2000, N 5




I. Madzhulis, A. Jakovics, V. Frishfelds

Latvian University, Faculty of Physics and Mathematics

Zeļļu iela 8, Riga, LV-1002, LATVIA

B. Nacke

Institute of Electroheat, University of Hanover

Wilhelm Busch Str. 4, Hanover D-30167, Germany

The lifetime of metallurgical furnaces is limited by accumulation of metal vapours in the pores of refractory material leading to remarkable change in conductivity of the refractory layer. We have derived macroscopic kinetic equations for accumulation of a self-interacting melt component in presence of high temperature gradient. The electric conductivity of the refractory layer and the electronic part of its heat conductivity during the melting process are calculated on the basis of a semiconductor model.


Induction electric furnaces are widely used in ferrous and non-ferrous metallurgy. Their lifetime is highly dependent on the interaction between the metal melt and the refractory layer of a crucible or a channel wall that is usually based on Al2O3 or MgO ceramics with a typical thickness of 5–20 cm.

The refractory material of the mentioned equipment possesses the following properties:

Accordingly, melt filtration (observed in iron-melting furnaces) and diffusion of metal vapour (e.g. zinc) take place in ceramic material contacting with melt. Their significance depends on the characteristic size of pores. Figure 1 presents a photo of a model experiment conducted with brass (35% Zn) kept for 55 hours at the temperature Tmelt = 1050–1100 ° C in a crucible with a thickness of 2.7 cm and porosity . The maximum mass fraction of zinc reaches 25% on the boundary of dark coloured ceramics layer (point B). At the same time, the detected concentrations of zinc directly at melt in the coloured layer of refractory material 0.2% (point A) and in periphery 0.4–0.8% (point C) are relatively small.

Fig. 1. Cross-section of a crucible after experiments on inductive melting

The growth in conductivity leads to appearance of induced Joule heat in the layer of refractory material, which considerably increases the heat flow to the inductor. As a result the equipment stops to operate and the refractory material of the crucible or the channel must be replaced. Therefore, it is of importance to predict accumulation of lightly fugitive components such as zinc so as to provide a stable and lasting utilisation of the non-ferrous melting furnaces.


Consider the system of uncoupled mesoscopic pores distributed with interval a along the x-axis. The occupation of impurities in pores C satisfies the master equation

. (1)

The flux of interacting impurities from one pore to another can be expressed as

, (2)

because the transition frequency from pore x to the neighbouring one x+a is proportional to the occupation of impurities in the former – C(x) and free sites in the latter – 1–C(x+a) (the pore is completely occupied at C(x) = 1). The exponential term arises from the demand that in the stationary case transition frequencies obey the Gibss distribution law:

, (3)

where e is the binding energy between two neighbouring impurity atoms (molecules); z is the maximum number of neighbouring impurities; k is the Boltzmann constant; T is the temperature; D U is the change in the system energy during transition; W0 is a constant representing transition frequency. The concentrations of impurities in neighbouring pores should satisfy the condition at . Then Eq.(1) gives the nonlinear diffusion equation

, , (4)

where Ea is the activation energy. Equation (4) will be used also for inhomogeneous temperature distribution neglecting the thermo-gradient effects. This could be assumed because the distance between pores (several m ms) is much greater than the mean free path of media. The effective diffusion coefficient can be negative in a certain range of concentrations below the critical temperature , and the dynamics becomes unstable. In this case we have a two-phase system. Considering the pore as a system of sites accessible for impurities that consists of two phases, the minimum of free energy in the mean field approximation neglecting the influence of phase boundaries is

, (5)

, , (6)

where N is the number of sites in the system; C = pCg+(1–p)Cl is the average occupation probability of sites in pores; Cg is the probability that the site is occupied in the gas phase, while Cl = Cg – in the liquid phase; p is the fraction of sites present in gas phase. Phase boundary effects would lead to appearance of the Laplace pressure depending essentially on the shape of pores. The diffusion process of metal vapour is much slower than the relaxation inside a single pore. Therefore, slight perturbation of the stationary free energy yields proportionality of the effective diffusion coefficient to the second derivative of F in respect to C (which is zero) or Cg. This can be interpreted in such a way that the gaseous phase alone moves among the pores. The system of kinetic equations takes a simple form assuming that the diffusion coefficient is independent of the impurity concentration and is roundly proportional to the porosity:

. (7)

The condition at the intersection point of one- and two-phase systems is (from equality of the chemical potential), where C is the occupation probability in a one-phase system. The diffusion of impurities alters the thermal properties of media. Thermal conductivity of zinc is much higher than that of oxide. Hence, the temperature distribution modifies and satisfies the equation dependent on C(x):

, (8)

where l Ce, l Zn(C,T) are thermal conductivities of ceramics and zinc; CCe, CZn are their heat capacities. The fraction of zinc in thermal conductivity of material l Zn(C,T) will be considered below. The heat connected with phase transition is neglected due to slow speed of the diffusion process. The quantity is equal to the boiling enthalpy per one molecule of zinc Q inside ceramics.


Let us assume that the diffusion of particles inside a porous material is much slower than their evaporation from the melt (diffusion-controlled process). In such a case the liquid-gas thermodynamic equilibrium condition is employable to evaluate the concentration of the particles on the inner surface of the material considering that C(0) does not depend on the process dynamics at the boundary layer.

In thermodynamic equilibrium the chemical potentials of atoms in the melt and porous material are mutually equal [1]. Assuming the melt as a binary system, the free energy in the molecular field approximation for atoms of two types A and B is:

, (9)

where N is the number of sites; nA, nB are the occupation probabilities of sites for atoms of both types; nA + nB = 1; z is the number of neighbours; e AA, e AB and e BB is binding energy between atoms A-A, A-B and B-B, respectively.

Considering that the fraction of B-type atoms in porous media is negligibly small, the free energy in ceramics is

, (10)

where C is the occupation probability of sites in pores of A-type atoms (impurities); e A is the interaction energy of impurities with porous media.

The equilibrium condition gives

. (11)

For a given composition of melt and C << 1, it becomes

, (12)

where Qmc is some effective melt-ceramics transition energy.

The boundary condition at the cooled surface is

, (13)

where x0 is the size of refractory layer; r is the reflection coefficient varying from 0 (freely penetrable surface) to 1 (impermeable surface). If the two-phase system is present at the cooled surface Cg < C < 1 – Cg, then average concentration C must be replaced by the concentration in gas fraction Cg in Eq. (13).


Low zinc concentration (less than 0.25 of volume) suggests that the conductivity is not metallic but depends on the number of free ionised electrons in media. It means that the conductivity of the layer at low temperatures is comparable with the intrinsic conductivity of material, because the zinc concentration is less than the percolation concentration of simple cubic lattice 0.31 [2] in the case of chaotic zinc distribution. Hence, closed macroscopic circuits do not form, which agrees with our assumption about uncoupled pores. Electrons contribute the main part to the conductivity of refractory material

, (14)

where t is the mean time between collisions of electrons with media atoms; e is the electron charge; m is its mass; a is the degree of ionisation for Zn atoms.

The experimental data [3] suggest that the mean time between collisions is inversely proportional to temperature, and Eq. (14) becomes:

, (15)

where s 0 is the electric conductivity of pure zinc at room temperature T0. The electron parts in electric and heat conductivities are related by the Wiedemann-Franz relationship useful for conductors in a wide range of temperatures:

, . (16)

Consequently, the electronic part of the heat conductivity in Eq. (8) is

, (17)

where is the heat conductivity of liquid zinc.

The concentration of free electrons can be calculated by a semiconductor model that works well at low concentrations of impurities. The charge neutrality condition in this approach gives the equation for chemical potential m [4] as

, (18)

where mc, mv are electron and hole effective masses in the conduction and valence bands, respectively; Eg is the energy band gap; Ed is the gap between the conduction band and the donor level; k is the wave vector; NA is Avogadro’s constant; r Zn is Zn density; MZn is the molar mass of Zn. The left-hand side of Eq. (18) represents the concentration of free electrons. For simplicity, in Eq. (18) it was assumed that each impurity atom is able to liberate one electron (instead of two for zinc). In order to calculate conductivity, the energy band gap Eg must be found. The concentration of free electrons for pure refractory material according to Eq. (18) is

, (19)

if Boltzmann’s statistics is used in the conduction and valence bands. The resistance is . The fit of the experimental data for the used Al2O3 material

Fig. 2. Resistance vs. temperature in reciprocal-log scale. Fit of experimental data

Fig. 3. Concentration of free electrons vs. Zn fraction in ceramics. The line after the break shows metallic behaviour differing from the semiconductor model

[5] in Fig. 2 gives Eg » 1.1 eV. The effective masses mc, mv will be taken equal to electron mass m, and the donor level is relatively close to conduction band Ed = 0.1 eV just for a qualitative analysis. The concentration of free electrons (scaled to the concentration of free electrons in metallic phase) at different temperatures and Zn concentrations are shown in Fig. 3. As can be seen, all impurities are ionised at low zinc concentration and ionisation degree a » 1. However, a < 1 at higher zinc concentrations. Strictly speaking, the semiconductor model is useless at high zinc concentrations because the band gap Eg significantly contracts and Zn impurities do not form a donor level anymore. At a very high zinc concentration, the conductivity must jump (shown by the break) up to that of metal and ionisation degree is again a » 1 (see Fig. 3.). Nevertheless, the effect is negligible as the zinc fraction by volume cannot exceed the porosity coefficient.

The solid curve in Fig. 3 shows the averaged concentration of free electrons versus weight-averaged Zn concentration in the material during the melting process in industrial furnace. This curve certainly goes between the isotherms of boundaries. As shown below, the ionisation degree a starts to decrease after a day till the pores are filled with zinc.


The validity of proposed zinc accumulation model (see Fig. 1) could be proved by simulation of the model experiment in the presence of a strong temperature gradient. The result in Fig. 4 is shown in assuming that the furnace has a radial symmetry. Zinc concentration sequentially increases in time (days) reaching the distribution very similar to the experimental one. The heat conductivity practically remains the same, thus the temperature distribution is almost stationary. The input data for simulations are shown in Table 1 except the porosity and the distance from inner to the outer surface. The values of heat capacities CZn, CCe and heat conductivities l 0Zn, l Ce are characteristic values of the used ceramics and liquid melt materials [5], while r, D(T0), Ea, Q, Qmc are model parameters. Both the model experiment and numerical simulations show the sharp zinc accumulation front forming a thin (» 5 mm) region with a high (> 0.05) zinc fraction.

Table 1

Input data for simulation of the melting process in the industrial furnace








inner surface temperature




outer surface temperature

x0 *



size of ceramics layer


3.46× 106


heat capacity of Zn


2.15× 106


heat vcapacity of ceramics

l 0Zn



heat conductivity of liquid Zn

l Ce



heat conductivity of ceramics




zinc fraction in melt

P *



porosity coefficient




reflection coefficient


2.7× 10–13


diffusion coefficient at 300 K




activation energy




boiling enthalpy of Zn in ceramics




melt-ceramics transition energy




energy band gap of ceramics




donor level

r Zn



Zn density




molar mass of Zn

* for the experimental furnace P =0.15 and x0 = 0.03 m.

Fig. 4. Experimental furnace with radial symmetry

Fig. 5. Behaviour of temperature and zinc concentration in the industrial furnace under strong temperature gradient

Figure 5 shows the behaviour of concentration and temperature in an industrial furnace applying the data from Table 1. In this case, the temperature slightly decreases during 500 days. This is due to the noticeable change in heat conductivity at a fixed temperature of the outer surface. The front of zinc accumulation (Fig. 5) shifts towards the inner surface of porous material depending on the change in temperature distribution.

The layer close to the outer surface contributes the main part to the integral resistance as can be seen in Fig. 6. The resistance is given in arbitrary units. Zn concentration is so small at time moment t = 0.0001 (days) that there is only intrinsic conductivity of semiconductor and the relationship between the concentration of free electrons and the zinc concentration is a >> 1. Practically all Zn impurities are ionised at t = 1 dominating over intrinsic free electrons and a » 1. However, Zn concentration is large enough at t = 100, and the level of chemical potential reaches the conduction band: a < 1.

Fig. 6. Distribution of resistance and ionisation degree for Zn in the industrial furnace at different time moments

Fig. 7. Drop in resistance and increase in heat transfer during zinc accumulation in material.
The upper graph is the initial phase of the process

Zinc accumulation process gradually slows down while the pores fill with zinc (see Fig. 7). Heat transfer through the material increases about one and a half times. The total resistance (from melt to the cooler) of the layer drops by five orders of magnitude during the zinc accumulation process, which agrees well with resistance measurements. The total resistance decreases mainly due to growth of zinc concentration in the cooled part of the refractory layer with a lower intrinsic conductivity (see Fig. 3, 6).


A mathematical model is constructed that allows one to analyse the diffusion processes of metal vapour and its accumulation in a porous material with given properties.

The model of electric resistance is added to examine the resistance change in refractory material at various regimes of temperature and zinc accumulation. Numerical simulations show that the resistance of refractory material decreases by several orders during zinc accumulation, especially in the cooled part of refractory material.

Experiments carried out in crucible furnaces qualitatively confirm the numerical results of zinc diffusion showing spatially arranged zinc accumulation. The introduced semiconductor model of resistance agrees well with sharp decrease in total resistance during the melting process.



  1. Landau L.D., Lifshitz E.M. (1976) Statistical Physics, vol.1 Moscow: Nauka 289–332 (in Russian).
  2. Essam J.W., Gaunt D.S., Guttmann A.J. (1978) J. Phys. A 11 1983–1990.
  3. Ashcroft N.W., Mermin N.D. (1979) Solid State Physics vol.1 Moscow: Mir 22–27 (in Russian).
  4. Anselm A.I. (1962) Introduction into Theory of Semiconductors. Moscow 199–216 (in Russian).
  5. Jakovics A., Jekabsons N., Actins A. (1997) Report for ABB Induction Furnaces Company Riga (in German).

Cinka akumulācija ugunsdrošajā slānī stipri nehomogēnā temperatūras laukā

I. Madžulis, A. Jakovičs, V. Frišfelds, B. Nacke

K o p s a v i l k u m s

Metāla kausēšanas krāšņu dzīveslaiku ierobežo metāla tvaiku akumulācija porās, kas noved pie būtiskām aizsargslāņa elektriskās vadītspējas izmaiņām. Ir iegūti makroskopiskie kinētiskie vienādojumi pašmijiedarbojoša cinka akumulācijai stipri nehomogēnā laukā. Aizsargslāņa vadītspēja un elektronu daļa siltumvadītspējai kausēšanas laikā ir aprēķināti, pamatojoties uz pusvadītāju modeli.