General, dimensionless expressions are derived for the parameters describing the thermo-mechanical behavior of the roll-strip system, on the basis of the boundary value problem associated with hot strip rolling. Then, it is shown that, by conducting process simulation with an integrated finite element process model, the dimensionless expressions may be transformed into various on-line models which may be applied to precision process set-up and control. The validity of the proposed approach is examined through comparison with predictions from finite element process simulation.KEY WORDS: finite element method; thermo-mechanical behavior; effective strain; non-dimensional analysis; hot strip rolling. (7) where DTϭT 2 ϪT 1 . Note that V 2 and T 2 may be precisely predicted from the FE process model described previously. It is to be noted that FЈ and PЈ represent the theoretical minimum (or very close to the theoretical minimum) of roll force and roll power, respectively, in the context of the assumed distribution of strip temperatures in the bite zone.
Dimensionless Expressions for the Process ParametersLet us consider a 2-D boundary value problem for the analysis of the rigid-plastic deformation of the strip, with the process geometry given in (18) where K, C 1 , C 2 , are constants that possess the same unit as s , T, and ē , respectively. Note that C 1 and C 2 are introduced since s is governed by non-dimensional T and ē , and therefore, their values may be chosen arbitrarily. In the present investigation, C 1 ϭ1°C and C 2 ϭ1 rad/s are assumed.Let us define the average values of the flow stress for the hypothetical mode of rolling, as follows: Selecting xϭCs 0j , where C is a prescribed constant, it follows from Eqs. (17) and (24) that˜˜,˜, Vol. 45 (2005) where N S represents ē, ė /w, s/w, as well as V ĩ /Rw. Now, let us consider the boundary value problem for the analysis of heat transfer in the strip, with the process geometry given in Fig. 2. • energy balance equation: (39) where Ñ s represents all the basic non-dimensional fields, which are, ē, ē˜/w, s, s 0j , and T/T 1 .It may be deduced from Eq. (39) that any, reasonably selected, dimensionless parameters that describe the thermomechanical behavior of the strip should, in general, be influenced by eight independent dimensionless parameters appearing in the right hand side of Eq. (39). Note that all of them represent design variables (variables to be prescribed by an engineer), except b˜3, since q s is unknown.For the work roll, the boundary value problem associated with the steady-state thermal behavior of the roll may be given, with the definition sketch shown in (42) Assuming a uniform roll cooling system (water is uniformly sprayed on the entire roll surface, except the roll-strip interface), it may be deduced from the boundary value problem that (43), which involves nine variables, may be reduced to a dimensionless form with five independent dimensionless variables, since four independent units (temperature, force, length, and time) are identified ...