B.Tech. Project Presentation 2012-13
Mathematical modeling and Experimental Determination ofGrade intermixing time and correlating grade intermixingtime with operating parameters for a single strand slabcasting tundish
By :
Ankit Karwa (Y9096)
Madhusudan Sharma (Y9312)
Guided by:
Prof. Dipak Mazumdar
Department of material science and Engineering
Indian Institute of Technology Kanpur
4/11/2013 1
Introduction
SECTION A: Experimental Part
SECTION B: Mathematical Modeling Part
4/11/2013 2
Introduction
SECTION A (Experimental Part)
What is Tundish?• tundish is a broad, open container with one or more holes in the
bottom
• used to feed molten metal into an ingot mould
• acts as buffer of hot metals while ladles are switched
• other uses are help in smoothing out flow and for cleaning the metal
4/11/2013 3
Introduction
Why it is important to calculate grade intermixing time?
• During the ladle change operation if the melt contained in the new
ladle is of different grade, the mixing of two grades starts as soon
as new ladle opened into tundish, which will result into products
having a varying composition.
• Time of intermixing of these two different grades is known as
Grade Intermixing time
• Product manufactured during this time period is of varying
composition so it is of no use, wastage of material
• Therefore it is necessary to calculate and minimize grade
intermixing time
4/11/2013 4
Experimental Setup
1. 28T Single strand industrial Tundish
• built in the laboratory using PLEXIGLAS®
• Geometric scale factor (λ= 0.4) used to scale down the industrial tundish
λ = Lmodel/Lactual
Qmodel = λ2.5Qactual
2. Buffer tank for storage and continuous supply of water
3. Electric pump to circulate water into tundish through inlet shroud
4. Flow meter to control the inflow rate of water
5. Salt, added to water to make it of different grade
6. Conductivity probe placed just above the outlet to measure the conductivity of water exiting the tundish
7. changing conductivity of the exiting water was read by a CyberScanTM
conductivity meter, interfaced with a computer
8. A manually operated stopper rod system is also placed over strand to ensure constant outflow rate
4/11/2013 5
Summary of work Done in Previous Semester
1. Calibration of flow meter
Q exp = 1.183Qtheo - 2.140
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90
Exp
eri
men
tal F
low
rate
(L
PM
)
Theoretical Flow rate (LPM)
Flow meter Calibration Curve for .4 scaled T28 Tundish
4/11/2013 6
Summary of work Done in Previous Semester
2. Relation b/w area of orifice and no. of turns given to knob of stopper rod
4/11/2013 7
17.66 26.6947.03
75.81
104.65129.82
180.55
227.43
262.39
326.47
y = 10.02x2 + 3.584x + 1.659
0
50
100
150
200
250
300
350
0 1 2 3 4 5 6
Are
a o
f o
rifi
ce (
mm
2)
No. of turns
No. of turns v/s Area of orifice (mm2)
Plot of no. of turns v/s Area of orifice (mm2)
Summary of work Done in Previous Semester
3.Grade Transition curve for different operating conditions:
Since the geometry of the tundish, the steady state operating bath height of liquid
in tundish and the number strands fixed consequently, intermixing time is
expected to be a function of following variables:
• Residual volume of older grade
• In-flow rate
• Out-flow rate
Three residual volume 23ltrs, 35ltrs, 46ltrs of salty water were considered
Three different In-flow rate conditions were considered
Total 9 different operating conditions and for each condition experiment was
performed three times therefore total 27 experiments were carried out.
4/11/2013 8
Summary of work Done in Previous Semester
Typical Grade intermixing curves for different operating condition
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200 1400
Co
nd
ucti
vit
y (
mS
) --
->
time (sec) --->
Grade intermixing curve for 23ltrs residual volume
Inflow condition
1
Inflow
Condition 2
Inflow condition
3
4/11/2013 9
Summary of work Done in Previous Semester
Evolution of grade intermixing time from grade transition curve
C95% = 0.05 (Cold − Cnew) + Cnew
the time at which the 5% deviation line intersects the grade transition curve reflects the 95% grade intermixing time.
4/11/2013 10
Results
0
50
100
150
200
250
300
350
1 2 3 Resi
du
al
Vo
lum
e
Avg. G
rad
e I
nte
rmix
ing t
ime
In-flow Condition
Variation of Grade intermixing time with In-flow conditions and residual
volume
Avg Grade Intermixing time
for Residual vol=23ltrs
Avg Grade Intermixing time
for Residual vol=35ltrs
Avg Grade Intermixing time
for Residual vol=46ltrs
4/11/2013 11
Current Semester Work
4/11/2013 12
Verification of working of Experimental Set-up
Performed
Old experimental condition for which experiment performed last semester
• Initial Residual Volume = 23ltrs ( .023m3 )
• Inflow Condition = condition no. 1
• Outflow rate = 40 LPM (.0067m3 )
Grade Intermixing time Obtained last semester (GITold): 233 sec
Grade Intermixing time Obtained this semester (GITcurrent): 245.67 sec
GITold ≈ GITcurrent
Experimental set-up can be used for further experiments
Operating Parameters
4/11/2013 13
Consideration of new Operating Parameters
• initial residual volume of water
5 residual volume are considered
0.023 m3 , 0.035m3, 0.046m3, 0.058m3, 0.069m3
• Outflow rate
40 LPM (0.0067 m3/s)
36LPM (0.0060 m3/s)
44LPM (0.0073 m3/s)
• Inflow Condition
3 different inflow conditions were considered
Using P&C on above mentioned condition gives a total of 45 different
operating Conditions
Operating Parameters
4/11/2013 14
5 different experiments were performed at steady state
bath depth of tundish, for these 5 experiments, 5 different
inflow rates were considered
So Total 150 Experiments ( 27 last sem and 123 this sem )
were performed for 50 different Condition and 3 times for
each condition
Experimental Procedure
History of in-flow conditions
0
10
20
30
40
50
60
70
80
90
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
In-F
low
rate
(L
PM
)
time (min)
In-flow condition 1
flow rate
Assuming t=6
is the time at
which bath
height reaches
its steady
state value
4/11/2013 15
Experimental Procedure
History of in-flow conditions
0
10
20
30
40
50
60
70
80
90
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
In-f
low
rate
(L
PM
)
time (min)
In-flow condition 2
Flow rate
4/11/2013 16
Experimental Procedure
History of in-flow conditions
0
10
20
30
40
50
60
70
80
90
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
In-f
low
rate
(LP
M)
time (min)
In-flow condition 3
flow rate
4/11/2013 17
Results and discussions
4/11/2013 18
0.0006
0.00067
0.00073
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
0.0230.035
0.046
0.058
0.069 Ou
tflo
w C
on
dit
ion
(m
3/s
)
Av
g. G
rad
e In
term
ixin
g t
ime
(sec
)
Residual Volume (m3)
Inflow Condition 1
Variation of GIT with residual volume at constant inflow condition
Results and discussions
4/11/2013 19
C 1
C 2
C 3
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
0.0230.035
0.0460.058
0.069 Infl
ow
Co
nd
itio
n
Av
g. G
rad
e In
term
ixin
g t
ime
(sec
)
Residual Volume (m3)
Outflow rate = .0006 m3/s
Variation of GIT with residual volume at constant outflow rate
Results and discussions
4/11/2013 20
Variation of GIT with outflow rate at constant inflow condition
.023
.035
0.046
0.058
0.069
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
0.00060.00067
0.00073
Res
idu
al V
olu
me
(m3/s
)
Av
g. G
rad
e In
term
ixin
g t
ime
(sec
)
outflow rate (m3/s)
Inflow Condition 1
Results and discussions
4/11/2013 21
Variation of GIT with outflow rate at constant Residual volume
C 1
C 2
C 3
0.00
50.00
100.00
150.00
200.00
250.00
300.00
0.0006
0.00067
0.00073
Infl
ow
Co
nd
itio
n
Av
g. G
rad
e In
term
ixin
g t
ime
(sec
)
Outflow rate (m3/s)
Residual Volume = .023 m3
4/11/2013 22
Results and discussions
Variation of GIT with inflow Condition at constant outflow rate
0.023
0.035
0.046
0.058
0.069
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
C 1C 2
C 3 resi
du
al
vo
lum
e (m
3)
Av
g. G
rad
e in
term
ixin
g t
ime
(sec
)
Inflow Condition
Outflow rate = .0006 m3/s
4/11/2013 23
Results and discussions
Variation of GIT with inflow condition at constant Residual volume
0.0006
0.00067
0.00073
0.00
50.00
100.00
150.00
200.00
250.00
300.00
C 1
C 2
C 3
Ou
tflo
w r
ate
(m
3/s
)
Av
g. G
rad
e In
term
ixin
g t
ime
(sec
)
Inflow Conditions
Residual Volume = .023 m3
Results and discussions
4/11/2013 24
Role of residual volume on intermixing time
Residual volume of the liquid has the strongest influence on
the grade intermixing time. As the residual volume of the
liquid in tundish decreased it is observed that the grade
intermixing time also decreased
Role of outflow rate on intermixing time
Outflow rate also has influence on grade intermixing time.
As outflow rate increases grade intermixing time decreases.
Role of inflow rate on intermixing time
grade intermixing time least depends on inflow rate as compared
to other operating parameter.
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 25
To represent grade intermixing time in terms of these operating parameter a mathematical equation has to be develop.
Use dimension analysis and regression method
operating variables considered
• Residual volume of liquid present in tundish (Vres)
• Inflow rate ( Qin)
• Outflow rate (Qout,T)
• Acceleration due to gravity (g)
For regression analysis we will need numerical value for inflow rate so we considered weighted avg. of inflow condition over intermixing time interval
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 26
Dimensional analysis
Dimensional analysis is used to represent a physical phenomenon in
terms of a mathematical equation between various measurable
dependent and independent quantities in a nondimensional format.
functional relationship between the dependent and independent
variables
τintmix = f (Vres, Qin, Qout,T ,g)
On the basis of the Raleigh’s method of the indices,
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 27
From the Buckingham’s π -theorem,
• three independent nondimensional π groups to represent the above
relationship in a dimensionless form.
The nondimensional equivalence of the Equation
f(π 1, π 2, π 3) = 0
By using the dimensional homogeneity the values of a, b, c and d
can be found and hence three π groups are determined and given as
π 1= , π 2= , π 3=
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 28
the functional relationship can be written in terms of dimensionless
groups as
Regression analysis carried out to find values of K, a and b.
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 29
Multiple nonlinear regression was carried out to find out
values of K, a and b
Equation obtained after regression analysis is
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 30
The fitness of the predicted model is shown in Figure,
by comparing actual measured dimensionless intermixing time with
the predicted dimensionless intermixing time
0
2
4
6
8
10
12
14
16
18
20
0 2 4 6 8 10 12 14 16 18 20
Dim
en
sio
nle
ss G
IT E
xp
.
Dimensionless GIT Predicted
Dimensionless GIT Experimental V/S Dimensionless GIT Predicted
R2 = 0.86
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 31
Correlation for intermixing time
Where,
τ int.mix = Grade Intermixing Time (Sec)
Qin = Inflow Rate (m3/s)
Qout = Outflow Rate (m3/s)
Vres = Residual Volume (m3)
Establishing Correlation b/w GIT
and operating parameters
4/11/2013 32
Validation of regression correlation Experimental Condition:
Residual Volume: 0.042m3
Inflow Condition: condition 3
Outflow rate: 0.00067m3/s
Experimental GIT obtained= 349.74 sec
Predicted GIT obtained= 372 sec
As predicted and Experimental Grade intermixing time are close so it
is observed that this predicted equation is giving result close to
experimental result.
Section B
Mathematical Modeling of Single Strand
Slab Casting Tundish & Simulations
4/11/2013 33
INTRODUCTION
4/11/2013 34
Mathematical Model for single strand slab
casting tundish
Strand mixing model
Calculate final composition distribution in the slab caused by
combined effects of:
• Transient mixing in the strand
• Solidification during grade change
Tundish mixing model
Seeks to improve above model by adding mixing in the tundish
Also known as “6 box model”
4/11/2013 35
Determines steel composition entering into the mold
Tundish Mixing Model & brief simulation
Fig: flow pattern &
different zones in
tundish
Fig: six box
model with
2 zones
1st zone
2nd zone
Q’p1
CP1 Q’m2
Summary of work Done in Previous
Semester
4/11/2013 37
Three Major boxes
• Mixing boxes
• Two mixing boxes are connected in series
• Each is well mixed, so maintain a uniform concentration equal to its
outlet concentration
• Plug flow boxes
• Delay the passage of new grade through the tundish
• Also make the eventual concentration change entering the mould
• Dead volume boxes
• Empirically dead zones must exist in tundishes
• Reduce the effective volume available for mixing and plug flow
Tundish Mixing Model & brief simulation
Tundish Mixing Model & brief simulation
Behavior of slab composition and bath depth during
ladle changeover operation
4/11/2013 39
On applying mass balance on both mixing boxes for an incompressible fluid, with well mixed assumption, yields
C is dimensionless concentration;
Transient volumes & flow rates
Volumes
Assumptions
1. In 2nd zone volume fraction decreases or increases in order to maintain its original volume during continuous increase in tundish volume so;
2. Total plug flow volume fraction, mixing volume fraction & dead volume fraction are constants
Tundish Mixing Model & brief simulation
& ---eq(1)
---eq(2)
fi = volume fraction of each box
Vi = volume of each box
Similarly;
Flow rates
• Inlet flow rate, Qin are related by satisfying the following overall mass
balance on any box out of 6 boxes assumed in the model:
• Following equations has been obtained on solving differential
equations for each box using eq(3)
Tundish Mixing Model & brief simulation
---eq(3)
Initial conditions
@ t = 0, Cp1 = Cm1 = Cm2 = 0
As,
Eq(1) is solved using “4th order Runge Kutta Integration Method”
iteratively & the concentration are:
Cm2(i+1) = CT ; as there is no mixing in plug flow box
Tundish Mixing Model & brief simulation
---Eq(4)
Tundish Mixing Model & brief simulation
0
10
20
30
40
50
60
70
80
0 200 400 600 800 1000 1200
co
nd
ucti
vit
y (
mS
) --
->
time (sec) --->
Modeled conductivity for 10% residual volume & condition 1
modelled
conductivity
Fig: conductivity(conc.) as a function of time using “6 box model”
Comparison
Results
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200
co
nd
ucti
vit
y (
mS
) --
->
time (sec) --->
comparison of experimental & modeled conductivity for 10% residual
volume, condition 1 & 40 lpm outflow
experimental
conductivity_10%_
80 to 40 LPM
modelled
conductivity_10%_
80 to 40 LPM
Type Grade transition time
Using Mathematical model 420 sec
Via experiments 233.67 sec
Results
Table:Grade intermixing time obtained experimentally & via mathematical modelling for
condition 1 with 10% residual volume
Comparison of grade intermixing time (95%) via Mathematical Model &
Experimentation
This show that extent of validity of the model is up to 55%.
But this has been done for 1 case only that time. The present work
consists the comparison of modeled conductivity with experimental one
with different conditions incorporated.
Summary of work Done in Current
Semester
4/11/2013 46
3 major assumptions made in the previous work to solve the differential
equations
Assumptions
1. In 2nd zone volume fraction decreases or increases in order to maintain its original
volume during continuous increase in tundish volume so;
2. Total plug flow volume fraction, mixing volume fraction & dead volume fraction
are constants
3. Dead volumes work together fd1 = fd2 = fd
Refined Major Assumptions included in
Present Work
Similarly;
Refined Major Assumptions included in
Present Work
Critical assumptions included to tune the curve finer & enhance the
validity of 6 box model
1) fm1 >> fm2. So it is assumed that mostly mixing occurs in the m1 box only.
So Cm1= Cm2 and Cm2 = CT = Cout so Cm1 = Cout
2) fp1, fp2, fm1, fm2 are required for transient mode but RTD was done for
steady state
3) In RTD, mean = peak; as steep curve obtained in the beginning.
4) All the volume fractions can’t be split in two parts experimentally
(fi = fi,1 + fi,2).
Iteration has been performed on the basis of assumptions made earlier to get
the best fit.
4/11/2013 48
RTD Experiment
Input: Pulse Input Tracer Material: Salt Water
Volume fractions computing
4/11/2013 49
0.0 0.5 1.0 1.5 2.0 2.50.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.050.12659
0.20715
0.299220.35676
0.42582C(d
l)
theta
C(dl)
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
(peak)
Figure: Non dimensional RTD curve
Comparison between Modeled &
Experimental conductivity
It is the residual volume that affects GIT significantly, so 5 cases studied for 5
different residual volumes
Case 1: Inflow condition 1: 80 to 40 lpm
Outflow condition: 40 lpm
Residual volume: 10% of steady state volume
4/11/2013 50
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200
co
nd
ucti
vit
ies
(mS
) --
-->
time (s) --->
comparison of experimental & modeled conductivity for 10% residual
volume, condition 1 & 40lpm outflow
Experimental
conductivity
modelled conductivity
Comparison between Modeled &
Experimental conductivity
Case 2: Inflow condition 2: linear variation
Outflow condition: 40 lpm
Residual volume: 15% of steady state volume
4/11/2013 51
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200
co
nd
ucti
vit
ies
(mS
) --
-->
time --->
comparison of experimental & modeled conductivity for 15% residual
volume, condition 2 & 40lpm outflow
Experimental
conductivity
modelled conductivity
Comparison between Modeled &
Experimental conductivity
Case 3: Inflow condition 2: linear variation
Outflow condition: 36 lpm
Residual volume: 20% of steady state volume
4/11/2013 52
0
5
10
15
20
25
30
35
40
45
0 200 400 600 800 1000
Co
nd
ucti
vit
ies
(mS
) --
-->
time (s) --->
comparison of experimental & modeled conductivity for 20% residual
volume, condition 1 & 36lpm outflow
Experimental
conductivity
modelled
conductivity
Comparison between Modeled &
Experimental conductivity
Case 4: Inflow condition 2: step function
Outflow condition: 36 lpm
Residual volume: 25% of steady state volume
4/11/2013 53
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400 500 600 700 800 900
Co
nd
ucti
vit
ies
(mS
) --
-->
time (s) --->
comparison of experimental & modeled conductivity for 20% residual
volume, condition 1 & 36lpm outflow
Experimental
conductivity
modelled
conductivity
Comparison between Modeled &
Experimental conductivity
Case 5: Inflow condition 2: 80 to 40 lpm
Outflow condition: 44 lpm
Residual volume: 30% of steady state volume
4/11/2013 54
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400 500 600 700 800 900
Co
nd
ucti
vit
ies
(mS
) --
-->
time (s) --->
comparison of experimental & modeled conductivity for 20% residual
volume, condition 1 & 36lpm outflow
Experimental
conductivity
modelled
conductivity
Comparison between Modeled &
Experimental conductivity
Table 8.3.2.1: Comparison of Grade intermixing time (GIT) calculated
via experiments & modelling
4/11/2013 55
Cases
Experimental
(average) GIT
(sec)
Modeled GIT
(sec)
Case 1 233.67 372
Case 2 287 385
Case 3 464.33 500
Case 4 539.33 263
Case 5 613 555
Clarifications for the graphs
4/11/2013 56
0
100
200
300
400
500
600
700
0 5 10 15 20 25 30 35
Esp
eri
men
tal v
s m
od
elle
d G
IT (
s) -
-->
Residual volume % --->
Experimental & Modelled GIT vs Residual volume
Experimental GIT (s)
Modelled GIT (s)
Clarifications for the graphs
Curves look to be fitted with experimental curves for low residual volumes
As the residual volume increases the conductivity varies with the time very
slowly in the beginning
Then follows the trend of variation similar to experimental one
can be explained on the basis of flow environment of the chemical species
As the residual volume increases pure water molecule initially takes
time to move
The same trend obtained experimentally thereafter to reach the outlet
Obstacles can be significantly represented by dead volume fraction.
Volume fractions obtained experimentally through RTD curves
4/11/2013 57
Volume fractions Values
Plug flow 0.09
Mixing 0.59
dead 0.32
Time delay
Two plug flow boxes in the “6 box model”
Responsible for delay of passage of new grade
Represented by t; t = t1 + t2
t2 is given by
Qp2 is taken as average of range of its values.
Now t1 is given as
Time delay for the case 1
Total time delay is very small as compared to grade intermixing time
4/11/2013 58
Avg
(Qp2)
Vp2 t2
(sec)
fp1
(t=0)
fp2
(t=0)
t1
(sec)
t
(sec)
0.804 1.755 2.1828 0.015 0.075 0.4365 2.619
STEP SIZE VARIATION
It is the time interval between any two measured values of bath depth of the
tundish
Grade intermixing time is also a function of step size
Not possible to have small step sizes (<= 5 sec) manually
Step size taken here is 15 seconds
Step size of 15 seconds is divided in suitable fractions and a linear
variation of volumes or bath depths is assumed in the original step size
4/11/2013 59
Step size (s) h (s) Modeled GIT (s)
15 30 385
5 10 435
3 6 429
Average experimental grade intermixing time = 287 s
Table: variation of modeled GIT with step size
STEP SIZE VARIATION
Adjustments of stopper rod needs to be automated to have small step size
4/11/2013 60
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200
Co
nd
uct
ivit
y (m
S) -
-->
time (s) --->
Effect of step size on modeled conductivity & comparison with experimental conductivity for "Case 2"
Experimental conductivity
modelled conductivity_step size_15
modelled conductivity_step size_5
modelled conductivity_step size_3
Figure: effect of step size on modeled conductivity
CONCLUSION
Experimentation
The residual volume of liquid has the strongest influence on GIT
Inflow conditions has least influence on GIT compared to other
operating variables
Outflow rate also has significant influence on GIT, GIT decreases as
outflow rate increases
GIT correlations with operating conditions for single strand 28T
industrial slab casting tundish
4/11/2013 61
CONCLUSION
Mathematical Modeling
By putting in more valid assumptions, refinement of modeled
conductivity curve is being done
Residual volume increases validity of the model (in terms of GIT)
increases
Increase in residual volume makes a move towards steady state
condition (or transient nature is reducing) & volume fractions are also
determined for steady state condition, hence modeled GIT reaches
towards experimental GIT
Apart from that, fluctuations from experimental curves also increase
Time delay due to plug flow boxes is negligible as compared to GIT
Variation in step size has a minute but visible impact on modeled
conductivity
4/11/2013 62
4/11/2013 63