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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
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

                                                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
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
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


                                        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
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
                                    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
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
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
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
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
Experimental Procedure
History of in-flow conditions



                                                         In-flow condition 2
                        90
                        80
                        70
   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   18
                                                              time (min)




                                                                           4/11/2013                                             16
Experimental Procedure
History of in-flow conditions


                                                   In-flow condition 3
                      90

                      80

                      70

                      60
  In-flow rate(LPM)




                      50

                      40
                                                                                                          flow rate
                      30

                      20

                      10

                       0
                           0   1   2   3   4   5    6    7     8       9    10   11   12   13   14   15
                                                        time (min)




                                                                     4/11/2013                                        17
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Summary of work Done in Previous
           Semester




                4/11/2013          37
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
Tundish Mixing Model & brief simulation
    Behavior of slab composition and bath depth during
    ladle changeover operation




                            4/11/2013                     39
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
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)
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
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”
Results
             Comparison

                                     comparison of experimental & modeled conductivity for 10% residual
                                                    volume, condition 1 & 40 lpm outflow
                         90

                         80
                                                                                                      experimental
                                                                                                      conductivity_10%_
                         70
                                                                                                      80 to 40 LPM
    conductivity (mS) --->




                         60

                         50

                         40

                                                                                                      modelled
                         30                                                                           conductivity_10%_
                                                                                                      80 to 40 LPM
                         20

                         10

                             0
                                 0        200       400        600          800   1000      1200
                                                          time (sec) --->
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.
Summary of work Done in Current
           Semester




               4/11/2013          46
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
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



                                                                                                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
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
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
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
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
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
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
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
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
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
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
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
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

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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
  • 16. Experimental Procedure History of in-flow conditions In-flow condition 2 90 80 70 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 18 time (min) 4/11/2013 16
  • 17. Experimental Procedure History of in-flow conditions In-flow condition 3 90 80 70 60 In-flow rate(LPM) 50 40 flow rate 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 time (min) 4/11/2013 17
  • 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
  • 33. Section B Mathematical Modeling of Single Strand Slab Casting Tundish & Simulations 4/11/2013 33
  • 34. INTRODUCTION 4/11/2013 34
  • 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
  • 37. Summary of work Done in Previous Semester 4/11/2013 37
  • 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”
  • 44. Results  Comparison comparison of experimental & modeled conductivity for 10% residual volume, condition 1 & 40 lpm outflow 90 80 experimental conductivity_10%_ 70 80 to 40 LPM conductivity (mS) ---> 60 50 40 modelled 30 conductivity_10%_ 80 to 40 LPM 20 10 0 0 200 400 600 800 1000 1200 time (sec) --->
  • 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.
  • 46. Summary of work Done in Current Semester 4/11/2013 46
  • 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
  • 63. 4/11/2013 63