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Rolling of Steel Plates in a Plate Rolling Mill

• satyendra
• February 8, 2021
• 1 Comment
• accelerated cooling, AGC, CVC, Flatness control, gauge control, HAGC, normalizing, Plate mill, Quenching and tempering, Rolling stand, Temperature controlled rolling, Thermo-mechanical cooling, TMCP,

Rolling of Steel Plates in a Plate Rolling Mill

Plate steel is defined as a flat, as-rolled or heat treated product of thickness of at least 5 mm and width of at least 1,200 mm. Plate steel is widely used steel product. It falls normally in the categories of carbon steel, high strength low alloy (HSLA) steel, and alloy steel. For structural applications, plate steel normally do not exceed 0.3 % carbon and 1.5 % manganese. Steel plates of higher thickness are needed for sky-scrapers, drilling rigs on the high seas, bridges with giant spans, slender wind turbines, pipelines, gasholders with enormous pressures, excavators, mobile cranes, container ships, and luxury liners etc. These applications need high quality and high strength in the plates for reliability.

The most important trend in the steel plate is the demand for plates with increasingly higher strengths, high toughness, hardness, and good weldability. Prime quality in terms of thickness, width, profile, flatness, rectangularity, and a homogeneous microstructure are likewise expected with all steel grades and even extreme dimensions. At the same time, ever closer tolerances are being demanded in respect of dimensional accuracy and flatness. For meeting of these requirements, the plate mills are required to have the capabilities of large range of product mix with a high proportion of high strength plates to be produced, and hence new plate mills are designed for a great variety of production technologies. In addition to this, unrestricted compilation of the rolling campaigns is called for with great variations in terms of thickness, width, and steel grades from one plate to the next without negative effects on the productivity (schedule-free rolling).

The requirements which are  normally made on heavy plate are (i) It is to possess the specified dimensions within narrow tolerances and with good flatness (the thicknesses can range from 5 mm to 500 mm and widths from around 1,200 mm to 5,500 mm), (ii) it is to possess the yield strength and tensile strength values needed by the designers (yield strengths ranging from around 235 MPa to above 1,100 MPa), (iii) it is to possess the toughness values needed by the designers even, in many cases, low temperatures toughness, (iv) it is to  possess good workability (such as deformability and weldability), and (v) if required, it is to possess resistance to corrosion resulting, for example, from attack by the hydrogen contained in H2S-bearing gases (sour gas), or a certain resistance to atmospheric corrosion (weathering resistance). For manufacturing steel plates for line pipes in sour gas service resistance to hydrogen induced cracking is the most important factor. For off shore marine application specially in those regions where temperatures falls very low during winter, steel plates need to have properties such as higher strength, larger plate thickness, and ultra-low temperature toughness (excellent weld-joint characteristics at low temperatures) etc. These properties are, in some cases, contradictory and have become achievable, in an extreme combination of alloying and processing technologies. In the recent past, several technologies for the property control of the steel plates have been developed  for the plate mills to meet the requirements of higher strength and toughness of the plates.

For meeting the demand of large range of products with a great proportion of high strength plates to be produced, modern plate mills are normally designed having capabilities for a great variety of production technologies. The production of high-strength plates has implications for all sections of a heavy-plate mill. Mill stands and plate cooling systems as well as all down-stream mill sections have to be designed such that high-strength plates can be produced and processed to get top-quality final products. This needs the plate mill to have improved rolling, cooling, levelling, and shearing technologies with a view to achieve high quality standards. The reproducible production processes and a high production rate also needs high degree of automation. Modern plate mills are designed to have integrated solutions including mechanical, electrical and automation equipment and hence the conditions for optimum process control.

Further, for the production of the heavy plate, there are requirements of qualified operators, efficient plant installations, and suitable control and instrumentation systems for all the process stages. Systematic procedures based on the up-to-date know-how and considerable quantities of energy (heating gases, and power for drive systems, etc.) are to flow into the rolling mill as input with the output consisting of, in addition to the plates, comprehensive process data registered for the purpose of quality monitoring and evaluation, and of test samples for mechanical and technological testing by the acceptance inspection personnel. Fig 1 shows a typical layout of a plate mill.

Fig 1 Typical layout of a plate mill 

Rolling mill equipment

The main equipments of the plate mill are given below.

Reheating furnaces are used for heating of the incoming inspected and conditioned slabs from the slab yard to be rolled. In the modern mills the walking beam type of reheating furnaces are preferred since they are energy efficient and ensure uniform heating of the slabs. The reheating furnace is required to have all the facilities for the waste heat recovery. It is also to be equipped with the combustion controls needed for the control of the slab temperature as well as other controls required for its efficient operation. Normally there are two or three furnaces. Each furnace is equipped with a charging and discharging device. A mathematical model is used to control the burners to achieve the optimum reheating curve for each steel grade. At the exit-side of the furnace, the heated slabs proceed through a roller table to the plate mill through a high pressure descaler.

Descaler unit is for the removal of the primary scale with the use high pressure water jets. For allowing constant impact pressure on the slab surface, a screw jack system is normally employed to adapt the top header position to the entry thickness of the slab.

The central element of the plate mill is made up of four-high rolling stands with process-computer control, on which rolling is performed in widening and elongating passes. The plate mill is either of a single stand configuration or with a two stands configuration. In case of two stand configuration, the two roll stands are known as roughing stand and the finishing stand. The stands have a rapid mechanical screw-down system in addition to the hydraulic roll-gap adjustment system. A vertical edger at the exit end of the stands ensures the precise setting of the plate width. The roll length decides the production of corresponding plate widths. The stands are equipped with high-power drive systems with three phase synchronous motors.

Rolling stands can have the well proven single-part mill housings or housings based on the multiple-part welded design concept. In case of the multiple part bolted mill housings, the finish-machined mill housing yokes and posts are connected by strong tension rods. This approach simplifies casting of the mill housing, and at the same time, controlled preloading of the tension rods ensures minor deformation of the mill housing under load and hence very good plate travel and excellent plate geometry. The mill housings are equipped with hydraulic automatic gauge control (HAGC), electro-mechanical or hydraulic screw down mechanisms and load cells. High pressure water descalers are installed on both rolling stands at entry and exit side in order to achieve optimal surface quality.

Hot leveller is installed before thermal processing equipment for the flattening of the steel plate before cooling. Cold leveller is installed after the thermo-processing section for the flattening of the plate to rectify the shape deteriorated by cooling for easy transfer to subsequent process. The levellers are normally of hydraulic, 4-high, 9-roll design with interchangeable cassettes.

Cooling system is needed for the development of the material properties of high-strength plates. It is normally designed to achieve high cooling rates (upto 80 deg C per second), necessary for high strength steel plates. It combines the spray cooling with the proven laminar cooling. Spray cooling is installed upstream of the laminar section. The spray cooling system attains very high cooling rates because of a powerful high-pressure water station in combination with special nozzles. For achieving good plate flatness also in the spray cooling at the highest cooling rates, pinch rolls are installed between the cooling headers to regulate the water flow onto the plate and thus improving the temperature distribution and the cooling efficiency.

The cooling system is separated into two zones for either direct quenching or accelerated cooling. The direct quenching system consists of a number of high pressure (5-bar) headers placed on the top and bottom of the roller table and separated by pinch rolls. The cooling rate ranges from 80 deg C per second to 3 deg C per second. The accelerated cooling system consists of a number of U-tube headers at the plate top and spray headers for the underside. The cooling rate ranges from 40 deg C per second to 2 deg C per second. The plate temperature and the proper cooling rate under cooling system are defined to ensure the metallurgic characteristics of the end products. The cooling model is based on the mapping of physical processes and controls the cooling process in such a way that the metallurgical properties in the plates are precisely achieved.

Shearing and finishing line is designed to side trim and cut to length plates normally upto 50 mm thick. All shears are usually of rocking type which ensure optimal final dimension tolerances and superior edge quality. The finishing line also includes cooling beds, ultrasonic inspection station, surface inspection beds, plate turning device, marking and stamping, plate piling and handling facilities.

Electrics and automation includes all basic and technological Level 1 and Level 2 automation systems, along with mathematical models for superior profile and flatness control, thus providing with a highly integrated and optimized automation system which ensures accurate and reproducible results in terms of product quality and improved plant efficiency.

Rolling process

The slabs after inspection are heated in a reheating furnace to temperatures of around 1,200 deg C which is suitable for plastic deformation of steel and hence for rolling of the steel in the rolling mill. High-pressure scale removal is performed prior to rolling.

The heated slab is then rolled in the plate mill. The plate mill is normally a four high reversing rolling mill with either a single stand configuration or with a two stands configurations. In case of two stand configuration, the two rolling stands are rouging mill stand and the finishing mill stand. The rolling stands normally have attached edger rolls for controlling the plate width. Plates are normally rolled to the prescribed thickness in the reversing rolling stand (i.e., repeatedly passing the plate back and forth through the roll stand) while progressively reducing the gap between the top and bottom rolls in a stepwise manner, and normally needs a number of rolling passes. The action of passing a plate through the roll gap is called a pass, and the amount of reduction of the plate thickness in each pass is called the rolling reduction. The thickness reduction during the rolling is distributed into several rolling passes. The process by which the number of passes and the rolling reduction in each pass from the slab thickness to the product thickness are decided in the rolling pass schedule. The finish rolling temperature affects the number of passes needed due to the material properties, where the cooler material gets harder.

In the case of normal thickness products (i.e., flat plates), the same thickness is obtained over the entire length by controlling the mill so that the gap between the top and bottom rolls does not change during a rolling pass.

For the rolling of the thin plate, the plate mill is required to be equipped with facilities for automatic shape control, flatness control, and gauge control. The mill needs an online gauge measurement instrument for thickness measurement. The rolling start and finish temperatures determine the process stability, where cooler material needs more rolling force than the hotter one. Hence, thin plate which has higher cooling rate than thick plate can make the rolling process unstable, especially for the low thickness plate where the temperature drop is high.

Mill stands and plate cooling systems as well as all downstream mill sections have to be designed such that high-strength plates can be produced and processed to obtain top-quality final products. Screw down and automatic gap control is the main parts of the rolling mill to adjust the roll gap in accordance with the set thickness. Each of the plate sizes has its own pass schedule calculation including the appropriate roll gap, roll force, and mill modulus.

The rolled plate is subjected to leveling in hot leveller before it enters the thermo-processing section and a cold leveller after the thermo-processing section. Good flatness of a steel plate is desired as during the process of cooling, flatness influences the distance for the water to collide with the steel plate and influences the flow of water on the steel plate. The function of the hot leveller installed before thermal processing equipment is to flatten the steel plate before cooling. On the other hand, the cold leveller installed after the thermo-processing section is intended to flatten the plate to rectify the shape deteriorated by cooling for easy transfer to subsequent process.

Thermo processing section is very important in the production and the processing of the steel plate since the final properties of the steel is achieved during the processing of the plate in this section. After the plate has been subjected to the needed thermo processing for achieving the desired properties of strength, hardness, and toughness, the steel plate is straightened again in the cold leveller and then the finishing activities such as shearing and cutting, sample cutting, testing and inspection and if required shot blasting and coating and dying are carried out as shown in the flowsheet. The plate is subjected to final inspection before its dispatch.

During testing all the tests needed as per the standards are required to be conducted for ensuring that the plates are conforming to the values specified in the standard with respect to the dimensions, dimensional tolerances, micro-structure, strength, hardness, and toughness. The rolling mill laboratory is to be equipped with all the needed testing and the inspection facilities so that the required testing and inspection of the rolled plates can be carried out. The flowsheet of rolling and processing of steel plate in a plate mill is shown in Fig 2.

Fig 2 Flowsheet of rolling and processing of steel plate in a plate mill

Technologies used for in plate mills

The production process for heavy plate includes many diverse potential combinations of process stages. During the rolling of plates, there are a large number of parameters which have impact on the quality of the plate. These include deformation of the rolling equipment, temperature, roll force, and rolling speed etc. Starting from defined steel compositions, metallurgical mechanisms which permit the achievement of the mechanical and technological properties are needed. These mechanisms are to be activated in a range of different process steps. The metallurgically relevant stages are which contribute not only to the shaping (geometry) of the heavy plate but also to its mechanical and technological properties, by means of modification of its structure. Some of the major technologies developed for the control of rolling parameters are given below.

The plate thickness control is carried out by automatic gauge control (AGC) system. In a typical system, the monitoring of AGC is done using a proximate gamma-ray thickness meter. The loads generated by rolling and the changes in the roll gap caused by those loads (roll flexure, mill housing deformation, etc.) are predicted by rolling load prediction and a gauge meter model, and the roll gap prior to rolling is set with high accuracy. Changes in the roll gap due to load fluctua­tions during rolling are corrected by AGC, reducing deviations in plate thickness, and the difference between the target thickness and the measured thickness imme­diately after rolling is fed back to the roll gap, making it possible to control the full length of the plate to the target thickness.

Achieving of high accuracy in the flat­ness (shape control) of plates is an important task during the rolling of plates in a plate mill. Strict control of the plate crown in each rolling pass is necessary in controlling the flatness of steel plates. In plate mills which do not have a shape control system, the larger part of plate crown control is performed using the work roll profile and control of the distribution of reduction in each pass. The technologies used for the shape control of the plates are continuous variable crown (CVC) along with work roll bending and back-up roll bending. The shape control technology also makes use of software/ sensors consisting of (i) a high-accuracy crown model which considers 3-dimensional deformation of the rolling material, (ii) proximate flatness sensor, and (iii) feed forward/ feed backward control based on measured flatness.

Austenitization, including homogenization and the dissolution of micro-alloying elements, occurs during heating of the slab upto a defined temperature within the 1,050 deg C to 1,200 deg C range. Depending on the temperature regime selected, a certain strengthening and grain refinement of the structure occurs during the rolling process and is further intensified as a result of structure transition and precipitation processes during the subsequent cooling phase, depending on rate of cooling. The plates are normally hot-stacked in the lower cooling range in order to ensure effusion of hydrogen. Defined structure modifications can be accomplished through the application of heat-treatment stages of the most diverse types.

Improved properties of plates are achieved by adopting different rolling technologies. These technologies are controlled rolling and accelerated cooling. In the conventional rolling process with no specific temperature requirements, also referred to as ‘normal rolling’, rolling is used purely as a shaping process. The slab heated to high temperatures is converted to the plate geometry in a rolling phase and cooling is accomplished in air.

In the normal rolling process for plate in the plate mills, there is no special temperature control of the rolling process. In this process, the heavy plate is delivered in non-heat-treated, or ‘as rolled’ condition without any further modification of the structure by means of heat-treating. However, a structure with a typical combination of properties of the rolled plate can be achieved by means of technological processes consisting of combination of treatment at specified temperatures and cooling. There are a number of technological processes employed in the plate mills for the rolling of plates to meet property requirements for different plate specifications. Fig 3 summarizes these processes.

Fig 3 Technological processes of plate rolling mill

Normalizing rolling consist of normal rolling with heat treatment consisting of heating the plate to austenitization (higher than Ac3, around 900 deg C)  temperature and then cooling in air. This is performed in correspondingly dimensioned furnaces either continuously (e.g. double walking beam furnace) or on a stationary basis (e.g. laterally chargeable furnace). The result is a structure consisting predominantly of polygonal ferrite and pearlite. With this treatment, higher yield strengths and tensile strengths can essentially be achieved for normalized steels only by means of higher alloying element contents. There are hence limits on the possible property combinations achievable in the heavy plate using this process. An equivalent state can be achieved by means of normalizing rolling, i.e., rolling with final deformation in the normalizing temperature range.

One of the variant of the normalizing rolling is with quenching and tempering. This consists of heating the plate to austenitization region (higher than Ac3) and then water quenching. This process is performed in a combination of a roller hearth furnace and a roller quench, or on a stationary basis in quenching boxes. Due to the extremely high rate of plate cooling, the result is a hard structure consisting predominantly of martensite and bainite. The toughness of the structure is increased by modifying the originally hard and brittle martensite zones by means of subsequent tempering (in a further roller hearth furnace), for example, at temperatures of around Ac1 minus 100 deg C, i.e., around 600 deg C. This gives a heat-treated structure with a combination of a still relatively high hardness or yield strength and tensile strength with a systematically adjusted toughness. Quenched and tempered steel plates are used in particular where requirements for strength or resistance to wear are especially high.

Another variant is the temperature controlled rolling. Controlled-rolling is widely practiced to increase strength and improve notch toughness of plate steel. It is a plate rolling practice which tailors the time-temperature deformation process by controlling the rolling parameters. The parameters of primary importance are (i) temperature for start of controlled-rolling in the finishing stand, (ii) the percentage reduction from start of controlled-rolling to the final plate thickness, and (iii) the plate finishing temperature. Controlled-rolling involves deformation at much lower finish rolling temperatures than hot rolling, normally in the range from 705 deg C to 815 deg C. In contrast, a normal hot-rolling practice takes advantage of the better hot workability of the material at higher temperatures. Hot-rolled plates are finished as quickly as possible, frequently at temperatures of 1,000 deg C and above. For controlled-rolling, a hold or delay is normally taken to allow time for the partially rolled slab to reach the desired intermediate temperature before start of final rolling.

Controlled-rolling practices (Fig 4) are designed specifically for use with micro-alloyed grades, which take advantage of the alloying element’s influence on recrystallization and grain growth, in combination with the specific reduction schedule. Because of practical considerations, primarily mill load and delay times, control-rolled plates are not normally produced above 25 mm thickness.

The term ‘controlled-finishing temperature rolling’ is used to differentiate from the term ‘controlled-rolling’. Controlled-finishing temperature rolling is a much less severe practice than controlled-rolling and is aimed primarily at improving notch toughness for plate of thickness upto 75 mm thick. The finishing temperatures in this practice (around 870 deg C) are higher than required for controlled-rolling. However, because heavier plates are involved, mill delays to reach the desired temperature are still encountered. By controlling the finishing temperature, fine-grain size can be obtained with resulting excellent notch toughness.

Accelerated cooling is a controlled-cooling cycle (water cooling to a temperature of around 540 deg C to 600 deg C, followed by air cooling) immediately after the final rolling operation (Fig 4). Accelerated cooling after either controlled rolling or controlled-finishing temperature rolling leads to additional structural refinement and, hence, an improved combination of properties. Accelerated cooling can improve properties of plates in the approximate thickness range of 12 mm through 100 mm.

 Fig 4 Conventional rolling, control rolling, and accelerated cooling processes

In the present day scenario, thermo-mechanical rolling is the most important production process for the manufacture of high strength plates. This process is used for meeting the demand for high yield and other strengths in large-diameter line-pipes (low wall thicknesses and high conveying pressures in the case of natural gas), combined with high toughness at low temperatures and good weldability. The thermo-mechanical rolling can be grouped together under the umbrella term ‘Thermo-Mechanical Control Process’ (TMCP). The essential difference with the other rolling processes described above is the fact that rolling is used not only as a shaping process but also systematically for the achievement of the specific combination of properties required. Thermo mechanical rolling can hence be defined as a process which aims at achieving a structure with a fine effective grain size, permits a favourable combination of service properties, and is tailored to the steel composition. The process is composed of a sequence of the following steps controlled in terms of time and temperature.

• Slab reheating in the reheating furnace is with a defined drop out temperature.
• Rolling is on the basis of a specified pass sequence with finish rolling in the non-recrystallizing austenite or (alpha plus gamma) two phase zone.
• Cooling is either in air or in the stack, or in accelerated form in the cooling line, down to a defined final cooling temperature.
• Possibly, additional heat treatment (tempering) is done.

This brief definition needs further explanation. The essential benefits of thermo mechanical rolling are based on the effects of micro-alloying, for example niobium, which achieves its full effect even at low content levels of 0.02 % to 0.05 %. Niobium retards or suppresses recrystallization of the austenite (reformation of the grains between the individual rolling passes). The deformation effect of a large number of passes at temperatures of around less than 850 deg C is thus accumulated, permitting the formation of very fine grains during transformation. During the course of the process, niobium forms carbo-nitride precipitations which block displacements in the atomic lattice and thus result in increases in yield strength and tensile strength.

These two effects of niobium can be exploited by means of process adaptation and make it possible to reduce alloying element contents and  carbon content to such an extent that high toughness values and good weldability can be achieved at identical or higher yield strength and tensile strength. The exploitation of strengthening mechanisms to best achieve the specified property profile by means of ‘microstructure breeding’ can be accomplished by means of an appropriate range of equipment in the rolling mill. Also, in this method, the temperatures are specified and set exactly for finish-rolling and for cooling in the plate-cooling systems.

The carefully targeted control of the above complex processes requires close interaction between the mechanical equipment and the automation systems.

Automation and process control

The heavy plate mill is controlled by the electrics, instruments and automation system. The electrics, instruments, and automation system include the technological measuring systems, instruments, sensory systems, Level 1 and Level 2 automation systems with process models, and the HMI (human machine interface) for the entire mill. The electrics include the complete drive technology with transformers, converters, main and auxiliary drives as well as the roller table motors.

The essential elements for the plate mill automation are (i) material tracking from the reheating furnace to the plate piler, (ii) the mill pacing for optimum throughput, (iii) the pass schedule calculation for the roughing and finishing stands, (iv) technological control systems such as width and thickness control for the mill stands, (v) set-up model, coolant volume control and edge masking for the plate cooling, and (vi) set-up model and hydraulic leveller-roll adjustment for the hot and cold plate levellers (drive control). The technological process models are crucial for productivity and product quality. The main technological models are the ‘pass schedule calculation’ model, ‘profile and flatness control’ model and the cooling and leveling models.

The complete operation of the plate mill is controlled by applying mathematical-physical models which precisely describe the various processes. Material tracking makes the logistics within the mill, that is, from the rolling mill to the finishing line, perfectly transparent. Combined with the pass schedule model as well as the profile and flatness control, it facilitates thermo-mechanical rolling in multi-plate operation. This ensures high productivity of the rolling mill. The rolling mill control desk normally has ergonomic design. The HMI systems are arranged to match the operator’s view, giving him a production oriented representation of the process showing all the relevant systems.

The plate rolling process goes through the several steps. Initially, a slab is reheated to recrystallization temperature (around 1,200 deg C) in the furnace, and it is rolled to a final target plate after a number of passes in the plate mill. Then, the microstructure of plate is controlled by the phase transformation of austenite during the cooling processes. After the slab’s extraction from the reheating furnace, the operation sequences in the rolling section are determined by a pass calculation algorithm, which calculates the sequences of rolling operations needed and predicts the characteristics of the plate after each pass. Fig 5 shows the rolling sequences in each pass which consists of three steps namely (i) the before calculation step, (ii) the real-time control step (or rolling phase), and (iii) the after calculation step.

Fig 5 Process control system of a plate mill

The plate rolling process is a complicated process with multiple variables, nonlinearity, and strong coupling. Because of the complexity of rolling environment, such as the changes of material constant, friction coefficient, surface roughness of roller, roll wear, oil film thickness, and lubrication condition, the set calculation results of the rolling force, rolling torque, front slide, and deformation resistance are different from the actual rolling process. The rolling force is the most important equipment parameter and technological parameter of the rolling mill, for it is the important basis of plastic processing technology, equipment optimization design, and process control. The calculation accuracy of the rolling force directly affects the setting accuracy of the rolling schedule. Besides, it is the key to make full use of the regulatory capacity of the thickness and the steel head.

As the conventional rolling force is calculated by the rolling force mathematical model based on experience and statistics, there are some defects in the process of using. Firstly, for the purposes of online control, the general mathematical model is simplified under certain assumptions, so it cannot provide sufficiently accurate predictive value. Secondly, because of the variation of the measurement errors and system characteristics, the parameter errors of model are also great. Therefore, in order to improve the accuracy of rolling force setting, adaptive and self-learning methods based on instant information are used to modify the model.

Rolling schedule plays an important role in the process of plate rolling production. And an excellent rolling schedule is the basic guarantee for the production capacity of rolling mill, for it can improve the quality of products. The medium and thick plate rolling schedule mainly includes the reduction (load) system, the speed system, the temperature system, and the roller type system. Based on the technical requirements of steel, raw material conditions, temperature conditions, and the actual situation of production equipment, rolling schedule design can make artificial calculation or computer calculation to determine the actual reduction, no-load roll gap, rolling speed, and other parameters with the use of mathematical formulas or charts; in the meanwhile, according to the adaptive correction and processing under condition of actual rolling, rolling schedule design can give full play to the equipment potential, increase production, guarantee quality, make operation easy, and make equipment safe.

For the development of the correct rolling schedule, a reasonable reduction (load) distribution is to be determined. Because of the characteristics of the plate rolling, whether it is the traditional optimization method or intelligent optimization method, the whole process of optimization is normally summarized as four steps namely (i) determination of the objective function of rolling load distribution according to the actual production conditions, (ii) determination of the constraint conditions according to the actual production conditions, (iii) choosing of the appropriate optimization method, and (iv) deriving the extreme value of the objective function and obtaining the process parameters when the objective function reaches its extreme value.