mathematical modelling of metallurgical processes with usage - Acta ...

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 308
MATHEMATICAL MODELLING OF METALLURGICAL PROCESSES
WITH USAGE OF THE THEORY OF INCREASING BODIES
Kochubey A.A. 1 , Syasev A.V. 1 , Vesselovskiy V.B. 1 , Mamuzych I. 2 , Makarenkov E.A. 1 ,
Scherbina E.A. 1
1
State University, Dnepropetrovsk, Ukraine
2
Faculty of Metallurgy, University of Zagreb, Croatia
Abstract
The mathematical model of increasing visco-elastic hollow of the barrel with a
fluid phase inside is offered. The material of the barrel in a fluid modular state represents
perfect fluid, and in solid state visco-elastic body. The analysis of temperature effect and
geometrical sizes the component on tight - strained state is conducted. The mathematical
model is designed and the calculation of a thermal state of rolls of hot-rolled steel is
conducted.
Key words: theory of increasing bodies, mathematical model, rolls of hot-rolled steel is
conducted, tight strained state
Introduction
Now theory of increasing bodies is on the stage of the becoming and is oriented
basically on problem solving, which one arise in practice in theory of facilities. However,
creation and development of new technologies in metallurgy (continuous casting of steel,
tube making and other parts of rotation with the help of a centrifugal casting of tubes etc.)
demands new methods development which allovs to decide, arising problems. Distinctive
feature of such problems is the availability: irregular temperature fields; mass and surface
power effects; existence, as a minimum of two phases of a stuff, which is object of research.
The development of a method of such problem solving is possible only in the supposition of
interaction interference of temperature and mechanical fields and results in necessity of
sharing of equations and laws of mechanics of a deformed solid body and theory of heat
conduction.
In the present paper, as against papers [1; 2], in which the problems about
crystallization of body of revolving on an external surface are reviewed, one of problems
arising in metallurgy is reviewed at manufacturing of tubes by a method of centrifugal to
pour problem about a crystallization of the circularbarrel with a fluid phase from inside.
Thermo–viscoelastic deforming of phase change, increasing in conditions of the barrel
The form for crystallization (steel mold) represents the hollow barrel of round cross
section with an inner radius a 0 and external b 0 . Before deformation the form is in state of
nature at the temperature of crystallizations of a melt is T 0 .
For simplicity we consider, that the form is a thin-wall cylindrical shell
b
0
− a
0

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 309
an internal side with radius a 1 . At the same time temperature of an external part is
instantaneously T < lowered up to value and the part of a stuff located in liquid
1
T 0
(a melt) passes in a firm modular state (solid phase). Thus, there is an escalating of stuff
owing to phase change, instead of owing to attachment of stuff from the outside, as it was
supposed in papers [4 - 6].
We shall introduce in the undeformed configuration the cylindrical coordinate
system of the axis (r, ϑ, z), which z coincides with an axis of the form, and unit vectors
designated by е r , е ϑ, е z . It is required to define temperature T castings, the law of motion of a
demarcation of phases a(t) and is tight - strained state of casting.
The temperature variation T − T
0 is considered small enough, so that heat expansion of
stuff can be neglected. We shall consider, that the behavior of a stuff of casting in liquid is
described by equations of state of ideal incompressible fluid [7], and in solid - equations of
state of an incompressible visco-elastic body in case of the linear law of a creep.
The mathematical formulation is reduced to a following set of equations
σ
1
−
1
ij
= p δ ij
∂ σ
∂ r
σ
= −
−σ
r , i r,
i ϑ,
i
r
( i = 1,2 )
σ
t,
r)
− σ ( t,
r)
= 2G
( t −τ*) ( ε ( t,
r)
− ε ( t,
)) −
r, 2( ϑ,2
2
r,2
ϑ,
2
r
−
t
∫
τ *
R(
t −τ
*, τ −τ
*) ( ε
r , 2
( τ , r)
− εϑ
,2
( τ , r))
dτ
∂ u&
u
ε r
+ ε ϑ
= 0 ,
r
&
& ε
r
= , & εϑ
= , T ′ + T ′ / r = 0
(1)
∂ r
r
with initial and boundary conditions
u r
( *( r)
, r) = ε ( τ *( r)
, r) = 0
τ
ϑ
, σ = 0,
σ = − p
T ( ) = , ( a()
t )
a 0
T 1
r r =a0
r r= a(
t )
T = T 0
, λ T ′ = ρ µ a(t & )
(2)
In the formulas (1), (2) and hereinafter index "1" corresponds to a stuff in liquid; an
index "2" - solid phase; p 1 - pressure arising in a melt; δ ij - components of a metric tensor in
an initial configuration; G ( t − *) – elastic-moment module of deformation;
2
τ
R ( t −τ *, τ −τ
*) – core of a relaxation of a visco-elastic stuff; τ * – time of a germing of a
stuff of a solid phase; λ – thermal conductivity; ρ – density of a melt; µ – latent heat of a
melting.
The problem (1), (2) was decided, using the approach which has been set up in
paper [2; 8]. In outcome the following determining ratio were obtained:

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 310
Law of motion of a demarcation of phases –
where
t
2
2
⎛ T ⎞
1
1 a*
( t)
a*
( )
⎜1
− − = lna*
( t)
−
T
⎟
⎝ 0 ⎠ 4 2
4
1 t
Λ
−
2
a0
ρ µ a(
t)
Λ = , a
*(
t)
=
λ T
a
0
0
(3)
temperature field –
T0
−T1
T = T1 +
ln( r a0)
(4)
ln( a(
t) / a )
tight-strained state –
u r
( t,
r )
D ( t ) − D ( τ * ( r ))
,
r
0
D(
t)
− D(
τ * ( r))
r
= − ε
r
( t,
r)
= εϑ
( t,
r)
=
2
t
* *
*
[ 2G
( t − τ *)( D ( t)
− D ( τ *)) − R( t − τ , τ − τ )( D( τ ) D ( τ
) dτ
]
2
σ
r
( t,
r)
− σ ϑ ( t,
r ) = −
2 s
∫ −
r
a0
t
2dr
⎡
*
*
* *
*
σ
r
( t,
r ) = ∫
⎢2G
S
( t − τ )( D () t − D ( τ
) − ( t τ τ τ )( D () τ D ( τ
) dτ
]
r
∫ − , − − . (5)
3
r ⎢
*
⎣
τ
The function D(t) is determined from an integral equation –
τ
where
t
H
1
( t)
D(
t)
−
∫
H
2
( t,
τ ) D(
τ ) =
0
p
a(
t)
2dr
H1(
t)
=
∫
2Gs
( t −τ *) ,
3
r
a0
H
t
2a&
( τ ) ⎡
⎤
t,
τ ) = ⎢2Gs
( t −τ
*) + ∫ R(
t − s,
τ − s)
ds⎥
a ( τ ) ⎢⎣
τ
⎥⎦
2
(
3
The numerical outcomes are submitted in a figure 1-6 at following values of main
specifications:
∂µ
( t,
τ )
−
R ( t,
τ ) = , ( , ) 2 ( ) ( )(1
γ ( t−τ )
µ t τ = G τ −ϕ
τ − e )
∂τ
( ) ( )С
G = const.,
− β τ
τ = 2G C + A e 0 = 0.38; А 0 =0.55; β = 0.015 h -1 ; γ = 0.1 h -1
ϕ
0 0

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 311
t
2,0
1,5
2
1,0
0,5
1
a * (t)
0,0
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
τ * ( a ) *
4,00
3,75
3,50
3,25
3,00
2,75
2,50
2,25
2,00
1,75
1,50
1,25
1,00
0,75
0,50
0,25
0,00
l = 0.25
l = 0.5
l = 0.75
l = 1
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
T 1 / T 0
Fig.1 The law of motion of a solidified front of
crystallization
Fig.2 Relation of time of a crystallization from
temperature and geometrical sizes of casting
In a figure 1 the graph
a = a
( * *
t
)
– law of motion of a solidified front of
crystally zation submitted at Λ = 1.5. The curve 1 corresponds to value T 1 /T 0 = 0.5, and
curve 2 – value T 1 /T 0 = 0.9. The outcomes of calculations demonstrate, that the temperature
variation of an environment renders essential influencing on process of a crystallization of
casting.
The schedules figured in a figure 2 illustrate relation of time of a crystallization of
casting to value of a temperature variation T 1 /T 0 at different fixed values of depth
l =
a
0
− a(t)
a
0
of the done pert. The outcomes of calculations demonstrate, that the change of the
geometrical sizes of casting exerts influence on duration of process of crystallization.
Namely: at increase of depth l walls of casting from value 0,25 up to value 0,75 at the
temperature of T 1 /T 0 = 0.5 the time of a crystallization of casting increases in 6.3 times, and
at the temperature of T1/T 0 =0.9 in 7 times, that confirms also temperature effect on time of
a crystallization. Besides, the temperature variation T 1 /T 0 , influences on duration of process
of crystallization and for separately taken wall thickness of casting. For example, at value of
depth l = 0. 5 the time indispensable for a crystallization of casting at the temperature of
T 1 = 0. 9 T 0 is in 4.5 times more, than in a case T1 = 0.5 T 0 . It speaks that, as well as in case
of a crystallization on an external surface (external escalating) the temperature variation
renders smaller influencing on duration of process of a crystallization, than change of the
geometrical sizes.
In a figure 3 the relation of time of a crystallization of casting to values of
temperature T 1 /T 0 for case of crystallization on an internal surface (internal escalating) –
continuous line and external – dotted curve is shown. These outcomes allow to make
following conclusions: the time of crystallization depends on its direction - at external
escalating this time will be more; the time of crystallization depends on value of temperature
T /T , and, than this relation is more, the more time of crystallization irrespective of a
1 0

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 312
τ * ( a * )
10,0
9,5
9,0
8,5
8,0
l = 0.5 внутр. наращивание
7,5
l = 0.5 внешн. наращивание
7,0
6,5
6,0
5,5
5,0
4,5
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
T 1 / T 0
0,0
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
0,25 D *
(t)
0,20
0,15
2
0,10
0,05
1
r*
0,00
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
Fig.3 Temperature effect of the form(shape) of
a on time of crystallization
Fig.4 Temperature effect of the form for radial
movings
direction of crystallization is and at tendency T 1 /T 0 to unit of the time of crystallization aims
at infinity.
For research of temperature effect T 1 of an external side on formation of
tight-trained state during a crystallization of casting we shall put numerical results in the
schedules (figure 4), illustrating relation to polar radius of a function D(t), describing radial
moving of points of cylindrical casting, at different values of temperature – the curve 1
corresponds to value T 1 /T 0 = 0.5, and curve 2 – value T 1 /T 0 = 0.9. From a figure 4 it is
visible, that at temperature rising the values of radial movings of points of casting increase
more than in 10 times.
Besides, in a figure 5 the law of motion of a demarcation of phases is submitted at
different directions of crystallization for miscellaneous values of temperature T 1 /T 0 . The
smooth
curve corresponds to value T 1 /T 0 =0.5, curve with points – value T 1 /T 0 =0.9. The
continuous lines fall into to a case of internal escalating, dashed - known solution at external
escalating [1].
The outcomes of calculations demonstrate, that at temperature rise T 1 from value
0.5T 0 up to value 0.9T 0 , the time indispensable for crystallization of all volume is from
inside augmented in 5 times, and at formation of casting with wall thickness, for example
l = 0.25 at the same temperature variation the time increases in 2.6 times. Such matchings
can be conducted and further in more details with the help of the schedules figured in a
figure 6, which enable to compare influencing depth l walls of casting on time of
crystallization at different directions of escalating and values of temperature T 1 /T 0 .
Continuous lines, as well as earlier - crystallization from inside, dashed - out of door.
From a figure 6 it is visible, that the time indispensable for manufacturing
(crystallization) of a part of the definite size at external escalating is much more, than at
in ternal. It speaks that the methods of internal escalating, at least, are more costeffective
from the point of view of costs of time. For example, to produce a part with final wall
thickness l = 0. 75 at the temperature of T 1 /T 0 = 0.45 by methods of external escalating, it is
required time in 2 times more, than at internal escalating. This difference will grow with
increase of temperature.
Thus, the time difference of crystallization will be unessential only

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 313
for parts, for which l ≤ 0. 25 . For remaining parts even at small temperatures, and it is not
always possible on technology, the internal escalating has advantage in time.
The analysis of the obtained outcomes demonstrates, that the temperature variation
of an external side and geometrical sizes of casting renders essential influencing on process
of crystallization of casting, and as a consequent and on tight-strained state in a part. So, for
example, at temperature rising not only the time of crystallization is rising, but also
essentially tight-strained state changes. Namely, at increasing of relative temperature T 1 /T 0
twice – the maximum shearing stresses on an internal part of casting (at external escalating)
increase in 28 times. As to movings, they also with increasing of temperature are increasing
approximately in 22 times at external escalating and more than in 10 times at internal
escalating.
a * ( t )
4
3
2
1
T 1 / T 0 = 0.5 внутр. наращивание
T 1 / T 0 = 0.9 внутр. наращивание
T 1 / T 0 = 0.5 внешн. наращивание
T 1 / T 0 = 0.9 внешн. наращивание
τ * ( a * )
0
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0
τ * ( a * )
4,00
3,75
3,50
3,25
3,00
2,75
2,50
2,25
2,00
1,75
1,50
1,25
1,00
0,75
0,50
0,25
0,00
l = 0.25 внутр. наращивание
l = 0.5 внутр. наращивание
l = 0.75 внутр. наращивание
l = 1 внутр. наращивание
l = 0.25 внеш. наращивание
l = 0.5 внеш. наращивание
l = 0.75 внеш. наращивание
l = 1 внеш. наращивание
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
T 1 / T 0
Fig.5 The law of motion of a solidified front for cases
of external and internal escalating of cylindrical
casting
Fig.6 Temperature effect on time of crystallization of
casting for different values of depth of done part
Thus, on time of crystallization apart from value of a temperature variation,
geometrical sizes a direction of crystallization has the essential influence.
Accoding to adopted assumptions the similar matchings on the one hand do
outcomes evaluation, but with other it is possible to speak about their veracity as they will
satisfactorily be agreeed with known computational and experimental data [9; 10]. The set
up approach can be used for the approximated solution of a problem on definition of
parameters of a centrifugal of tubes (pressure in a reshaped tube, law of change of movings
on depth in velocity function of escalating of a stuff, law of motion of a solidified front and
esc.). Thus, we leave out such essential factors, as nonstationary of a temperature field,
thermoexchange with environment and some other. Such simplified statement makes
possible reliable obtaining of numerical outcomes, though they are approximated from the
point of view of entirety of the description of process.
Mathematical modelling and Calculation of a thermal state of rolls of hot-rolled steel
Now the main way of effecting of a plate, is roll. During winding of rolls of flat
hot-rolled bars, reeled off on wire-winding machines of broad-strip mills temperature of

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 314
them, as a rule, is nonconstant. The temperature of rolled bands depends on properties of a
deformed stuff, rolling speed, amount of reduction, conditions of lubrication and roll
cooling, rolls and number of other factors. Influencing of a thermal factor the known
techniques of calculation of pressure in rolls leave out [11; 12].
The problem is decided at the following suppositions. The slip of convolutions
rather one another misses. The orbits of a band in a roll are esteemed as concentric rings.
Pressure and temperature is considered varied from an orbit to an orbit. Depth and the
elastic properties of each orbit can be different. The condition of coupling of conjugated
surfaces of adjacent orbits of a band in a roll will be written:
u
H
i
B
/
= ui
+ 1
− δ ( qi
, Ti
)
(6)
Where B - notation of an internal surface,
H - outside surface,
⎛ ′
δ q , T ⎞ - value of a gap between і and (і + 1) orbits,
⎜ i
⎝
′
i
i
⎟
⎠
q , - pressure and temperature on a contact of orbits.
i
T i
Let's consider multilayer winding shown in a figure 7. Here E n and d n represent a
Young's modulus and depth of a cylindrical layer arranged between radiuses r n-1 and r n. The
displacement U (r) for a cylindrical layer at set values of pressure on internal r = r n and
external r = r n+1 radiuses is determined under the formula:
U
2 2
2 2
1 ⎧ P ⎫
nrn
− Pn
+ 1rn
+ 1
rn
rn
+ 1
Pn
− Pn
+ 1
( r)
= ⎨( 1−ν ) r
+ ( + ν )
2 2
2 ⎬ (7)
En+
1 ⎩ rn
+ 1
− rn
r rn
+ 1
− rn
⎭
n, n+ 1
1
2
r
n+ 1
= rn
+ dn+1
(8)
We consider, that the basic material is isotropic, σ Z = 0 (flat state of stress) ,
ν = 0.33 – Poisson's constant. Unknown quantities here are the pressure P .
n
U
2 2
2 2
1 ⎧ Pn
rn
− Pn
+ 1rn
+ 1
rn
rn
+ 1
Pn
− P ⎫
n+
1
( r)
= ⎨( 1−ν
) r
+ ( + ν ) ⎬ −δ
( P)
2 2
2
(9)
En+
1 ⎩ rn
+ 1
− rn
r rn
+ 1
− rn
⎭
n, n+ 1
1
2
r
n+ 1
= rn
+ dn+1
(10)
σ
r
ε
r
=
E
r
ν
21
− σ
t
,
E
t
∂u
ε
r
=
(11)
∂r
1
ε
t
= ( −ν
21σ
r
+ σ
t
),
E
t
u
ε
t
=
r

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 315
Accoding to the solution of a problem of Lame about pressure in a tube under
operating of internal and external pressure the value of movings of an internal surface of a
ring-type orbit of a band with allowance for of temperature strains, is determined with the
help of a following equation:
u
2 2
B 1−
µ
i
qi− 1ri
−1
− qiri
1−
µ
i
qi−
1
− qi
2
i
= r +
r r + r
2 2 i−1
α
2 2 i−1
i T i−1
Ei
ri
− ri
−1
Er
ri
− ri
−1
∆T
(12)
where E – a modulus of elasticity of a stuff of a band i of an orbit
i
µ
i
– Poisson's constant
r – an inner radius i of an orbit
i −1
r – outside radius
i +1
r – current radius
q
i − 1
– internal pressure which is operational on i an orbit
q
i
– external pressure
α – a factor of linear temperature dilating
T
∆ T – a temperature variation (Fig.7)
Taking
into account a cylindrical symmetry and limiting by consideration of
distribution(propagation) of heat in a radial direction, we shall record a general view of a
heat conduction equation for multilayer barrels with allowance for of internal heat sources in
the form [13]:
∂( ρ c m
T ) 1 ∂T
∂ ∂T
m ( )
*
= λ
+ , m = 1 ,
m
+ λm
ω
m
M
∂τ
r ∂r
∂r
∂r
(13)
where
m = 1,...M
M
t
r
ω * m
m
ρ m
– number of a layer,
– quantity of orbits,
– time,
– coordinate,
– source intensity (sink) of heat for m of a layer,
λ – thermal conductivity,
– density of a stuff m of a layer.
On internal and external surfaces of a roll the unitized boundary conditions are set:
*
[ f0( τ ) M
0T
r =
]
* ∂T1
*
α
0λ1
r = r
= h
0 0α1
−
1 r
(14)
0
∂r
α
[ f1()
M1TM
]
* ∂ TM
*
M r= r
= h
M M
−
*
1λ
1α
τ
∂r
r=r M
(15)

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 316
Fig.7 Schemes of engagement of orbits of a band in a roll:
а) Model of a roll with forced cooling; б) approach of conjugated surfaces under operating of load; в) the
computational scheme
In (14), (15) f l
( τ ), ( l = 0 ,1 ) – boundary functions, which depending on
a type of boundary conditions are a surface temperature, ambient temperature by a heat
flow;
α
1
, α
M
– heat-transfer coefficients of internal and external surfaces. Supposing in
* * *
(10), (11) factors α h , M equal >, we shall have boundary conditions
l
,
l l
I, II, III of a kind and their different combinations. The distribution of temperature in an
initial instant looks like:
T
*
τ = 0
Tm
( r)
(16)
m
=
Contact between orbits imperfect. The connection between thermal and elastic
problems implements through factors of thermal resistance, which are calculated under the
formula [14]:
R (17)
−0,28
V
= 0,96/ r1 (2,6 + qi
)
General view of conditions on joints with allowance for of contact thermal
resistance and physico-chemical transformations accompanying with allocation (absorption)
of a heat:
⎧ + − ∂Tm
+
⎪Tm
−Tm
= Rm
( λm
)
∂r
⎨
⎪ ∂Tm
+ ∂Tm
−
( λ ) − ( ) = , m = 2,3,... M −1
m
λm
ωm
⎩ ∂r
∂r
(18)
where R m – value of a factor of contact thermal resistance for m of the joint

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 317
ω
m
– value of a source intensity (sink) of heat on m the joint Here index "+",
"-" indicates value of temperature and heat flow of adjoining surfaces
resistance is determined from ratio [13; 14].
For the solution of a problem (6) - (18) we use a method of finite differences. Let's
take for flexons on transversal coordinate a difference approximation of the second order of
accuracy rather ∆
rm
( ∆
rm
–integration step on coordinate for m of a layer). For maiden
derivative approximating by central differences of the second order of accuracy rather ∆
rm
.
For derivative on time we use approximating the maiden order of accuracy rather ∆ τ . The
difference equations of definition of values of temperature both for internal points of the
multilayer barrel, and for joints are adduced to a unified standard kind in the form of a set of
equations with a scalar matrix:
A T B T + C T = D ,
i i+ 1
+
i i i i−1
i
i = i b
÷ ie
the factors of which are calculated under the formulas for internal points of each orbit and
under the formula for joints between adjoining surfaces of adjacent orbits. Here i b
–
sequence number of an initial checkout of an incremental grid of an axis Oz, i
e
– numb er of
a final unit.
According to known values q
i , j tangential pressure σ i, j can be determined with
the help of an equation expressing an equilibrium condition of an orbit:
qiri
− q r
σ
i
=
r − r
i+
1
i+
1 i+
1
i
For check of algorithm the calculations on definition of tight-strained state of a roll
on a reel block were conducted. In a figure 8 design values of pressure of a roll on a reel
block are compared to experimental outcomes of activity [11].
The outcomes of numerical calculation correspond to the following elastic and
geometrical characteristics of stuffs of a band and reel block: L = E δ = 2100 kg/mm 2 , µ =µ δ
= 0.3, λ = 0.7, d=250mm, h=1mm, s=100МPа. The analysis of outcomes, reduced in a
figure 2 demonstrates, that at calculation of pressure in rolls disregarding changes of value
of gaps the pressure on a barrel with increasing of quantity of the reeled orbits continuously
increases. The outcomes receive to high.
The calculations executed (made) with allowance for of actual conditions of contact
interplay of surfaces of a band in a roll (with allowance for of gaps), demonstrate, that with
growth of quantity of orbits the value of pressure is increased up to definite (critical) value,
then practically does not change. It is conditioned by that the efforts from operating
subsequent (after a critical number) orbits almost are completely expended on change of
gaps between orbits. The obtained computational relations of pressure on a reel block from
quantity of the reeled layers of a band to the full will be agreeed experimental data [14].

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 318
The outcomes of a numerical solution are illustrated in a figure 9 by the way distributions of
temperature on depth of an orbit for concrete period (figure 9а) and relation of temperature
to time under different conditions of heat convection with environment (Fig.9b)
The distribution of temperature on depth of an orbit for concrete period is shown in
a figure 9а. On this figure a continuous line the outcomes obtained with allowance for of a
a)
Fig. 8 Experimental and computational relations of pressure of a roll on a reel
block from quantity of orbits at winding with a constant tension:
1- computational relations of pressure of a roll on a reel block from quantity of orbits at winding (disregarding of
changes of turn-to-turn gaps); 2- computational relations of pressure of a roll on a reel block (with allowance for
changes of gaps); 3,4- experimental relations of pressure of a roll on a reel block [11].
a)

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 319
b)
Fig.9 Influencings of the environmental conditions of thermoexchange on a temperature field of
а) relation of temperature to time at different depth of winding
b) a temperature Variation Т in a median convolution which is removed from a barrel
factor of thermal resistance (FTR) and crumpling of irregularities are rotined; a shaped line
is similar to a case R T =0 (ideal thermal contact). Curves in a figure 9а indicated in digit 1–
correspond to boundary conditions of 1-st kind, i.e. to absence of thermoexchange of an
external orbit with environment (Т с =200С); 2 and 3 – boundary conditions of III kind, 2 –
heat-transfer coefficient α =20 Wt / m 2 K; 3 – α =20 Wt / m 2 K. It is clear, that not taking
into account FTR results in essential underestimation of a temperature field of a roll, the
increasing of intensitry of thermoexchange with environment essentially accelerates a
cooling of a roll. The temperature variation Т in a median convolution, which is removed
from a barrel, is shown in a figure 9б. It is clear, that on external 150 orbit the heavy
gradient of temperature on depth of an orbit of winding is watched. The curves 1-5
correspond to time of winding 1-0,001с; 2-0,109с; 3-0,122с; 4-0,135с, 5-0,177с. On 50-th
and 100-th orbits temperature remains invariable irrespective of time and depth of winding a
curve 6.
Literature
[1] Arutyunyan, N. H., Drozdov A. D., Naumov V. A.: Mechanics of increasing
viscoelastic-plastic bodies, 1987, M. Science, p. 335
[2] Syasev A. V.: Formation of tight-strained state of cylindrical casting in conditions of
phase change// Visnyk DSU, 2001, Mechanics, B.2, Edition 4
[3] Yudin S. B., Levin M. M., Rozenfeld S. E.: A spun casting, 1972, Machine industry,
Moscow, p. 280
[4] Arutyunyan N. H., Kolmanovskiy V. B.: The theory of a creep of inhomogeneous
bodies, 1983, M. Science, p. 289
[5] Syasev A. V., Kochubey A. A.: Peculiarities of pressure distribution in the visco-elastic
round cylinder under the inner increasing as a model of from central casting of tubes //
Visnyk DNU, 2001, Mechanics, B.2, edition 4
[6] Kochubey A. A., Syasev A. V., Makarenkov E. A.: Continuous Escalating of Body of
Revolutions from a Visco-elastic Stuff in a Case of the Non-linear Law of a Creep, 41
(2002) 3, Metallurgy, 224
[7] Syasev A. V., Kochubey A. A. Mathematical modelling of centrofugal casting of tubes,
Mathematical modelling, 2001, №6, p. 59-63

Acta Metallurgica Slovaca, 8, 2002, 3 (308 - 320) 320
[8] Syasev A. V., Kochubey A. A., Mamuzic I.: Mathematical Model of Growth of
Diphasic Cylindrical Bodies with a Fluid Phase on the Inside, Metallurgy, 41 (2002) 3,
p. 227
[9] Esman R. I., Bahmat V. А., Korolyev V. M.: A thermal physics of foundry processes,
Belaruskaya Science, Minsk, 1998, p. 144
[10] Shmrga L.: A solidification and crystallization of steel ingots, Metallurgy, 1985, p. 248
[11] Mazur V.L.: Effecting of a sheet with a high-performance surface, Engineering, Kiev,
1982, p. 166
[12] Pimshteyn P.G., Zhukov V.N.: Calculation of pressure in the multilayer barrel with
allowance for of features of a contact of layers // of a Problem of strength, 1977, №5, p.
71 - 77
[13] Vesselovskiy V.B.: Application of computational methods of thermal modes of flight
vehicles for research and intensification of master schedules, Heat-mass exchange, B.10,
part 2, ITMT НАS Belarus, Minsk, 1996, p. 62 – 66
[14] Vesselovskiy V.B., Ostrovskaya A.V., Beluy N.I., Lyashenko V.I.: Contact thermal
resistance in members of designs, Heat-mass exchange - Heat conduction and problems
of optimization, B.3, ITMT НАS Belarus, Minsk, 2000, p. 91 – 98