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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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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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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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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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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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(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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(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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(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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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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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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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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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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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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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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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::::
Fig 26: temperature decrement b/w stand 2 and stand 3 for A12010 by regression model
12 14 16 18 20 22 2412
14
16
18
20
22
24
Tem
peraturedifferenceby
regressionm
ode
l
actual Temperature difference5 10 15 20 25
-5
-4
-3
-2
-1
0
1
2
3
4
5
Residual Case Order Plot
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:
Fig 27: temperature decrement b/w stand 3 and stand 4 for A12010 by regression model
14 16 18 20 22 24 2614
16
18
20
22
24
26
28
T
em
pera
ture
difference
by
regressio
n
m
odel
actual Temperature difference5 10 15 20 25
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Residual Case Order Plot
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
peraturedifferenceby
regressionmo
del-->
Actual Temperature difference -->5 10 15 20 25
-1.5
-1
-0.5
0
0.5
1
Residual Case Order Plot
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
Fig 29: temperature decrement b/w stand 5 and stand 6 for A12010 by regression model
21 22 23 24 25 26 27 28 29 30 3121
22
23
24
25
26
27
28
29
30
31
Tem
peraturedifferenceby
regressionm
odel
actual Temperature difference
5 10 15 20 25
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Residual Case Order Plot
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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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