Numerical study of glass fining in a pot melting space with different melt-flow patterns

Journal of Non-Crystalline Solids 361 (2013) 47–56

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Numerical study of glass fining in a pot melting space with different melt-flow patterns Marcela Jebavá ⁎, Lubomír Němec Laboratory of Inorganic Materials, Joint Workplace of the Institute of Chemical Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic, and the Institute of Rock Structure and Mechanics ASCR, v.v.i., V Holešovičkách 41, 182 09 Prague 8, Czech Republic

a r t i c l e

i n f o

Article history: Received 2 July 2012 Received in revised form 18 October 2012 Available online 24 November 2012 Keywords: Glass fining; Glass-melt circulation; Bubble growth; Pot furnace; Mathematical modelling

a b s t r a c t A numerical model of glass-melt flow and bubble removal was applied in a pot melting space with a different character of melt flow. The linear-temperature gradients with a higher temperature either near the space wall or in the center of the melt level were put on the melt level to simulate the melt circulations, corresponding to melt heating through the pot wall and from the top, respectively. The removal times of the small bubbles have been calculated at different intensities of evoked melt circulations and compared with the values attained in the quiescent glass melt. The results have shown that the melt circulations increased the rising time of bubbles and consequently led to higher energy consumption and lower melting performance. The impact of melt circulations became stronger with the increasing intensity of the circulations and weakened with the increasing bubble growth rate. The removal times of critical bubbles were generally lower in the case of heating from the top and the relevant values obtained in the space at very low bubble growth rates and relatively high melt circulation velocities attained even lower values than in the quiescent melt. The low values of the bubble fining times were described by the characteristic shapes of the critical bubble trajectories. The case may be utilized for the fining of special glasses without addition of fining agents. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Discontinuous glass melting in pot-melting furnaces is characterised by a considerably high energy consumption caused mostly by the fact that both the melt and melting furnace undergo the heating and working cycles at apparently different temperatures. The main portion of the energy is nevertheless consumed during the heating and melting period and consequently an improvement of the melting processes would beneficially impact the total energy usage. One of the energy-determining melting processes is fining, which is slow owing to the considerably low bubble-rise velocity in the viscous melt, thick melt layers and the existence of melt-flow patterns in the space. In the past, the rate of the fining process has been extensively examined through laboratory experiments, with the important role of the fining agents, as well as the temperature, pressure and glass composition being defined [1–7]. The mathematical models of bubble behaviour in a quiescent glass melt were constituted [8–11] and subsequently applied in continual glass melting furnaces [12–15]. However, the quantitative expression of the impact of the melt flow character on the rate of the bubble removal (fining) process was realised neither in continuous nor discontinuous (pot melting) spaces. In their recent works [16–21], the authors

⁎ Corresponding author. Tel.: +420 220 445 195. E-mail address: [email protected] (M. Jebavá). 0022-3093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnoncrysol.2012.10.029

and colleagues have dealt with the impact of glass flow character on bubble removal in a continuous glass melting channel and found both advantageous and disadvantageous types of natural melt flows in terms of the bubble removal process. In a discontinuous pot furnace, the glass-melt flow is determined by the radial temperature gradients between the pot wall and the inside. The arising vertical circulations of the glass melt resemble the transversal melt circulations set in a horizontal continuous channel, where the bubble should pass a part of its trajectory against the downward melt flow, and its rising to the level is thus hindered [21]. In a previous work [22], the authors derived a simplified model of bubble behaviour in a rotating glass melt with the aim of revealing the fundamental features of mutual interactions between bubbles and vertically rotating melt. The conception of the space utilisation applied in [17–21] was also utilisable to describe the problem of bubble removal. The heating of the melt through the space wall or from the top was simulated by means of temperature gradients put on the melt level. The modelling results have shown that the bubble fining times under isothermal conditions and in a vertically rotating glass melt are higher than those in a quiescent melt. The bubble removal was retarded by an increasing intensity of melt circulations, whereas the increasing bubble growth rate enhanced the fining process. At higher bubble growth rates, the melt rotation intensity barely influenced the rate of the fining process. The results generally provided evidence of a more intensive fining process under conditions simulating the heating of the glass from the top. This work is at first focused on the qualitative comparison

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Fig. 1. a, b. The schematic picture of two simulated cases of glass melting in the pot. Left: the melt is heated through the wall. Right: the melt is heated from the top.

of the simplified mathematical model with a numerical one, which works under non-isothermal conditions and applies linear temperature gradients between the space wall and melt to evoke melt circulations. The qualitative agreement or difference between both models would provide a reliable chance to use the available simplified model for assessing fining efficiency should there be interest. However, the principal goal of the work is the calculation of more precise quantitative relations between the fining efficiency, melt circulation intensity, and bubble growth rate. 2. Theoretical For an isothermal or almost isothermal process without energy recycling, the melting performance and specific energy consumption of the fining process can be described by the following equations [17–21]: V V V_ ¼ ¼ u τG τ Fref F

ð1Þ

L L H_ τFref 1 H_ τ G 0 T T ¼ HM þ ; HM ¼ HM þ ρV ρV uF

ð2Þ

where V_ [m3/s] is the melting performance of the volume of the melting space V [m3], τG is the mean residence time of the glass melt in the space, τG ¼ V=V_ , τFref [s] is the bubble removal time for the bubble of the critical 0 size in a quiescent molten glass which started from the space bottom, HM T [J/kg] is the specific energy consumption, HM [J/kg] is the specific theoretical heat necessary for the chemical reactions, phase transitions and heating of both the batch and melt to the melting temperature T [K], L H_ [J/s] is the total heat flux across the space boundaries, ρ [kg/m3] is the glass density. In analogy with the space utilisation in a continuous space, the space utilisation in the pot furnace expresses the virtual dead space and the ratio between the fining time of the critical bubble in the quiescent and circulating melts: τFref uF ¼ ð1−mvirt Þ ; u≥0: τFcrit

ð3Þ

Table 1 The composition of the TV-glass used for modelling, in wt.%. SiO2 Al2O3 MgO CaO SrO

60.6 2.0 0.5 1.0 5.0

BaO PbO ZrO2 Na2O K2O

11.2 2.2 1.35 7.6 7.4

CeO2 TiO2 Sb2O3 Fe2O3

0.3 0.2 0.6 0.05

In the pot furnace, τFcrit is the fining time of the critical bubble with flow patterns. The melting performance of the pot, given by Eq. (1), is then described by the effective melting performance. The value of the fraction of dead space, mvirt, defined in [17–21] has a zero value for the discontinuous melting space, because mvirt = 1 − τFcrit/τG, but τG is equivalent to τFcrit if the process is discontinuous. The space utilisation is then simply given by the ratio: uF ¼

τFref τ Fcrit

ð4Þ

The value of τFref can implicitly be calculated from the following equation [17,20,21]: h0 ¼

! a_ 2 τ 3Fref 2gρ 2 _ 2Fref þ a0 τ Fref þ a0 aτ : 9η 3

ð5Þ

When η [Pa·s] and ρ are the glass melt viscosity and density, h0 is the height of the glass melt layer in the pot, a0 is the initial bubble radius and a_ is the growth rate of bubble radius. Possibly, the simplified Eq. (5a) could be used for calculation of τFref when a0 → 0:  τFref ¼

27h0 η 2gρa_ 2

1=3

:

ð5aÞ

If the value of τFcrit is expressed in analogy with Eq. (5a) and by using the average values of glass density and viscosity, we have:  τFcrit ¼

27hvirt η 2gρa_ 2

1=3

;

ð6Þ

where hvirt is the virtual height, which is the vertical distance passed by the critical bubble with respect to the flowing glass melt in time τFcrit.

Table 2 The temperature dependences of the most important quantities of the TV glass melt, i.e. viscosity, density and thermal conductivity. T in K. Kinematic viscosity ν (m2/s) Density ρ (kg/m3) Thermal conductivity λ (W/(m·K))

ν = exp(−11.50 + 6144.57/(T − 710.64)) ρ = 2790 − 0.2378 ⋅ T λ = 2 + 1.5 ⋅ 10−8 ⋅ T3

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Fig. 2. The typical picture of temperature and velocity distribution in the vertical central section through the melting pot when the melt is heated through the wall. The average temperature in the space is 1573.15 K (1300 °C), gradT = 200 K/m, the length of streamlines is 60 s, the temperature field is in the range of 1523.15–1653.15 K (1250–1380 °C).

The value of space utilisation may then be alternatively expressed with the help of the virtual height: uF ¼

τFref ¼ τ Fcrit



h0 hvirt

1=3

:

ð7Þ

The application of the value of hvirt provides the values of space utilisation with a physical meaning. The relations and values describing the fining time of the critical bubble at the limit values of either bubble growth rate or melt circulation intensity are presented in [22]. Note that the critical bubbles of constant size in the circulating viscous melt cannot reach the glass level. 3. Calculation conditions The numerical mathematical model Glass Furnace Model (GFM) [23] was applied for the calculations (unsteady three-dimensional melt flow derived from general laws of mass and momentum conservation based on assumption of molten glass taken as an incompressible Newtonian fluid, model solved using the finite element numerical method). The melting space was represented by a cylindrical pot; its inner diameter was 1 m and the height of the melt in the pot was 0.5 m. The average glass melt temperature was always maintained at 1573.15 K (1300 °C) to enable a comparison of the individual cases. The linear temperature gradients with higher temperature either in the centre or on the inner wall of the pot were put on the melt level to simulate the heating from the top or through the wall as is clear from Fig. 1. The value of the temperature gradients was in the interval of 25–300 K/m; the walls and bottom were insulated. TV-glass was

used as the model glass as for most previous modelling cases. Its composition is provided in Table 1, whereas the relevant glass properties are given in Table 2. The bubble-radius growth rates at the average temperature were mostly in the interval of 10−9–10−6 m/s; these were used for the characterisation of single calculation cases. 10000 bubbles of the initial radius of 5 × 10−5 m were scattered all over vertical central sections through the channel in order to cover any potential bubble starting position in the section. The size of 5 × 10−5 m was assessed as the minimum size of bubbles in the glass fining stage according to inspection of the laboratory tests. The growth and trajectories of all bubbles in the melt were calculated until the bubbles attained the glass level. The critical bubble was characterised by the maximum value of the fining time, τFcrit. The bubbles of larger sizes provide a similar picture of the fining process but their maximal fining times, τFcrit, were always shorter than these obtained for minimal bubbles. 4. Results of modelling The typical picture of the temperature and velocity distributions of the melt in the melting pot heated through the wall is presented in Fig. 2. Two circulation regions are apparent with a downward stream of the melt in the pot axis. The trajectories of 1 or 2 long-term bubbles, involving the critical one, at three horizontal temperature gradients and bubble growth rates are shown in Fig. 3a. The critical trajectories provide an important information on the bubble removal process at given conditions. The starting points of the most critical trajectories are located not too far or near the centre of the melt rotation and the trajectories show an increasing number of rotations with the decreasing growth rate of the bubble radius and with the growing intensity of circulations. The shapes of trajectories correspond

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Fig. 3. a, b. a) Trajectories of the 1–2 slowest (the absolutely slowest is critical) bubbles for the case of heating through the wall. The values of horizontal temperature gradients 25 K/m and 200 K/m were applied at the different values of growth rates of bubble radius, a_ = 2 × 10−8, 4 × 10−8 and 5 × 10−7 m/s. The value of 5 × 10−7 m/s characterises the effective bubble removal in a quiescent glass melt. b) The comparison of the critical-bubble trajectories in the pot for gradT = 25 K/m, a_ = 1 × 10−7 m/s (left) and the YZ projection of the critical bubble trajectory in the horizontal channel with the transversal temperature gradient 25 K/m, a_ = 3 × 10−7 m/s [21] (right). Starting positions of the critical bubbles are depicted by black points.

qualitatively to the critical trajectories found in the horizontal continual channel with the transversal temperature gradient [21] (see the comparison in Fig. 3b). Alternatively, the starting points of the slowest bubbles are located near the melt level and the relevant bubbles are pulled straightforwardly down by the central downward flow of the melt. However, as soon as the bubble attains a sufficiently large size near the bottom, it overcomes the downstream of the melt and rises back to the glass level. The dependence between the removal time of the critical bubble (i.e. its fining time) and the growth rate of the bubble radius at average temperature is then provided in Fig. 4 for different intensities of the melt circulation (given by the different values of the horizontal temperature gradients). The lowest curve in the figure represents the bubble fining time in the quiescent

glass melt (the reference removal times of the minimal bubbles, τFref). In all of the cases, the removal times of the critical bubbles are higher than the relevant reference times and the difference grows with both the growth rate of the critical bubbles and the intensity of melt circulation, given by the value of the temperature gradient. The latter dependence is demonstrated also by Fig. 5, where the values of τFref against gradT are plotted. In order to describe the character of the melt circulation in detail, the average angular velocities of the melt in four quadrants, as demonstrated by Fig. 2, and the total average angular velocity of the melt were calculated. The values are given in Table 3. A slightly different picture of melt flow is provided by Fig. 6, which is valid for pot heating from the top. Top heating is the usual way of

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Table 3 The absolute values of the average angular velocities of the melt in four regions of the  W — the vertical angular velocity in the wall region, ω  C — the vermelt rotation body: ω  U — the horizontal angular velocity in the tical angular velocity in the central region, ω  B — the horizontal angular velocity in the bottom region, ω  — upper (level) region, ω the total average angular velocity. The case of heating through the wall. gradT (K/m) 25 50 100 150 200

Fig. 4. The dependence between the critical bubble removal time and the growth rate of the bubble radius at the different values of the horizontal temperature gradients. The case of heating through the wall.

 W (rad/s) ω −3

1.47 × 10 2.35 × 10−3 2.25 × 10−3 2.93 × 10−3 2.55 × 10−3

 C (rad/s) ω −3

3.04 × 10 4.20 × 10−3 7.37 × 10−3 8.42 × 10−3 1.27 × 10−2

 U (rad/s) ω −3

2.33 × 10 3.14 × 10−3 4.60 × 10−3 4.85 × 10−3 7.84 × 10−3

 B (rad/s) ω −4

9.52 × 10 1.44 × 10−3 2.36 × 10−3 2.87 × 10−3 2.51 × 10−3

 (rad/s) ω 1.95 × 10−3 2.78 × 10−3 4.14 × 10−3 4.76 × 10−3 6.40 × 10−3

relatively complicated dependence between both quantities surprisingly reveals that the bubble removal times in the rotating melt are in some cases even lower than the reference values in the quiescent melt are. The average values of the melt angular velocities in four quadrants and the total average velocity of the melt are then plotted in Table 4. 5. Discussion of results 5.1. The case of heating through the wall

melting in gas-fired furnaces with one or more pots. The spring zone and the maximal temperatures are set in the centre of the melt level and the vertical pot axis is surrounded by the upward stream of the melt. At least three types of critical bubble trajectories are apparent from Fig. 7. At low temperature gradients and higher growth rates of the bubble radii, the starting points of critical bubble trajectories are mostly located in the close-to wall region, where the melt flows downwards, as shown in Fig. 7a. The spiral critical trajectories with a varying number of rotations and with starting points near the centre of the melt rotation are typical for the higher temperature gradients and medium values of the bubble-growth rates (see Fig. 7b–e). At very low values of the growth rates of the critical bubbles, the critical trajectories start near the bottom, and subsequently the bubbles are lifted to the melt level by the central upward flow of the melt (Fig. 7f). The efficiency of the fining, given by the dependence between the removal time of the critical bubble and the value of the growth rate of the bubble radius, is demonstrated by Fig. 8. The

Fig. 5. The increase of the critical bubble removal time with the growing intensity of the melt circulation (given by gradT) at different values of the growth rates of bubble radius. The case of heating through the pot wall.

The results of the modelling in the case of the pot heating through the wall qualitatively confirm the conclusions obtained by the simple analytical model [22]. The removal times of the critical bubbles evidently decrease with the increasing values of the growth rates of the bubble radius and with the decreasing temperature gradient, as is apparent from Figs. 4 and 5. The removal times are permanently considerably higher in comparison with bubble removal in the quiescent glass melt. At values of bubble growth rates higher than about 5 × 10 −7 m/s, the character of the glass flow is going to play a minor role for fining and the values of bubble removal times approach the values attainable for the bubbles rising in the quiescent melt (see Fig. 4). The important function of the fining agents, providing relevantly high bubble growth rates, is thus obvious also under the conditions of a circulating melt. The negative impact of the growing melt circulation intensity, obvious from Figs. 4 and 5, strikes both types of critical trajectories. As Table 3 shows, the central downward flow of the melt,  C , represents the part of the circulation flow with the highest velocω ity of the melt and therefore plays the role of a significant obstacle for the bubbles starting and rising in the central part of the pot. The higher the temperature gradient is (the melt rotation intensity), the higher the growth rate of the bubble radius, at which the vertical critical trajectory still exists, should also be (see Table 5). The further hindering effect of melt rotation is indicated by the growing number of circulations along the second type of trajectory of the critical bubble as shown by Fig. 3a. The bubble trajectory shifts to the pot axis, because the melt velocity in the upper part of the bubble spiral trajectory is higher than in the lower one. Sufficiently large bubbles then reach the glass level in the central part of the glass level. The bubbles in the intensively rotating glass melt without any fining agent (the average bubble growth rates are lower than about 1 × 10 −7 m/s) are characterised by the fully developed spiral trajectories and gradually almost copy the circulations of the melt. The appearance of the fully developed spiral critical trajectories is signalled by the values of bubble growth rates growing with the increasing value of the temperature gradient, as is clear from Table 5. The concentric shapes of the critical trajectories at the high melt rotation velocities and very low values of the growth rates of bubble radii in Fig. 3a show that there is almost the equilibrium between the forces raising the bubble to the level and dragging it down to the bottom. This fact leads to an enormous number of circulations of the slowly enfolding critical bubble trajectory (see Fig. 3a, gradT = 200 K/m) and is reflected by the

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Fig. 6. The typical picture of temperature and velocity distribution in the vertical central section through the melting pot when melt is heated from the top. The average temperature in the space is 1573.15 K (1300 °C), gradT = 200 K/m, the length of streamlines is 60 s, the temperature field is in the range of 1523.15–1653.15 K (1250–1380 °C).

high values of the removal times of the critical bubbles (Fig. 4), by the very low values of space utilisation, uF, as well as by the very high values of the virtual heights, hvirt, in Fig. 9a,b. The values of hvirt approaching h0 (uF is approaching 1) at the highest values of the bubble growth rates demonstrate on the contrary the independence of the bubble removal time on the intensity of melt rotation. The long vertical distances, hvirt, passed by bubbles with respect to the rotating glass melt at higher temperature gradients and low bubble growth rates give evidence of the dragging-down effect of the melt rotation on the bubble rise to the glass level. 5.2. The case of heating from the top 5.2.1. The values of bubble removal times, space utilisation and virtual height The more usual case of heating from above (which represents the classical heating of furnaces with one or more melting pots) is provided by the results presented in Fig. 8. The dependence between the removal time of the critical bubbles and the growth rate of the bubble radius shows a qualitative agreement with the analytical model [22] at higher bubble growth rates. However, if the bubble growth rates are low and the temperature gradients, i.e. the average melt angular velocities, are sufficiently high, the values of the bubble removal times are even lower than the reference ones in the quiescent melt. A corresponding behaviour is shown by the calculated values of the space utilisation, uF, and the virtual height, hvirt, in Fig. 10a,b. In the mentioned region of the bubble growth rates and circulation intensities, the space utilisation attains values greater than 1 and the values of hvirt b h0. Note that the values of uF > 1 were never found in the continual melting space [17,20,21], because in the continual space virtual dead space always exists, mvirt > 0 (see Eq. (3)).

5.2.2. The shapes of critical trajectories as indicators of the fining process The reason for this unexpected but potentially beneficial behaviour should be revealed with the help of an examination of the relevant critical bubble trajectories. Table 6 and Fig. 7 show that the shape of the critical bubble trajectories changes with both the temperature gradient (the melt circulation intensity) and the bubble growth rate. It is possible to distinguish between four types of critical trajectories, whose occurrence and shapes relate to the values of τFcrit, uF and hvirt. The first type of the critical trajectory, a vertical bubble trace with its starting point near the wall and the space bottom (see Fig. 7a),  W in overcomes the downstream of the melt (see the values of ω Table 4) and disappears at the lower values of bubble growth rates. The last occurrence of the trajectory of the vertical type is demonstrated by Table 6 in column 2. The virtual height, hvirt, grows in this region with the decreasing growth rate of the critical bubble and with the increasing value of the temperature gradient, as shows Fig. 10b. The increase of the hvirt is reflected by the slow decrease of the space utilisation, uF, in Fig. 10a. The increasing velocity of the downward flow of the melt is the cause of the unfavourable changes of both quantities. The critical spiral bubble trajectories develop as the second type of the critical trajectory (see Fig. 7b,c,e) at lower values of the bubble growth rates, and the virtual height subsequently attains its maximal value, hvirtmax (see the maxima in Fig. 10b). The values of the bubble growth rate at which the virtual heights of the developing spiral trajectories reach their maximal values grow with the growing temperature gradient as presented in column 3 in Table 6 and Fig. 10b. The increase of the virtual height and the corresponding decrease of uF (showing a flat minimum at hvirtmax) lead to the increase of the values of bubble removal times in Fig. 8. The corresponding critical bubbles

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Fig. 7. a–f. The trajectories of critical bubbles for the case of heating from the top. The values of the horizontal temperature gradients 25 K/m and 200 K/m were applied at different values of the growth rates of the bubble radius, a_ = 1 × 10−8, 4 × 10−8 and 5 × 10−7 m/s. Starting positions of the critical bubbles are depicted by black points.

start slightly above the centre of the melt rotation when both the melt angular velocity and bubble rising velocity are small. Owing to this rotation asymmetry, the bubble moves with a higher velocity along the upper part of its trajectory cycle (to the space wall) than it does in the lower part (to the centre). As a result of that, the bubble spiral trajectory develops towards the wall of the space (see Fig. 7e), where bubble rising to the level is hindered by the downward flow of the melt. The fact is then demonstrated by the increasing values of the bubble

removal times τFcrit, by the high values of hvirt, and by the decreasing values of uF (see Figs. 8 and 10a,b). The small critical bubbles develop almost concentric trajectories of the third type (see Fig. 7d) when the growth rate of the critical bubble further decreases. The third type of critical spiral bubble trajectories shows a lower number of rotations in comparison with the case of heating through the wall (see Fig. 3a, gradT = 200 K/m) and becomes unfolded owing to bubble rise. The unfolding character of the bubble trajectory is then marked by a higher temperature and consequently by a higher bubble rising velocity, in the close-to-level part of the spiral trajectory (see the temperature field of the melt in Fig. 6). Important feature of the trajectory development with the decreasing bubble growth rate is the shift of the rotations from the downward flow of the melt near the wall to the upward flow in the centre of the space (see the example of the trajectory development in Fig. 11). The upward flow of the melt supports the bubble rising to the melt level. This fact is reflected by the decrease of the virtual height, hvirt,

Table 4 The absolute values of the average angular velocities of the melt in four regions of the  W — the vertical angular velocity in the wall region, ω  C — the vermelt rotation body: ω  U — the horizontal angular velocity in the tical angular velocity in the central region, ω  B — the horizontal angular velocity in the bottom region, ω  — upper (level) region, ω the total average angular velocity. The case of heating from the top.

Fig. 8. The dependence between the critical bubble removal time and the growth rate of the bubble radius at different values of the horizontal temperature gradients. The case of heating from the top.

gradT (K/m)

 W (rad/s) ω

 C (rad/s) ω

 U (rad/s) ω

 B (rad/s) ω

 (rad/s) ω

25 50 100 150 200 300

9.95 × 10−4 2.21 × 10−3 2.89 × 10−3 3.46 × 10−3 5.70 × 10−3 6.02 × 10−3

1.55 × 10−3 1.94 × 10−3 2.80 × 10−3 3.60 × 10−3 3.19 × 10−3 4.35 × 10−3

1.22 × 10−3 2.39 × 10−3 5.15 × 10−3 6.12 × 10−3 7.04 × 10−3 7.90 × 10−3

6.20 × 10−4 1.00 × 10−3 1.33 × 10−3 1.58 × 10−3 1.87 × 10−3 3.23 × 10−3

1.09 × 10−3 1.88 × 10−3 3.04 × 10−3 3.69 × 10−3 4.45 × 10−3 5.37 × 10−3

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Table 5 Values of the growth rates of bubble radii indicating a change of the character of the critical bubble trajectories. The case of heating through the wall. gradT (K/m)

25 50 100 150 200

The growth rate of the bubble radius (m/s) The last vertical trajectory (in the space centre)

The spiral shape is fully developed

1 × 10−8 5 × 10−8 2 × 10−7 2 × 10−7 5 × 10−7

2 × 10−8 4 × 10−8 5 × 10−8 6 × 10−8 6 × 10−8

and by the slow increase of the space utilisation, uF, (for gradT = 150 K/m, see Fig. 10a,b, a_ = 1 × 10 −7–5 × 10 −9 m/s). The values of hvirt in the region sink gradually to the values of h0, and their positions at hvirt = h0 are shifted to the lower values of the bubble growth rate with the decreasing value of the temperature gradient (see column 4 in Table 6). Subsequently, the values of hvirt permanently decrease and uF increase. Particularly the cases with higher temperature gradients (gradT = 150–300 K/m) are accompanied by an increase of space utilisation and by the values of the bubble removal times, which are even lower than the reference values, τFref (see Figs. 8 and 10a for a_ around 1 × 10 −8 m/s). The lift effect of the melt flow on the bubble

Fig. 10. a, b. The values of the space utilisation, uF, and the virtual height, hvirt , calculated according to Eqs. (4) and (7), as a function of the average growth rate of the bubble radius for the case of melt heating from the top.

rising is obvious in the above-mentioned cases; the bubble removal times are reduced to about 60% of the reference value in the quiescent glass melt and at a bubble growth rate of around 1 × 10 −8 m/s. The critical trajectories in the region of the lowest bubble growth rates undergoes a further substantial change to their forth type. Figs. 7f and 12 show that the critical bubble trajectories are now characterised by starting points near the bottom. The critical bubbles move along the bottom to the space centre for most of their life and subsequently fully utilise the lifting effect of the central upward flow of the melt. With the growing temperature gradient, this G. Table 6 Values of the growth rates of the bubble radii indicating the character of the vertical bubble trajectories. The case of heating from the top.

Fig. 9. a, b. The values of the utilisation of the space, uF, and the virtual height, hvirt, calculated according to Eqs. (4) and (7), as a function of the average growth rate of the bubble radius for the case of melt heating through the wall.

gradT (K/m)

The growth rate of the bubble radius (m/s) The last vertical trajectory

hvirtmax

uF = 1 hvirt = h0

The first G. trajectory

25 50 100 150 200 300

7.0 × 10−8 3.0 × 10−7 7.0 × 10−7 8.0 × 10−7 8.0 × 10−7 1.1 × 10−6

2.5 × 10−9 1.9 × 10−8 6.0 × 10−8 1.0 × 10−7 1.7 × 10−7 3.5 × 10−7

2.0 × 10−10 2.0 × 10−9 7.0 × 10−9 1.8 × 10−8 2.5 × 10−8 4.5 × 10−8

– 1.0 × 10−10 3.0 × 10−9 6.0 × 10−9 1.0 × 10−8 3.5 × 10−8

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Fig. 11. The development of the critical-bubble trajectory for the case gradT = 150 K/m and for a_ between 1 × 10−7 m/s (hvirtmax) and 0 m/s. The case of heating from the top.

type of the critical trajectories sets up at higher values of the bubble growth rates as is seen from Table 6, column 5. The values of hvirt further decrease and the values of uF increase with the decreasing values of the growth rates of bubble radii, and the character of the forth type critical trajectories remains unchanged up to a_ = 0. The space utilisation grows in the entire range of the bubble growth rates, and the bubble removal times are much lower than the reference ones. However, the absolute values of the bubble removal times even at very high values of temperature gradients are still high for the practical application of fining in the circulating melt without fining agents. The high sensitivity of the results to the practically attainable values of the bubble growth rates may also be an obstacle for practical use.

The presented results indicate that the type of the critical trajectory is a function of the relation between the melt circulation velocity and the bubble rising velocity. This is significant particularly for the case of heating from above, where the transition from one to another type of the critical trajectory is bound to the trends of the development of both the virtual height and space utilisation. 5.2.3. The consequence of the numerical study for the bubble removal process in glass melts without any fining agent The case of heating the melt from above reveals an interesting situation when the slowly growing bubbles are removed from the circulating glass melt earlier than from the quiescent melt. The fact that

Fig. 12. The character of bubble critical trajectories in the case of heating from the top and for the temperature gradients 150, 200 and 300 K/m at the values of the bubble-growth rates a_ = 3 × 10−8, 1 × 10−8 and 5 × 10−9 m/s.

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even the bubbles starting from the bottom of the space with the circulating melt show lower removal times than the same bubbles starting from the bottom in the quiescent melt clearly presents the existence of the lift effect of the circulating melt under the conditions of heating from above. Both the character of melt flow and the temperature distribution in the space bring about the effect; however, the character of the glass flow appears more significant. The effect, taking place at the growth rates of bubble radius around 10 −8 m/s and at higher temperature gradients, is here described in detail by the character of the critical bubble trajectories and by the values of the virtual heights, hvirt. If the inequality hvirt b h0 is valid, the critical and some other bubbles starting near the space bottom utilise the lift effect of the rotating glass melt. This is clearly seen in the case of the critical bubble trajectories in Figs. 7f and 12, but the effect is present in all the cases when hvirt b h0. It is interesting that the velocity and temperature distribution in the space heated through the wall does not provide conditions for accelerated bubble removal. The favourable lift effect could facilitate the fining of some glass melts which must not be contaminated by current oxidation-reduction fining agents. The bubbles in such viscous glass melts retain almost constant sizes and can only hardly be removed by the buoyancy force. Some experiments show [24,25] that the growth rates of the bubble radii in the industrial glass melts without addition of fining agents move around 10 −8 m/s as presented in Table 7. The anticipated effect can be expected to be more pronounced in the real spaces, characterised by small or medium temperature gradients (see a comparison of the values of τFcrit at 150–300 K/m with those at gradT = 25–50 K/m for a_ around 10 −8 m/s in Fig. 8). 6. Conclusion The numerical modelling of bubbles of the critical radius in the model glass melting pot proves a significant impact of the character of the melt flow on bubble removal. The temperature gradients, set up in the melting pot according to the type of pot heating, bring about the circulation flows upon which the bubble rising is superimposed. In most cases, the resulting complex of critical bubble trajectories shows longer residence (fining) times than the vertical trajectories of critical bubbles drawn by its rising in a quiescent melt. This is caused by the fact that bubbles in the former case have to overcome not only the distance between their starting point and the melt level, but they also move with the circulating melt. Since the bubble in a circulation flow spends a longer average time in the downward than in the upward flowing melt (owing to bubble ascension), the distance for bubble rising, as well as the bubble removal time, are prolonged. The case of heating from above is characterised by the central upward flow of the melt; nevertheless, the case shows much lower values of the bubble removal times, when the melt circulation intensity (given here by the values of temperature gradients) is sufficiently high and the bubble growth rate is sufficiently low, than were expected. The case is

Table 7 The average growth rates of bubble radius in glasses melted without any fining agents or at lower temperatures where fining agents do not work. Lead–silica glass

Soda–lime–silica glass

TV glass

Float glass

No fining agents 1200 °C [24] 5 × 10−8 m/s

No fining agents 1400 °C [25] 3.3 × 10−9 m/s

1300 °C [17]

1150 °C

1.2 × 10−8 m/s

9.4 × 10−9 m/s

characterised by the unfolding spiral shape or by a simple, almost vertical shape of the critical bubble trajectories and is described by the values of the space utilisation and virtual height. The values of the virtual heights of these critical bubbles are lower than the height of the molten glass layer is. The lift effect of the circulating melt on the rise of the bubbles is obvious. The fining process of the special glasses without the addition of fining agents could thus be enhanced. Acknowledgement This work has been supported by the Technology Agency of the Czech Republic in the project no. TA01010844 ‘New glasses and their technologies’ and by the EU project of the Cross-Border Cooperation CR–SR, no. 22410420008 ‘White-Carpathian Glass Research, Development and Educational Base’. References [1] M. Cable, Glass Technol. 1 (1960) 139. [2] M. Cable, Glass Technol. 1 (1960) 144 (ibid. 2 (1961) 60, ibid. 2 (1961) 151). [3] M. Cable, A.R. Clarke, M.A. Haroon, Glass Technol. 9 (1968) 101 (ibid. 10 (1969) 15, ibid. 11 (1970) 48). [4] C.H. Greene, D.R. Platts, J. Am. Ceram. Soc. 52 (1969) 106. [5] H.O. Mulfinger, Glastechn. Ber. 45 (1972) 238. [6] A.K. Lyle, in: Travaux IVe Congress International du Verre, Paris, France, 1956, pp. 93–102. [7] L. Němec, Glass Technol. 15 (1974) 153. [8] L. Němec, Glass Technol. 21 (1980) 134. [9] M. Cable, J.R. Frade, Glastechn. Ber. 60 (1987) 355. [10] P.I.K. Onorato, M.C. Weinberg, J. Am. Ceram. Soc. 64 (1981) 676. [11] R.C.G. Beerkens, Glastechn. Ber. 63K (1990) 222. [12] B. Balkanli, A. Ungan, Glass Technol. 37 (1996) 164. [13] A. Ungan, Glastechn. Ber. 63K (1990) 19. [14] J. Matyáš, L. Němec, Glass Sci. Technol. 76 (2003) 71. [15] K. Oda, M. Kaminoyama, J. Ceram. Soc. Jpn. 117 (2009) 736. [16] L. Němec, M. Jebavá, P. Cincibusová, Ceramics-Silikáty 50 (2006) 140. [17] L. Němec, P. Cincibusová, Ceramics-Silikáty 52 (2008) 240. [18] L. Němec, P. Cincibusová, Ceramics-Silikáty 53 (2009) 145. [19] M. Polák, L. Němec, Ceramics-Silikáty 54 (2010) 212. [20] M. Polák, L. Němec, J. Non-Cryst. Solids 357 (2011) 3108 (ibid. 358 (2012) 1210). [21] P. Cincibusová, Ph.D. Thesis, Prague, 2010, pp. 40, 58, 95. [22] M. Jebavá, L. Němec, Ceramics-Silikáty 55 (2011) 232. [23] Glass Furnace Model: User's Guide (version 4.12). Glass Service, Inc., Vsetín, Czech Republic, 2009. [24] J. Kloužek, L. Němec, Glastechn. Ber. Glass Sci. Technol. 73 (2000) 329. [25] L. Němec, J. Am. Ceram. Soc. 60 (1977) 436.