The equation of temperature field for the beginning phase of the continuous ingot forming and its applying in practice

19. 1994

Solidification of Metais and Alloys

Knepnięcie Metali i Stopów PL ISSN 0208-9386

THE EQUATION OF TEMPERATURE FIELD FOR THE BEGINNING PHASE

OF THE CONTINUOUS INGOT FORMING AND ITS APPL YING IN PRACTICE

MAŁGORZATA BIEDROŃSKA

RADOSŁA

W GRZYMKOWSKI

Silesian Technical University, Gliwice

The work concerns one of the fundamental problems associated with the process of the continuous casting, which are the beat ocenrences taking płace in the beginning phase of the ingot forming. lt is derived the equations, which ałlow to determine in what clistance of surface of the liquid metal mirror the processof cuticłe's growing begins. These equations indudes aU the essential parameters of the process and can be used in the analysis of the beat effects taking place in the ingot, in the

initiał phase o f i ts forming.

lntroduction

Among many factors determining the moment of the beginning of the cuticle forming process, the following parameters have the essential influence:

- pouring temperature, - fullering velocity,

- kind of the processing materiał,

- ingot's dimensions,

- parameters determining the process o f beat exchange between the ingot and surrounding its medium.

At the same time, it is appeared the question what is the dependence between these parameters and what is their influence over the process.

12 M. Biedrońska, R. Grzymkowski

The derived in the paper equations includes all the essential parametersof the process and can be used in the analysis of the effects taking place in the ingot in the beginning phase of its forming, among other, these equations allow to determinate, in what distancc of surface of the liquid metal mirror, the process of cuticle's growing begins (distance ś

on Fig. l).

/

/

Fig. l. The modelled area

The Mathematical Model

/

/

liquid phase solid (cuticl e )

It is considered flat ingots of 2R tJ1ickness produced on tJ1e vertical continuous casting machine. Witll assumed geometry of tlle areas, the smali conductivity in tlle direction of tlle ingot fullering and more distant side walls, tlle faiłure-free work of tlle machinc causes tJ1e generation of tlle pseudo-stacionary temperature field in tlle ingot-crystallizer system, whicb, in tlle longitudinal section of tlle liquid phase, above tlle cuticle (area

rr

on Fig. l), in immovable coordinate system, in tlle connection willi solidified ingot, is described by tJ1e equation

The Equation oj Temperatu re Field ...

ar

o2T

o

^:s; z < )::. •

w - = Q [ -

o<

^{X< R ':>}

oz ox2

instead in the area, in which the cuticle appears (area Q") by the equations

2JT 2J

²T;

w -

oz

' -

-

a·

^l

- -

ox 2 O <

x

< R

z

> ~;· ^{i = l, 2}

13

(la)

(l b)

where x, z are space coordinates, T, (x, z) - temperature, respectively of liquid phase (i= l) and solid phase (i = 2), w - fullering velocity and ai- thermal diffusivity coefti- cients. The equations (la) and (l b) are parabalie equations, in whicb the coordinate z ser- ves as the time and the wbole of the problem is completed by the boundary conditions:

- in the symmetry axis o f the ingot

aT

= o

^x

= o

ox

- on the surface of the ingot, according to the possessed informations

-A; oT; =

a

(T; - r ) x = R ox

A; aT; =

ą

^{x = R} ox

wbereas i = l w hen O < z < ś

·

^{and i}

=

^{2 w ben z} ~ Ę, -on the surface of pbases division

and the initial condition:

oT1 oT2

-At oxt + A2 ox = Y2 Kąl'(z) x = <p(z) z ~ ś

·

T₁

= f z =

O, O :s;

x

:s; R

(2)

(3a)

(3b)

(4a)

(4b)

(5)

wbere T⁰ is the pouring temperature, A - the thermal conductivity coefficient,

a

is the coefficient of beat penetration, q - is the beat stream, T~ - the temperature of the envi- ronment, and the function x

=

<p( z), z ~ Ę, • describes the position o f the pbases division's limit, wbere <p(Ę,')

=

B.

14 M. Biedrońska. R. Grzymkowski

The Solution of the Problem for the Area above the Cuticle

Le t assume, that on the level z

= s

^<

s*,

^{the temperatu re field T1 (x, z) in the in g ot}

section can be described by

(6)

where <P (x) is unknown function of suitahle class. After Ule dillerence approximation o f Ule left s ide or Ule equation (l) and including Ule condition (5) we o b tai n

T] (x,

S ) - ro . .

^~

w

S

⁼ a 1 T1 (x, ":>) 0 < x < R ⁽⁷⁾

From Ule last equation after including Ule fonnula (6) and after Ule easy transformations we obtain

(8) The genera l integral of Ule equation (8) has t11c form

<jl(x) = Aexp (

J a~·Ę x )

^{+ Bexp (-}

J

^a1

1

1

S x )

^{O < x < R (9)}

lt means, that Ule temperature field in UJe ingot section. on Ule level z =

s

^{is described}

by the function

.

( ~ )

^.

( ~ )

7J(x.s) = Aexp vals x + 1/exp -vals x + 7-{l O<x < R ⁽¹⁰⁾

which after including or Ule condition (2) has t11e fom1

T] (X,

S )

⁼

Dcl! (Ja\; s ^{X )}

⁺

ro o

^<

X

^<

R

^{(l l)}

To calculale Ule constant D we use UJe conditions (3), which lead to equation

The Equazion of Temperature Field ... 15

when U1e coefticient of heat penetration a, between Ule ingot and Ule erystallizer is known (condition (3a)), or to Ule equation

= q

when tilis heat exchange is characterized by Ule stream q (condition (3b)).

From Ule last equation we obtain

D

a

(T~ - "f'J)

!t means, U1at U1e function describing Ule temperature tield

D .. = - _ _ _ ..,!.ą _ _ - , -

Ą gis h ( Ja 7 ~ R )

in U1e ingot section, on Ule level z

= s

have respectively, Ule form

T, (X,

S)

^{a (T"' - 'f'J)}

c h ( J a·:~ x)

+"f'J

o<

^{X< R}

(13)

(14a)

(l4b)

(l5a)

(15b)

according to tilis, what information about Ule beat exchange between Ule ingot and Ule erystallizer we use.

The Relationships between the Parameters

of the Process and the Beginoing Position of the Cuticle

The process of cuticle's forming begins in Ule moment when the temperature of ingot's surface reaches Ule crystalłization tcmperature T•. Le t assume, iliat on Ule Ie vel z =

S •

^Ule

temperature o f ingot's surface bas reached Ule temperature T•. This assuming Ieads to equations:

16 M. Biedrońska. R. Grzymkowski

r

r

after easy transformalians we receive

and

a (T" - T~)

At (T^{0 -} T")

( 16a)

(l6h)

( 17a)

(l 7b)

The last equations include al l t11e essential parametersof t11e process. determining tJJC IJCat exchange in t11e beginning phase of continuous casting fonning. and can he uscd in t.hc practice, among otJ1ers. tJ1ese equations allow to detennine in what distance or surface of t11e liquid metal mirror, t11e process of cuticle's growing begins.

The Example of the Calculations

It was considered tJ1e cast steel ingot of plate's shape and tJ1ennophysical panuncters:

/.. =

30 Wm/K, a= 0.0000056 m²/s. Among otllers, i ^t was analysed t11e influence or ingot's tl1ickness R and tlle initial pauring temperature T⁰ for t11e initial cuticle's position.

Jn tJ1e calculations. it was assumed tllat w = 0.02 m/s, T = 30°C, a = 1200 W/mK.

q= l 800 000 W 1m².

The effects of calculations were graphic illustrated on Figures 2 and 3.

2.00

1.00 -

0.00

jdtstance, mm

l

l o

o l

o o

.

The Equation ofTemperature Field ...

4.00

1ctistance J mm

2.00

R =O 2 m R

=

^0.05 m •

• •

. . . ^{. .}

. ..

. ·

•

pounng temp , •c

0.00

l

^{i '}

1520 00 1560.00 1600.00 1520 00

4.00

2 00

0.00

1dtstance, mm

R = O 75 rn

o

. .

.

l ~ i ^{l l}

.· _,

^,

. ^.

. .

pounng temp , •c

; i ! i

4 (1(1

; (l(,l

0.00

1520.00 1560.00 1600.00

:distance,. mm

l

R =O 1 m

l •

1520.00 Fig. 2. The effect of calculation for the beat stream q

17

pounng temp ₁ •c

1 l ! ł

l ~60.00

•

o

l()( l (,l (l {l

pauring temp ,. "C

i 1

1560.00 1600.00

18 M. Biedrońska, R. Grzymkowski

2.00 _{distance, mm} 4.00

-=jd1slance, mm

-i

--1

~

'

~

1.00 : ^2.00

'

~

~ R = 0.05 m OC R = 0.2 m

c l

c c

~

o

'

0.00 ^o pouring temp 1 ·C

0.00

l

pounng temp , 'C - Ti-,1-r rr r l

l'

^{l i f f l 1 t 1}

--rrr

^{1 l l l l l}

l

^{l l l l l 1 l l l} l

1520.00 1560.00 1600.00 1520.00 1560.00 1600.00

2.00

1.00

0.00

-~ dislance

1 mm _,

'i l -l

j

- -

-~

]

~

R =O 75 m

:

~-! l

··t

o

'

ł>~-

'

o

=1

^{o o" o} pouring temp., C

--t- rTTTTTT r r r n l ! l l l 1 1

4.00

2.00

0.00

1520.00 1560.00 1600.00

j d1stance , mm -;

=

i

\

^{R=01 rn}

=1

_-l

o o o

o o

o o

o o o o

j

^pouring

-+-n~.-, r-1 .,t ,_,..1__,1_',...,-=r-,...~~~

1520.00 1560.00 1600.00 Fig. 3. Tbe effect of calculation for tbe beat penetration coefficient a

The Equation o f Temperature Field ...

Streszczenie

RÓWNANIE POLA TEMPERATURY DLA POCZĄTKOWEJ FAZY FORMOWANIA SIĘ WLEWKA CIĄGŁEGO

I JEGO ZASTOSOWANIE W PRAKTYCE

Skoncentrowano się na jednym z podstawowych problemów,

związanych z procesem ciągłego odlewania, jakim są zjawiska cieplne

zachodzące w początkowej fazie formowania się wlewka. Wyprowa- dzono między innymi równania, pozwalające określić w jakiej odległości

od powierzchni zwierciadła ciekłego metalu rozpoczyna się proces narastania naskórka. Wyprowadzone równania wiążą ze sobą wszystkie istotne parametry procesu i mogą być wykorzystane do analizy zjawisk cieplnych, zachodzących we wlewku w początkowej fazie jego formowania.

19