of the wraps

(12)

for points on the outer surface of the wraps

(13)

r₀ represents the radius of the bar point closest to the center of the coil.

3. Position of the coiled bar for uncoiling is achieved by the following 180º rotation:

(14)a

(14)b

4. Uncoiling is controlled by expressions similar to those in numerals 1 and 2 above.

The Lagrangian approach is used to keep track of the evolving temperature field in a moving and shape changing domain according to expressions in Eq.6 through 14.

The Contact Problem

Of major concern in the model is the numerical handling of the thermal contact which arises as a result of adjacent wraps coming into contact during coiling and losing contact during uncoiling. This issue which appears in many applications has been treated in a number of different ways by various authors, Nakajima (1995), Sridhar and Yovanovich (1996) and Tseng and Wang (1996), just to name a few. In all these works some form of actual contact is used in the modelling of the interfacial heat transfer process. A totally different approach was used to deal with the thermal contact aspect of the problem based on the following considerations.

First, it is assumed that the outer surface of any wrap is at an infinitesimal distance from the inner surface of the next outer wrap ( zero distance in the computations ).

Second, the surfaces facing each other are assumed to exchange heat according to a film coefficient of heat transfer which is made consistent with the thermal contact conductance. In effect, the contacting surfaces are assumed to exchange heat according to a boundary condition of the form:

(15)

where h_tccfis defined as the thermal contact conductance consistent film coefficient T, is the heat emmiting surface temperature, and T_¥ is the heat receiving surface temperature. The form of Eq.(15) is completely analogous to the form in Eq.(2), and so the complex thermal contact problem has been reduced to a simple boundary condition. Specific values of n_tccf are obtained from published results for TCC values, see for instance (Barzelay et al., 1955). It should be noted that the proposed approach is at least macroscopicaly consistent with the heat transfer process at contact.

Numerical Results and Discussion

A double precision FORTRAN code FETAHBRF.FOR ( finite element thermal analysis hot bar rougher to finisher ) was developed, tested and run off-line in a VAX 4000/200. Hollander (1967) reports, from experimental measurements, that a 0.027 m thick carbon steel bar exposed to the environment loses temperature at an average rate of 2.256 ºK/s in the range of 1393.0 ºK-1203.0 ºK to be compared with an everage rate of 2.289 ºK/s for the same range using the present approach. For the coiled portion of a 0.0254 m thick bar, Stelco reports an average loss of temperature of 0.0556 ºK/s to be compared with 0.0544 ºK/s using the present approach.

The particular speed schedule of the bar used in the computations illustrated here is shown in Fig. 2. The material properties used in the calculations correspond to the Rimming 0.06% carbon steel and their dependence on temperature is shown in Table 1 below.

Table 1 Thermal properties of the Rimming 0.06% carbon steel.

These thermal properties were used in the code in the form of minimum squares fitted polynomials. Figure 3 shows the calculated evolution of temperature for a 0.0254 m thick 85.34 m long bar, in three groups of three curves each, corresponding to nine nodal points, based on the initial temperature distribution shown in Fig.4. Although the actual computations required 268 time steps Fig. 3 shows 34 time points only, for clarity.

Fig. 3 Temperature evolution at nine bar selected points based on the off-line solution.

The upper set corresponds to three points located in the upper surface, middle plane and lower surface respectively at a distance of 0.305 m of the, initially, head end of the bar. The middle set corresponds to three points similarly located through the thickness as the ones just mentioned except that they are located in the center, lengthwise, of the bar. The lower set corresponds to three points also similarly located as the ones already mentioned except that they belong to the , initially, tail end of the bar, and at 0.305 m of the end.

Note that as the bar is coiled and uncoiled, what was originally the head end of the bar becomes the tail end, and viceversa. The value of for the contact conductance was used from (Barzelay et al., 1955). This value corresponds to a surface roughness of 2.54 mm at an interface pressure of 5.2 atm. and 477.0 ºK.

Each set of three curves exhibits four sectors indicated in Fig. 3.

1. The typical initial exponential temperature loss for the uncoiled parts.

2. A short transition sector corresponding to the process of "temperature soaking" through the thickness of the bar. This sector coincides with actual coiling of a given sector of the bar.

3. A third sector where the temperature within the thickness of the wrapped bar is practically uniform, and temperature loss is reduced to an exponential curve with large relaxation time.

4. The last sector is similar to the first one in that there is full exposure to heat loss to the environment and corresponds to uncoiled parts of the bar.

Note that the length of the sectors depends on the position of the points along the bar. For instance, points at the head end (originally) of the bar remain coiled longer than any other set of points on the bar, since they are coiled first and uncoiled last. As a result sector 3 for these points (upper set of curves in the figure) is longer than it is for any other set of points in the bar.

On Line Fast Temperature Prediction System OFTPS

Since the numerical solution, in the hot mill process controlling computers, takes more time than the physical process itself it was necessary to develop an on-line fast temperature procedure which could be used during the process itself. The OFTPS as described in this work represents such a device.

The OFTPS is based on the off-line solution of the described model for 5 different thickness (0.0125 m, 0.01875 m, 0.025 m, 0.03125 m and 0.037 m) covering the range of most used bar thicknesses at rougher exit, and it predicts the temperature at the nine points described above. For this purpose the curves in Fig. 3 are divided, as already explained in the previous section, in four identifyable sectors, for each thickness. For each sector, corresponding to each curve, and for each thickness a least squares approximating polynomial giving the temperature as a function of time was developed in the form

(16)

A total of 180 polynomials were generated and chosen so that the least squares norm is minimized. For some particular sectors, lower degree polynomials than indicated in Eq.(16) were used, since due to the small variation of the temperature in these sectors higher degree polynomials were unnecesary, sector three for the upper set of three curves in Fig. 3, for example.

The estimated temperature T(x_j ,t) for point x_j and time t from a given initial temperature T in (determined through pyrometers in the mill ) is obtained by adding to the initial temperature the sum of temperature changes D T_i calculated from the appropriate polynomials, in the following form:

(17)

Each of the four temperature changes correspond to each of the four sectors in Fig. 4.

The OFTPS system is based on using expression (17) in connection with the following steps.

1. Given an initial temperature T in for a given point in the bar (actually determined through a pyrometer appropriately located in the mill), an initial fictitious time, t in, using the appropriate polynomial ( form in Eq.16 above) for sector 1 is calculated through a standard Newton-Raphson procedure. This is illustrated in Fig. 5, where the first sector of the head bottom temperature curve from Fig. 3 is reproduced.

Fig. 5

2. A final time for the first sector, t sf, is calculated on the basis of the elapsed time due to the operational speed schedule of the coilbox, Fig. 2.

3. Corresponding to this final time a final temperature for the sector, T sf, is computed using Eq.(16) again, see Fig. 5.

4. The change of temperature corresponding to the first sector, as an ilustration, is then simply

(18)

The other three temperature changes calculations in expresion (17) above, are performed in a totally similar fashion, keeping in mind that the final time and final temperature of a given sector calculation are the initial time and the initial temperature of the next sector calculation.

For thicknesses other than the five specified above linear interpolation among the appropriate polynomials is used.

A sample of the polynomial coeffcients and expressions determined and incorporated in the OFTPS are given below for 0.025 m thick bars:

For the curve corresponding to a point in the bottom surface of the bar, head end, in sector 1

(19)

For the curve corresponding to a point in the middle plane of the bar, head end, in sector 3

(20)

Clearly, the proposed temperature prediction system is independent of the initial fictitious time tin.

The 180 polynomial expressions of the type indicated in Eq.(16), and illustrated in Eqs. (19) and (20) have been programmed into hot mill controlling software. This software uses on-line evaluation of these polynomial expressions to predict bar temperature at the entry of the finishing stads in order to contribute to make finishing stands operating parameter presetting (part of the process operation control) prior to rolling the oncomming bar.

The OFTPS is operational in hot mills instalations.

Conclusions

A novel and efficient temperature prediction mathematical model and the corresponding finite element solution is presented for manufacturing processes requiring coiling. The definition of a film coefficient of heat transfer consistent with the TCC to treat the complex contact problem results in a particularly efficient computational scheme since the need for remeshing and node renumbering at each computational coiling time step is eliminated. This novel definition also provides the added advantage of producing the minimum possible bandwith of the resulting equations with the clear advantage of minimizing CPU time. Herein lies the computational advantage of the present approach, in this particular regard, over previous ones which, typically, treat the contact problem in more traditional fashion.

The value of was used for lack of specific experimental values of the TCC pertaining our particular application. The solution strategy would not be affected by the particular values of the TCC used, only the length of sector 2 in Fig. 3 would be modified. Indeed, sector 2 would be longer for smaller values of the TCC (diminished heat transfer between contacting surfaces), and shorter for larger values of the TCC ( enhanced heat transfer between contacting surfaces).

An on-line two-dimensional fast temperature prediction system based on the model and the solution is presented as well.

We conclude by noting that a three-dimensional solution for the type of coiling devices described in this work can be found in (Ayala and Troyani, 1998), and by reporting that different numerically efficient strategies are being examined at present to extend the existing two-dimensional OFTPS to a three-dimensional version.

Manuscript received: October 1998; Technical Editor: Angela Ourívio Nieckele

• Ayala O., Troyani N., 1998, "Extensión a Tres Dimensiones de la Simulación Numérica del Coilbox por Elementos Finitos", Proc. IV Congreso Internacional de Métodos Numéricos en Ingeniería y Ciencias Aplicadas, Ciudad Guayana, Venezuela, pp TC28-TC36.
• Barzelay, M. E.,Tong, K. N., Holloway G. F., 1955, NACA, Tech. Note 3295.
• Becker, E. B, Carey, G. F.,and Oden J. T., 1981, "Finite Elements: An Introduction", Chap. 5, Prentice Hall.
• Hollander, F., 1967, "A Model to Calculate the Complete Temperature Distribution in Steel During Hot Rolling", AISE Year Book , pp 46-78.
• Nakajima K., 1995, "Thermal Contact Resistance Between Balls and Rings of Bearings under Axial, Radial and Combined Loads", Journal of Thermophysiscs and Heat Transfer, Vol. 9, N. 9, pp 88-95.
• Ozisik, M. N.,1993, "Heat Conduction", Chap. 1, 2^nd Ed., Wiley.
• Smith, W., 1981 The Coilbox: "A New Approach to Hot Strip Rolling", AISE Year Book , pp 432-436.
• Sridhar, M. R.,Yovanovich, M. M., 1996, "Elastoplastic Contact Conductance Model for Isotropic Conforming Rough Surfaces and Comparison with Experiments", Journal of Heat Transfer-Transactions of the ASME, Vol. 118, pp 3-9.
• Troyani, N., 1996, "Nonlinear Geometrically Adaptive Finite Element Model of the Coilbox", Numerical Heat Transfer, Part A, pp 849-858.
• Tseng, A. A., Wang S. R., 1996, Effects of Interface Resistance on Heat Transfer in Steel Cold-Rolling, Steel Research, Vol. 67, pp 44-51.

Publication Dates

• Publication in this collection
  11 Oct 2001
• Date of issue
  Dec 1999

History

• Received
  Oct 1998