The power density in glass melts has been studied at different arrangements of electrodes in all-electric melting furnaces

The power density in glass melts has been studied at different arrangements of electrodes in all-electric melting furnaces. Bottom, top and plate electrodes were arranged in the model furnace in hexahedron form of about 1 metre along its edges. The results of mathematical modelling showed that there is a very close relationship between the distribution of power density in glass melt and the temperature field and, therefore, by means of suitable arrangement of electrodes it is possible to influence the intensity of convective currents of the glass melt. From evaluated dependencies of power density distribution near the tips of electrodes, it follows that in the case of rod electrodes, the power density decreases on increasing the length of the electrodes.

Opposite behaviour happens with plate electrodes because the power density distribution in the centre of the basin between the electrodes increases on increasing the distance of the electrodes from the bottom of the furnace. The volumes of glass melt surrounding the electrodes where the power densities are superior to pmean (60,000 W.m-3) have also been evaluated by means of mathematical modelling. The volumes are very small with regards to the total volume of the furnace and do not exceed 22 per cent. Mathematical modelling of glass melting furnaces by means of the CFD programme Fluent leads to acceptable computational subservience to the study of power density distribution in all-electric melting furnaces.

The possibility...

Opposite behaviour happens with plate electrodes because the power density distribution in the centre of the basin between the electrodes increases on increasing the distance of the electrodes from the bottom of the furnace. The volumes of glass melt surrounding the electrodes where the power densities are superior to pmean (60,000 W.m-3) have also been evaluated by means of mathematical modelling. The volumes are very small with regards to the total volume of the furnace and do not exceed 22 per cent. Mathematical modelling of glass melting furnaces by means of the CFD programme Fluent leads to acceptable computational subservience to the study of power density distribution in all-electric melting furnaces.

The possibility...

The power density in glass melts has been studied at different arrangements of electrodes in all-electric melting furnaces. Bottom, top and plate electrodes were arranged in the model furnace in hexahedron form of about 1 metre along its edges. The results of mathematical modelling showed that there is a very close relationship between the distribution of power density in glass melt and the temperature field and, therefore, by means of suitable arrangement of electrodes it is possible to influence the intensity of convective currents of the glass melt. From evaluated dependencies of power density distribution near the tips of electrodes, it follows that in the case of rod electrodes, the power density decreases on increasing the length of the electrodes.

Opposite behaviour happens with plate electrodes because the power density distribution in the centre of the basin between the electrodes increases on increasing the distance of the electrodes from the bottom of the furnace. The volumes of glass melt surrounding the electrodes where the power densities are superior to pmean (60,000 W.m-3) have also been evaluated by means of mathematical modelling. The volumes are very small with regards to the total volume of the furnace and do not exceed 22 per cent. Mathematical modelling of glass melting furnaces by means of the CFD programme Fluent leads to acceptable computational subservience to the study of power density distribution in all-electric melting furnaces.

The possibility to predict the operation characteristics of all-electric glass melting furnaces is of pivotal importance when electricity is used for glass melting. The determination of the most convenient electrode configuration represents one of the most important steps in the design of the glass melting tank furnace and, specifically, of its melting compartment, because this configuration influences the following two factors of great importance for furnace operations:

a) technological conditions – consisting in the setting of the required temperature field in the glass melt which, in turn, results in the creation of a suitable flow pattern in the tank;

b) electrical conditions – consisting of a uniform loading of all installed electrodes and all phases of the supply source, as well as in adequate values of voltages and resistances between the electrodes.

The first factor is very important because – as obvious from the principle of the electric melting of glass – the electric power is transformed into thermal energy directly in the glass melt, which functions as an ohmic resistance generating the Joule heat. In this way, a local convection flow can be developed in the glass melt by arranging the electrodes conveniently; the convection flow exerts a favourable effect on the rate of the glass melting process as well as on the final glass quality because the glass is homogenized more thoroughly.

The distribution of the power density released by the electric current in the glass melt, i.e. the distribution of the power density within the glass bath must be known for each proposed configuration of heating electrodes if the temperature distribution in the tank is to be determined. The intensity of the convection flow of molten glass is then assessed on the basis of the temperature distribution.

The above problems were investigated by a variety of workers in the past because it is important to know the distribution and magnitude of the power density in the glass melt at different electrode arrangements in the tank when all-electric furnaces are designed. The work carried out by Stanêk and Curran can be regarded as fundamental in this field. By using a method of physical modelling, Stanêk studied the amount of energy released in the glass and the glass temperature distribution for four basic configurations of electrodes. The data obtained enabled to visualize the amount of energy released and the temperature in a model liquid in the cross-section tank of a physical model and to formulate a statement according to which the amount of released energy corresponds to the temperature distribution.

Curran investigated the distribution of energy released in the glass melt by using a 2-D mathematical model under iso-thermal conditions when the following equation is valid because of the validity of Joule’s law governing the generation of heat released by the electric current passing through a conductive medium (glass melt) and Ohm’s law for the 3-D conductor:

The data published by Stanêk and Curran, i.e. the distribution of the power density as well as the temperature field, were in agreement.

At present, both hardware and, particularly, software, e.g. CFD FLUENT (V6.3.26) expanded by a MHD module[4, 5] are available. The above-mentioned software gives the possibility to find solutions for 3-D mathematical models yielding the distribution of physical fields in molten glass melted by the Joule heat under non-isothermal conditions. The present contribution focuses on calculations yielding the distributions of temperature and power density fields in a furnace of a simple shape at various configurations of three types of electrodes under the following conditions:

a) Configuration of the 3D mathematical model. The investigation into the effect of the heating electrode configuration on the amount and distribution of electric power was carried out by using a furnace having the shape of a cube with an edge 1 metre long. Two rod electrodes of various lengths were inserted into the tank through the tank bottom (so-called bottom electrodes) or through the tank crown (top electrodes). The third type of the furnace was equipped with two plate electrodes and the distance between the lower edge of the electrodes and the tank bottom was changed in a step-by-step manner. Furnace operation was investigated at zero pull and a constant power input was supplied to all electrode pairs. The configurations of the investigated furnaces equipped with the bottom, top and plate electrodes are shown in Figures 1, 2 and 3.

b) Arrangement and parameters of heating electrodes. Furnace with bottom and top electrodes: type of electrodes: rod; electrode diameter: 0.06 metres; electrode length: 0.2-0.8 metres. Furnace with plate electrodes: type of electrode: plate-shaped; electrode dimensions: 0.8 x 0.2 x 0.01 metres (length x height x width).

c) Glass and its properties. Glass type: TV glass (faceplate); viscosity: log (log (η)) = 7.0922 – 2.348 log T [dPa.s; K]; density: ρ = 2790 – 0.2378 T [kg.m-3 ; K]; electric resistivity: log ρel = - 1.5235 + 3874.5/T [Ohm.cm; K]; effective thermal conductivity: λeff = 2.0 + 1.5 e–8 T3 [W.m-1.K-1; K]; heat capacity: cp = 1200 J. kg-1.K-1; coefficient of the thermal volume expansion: β = 1.005 e-4 K-1; reference temperature: Tr = 1623 K.

d) Selected boundary conditions: Sidewalls and bottom: zero heat losses; glass surface: heat transfer by convection, α = 42.553 W. m-2.K-1; Ta = 313.15 K; electrode input: 60 kW.

The mathematical model of the investigated furnace provided sets of data containing the distributions of the temperature and power density in the whole volume of the furnace. The numerical values of temperatures and power densities in selected gradients, as well as other parameters (for instance, the volume of the glass melt with a power density superior to the mean power density in the furnace), have been calculated:

a) Distribution of the power density and temperature in the horizontal gradients at the following levels: (Figure 4)

- 0.05 metres above the electrode tips – for the bottom electrodes;

- 0.05 metres below the electrode tips – for the top electrodes;

- 0.05 metres above the top edge of the electrode – for the plate electrodes.

b) Power density distribution in the central vertical gradients at the electrode tip level of bottom and top electrodes or at the top-edge level of plate electrodes in dependence on the length of bottom and top electrodes and the distance of plate electrodes from the tank bottom. (Figure 5)

c) Dependence of the depth in the central vertical gradient with the maximum power density on the electrode length or the distance of plate electrodes from the tank bottom. (see Figure 6)

d) Volume of the glass melt with the power density superior to pmean (60,000 W.m-3).

The values of the volume of the glass melt with the power density superior to pmean (60,000 W.m-3) are given in Table 1.

The results yielded by the 3-D non-isothermal mathematical model set up with the aid of CFD programme modules of Fluent software corroborated the conclusions of investigations carried out with the aid of isothermal models in previous years. Nevertheless, the 3-D mathematical model applied within the framework of the present investigation represents a very extensive mathematical apparatus yielding a considerable wealth of data that offer completely new perspectives as regards the solution of the given problem.

First, the distribution of the power density and temperatures in horizontal gradients at levels close above the electrodes were evaluated on the basis of the acquired data. The calculated and plotted data reveal a very intimate relationship between power density distribution and the temperature field in the glass melt. Both dependences exhibit pronounced peaks in the electrode zones for all examined alternatives. It can therefore be declared that the intensity of the glass convection flow can be influence by a convenient arrangement of electrodes in the tank.

The next step focused on the evaluation of the power density distribution in the central vertical gradient at the level of cylindrical electrode tips as well as at the top-edge level of plate electrodes.

As regards the rod electrodes, the power density in the central gradient drops with the increasing length of electrodes. This phenomenon is due to the fact that the glass volume in the close vicinity of electrodes where the maximum amount of the electric energy may be released also increases with the increasing electrode length. A quite opposite situation occurs in the case of plate electrodes: the amount of energy released in the centre of the tank grows with the increasing distance between the electrodes and the tank bottom. The results showing the dependence of the depth in the central vertical gradient with the maximum power density on the electrode length are in a perfect agreement with the above findings.

Attention has also been paid to the calculation of the glass volume in which the released energy is superior to the mean power density. It is evident from the data in Table 1 that a predominant part of energy is released in the glass melt in the close vicinity of electrodes because the power density superior to the mean value calculated for the whole tank could be detected in the glass melt whose volume ranges from 10 to 20 per cent of the total tank content. This statement is valid for both bottom and top electrodes. As regards the plate electrodes, the glass volume with p > pmean does not vary in dependence of the electrode position in the tank; it amounts to about 20 per cent.

The following conclusions can be drawn on the basis of the above findings:

1. the power density distribution in the glass melt shows typical ocal characteristics (temperature, power density);

2. the above characteristics can be influenced substantially by electrode arrangement in the furnace as well as by the type and dimensions of the electrodes;

3. local characteristics also fundamentally influence the glass flow pattern in the tank;

4. the above characteristics must be identified if efficient and high-performance all-electric tank furnaces are to be designed;

5. the application of the method of mathematical modeling to the identification of the above characteristics is extremely useful, for instance, CFD Fluent software expanded by suitable user’s functions targeted at the solution of magneto-hydro dynamic phenomena in a conductive medium;

6. the necessity to pay attention to the transfer of released energy in the direction from the heating electrodes to the surrounding glass melt remains a very important tasks of furnace designers.

1. Stanêk J. et al: Glass Technology Vol.10, (1969), p. 43

2. Curran R. L.: IEEE Trans. Ind. Gen. Appl. Vol. IGA-7, (1971), p. 116

3. Curran R. L.: IEEE Trans. Ind. Appl. Vol.IA-9, (1973), p. 348

4. Fluent 6.3, Documentation, User’s Guide, 2006 (http://www.fluent-users.com)

5. Fluent 6.3, Magnetohydrodynamics (MHD) Module Manual, 2006 (http://www.fluentusers.com)

This work was a part of the project No 2A-1TP1/063, “New glasses and ceramic materials and advanced concepts of their preparation and manufacturing”, realized under financial support of the Ministry of Industry and Trade and also a part of the research programme MSM 6046137302 “Preparation and research of functional materials and material technologies using micro and nanoscopic methods” realized under financial support of the Ministry of Education, Youth and Sports.

Opposite behaviour happens with plate electrodes because the power density distribution in the centre of the basin between the electrodes increases on increasing the distance of the electrodes from the bottom of the furnace. The volumes of glass melt surrounding the electrodes where the power densities are superior to pmean (60,000 W.m-3) have also been evaluated by means of mathematical modelling. The volumes are very small with regards to the total volume of the furnace and do not exceed 22 per cent. Mathematical modelling of glass melting furnaces by means of the CFD programme Fluent leads to acceptable computational subservience to the study of power density distribution in all-electric melting furnaces.

The possibility to predict the operation characteristics of all-electric glass melting furnaces is of pivotal importance when electricity is used for glass melting. The determination of the most convenient electrode configuration represents one of the most important steps in the design of the glass melting tank furnace and, specifically, of its melting compartment, because this configuration influences the following two factors of great importance for furnace operations:

a) technological conditions – consisting in the setting of the required temperature field in the glass melt which, in turn, results in the creation of a suitable flow pattern in the tank;

b) electrical conditions – consisting of a uniform loading of all installed electrodes and all phases of the supply source, as well as in adequate values of voltages and resistances between the electrodes.

The first factor is very important because – as obvious from the principle of the electric melting of glass – the electric power is transformed into thermal energy directly in the glass melt, which functions as an ohmic resistance generating the Joule heat. In this way, a local convection flow can be developed in the glass melt by arranging the electrodes conveniently; the convection flow exerts a favourable effect on the rate of the glass melting process as well as on the final glass quality because the glass is homogenized more thoroughly.

The distribution of the power density released by the electric current in the glass melt, i.e. the distribution of the power density within the glass bath must be known for each proposed configuration of heating electrodes if the temperature distribution in the tank is to be determined. The intensity of the convection flow of molten glass is then assessed on the basis of the temperature distribution.

The above problems were investigated by a variety of workers in the past because it is important to know the distribution and magnitude of the power density in the glass melt at different electrode arrangements in the tank when all-electric furnaces are designed. The work carried out by Stanêk and Curran can be regarded as fundamental in this field. By using a method of physical modelling, Stanêk studied the amount of energy released in the glass and the glass temperature distribution for four basic configurations of electrodes. The data obtained enabled to visualize the amount of energy released and the temperature in a model liquid in the cross-section tank of a physical model and to formulate a statement according to which the amount of released energy corresponds to the temperature distribution.

Curran investigated the distribution of energy released in the glass melt by using a 2-D mathematical model under iso-thermal conditions when the following equation is valid because of the validity of Joule’s law governing the generation of heat released by the electric current passing through a conductive medium (glass melt) and Ohm’s law for the 3-D conductor:

The data published by Stanêk and Curran, i.e. the distribution of the power density as well as the temperature field, were in agreement.

**Modelling**At present, both hardware and, particularly, software, e.g. CFD FLUENT (V6.3.26) expanded by a MHD module[4, 5] are available. The above-mentioned software gives the possibility to find solutions for 3-D mathematical models yielding the distribution of physical fields in molten glass melted by the Joule heat under non-isothermal conditions. The present contribution focuses on calculations yielding the distributions of temperature and power density fields in a furnace of a simple shape at various configurations of three types of electrodes under the following conditions:

a) Configuration of the 3D mathematical model. The investigation into the effect of the heating electrode configuration on the amount and distribution of electric power was carried out by using a furnace having the shape of a cube with an edge 1 metre long. Two rod electrodes of various lengths were inserted into the tank through the tank bottom (so-called bottom electrodes) or through the tank crown (top electrodes). The third type of the furnace was equipped with two plate electrodes and the distance between the lower edge of the electrodes and the tank bottom was changed in a step-by-step manner. Furnace operation was investigated at zero pull and a constant power input was supplied to all electrode pairs. The configurations of the investigated furnaces equipped with the bottom, top and plate electrodes are shown in Figures 1, 2 and 3.

b) Arrangement and parameters of heating electrodes. Furnace with bottom and top electrodes: type of electrodes: rod; electrode diameter: 0.06 metres; electrode length: 0.2-0.8 metres. Furnace with plate electrodes: type of electrode: plate-shaped; electrode dimensions: 0.8 x 0.2 x 0.01 metres (length x height x width).

c) Glass and its properties. Glass type: TV glass (faceplate); viscosity: log (log (η)) = 7.0922 – 2.348 log T [dPa.s; K]; density: ρ = 2790 – 0.2378 T [kg.m-3 ; K]; electric resistivity: log ρel = - 1.5235 + 3874.5/T [Ohm.cm; K]; effective thermal conductivity: λeff = 2.0 + 1.5 e–8 T3 [W.m-1.K-1; K]; heat capacity: cp = 1200 J. kg-1.K-1; coefficient of the thermal volume expansion: β = 1.005 e-4 K-1; reference temperature: Tr = 1623 K.

d) Selected boundary conditions: Sidewalls and bottom: zero heat losses; glass surface: heat transfer by convection, α = 42.553 W. m-2.K-1; Ta = 313.15 K; electrode input: 60 kW.

ResultsResults

The mathematical model of the investigated furnace provided sets of data containing the distributions of the temperature and power density in the whole volume of the furnace. The numerical values of temperatures and power densities in selected gradients, as well as other parameters (for instance, the volume of the glass melt with a power density superior to the mean power density in the furnace), have been calculated:

a) Distribution of the power density and temperature in the horizontal gradients at the following levels: (Figure 4)

- 0.05 metres above the electrode tips – for the bottom electrodes;

- 0.05 metres below the electrode tips – for the top electrodes;

- 0.05 metres above the top edge of the electrode – for the plate electrodes.

b) Power density distribution in the central vertical gradients at the electrode tip level of bottom and top electrodes or at the top-edge level of plate electrodes in dependence on the length of bottom and top electrodes and the distance of plate electrodes from the tank bottom. (Figure 5)

c) Dependence of the depth in the central vertical gradient with the maximum power density on the electrode length or the distance of plate electrodes from the tank bottom. (see Figure 6)

d) Volume of the glass melt with the power density superior to pmean (60,000 W.m-3).

The values of the volume of the glass melt with the power density superior to pmean (60,000 W.m-3) are given in Table 1.

**Discussion of results**The results yielded by the 3-D non-isothermal mathematical model set up with the aid of CFD programme modules of Fluent software corroborated the conclusions of investigations carried out with the aid of isothermal models in previous years. Nevertheless, the 3-D mathematical model applied within the framework of the present investigation represents a very extensive mathematical apparatus yielding a considerable wealth of data that offer completely new perspectives as regards the solution of the given problem.

First, the distribution of the power density and temperatures in horizontal gradients at levels close above the electrodes were evaluated on the basis of the acquired data. The calculated and plotted data reveal a very intimate relationship between power density distribution and the temperature field in the glass melt. Both dependences exhibit pronounced peaks in the electrode zones for all examined alternatives. It can therefore be declared that the intensity of the glass convection flow can be influence by a convenient arrangement of electrodes in the tank.

The next step focused on the evaluation of the power density distribution in the central vertical gradient at the level of cylindrical electrode tips as well as at the top-edge level of plate electrodes.

As regards the rod electrodes, the power density in the central gradient drops with the increasing length of electrodes. This phenomenon is due to the fact that the glass volume in the close vicinity of electrodes where the maximum amount of the electric energy may be released also increases with the increasing electrode length. A quite opposite situation occurs in the case of plate electrodes: the amount of energy released in the centre of the tank grows with the increasing distance between the electrodes and the tank bottom. The results showing the dependence of the depth in the central vertical gradient with the maximum power density on the electrode length are in a perfect agreement with the above findings.

Attention has also been paid to the calculation of the glass volume in which the released energy is superior to the mean power density. It is evident from the data in Table 1 that a predominant part of energy is released in the glass melt in the close vicinity of electrodes because the power density superior to the mean value calculated for the whole tank could be detected in the glass melt whose volume ranges from 10 to 20 per cent of the total tank content. This statement is valid for both bottom and top electrodes. As regards the plate electrodes, the glass volume with p > pmean does not vary in dependence of the electrode position in the tank; it amounts to about 20 per cent.

**Conclusions**The following conclusions can be drawn on the basis of the above findings:

1. the power density distribution in the glass melt shows typical ocal characteristics (temperature, power density);

2. the above characteristics can be influenced substantially by electrode arrangement in the furnace as well as by the type and dimensions of the electrodes;

3. local characteristics also fundamentally influence the glass flow pattern in the tank;

4. the above characteristics must be identified if efficient and high-performance all-electric tank furnaces are to be designed;

5. the application of the method of mathematical modeling to the identification of the above characteristics is extremely useful, for instance, CFD Fluent software expanded by suitable user’s functions targeted at the solution of magneto-hydro dynamic phenomena in a conductive medium;

6. the necessity to pay attention to the transfer of released energy in the direction from the heating electrodes to the surrounding glass melt remains a very important tasks of furnace designers.

**References**1. Stanêk J. et al: Glass Technology Vol.10, (1969), p. 43

2. Curran R. L.: IEEE Trans. Ind. Gen. Appl. Vol. IGA-7, (1971), p. 116

3. Curran R. L.: IEEE Trans. Ind. Appl. Vol.IA-9, (1973), p. 348

4. Fluent 6.3, Documentation, User’s Guide, 2006 (http://www.fluent-users.com)

5. Fluent 6.3, Magnetohydrodynamics (MHD) Module Manual, 2006 (http://www.fluentusers.com)

**Acknowledgements**This work was a part of the project No 2A-1TP1/063, “New glasses and ceramic materials and advanced concepts of their preparation and manufacturing”, realized under financial support of the Ministry of Industry and Trade and also a part of the research programme MSM 6046137302 “Preparation and research of functional materials and material technologies using micro and nanoscopic methods” realized under financial support of the Ministry of Education, Youth and Sports.

expand