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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

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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. 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 KanpurGuided by: By :Prof. Dipak Mazumdar Ankit Karwa (Y9096) Madhusudan Sharma (Y9312) 4/11/2013 1
2. 2. Introduction SECTION A: Experimental PartSECTION B: Mathematical Modeling Part 4/11/2013 2
3. 3. IntroductionSECTION 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. 4. IntroductionWhy 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. 5. Experimental Setup1. 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 = λ2.5Qactual2. Buffer tank for storage and continuous supply of water3. Electric pump to circulate water into tundish through inlet shroud4. Flow meter to control the inflow rate of water5. Salt, added to water to make it of different grade6. Conductivity probe placed just above the outlet to measure the conductivity of water exiting the tundish7. changing conductivity of the exiting water was read by a CyberScanTM conductivity meter, interfaced with a computer8. A manually operated stopper rod system is also placed over strand to ensure constant outflow rate 4/11/2013 5
6. 6. Summary of work Done in Previous Semester1. 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. 7. Summary of work Done in Previous Semester2. 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. 8. Summary of work Done in Previous Semester3.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 rateThree residual volume 23ltrs, 35ltrs, 46ltrs of salty water were consideredThree different In-flow rate conditions were consideredTotal 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. 9. Summary of work Done in Previous SemesterTypical 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. 10. Summary of work Done in Previous SemesterEvolution of grade intermixing time from grade transition curveC95% = 0.05 (Cold − Cnew) + Cnewthe time at which the 5% deviation line intersects the grade transition curve reflects the 95% grade intermixing time. 4/11/2013 10
11. 11. Results Variation of Grade intermixing time with In-flow conditions and residual volume 350 Avg Grade Intermixing time 300 for Residual vol=23ltrsAvg. 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. 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. 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 consideredUsing P&C on above mentioned condition gives a total of 45 different operating Conditions 4/11/2013 13
14. 14. Operating Parameters5 different experiments were performed at steady statebath depth of tundish, for these 5 experiments, 5 differentinflow rates were consideredSo Total150 Experiments ( 27 last sem and 123 this sem )were performed for 50 different Condition and 3 times foreach condition 4/11/2013 14
15. 15. Experimental ProcedureHistory 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. 16. Experimental ProcedureHistory 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. 17. Experimental ProcedureHistory 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. 18. Results and discussions Inflow Condition 1 800.00 700.00 600.00Avg. 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. 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) 0.069Variation of GIT with residual volume at constant outflow rate 4/11/2013 19
20. 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. 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. 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 ConditionVariation of GIT with inflow Condition at constant outflow rate 4/11/2013 22
23. 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 ConditionsVariation of GIT with inflow condition at constant Residual volume 4/11/2013 23
24. 24. Results and discussions Role of residual volume on intermixing timeResidual 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 timeOutflow rate also has influence on grade intermixing time.As outflow rate increases grade intermixing time decreases. Role of inflow rate on intermixing timegrade intermixing time least depends on inflow rate as compared to other operating parameter. 4/11/2013 24
25. 25. Establishing Correlation b/w GITand 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. 26. Establishing Correlation b/w GITand operating parameters Dimensional analysisDimensional 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. 27. Establishing Correlation b/w GITand 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 asπ 1= , π 2= , π 3= 4/11/2013 27
28. 28. Establishing Correlation b/w GITand 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. 29. Establishing Correlation b/w GITand operating parametersMultiple nonlinear regression was carried out to find out values of K, a and bEquation obtained after regression analysis is 4/11/2013 29
30. 30. Establishing Correlation b/w GITand 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 18Dimensionless 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. 31. Establishing Correlation b/w GITand 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. 32. Establishing Correlation b/w GITand operating parameters Validation of regression correlation Experimental Condition: Residual Volume: 0.042m3 Inflow Condition: condition 3 Outflow rate: 0.00067m3/sExperimental GIT obtained= 349.74 secPredicted GIT obtained= 372 secAs 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. 33. Section BMathematical Modeling of Single Strand Slab Casting Tundish & Simulations 4/11/2013 33
34. 34. INTRODUCTION 4/11/2013 34
35. 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. 36. Tundish Mixing Model & brief simulation  Determines steel composition entering into the mold Fig: flow pattern & different zones in tundish 2nd zone Q’p1Fig: six boxmodel with CP1 Q’m22 zones 1st zone
37. 37. Summary of work Done in Previous Semester 4/11/2013 37
38. 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. 39. Tundish Mixing Model & brief simulation Behavior of slab composition and bath depth during ladle changeover operation 4/11/2013 39
40. 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. 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. 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. 43. Tundish Mixing Model & brief simulation Modeled conductivity for 10% residual volume & condition 1 80 70 60conductivity (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. 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. 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. 46. Summary of work Done in Current Semester 4/11/2013 46
47. 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. 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. 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. 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. 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. 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. 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. 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. 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. 56. Clarifications for the graphs Experimental & Modelled GIT vs Residual volume 700 600Esperimental 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. 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. 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 timeAvg 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. 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. 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. 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. 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. 63. 4/11/2013 63