Mathematical modeling and Experimental Determination of Grade intermixing time and correlating grade intermixing time and operating parameters for a single strand slab casting tundish
The document summarizes a project presentation on mathematical modeling and experimental determination of grade intermixing time in a single strand slab casting tundish.
Key points:
- Experiments were conducted on a scaled-down physical model of an industrial tundish to measure grade intermixing time under different operating parameters.
- Over 150 experiments were performed under 50 different conditions by varying residual volume, inflow rate, and outflow rate.
- Results show grade intermixing time decreases with decreasing residual volume and increasing outflow rate. It depends least on inflow rate.
- Dimensional analysis and regression analysis are being used to develop a mathematical correlation between grade intermixing time and the operating parameters.
Polkadot JAM Slides - Token2049 - By Dr. Gavin WoodJuan lago vázquez
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Mathematical modeling and Experimental Determination of Grade intermixing time and correlating grade intermixing time and operating parameters for a single strand slab casting tundish
1. B.Tech. Project Presentation 2012-13
Mathematical modeling and Experimental Determination of
Grade intermixing time and correlating grade intermixing
time with operating parameters for a single strand slab
casting tundish
Department of material science and Engineering
Indian Institute of Technology Kanpur
Guided by: By :
Prof. Dipak Mazumdar Ankit Karwa (Y9096)
Madhusudan Sharma (Y9312)
4/11/2013 1
2. Introduction
SECTION A: Experimental Part
SECTION B: Mathematical Modeling Part
4/11/2013 2
3. 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
4. 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
5. 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
6. Summary of work Done in Previous Semester
1. Calibration of flow meter
Q exp = 1.183Qtheo - 2.140
Flow meter Calibration Curve for .4 scaled T28 Tundish
100
90
Experimental Flow rate (LPM)
80
70
60
50
40
30
20
10
0
0 10 20 30 40 50 60 70 80 90
Theoretical Flow rate (LPM)
4/11/2013 6
7. Summary of work Done in Previous Semester
2. Relation b/w area of orifice and no. of turns given to knob of stopper rod
No. of turns v/s Area of orifice (mm2)
350
326.47
300
y= 10.02x2 + 3.584x + 1.659
Area of orifice (mm2)
250 262.39
227.43
200
180.55
150
129.82
100 104.65
75.81
50 47.03
17.66 26.69
0
0 1 2 No. of turns
3 4 5 6
Plot of no. of turns v/s Area of orifice (mm2)
4/11/2013 7
8. 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
9. Summary of work Done in Previous Semester
Typical Grade intermixing curves for different operating condition
Grade intermixing curve for 23ltrs residual volume
90
80
Inflow condition
1
70
Conductivity (mS) --->
60
50
Inflow
40 Condition 2
30
20
10 Inflow condition
3
0
0 200 400 600 800 1000 1200 1400
time (sec) --->
4/11/2013 9
10. 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
11. Results
Variation of Grade intermixing time with In-flow conditions and residual
volume
350
Avg Grade Intermixing time
300
for Residual vol=23ltrs
Avg. Grade Intermixing time
250
Avg Grade Intermixing time
for Residual vol=35ltrs
200
Avg Grade Intermixing time
150 for Residual vol=46ltrs
100
Residual Volume
50
0
1 2 3
In-flow Condition
4/11/2013 11
12. Current Semester Work
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
4/11/2013 12
13. Operating Parameters
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
4/11/2013 13
14. Operating Parameters
5 different experiments were performed at steady state
bath depth of tundish, for these 5 experiments, 5 different
inflow rates were considered
So Total150 Experiments ( 27 last sem and 123 this sem )
were performed for 50 different Condition and 3 times for
each condition
4/11/2013 14
15. Experimental Procedure
History of in-flow conditions
In-flow condition 1
Assuming t=6
90
is the time at
80 which bath
height reaches
70 its steady
state value
In-Flow rate (LPM)
60
50
40
flow rate
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
time (min)
4/11/2013 15
18. Results and discussions
Inflow Condition 1
800.00
700.00
600.00
Avg. Grade Intermixing time (sec)
500.00
400.00
300.00
200.00
Outflow Condition (m3/s)
100.00
0.00073
0.00
0.00067
0.023
0.035
0.046 0.0006
Residual Volume (m3) 0.058
0.069
Variation of GIT with residual volume at constant inflow condition
4/11/2013 18
19. Results and discussions
Outflow rate = .0006 m3/s
800.00
Avg. Grade Intermixing time (sec)
700.00
600.00
500.00
400.00
300.00
200.00
Inflow Condition
100.00
C3
0.00
C2
0.023
0.035 C1
0.046
0.058
Residual Volume (m3) 0.069
Variation of GIT with residual volume at constant outflow rate
4/11/2013 19
20. Results and discussions
Inflow Condition 1
Avg. Grade Intermixing time (sec)
800.00
700.00
600.00
500.00
400.00
300.00 0.069
Residual Volume (m3/s)
200.00 0.058
0.046
100.00
.035
0.00
.023
0.0006
0.00067
0.00073
outflow rate (m3/s)
Variation of GIT with outflow rate at constant inflow condition
4/11/2013 20
21. Results and discussions
Residual Volume = .023 m3
Avg. Grade Intermixing time (sec) 300.00
250.00
200.00
150.00
100.00
Inflow Condition
C3
50.00
C2
0.00
0.0006 C1
0.00067
0.00073
Outflow rate (m3/s)
Variation of GIT with outflow rate at constant Residual volume
4/11/2013 21
22. Results and discussions
Outflow rate = .0006 m3/s
800.00
Avg. Grade intermixing time (sec) 700.00
600.00
500.00
400.00
residual volume (m3)
300.00
0.069
200.00 0.058
0.046
100.00
0.035
0.00
0.023
C1
C2
C3
Inflow Condition
Variation of GIT with inflow Condition at constant outflow rate
4/11/2013 22
23. Results and discussions
Residual Volume = .023 m3
300.00
Avg. Grade Intermixing time (sec)
250.00
200.00
150.00
100.00
0.00073
50.00
Outflow rate (m3/s)
0.00067
0.00
C1 0.0006
C2
C3
Inflow Conditions
Variation of GIT with inflow condition at constant Residual volume
4/11/2013 23
24. Results and discussions
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.
4/11/2013 24
25. Establishing Correlation b/w GIT
and operating parameters
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 4/11/2013 25
26. Establishing Correlation b/w GIT
and operating parameters
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,
4/11/2013 26
27. Establishing Correlation b/w GIT
and operating parameters
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=
4/11/2013 27
28. Establishing Correlation b/w GIT
and operating parameters
the functional relationship can be written in terms of dimensionless
groups as
Regression analysis carried out to find values of K, a and b.
4/11/2013 28
29. Establishing Correlation b/w GIT
and operating parameters
Multiple nonlinear regression was carried out to find out
values of K, a and b
Equation obtained after regression analysis is
4/11/2013 29
30. Establishing Correlation b/w GIT
and operating parameters
The fitness of the predicted model is shown in Figure,
by comparing actual measured dimensionless intermixing time with
the predicted dimensionless intermixing time
Dimensionless GIT Experimental V/S Dimensionless GIT Predicted
20
18
Dimensionless GIT Exp.
16
14 R2 = 0.86
12
10
8
6
4
2
0
0 2 4 6 8 10 12 14 16 18 20
Dimensionless GIT Predicted
4/11/2013 30
31. Establishing Correlation b/w GIT
and operating parameters
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)
4/11/2013 31
32. Establishing Correlation b/w GIT
and operating parameters
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.
4/11/2013 32
35. 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
36. Tundish Mixing Model & brief simulation
Determines steel composition entering into the mold
Fig: flow pattern &
different zones in
tundish
2nd zone
Q’p1
Fig: six box
model with CP1 Q’m2
2 zones
1st zone
38. Tundish Mixing Model & brief simulation
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
39. Tundish Mixing Model & brief simulation
Behavior of slab composition and bath depth during
ladle changeover operation
4/11/2013 39
40. Tundish Mixing Model & brief simulation
On applying mass balance on both mixing boxes for an incompressible fluid, with
well mixed assumption, yields
& ---eq(1)
C is dimensionless concentration; ---eq(2)
Transient volumes & flow rates
Volumes
fi = volume fraction of each box
Vi = volume of each box
Assumptions
1. In 2nd zone volume fraction decreases or increases in order to maintain its original
volume during continuous increase in tundish volume so;
Similarly;
2. Total plug flow volume fraction, mixing volume fraction & dead volume fraction
are constants
41. Tundish Mixing Model & brief simulation
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:
---eq(3)
• Following equations has been obtained on solving differential
equations for each box using eq(3)
42. Tundish Mixing Model & brief simulation
Initial conditions
@ t = 0, Cp1 = Cm1 = Cm2 = 0
As, ---Eq(4)
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
43. Tundish Mixing Model & brief simulation
Modeled conductivity for 10% residual volume & condition 1
80
70
60
conductivity (mS) --->
50 modelled
conductivity
40
30
20
10
0
0 200 400 600 800 1000 1200
time (sec) --->
Fig: conductivity(conc.) as a function of time using “6 box model”
45. Results
Comparison of grade intermixing time (95%) via Mathematical Model &
Experimentation
Type Grade transition time
Using Mathematical model 420 sec
Via experiments 233.67 sec
Table:Grade intermixing time obtained experimentally & via mathematical modelling for
condition 1 with 10% residual volume
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.
47. Refined Major Assumptions included in
Present Work
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;
Similarly;
2. Total plug flow volume fraction, mixing volume fraction & dead volume fraction
are constants
3. Dead volumes work together fd1 = fd2 = fd
48. 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
49. RTD Experiment
Input: Pulse Input Tracer Material: Salt Water
Volume fractions computing
C(dl)
1.05
d e m o 0.12659m o
d e d e m o d e m o d e m o
1.00
(peak)
d e m o
0.20715
d e m o d e m o d e m o d e m o
0.95
d e m o d e m o d e m o d e m o d e m o
0.90
0.29922
C(dl)
0.35676
0.85 d e m o d e m o d e m o d e m o d e m o
0.42582
0.80
d e m o d e m o d e m o d e m o d e m o
0.75
d e m o d e m o d e m o d e m o d e m o
0.70
0.65
0.0 0.5 1.0 1.5 2.0 2.5
theta
Figure: Non dimensional RTD curve
4/11/2013 49
50. 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
comparison of experimental & modeled conductivity for 10% residual
volume, condition 1 & 40lpm outflow
90
80
Experimental
conductivities (mS) ---->
70
conductivity
60
50
40 modelled conductivity
30
20
10
0
0 200 400 600 800 1000 1200
time (s) --->
4/11/2013 50
51. Comparison between Modeled &
Experimental conductivity
Case 2: Inflow condition 2: linear variation
Outflow condition: 40 lpm
Residual volume: 15% of steady state volume
comparison of experimental & modeled conductivity for 15% residual
volume, condition 2 & 40lpm outflow
90
80
conductivities (mS) ---->
70
60 Experimental
conductivity
50
40
modelled conductivity
30
20
10
0
0 200 400 600 800 1000 1200
time --->
4/11/2013 51
52. Comparison between Modeled &
Experimental conductivity
Case 3: Inflow condition 2: linear variation
Outflow condition: 36 lpm
Residual volume: 20% of steady state volume
comparison of experimental & modeled conductivity for 20% residual
volume, condition 1 & 36lpm outflow
45
40
Conductivities (mS) ---->
35
Experimental
30 conductivity
25
20
modelled
15 conductivity
10
5
0
0 200 400 600 800 1000
time (s) --->
4/11/2013 52
53. Comparison between Modeled &
Experimental conductivity
Case 4: Inflow condition 2: step function
Outflow condition: 36 lpm
Residual volume: 25% of steady state volume
comparison of experimental & modeled conductivity for 20% residual
volume, condition 1 & 36lpm outflow
45
40
Conductivities (mS) ---->
35
Experimental
30 conductivity
25
20
modelled
15 conductivity
10
5
0
0 100 200 300 400 500 600 700 800 900
time (s) --->
4/11/2013 53
54. 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
comparison of experimental & modeled conductivity for 20% residual
volume, condition 1 & 36lpm outflow
45
40
Conductivities (mS) ---->
35
Experimental
30 conductivity
25
20
modelled
15 conductivity
10
5
0
0 100 200 300 400 500 600 700 800 900
time (s) --->
4/11/2013 54
55. Comparison between Modeled &
Experimental conductivity
Table 8.3.2.1: Comparison of Grade intermixing time (GIT) calculated
via experiments & modelling
Experimental Modeled GIT
Cases (average) GIT (sec)
(sec)
Case 1 233.67 372
Case 2 287 385
Case 3 464.33 500
Case 4 539.33 263
Case 5 613 555
4/11/2013 55
56. Clarifications for the graphs
Experimental & Modelled GIT vs Residual volume
700
600
Esperimental vs modelled GIT (s) --->
500 Experimental GIT (s)
400 Modelled GIT (s)
300
200
100
0
0 5 10 15 20 25 30 35
Residual volume % --->
4/11/2013 56
57. 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
Volume fractions Values
Plug flow 0.09
Mixing 0.59
dead 0.32
4/11/2013 57
58. 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
Avg Vp2 t2 fp1 fp2 t1 t
(Qp2) (sec) (t=0) (t=0) (sec) (sec)
0.804 1.755 2.1828 0.015 0.075 0.4365 2.619
4/11/2013 58
59. 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
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
4/11/2013 59
60. STEP SIZE VARIATION
Adjustments of stopper rod needs to be automated to have small step size
90 Effect of step size on modeled conductivity & comparison with
experimental conductivity for "Case 2"
80
70
60
Experimental conductivity
Conductivity (mS) --->
50
modelled conductivity_step size_15
40
modelled conductivity_step size_5
30
modelled conductivity_step size_3
20
10
0
0 200 400 600 800 1000 1200
time (s) --->
Figure: effect of step size on modeled conductivity
4/11/2013 60
61. 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
62. 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