Patent Publication Number: US-2021189515-A1

Title: Method for thermo-mechanically controlled process for high toughness beam production

Description:
FIELD 
     The present invention relates generally to the field of rolling steel sections and beams from cast beam blanks produced using DRI route and, in particular, to a thermo-mechanically controlled process for high toughness beam production. 
     BACKGROUND 
     Universal columns, also called I-beams or H-beams including high-grade jumbo universal columns are used in industries which require high toughness properties as per the BS EN10025-2:2004 standard including grades S355 JR, J0 and J2. 
     Different dimensions for the universal columns (UC) are available including large columns “UC 356×406” with different weight/mt (467, 509, 551, 592 and 634) which have difference flange thicknesses up to 77 mm. UC 356×406 flange thickness ranges between 58 mm and 77 mm. In particular, UC 402 has a flange thickness of 58 mm; UC 404 has a flange thickness of 62.7 mm; UC 406 has a flange thickness of 67.5 mm; UC 408 has a flange thickness of 67.6 mm; UC 410 has a flange thickness of 72.3 mm; and UC 412 has a flange thickness of 77 mm. 
     Rolled UC are formed from a metal cast near shape intermediate product, called a blank, that is subject to a deformation process or “rolling”. During rolling the blank is passed through one or more pairs of rolls to reduce its thickness and make its thickness uniform. When the temperature of the blank is above the metal&#39;s non recrystallization temperature, the process is known as recrystallization-controlled rolling. When the temperature of the blank is below the metal&#39;s non-recrystallization temperature, the process is known as conventional controlled rolling. 
     Despite advances in the art, production of high toughness grade beams and columns remain a challenge, especially as it relates to high value grade beams (Jumbo Beams). Production of those high toughness grades (JR/J0/J2) is challenging due to at least the following reasons: low reduction ratio (increment of the metric weight), reduction is only about 38.5%; failure and inconsistency of the mechanical properties for heavy and jumbo sections due to heterogeneous microstructure; surface defects due to low reduction and roll pass design limitation; limitations in the production of special grades (W/mt); Cost of alloying elements ($/ton); and weldability properties due to high alloy rate which increase the carbon equivalent (CE). 
     Many have attempted to find solutions to these problems through cooling of beams. These prior art solutions include the use of quench and self-tempering (QST) systems and beam cooling systems. The prior art solution tend to require capital expenditure and have high operating expenses. 
     There exists a need for alternative methods and processes to ensure high quality grade UC. 
     SUMMARY 
     An object of the present invention is to provide a thermo-mechanically controlled process for high toughness beams production. In accordance with an aspect of the specification, there is provided a method to obtain a high strength and high toughness yield during production of steel beams, the method comprising: at a tandem mill, rolling a steel beam blank above a non-recrystallization temperature, the beam blank having an austenite grain structure to obtain a rolled beam; rolling the rolled beam below the non-recrystallization temperature to obtain critical strain accumulation for increased austenite grain refinement, wherein the non-recrystallization temperature (Tnr) is determined by: 
         Tnr= 887+464*(% C)+6645*(% Nb)−664*√{square root over (% Nb)}+732*(% V)−230*√{square root over (% V)}+890*(% Ti)+363*(% Al)−357*(% Si)
 
     wherein C: Carbon content in steel (in wt %), Nb: Niobium content in steel (in wt %), V: Vanadium content in steel (in wt %). Ti: Titanium content in steel (in wt %), Al: Aluminium content in steel (in wt %), Si: Silicon content in steel (in wt %). 
     In accordance with another aspect of the specification, there is provide a computer implemented method for execution at a data storage device, the method comprising providing at least one input parameter in relation to beam rolling; outputting at least one rolling parameter for beam rolling to achieve target metallurgical properties. 
    
    
     
       BRIEF DESCRIPTIONS OF THE DRAWINGS 
       The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. Embodiments of the invention will now be described, by way of example only, by reference to the attached Figures, wherein: 
         FIG. 1  depicts specifications for high quality direct reduced iron. 
         FIG. 2  depicts size dimensions for the BB3b beam blank. 
         FIG. 3  depicts an example heavy section mill configuration. 
         FIG. 4  depicts various shapes and sizes of the UC 356×406 large columns. 
         FIG. 5  displays example required toughness properties under the BS EN10025-2:2004 standard. 
         FIG. 6A  depicts the heterogeneous microstructure for some products of the prior art. 
         FIG. 6B  depicts the surface defects for some products of the prior art. 
         FIG. 6C  depicts a graph of the percentage of casting defects as a function of the reduction ratio of the prior art. 
         FIG. 7  depicts an example strategic target. 
         FIG. 8  depicts an example conceptual design of a two-step controlled rolling process. 
         FIG. 9  depicts a schematic summarizing the structure of an example model of the invention disclosed herein. 
         FIG. 10  depicts a scheme showing the interaction between the driving force for recrystallization and the pinning force exerted to precipitates. 
         FIG. 11  depicts stress-strain curve typical of dynamic recrystallization with the main singular strains. 
         FIG. 12  depicts an example metallurgical model output. 
         FIG. 13  depicts example metallurgical model recommended process parameters settings. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention provides processes and methods for the production of high-quality universal columns including high grade jumbo universal columns from casted beam blanks, and in particular the bigger flange beams or so-called “jumbo beams”. Methods for controlling the rolling process to produce rolled structural steel products meeting specific standards, including European Standard BS EN10025-2:2004 are also provided. In some embodiments, the process is configured to produce products meeting grades S355 JR, S355 J0, and S355 J2. 
     Products of one embodiment of the process of the invention include universal columns (UC). The final products after the last finishing stand include large columns “UC 356×406” with different weight/mt (467, 509, 551, 592 and 634) which have difference flange thicknesses up to 77 mm. In some embodiments, the process produces UC 356×406 of different shapes and sizes, with flange thicknesses ranging between 58 mm and 77 mm including UC 402 with a flange thickness of 58 mm; UC 404 with a flange thickness of 62.7 mm; UC 406 with a flange thickness of 67.5 mm; UC 408 with a flange thickness of 67.6 mm; UC 410 with a flange thickness of 72.3 mm; and UC 412 with a flange thickness of 77 mm. 
     The present invention further provides methods of determining process parameters for rolling and/or chemistry including alloying elements to achieve target metallurgical properties and/or production cost. Parameters for rolling include temperature and required reduction and number of passes. 
     Direct reduced iron (DRI) may be used as the main source of raw material in processes and methods of the invention. Optionally, the DRI has a composition as shown in  FIG. 1 . The DRI is used to produce molten alloy steel having a specific chemistry and cast into beam blanks (a near-shape intermediate product) that are ready for the deformation process “rolling”. The blanks used to produce high value grade beams, also referred to herein as Jumbo Beams by the methods and processes of the invention generally have the dimensions shown in  FIG. 2  and are referred to as BB3b beam blank. 
     In some embodiments of the invention, there is provided a process for optimizing the chemistry of the blanks including levels of micro-alloying elements. 
     The BB3b beam blank are processed in the rolling mill which is configured with a re-heating furnace (RF) to re-heat the beam blanks, one breakdown mill (BD) and three tandem mills (TDM) (roughing, edging and finishing stands), followed by a cooling bed (CB). 
     The heavy section mill configuration is presented in  FIG. 3 , with reheating furnace  302 , breakdown mill  304 , tandem mill  306  and cooling bed  308 . 
     In some embodiments, the invention provides a thermomechanical controlled process (TMCP) that utilizes an integrated metallurgical model to control the rolling process in a heavy section mill to solve one or more problems of the prior art methods. The integrated model is used to determine the required controlled temperatures for the controlled rolling process, the required reduction/number of passes for the TMCP, all of the metallurgical calculations during rolling (e.g., strain type, grain size, recrystallization type, recrystallization controlled rolling (RCR), and conventional controlled rolling (CCR)) based on, in part, chemistry including micro-alloy content of the blanks. 
     Micro-alloying elements such as niobium (Nb), vanadium (V), and titanium (Ti) are commonly used to promote the effect of grain refinement by retarding or inhibiting austenite grain growth during controlled rolling. 
     As conceptual design considerations, these types of grades require a product of high strength and high toughness. To increase strength, the yield strength (YS), tensile strength (TS) and hardness is be improved. The main control parameter for increase strength is the grain refining or the grain size. Accordingly, in some embodiments the methods of the invention provide control of grain refining or grain size. 
     The strengthening mechanism of any high strength steel product depends mainly on the precipitation and grain size which affects mainly about 73% of the total strength of the product. Addition of vanadium (V) and aluminum (Al) affects the precipitation percentage while addition of niobium (Nb) and titanium (Ti) affect the grain size and the homogeneity of the microstructure. 
     The grain homogeneity is controlled by the reduction ratio, controlled temperature and Nb—Ti addition. In embodiments, where the reduction ratio is limited, the percentage of addition of Nb or Ti and/or the required controlled temperature are optimized as shown in  FIG. 7 . The invention therefore in some embodiments provides for the optimization of these parameters using metallurgical model of  FIG. 9 . 
     TMCP has been mainly adopted in the production of HSLA steels by hot rolling mills where a typical schedule takes a beam containing coarse and non-uniform as-cast austenite grain structure, breaks down and refines the structure via repeated deformation and dynamic and static recrystallization above the Tnr temperature. This typically takes place at the initial stage of the hot rolling process in the roughing stands and is referred to as recrystallization-controlled rolling (RCR). 
     Upon completing the RCR, the transfer bar is dispatched from the roughing stands to the finishing stands. A Conventional Controlled Rolling (CCR) will be used to roll the transfer beam below the Tnr temperature, so that critical strain accumulation can be achieved for further austenite grain refinement. 
       FIG. 8  details the two-step controlled rolling (TMCP) in the tandem mill (ultra-flexible reversing or UFR) of the invention. A main focus of this process will be on the beam flange. The process includes rolling a beam blank having an austenite grain structure above a non-recrystallization temperature; and rolling the beam below the non-recrystallization temperature to obtain critical strain accumulation for increased austenite grain refinement, where the non-recrystallization temperature is determined based on chemical composition of the blank. 
     The metallurgical model described herein and used by the methods of the invention determines the required chemistry and/or controlled temperature settings for enhancing the metallurgical behavior of the rolled product. Inputs into the model include roll diameter, rolling temperature, reduction/stand, etc. The model is compatible with different multiple mixes of the alloying elements such as Nb—Ti, Ti—V, and the like. 
     The developed model is divided into a set of sub models to predict the metallurgical behavior of the product during reheating, rolling/deformation and cooling. The inputs into the method include the chemical composition, the thickness/stand, roll force/stand, roll diameter/stand, roll speed/stand and deformation temperature/stand. 
     The outputs of the method include: Reheating sub-model: includes the needed reheating temperature for complete solubility of microalloy (niobium, vanadium and titanium) carbides and nitrides; Rolling sub-model: includes the deformation, total strain, strain rate, mean flow strength, type of recrystallization, softening between stands, time for precipitation of niobium carbides and nitrides, grain size/stand and grain growth between stands and final austenite grain size; and Cooling and mechanical properties sub-model: includes the cooling rate, ferrite grain size after transformation, RCR percentage, CCR percentage which affects the final microstructure and subsequently the mechanical properties mainly the toughness. Impacts of changes in chemistry and process are optionally assessed in silico. 
       FIG. 9  depicts a schematic diagram summarizing the structure of the developed model. The model studies the effect of changing rolling process parameters (primarily the chemical composition and finishing temperature) on the metallurgical and mechanical properties of the material to be rolled. The effect of the reduction pattern is also taken into consideration (i.e., the effect on fraction softening/stand and grain size/stand, as well as recrystallization type). Accordingly, in some embodiments, impact of changing one or more rolling process parameters is assessed in silico. 
     The model predicts the type of recrystallization in each stand during rolling of different steel groups (such as Nb, Ti and V). It focuses on the transition zone between static and metadynamic recrystallization, which was not taken into consideration previously in literature. The model calculates the mean recrystallized grain size and the material grain size, which are not recrystallized during reduction. The model may be used fora wide range of chemical compositions for the same steel group. Accordingly, in some embodiments, changes in chemical compositions on mean recrystallized grain size are assessed in silico and chemical composition is optimized based on the in silico studies. 
     The addition of any micro alloying element requires applying controlled rolling to be a benefit for the product mechanical properties. 
     The controlled rolling can be applied by controlling the main rolling process parameters (speeds, reductions, temperatures and strain rate). Faster rolling speeds accompanied with high reductions are advantageous for enhancing dynamic recrystallization kinetics for austenite grain refinement due to higher stain rates. Another beneficial effect is that shorter inter-pass times reduces the premature strain induced precipitations of Nb(C, N) and VN, therefore more soluble niobium and vanadium are available for precipitations during the post-deformation transformation which occurs during rolling and also during cooling at the run-out table. 
     Refining the ferrite grain size improves both strength and toughness of the product. The methods of the invention through controlling rolling strain, strain rate, speed and temperature the resulted mechanical properties of the product are improved. In some embodiments, the reduction at first passes is increased for refining the as cast grains and also to decrease in a small portion the beam edge cracks (to be within permissible standard limits). Using niobium, vanadium and titanium micro-alloys is recommended for grain refinement during rolling but the casting parameters should as well be controlled. 
     The determination of the non-recrystallization temperature (Tnr) is a crucial step in designing controlled rolling schedules, because it defines the temperature below which strain is accumulated in the austenite. This temperature is a result of the interaction between deformation, recrystallization and precipitation. The time for precipitation is predicted using the following equation: 
         Tnr= 887+464*(% C)+6645*(% Nb)−664*√{square root over (% Nb)}+732*(% V)−230*√{square root over (% V)}+890*(% Ti)+363*(% Al)−357*(% Si)   (Eq. 1)
 
     Where: T nr : Recrystallization stop temperature (° C.), C: Carbon content in steel (in wt %), Nb: Niobium content in steel (in wt %), V: Vanadium content in steel (in wt %). Ti: Titanium content in steel (in wt %), Al: Aluminium content in steel (in wt %), Si: Silicon content in steel (in wt %). 
     If the absolute deformation temperature is lower than the non-recrystallization temperature (T def &lt;T nr ) then the pinning force exerted by the precipitates overcomes the stored energy of deformation, as depicted in  FIG. 10 . 
     Fine precipitation of aluminum nitrides will then inhibit further recrystallization during the remainder of the processing. Due to the strain induced inside the material the value of strain and the material flow strength will be changed and thus the rolling force will be affected. 
     The final ferrite grain size (da) can be calculated as follows: 
         d   α   0 =2.5+3* T°   1/2 +20*[1−exp(−1.5*10 −1   *d γ)]  (Eq. 2)
 
         d   α   =d   α   0 *(1−0.45*ε a   1/2 )  (Eq. 3)
 
     Where: d α : Ferrite grain size (μm); T°: Cooling rate (° C./Sec); d α   o : Initial ferrite grain size (μm); ε a : Accumulated strain at final pass; and d γ : Austenite grain size before last stand (μm). 
     According to Eq. 2, to get finer ferrite grain size “d α   o ” the final austenite grain size (d γ ) after rolling should also be fine and the cooling rate should be controlled. In order to quantify the effect of any retained strain “ε a ” present in the austenite; another equation (Eq. 3) is also necessary or required “d α ”. 
     To obtain fine and homogeneous final austenite grain size after final stand three points are optimized: 1. Accumulated strain at last stands; 2. Type of recrystallization/stand and recrystallized grain size; and 3. Precipitation kinetics of niobium precipitates. 
     The accumulated strain “ε ai ” depends on the fraction softening taking place in previous stand and the total strain calculated in the considered stand. 
       ε a     i   =ε i   +K *(1− X   i-1 )*ε i-1   (Eq. 4)
 
     Where: ε i   a : Accumulated strain for stand; Fraction softening from previous stand; ε i-1 : Total strain from previous stand; ε i : Total strain for needed stand; K: constant (=1 for shorter interpass time “low rates of recovery” while for longer interpass time (k=0.5)). 
     The total strain/stand is calculated as follows: 
       ε i =ε h +ε r   (Eq.5)
 
     Where: ε r : Redundant strain; ε h : Effective strain; and ε i : Total strain at needed stand. 
     The effective strain depends on the entry thickness and exit thickness/stand ( FIG. 3.1 ), the values of these two thicknesses will change after skipping any mill stand (von Mises yield criteria). 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     h 
                   
                   = 
                   
                     
                       2 
                       
                         3 
                       
                     
                      
                     Ln 
                      
                     
                       
                         h 
                         0 
                       
                       
                         h 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: ε h : effective strain; h 1 : exit thickness (mm); and h 0 : entry thickness (mm). 
     The redundant strain is affected by the change in the entry and exit thicknesses for the needed mill stand, the rolling contact angle and the flattened roll radius. 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     t 
                   
                   = 
                   
                     
                       
                         h 
                         0 
                       
                       - 
                       
                         h 
                         1 
                       
                     
                     
                       4 
                       * 
                       
                         
                           
                             4 
                             * 
                             
                               R 
                               2 
                             
                             * 
                             
                               
                                 sin 
                                 2 
                               
                                
                               
                                 ( 
                                 α4 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     h 
                                     0 
                                   
                                   - 
                                   
                                     h 
                                     1 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             4 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: ε r : Redundant strain; α: Bite/contact angle (deg) (rad=(deg*180)/π); R′: Flattened roll radius (mm); h 1 : exit thickness (mm); h 0 : entry thickness (mm). 
     As a result, it is preferred to increase the accumulated strain to obtain finer austenite grains after rolling. 
     The recrystallization type plays an important role for obtaining fine and homogenous grains. To obtain finer and homogeneous grains, the type of recrystallization should go toward metadynamic recrystallization or at least the transition zone between metadynamic and static recrystallization type. 
     In some cases, the recrystallization is static depending on different hot rolling process parameters (reduction pattern, deformation temperature, final product thickness, etc.). If the recrystallization type is static then there should be enough interpass time to complete the recrystallization before another deformation takes place in the next stand. 
     Before calculating the recrystallized grains, the time needed for recrystallization and the fraction softening percentage for each stand should be calculated first. To calculate the time for recrystallization and the fraction softening percentage for each stand, different strain types/stand (see  FIG. 11 ) should be calculated first. 
     The addition of titanium to niobium and vanadium steels will slightly affect the calculation of the fraction softening and the recrystallized grains. The peak strain is affected by the change in titanium content and thus will control the recrystallization type/stand 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     p 
                   
                   = 
                   
                     2.8 
                     * 
                     
                       10 
                       
                         - 
                         4 
                       
                     
                     * 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             20 
                             * 
                             
                               [ 
                               Nb 
                               ] 
                             
                           
                           + 
                           
                             0.02 
                              
                             
                               [ 
                               Ti 
                               ] 
                             
                           
                         
                         ) 
                       
                       1.78 
                     
                     * 
                     
                       d 
                       e 
                       0.5 
                     
                      
                     
                       Z 
                       0.17 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: ε p : Peak strain; d o : Austenite grain size from previous stand (μm); Nb: Niobium content in steel (in wt %); Ti: Titanium content in steel (in wt %); and Z: Zener Hollomon parameter. 
     
       
         
           
             
               
                 
                   Z 
                   = 
                   
                     
                       ɛ 
                       0 
                     
                     * 
                     exp 
                      
                     
                       { 
                       
                         
                           Q 
                           def 
                         
                         
                           [ 
                           RT 
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: Z: Zener Hollomon parameter; Q def : Apparent activation energy for deformation; ε°: Strain rate/stand (s−1); and R: Universal gas constant (=8.31 J/mol ° K). Note that the Zener-Hollomon parameter is used to help describe high temperature creep strain of a material such as steel. 
     The critical strain depends on the effective niobium percentage and the peak strain. The effective niobium percentage depends on the niobium content, silicon content and manganese content in steel, the critical strain “ε c ” can be calculated as follows: 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     c 
                   
                   = 
                   
                     
                       ( 
                       
                         0.8 
                         - 
                         
                           13 
                           * 
                           
                             [ 
                             
                               NB 
                               eff 
                             
                             ] 
                           
                         
                         + 
                         
                           
                             112 
                              
                             
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                                 NB 
                                 eff 
                               
                               ] 
                             
                           
                           2 
                         
                       
                       ) 
                     
                     * 
                     
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                       p 
                     
                      
                     
                         
                     
                      
                     and 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     NB 
                     eff 
                   
                   = 
                   
                     
                       [ 
                       NB 
                       ] 
                     
                     - 
                     
                       
                         [ 
                         Mn 
                         ] 
                       
                       120 
                     
                     + 
                     
                       
                         [ 
                         Si 
                         ] 
                       
                       94 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: ε c : Critical strain; ε p : Peak strain; Nb: Niobium content in steel (in wt %); Mn: Manganese content in steel (in wt %); Si: Silicon content in steel (in wt %). 
     There is a relation between transition strain and critical strain. This relation has been described by many researchers and it differs according to the steel group. For multiple microalloyed steels, the transition strain “ε T ” an be calculated as follows: 
       ε T =2.2*ε c   (Eq. 12)
 
     Where: ε T : Transition strain and ε c : Critical strain. 
     After the calculation of different strain types/stand, the time for recrystallization and fraction softening percentage/stand should be calculated. Subsequently the recrystallized grains/stand is calculated. There are three cases for recrystallization types as follows: 
     The addition of titanium will affect the calculation of the time required for static recrystallization according to the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       t 
                       
                         0.5 
                          
                         SRX 
                       
                     
                     = 
                     
                       9.92 
                       * 
                       
                         10 
                         
                           - 
                           11 
                         
                       
                       * 
                       
                         d 
                         c 
                       
                       * 
                       
                         ? 
                       
                       * 
                       
                         ? 
                       
                       * 
                       
                         exp 
                          
                         
                           ( 
                           
                             180000 
                             RT 
                           
                           ) 
                         
                       
                       * 
                       
                         exp 
                          
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   275000 
                                   T 
                                 
                                 - 
                                 185 
                               
                               ) 
                             
                             * 
                             
                               ( 
                               
                                 
                                   [ 
                                   Nb 
                                   ] 
                                 
                                 + 
                                 
                                   0.374 
                                    
                                   
                                     [ 
                                     Ti 
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ? 
                     
                      
                     
                       indicates text missing or illegible when filed 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     13 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: ε a : Accumulated strain/pass; ε°: Strain rate/stand (s −1 ); R: Universal gas constant; T: Absolute temperature/stand (° K); d 0 : Austenite grain size from previous stand (μm); Nb: Niobium content in steel (in wt %); Ti: Titanium content in steel (in wt %). 
     The fraction softening for metadynamic recrystallization depends on the change in the interpass time between stand and the time for metadynamic recrystallization. 
         X   SRX =1−exp[−0.693(( t/t   0.5SRX )]  (Eq. 14)
 
     Where: X SRX : Fraction softening for static recrystallization; t: Interpass time between stands (sec); t 0.5SRX : Time for 50% static recrystallization (sec). 
     For a static recrystallized grain size: 
     
       
         
           
             
               
                 
                   
                     d 
                     SRX 
                   
                   = 
                   
                     1.4 
                     * 
                     
                       
                         ( 
                         
                           d 
                           o 
                           0.67 
                         
                         ) 
                       
                       
                         ɛ 
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     15 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: d SRX : Static recrystallized grain size (μm); d 0 : Austenite grain size from previous stand (μm); ε a : Accumulated strain/pass. 
     For fully metadynamic recrystallization type (when εa&gt;εc and εT): 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         t 
                         
                           0.5 
                            
                           MDRX 
                         
                       
                       = 
                       
                         1.77 
                         * 
                         
                           10 
                           
                             - 
                             5 
                           
                         
                         * 
                         
                           ? 
                         
                         * 
                         
                           exp 
                            
                           
                             ( 
                             
                               153000 
                               RT 
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       
                         ? 
                       
                        
                       
                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     16 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: ε°: Strain rate/stand (s −1 ); T: Absolute temperature/stand (° K); and R: Universal gas constant. 
     The fraction softening for metadynamic recrystallization depends on the change in the interpass time between stand and the time for metadynamic recrystallization. 
         X   MDRX =1−exp[−0.693(( t/t   0.5MDRX )]  (Eq. 17)
 
     Where: X MDRX : Fraction softening for metadynamic recrystallization; t: Interpass time between stands (sec); and t 0.5MDRX : Time for 50% metadynamic recrystallization (sec). 
     For the metadynamic recrystallized grain size: 
     
       
         
           
             
               
                 
                   
                     d 
                     MDRX 
                   
                   = 
                   
                     39700 
                     * 
                     
                       
                         [ 
                         
                           
                             ɛ 
                             o 
                           
                           * 
                           
                             exp 
                              
                             
                               ( 
                               
                                 372000 
                                 RT 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       
                         - 
                         0.42 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     18 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: d MPRX : Metadynamic recrystallized grain size (μm); R: Universal gas constant; ε°: Strain rate/stand (s −1 ); and T: Absolute temperature/stand (° K). 
     This type of recrystallization in hot rolling mill plants enhances grain size refining and helps adjusting the most critical rolling parameters (reduction pattern, strain, strain rate and deformation temperature) towards metadynamic recrystallization more than static recrystallization leading to finer austenite grains during rolling. As the main aim is to get finer and homogeneous grains after rolling. 
     To calculate the fraction softening for transition zone between static and metadynamic recrystallization, the fraction softening for static recrystallization and metadynamic recrystallization should be calculated first and then the addition of them gives the actual fraction softening percentage as described below: 
         X   MDRX+SRX   =X   SRX   +X   MDRX   (Eq. 19)
 
     Where: X MDRX+SRX =Fraction softening for dynamic and static recrystallization (transition zone); X MDRX =Fraction softening for dynamic recrystallization; and X SRX =Fraction softening for static recrystallization. 
     The fraction softening for metadynamic recrystallization depends on the final recrystallized fraction for metadynamic recrystallization, the interpass time between stands and the recrystallization time for metadynamic recrystallization. 
         X   MDRX   =X   f   MDRX *(1−exp[−0.693(( t/t   0.5MDRX )])  (Eq. 20)
 
     Where: X MDRX : Fraction softening for metadynamic recrystallization; X f   MDRX : Final recrystallized fraction for metadynamic recrystallization; t: Interpass time between stands (sec); and t 0.5MDRX : Time for 50% metadynamic recrystallization (sec). 
     To calculate the final recrystallized fraction for metadynamic recrystallization, the critical strain, accumulated strain and transition strain should be calculated. 
     
       
         
           
             
               
                 
                   
                     X 
                     f 
                     MDRX 
                   
                   = 
                   
                     
                       
                         ɛ 
                         a 
                       
                       - 
                       
                         ɛ 
                         c 
                       
                     
                     
                       
                         ɛ 
                         T 
                       
                       - 
                       
                         ɛ 
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     21 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: X f   MDRX : Recrystallized fraction for metadynamic recrystallization; ε a : Accumulated strain/stand; ε c : Critical strain/stand; and εT: Transition strain/stand. 
     Time for metadynamic recrystallization (t0.5DRX) described before. In case of static recrystallization; the fraction softening depends on the final recrystallized fraction for static recrystallization, the interpass time between stands and the recrystallization time for static recrystallization. 
         X   SRX   =X   f   SRX *(1−exp[−0.693(( t/t   0.5SRX )])  (Eq. 22)
 
     The final recrystallized fraction for static recrystallization depends on final recrystallized fraction for metadynamic recrystallization: 
         X   f   SRX =1− X   f   MDRX   (Eq. 23)
 
     Where: X f   SRX : Recrystallized fraction for static recrystallization and X f   MDRX : Recrystallized fraction for metadynamic recrystallization. 
     The recrystallization time for static recrystallization (t 0.5SRX ) can be calculated. According to the model both the recrystallized grains for static recrystallization and metadynamic recrystallization should be calculated first. 
         d   MDRX+SRX   =d   SRX   *X   SRX   +d   MDRX   *X   MDRX   (Eq. 24)
 
     Where: d MDRX+SRX : Metadynamic and static recrystallized grain size (μm); d MDRX : Metadynamic static recrystallized grain size (μm); d SRX : Static recrystallized grain size (μm); X MDRX : Metadynamic recrystallization fraction softening (%); and X SRX : Static recrystallization fraction softening (%). 
     As a result, through optimization of different rolling process parameters (e.g. strain rate, deformation temperature, reduction pattern, interpass time, etc.) the type of recrystallization in each stand could be changed to get finer grains and avoid grain coarsening. This will give fine austenite grain size after deformation. 
     In multipass rolling, it is possible to have partial recrystallization after one pass, if the time between passes is not long enough for complete recrystallization. This introduces a mixed microstructure before the next deformation pass. In this case the partially recrystallized microstructure is described by the average grain size. In this study; the mean recrystallized grains and the un-recrystallized grain size have been calculated. 
     Irrespective of the softening mechanism involved (static or metadynamic), partially recrystallized microstructures are characterized by the recrystallized grain size, “d r ”. 
     It was predicted that if site saturation holds there are no shape changes and the distribution of recrystallized grains remains stable during recrystallization and no grain coarsening takes place, the evolution of the mean recrystallized grain size “d r ” with time is fairly described by the following relationship: 
         d   r   =d   rex   *X   rex   1/3   (Eq. 25)
 
     Where: d r : Mean recrystallized grain size (static “SRX”, metadynamic “MDRX” or transition “SRX&amp;MDRX”) (μm); d rex : Final recrystallized grain size (static “SRX”, metadynamic “MDRX” or transition “SRX&amp;MDRX”) (μm); and X rex : Fraction softening for recrystallization type (static “SRX”, metadynamic “MDRX” or transition “SRX&amp;MDRX”) (%). 
     “d rex ” represents the final recrystallized grain size calculated for the corresponding post-dynamic softening mechanism whether it is static, metadynamic or transition between static and metadynamic recrystallization. 
     The effective size of unrecrystallized grains “du” can be described with the help of the following expression: 
         d   u =1.06* d   o *exp(−ε a )*(1− X   rex ) 1/3   (Eq. 26)
 
     Where: d u : Unrecrystallized grain size (static “SRX”, metadynamic “MDRX” or transition “SRX&amp;MDRX”) (μm); X rex : Fraction softening for recrystallization type (static “SRX”, metadynamic “MDRX” or transition “SRX&amp;MDRX”) (%); ε a : Accumulated strain/pass; and d o : Austenite grain size from previous stand (μm). 
     After recrystallization, the precipitation kinetics are an important tool to control the grain refinement during rolling. The selection of the precipitation time during rolling depends on the value of the deformation temperature/stand. If the deformation temperature is lower than the non-recrystallization temperature then precipitation starts, but if it was higher, recrystallization will continue. 
     The determination of the non-recrystallization temperature (Tnr) is a crucial step in designing controlled rolling schedules, because it defines the temperature below which strain is accumulated in the austenite. 
     If the absolute deformation temperature is lower than the non-recrystallization temperature (T def &lt;T nr ) then the pinning force “F pin ” exerted by the precipitates overcomes the stored energy of deformation “F rex ” (see  FIG. 10 ). But if F pin &gt;F rex  recrystallization will stop completely. This is the case when strain induced precipitation takes place at the same time as recrystallization. 
     Fine precipitation of microalloyed carbides and nitrides will then inhibit further recrystallization during the remainder of the processing. The strain induced inside the material increases Luder bands inside the grains and this will result in more fine and homogeneous grains after austenite to ferrite transformation. 
     The precipitation time is affected by the change in strain rate, interpass time, deformation temperature, super-saturation ratio, accumulated strain and niobium content in steel. 
     
       
         
           
             
               
                 
                   
                     t 
                     
                       0.05 
                        
                       p 
                     
                   
                   = 
                   
                     5.3 
                     * 
                     
                       10 
                       
                         - 
                         7 
                       
                     
                     * 
                     
                       
                         [ 
                         Nb 
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     * 
                     
                       ɛ 
                       a 
                       
                         - 
                         1 
                       
                     
                     * 
                     
                       
                         [ 
                         
                           
                             ɛ 
                             o 
                           
                           * 
                           
                             ( 
                             
                               exp 
                                
                               
                                 341000 
                                 RT 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         - 
                         0.5 
                       
                     
                     * 
                     exp 
                      
                     
                       270000 
                       RT 
                     
                     * 
                     
                       exp 
                        
                       
                         ( 
                         
                           
                             1.3 
                             * 
                             
                               10 
                               10 
                             
                           
                           
                             
                               T 
                               3 
                             
                             * 
                             
                               
                                 ln 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     s 
                                   
                                   ) 
                                 
                               
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     27 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: Nb: Niobium content in steel (in wt %); ε a : Accumulated strain/pass; d o : Grain size after static or dynamic recrystallization (μm); R: Universal gas constant; T: Absolute temperature/stand (° K); ε°: Strain rate/stand (s −1 ); and K s : Super-saturation ratio. 
     The super saturation ratio is calculated from an empirical relation which is affected by the change of niobium content in steel, nitrogen content, carbon content and interpass time. 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         k 
                         s 
                       
                       = 
                       
                         
                           
                             [ 
                             Nb 
                             ] 
                           
                           * 
                           
                             
                               [ 
                               
                                 C 
                                 + 
                                 
                                   
                                     12 
                                     14 
                                   
                                    
                                   N 
                                 
                               
                               ] 
                             
                             sol 
                           
                         
                         
                           ? 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       
                         ? 
                       
                        
                       
                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     28 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: N: Nitrogen content in steel (in wt %); C: Carbon content in steel (in wt %); T PASS : Absolute pass (deformation) temperature (° C.); Nb: Niobium content in steel (in wt %); and Sol: Soluble. 
     The change in any rolling process parameter affects the metallurgical behavior during rolling/deformation and consequently affects the material final mechanical properties. The finishing temperature is an important rolling process parameter. 
     For hot rolling there is a fixed rule that volume before and after the mill stand should be constant for safe rolling process. When the finishing temperature changes with constant entry beam temperature, the material speed will change taking into consideration that the final strip thickness is kept constant: 
         V   2 =( V   1   *T   1 )/ T   2   (Eq. 29)
 
     Where: V 2 : New material speed (m/sec); V 1 : Old material speed (m/sec); T 2 : Target finishing temperature (° C.); and T 1 : Old finishing temperature (° C.). 
     Consequently, the rolling mill rotational speed/stand and strain rate/stand may change. The strain rate depends on the roll rotational speed, the roll contact/biting angle and the effective strain/stand. 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     o 
                   
                   = 
                   
                     
                       
                         ɛ 
                         h 
                       
                       * 
                       2 
                       * 
                       π 
                       * 
                       U 
                     
                     
                       α 
                       * 
                       60 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     30 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: ε°: strain rate (s −1 ); α: Bite/contact angle (radian) (deg=(rad*π)/180); U: Roll rotational speed (r.p.m); and ε h : Effective strain. 
     When the strain rate changes; the calculated material flow strength/stand will change and the Zener Holloman parameter also will change. This will affect the calculation of the peak strain, critical strain and transition strain which consequently affect the recrystallization type/stand. 
     As described before, the metadynamic recrystallization fraction softening is affected by the change in strain rate and interpass time between stands. Thus, with any change in the finishing temperature the interpass time between stands and the metadynamic recrystallization fraction softening will change. Moreover, the transition zone between static and metadynamic recrystallization will be affected too. 
     Consequently, the calculated mean recrystallized grains for metadynamic recrystallization will change as it depends on the change in strain rate. The recrystallized grains for transition zone between static and metadynamic recrystallization will be affected also. 
     The change in the speed will affect the precipitation kinetics as the time for precipitation depends on the strain rate, interpass time, deformation temperature, super-saturation ratio, accumulated strain and niobium content in steel. 
     In addition, the change in the finishing temperature will affect the calculation of the cooling rate which therefore will affect the calculation of the final ferrite grain size after transformation. 
         T °=[( T   Finishing   −T   Coiling )* V   Exit ]/ L   cool   (Eq. 31)
 
     Where: T°: Cooling rate (° C./Sec); T Finishing : Finishing temperature (° C.); T Coiling : Cooling temperature (° C.); V Exit : Strip exit speed after final stand (m/sec); and L cool : Length of cooling area (mt). 
     For more grain refinement during rolling the reduction pattern for all mill stands are optimized to get higher dynamic recrystallization at 1st stands before precipitation of microalloyed elements precipitates (carbides and nitrides). 
     To optimize the reduction pattern for mill stands and increase possibility of dynamic recrystallization, some reduction passes should be reduced. The accumulated strain at 1st stands should be higher than the critical strain and may exceed the transition strain to get fully metadynamic recrystallization (as much as possible). 
     The calculation of accumulated strain/stand has been described previously. The relative reduction per stand is calculated as follows. 
         r =(( A   0   −A   1 )/ A   0 )  (Eq. 32)
 
     Where: r: relative reduction (%); A 1 : exit flange area (mm 2 ); and A 0 : entry flange area (mm 2 ). 
     And the interpass time between stands is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     t 
                     sp 
                   
                   = 
                   
                     ( 
                     
                       
                         I 
                         d 
                       
                       
                         V 
                         e 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     33 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: I d : Interstand distance (m); t ip : Interpass time (sec); and V e : Exit speed (m/sec). 
     The exit speed per stand depends on the work tangential speed, the flattened radius, the exit thickness from mill stand and the neutral point contact angle. 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       e 
                     
                     = 
                     
                       
                         V 
                         r 
                       
                       * 
                       
                         [ 
                         
                           1 
                           + 
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       R 
                                     
                                     
                                       h 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 * 
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ϕ 
                                   n 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                         ) 
                       
                       * 
                       
                         ( 
                         
                           1 
                           - 
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               n 
                             
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     34 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: V e : Exit speed (m/sec); V r : Work roll tangential speed (m/sec); φ n : Neutral point contact angle (radian) (deg=(rad*π)/180); R′: Flattened roll radius (mm); and h 1 : exit thickness (mm). 
     The work rolls tangential speed is affected by the change in roll radius and roll rotational speed: 
         V   t =2*π* d/ 2* U   (Eq. 35)
 
     Where: d: Roll diameter (mm) and U: Roll rotational speed (r.p.m). 
     The neutral point contact angle depends on the change in relative reduction, exit thickness and flattened work roll radius. 
     
       
         
           
             
               
                 
                   
                     ϕ 
                     n 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             h 
                             1 
                           
                           R 
                         
                         ) 
                       
                       
                         1 
                         / 
                         2 
                       
                     
                     * 
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               π 
                               8 
                             
                             * 
                             
                               
                                 ( 
                                 
                                   
                                     h 
                                     1 
                                   
                                   R 
                                 
                                 ) 
                               
                               
                                 1 
                                 / 
                                 2 
                               
                             
                             * 
                             
                               ln 
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   r 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           * 
                           
                             ( 
                             
                               
                                 
                                   tan 
                                   
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     
                                       1 
                                       - 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 1 
                                 / 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     36 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: R′: Flattened roll radius (mm); h 1 : exit thickness (mm); and r: relative reduction (%). 
     As a result of higher accumulated strain/stand, the material will tend to dynamically and metadynamically recrystallize; the dynamic recrystallization is calculated as follows: 
         X   DRX =1−exp[−ρ(ε a −ε c ) n ]  (Eq. 37)
 
     Where: X DRX : Fraction softening for dynamic recrystallization; ε a : Accumulated strain/pass; ε cDRX : Critical strain/pass for dynamic recrystallization; n: Exponent (=1.5 approximately); β: Constant (If (ε°) Strain rate/stand &lt;0.2 then β=1.07 else 0.89); and ε°: Strain rate/pass (s −1 ). 
     As shown in Eq. 33; when the accumulated strain is increased more than the critical strain, the dynamic recrystallization will increase followed by an increase in the metadynamic recrystallization which finally leads to softer/finer grains and higher mechanical properties for the rolled material. 
     In this invention; this activation energy is not kept constant. It changes according to the change in the chemical composition of the produced steels and this gives accurate results for the metallurgical calculations during rolling. 
     The activation energy depends on the change of niobium, molybdenum content, nickel and chromium contents. 
         Q   def =297+641*[Nb %]+123*[Mo %]+1*[Ni %]−111[Cr %]  (Eq. 38)
 
     Where: Nb: Niobium content in steel (in wt %); Mo: Molybdenum content in steel (in wt %); Ni: Nickel content in steel (in wt %); and Cr: Chromium content in steel (in wt %). 
     The change in activation energy for hot deformation affects the Zener Holloman parameter (eq.9) which indeed affects the recrystallization type/stand and recrystallized grain/stand. 
     In the reheating stage; to obtain the optimal results of the microalloy additives (carbides and nitrides), it must be ensured that complete solubility of those precipitates inside the reheating furnace takes place. Accordingly, the setting of the reheating temperature is very important parameter to achieve the required metallurgical behavior. 
     Adding vanadium will require the setting of reheating temperature in the reheating furnace. When vanadium content increases the reheating temperature should increase to ensure complete solubility of the vanadium precipitates (carbides and nitrides inside the reheating furnace). The reheating temperature for different vanadium compounds can be estimated using the following formulas: 
     
       
         
           
             
               
                 
                   
                     Trh 
                     VN 
                   
                   = 
                   
                     8330 
                     
                       3.40 
                       - 
                       
                         log 
                          
                         
                           ( 
                           
                             
                               [ 
                               V 
                               ] 
                             
                             * 
                             
                               [ 
                               N 
                               ] 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     39 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     Trh 
                     VC 
                   
                   = 
                   
                     9500 
                     
                       6.72 
                       - 
                       
                         log 
                          
                         
                           ( 
                           
                             
                               [ 
                               V 
                               ] 
                             
                             * 
                             
                               [ 
                               C 
                               ] 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     40 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     Trh 
                     V 
                   
                   = 
                   
                     Max 
                      
                     
                       ( 
                       
                         
                           
                             T 
                             rhVC 
                           
                           &amp; 
                         
                          
                         
                             
                         
                          
                         
                           T 
                           rhVN 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     41 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: Trhv: Reheating temperature for vanadium precipitates (° K),(° C.=(° K)−273)); T rh vc: Reheating temperature for Vanadium carbides (° K),(° C.=(° K)−273)); and T rhVN : Reheating temperature for Vanadium nitrides (° K),(° C.=(° K)−273)). 
     While for niobium precipitates, the formula is as follows: 
     
       
         
           
             
               
                 
                   Trh 
                   = 
                   
                     
                       
                         838 
                         * 
                         
                           
                             ( 
                             Mn 
                             ) 
                           
                           0.246 
                         
                       
                       - 
                       
                         1730 
                         * 
                         
                           
                             ( 
                             Si 
                             ) 
                           
                           0.584 
                         
                       
                       - 
                       6440 
                     
                     
                       
                         log 
                          
                         
                           [ 
                           
                             
                               ( 
                               Nb 
                               ) 
                             
                             * 
                             
                               ( 
                               
                                 C 
                                 - 
                                 
                                   
                                     12 
                                     14 
                                   
                                    
                                   N 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       - 
                       2.26 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     43 
                   
                   ) 
                 
               
             
           
         
       
     
     Where: T rhTi : Reheating temperature for Ti precipitates (° K),(° C.=(° K)−273)); C: Carbon content in steel (in wt %); N: Nitrogen content in steel (in wt %); Nb: Niobium content in steel (in wt %); Mn: Manganese content in steel (in wt %); and Si: Silicon content in steel (in wt %). 
     Referring now to  FIG. 12 , it is observable for certain chemistry, as an example for the metallurgical model output, the size and grade, the temperature profile of the product VS the rolling passes and in each pass. The strain and strain rate may be noticeable. 
     It may be seen that in some passes, the product was rolled under the TNR, which is controlled rolling. 
     The model recommends the optimum rolling process parameters required to enhance the metallurgical behavior and subsequently enhance the mechanical properties mainly the toughness properties, as seen in  FIG. 13 . 
       FIG. 13  depicts some results of the model runs, where the selection of the chemistry and the controlled temperature affect the metallurgical behavior and the resultant ACTUAL YS, TS and Toughness. 
     The scope of the claims should not be limited by the embodiments set forth in the above but should be given the broadest interpretation consistent with the description as a whole.