2002, N 3






J. Kaupužs, A. Jakovičs


Department of Physics, University of Latvia
Zeļļu iela 8,
Rīga, LV-1002, LATVIA


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 [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 build-up 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 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 [1]. 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.




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

.                                                                        (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


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

.                                                                                                     (7)



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.




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

.                                                                                      (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


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).




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.,



 .                                                                                                                            (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 oC:

   for  Al2O3
(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 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.

3.        Rolov B., Ivin V., Kuzovkov V. (1979) Statistics and Kinetics of Phase Transitions in Solid Riga.

4.        Nicolis G., Prigogine I. (1989) Employing Complexity New York: Freeman.

5.        Lifshitz I., Slyozov V. (1961) J. Phys. Chem. Solids 19, 35.




J. Kaupužs, A. Jakovičs


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.