LATVIAN JOURNAL OF PHYSICS AND TECHNICAL SCIENCES
2002, N 3
STABILITY CONDITIONS AT THE
ON THE SURFACE OF A REFRACTORY
Department of Physics,
Zeļļu iela 8,
A phenomenological model has been developed to describe the build-up formation and to study the stability conditions for a uniform growth of the built-up layer. Based on suitable approximations, analytical formulae have been derived allowing the stability region in the concentration-temperature plane to be defined depending on the initial roughness of the growing surface.
A typical life-time contraction of the inductor in an induction channel furnace (ICF) caused by build-up formation is from several weeks up to a year. The dangerous zones of the ICF where build-ups mostly form are :
- 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 build-up formation have been reported in , 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 build-up 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 build-up after dismounting the channel, on the velocity
Fig. 1. A compact thin layer and a fractally structured thick layer built-up in an ICF channel
at the melting of ferrous materials
and temperature distributions obtained under laboratory conditions and on numerical experiments . A typical view of a built-up 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 build-up formation that intensifies the process and contracts the channel’s life-time. From the technological point of view it is especially important to know the conditions under which the character of build-up growth changes from compact or uniform to fractal.
2. THE MODEL
To study the stability of a uniform build-up formation we will consider a simple model. It is illustrated in Fig. 2. The surface separates the solid built-up phase () from the molten metal (). The liquid phase (e.g. Fe, Al) contains a built-up substance (e.g. MgO, Al2O3) 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 semi-spherical 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 built-up 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 built-up 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
Here , and is the linear size of a cubic elementary cell with volume . In the thermodynamic equilibrium there holds
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
Equation (3) should be solved together with the diffusion equation for the impurity concentration
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
According to the discussion in the beginning of this section, the boundary conditions of the concentration layer are
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 dendrite-like built-up 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
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 , 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:
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
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
In the general case of interaction between A (basic substance) and B (impurity) particles we have
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 z-axis is given by the solution of Eq. (4) at (owing to the symmetry), while at infinitely large distance from the bulge we have
Thus, at the actual x and y values, the kinetic equation (3) and the balance condition (5) at the surface read:
from which we obtain
Here and . If the bulge is small as compared with the thickness of the concentration layer, i.e. , then we may assume that
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
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:
as consistent with Eqs. (14)–(16), where is the oversaturation and
is the equilibrium concentration for a flat surface.
Assuming that mm and , at we have roughly
for (J, )
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/ceq and radius r for the growth of a flat built-up layer (solid line – Al2O3, 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 oC and T = 1800 oC.
Fig. 4. The stability and unstability regions for the growth of a flat built-up layer depending on the radius r and the concentration (in mass percents) (solid line – Al2O3, 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
where , defined by Eq. (2), is the equilibrium concentration of a curved surface with radius r. In any case, we can write
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.,
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
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
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 oC:
(J, ) ,
(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 oC and T = 1800 oC.
1. A phenomenological model has been developed to describe the build-up formation and to study the stability conditions for a uniform growth of the built-up layer.
2. Based on suitable approximations, analytical formulae have been derived allowing the stability region in the concentration-temperature 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 build-up formation in the induction channel furnaces Magnetohydrodynmics, 30 247–258.
2. Drewek R. (1995) Verschleissmechanismen in Induktions-Rinnenoefen fuer Gusseisen und Aluminium Duesseldorf: VDI Verlag 33–54.
B., Ivin V., Kuzovkov V. (1979) Statistics
and Kinetics of Phase Transitions in Solid
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, Al2O3) 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.