LATVIAN JOURNAL OF PHYSICS AND
TECHNICAL SCIENCES
2002, N 3
STABILITY CONDITIONS AT THE
BUILDUP FORMATION
ON THE SURFACE OF A REFRACTORY
Department of Physics,
Zeļļu iela 8,
A phenomenological
model has been developed to describe the buildup formation and to study the
stability conditions for a uniform growth of the builtup layer. Based on
suitable approximations, analytical formulae have been derived allowing the
stability region in the concentrationtemperature plane to be defined depending
on the initial roughness of the growing surface.
1. INTRODUCTION
A typical lifetime
contraction of the inductor in an induction channel furnace (ICF) caused by
buildup formation is from several weeks up to a year. The dangerous zones of
the ICF where buildups mostly form are [1]:

the
part of the channel near its openings;

the
throat zone of the furnace;

the
top of the inductor between the channel’s outlets;

the in and outlets of the furnace.
Especially disadvantageous
situations in terms of buildup formation have been reported in [2], e.g. for

the melting of Fe with a high MgO concentration (> 2%);

the
underheating of the melt at weekends;

the rapid development of fractal structures at the
surface of the refractory.
It is impossible to investigate
experimentally the process of buildup formation in industrial devices, and
development of theoretical models is therefore of prime importance. These can
be based on the analysis of the melt and the structure of deposits, on
measurements of the geometry of a buildup after dismounting the channel, on
the velocity
Fig. 1. A compact thin
layer and a fractally structured thick layer
builtup in an ICF channel
at the melting of ferrous materials
and temperature distributions obtained
under laboratory conditions and on numerical experiments [1]. A typical view of a builtup
layer is given in Fig. 1 – a thin layer of oxides on the surface of the
refractory, which is compact (of a high density), and an outer layer, which is
porous (of a low density) and has a fractal structure with impurities in the
form of gaseous and metallic droplets. It is the fractal type of buildup
formation that intensifies the process and contracts the channel’s lifetime.
From the technological point of view it is especially important to know the
conditions under which the character of buildup growth changes from compact or
uniform to fractal.
2. THE MODEL
To study
the stability of a uniform buildup formation we will consider a simple model. It is illustrated in Fig. 2. The surface
_{} separates the solid
builtup phase (_{}) from the molten metal (_{}). The liquid phase (e.g. Fe, Al) contains a builtup
substance (e.g. MgO, Al_{2}O_{3}) as an impurity with concentration _{}. Here the concentration is defined as the number of
particles (molecules) per volume. It is assumed that the relative concentration
(i.e. normalized to that of basic substance particles) is low. The surface is
flat except the region around _{} where a semispherical
bulge of radius r is located. The
main problem is to find the conditions under which the bulge develops or
disappears during the process of impurity deposition. We assume that the rate
of the process is limited by diffusion of impurity through the concentration boundary layer of thickness _{}, as shown in Fig. 2. Inside this layer, the mass transport
perpendicular to the surface is governed by diffusion. The figure also shows a
liquid flow parallel to the surface.
Fig.
2. A schematic
illustration of the model
Both the concentration
and the temperature are assumed to be constant outside the concentration
boundary layer where the liquid flow is strongly turbulent. The growth kinetics
of the builtup layer is described by a lattice model. In this model, one
lattice site is associated with one molecule of the impurity substance. The
lattice is present in the solid phase and can formally be extended to the liquid phase. An elementary
volume, _{}, is associated with one lattice site, where _{} is the concentration
of molecules in the solid phase, M is
the molar mass of the builtup substance, _{} is its density, and _{} is the Avogadro
number. Each molecule on the surface can detach from the solid phase with the
probability per time _{}. Each molecule appearing in an elementary volume _{} around any of the
lattice sites located in the liquid phase in the close vicinity of the surface _{} can attach to the
solid phase with the probability per time _{}. According to this model, the development of the surface in
time is described by the kinetic equation
_{}. (1)
Here _{}, _{} and _{} is the linear size of
a cubic elementary cell with volume _{}. In the thermodynamic equilibrium there holds
_{}, (2)
where _{} is the variation in the system energy if one impurity particle (molecule) is subtracted from a growing surface with curvature radius r. The latter relation agrees with the principle of detailed balance [3, 4]. We have assumed here that the variation in the thermodynamic potential is equal to that in the system energy, which is true in the actual case of small concentration c at the fixed total volume of the system. According to (1) and (2), we have
_{}. (3)
Equation (3) should be solved
together with the diffusion equation for the impurity concentration
_{} , (4)
where v is the velocity of the liquid flow and D is the impurity diffusion coefficient in the liquid phase. The
particle conservation (balance) condition at the surface reads as
_{}. (5)
According to
the discussion in the beginning of this section, the boundary conditions of the
concentration layer are
_{} (6)
where _{} and _{} are the impurity
concentration and the temperature in the bulk of molten metal. In the first
approximation we assume that the temperature on the surface _{} is constant, i.e., _{}.
Our aim is to derive
stability conditions for a flat growing surface. If the intensity of the
surface growth at _{} exceeds that at _{}, there develops a bulge (shown in Fig. 1), which means that
the flat surface is unstable with respect to the initial roughness
characterized by grains of radius r.
The development of these grains results in a distortion of the flat profile of
the growing surface, i.e., in the formation of a porous or dendritelike
builtup layer. Therefore, the stability condition at which such a bulge is
tending to dissolve (i.e., the surface is tending to become ideally flat) reads
as
_{}. (7)
3. ESTIMATION OF THE ENERGY e (r)
For simplicity, let us first assume that the
impurity particles only interact with each other. Then, by analogy with the
classical nucleation theory [3], the binding energy of a nucleus of radius r can be written as the sum of the bulk
(index b) and surface (index s) terms:
_{}, (8)
where _{} is the total number of
particles in a spherical nucleus, _{} is the number of
particles on the surface, and _{} is the size of the
unit cell. For large enough n we have
_{} , (9)
where _{}. According to our lattice model _{} and _{}, where _{} is the energy of one
bond, l and _{} are the average
numbers of the nearest impurity molecules in the bulk of solid and on the
surface, respectively. The numbers l
and _{} depend on the lattice
symmetry and the microscopic structure of the surface. Thus, Eq.(9) becomes
_{}. (10)
In the
general case of interaction between A (basic substance) and B
(impurity) particles we have
_{}, (11)
where _{} is the binding energy
between two particles, X and Y.
Equation (11) is
consistent with a simple counting of bonds when one impurity particle is moving
from the solid phase to the liquid one with the same lattice structure.
Besides, it is assumed that the impurity particles have only basic substance
atoms at the neighbouring sites in the liquid phase, which is consistent with
our assumption that the concentration c
is small.
4. ESTIMATION OF THE
CRITICAL RADIUS IN THE CASE OF v = 0
At large
times t, the impurity concentration c is given by the stationary solution of
Eq. (4) if the growth on the surface is relatively slow. This condition is
satisfied in the actually considered limit of small concentrations. At the zero
velocity along the surface, i.e. _{}, the linear stationary concentration profile along zaxis is given by the solution of
Eq. (4) at _{} (owing to the
symmetry), while at infinitely large distance from the bulge _{} we have
_{}. (12)
Thus, at
the actual x and y values, the kinetic equation (3) and the balance condition (5) at
the surface read:
_{}, (13)
from which we obtain
_{}. (14)
Here _{} and _{}. If the bulge is small as compared with the thickness of the
concentration layer, i.e. _{}, then we may assume that
_{}, (15)
where _{} is the thickness of an
unperturbed (i.e., a totally flat) layer (see Fig. 2). Since a very small
bulge consisting of few atoms is of no interest for us, we will have _{}, which means that the exponent in (14) can be approximated
as
_{}. (16)
The diffusion coefficient can be represented as
_{} where n is the characteristic frequency of atom
oscillations, i.e., the frequency of the attempts to overcome the potential
(diffusion) barrier _{}. Similarly, the attachment frequency _{} is representable as _{} where _{} is the barrier of the
attachment reaction. One may assume that _{} is smaller than or
comparable with _{}, since both the diffusion and the attachment reaction are
related to a similar spatial rearrangement of particles, but in the case of the
attachment reaction the attraction forces between particles in the crystalline
phase tend to lower the potential barrier. According to this, _{} is a very small
quantity of the order of _{}. Therefore, the stability condition (7) at _{} reads:
_{}, (17)
as consistent with Eqs. (14)–(16),
where _{} is the oversaturation
and
_{} (18)
is the equilibrium concentration for a flat
surface.
Assuming that _{}mm and _{}, at _{} we have roughly
_{} for
_{} (_{}J, _{} )
and
_{} for
_{} (_{}J, _{} ) .
The stability region _{} for both cases,
depending on the relative oversaturation _{}, is shown in Fig. 3.
Fig. 3. The
stability and unstability regions depending on the relative oversaturation Dc/c_{eq} and radius r for the growth of a flat builtup
layer (solid line – Al_{2}O_{3}, dashed line – MgO; _{})
The same, but depending on the
impurity content of the liquid Fe expressed as mass percentage is shown in Fig.
4 for two temperatures, T = 1550 ^{o}C
and T = 1800 ^{o}C.
Fig. 4. The stability and
unstability regions for the growth of a flat builtup layer depending on the
radius r and the concentration (in
mass percents) (solid line – Al_{2}O_{3}, dashed line – MgO; v = 0).
5. ESTIMATION OF THE
CRITICAL RADIUS IN THE CASE OF _{}
The linear
approximation (12) is the stationary solution of diffusion equation (4) at _{} for _{} and _{}, as well as at _{} in a special case of a flat homogeneous
surface, i.e., this approximation in any case is meaningful at a far distance_{} from the bulge. Thus, according to (12) and (13), the
concentrations _{} and _{} on the surface at _{} and _{}, respectively, are given by
_{}, (19)
_{}, (20)
where _{}, defined by Eq. (2), is the equilibrium concentration of a
curved surface with radius r. In any
case, we can write
_{}, (21)
where the factor f depends on v. Obviously, we have _{} at _{}. At _{}, one should expect_{} owing to the strong flow along the surface. From (3) and
(19)–(21) we obtain an equation for the critical radius where _{} values are equal at _{} and _{}, i.e.,
_{} (22)
where
_{} . (23)
The factor f is dimensionless, therefore it should be a function of a
dimensionless argument composed of physical quantities responsible for the
concentration difference _{}. On the one hand, the diffusion causes a mass transport
perpendicular to the surface, which tends to maintain some difference between
the concentrations at _{} and _{}. On the other hand, the flow along the surface tends to
smear out the concentration profile _{}. Therefore, f
should be a function of the ratio of parallel and perpendicular fluxes at a
characteristic distance, _{}, from the surface, i.e., a function of the dimensionless
argument
_{}.
The linear
approximation _{} or _{} provides a
qualitatively correct behavior of the function f. Taking also into account that _{}, we obtain a simplified equation of the form
_{}, (24)
where _{} and _{}.
At _{} Eq.(24)
gives (17). An estimation of the order of magnitude shows that in a typical
case of the induction channel furnaces there holds_{} (_{} or even larger), so that the transport parallel to the
surface changes the result (17) rather strongly. Taking into account that _{}, we obtain an approximate solution of Eq. 24) as
_{}, (25)
where _{} is the velocity at the
boundary of the laminar sublayer of thickness _{} and _{}.
We have used the approximation
_{}, which makes sense if _{} is coniderably
smaller than _{} and _{}.
To estimate the order of _{}magnitude expected in our application, we have set _{}, _{}, _{}, and _{}. This yields at T = 1600 ^{o}C:
_{} for Al_{2}O_{3}
(_{}J, _{} ) ,
_{} for MgO
(_{}J, _{} ) .
Fig.
5. The same as in Fig. 3, but with nonzero flow along the surface, v > 0
In Fig. 5 we have
shown the corresponding stability region in the _{}and concentration (the content in mass percent
relative to the liquid Fe) plane at different temperatures, T = 1550 ^{o}C and T = 1800 ^{o}C.
6. CONCLUSIONS
1.
A
phenomenological model has been developed to describe the buildup formation
and to study the stability conditions for a uniform growth of the builtup
layer.
2.
Based
on suitable approximations, analytical formulae have been derived allowing the
stability region in the concentrationtemperature plane to be evaluated
depending on the initial roughness of the growing surface.
3.
A
further problem is to test numerically our analytical approximations, in
particular the validity of our description by the phenomenological function f, and to make a quantitative estimation
of the parameters _{} and _{}.
1.
Bethers
U., Jakovics A., Jekabsons N., Madzulis I., Nacke B. (1994) The theoretical
investigation of the conditions of the buildup formation in the induction
channel furnaces Magnetohydrodynmics,
30 247–258.
2.
Drewek
R. (1995) Verschleissmechanismen in
InduktionsRinnenoefen fuer Gusseisen und Aluminium Duesseldorf: VDI Verlag
33–54.
3.
Rolov
B., Ivin V., Kuzovkov V. (1979) Statistics
and Kinetics of Phase Transitions in Solid
4.
Nicolis
G.,
5.
IZGULSNĒJUMU
AUGŠANAS STABILITĀTES NOSACĪJUMI
UZ UGUNSDROŠĀ MATERIĀLA VIRSMAS
K o p s a v i l k u m s
Izveidots vienkāršots modelis metālu oksīdu (piem., MgO, Al_{2}O_{3}) izgulsnēšanās no metālu (piem., Fe, Al) kausējuma procesa kinētikas aprakstam. Iegūti tuvinātie nosacījumi pārejai no slāņaina uz fraktālu augšanas režīmu. Parādīts, ka slāņainas kristalizācijas apgabals ir būtiski atšķirīgs dažādiem piemaisījumu materiāliem un procesa stabilitātei kritiskais izauguma izmērs būtiski samazinās, pieaugot piemaisījuma koncentrācijai un kausējuma plūsmas raksturīgajam ātrumam, bet palielinās, pieaugot kausējuma temperatūrai. Iegūtie rezultāti kvalitatīvi atbilst prakses novērojumiem industriālās indukcijas kanālkrāsnīs un izveidotā pieeja ļauj prognozēt un analizēt, piem., aktīvās zonas (kanāla) aizaugšanu tajās.