Mathematical modeling and Experimental Determination of Grade intermixing time and correlating grade intermixing time and operating parameters for a single strand slab casting tundish

B.Tech. Project Presentation 2012-13

Mathematical modeling and Experimental Determination ofGrade intermixing time and correlating grade intermixingtime with operating parameters for a single strand slabcasting tundish

By :

Ankit Karwa (Y9096)

Madhusudan Sharma (Y9312)

Guided by:

Prof. Dipak Mazumdar

Department of material science and Engineering

Indian Institute of Technology Kanpur

4/11/2013 1

Introduction

SECTION A: Experimental Part

SECTION B: Mathematical Modeling Part

4/11/2013 2

Introduction

SECTION A (Experimental Part)

What is Tundish?• tundish is a broad, open container with one or more holes in the

bottom

• used to feed molten metal into an ingot mould

• acts as buffer of hot metals while ladles are switched

• other uses are help in smoothing out flow and for cleaning the metal

4/11/2013 3

Introduction

Why it is important to calculate grade intermixing time?

• During the ladle change operation if the melt contained in the new

ladle is of different grade, the mixing of two grades starts as soon

as new ladle opened into tundish, which will result into products

having a varying composition.

• Time of intermixing of these two different grades is known as

Grade Intermixing time

• Product manufactured during this time period is of varying

composition so it is of no use, wastage of material

• Therefore it is necessary to calculate and minimize grade

intermixing time

4/11/2013 4

Experimental Setup

1. 28T Single strand industrial Tundish

• built in the laboratory using PLEXIGLAS®

• Geometric scale factor (λ= 0.4) used to scale down the industrial tundish

λ = Lmodel/Lactual

Qmodel = λ2.5Qactual

2. Buffer tank for storage and continuous supply of water

3. Electric pump to circulate water into tundish through inlet shroud

4. Flow meter to control the inflow rate of water

5. Salt, added to water to make it of different grade

6. Conductivity probe placed just above the outlet to measure the conductivity of water exiting the tundish

7. changing conductivity of the exiting water was read by a CyberScanTM

conductivity meter, interfaced with a computer

8. A manually operated stopper rod system is also placed over strand to ensure constant outflow rate

4/11/2013 5

Summary of work Done in Previous Semester

1. Calibration of flow meter

Q exp = 1.183Qtheo - 2.140

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90

Exp

eri

men

tal F

low

rate

(L

PM

)

Theoretical Flow rate (LPM)

Flow meter Calibration Curve for .4 scaled T28 Tundish

4/11/2013 6


2. Relation b/w area of orifice and no. of turns given to knob of stopper rod

4/11/2013 7

17.66 26.6947.03

75.81

104.65129.82

180.55

227.43

262.39

326.47

y = 10.02x2 + 3.584x + 1.659

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6

Are

a o

f o

rifi

ce (

mm

2)

No. of turns

No. of turns v/s Area of orifice (mm2)

Plot of no. of turns v/s Area of orifice (mm2)


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


Typical Grade intermixing curves for different operating condition

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200 1400

Co

nd

ucti

vit

y (

mS

) --

->

time (sec) --->

Grade intermixing curve for 23ltrs residual volume

Inflow condition

1

Inflow

Condition 2

Inflow condition

3

4/11/2013 9


Evolution of grade intermixing time from grade transition curve

C95% = 0.05 (Cold − Cnew) + Cnew

the time at which the 5% deviation line intersects the grade transition curve reflects the 95% grade intermixing time.

4/11/2013 10

Results

0

50

100

150

200

250

300

350

1 2 3 Resi

du

al

Vo

lum

e

Avg. G

rad

e I

nte

rmix

ing t

ime

In-flow Condition

Variation of Grade intermixing time with In-flow conditions and residual

volume

Avg Grade Intermixing time

for Residual vol=23ltrs





4/11/2013 11

Current Semester Work

4/11/2013 12

Verification of working of Experimental Set-up

Performed

Old experimental condition for which experiment performed last semester

• Initial Residual Volume = 23ltrs ( .023m3 )

• Inflow Condition = condition no. 1

• Outflow rate = 40 LPM (.0067m3 )

Grade Intermixing time Obtained last semester (GITold): 233 sec

Grade Intermixing time Obtained this semester (GITcurrent): 245.67 sec

GITold ≈ GITcurrent

Experimental set-up can be used for further experiments

Operating Parameters

4/11/2013 13

Consideration of new Operating Parameters

• initial residual volume of water

5 residual volume are considered

0.023 m3 , 0.035m3, 0.046m3, 0.058m3, 0.069m3

• Outflow rate

40 LPM (0.0067 m3/s)

36LPM (0.0060 m3/s)

44LPM (0.0073 m3/s)

• Inflow Condition

3 different inflow conditions were considered

Using P&C on above mentioned condition gives a total of 45 different

operating Conditions

Operating Parameters

4/11/2013 14

5 different experiments were performed at steady state

bath depth of tundish, for these 5 experiments, 5 different

inflow rates were considered

So Total 150 Experiments ( 27 last sem and 123 this sem )

were performed for 50 different Condition and 3 times for

each condition

Experimental Procedure

History of in-flow conditions

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

In-F

low

rate

(L

PM

)

time (min)

In-flow condition 1

flow rate

Assuming t=6

is the time at

which bath

height reaches

its steady

state value

4/11/2013 15



0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

In-f

low

rate

(L

PM

)

time (min)

In-flow condition 2

Flow rate

4/11/2013 16



0

10

20

30

40

50

60

70

80

90

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

In-f

low

rate

(LP

M)

time (min)

In-flow condition 3

flow rate

4/11/2013 17

Results and discussions

4/11/2013 18

0.0006

0.00067

0.00073

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

0.0230.035

0.046

0.058

0.069 Ou

tflo

w C

on

dit

ion

(m

3/s

)

Av

g. G

rad

e In

term

ixin

g t

ime

(sec

)

Residual Volume (m3)

Inflow Condition 1

Variation of GIT with residual volume at constant inflow condition


4/11/2013 19

C 1

C 2

C 3

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

0.0230.035

0.0460.058

0.069 Infl

ow

Co

nd

itio

n

Av

g. G

rad

e In

term

ixin

g t

ime

(sec

)

Residual Volume (m3)

Outflow rate = .0006 m3/s

Variation of GIT with residual volume at constant outflow rate


4/11/2013 20

Variation of GIT with outflow rate at constant inflow condition

.023

.035

0.046

0.058

0.069

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

0.00060.00067

0.00073

Res

idu

al V

olu

me

(m3/s

)

Av

g. G

rad

e In

term

ixin

g t

ime

(sec

)

outflow rate (m3/s)

Inflow Condition 1


4/11/2013 21

Variation of GIT with outflow rate at constant Residual volume

C 1

C 2

C 3

0.00

50.00

100.00

150.00

200.00

250.00

300.00

0.0006

0.00067

0.00073

Infl

ow

Co

nd

itio

n

Av

g. G

rad

e In

term

ixin

g t

ime

(sec

)

Outflow rate (m3/s)

Residual Volume = .023 m3

4/11/2013 22


Variation of GIT with inflow Condition at constant outflow rate

0.023

0.035

0.046

0.058

0.069

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

C 1C 2

C 3 resi

du

al

vo

lum

e (m

3)

Av

g. G

rad

e in

term

ixin

g t

ime

(sec

)

Inflow Condition

Outflow rate = .0006 m3/s

4/11/2013 23


Variation of GIT with inflow condition at constant Residual volume

0.0006

0.00067

0.00073

0.00

50.00

100.00

150.00

200.00

250.00

300.00

C 1

C 2

C 3

Ou

tflo

w r

ate

(m

3/s

)

Av

g. G

rad

e In

term

ixin

g t

ime

(sec

)

Inflow Conditions

Residual Volume = .023 m3


4/11/2013 24

Role of residual volume on intermixing time

Residual volume of the liquid has the strongest influence on

the grade intermixing time. As the residual volume of the

liquid in tundish decreased it is observed that the grade

intermixing time also decreased

Role of outflow rate on intermixing time

Outflow rate also has influence on grade intermixing time.

As outflow rate increases grade intermixing time decreases.

Role of inflow rate on intermixing time

grade intermixing time least depends on inflow rate as compared

to other operating parameter.

Establishing Correlation b/w GIT

and operating parameters

4/11/2013 25

To represent grade intermixing time in terms of these operating parameter a mathematical equation has to be develop.

Use dimension analysis and regression method

operating variables considered

• Residual volume of liquid present in tundish (Vres)

• Inflow rate ( Qin)

• Outflow rate (Qout,T)

• Acceleration due to gravity (g)

For regression analysis we will need numerical value for inflow rate so we considered weighted avg. of inflow condition over intermixing time interval



4/11/2013 26

Dimensional analysis

Dimensional analysis is used to represent a physical phenomenon in

terms of a mathematical equation between various measurable

dependent and independent quantities in a nondimensional format.

functional relationship between the dependent and independent

variables

τintmix = f (Vres, Qin, Qout,T ,g)

On the basis of the Raleigh’s method of the indices,



4/11/2013 27

From the Buckingham’s π -theorem,

• three independent nondimensional π groups to represent the above

relationship in a dimensionless form.

The nondimensional equivalence of the Equation

f(π 1, π 2, π 3) = 0

By using the dimensional homogeneity the values of a, b, c and d

can be found and hence three π groups are determined and given as

π 1= , π 2= , π 3=



4/11/2013 28

the functional relationship can be written in terms of dimensionless

groups as

Regression analysis carried out to find values of K, a and b.



4/11/2013 29

Multiple nonlinear regression was carried out to find out

values of K, a and b

Equation obtained after regression analysis is



4/11/2013 30

The fitness of the predicted model is shown in Figure,

by comparing actual measured dimensionless intermixing time with

the predicted dimensionless intermixing time

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10 12 14 16 18 20

Dim

en

sio

nle

ss G

IT E

xp

.

Dimensionless GIT Predicted

Dimensionless GIT Experimental V/S Dimensionless GIT Predicted

R2 = 0.86



4/11/2013 31

Correlation for intermixing time

Where,

τ int.mix = Grade Intermixing Time (Sec)

Qin = Inflow Rate (m3/s)

Qout = Outflow Rate (m3/s)

Vres = Residual Volume (m3)



4/11/2013 32

Validation of regression correlation Experimental Condition:

Residual Volume: 0.042m3

Inflow Condition: condition 3

Outflow rate: 0.00067m3/s

Experimental GIT obtained= 349.74 sec

Predicted GIT obtained= 372 sec

As predicted and Experimental Grade intermixing time are close so it

is observed that this predicted equation is giving result close to

experimental result.

Section B

Mathematical Modeling of Single Strand

Slab Casting Tundish & Simulations

4/11/2013 33

INTRODUCTION

4/11/2013 34

Mathematical Model for single strand slab

casting tundish

Strand mixing model

Calculate final composition distribution in the slab caused by

combined effects of:

• Transient mixing in the strand

• Solidification during grade change

Tundish mixing model

Seeks to improve above model by adding mixing in the tundish

Also known as “6 box model”

4/11/2013 35

Determines steel composition entering into the mold

Tundish Mixing Model & brief simulation

Fig: flow pattern &

different zones in

tundish

Fig: six box

model with

2 zones

1st zone

2nd zone

Q’p1

CP1 Q’m2

Summary of work Done in Previous

Semester

4/11/2013 37

Three Major boxes

• Mixing boxes

• Two mixing boxes are connected in series

• Each is well mixed, so maintain a uniform concentration equal to its

outlet concentration

• Plug flow boxes

• Delay the passage of new grade through the tundish

• Also make the eventual concentration change entering the mould

• Dead volume boxes

• Empirically dead zones must exist in tundishes

• Reduce the effective volume available for mixing and plug flow



Behavior of slab composition and bath depth during

ladle changeover operation

4/11/2013 39

On applying mass balance on both mixing boxes for an incompressible fluid, with well mixed assumption, yields

C is dimensionless concentration;

Transient volumes & flow rates

Volumes

Assumptions

1. In 2nd zone volume fraction decreases or increases in order to maintain its original volume during continuous increase in tundish volume so;

2. Total plug flow volume fraction, mixing volume fraction & dead volume fraction are constants


& ---eq(1)

---eq(2)

fi = volume fraction of each box

Vi = volume of each box

Similarly;

Flow rates

• Inlet flow rate, Qin are related by satisfying the following overall mass

balance on any box out of 6 boxes assumed in the model:

• Following equations has been obtained on solving differential

equations for each box using eq(3)


---eq(3)

Initial conditions

@ t = 0, Cp1 = Cm1 = Cm2 = 0

As,

Eq(1) is solved using “4th order Runge Kutta Integration Method”

iteratively & the concentration are:

Cm2(i+1) = CT ; as there is no mixing in plug flow box


---Eq(4)


0

10

20

30

40

50

60

70

80

0 200 400 600 800 1000 1200

co

nd

ucti

vit

y (

mS

) --

->

time (sec) --->

Modeled conductivity for 10% residual volume & condition 1

modelled

conductivity

Fig: conductivity(conc.) as a function of time using “6 box model”

Comparison

Results

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200

co

nd

ucti

vit

y (

mS

) --

->

time (sec) --->

comparison of experimental & modeled conductivity for 10% residual

volume, condition 1 & 40 lpm outflow

experimental

conductivity_10%_

80 to 40 LPM

modelled

conductivity_10%_

80 to 40 LPM

Type Grade transition time

Using Mathematical model 420 sec

Via experiments 233.67 sec

Results

Table:Grade intermixing time obtained experimentally & via mathematical modelling for

condition 1 with 10% residual volume

Comparison of grade intermixing time (95%) via Mathematical Model &

Experimentation

This show that extent of validity of the model is up to 55%.

But this has been done for 1 case only that time. The present work

consists the comparison of modeled conductivity with experimental one

with different conditions incorporated.

Summary of work Done in Current

Semester

4/11/2013 46

3 major assumptions made in the previous work to solve the differential

equations

Assumptions

1. In 2nd zone volume fraction decreases or increases in order to maintain its original

volume during continuous increase in tundish volume so;

2. Total plug flow volume fraction, mixing volume fraction & dead volume fraction

are constants

3. Dead volumes work together fd1 = fd2 = fd

Refined Major Assumptions included in

Present Work

Similarly;

Refined Major Assumptions included in

Present Work

Critical assumptions included to tune the curve finer & enhance the

validity of 6 box model

1) fm1 >> fm2. So it is assumed that mostly mixing occurs in the m1 box only.

So Cm1= Cm2 and Cm2 = CT = Cout so Cm1 = Cout

2) fp1, fp2, fm1, fm2 are required for transient mode but RTD was done for

steady state

3) In RTD, mean = peak; as steep curve obtained in the beginning.

4) All the volume fractions can’t be split in two parts experimentally

(fi = fi,1 + fi,2).

Iteration has been performed on the basis of assumptions made earlier to get

the best fit.

4/11/2013 48

RTD Experiment

Input: Pulse Input Tracer Material: Salt Water

Volume fractions computing

4/11/2013 49

0.0 0.5 1.0 1.5 2.0 2.50.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.050.12659

0.20715

0.299220.35676

0.42582C(d

l)

theta

C(dl)

d e m o d e m o d e m o d e m o d e m o






(peak)

Figure: Non dimensional RTD curve

Comparison between Modeled &

Experimental conductivity

It is the residual volume that affects GIT significantly, so 5 cases studied for 5

different residual volumes

Case 1: Inflow condition 1: 80 to 40 lpm

Outflow condition: 40 lpm

Residual volume: 10% of steady state volume

4/11/2013 50

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200

co

nd

ucti

vit

ies

(mS

) --

-->

time (s) --->


volume, condition 1 & 40lpm outflow

Experimental

conductivity

modelled conductivity



Case 2: Inflow condition 2: linear variation



4/11/2013 51

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200

co

nd

ucti

vit

ies

(mS

) --

-->

time --->



Experimental

conductivity

modelled conductivity



Case 3: Inflow condition 2: linear variation



4/11/2013 52

0

5

10

15

20

25

30

35

40

45

0 200 400 600 800 1000

Co

nd

ucti

vit

ies

(mS

) --

-->

time (s) --->



Experimental

conductivity

modelled

conductivity



Case 4: Inflow condition 2: step function



4/11/2013 53

0

5

10

15

20

25

30

35

40

45

0 100 200 300 400 500 600 700 800 900

Co

nd

ucti

vit

ies

(mS

) --

-->

time (s) --->



Experimental

conductivity

modelled

conductivity



Case 5: Inflow condition 2: 80 to 40 lpm



4/11/2013 54

0

5

10

15

20

25

30

35

40

45

0 100 200 300 400 500 600 700 800 900

Co

nd

ucti

vit

ies

(mS

) --

-->

time (s) --->



Experimental

conductivity

modelled

conductivity



Table 8.3.2.1: Comparison of Grade intermixing time (GIT) calculated

via experiments & modelling

4/11/2013 55

Cases

Experimental

(average) GIT

(sec)

Modeled GIT

(sec)

Case 1 233.67 372

Case 2 287 385

Case 3 464.33 500

Case 4 539.33 263

Case 5 613 555

Clarifications for the graphs

4/11/2013 56

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30 35

Esp

eri

men

tal v

s m

od

elle

d G

IT (

s) -

-->

Residual volume % --->

Experimental & Modelled GIT vs Residual volume

Experimental GIT (s)

Modelled GIT (s)

Clarifications for the graphs

Curves look to be fitted with experimental curves for low residual volumes

As the residual volume increases the conductivity varies with the time very

slowly in the beginning

Then follows the trend of variation similar to experimental one

can be explained on the basis of flow environment of the chemical species

As the residual volume increases pure water molecule initially takes

time to move

The same trend obtained experimentally thereafter to reach the outlet

Obstacles can be significantly represented by dead volume fraction.

Volume fractions obtained experimentally through RTD curves

4/11/2013 57

Volume fractions Values

Plug flow 0.09

Mixing 0.59

dead 0.32

Time delay

Two plug flow boxes in the “6 box model”

Responsible for delay of passage of new grade

Represented by t; t = t1 + t2

t2 is given by

Qp2 is taken as average of range of its values.

Now t1 is given as

Time delay for the case 1

Total time delay is very small as compared to grade intermixing time

4/11/2013 58

Avg

(Qp2)

Vp2 t2

(sec)

fp1

(t=0)

fp2

(t=0)

t1

(sec)

t

(sec)

0.804 1.755 2.1828 0.015 0.075 0.4365 2.619

STEP SIZE VARIATION

It is the time interval between any two measured values of bath depth of the

tundish

Grade intermixing time is also a function of step size

Not possible to have small step sizes (<= 5 sec) manually

Step size taken here is 15 seconds

Step size of 15 seconds is divided in suitable fractions and a linear

variation of volumes or bath depths is assumed in the original step size

4/11/2013 59

Step size (s) h (s) Modeled GIT (s)

15 30 385

5 10 435

3 6 429

Average experimental grade intermixing time = 287 s

Table: variation of modeled GIT with step size

STEP SIZE VARIATION

Adjustments of stopper rod needs to be automated to have small step size

4/11/2013 60

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200

Co

nd

uct

ivit

y (m

S) -

-->

time (s) --->

Effect of step size on modeled conductivity & comparison with experimental conductivity for "Case 2"


modelled conductivity_step size_15



Figure: effect of step size on modeled conductivity

CONCLUSION

Experimentation

The residual volume of liquid has the strongest influence on GIT

Inflow conditions has least influence on GIT compared to other

operating variables

Outflow rate also has significant influence on GIT, GIT decreases as

outflow rate increases

GIT correlations with operating conditions for single strand 28T

industrial slab casting tundish

4/11/2013 61

CONCLUSION

Mathematical Modeling

By putting in more valid assumptions, refinement of modeled

conductivity curve is being done

Residual volume increases validity of the model (in terms of GIT)

increases

Increase in residual volume makes a move towards steady state

condition (or transient nature is reducing) & volume fractions are also

determined for steady state condition, hence modeled GIT reaches

towards experimental GIT

Apart from that, fluctuations from experimental curves also increase

Time delay due to plug flow boxes is negligible as compared to GIT

Variation in step size has a minute but visible impact on modeled

conductivity

4/11/2013 62

4/11/2013 63