microsoft word - project 1 final

8/4/2019 Microsoft Word - Project 1 Final

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Tata Steel for

Company Limited, is the

annual crude steel capac

Indian Parsi Businessma

operating in the year 191

India in terms of domesti

Global 500, it is based i

Group of companies. Tat

most profitable compan

place in Indian businessunique concepts like 8-

system for the first ti

industrialization process.

implemented by Tata b

Indian employees.

erly known as TISCO and Tata

world's seventh largest steel c

ity of 31 million tones. It was es

Jamsetji Tata in the year 190

2. It is the largest private sector s

c production. Currently ranked 4

Jamshedpur, Jharkhand, India. I

Steel is also India's second-larg

in private sector. Tata Steel h

history, because it has introducour working days, leave with p

e in India and the first player

In the later part the concept

came lawful and compulsory

Iron and Steel

mpany with an

tablished by an

and it started

teel company in

0th on Fortune

is part of Tata

st and second-

lds a very vital

ed some of theay and pension

to start rapid

s invented and

ractice for the


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Tata Steel is the largest, flagship company of the Tata Group of

companies, headquartered in Mumbai, India. Established in 1907, Tata

Steel is Asia's first and India's largest private sector steel company. Tata

Steel is among the lowest cost producers of steel in the world and one ofthe few select steel companies in the world that is EVA-positive (Economic

Value Added). Concerns over the availability of iron ore and coal, and the

resultant volatility in prices, meant that most Indian steel producers had to

integrate backwards in order to have greater access and pricing power over

these commodities. Tata Steel has its own iron ore, coal and chrome mines

and reserves (on long term leases from the Government of India), andhence is largely self-sufficient in most critical raw materials. Its main plant

is located in Jamshedpur, Jharkhand, with its recent acquisitions; the

company has become a multinational with operations in various countries.

Its captive raw material resources and the state-of-the-art 7 MTPA (million

tonne per annum) plant at Jamshedpur have given it a competitive edge.

The Company plans to enhance its capacity, manifold through organic

growth and investments. Tata Steel's products are targeted at the quality

conscious auto sector and the burgeoning construction industry. With wire

manufacturing facilities in India, Sri Lanka and Thailand, the Company

plans to emerge as a major global player in the wire business. Tata Steel's

products include hot and cold rolled coils and sheets, galvanized sheets,

tubes, wire rods, construction rebars, rings and bearings. In an attempt to

'discommodities' steel, the company has introduced brands like Tata

Steelium (the world's first branded Cold Rolled Steel), Tata Shakti

(Galvanized Corrugated Sheets), Tata Tiscon (re-bars), Tata Bearings, Tata

Agrico (hand tools and implements), Tata Wiron (galvanized wire

products), Tata Pipes (pipes for construction) and Tata Structural


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(contemporary construction material). The company has launched the

Customer Value Management initiative with the objective of creating

complete understanding of customer problems and finding solutions

jointly. The company's Retail Value Management addresses the needs ofdistributors, retailers and end consumers. The company has also launched

India's first steel retail store steel junction - for making steel shopping a

happy and memorable experience. Tata Steels profitability ranks among

the best in the industry.

Apart from the main steel division, Tata Steel's operations are

grouped under strategic profit centers like tubes, growth shop (for its steel

plant and material handling equipment), bearings, Ferro alloys and

minerals, rings, agrico and wires. Tata Steel's products include hot and

cold rolled coils and sheets, tubes, wire rods, construction bars, structural,

forging quality steel, rings and bearings.

Tata Steel has set an ambitious target to achieve a capacity of

100 million tonne by 2015. Tata Steel is planning a 50-50 balance

between greenfield facilities and acquisitions.

Overseas acquisitions have already added up to 21.4

million tonne, which includes Corus production at 18.2 million tonne,

Natsteel production at two million tonne and Millennium Steel production

at 1.2 million tonne. Tata is looking to add another 29 million tonnes

through the acquisition route.


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Tata Steel has

and outside which includ

1. 6 million

2. 6.8 millio3. 5 million

4. 3-million

5. 2.4-millio

6. 5 million

7. 4.5 millio

Flow

lined up a series of greenfield

s:

onne plant in Orissa (India)

tonne in Jharkhand (India)onne in Chhattisgarh (India)

tonne plant in Iran

n tonne plant in Bangladesh

onne capacity expansion at Jams

tonne plant in Vietnam

iagram 1: TATA Steel Plant Layout

rojects in India

edpur


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Process Fl

The multiple st

machines in a steel plant.

four-high configuration.

controllers, work roll ben

present in these complex

ow Diagram 2: Hot Strip Mill Operation

and Finishing mill is one of the m

It consists of 6 or 7tandem stan

evices such as loopers, automati

ers, descalers, coilboxes and oth

machines.

st productive

s, each in a

gauge

ers areusually


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Most of the metallic materials produced commercially undergo

at least one hot deformation stage during their fabrication. Suchprocessing leads to the production of plates, strips, rods, pipes, etc. at

low cost when compared to the cold deformation / annealing route.

Comprehensive study of the metallurgical phenomena associated with hot

deformation has considerable potential application in the control of

industrial processes. However, investigations in the hot deformation field

usually require appreciable effort and specialized equipment. The

temperatures involved for most metals, especially steels, make the directobservation of mechanisms very difficult. Most steels are ferritic at

ambient temperatures and hot deformation is, by definition, performed in

the austenite phase. Therefore, study of the metallurgical mechanisms

taking place during the hot deformation of steels involves a great deal of

creativity, imagination and hard work.

The topic selected in this study is the mathematical modeling of

flow stress and microstructure during the hot strip rolling of steels. This

includes such microstructural aspects as hardening and softening. Here,

the niobium microalloyed steels play the main role, but plain C-Mn steels

are also of interest and a few multiply-alloyed grades as well. Special

attention is given to the large-scale softening process concurrent with

deformation known as dynamic recrystallization and its occurrence during

industrial hot strip rolling. Some laboratory work was also performed in

this investigation. However, most of the study is based on data supplied

by TATA STEEL. These data are considered to come from the laboratory of

"real life" and are analyzed in some detail here.


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Definition:Definition:Definition:Definition:

Flow stress is d

required to continue defo

The mean flow stress (MF

strain curve for the strain

CalculatioCalculatioCalculatioCalculation:n:n:n:With reference

o and 1 is calculated as

Fig 1:

Although the Mstress-strain curves, in w

integration, the situation

description of the MFS is

be required for the rollin

efined as the instantaneous value

rming the material - to keep the

) is defined as the area under a g

interval selected.

to the following figure the MFS b

follows:

raphical representation of the MFS.

FS calculation is fairly simple in tich case numerical methods can

is different during rolling. A "mec

ore complex than that describe

calculations and is described bel

of stress

etal flowing.

iven stress

tween strains

e case ofbe used for the

anical"

above, but will

w.


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Recrystallizatio

replaced by a new set of

the original grains have b

usually accompanied by a

material and a simultane

may be introduced as a d

A precise defin

is strongly related to sevegrain growth. In some cas

which one process begins

recrystallization as: "... th

deformed material by the

boundaries driven by the

boundaries are those with

process can be differentiaboundaries do not migrat

only due to the reduction

during or after deformati

treatment, for example).

termed static.

Fig.2: Recrystallization of a me

n is a process by which deformed

ndeformed grains that nucleate a

en entirely consumed. Recrystalli

reduction in the strength and ha

us increase in the ductility. Thus,

liberate step in metals processin

ition of recrystallization is difficul

ral other processes, most notablyes it is difficult to precisely defin

and another ends. Doherty et al.

formation of a new grain struct

formation and migration of high

stored energy of deformation. Hi

greater than a 10-15 misorient

ted from recovery (where high ane) and grain growth (where the dr

in boundary area). Recrystallizati

n (during cooling or a subsequen

he former is termed dynamic whi

tallic material (a b) and crystal grains

grains are

nd grow until

zation is

dness of a

the process

.

as the process

recovery andthe point at

(1997) defined

re in a

ngle grain

h angle

tion" Thus the

le grainiving force is

n may occur

t heat

le the latter is

growth (b c d).


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Types of Recrystallization.Types of Recrystallization.Types of Recrystallization.Types of Recrystallization.

1) Static recrystallization

2) Dynamic recrystallization

Static Recrystallization:Static Recrystallization:Static Recrystallization:Static Recrystallization:

Static recrystallization (SRX) is a softening mechanism that

occurs commonly during the hot working of steels; this process involves

the migration of high angle boundaries, which annihilates dislocations and

releases stored energy in this way. In a HSM, it frequently occurs betweenpasses, after the deformation and during the interpass time. The driving

force is the energy stored previously in the material in the form of

dislocations.

As static recrystallization is a growth transformation its kinetics

follows the equation given below:-

X= 1X= 1X= 1X= 1---- exp(exp(exp(exp(----b.tb.tb.tb.tnnnn)))) ---(1)

WhereX is the fraction transformed,

b is a constant that depends on the nucleation and growth rates,

t is the time, and

n is the time exponent.

Dynamic RDynamic RDynamic RDynamic Recrystallization:ecrystallization:ecrystallization:ecrystallization:

If a material is worked, that is, deformed at the same time as it

is hot, above the recrystallization temperature, the material will not work

harden, but will recrystallize at the same time it is being worked. This is

dynamic recrystallization. It is called hot working. In this case hot is

relative to the recrystallization temperature, not absolute temperature. A


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metal can be red hot but still be cold worked because it s below its

recrystallization temperature.

DRX occur during deformation when the applied strain exceeds

the critical strain, for the initiation of DRX. At high strain rates, the work

hardening is "balanced" by the rapid DRX softening that takes place,

causing a peak (at a strain%),followed by a drop in stress. After a certain

time (strain), the generation and annihilation of dislocations reach

equilibrium and the material can be deformed without any further increase

or decrease in stress; this is referred to as attaining the steady state stress.

The occurrence of DRX causes large scale and rapid softening.

When the strain rate is reduced sufficiently, the stress-strain curve

generally displays a "cyclic" or multiple peak behavior.

Knowledge of the critical strain for the initiation of dynamic

recrystallization (DRX) is a requirement for prediction of the operating

static softening mechanism during a given hot working interpass period in

this way, the rapid softening and intense grain refinement caused by DRX

can considerably modify the work hardening, and therefore the load

behavior in the following pass.

Metadynamic recrystallization:Metadynamic recrystallization:Metadynamic recrystallization:Metadynamic recrystallization:

Metadynamic recrystallization (MDRX) basically results from

continued growth after unloading of the nuclei formed during deformation.

This situation is generally observed in hot deformation schedules when the

reductions applied do not reach the steady-state regime, but nevertheless

attain or exceed the peak strain. Once deformation is interrupted, thenuclei formed dynamically grow statically during the Interpass time. Like

DRX, MDRX is known to involve rapid kinetics, sometimes attributed to the

absence of the incubation period normally required for nucleation.


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Need of modeling:Need of modeling:Need of modeling:Need of modeling:

BYBYBYBY

Flow chart 3: Need of MFS modeling

Thevariation of MFS value from pass to pass describes themicrostructural changes that occur during rolling such as Recystallization

and Strain accumulation. Knowledge of microstructural changes will be

helpful in optimizing the passing schedule. By comparing the mean flow

stresses from strip to strip, the causes to variations in mechanical

properties between strips may be identified.

Desired Mechanical

Properties

Knowledge of

Microstructure

Correlating the

microstructure with

Macroscopic changes

Observation Of

Microstructure at every

stand

MFS measurement by

MFS models

Difficult and requires

time

Requires

OR

By


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MFS Models:MFS Models:MFS Models:MFS Models:

Different para

(i) By Chemistry.(ii) By Rolling Para

(iii) By Recrystalliz

Models byModels byModels byModels by Chemistry:Chemistry:Chemistry:Chemistry:

Table 1 : MF

eters which can be used for MFS

eters.

tion (Static and Dynamic).

models based on chemistry of steel [2

modeling are:

]


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Models by Rolling Parameters:Models by Rolling Parameters:Models by Rolling Parameters:Models by Rolling Parameters:[3](i) OROWANS THEORY

(ii) SIMS THEORY

(iii) BLAND & FORD THEORY

(iv) FORD & ALEXANDER

(v) TSELIKOV

Orowans TheoryOrowans TheoryOrowans TheoryOrowans Theory:

The equation developed does not appear to yield analytical

solution. Alexander gave the complete solution to his equation using

fourth order R-K method. Due to its complexity and the need for

numerical integration to describe the non-uniform deformation associated,

other theories have been proposed. The model assumed variable

coefficients of friction.

Sims Theory:Sims Theory:Sims Theory:Sims Theory:

Sims considered that sticking friction occurs between the work

roll and the work piece, resulting in a simpler calculation and generation of

a analytical solution to MFS.

It is not straight forward or easy to determine the flow stress in

the roll gap; this is because of the complexity and inhomogeneity of the

deformation. Numerical methods such as finite elements have been use

and which can provide a detailed description of the strain distribution.

However, this method requires long compute times and simple

"mechanical" methods are desirable in most applications. The first

complete and comprehensive calculation of the roll pressure distribution

was described by Orowan. However, due to its complexity and the need

for numerical integration describe the non uniform deformation

associated with the assumption of a variable fiction coefficient, other

theories have been proposed, like the one developed by Sims.


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Sims' approach

assumptions to allow for

integration needed in Or

action occurs between th

simpler calculation that i

each pass requires knowl

Diameter and rolling forc

Where P is the roll for

R is the roll rad

W is the strip w

H is the entry t

H is the exit thi

Q is a geometri

Note that the

units and the factor 2/3

geometrical factor is calc

which shows the geometr

Fig

for the calculation of roll force a

an analytical solution of the num

wan's theory. Also, Sims conside

e work roll and the Work piece, re

widely used. Sims calculation o

edge of the strip width, thickness

e. The MFS equation is shown bel

e,

ius,

idth

ickness,

ckness, and

cal factor (defined below).

FS is expressed in equivalent (vo

is included to allow for plane str

lated with reference to the follow

of the roll gap during strip rolli

: Geometry of strip/roll contact.

d torque made

rical

ed that sticking

sulting in a

The MFS in

, and work roll

w:

Mises) stress

in. The

ing Figure,

g.

--- (2


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The normal roll pressure s and the yield strength in plane strain

compression k are related from the neutral point (((() to the exit point ()

as given by:

From the roll gap entrance ( = ( = ( = ( = ))))to the neutral point by,,,,

Where is a generic angle in the working zone. For small

angles, the differences between normal and vertical roll pressures are

negligible. Under this assumption, the neutral point angle, , can be

determined by settingssss = s= s= s= s and rearranging:

Where

r is the reduction, defined by: r ==== (H-h)/H....

The thickness atatatat the neutral point is defined by:

Geometric factor Q is finally defined by:

--- (3

--- (4

--- (5

--- (6

--- (7


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The Sims model can also use the elastically flattened work rolls

radius, which is defined by R' according to the relation derived by

Hitchcock. The flattened Radius R' is given by:

Where his the height reduction defined by h= (H-h), Pis

the roll pressure, w is the strip width and C isan elastic term specified by:

Here is Poisson's ratio and E is Young's modulus for the work

roll material. The MFS cane now calculated with the aid of main MFSequation.

A simplification of the Sims model was proposed by Ford and

Alexander that is sometimes used in on-line models because of its

computational simplicity. Their Equation for the "mean shear yield stress"

(MSYS) is shown below:

Where P is the roll force, H and h arethe entry and exit

thicknesses, respectively, as before, and R is the work roll radius.

Note that the Sims approach refers to the von Mises mean flow

stress, while the F&A method calculate the "mean shear yield stress". Therelation between the Sims von Mises and the F&A MSYS is therefore:

--- (8

--- (9

--- (10

--- (11


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Prediction of th

important tool for the imsimulations. The control

also important as it is the

(MFS) and consequently, t

function of the inverse ab

the main microstructural

followed by MDRX, strain

Knowledge of these eventfor the control of hot stri

Fig 4: Representation

e rolling force during finish rollin

rovement of rolling schedules byf microstructural evolution durin

key factor that influences the me

he rolling load. Analysis of the M

solute temperature can lead to id

hanges taking place. These inclu

accumulation, and phase transfor

s is vital for the development of amill.

f mfs as a function of inverse absolute t

is an

off-linehot rolling is

n flow stress

S behavior as a

ntification of

e SRX, DRX

mation.

ccurate models

emperature.


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1)Recrystallization conRecrystallization conRecrystallization conRecrystallization con

This approach i

seamless tubes, where th

. The high rolling temper

recrystallization to take pl

are added, during recryst

schedule,,,, but preventing

recrystallization is compladded lead to the precipit

dispersion prevents the o

conditions are not suited

mill load limitations mak

process should be emplo

to produce relatively fine

Fig 5

trolled rolling(RCR)trolled rolling(RCR)trolled rolling(RCR)trolled rolling(RCR)

s normally used for thick plates a

rolling loads are near the upper

tures involved (above 950C)cau

lace between passes. For this pur

llization to go to completion all

raingrowth from taking place w

te well before the next pass.... Theation of Tin during continuous ca

currence of extensive grain grow

to producing the finest grain size

this approach necessary in some

ed in association with fast coolin

erritic grain sizes after transform

: Recrystallized controlled Rolling

nd thick-walled

limit of the mill

e full

ose, Ti and V

long the

en

low levels of Tisting. This fine

th.... These

; nevertheless,

cases.... The RCR

rates in order

ation.


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2)Conventional controller rolling(CCR)Conventional controller rolling(CCR)Conventional controller rolling(CCR)Conventional controller rolling(CCR)

The main purpose of CCR is to produce a work hardened

austenite after the last stand in order to increase the number of nucleation

sites for the austenite-to-ferrite transformation. This leads to production

of the fullest ferrite grain sizes, improving mechanical properties such as

the toughness and yield strength. The application of final passes in the

austenite + ferrite region is sometimes also desired. In this case, the

transformed ferrite will work harden and the remaining austenite will

transform to a undeformed ferrite ....Also, the ferrite present at this stage

leads to lower loads in the mill because it is softer than austenite, thus

compensating for the load increase associated with the decrease in

temperature .... The fine microstructures formed in this way are responsible

for yield strength and toughness improvements in the hot rolled steel.

This approach generally involves the use of high reheat

temperatures so as to dissolve the microalloying elements Nb and V

completely in the austenite. Then, roughing is carried out at temperatures

above the Tm,,,, allowing full softening between passes and keeping the

microalloying elements in solution. Finally,,,, finishing is applied to flatten or"pancake" the austenite grains at temperatures below the Tnr. The effect of

the microalloying element (usually Nb) is crucial here. The solute drag

acting on the moving grain boundaries and the particle pinning (after the

precipitation of Nb(C, N) retard or even prevent the occurrence of

recrystallization. However, in some cases, the strain accumulation caused

by this process can trigger DRX,,,, followed by metadynamic recrystallization

(MDRX),,,, leading to rapid softening between passes. In terms ofmathematical modeling, if there is no precipitation and the accumulated

strain exceeds the critical strain, DRX is initiated, often causing full and

fast softening.


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Fig

3)Dynamic recrystallizDynamic recrystallizDynamic recrystallizDynamic recrystalliz

This type of pr

passes during the rolling

large single strains to th

methods will allow the to

initiation of DRX. The for

temperatures. The latterthe1st passes, after the s

Some of the benefits of t

caused by DRX when hig

applied (single peak beh

Fig 7: Dyn

6: Conventional Controlled Rolling

d controlled rollingd controlled rollingd controlled rollingd controlled rolling

cess consists of inducing DRX in

schedule. This can be done eithe

material or via strain accumulati

tal strain to exceed the critical str

mer can be applied in the initial p

can occur at relatively low tempertrain has accumulated in the prev

is approach involve the intense

strain rates are employed and la

vior in the stress-strain curve)

mic Recrystallization Controlled Rolling

one or more

by applying

n. Both

ain for the

asses at high

atures, inious passes.

rain refinement

rge strains are


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DATA ANALYSISUsing

MATLAB


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The MFS calculation here is done according to Sims model,

which is mentioned in the section . MFS result from the mentioned model

is plotted against the 1000/T. However, the derived MFS's are normalized

because, in strip rolling, the reductions are applied at various strains and

strain rates; this makes it difficult to compare MFS values from one stand

to the next. It is therefore useful, to correct those to constant values of

strain and strain rate by using the expression:

MFSMFSMFSMFScorrcorrcorrcorr = MFS= MFS= MFS= MFSsimssimssimssims x (0.4/x (0.4/x (0.4/x (0.4/passpasspasspass))))0.210.210.210.21 xxxx (5/(5/(5/(5/passpasspasspass))))0.130.130.130.13

These exponents 0.21 and 0.13 are strain sensitivity and strain rate

sensitivity, respectively.

These values of Simscorr are plotted against the inverse of

temperature to give the better idea of temperature history dependence of

the MFS during strip rolling. Normalized MFS is plotted versus 1000/T for

several strips to determine MFS values in high temperature region, i.e.

where SRX occurs. Plot is drawn for the MFS values at 6 six passes of

several slabs of plain carbon steel of grade A12010. All the 6 passes are

the part of finishing mill in Hot Strip Mill (HSM) and have taken place at 6

different stands with varying Roll force, Roll Radius and other geometrical

parameters but most importantly at different strain and strain rates at each

stand.

--- (12


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Fig 8: MFS plots of plain carbon steel: A12010 by Sims and corrected Sims model

Figure gives the quite clear picture of, role or effect of strain

and strain rate terms in the calculation of MFS. As the MFS values areinversely proportional to the strain and strain rates, thus MFS by Sims

corrected formula are lower than those calculate by original Sims model.

Lines with black color give the average values or the mean trend for the

variation of MFSsims and MFScorr.

0.76 0.78 0.8 0.82 0.84 0.86 0.88100

150

200

250

1000/T (k-1)-->

M

F

S

(M

P

a)--->

MFS variation with Temperature by Sims' model

MFS Sims' cor

MFS Sims'


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From the figure, it is clear that at higher temperature values or

for low 1000/T values, MFS values are quite low for both the models. This

region of lower MFS is said to be region of Static Recrystallization (SRX).

These zones represents that at high temperature static recrystallization

takes place in the slab after coming out of stand which softens the steel

and results into lower values of MFS.

Examining the above plot, we can clearly see that at first

pass (high temperature side) there is a low slope region, where SRX

occurs. The higher temperature permit full softening during the interpass

time. After pass 2, the lower temperature does not permit full softening,

leading to the accumulation of some strain. This accumulation then leadsto the onset of DRX (as there is no precipitation). In the above plot we cant

observe the DRX region clearly this is because, the strain in the slabs in the

final passes are also not very high. As DRX demands a optimum increase in

the strain values and in this particular case these values are not reached to

the level where we can see a great amount of softening due to DRX, but

still we can see a small decrease in the slope of curve which shows that

there is some softening occurring at that strain.


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Original Misaka model for MFS is based on the strain,

strain rate, temperature of process and chemistry of steel which includes

only Carbon. This model has no effect of the other constituents of the steelgrade such as Mn, V, Nb, Ti and etc. Several works including Misaka

himself predicted a multiplying factor to the original Misaka equation to

consider the strengthening effect of other alloying elements.

Fig 9: MFS of plain carbon steel A12010 by different models base on chemistry of steel

0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85

100

110

120

130

140

150

160

170

180

1000/Temperature (k-1)-->

M

FS(

M

Pa)-->

MFS MODELS BASED ON CHEMISTRY OF STEEL

original Miska

Misaka et al.

Minami et al.

Mirihata et a.

Poliak et al.


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Plot of MFS vs. 1000/T is drawn for the different MFS

models proposed, based on the chemistry of steel for comparing the

deviation in MFS values. This plot is for the Plain C-Mn steel A12010. Forall models MFS values are quite low for the initial stages where

temperature is high or the softening range but as the temperature

decreases we can see the increment in the MFS which may be the result of

strain accumulation.

Original Misaka model doesnt consider the effect of other

alloying elements except C but for other models as Misaka and Poliak do

consider the solution hardening effect of Mn, Nb, Al and etc. Thus the MFS

values for these models are greater than those calculated by original

Misaka model.

Other models which are showing lower values because they

have considered the strengthening effect of other alloying elements and

are not best applicable for the plain C-Mn steel. These models can show

the better results for more micro alloyed steel grades.


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As there is no allowance in the original Misaka equation for the

occurrence of solution strengthening strain accumulation and DRX, whichlead to its deviation from MFS by Sims as these effects cause increase and

decrease in the MFS respectively. Modifications in this original Misaka

equation cover the different chemical composition as well as the

occurrence of DRX. To examine the difference in two models: Sims and

Misaka, data for steel: A12010 were plotted for MFS by Sims and Misaka

vs. inverse of temperature.

Fig 10: Variation of MFS of plain carbon steel A12010 calculated by Sims and Misaka

0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85110

120

130

140

150

160

170

180

190

200

210

1000/T (k-1) -->

M

FS

(M

P

a)-

->

VARIATION IN MFS BY SIMS & MISAKA

0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.850

10

20

30

40

50

60

70

1000/T (k-1) -->

errorinM

P

a

-->

VARIATION OF ERROR

sims

Misaka


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it is apparent from the plot that in the higher temperature

region or during initial passes, steel slab experiences full softening due to

SRX, this is the reason why plots or MFS by two models : Sims and Misaka

are so close. Lower temperature region (final passes) shows the decrease

in MFS Misaka as it considers the effect of DRX due to high amount of

strain accumulation during the higher passes.

However, even the improved Misaka equation is unable to

specify the MFS behavior in the high slope regions i.e. in the last few

passes of rolling. This behavior can be attributed to incomplete SRX,

leading to strain accumulation.

This variation in the two models can be clearly observed by the

plot of error against inverse of temperature, which shows that the

deviation between the two is not much during higher temperature range or

softening range. Variation of error between the initially increases and then

decreases when the MFS by Sims decreases due to the strain accumulation

before the DRX and MDRX occurs and MFS Sims again increase to result

into higher error in the lower temperature region.


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(i)(i)(i)(i) Plain Carbon Steel:Plain Carbon Steel:Plain Carbon Steel:Plain Carbon Steel:It is common to find Mn level varying from 0.2 to 1.5 % in plain

C-Mn steels. Therefore, the solution strengthening effect due to Mn has to

be taken into account and thus MFS obtained from the Misaka equation

matches the initial MFS calculation of MFS by Sims model at higher

temperature range.

Fig 11: Variation of MFS of plain carbon steel A12010 calculated by Sims and Misaka

0.76 0.78 0.8 0.82 0.84 0.86 0.88100

120

140

160

180

200

220

240

1000/T (k-1)-->

M

FS(MP

a)

-->

Sims vs Misaka for Plain Carbon Steel A12010

MFS Sims' corr

MFS Misaka


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(ii)(ii)(ii)(ii) Niobium steelNiobium steelNiobium steelNiobium steel

Niobium is a very important alloying element because it can not

only restrain the growth of austenite grain, but also inhibit austenite

recrystallization, so controlled rolling and controlled cooling technology is

very effective to increase the strength and toughness of the steels

containing Niobium [4].

During hot strip rolling of Nb steels, Nb in solid solution retards

recrystallization due to solute drag and at lower temperatures strain-

induced precipitation of Nb(C, N) may occur which effectively retardrecrystallization [4]. Nb impedes the movement of grain and sub grain

boundaries causing a retardation of recrystallization. The retarding effect

on recrystallization depends on both Nb in solid solution and on

precipitated Nb(C, N) [5]. This effect of Nb results into increase in the no

recrystallization temperature.

Because of the presence of Nb in the solution leads to the

retardation of SRX, it has been proposed that DRX also occurring during

hot strip rolling, particularly in Nb microalloyed steels. The accumulation

of retained work hardening leading to the initiation of DRX during the hot

working of Nb steels was first detected in strip mill simulation [4].


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Fig 12: Variation of MFS of Niobium Steel A19002 calculated by Sims and Misaka model

MFS values for low carbon Niobium steel: A19002 were plotted

against inverse of temperature. Plot analysis shows that during early

passes, MFS values calculated by Sims are not matching with MFS values

by Misaka, this may be solid solution strengthening effect of Nb present in

the steel. This deviation is seen to be decreased while MFS values by Simsdecreases during the initial strain accumulation stages.

In the lower temperature range or final passes we can observe

the great increase in the MFS by Sims the reason to which may be the

precipitation of precipitates going to precipitate out at lower temperature

during last passes.

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9100

120

140

160

180

200

220

240

260

1000/T (k-1

)-->

M

F

S

(M

P

a)

-->

Sims vs Misaka for Niobium Steel A19002

ms cor

MFS Misaka


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34

Plain CPlain CPlain CPlain C----Mn steel vsMn steel vsMn steel vsMn steel vs.... Nb micro alloyed steel:Nb micro alloyed steel:Nb micro alloyed steel:Nb micro alloyed steel: A12010 vs.A12010 vs.A12010 vs.A12010 vs. A19002A19002A19002A19002

Difference in the MFS behavior of steel due to Nb addition

can be observed by examining the 2 steel grades with almost same

chemistry except Nb. Thus for this study, plots of steel grades: A12010

and A19002 are plotted with their MFS values by Sims and Misaka against

inverse of absolute pass temperature.

Chemistry of 2 steel grades to be analyzed:

Steel GradeSteel GradeSteel GradeSteel Grade CCCC MnMnMnMn AlAlAlAl NbNbNbNb VVVV TiTiTiTi CrCrCrCr CuCuCuCu

A12010A12010A12010A12010 0.1542 0.8297 0.052 0 0 0 0.0217 0.0045

A19002A19002A19002A19002 0.1479 0.811 0.016 0.0112 0 0 0.0214 0.0038

Table 2: chemical composition of 2 steel grades: A12010 and A19002

From the above table it is clear that 2 steel grades to be

analyzed have almost same chemistry except the Nb content which is nil in

the A12010 steel grade but there is around 0.0112 wt % of Nb in the Nb

steel A19002.

Point on the Sims curve where the curve deviates from a regular

pattern and the slope increases steeply is known as no recrystallizationpoint and the temperature at this point is mentioned as Tnr. This

temperature is to be compared for 2 different steel grades. This point

gives the temperature after which no recrystallization will occur.


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Fig 13: Comparison of MFS by Sims and Misaka, for Plain Carbon Steel and Niobium Steel.

0.76 0.78 0.8 0.82 0.84 0.86 0.88100

120

140

160

180

200

220

240

1000/T (k-1)-->

M

FS

(M

Pa)

-->

Sims vs Misaka for Plain Carbon Steel A12010

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9100

120

140

160

180

200

220

240

260

1000/T (k-1

)-->

M

FS

(M

Pa)

-->

Sims vs Misaka for Niobium Steel A19002

ms cor

MFS Misaka

MFS Sims' corr

MFS Misaka


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36

Examining the above2 plots we can easily make out the

following points:

(a) For all the curves of steel: A19002 deviation on the curve for

steep slope (no recrystallization point) has started at the lower values of(1000/temperature) or at higher temperatures. As it is clear from the plot

that for plain C-Mn steel no recrystallization point is starting around 0.83

while for Nb steel it is started around 0.82, this gives the clear cut idea

that Tnr, increases after the addition to the Nb to the plain C-Mn steel.

(b) In the softening range, error between the 2 models is less for

the C-Mn steel as compared to the Nb steel which may be due to solidsolution strengthening of Nb.


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Effect of Nb addition on

the recrystallization behavior of plain

C-Mn steel has been observed in the previous section, this section deals

with the effect of increasing Nb content in the Nb steel. This effect can be

examined by considering the different grades of steel with varying Nb

concentration. Nb steel grades as J10007, E07000 and E08000 are

considered for this study, for these grades Nb concentration changes from

0.0075 to 0.0223.

Steel GradeSteel GradeSteel GradeSteel Grade CCCC MnMnMnMn AlAlAlAl NbNbNbNb VVVV TiTiTiTi CrCrCrCr CuCuCuCu

J10007J10007J10007 J10007 0.0842 1.063 0.0344 0.0075 0 0 0.0239 0.0047

E07000E07000E07000E07000 0.071 0.552 0.039 0.0117 0 0 0.025 0.005

E08000E08000E08000E08000 0.0664 0.901 0.0424 0.0223 0 0 0.0239 0.0045

Table 3: chemical composition of 3 Nb steel grades

For the unbiased study of MFS for the above mentioned steel

grades the slabs to be analyzed should be rolled to the same final

thickness so that the effect of strain and, strain rates on the MFS

calculations can be neutralised. Because of this same strain for all the

slabs varying MFS for the various steel grades depends upon the varying

Nb concentration. For this purpose the steel slabs data is divided into 3

categories based on the final thickness of the slab:

Category 1: 3.64 to 4.04 mm

Category 2: 4.55 to 4.90 mm


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Category 1: (3.64mmCategory 1: (3.64mmCategory 1: (3.64mmCategory 1: (3.64mm 4.04mm)4.04mm)4.04mm)4.04mm)

For this category the final thickness of the slab varies from

3.64 to 4.04 mm. Data available for this thickness category is just for 2

steel grades: J10007 and E07000. For each pass, several MFS values are

available for which average is taken and plotted against the inverse of

average pass temperature.

Fig 14: Effect of Increasing content of Nb on MFS and Tnr

0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85100

120

140

160

180

200

220

1000/T (K-1) -->

M

F

S

(M

P

a

)

-->

EFFECT OF INCREASING NIOBIUM CONTENT

J10007

E07000


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39

This plot gives the clear idea about the effect of increasing Nb

content in the steel. MFS values for the Nb rich E07000 steel grade are

slightly higher in the higher temperature prior to the start of strain

accumulation; the reason to this is increasing solid solution strengthening

due to increasing Nb content.

Apart from the increasing MFS due to increasing Nb content,

effect on recrystallization is also examined from the above plot. As it was

mentioned earlier that the presence of Nb resist the austenitic

recrystallization due to solute drag and at lower temperatures strain-

induced precipitation of Nb(C,N) may occur which effectively retard

recrystallization . Nb impedes the movement of grain and sub grain

boundaries causing a retardation of recrystallization.

This retardation of recrystallization in the steel can be clearly

observed by the increase in the recrystallization temperature Tnr in the

E0700 steel grade, early occurrence of high slope in the MFS curve of

E07000 steel is proof to this. Thus we can make out that the

recrystallization stops at higher temperature for steel with more amount ofNb.

Category 2: (4.55mm to 4.90 mm)Category 2: (4.55mm to 4.90 mm)Category 2: (4.55mm to 4.90 mm)Category 2: (4.55mm to 4.90 mm)

For this category the final thickness of slab varies from

4.55mm to 4.90 mm. for this category data is available for all the 3 steelgrades. Average MFS values for each pass is plotted against the inverse of

average pass temperature.


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40

Fig 15: Effect of Increasing content of Nb on MFS and Tnr

From this plot also above mentioned effect of increasing Nb

concentration can be seen clearly;

MFS values increases from minimum for J10007 to the maximum forE08000 and also the stop recrystallization temperature increases with the

increasing Nb concentration.

0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85100

150

200

250

1000/T (k-1) -->

M

F

S

(M

P

a)-->

EFFECT OF INCREASING Nb CONTENT : CATEGORY 2

J10007

E07000

E08000


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41

(i)(i)(i)(i) Need of Optimization.Need of Optimization.Need of Optimization.Need of Optimization.

This part of the paper gives the optimization of models based

on chemistry of steel such as: Misaka et al., Minami et al., Mirihata et al.,

Poliak et al., Kang et al. and Bruna et al. to a derive a model similar to

above mentioned models but having least deviation to the MFS values

given by Sims model.

These above mentioned models give the MFS value depending

upon the Temperature, strain, strain rate and concentration of micro

allying elements and carbon. However for the same values of above

mentioned variables also, there is a difference in MFS reading for all

models and there is an error or deviation to the MFS calculated by Sims

model. This difference is due to the different coefficients multiplied to theconcentration of different micro alloying elements. For the least deviation

from practically calculated MFS, these coefficients have to be optimized at

a particular temperature and rolling parameters for particular steel grade.

So there is a need of optimized equation based on chemistry of

steel which may give the least error while calculating the MFS taking MFS

by Sims model as reference.


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42

(ii)(ii)(ii)(ii) General model for MFS based on chemistry of steel:General model for MFS based on chemistry of steel:General model for MFS based on chemistry of steel:General model for MFS based on chemistry of steel:In all models based on chemistry of steel, the original Misaka

part is multiplied as such and the rest of the part is dependent on micro

alloying elements concentration. This multiplication factor takes the effectof solid strengthening by micro alloys into account. Coefficients for

different micro alloying elements is different for all models which have to

be optimized for getting better approximation and least deviation from

MFS calculated by Sims model.

Taking the above concentration into account, we can write the

general equation for MFS optimized in the following form:

MFSMFSMFSMFSoptoptoptopt ====[original[original[original[original MisakaMisakaMisakaMisaka]]]] (A+B(A+B(A+B(A+B [Mn]+C[Mn]+C[Mn]+C[Mn]+C [Nb]+D[Nb]+D[Nb]+D[Nb]+D [Al]+E[Al]+E[Al]+E[Al]+E [V])[V])[V])[V]) --(13)

Thus for getting optimized model for MFS, we have to

optimized these coefficients (A, B, C, D and E). The optimal values for A, B,

C and other will result in the least deviation of MFS from MFS practicallycalculated in rolling mill by Sims model. Misaka model is a good

prediction for C-Mn steel if full SRX occurs in the steel while it has no

provision for controlling MFS with changing values if strain accumulation

and DRX occurs there. This variation in the coefficients will help to get the

close results in the lower temperature or high strain rate range.


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43

(iii)(iii)(iii)(iii) Optimization Criterion:Optimization Criterion:Optimization Criterion:Optimization Criterion:

Optimization is to maximize or minimize some variable

function which is dependent on one or many independent variables. Thesevalues to be assigned to these variables have some constraints or

requirements which have to be followed and following these limits only

independent variables are adjusted to get the optimized results. These

requirements can be given in the form of equation known as constraint

equations.

In this study we have to optimize the MFS model based on

chemistry to get the close results to MFS by sims model or we can say that

there is need on minimizing the error between MFS Misaka and MFS Sims .

This error function can be written as:

Error= l MFSoptError= l MFSoptError= l MFSoptError= l MFSopt MFS Sims lMFS Sims lMFS Sims lMFS Sims l -------- 14)

Where MFSopt comes from the equation 2 mentioned above,

coefficients A, B, C are independent variables which have to be

optimized to give the minimum error or optimized results. Constraintequations for these coefficients can be derived from the previous MFS

models designed by Misaka, Minami, Mirihata, Poliak and etc. The value of

the coefficients in all these models varies within a range. Smallest and

largest limit is calculated for each coefficient and is considered as the

constraint equation.


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(iv)(iv)(iv)(iv) Constraint Equation:Constraint Equation:Constraint Equation:Constraint Equation:

Analyzing the models we can sort out that the value of all

coefficients multiplied to micro alloys concentrations vary between certaininterval. Value of coefficients A, B, C, D and E also varies between a certain

intervals which give constraint equations.

Constraint Equations:

1.0.6 A 1.2 ---(15)

2.0.1 B 0.2 ---(16)

3.0.5 C 4.54 ---(17)4.0.05 D 0.06 ---(18)

Where coefficients belong to:

A Independent

B Mn

C Nb

D Al


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Optimization Steps:Optimization Steps:Optimization Steps:Optimization Steps:

Flow chart 4: Optimization steps

General Equation forGeneral Equation forGeneral Equation forGeneral Equation for

MFS by chemistryMFS by chemistryMFS by chemistryMFS by chemistry

MFS by Sims modelMFS by Sims modelMFS by Sims modelMFS by Sims model

Error functionError functionError functionError function

formulationformulationformulationformulation

Constraint EquationsConstraint EquationsConstraint EquationsConstraint Equations

fcoefficients

MFS models byMFS models byMFS models byMFS models by

ChemistryChemistryChemistryChemistry

Optimized Model ofOptimized Model ofOptimized Model ofOptimized Model of

MFSMFSMFSMFS

Data forData forData forData for

OptimizationOptimizationOptimizationOptimization

Optimization usingOptimization usingOptimization usingOptimization using

MATLABMATLABMATLABMATLAB

OptimizedOptimizedOptimizedOptimized

coefficientscoefficientscoefficientscoefficients


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46

FMINCON finds a constrained minimum of a function of several

variables. FMINCON attempts to solve problems of the form:

min F(X) subject to: A X


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47

MATLAB PROGRAM FOR OPTIMIZATION:MATLAB PROGRAM FOR OPTIMIZATION:MATLAB PROGRAM FOR OPTIMIZATION:MATLAB PROGRAM FOR OPTIMIZATION:

clc;clear all;

C= input (carbon percentage in steel);Mn = input (Mn percentage in steel);Al= input (Al percentage in steel);Cr= input (Cr percentage in steel);Nb= input (Nb percentage in steel);Ti= input (Ti percentage in steel);

Cu= input (Cu percentage in steel);T= input ('temperature of observations');

mfscorr= input ('values of MFS by sims corrected');mfsmisakao= input ('values of MFS by Misaka original');

mfsmisaka= input ('values of MFS by Misaka');observation = (length(mfscorr));for i=1:observation

x0=[0.6 0.1 0.05 0.5];lb=[0.6 0.1 0.05 0.5];

ub=[1.2 0.2 0.06 4.54 ];f=@(x) abs(mfscorr(i)-(mfsmisakao(i)*(x(1)+(Mn*x(2))+(x(3)*Al)+(0.128*Cr)+(0.3*Cu));[x]=fmincon(f,x0,[],[],[],[],lb,ub,[])mfsopt(i)=mfsmisakao(i)*(x(1)+(x(2)*Mn)+(x(3)*Al)+(0.128*Cr)+(0.3*Cu)+(x(4)*Nb));

e(i)=(abs(mfscorr(i)-mfsopt(i)));em(i)=(abs(mfscorr(i)-mfsmisaka(i)));

t(i)=1000/T(i);

endj=1;

l=length(t)for k=1:(l/6)

for i=1:6MFSmisaka(i)=mfsmisaka(j);


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MFSopt(i)=mfsopt(j);MFScorr(i)=mfscorr(j);tp(i)=t(j);

E(i)=e(j);

Em(i)=em(j);j=j+1;

endsubplot(1,2,1)

plot(tp,MFScorr,'g*-');hold on;

plot(tp,MFSmisaka,'b*-')

plot(tp,MFSopt,'r*-')xlabel ('1000/T')ylabel ('MFS')subplot(1,2,2)scatter(tp,E,'b')hold on;scatter(tp,Em,'r')

xlabel ('1000/T')ylabel ('error in MPa')

end


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(i)(i)(i)(i) CCCC----Mn steel:Mn steel:Mn steel:Mn steel:For plain carbon steel as A12010, only coefficients A, B and C

are considered while framing the general equation for MFS in terms ofcomposition of steel. Coefficients attached to Cr and Cu are taken constant

as there is not much change seen in them while observing the original MFS

equations.

Fig 16: MFS Results with minimum error using Optimized model for C-Mn :A12010 steel

0.76 0.78 0.8 0.82 0.84 0.86 0.88100

150

200

250

1000/T (k-1

) -->

M

F

S

(M

P

a

)

-->

Comparison b/w MFS calculated by Sims,Misaka & Optimized

0.76 0.78 0.8 0.82 0.84 0.86 0.880

20

40

60

80

100

1000/T (k-1

) -->

e

rror

in

M

P

a

-->

Error from 2 models : Misaka & Optimized

MFS opt

MFS misaka

MFS sims corr

5 error b/w MFS sims corr & MFS optimized

error b/w MFS sims corr & MFS misaka


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(ii) Nb Steel:While for the case of Nb alloyed steel, coefficient of Nb is varied

along a long interval as it is done in the original equations. To give better

results in the lower temperature region, value of coefficient A is variedover a larger range as compared to the previous case of plain carbon steel.

Fig 17: MFS Results with minimum error using Optimized model for C-Mn :A12010 steel

In both the cases MFS values from optimized model are very

close to MFS by Sims resulting into the minimum error, almost zero. For

most of the portion MFS optimized is just overlapping the MFS Sims

corrected, except some in the lower temperature.

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9100

150

200

250

300

1000/T (k-1

) -->

M

F

S

(M

P

a)-->

Comparison b/w MFS by Sims, Misaka & optimized

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.90

50

100

150

1000/T (k-1

) -->

errorin

M

P

a

-->

Error from 2 models : Misaka & Optimized

MFS opt

MFS misaka

MFS sims corr

5 error b/w MFS sims corr & MFS optimized

error b/w MFS sims corr & MFS misaka


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Variation of Optimized coefficients:Variation of Optimized coefficients:Variation of Optimized coefficients:Variation of Optimized coefficients:

As optimization of MFS model is done at each

temperature, thus we get different optimized coefficients for optimized

model for every calculation. This variation of coefficients is plotted againstinverse of temperature.

(i)(i)(i)(i) CCCC----Mn steelMn steelMn steelMn steel

Fig 18: Variation of coefficients with temperature for Optimized model of C-Mn :A1201

steel

0.76 0.78 0.8 0.82 0.84 0.86 0.880

0.2

0.4

0.6

0.8

1

1.2

1.4

1000/T (k-1) -->

coeffici

ents

value-->

Variation of Coefficients

A: Independent

B: Mn

C:Al


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(ii)(ii)(ii)(ii) Nb microNb microNb microNb micro alloyed steelalloyed steelalloyed steelalloyed steel

Fig 19 : MFS Results with minimum error using Optimized model for Nb steel: A19002

For Nb steel extra coefficient D, attached to Nb concentration is

also plotted. This coefficient is allowed to vary from 0.5 to 5 during the

optimization process, but its most of the values are varying between the

interval (1, 2).

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1000/T (k

-1

) -->

coefficients'value

-->

Variation of Coeffic ients -->

A: Independent

B: Mn

C: Al

D: Nb


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53

From figure 14 and 15, it can be observed that the Optimized

coefficients vary with the temperature with a regular pattern. Observing

this pattern closely, we can find out 2 cluster of variation. For 1000/T

value less than 0.83, there is a regular variation in value of coefficients but

for further values, we can observe this variation is following another trend.

Thus we can find out the 2 different equations for the variation of

coefficients with varying Temperature.

MATLAB CodeMATLAB CodeMATLAB CodeMATLAB Code (for A19002 Nb steel):

clc;clear all;

C= input (carbon percentage in steel);Mn = input (Mn percentage in steel);

Al= input (Al percentage in steel);Cr= input (Cr percentage in steel);Nb= input (Nb percentage in steel);

Ti= input (Ti percentage in steel);Cu= input (Cu percentage in steel);

T=input('temperature');mfscorr=input('MFS sims corrected');

mfsmisakao=input('misaka original');mfsmisaka=input('misaka');observation =(length(mfscorr));

l=1;m=1;for i=1:observationx0=[0.6 0.1 0.001 0.5];lb=[0.6 0.1 0.05 0.5 ];ub=[1.2 0.2 0.06 5];f=@(x) abs(mfscorr(i)-(mfsmisakao(i)*(x(1)+(Mn*x(2))+(x(3)*Al)+(0.128*Cr) +(0.3*Cu)

+(x(4)*Nb) )));


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[x]=fmincon(f,x0,[],[],[],[],lb,ub,[])t(i)=1000/T(i);if (t(i)


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55

x32=xc32(1).*(t2)+xc32(2);x42=xc42(1).*(t1)+xc42(2);l=1;m=1;

for i=1:observation

if (t(i)


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xlabel ('1000/T')ylabel ('MFS')subplot(2,1,2)

[tpp,index]=sort(t);

E1=e(index);plot(tpp,E1,'b')xlabel ('1000/T')ylabel ('error in MPa')

end

MATLAB Plots:MATLAB Plots:MATLAB Plots:MATLAB Plots:

(i)(i)(i)(i) CCCC----Mn steel:Mn steel:Mn steel:Mn steel:

0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1000/T (k-1)-->

valueofcoefficientA

-->

y = 0.26*x + 0.59 optimized value

Regression


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Fig 20: best fit curve for the variation of Coefficient A for C-Mn steel :A12010

Fig 21: best fit curve for the variation of Coefficient B for C-Mn steel :A12010

0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1000/T (K-1)-->

valueofcoe

fficientA-

->

y = 5.7*x - 3.8 optimized

Regression

0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.840.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

1000/T (k-1)-->

valueofcoefficientB

-->

y = 0.043*x + 0.099 Optimized

regression

0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91

0.12

0.14

0.16

0.18

0.2

0.22

1000/T (k-1)-->

value

ofcoef

ficientB

-->

y = 0.95*x - 0.64

Optimized

Regression


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58

Fig 22: best fit curve for the variation of Coefficient C for C-Mn steel :A12010

Now from the above plots and best fitted equations, we can get

2 different sets of equations for calculating coefficients to be used for

calculating MFS by chemistry of steel.

(i) For (1000/Temperature) < 0.83

0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.840.051

0.052

0.053

0.054

0.055

0.056

0.057

0.058

0.059

0.06

0.061

1000/T (k-1)-->

valueofcoefficien

tC-

->

y = 0.0042*x + 0.05 Optimized

Regression

0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.910.051

0.052

0.053

0.054

0.055

0.056

0.057

0.058

0.059

0.06

0.061

1000/T (k-1)-->

valueofcoe

fficientC-

->

y = 0.094*x - 0.023

Optimized

Regression


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59

a. A= 0.26 x (1000/T) +0.59 ---(19)

b. B= 0.043 x (1000/T) +0.099 ---(20)

c. C= 0.0042 x (1000/T) +0.05 ---(21)

(ii) For (1000/Temperature) > 0.83

a.A= 5.7 x (1000/T) -3.8 ---(22)

b.B= 0.95 x (1000/T) - 0.64 ---(23)

c. C= 0.094 x (1000/T) -0.023 ---(24)

Now these equations are used to calculate the different coefficients at

different temperature values and are further used to calculate the MFS by

chemistry of steel. This MFS calculated is examined and plotted against

inverse of temperature along with the MFS calculated by Sims. Error

between the 2 MFS is calculated to examine the applicability of these

coefficients equations.

MFS model for plain C-Mn steel:

Temperature > 1200 k

MFS = Original Misaka x (0.2544 x (1000/T) +0.5913) + (Mn x (0.0424 x

(1000/T) +0.0986)) + (( 0.0042 x (1000/T) +0.0499) x Al) +

(0.128Cr) +(0.3Cu) ---(25)

Temperature < 1200 k

MFS = Original Misaka x ((5.6823 x (1000/T) -3.8628) + (Mn x (0.9471 x(1000/T) - 0.6438)) + ((0.0937 x (1000/T) -0.0235) x Al) +

(0.128Cr) + (0.3Cu) ---(26)


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Fig 23: MFSregression and its deviation from MFS Sims corr for C-Mn steel :A12010

Above plot gives the MFS (MFSreg) calculated by equation similar

to Misaka with the coefficients attached to alloying elements

concentration changing according to the equations (9-14). Analyzing the

plots we can see that MFSreg is close to the MFSsims almost for all the

temperature ranges and have maximum error of around 30 MPa which is

very less as compared to the error between original Misaka and Sims.

Most importantly it is good observe that the error is almost constant.

0.76 0.78 0.8 0.82 0.84 0.86 0.88100

150

200

250

1000/T (k-1) -->

M

F

S

(M

Pa

)-->

0.76 0.78 0.8 0.82 0.84 0.86 0.880

10

20

30

40

1000/T (k-1) -->

errorin

M

P

a

-->

MFS regression

MFS corr


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Nb steel:Nb steel:Nb steel:Nb steel:

Temperature > 1200 k

MFS = original Misaka ((1.2010 x (1000/T) -0.0603)+( 0.2002 x

(1000/T) - 0.0101) x Mn +( 0.0198 x (1000/T) +0.0392) x Al )+(12.5494x(1000/T) 8.6947) x Nb ) + (0.128Cr) +(0.3Cu)

Temperature < 1200 k

MFS = original Misaka ((3.2843 x (1000/T) -1.6521)+( 0.5473 x

(1000/T) - 0.2753) x Mn +( 0.0542 x (1000/T) +0.0129) x Al )

+(55.9623x(1000/T) 43.7398) x Nb ) + (0.128Cr) +(0.3Cu)

Fig 24: MFSregression and its deviation from MFS Sims corr for Nb steel :A19002

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9100

150

200

250

300

1000/T (k-1) -->

M

F

S

(M

P

a)-->

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.90

10

20

30

40

50

60

70

1000/T (k -1) -->

error

in

M

P

a

-->


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62

Rolling a new steel grade, will require the first idea of Roll force

to be used to design a needed Rolling atmosphere and get the desired

microstructural results. MFS determination is required for this, which can

be fitted into the Sims equation to find out Roll Force. Misaka equation is

used for calculating the MFS for these new steel grades using chemical

composition and absolute rolling temperature. This calculated MFS is then

back fitted into the MFS equation by Sims to find out the Roll force.

This section of paper deals with the calculation of Roll force forthe plain C-Mn steel: A12010, using the regressed MFS model developed

in the previous section and then it is compared with the actual Roll force

used. Revised Misaka equation used for this calculation is given in 2 parts:

Equation 1:Equation 1:Equation 1:Equation 1: Temperature > 1200 k

MFS = Original Misaka x (0.2544 x (1000/T) +0.5913) + (Mn x (0.0424 x(1000/T) +0.0986)) + (( 0.0042 x (1000/T) +0.0499) x Al) +

(0.128Cr) +(0.3Cu) ---(27)

Equation 2Equation 2Equation 2Equation 2: Temperature < 1200 k

MFS = Original Misaka x ((5.6823 x (1000/T) -3.8628) + (Mn x (0.9471 x

(1000/T) - 0.6438)) + ((0.0937 x (1000/T) -0.0235) x Al) +(0.128Cr) +(0.3Cu) ---(28)


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Observation Table:Observation Table:Observation Table:Observation Table:

slab

num.

Temperature Eq.

applied

MFS by revised

Misaka

Roll Force

Calculated

Actual

Roll

Force

Error

1 1309.65 1 118.833079 11.3577956 12.46 1.1022041 1286.81 1 126.784033 10.78515625 12.51 1.72484375

1 1263.63 1 140.614695 10.02162208 10.72 0.69837792

1 1238.4 1 146.494352 8.597858288 10.48 1.88214172

1 1210.93 1 147.225849 6.152436363 8.18 2.02756364

1 1181.26 2 172.087169 5.603219072 6.02 0.41678093

2 1272.03 1 117.868809 15.41418188 19.3 3.88581812

2 1244.71 1 129.832289 14.46243003 18.49 4.0275698

2 1225.01 1 143.148339 14.11343195 16.64 2.52656806

2 1204.59 1 148.747126 11.67224689 15.15 3.477753112 1180.81 2 177.169622 9.817497289 10.86 1.0425027

2 1154.27 2 192.545211 8.384089983 8.08 -0.30409

3 1282.43 1 119.482278 16.0739462 17.78 1.7060538

3 1255.09 1 130.711243 14.65671889 18.52 3.86328111

3 1235.09 1 142.70511 13.90773395 16.76 2.8522660

3 1214.34 1 154.869648 12.42278094 14.91 2.48721907

3 1190.32 2 173.113607 9.6755656 10.83 1.1544344

3 1163.68 2 193.670212 8.684389174 8.85 0.16561083

4 1289.97 1 114.245007 15.19473976 17.51 2.315260244 1262.95 1 121.495359 12.96557069 17.63 4.66442930

4 1242.6 1 137.219763 13.03110823 16.16 3.12889177

4 1221.37 1 144.801179 11.26494867 13.78 2.51505133

4 1197.25 2 160.967859 9.04890326 11.1 2.0510967

4 1170.61 2 178.755311 8.14263722 9.03 0.8873628

5 1280.55 1 120.6052025 12.52551539 14.28 1.7544842

5 1256.18 1 128.7370189 10.38804408 13.06 2.6719552

5 1242.74 1 141.7723088 9.928720039 11.15 1.22127991

5 1228.04 1 150.9967475 8.768288437 10.79 2.021711565 1209.41 1 149.3410288 6.314884225 8.94 2.62511577

5 1187.71 2 170.6318812 5.732091519 7.09 1.35790848

Table 4: Roll Force by Revised Misaka and error from the actual


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While the slab enters the finishing mill in the stand one,

temperature of slab is very well known to us but after that we just calculate

the temperature of slab according to the model proposed and finally when

the slab comes out of the last stand, we can measure its temperature. This

knowledge of temperature after last stand is very important and it should

be very close to the upper critical temperature (Ar3) temperature for the

steel grade.

As the transformation of steel microstructure takes place onlywhen the steel is cooled below Ar3, prior to the attainment of this

temperature steel slab is just cooled to bring down the temperature which

just a waste of energy. If the temperature variation between the stands is

controlled such that the final temperature of the slab coming out of

finishing mill is having the temperature almost very close to Ar3 then this is

the most productive condition with least energy wastage. Energy savings

are mostly due to the controlled heating of slab before it enters thefinishing mill which is adjusted in such a way that the temperature of slab

finally coming out of finishing mill is very close to Ar3 temperature.

For this a model or equation is to be formed for the calculation

of decrease in temperature from stand to stand which depends upon the

various variable and constant parameters. The parameters which will

control this temperature decrement are:

(a)Strain rate:Strain rate:Strain rate:Strain rate: the strain induced in the slab during the pass at

particular stand and the time required for this deformation both are

used for the determination of this temperature decrement. These

both the parameters are very well covered by a single parameter i.e.

strain rate.


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(b)Residence time:Residence time:Residence time:Residence time: Interstand residence time which represents the time

interval for open exposure to the coil between the 2 stand rolls. This

really controls the temperature loss as more the residence time, more

will be the temperature decrement.

(c)Initial temperature:Initial temperature:Initial temperature:Initial temperature: Initial temperature of the slab before entering

into the stand decides the rate of temperature loss as more the

temperature difference b/w slab and atmosphere higher will be the

rate of heat transfer from slab to open atmosphere.

(d)(d)(d)(d)WidWidWidWidth/ thickness ratio:th/ thickness ratio:th/ thickness ratio:th/ thickness ratio: Ratio is directly related to the ease ofradiating heat from the slab as more the surface area more easy will

be the temperature reduction while it is inversely proportional to the

volume or thickness of slab.

(e) Interstand coolingInterstand coolingInterstand coolingInterstand cooling : between all the stands, cooling is provided using

the water sprays which can be very important parameter for the

decrease in the temperature of slab between the 2stands.

(f)Coefficient of thermal conductivity of steel slabCoefficient of thermal conductivity of steel slabCoefficient of thermal conductivity of steel slabCoefficient of thermal conductivity of steel slab: this is a constant

quantity for a particular steel grade and describes the rate of heat

transfer from the metal slab to the rolling machinery.

Last parameter with several other parameters such as specific

heat of rolling machinery and etc can be taken as constant values as these

values do not change for a rolling process of one steel grade. Interstand

cooling rate can be variable from stand to stand but can be taken as a

constant for the cooling between particular 2 stands.


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Considering above parameters a general equation for the

temperature decrement between two stands can be given as:

T = aT = aT = aT = a + b+ b+ b+ b ttttresresresres + c+ c+ c+ c TTTT1111 + d+ d+ d+ d (W/H) +eW/H) +eW/H) +eW/H) +e ---(29)

Where:

: Strain rate

Tres : Residence time between two stands

T1 : Initial temperature of slab

(W/H) : Width / average thickness ratio.

The regression of the provided data to find out the coefficients

a, b, c, d and e is done by using MATLAB with the help of function regressregressregressregress

, the MATLAB code to this problem is:

MATLAB Program:

clc;clear all;for i=1:5

Tenter= input ('value of temperature of slabs on entering into stand');Texit= input ('value of temperature of slabs on exit out of stand');tt=(Tenter-Texit);

sr= input ('value of strain rates at stand');restime= input ('value of Residence time between stand 1 & stand 2');

wh= input ('ratio of width to thickness for slabs considered')if i==1

[X]=[ones(length(tt),1),(sr)',(restime)',(Tenter)',(wh)'];[c,ibeta,res,Ires,stats]=regress(tt',X,0.05)treg=c(1)+(c(2).*sr)+(c(3).*restime)+(c(4).*Tenter)+(c(5).*wh);

else[X]=[ones(length(tt),1),(sr)',(restime)',(Tenter)',(wh)',t'];[c,ibeta,res,Ires,stats]=regress(tt',X,0.05)


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treg=c(1)+(c(2).*sr)+(c(3).*restime)+(c(4).*Tenter)+(c(5).*wh)+(c(6).*t);endsubplot(1,2,1)

scatter(tt,treg,'filled')

hold on;t=tt;plot(t,tt)ylabel( 'Temperature difference by regression model')

xlabel('actual Temperature difference')subplot(1,2,2)

rcoplot(res,Ires)

pauseend

data for temperature decrement between the first two rolls of

the finishing mill is used to find the regression model to keep the

interstand cooling rate constant. The data includes the rolling of 29 slabs

of plain C-Mn steel: A12010 between first two stands.

Regression results and plot:Regression results and plot:Regression results and plot:Regression results and plot:

(i)(i)(i)(i) CCCC----Mn steel : pass1Mn steel : pass1Mn steel : pass1Mn steel : pass1 Regression ResultRegression ResultRegression ResultRegression ResultEquation formed for the calculation of temperature decrement

of Plain carbon steel between first two stand is:

T = -(1.0946 )-(4.1458 tres) +(0.0966 T1)+(0.8646x (W/H))

-113.8650 ---(30)


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PLOT:PLOT:PLOT:PLOT:

Fig 25: temperature decrement b/w stand 1 and stand 2 for A12010 by regression model

The plot shows the scattered points which represents the

temperature decrement between stand 1and 2 calculated by regression

model formed after regression. The plot present on the right side gives the

residuals between the regressed data and the actual data. The lines with

the green are representing that the regression is having 95% confidencelimit while the red one is outside this limit. As most of the data is lying

inside 95 % confidence limit thus the proposed model is quite acceptable.

15 20 25 30 3515

20

25

30

35

T

em

perature

difference

by

regression

m

odel

actual Temperature difference5 10 15 20 25

-6

-4

-2

0

2

4

6

8

10

Residual Case Order Plot

R

esiduals

Case Number


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(ii)(ii)(ii)(ii) CCCC----Mn steel : Pass 2Mn steel : Pass 2Mn steel : Pass 2Mn steel : Pass 2 Regression ResultRegression ResultRegression ResultRegression ResultAs the regression equation for the temperature decrement has

been calculated for pass 1, the next step is the temperature decrement

equation for further passes in the finishing mill. For this calculation,temperature decrement in the previous stand is also taken as the judging

parameter for the calculation of temperature decrement. Thus the general

equation for this decrement can be written as:

T = a + b tres + c T1 + d (W/H) +e (T) +f ---(31)

Where T is temperature decrement in the previous stand.

EquationEquationEquationEquation::::

T = -(0.0262 ) - (0.6641 tres )+ (0.1180 T1 )+ ( 0.1344 (W/H)

+ (0.1347 xT) -139.2790 ---(32)

PlotPlotPlotPlot::::


12 14 16 18 20 22 2412

14

16

18

20

22

24

Tem

peraturedifferenceby

regressionm

ode

l


-5

-4

-3

-2

-1

0

1

2

3

4

5


R

esiduals

Case Number


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(iii)(iii)(iii)(iii) CCCC----Mn steel : Pass 3Mn steel : Pass 3Mn steel : Pass 3Mn steel : Pass 3 Regression resultRegression resultRegression resultRegression resultEquationEquationEquationEquation

T = -( 0.0917

) - (2.5864

tres )+ (0.0046

T1 ) - (0.0774

(W/H)+ ( 1.0129 xT) +11.2039 ---(33)

Plots:Plots:Plots:Plots:


14 16 18 20 22 24 2614

16

18

20

22

24

26

28

T

em

pera

ture

difference

by

regressio

n

m

odel


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


R

esiduals

Case Number


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(iv)(iv)(iv)(iv) CCCC----Mn steel : Pass 4 Regression resulMn steel : Pass 4 Regression resulMn steel : Pass 4 Regression resulMn steel : Pass 4 Regression resulttttEquationEquationEquationEquation

T = -( 0.0707

) - (2.4051

tres ) + (0.0159

T1 ) - (0.0007

(W/H)+ (0.8960) x T) -7.2319 ---(34)

PlotsPlotsPlotsPlots

Fig 28: temperature decrement b/w stand 4 and stand 5 for A12010 by regression mode

18 20 22 24 26 28 3018

20

22

24

26

28

30

Tem


regressionmo

del-->

Actual Temperature difference -->5 10 15 20 25

-1.5

-1

-0.5

0

0.5

1


R

esiduals

-->

Case Number -->


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(v)(v)(v)(v) CCCC----Mn steel : Pass 5Mn steel : Pass 5Mn steel : Pass 5Mn steel : Pass 5 :::: Regression resultRegression resultRegression resultRegression resultEquationEquationEquationEquation

T = -( 0.0171

) - (0.9302

tres ) - (0.0076

T1 ) +( 0.0132

(W/H))+ ( 0.8455 x T) + 14.9623 ---(35)

PlotPlotPlotPlot


21 22 23 24 25 26 27 28 29 30 3121

22

23

24

25

26

27

28

29

30

31

Tem


regressionm

odel

actual Temperature difference

5 10 15 20 25

-0.4

-0.2

0

0.2

0.4

0.6

0.8


R

esiduals

Case Number


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OUTPUT TABLE:OUTPUT TABLE:OUTPUT TABLE:OUTPUT TABLE:

Slab

number

Temperature decrements

Stand 1 to stand 2 Stand 2 to stand 3 Stand 3 to stand 4

Actual By

regression

Actual By

Regression

Actual BY

regression

1 16.91 16.32 17.7 17.41 18.71 18.45

2 17.4 16.96 19.13 18.53 20.75 20.99

3 20.54 18.55 20.06 20.23 22.51 21.75

4 23.73 25.55 20.35 21.51 23.84 24.06

5 28.49 28.25 23.09 21.6 25.19 26.1

Table 5: Temperature decrement for slab of A12010 stand by stand

Slab

number

Temperature decrements

Stand 4 to stand 5 Stand 5 to stand 6

Actual By

regression

Actual By

Regression

1 23.07 23.14 25.02 25.02

2 24.21 24.44 26.71 27.433 25.43 24.9 27.23 27.23

4 25.94 25.94 30.78 30.78

5 27.22 27.47 29.67 28.98


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All mathematical models based on chemical composition of

steel and Rolling parameters for predicting MFS were compiled. Plant datafrom the HSM section of TATA STEEL was taken and then analyzed and

statistically operated to draw out the following conclusions:

1.MFS models by Sims and Misaka were compared for different

temperature range and passes , both the models are very close to

each other in the softening or high temperature range while there is

large deviation between the two in the lower temperature region

during final passes.

2.Effect of Nb microalloying on the MFS values of different steel grades

was studied and found that there is increase in the MFS values on Nb

addition due to its solution strengthening effect. Variation in the no

Recrystallization temperature was also observed which is increased asthe Nb concentration increases through the different grades.

3.Misaka model was optimized using MATLAB programming close to

the MFS values by Sims model resulting into varying optimized

equations from temperature to temperature to give excellent

prediction of MFS. Values of coefficients attached MFS model bychemistry can be observed varying over a temperature range.

4.Variation of these coefficients was observed to be 2 clusters in the

whole temperature range during rolling operation. Thus this variation

is tried to be best fitted by using a linear relation of these coefficients


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75

with temperature. 2 equations for each coefficient is formed in the

two temperature range ( 1200 k) which are finally

collected together to give the final equation to predict MFS. MFS

results from this revised equation are finally compared to the MFS bySims and found that the error is constant throughout the Rolling

process and reaches maximum to around 30-35 MPa which is quite

acceptable as compared to error between Misaka and Sims which

goes up to 80-90 MPa.

5.Roll Force is calculated by putting MFS calculated from the revisedEquation in the Sims formula. This Roll force is then finally compared

with the actual Roll force used during the process and found that

error is maximum during the middle passes (pass 3 or Pass 4) while

during early and final passes, error is maximum to 1 kilo tone.

6.Readings of temperature decrements through the Rolling stands in

finishing mill are finally regressed taking Strain rate, width by height

ratio for slab, residence time between the 2 stands and initial

temperature of slab as the controlling parameters. Results shows that

the regression model grows more accurate and more predictable if

we consider the temperature decrement in the previous stand also as

a controlling parameter.


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[1] Folio Sicilian Jr. Thesis on Mathematical Modeling of the Hot Strip

Rolling of Nb Microalloyed Steels, 1999, p. 1-63.[2] Folio Sicilian, Oswald Marini, Roberto G. Bruna, p.3.

[3] L. Petty KARJALAINEN, T. Terrence, M. MACCAGNO and John J. JONAS,

1995, p .3.

[4] Linda Lasses, Groan Newberg and Ulrika Berggren, 1996, p.7.

[5] Ningbo Yu, Shandong Wang, Xinhua Liu and Gooding Wang, Materials

and Science Engineering , 2004, p.1.

[6] M. Piertrzyk, John G. Lenard, Thermal-Mechanical Modelling of theFlat Rolling Process, 1991, p.53-63.

[7] www.wikipedia.org

[8] www.tatasteel.com

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