Modelling of the controlled melt flow in a glass melting space — Its melting performance and heat losses

Journal of Non-Crystalline Solids 430 (2015) 52–63

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Modelling of the controlled melt flow in a glass melting space — Its melting performance and heat losses Marcela Jebavá ⁎, Petra Dyrčíková, Lubomír Němec Laboratory of Inorganic Materials, Joint Workplace of the University of Chemistry and Technology Prague, Technická 5, 166 28, Prague 6, and the Institute of Rock Structure and Mechanics of the 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 13 July 2015 Received in revised form 20 August 2015 Accepted 22 August 2015 Available online 5 October 2015 Keywords: Glass melt flow; Mathematical modelling; Energy distribution; Space utilization; Melting performance

a b s t r a c t The factors influencing the character of the melt flow were defined and examined in a model glass melting space. The batch blanket was simulated by an inflowing glass melt with sand particles and bubbles and the heating elements by the defined volumes of the melt where heat was evolved. The character of the melt flow was set up by a proper arrangement of the heating elements in the space. The sand dissolution and the bubble removal were modelled in the space; the space utilization, melting performance, and heat losses were calculated. The required character of the melt flow was brought about by the energy evolution in the region of the longitudinal space axis and by the supply of a substantial part of energy to the region beneath the inflowing melt. The results of the modelling have confirmed that the suitable flow character in the space was a helical-like flow, which was attained by the combination of an almost uniform forward flow with imposed transversal melt circulations. High values of the space utilization, several times higher melting performance and proportionally lower specific heat losses were acquired when compared with the values attained under conditions simulating the melt flow in industrial melting furnaces. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The industrial glass melting space represents a continual reactor in which the homogenization phenomena, namely the dissolution of both solid and liquid inhomogeneities and bubble removal, should be completely and economically accomplished. No wonder so much effort was devoted to the examination of melting kinetics [1–12]. Later, mathematical modelling was an excellent tool for phenomena investigation following also under the condition of the flowing melt [13–16]. Thus, the quality of the melt flow character with respect to the quality of the entire melting process could be assessed as well and the question of the optimal character of the melt flow in the glass melting space arose. Cooper [17] was one of the first who discussed the role of longitudinal and transversal melt circulations present in the horizontal melting space and referred to both positive and negative impacts of natural convection on the course of homogenization melting phenomena. However, the question of the optimal melt flow remained unresolved. Theoretically, the optimal character of the melt flow may be estimated from the fundamental requirements of the melting process and from the properties of the standard types of liquid flow such as plug flow and ideal mixer. The requirement of homogenization phenomena on the flow character can be briefly summarized as follows: no regions of lazy or longitudinally circulating melt (dead spaces), no regions of ⁎ Corresponding author. E-mail address: [email protected] (M. Jebavá).

http://dx.doi.org/10.1016/j.jnoncrysol.2015.08.039 0022-3093/© 2015 Elsevier B.V. All rights reserved.

overprocessing (regions where the homogenization is already accomplished), adequate time for the homogenization phenomena, and sufficient mixing ability to enhance dissolution phenomena. Both plug flow and mixer fulfil the requirement of zero dead spaces. No regions of overprocessing arise in the plug flow, but no mixing ability is available here. On the contrary, the ideal mixer provides the maximal mixing ability of the melt; nevertheless, its broad residence time distribution curve — starting at the zero residence time — excludes the mixer from the consideration. Regardless of the zero mixing ability, the plug flow is the primitive base for the optimal flow character; its realistic accomplishment is the uniform isothermal flow. When both the dissolution and the bubble removal have an adequate and predictable chance to be realized, no dead spaces exist. However, the practical realization of the flow should encounter problems in glass melting spaces with a slow working flow and heterogeneous temperature distribution. The horizontal temperature gradients as well as heating from the bottom can cause melt circulation and large temperature gradients with higher temperature near the level enlarge the quality differences between melt trajectories and, consequently, the space of overprocessing increases. In spite of that, uniform flow remains a chance for the melting process and should be tested. Another way of efficiently organizing the flow consists in the combination of a forward plug flow with the melt mixing perpendicularly to the main flow which represents a theoretical solution for melting phenomena (quasi-plug flow) [18], but practically does not solve bubble removal. The helical flow resulting from the superposition of slow transversal melt circulations to the uniform longitudinal

M. Jebavá et al. / Journal of Non-Crystalline Solids 430 (2015) 52–63

flow appears a realistic variant of the suitable flow character for both phenomena. The helical flow at least partially preserves a good homogenization effect and, in addition, it seems to fulfil also the requirements of efficient bubble removal. To set up the helical character of the melt flow has been the subject of several patents [19–23], but the existence of adequate transversal circulations, as well as the effect of the controlled melt flow on the course of homogenization phenomena, could not be proved under the conditions of industrial operation. The problem that arose of the quantitative evaluation of the melt flow character was dealt with through the introduction of a new quantity called utilization of the space [24–28]. The utilization of the space represents the ratio between the time necessary to accomplish the given homogenization phenomenon under conditions of a quiescent melt and the time consumed in a continual space with a given character of the melt flow. Dead and overprocessing spaces are involved and may be obtained by the mathematical modelling of the relevant homogenization phenomena in the space. The values of the space utilization can be included in the relations for the melting performance and specific energy consumption of the process in the melting space. The values of the space utilization for both phenomena under conditions of the plug flow are 1, for the sand dissolution, they are 0.445, and for the removal of linearly growing bubbles, they are 0.666 in the orthogonal channel with the isothermal uniform flow [27]. The fact of the helical-like character of flow as the optimal variant of flow for both sand dissolution and bubble removal was then proved with the help of the space utilization quantity in the cited modelling studies. The optimal character of the melt flow was determined as a function of the ratio between the preset transversal and longitudinal temperature gradients in the melt; the maximal utilization values were mostly 0.6–0.8 at the gradient ratio between 5 and 10. The results have shown that the melting performance can increase (and the energetic losses decrease) even several times if the helical-like flow were set in the melting space. Now, more realistic conditions are demanded — particularly the sources of energy should be used instead of merely setting temperature gradients to show whether the optimal flow conditions will be realizable in practice. If energy sources are applied to simulate real melting, the resulting character of the melt flow should strongly depend on their local positions in the melting space. This work is focused on the definition of the fundamental factors affecting the establishment of the helical-like melt flow or potentially the uniform flow in a model melting space with a simulated batch blanket and with sources of energy. Further, the work deals with the spatial energy distribution as the main factor determining the character of the flow and, consequently, the utilization of the space, melting performance, and specific heat losses of the model melting space. 2. Theoretical part The resulting character of the melt flow particularly depends on the horizontal distribution of the supplied energy in the space. Let us consider an orthogonal horizontal space for continual melting. The energy needed for the process primarily involves the theoretical specific heat HTM, i.e. the energy for batch reactions, phase and modification transitions and for heating of the contents to the space exit temperature. This part of the energy is dominant and should be delivered just in the batch blanket and its vicinity. The second part of the energy needed ·L represents the heat losses through boundaries, the relevant heat flux H . ·tot In a simple space with inner sources of energy, the total heat flux H is given by: · · · Htot ¼ HTM MþHL ðkJ=s ¼ kWÞ; · where M is the mass melting performance (kg/s).

ð1Þ

53

If the fraction of the space surface corresponding to the region with the batch blanket is ξ, Eq. (1) can be written as: · · · · Htot ¼ H TM MþξHL þ ð1−ξÞHL :

ð2Þ

Thus, the needed heat flux into the input part of the furnace amounts · · to HTM MþξHL and the heat flux to cover the heat losses in the following · part of the space with the free level is ð1−ξÞHL, If the energy distribution for Eq. (2) is valid, the space occurs in a balanced state from the point of view of energy delivery, no global temperature differences arise between both parts of the space and no natural longitudinal circulations would be expected. In the unbalanced state, however, longitudinal melt circulations develop according to the longitudinal temperature gradient that arises and the adjustment of another type of flow, such as the helical-like flow, would be much more difficult. The situation is schematically presented in Fig. 1. If Eq. (2) is valid, the actual amount of heat delivered in the region · · of free level, Hlevel , is equivalent to ð1−ξ ÞHL and the longitudinal component of the melt velocity will have a parabolic profile as curve 1 shows — a uniform flow will result. On the other hand, if an inequality will hold, longitudinal circulations will develop as curves 2 and 3 demonstrate. The current industrial case is noticeably unbalanced because the actual amount of energy delivered directly in the batch region is lower · than the one needed (being currently around 0.5 Htot ) and a strong longitudinal circulation develops with the backflow near the melt level (curve 2 in Fig. 1) as was observed in practice and proved by mathematical modelling. Consequently, it would be difficult to set up sufficiently intensive transversal circulations in the space leading to a helical-like flow. Particularly, when the amount of energy available for transversal circulations below the free level is low, only a part of heat losses is involved (Eq. (1), the third term). Therefore, it is necessary to examine if all the factors are able to increase the intensity of the transversal circulations merely to attain a helical-like character of flow, especially in spaces with unbalanced energy distribution as well as to modify the helical-like melt flow favourably. The potential factors affecting the process may be defined as follows: 1) 2) 3) 4)

The horizontal energy distribution. The vertical energy distribution. The space insulation. The interruption of the symmetry of the imposed transversal circulations. 5) The mechanical support of transversal circulations. This article deals with the effect of horizontal and vertical energy distributions in the model space on the established character of the melt flow. The values of the relevant quantities — the space utilization, mass melting performance, and specific heat losses — are used. A definition of the space utilization is needed. The utilization of the continual space, uH (index H designates the relevant homogenization process, sand dissolution or bubble removal) expresses the relation between the reference homogenization time in a quiescent melt, τHref, and the mean residence time of the melt in the space under critical conditions, τG [25–28]. Thus, τG is the time needed for the same process realized in the continuous space with flow patterns, being given by the ratio between the space volume and volume flow rate (see Eq. (3)). The critical state describes the situation when the first particle of the critical size, sand or bubble, coming from the set of the examined particles attains either space output (sand particle) or the melt level (bubble). The slower of the two parallel phenomena is the controlling one. The uH value is defined as follows: uH ¼

τHref V ; τG ¼ · ; u ∈ h0; 1i; τG V

ð3Þ

54

M. Jebavá et al. / Journal of Non-Crystalline Solids 430 (2015) 52–63

· · Fig. 1. The schematic representation of the longitudinal character of the melt flow as a function of the horizontal distribution of energy in the melting space. Hinput and Hlevel are actual heat · fluxes of heat delivered in both space regions, their sum, Htot , is practically invariable at a constant average temperature in the entire space.

· where V is the volume of the space (m3) and V is the volume flow rate (melting performance or pull rate) (m3/s). If the controlling phenomenon is sand dissolution, τΗref = τDave, whereas τΗref = τFref is valid when bubble removal is controlling. The quantity τFref is the fining time which the critical bubble needs to rise the distance h0 in a quiescent liquid at the average temperature in the space. The details of the τFref calculation are given in [26], equation (6). τDave is the average sand dissolution time. Both values of the space utilization — uF for fining and uD for sand dissolution — may be involved in the expressions for the heat losses of the process and for the melting performance. If both homogenization phenomena are considered to be parallel, the less efficient phenomenon is the controlling one. The specific heat losses of the space through boundaries decrease and the performance of the glass melting process increases with space utilization: H LM ¼

· HL τ Dave 1 ρV uD

ð4aÞ

· HL τ Fref 1 ; ρV u F

ð4bÞ

or H LM ¼

· VuD V¼ τDave

ð5aÞ

or · Vu F ; V¼ τFref

ð5bÞ

· where HLM are the specific heat losses (J/kg), HL is the total heat flux across the space boundaries (J/s) and ρ is the glass density (kg/m3). The former expressions in Eqs. (4a), (4b), (5a) and (5b) are valid for the sand dissolution as the controlling phenomenon and the latter ones for the controlling phenomenon being bubble removal. The space utilization for the sand dissolution may be expressed with the assistance of two fractions of dead spaces — the fraction of dead space for the melt flow mG and the fraction of the space of the sand dissolution mD. Similarly, the space utilization for the bubble removal can be expressed through the fraction of virtual dead space for bubble

removal mvirt and virtual bubble rising distance hvirt [28]. The mentioned quantities provide a more detailed view of the character of the melt flow. Only the values of the space utilization will be applied in this work for the evaluation of the flow efficiency because a predictable character of the flow was set up in the modelling experiments. If the bubble nucleation on the quartz particles does not occur, both phenomena run simultaneously. So, the less efficient phenomenon becomes the controlling one. When the melting conditions are varied and the sand dissolution is the controlling phenomenon, the effect of flow changes, described by the value of the space utilization uD, may be separated from the effect of different time-temperature histories of particle trajectories described by the value of τDave (see Eqs. (4a) and (5a)). If the bubble removal controls the melting, both effects are separable only when the average temperature in the space varies (at a constant average temperature, the value of τFref is constant). The quantities defined by Eqs. (3)–(5b) may be acquired by modelling the critical state of bubble removal or sand dissolution (the critical particle disappears just at the output) in the melting space with adjusted flow patterns. Here, the demanded flow patterns have been set by proper energy distribution in the model melting space. 3. Modelling conditions and procedure The model melting space was chosen for calculations corresponding to the small horizontal melting furnace as is clear from Fig. 2. Three layers of refractory materials formed the walls and bottom of the space. The free level of glass was insulated (the heat flux through the level was zero). The melt heating was ensured by the initial volume heating condition. When using this boundary condition, the heat necessary is evolved in a defined volume of the melt and serves as an inner source of energy. The use of the volume heating condition allows greater variability of the heating system and purifies the obtained results from the side effects accompanying the application of real heating elements (electrodes). The applied central longitudinal heating volumes of glass designated as heating element 1 and heating element 2 are depicted in Fig. 2. The same heating elements but with a height of 0.3 and 0.05 m were used in the selected cases. The bottom heating by two horizontal melt plates having the same length as elements 1 and 2 and a thickness of 0.04 m were also used in several cases. The input of the melt containing sand particles and bubbles was applied instead of considering the batch blanket. The reason is that the batch model is not able to preset the relevant batch conversion capacity in a very broad range of the critical

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55

Fig. 2. Scheme of the model melting space with heating elements. Inner dimensions: length of 6.77 m (6.225 m to the transversal barrier from refractory material), width of 2 m, the height of the glass is 1 m. Dimensions of the heating element 1: length of 2.023 m, width of 0.04 m, height of 0.6 m. Dimensions of the heating element 2: length of 4.197 m, width of 0.04 m, height of 0.6 m. Dimensions of the melt input through the level: length of 2.023 m, width of 2 m. Temperature of inflowing glass melt: 950 °C. Average temperature in the space: 1420 °C.

melting performance achieved in the melt with different melt flow characteristics. Nevertheless, the authors believe that the accepted simplifications do not hinder the ability to specify the principles controlling the relation between the energy distribution and the overall character of the melt flow. The input melt velocities had a half parabolic profile in the X-direction and the full parabolic profile in the Y-direction and the output melt velocities had a full parabolic profile. The non-slip conditions, i.e. the zero melt velocities, were assumed on the solid boundaries (walls) and the slip condition on the free glass level. The float type of glass was chosen as the model glass. The temperature dependence of the glass density and kinematic viscosity in the temperature interval of 900–1800 °C have the form [30]: ρ ¼ 2558:1−0:1385ðt þ 273:15Þ


 3 kg=m ;

ð6Þ

   2  11952 m =s : ν ¼ exp −15:452 þ t−206:5

ð7Þ

The temperature dependences of the average sand dissolution rates · and the average growth rate of the bubble radius a in the temperature interval of 1100–1500 °C are as follows [30]: h i · −1:562  10−18 a¼ exp 1:53  10−2 ðt þ 273:15Þ ðm=sÞ; 2 " # · 1 3:804  105 2:608  108 ðm=sÞ; þ a¼ exp 120:343− 2 t þ 273:15 ðt þ 273:15Þ2

ð8Þ ð9Þ

· where a is the particle radius. Both values of a were approximated by the zero values in the temperature interval of 950–1100 °C. Only a small error results from this approximation owing to the very low dissolution

or growth rates of particles in the given temperature interval and the rapid glass heating behind the input. The semiempirical model of bubble behaviour was applied which takes into account partial bubble dissolution at decreasing temperature [12]. Bubbles with an initial radius of 5 × 10−5 m were chosen as the smallest ones (experience from laboratory melts) and the sand particles of 5 × 10−4 m as the maximal ones (from the sieve analysis). The history of around 104 bubbles and sand particles with regularly located starting points at the level of the input were followed by using the average rates of sand dissolution and the average bubble growth rates applied in the semiempirical model of bubble behaviour. The later bubble nucleation inside of the space was not considered. When calculating the sand trajectories, the velocity components of the melt flow were used whereas the vx and vy components of the melt velocity and the sum of the vz component with the instant bubble rising velocity according to the Stokes' law were used to calculate the bubble trajectories. The trajectories and sand particles radii were calculated with the time step of maximally 1 s; the time step for the bubble radius calculation was 0.1 s. The time and temperature dependent values of the sand particle and bubble radius developments were experimentally obtained. To reduce the errors of trajectory calculations as much as possible, the dense net of calculation grid points was used with an average distance of 2.5 cm between grid points. The critical particle, its critical trajectory, and controlling phenomenon were characterized by the fact that the critical particle was the first one to reach the location above the transversal refractory wall (barrier) before the output. The accomplished controlling phenome· non determined the final mass melting performance M , the mean residence time of the melt in the space τG, and the average sand dissolution time τDave. The reference bubble removal time τFref was calculated from equation (6) in [26]. The values of τG and τDave or τFref applied to Eq. (3) provide the space utilization uD or uF.

Table 1

· · The summary results of calculations when increasing the value of 100Hinput =Htot . With the exception of case 1, central longitudinal heating is applied. · H

Case

1 2 3 4 5 6 7 8 9 10

· Hinput · Htot

u

τDave, τFref

τG

· M

HLM

100 ·glass1

100

(%)

(%)

(controlling)

(s)

(s)

(kg/s)

(t/day)

(kJ/kg)

Ref. case 0/100 33/66 50/50 66/33 75/25 80/20 85/15 90/10 100/0

46.5 20 59 76 82 90 93 95 96.5 100

uD = 0.086 uD = 0.095 uD = 0.13 uD = 0.18 uD = 0.23 uD = 0.26 uD = 0.30 uD = 0.35 uF = 0.36 uD = 0.31

τDave = 2471 τDave = 2736 τDave = 2492 τDave = 2414 τDave = 2286 τDave = 2282 τDave = 2147 τDave = 1967 τFref = 1871 τDave = 1707

28,904 28,905 19,269 13,444 9798 8706 7119 5635 5276 5548

1.0 1.0 1.5 2.15 2.95 3.32 4.06 5.13 5.48 5.21

86.4 86.4 129.6 185.8 254.9 286.8 350.8 443.2 473.5 450.1

331.2 325.3 216.9 151.0 109.9 97.4 79.8 63.6 59.8 63.2

Hglass2

56

M. Jebavá et al. / Journal of Non-Crystalline Solids 430 (2015) 52–63

Table 2 The summary results of calculations when the character of heating is varied.

Case

100

· Hinput · Htot

(%) 1 2 3 4 5 6 7 8 9 10 11

93 93 39 81 95.8 100 100 100 93 93 93

Central longit. heating, h = 0.3 m Central longit. heating, h = 0.05 m Bottom heating Bottom heating Bottom heating Bottom heating Planar heating, 0.6 m Planar heating, 0.8 m Bottom and central longit. heating Central longit. heating 0.6 + 0.05 Central longit. heating 0.6 + 0.05

HLM

τG

(controlling)

(s)

(s)

(kg/s)

(t/day)

(kJ/kg)

uD = 0.32 uF = 0.32 uD = 0.095 uD = 0.069 uF = 0.18 uD = 0.085 uD = 0.094 uD = 0.12 uD = 0.26 uF = 0.31 uD = 0.40

τDave = 2108 τFref = 1896 τDave = 2404 τDave = 2217 τFref = 1896 τDave = 2046 τDave = 2084 τDave = 1456 τFref = 1882 τDave = 1896 τDave = 2548

6645 5929 22,788 32,130 10,533 24,087 22,170 12,133 7226 6116 6370

4.35 4.88 1.35 0.90 2.52 1.20 1.30 2.40 4.00 4.72 4.55

375.8 421.6 116.6 77.8 217.7 103.7 112.3 207.4 345.6 407.8 393.1

74.7 66.8 244.8 363.0 130.6 273.0 216.9 136.7 82.6 68.8 71.7

The following procedure was applied to examine the effect of the energy distribution in the model space on the character of the melt · flow. The heat flux brought by the inflowing melt Hglass input can be determined from the equation: · · Hglass input ¼ cM ΔtMð J=sÞ;

· M

τDave, τFref

u Heating arrangement

of 0.2 m and at a distance of 5.525 m from the inner side of the front wall. The temperature of the inflowing melt was here kept at 1250 °C, so the resulting amount of energy supplied in the space inflow region was 46% of the total energy amount and roughly corresponded to the typical industrial case.

ð10Þ

· where M is the mass flow rate (melting performance). The specific heat capacity cM of the float glass in the interval of 293–2073 K is described by the equation [30]: cM ¼ 905:47 þ 0:41714ðt þ 273:15Þ−4:1098  10−5 ðt þ 273:15Þ2 ð11Þ ð J=ðkg  KÞÞ: The temperature of the inflowing melt was estimated as 950 °C. According to Eq. (10), the total amount of energy brought by the melt depends on its flow rate. The heat flux supplied into the space by heating elements 1 and 2 to attain and keep an average temperature of 1420 °C was distributed between the region under the inflowing melt and the region under the free level. The part of energy applied in · the inflow region, namely Hglass1 , (heating element 1, see Fig. 2) was set from 0 to 100% of the heat flux applied in the respective calculation · cases, and the other part, namely Hglass2 (heating element 2), was used in the region under the free level. The overall heat flux supplied in the inflow region in the respective calculation cases was then · · · Hinput ¼ Hglassinput þ Hglass1 and the heat flux in the region under free · · level was then Hlevel ¼ Hglass2 (see Fig. 1). The fraction of energy supplied in the melt inflow region can individually be expressed with the help of · · · · · · heat fluxes as Hinput =Htot where Htot ¼ Hglassinput þ Hglass1 þ Hglass2 . The reference case approximately corresponding to the industrial one was modelled by the transversal heating volume between the sidewalls located at the bottom and having a height of 0.3 m, a thickness

4. Results of modelling The numerical results relevant for the modelling evaluation are summarized in Tables 1 and 2. The values corresponding to the controlling phenomenon are presented. The development of glass flow character under the conditions of energy redistribution is demonstrated in the following figures. The longitudinal and transversal sections through the melting space at selected Y and X coordinates are presented, in which the projections of the melt trajectories (their length corresponds to the distance travelled by the melt in 60 s) are presented with a depiction of the main flow direction and spring point (if present). Fig. 3 provides the picture of the glass flow in the reference case which approximately simulates the flow patterns obtained by the modelling of industrial melting furnaces. The apparent longitudinal circulations of the melt with the back flow along the level and spring point close to the space exit were observed. The transversal section provides a flow picture without noticeable transversal flows. Fig. 4 brings the relevant projections of the critical trajectories. As expected, the substantial part of the critical trajectories is located in the melt forward flow close to the bottom. The value of the space utilization 0.085 (case 1 in Table 1) roughly corresponds to the values between 0.05 and 0.06 found in an industrial melting furnace [31]. The following figures show the glass flow patterns when the heating through the central longitudinal plate of glass was applied (see Fig. 2) which supports the transversal circulations of the melt and, consequently, the resulting helical-like flow. Fig. 5 presents the case with 59% of the total energy supplied into the melt inflow region and Fig. 6 the relevant critical trajectories of the sand particles.

Fig. 3. The character of the longitudinal and transversal melt flow patterns marked by the streamlines of the melt in the reference case simulating the flow in an industrial furnace. The central longitudinal section through the space (XZ plane) and the transversal section at the distance of 4.6 m from the inner side of the front wall (YZ plane). The arrows → and ← highlight the main directions of the melt flow, the arrow ↓ points at the spring point position. 46% of the total energy is supplied to the input region. Temperature scale is in °C.

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Fig. 4. The projections of two critical trajectories of the sand particles (sand dissolution is the controlling phenomenon) into the XY and XZ planes. In the reference case, 46% of the total energy is supplied to the input region.

The overall character of the melt flow in Figs. 5 and 6 appears similar to the previous case, but less noticeable transversal circulations are already manifest in Fig. 5. The value of the space utilization is higher than in the reference case (Figs. 3–4, Table 1). The intensity of the longitudinal circulations of the melt decreases (see below) and the position of the spring point shifts to the front wall with the increasing amount of energy supplied into the melt inflow region. The critical

sand trajectories remain in the melt forward flow near the bottom. The character of the melt flow at 93% of the total energy supplied in the melt inflow region is presented in Figs. 7 and 8; the space utilization considerably increases (case 7 in Table 1). · · Subsequently, the further slight increase of the value of100Hinput =Htot to 95% brings a substantial rearrangement of the flow patterns, which is apparent from Fig. 9. The spring point shifts substantially closer to the

Fig. 5. The character of the longitudinal and transversal melt flow patterns marked by the streamlines of the melt when the melt is heated along the central longitudinal axis of the space. The longitudinal section through the space corresponding to ¼ or ¾ of the space width (XZ plane) and the transversal section at the distance of 4.6 m from the inner side of the front wall (YZ plane). The arrows → and ← highlight the main directions of the melt flow, the arrow ↓ points at the spring point. 59% of the total energy is supplied to the input region.

Fig. 6. The projections of two critical trajectories of the sand particles (sand dissolution is the controlling phenomenon) into the XY and XZ planes. 59% of the total energy is supplied to the input region.

58

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Fig. 7. The character of the longitudinal and transversal melt flow patterns marked by the streamlines of the melt when the melt is heated in the central longitudinal axis of the space. The longitudinal section through the space corresponding to ¼ or ¾ of the space width (XZ plane) and the transversal section at the distance of 4.6 m from the inner side of the input front wall (YZ plane). The arrows → and ← highlight the main directions of the melt flow, the arrow ↓ points at the spring point. 93% of the total energy is supplied to the input region.

front wall, the forward melt flow occurs along the level behind the spring point, but the forward flow near the bottom is still manifest. The weak backward flow is observable in the middle part between both forward flows. The complex distribution of the longitudinal velocities of the melt is the result of the fact that the shift of the spring point to the input forms the forward flow along the level but the strong bottom forward flow of the relatively cool glass remains preserved. The critical sand particle trajectory stays in the bottom forward flow. · · The further increase of the value of 100Hinput =Htot to 96.5% preserves the picture of the longitudinal component distribution with two forward flows, but the critical bubble trajectory (bubble removal becomes the controlling phenomenon) shifts to the level (see Fig. 10). This and the previous variant show the maximum values of the space utilization

(see Table 1) despite only a very small amount of energy is available for the development of transversal circulations in both cases. If the entire energy is supplied in the input region, the overall character of the melt flow approaches the uniform forward flow, but the weak back flow is apparent near the exit and bottom, as Fig. 11 shows. The spring point shifts to the border of the melt input. Almost no transversal circulations are observed in the YZ section through the space. The critical sand trajectory stays in a relatively fast forward flow along the level. Also the planar-like heating of the melt by two glass layers on the space bottom was applied in several cases. Fig. 12 shows the melt flow character in the case when 81% of energy is located in the melt inflow region. The longitudinal velocity component shows the expected flow

Fig. 8. The projections of two critical trajectories of the sand particles (sand dissolution is the controlling phenomenon) into the XY and XZ planes. 93% of the total energy is supplied to the input region.

Fig. 9. The character of the longitudinal and transversal melt flow patterns marked by the streamlines of the melt when the melt is heated in the central longitudinal axis of the space. The longitudinal section through the space corresponding to ¼ or ¾ of the space width (XZ plane) and the transversal section at the distance of 4.6 m from the inner side of the front wall (YZ plane). The arrows → and ← highlight the main directions of the flow, the arrow ↓ points at the spring point. 95% of the total energy is supplied to the input region.

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59

Fig. 10. The projections of a few critical trajectories of bubbles (fining is the controlling phenomenon) into the XY and XZ planes. 96.5% of the total energy is supplied to the input region.

Fig. 11. The character of the longitudinal and transversal melt flow patterns marked by the streamlines of the melt when the melt is heated in the central longitudinal axis of the space (XZ plane). The longitudinal section through the space corresponding to ¼ or ¾ of the space width (XZ plane) and the transversal section at the distance of 4.6 m from the inner side of the front wall (YZ plane). The arrows → and ← highlight the main directions of the flow, the arrow ↓ points at the spring point. 100% of the total energy is supplied to the input region.

patterns forming the longitudinal circulation with the backflow along the level. However, multiple circulation flow patterns (Bernard's circulations) are seen in the transversal section through the space. The critical trajectory of sand as a controlling phenomenon runs near the space bottom and its curly character is caused by the mentioned flow patterns. The space utilization is low, as Table 2 case 4 shows. The remaining three examined variants with planar-like heating and 39, 95.8 and 100% of energy situated in the melt inflow region showed the same tendency in the development of the longitudinal melt velocity component as the previous group with central longitudinal heating, but the values of the space utilization were always several times lower. The reason for the low utilization values consists in the absence of extended transversal melt circulations developed in the previous group. However, if the planar heating in the inflow region is combined with the central longitudinal heating in the region under the free level, the transversal

circulations develop in the below level region and the space utilization appreciably increases, as case 9 in Table 2 shows. The vertical distribution of energy slightly affects the melt flow character and, consequently, the melting efficiency as well. When the central longitudinal heating is applied, the shift of energy to the space bottom causes only a very slight increase of the space utilization, but the melting performance grows by 20% (see case 7 in Table 1 and case 2 in Table 2). The temperature in the bottom forward flow increases by about 6 °C in the region under the free level if the height of the longitudinal heating volume is only 0.05 m. The effect of the vertical energy distribution in the region of the inflowing melt appears less clear. The planar-like heating by the thin glass volumes situated at 0, 0.6 and 0.8 m above the bottom and with the entire energy situated in the inflow region (cases 6–8 in Table 2) gives some indication. Its application shows the expected longitudinal circulations with the forward

Fig. 12. The character of the longitudinal and transversal melt flow patterns marked by the streamlines of the melt when the planar-like heating is applied by two melt plates on the space bottom. The central longitudinal section through the space (XZ plane) and the transversal section at the distance of 4.6 m from the inner side of the front wall (YZ plane). The arrows → and ← highlight the main directions of the flow, the arrow ↓ points at the spring point. 81% of the total energy is supplied into the input region.

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Fig. 13. The character of the longitudinal and transversal melt flow patterns marked by the streamlines of the melt when the planar-like heating is applied by the horizontal thin glass volume located in the melt inflow region 0.8 m above the bottom. The central longitudinal section through the space (XZ plane) and the transversal section at the distance of 4.6 m from the inner side of the front wall (YZ plane). The arrows → and ← highlight the main directions of the flow. 100% of the total energy is supplied into the input region.

flow along the level, but the character of the melt flow with two forward flows develops in the case with planar-like heating at 0.8 m above the bottom. The other forward flow arises here near the bottom whereas the backflow evolves in the middle part of the space (see Fig. 13). Such a type of flow obviously arises when a high portion of the energy is supplied in the inflow region and near the level (compare with Fig. 9). The space utilization in this run of calculations slightly increases with the energy shift to the level, but the 100% increase of the mass melting performance found in the last case is caused by the decrease of τDave rather than by an increase of the space utilization. This is evidently because most of trajectories (involving the critical one) run through the region of high temperatures near the melt level. As compared to the energetically analogical case with the central longitudinal heating (case 10 in Table 1), the planar heating appears much less effective. Four variations are further compared when the effect of the vertical energy distribution in the inflow region with the central longitudinal heating is examined: case 7 in Table 1 and cases 2, 10, and 11 in Table 2. Better results are achieved in cases where the heating is shifted to the bottom in at least one part of the space. 5. Discussion of results The modelling results show that the character of the melt flow determines the utilization of the space for the melting phenomena and affects the time–temperature history of the phenomena. The melt flow quality thus becomes an important aspect of the glass melting process. According to the previous modelling results [25–26,28–29], the superposition of the working flow, naturally existing longitudinal circulations, and transversal melt rotations imposed can result in a helical-like flow which enhances the originally low utilization of the melting space and mostly improves the time–temperature history of the process. The aim of this work was to distribute the supplied energy — as the principal driving force of natural convection — in such a manner that the needed helical-like character of the flow with high space utilization will be achieved. This set-up should adequately affect the values of the melting performance and the specific heat losses. The results of mathematical modelling from Table 1 are graphically presented in Fig. 14. The almost parallel increase of the mass melting performance and the space utilization with the fraction of energy supplied to the melt inflow region is clear. The values of the specific heat losses are small owing to insulation of the space level (the necessary boundary condition), but adequately decrease with the increase of the mass melting performance. The sand dissolution is the controlling phenomenon in all the · · examined cases with the exception of Hinput =Htot = 0.965. The values of uD are nevertheless applied in all cases, because both phenomena are · · terminated almost simultaneously at Hinput =Htot = 0.965. The increase of the mass melting performance and decrease of the specific heat losses correspond to Eqs. (4a)–(5b). The beneficial values of the space utilization · · are achieved at values of Hinput =Htot N0:8; hence, the decisive energy

should be located in the input region. The observed slight decrease · of the space utilization and corresponding changes of M and HLM at · · Hinput =Htot →1 can be clarified by the change of the overall melt flow character. The character of the important longitudinal component of the melt flow presented in Figs. 5 and 7 is preserved up to about · · Hinput =Htot = 0.93 and qualitatively corresponds to the picture of the melt flow in industrial melting spaces. However, the melt flow character · · dramatically changes at a slightly higher value of Hinput =Htot = 0.95. A flow pattern with two forward flows and one backward flow sets up (see Fig. 9): The forward flows are at the level and at the bottom, and the backward flow appears approximately in the middle of the space. This type of flow is characterized by the maximum utilization value, by the maximum melting performance, and minimum specific heat losses · · (Table 1). The critical trajectories elevate at Hinput =Htot = 0.965 from the bottom to the level due to a shift of the spring point towards the input as shown in Fig. 10. The effect of the energy transfer on the space utilization is still significant despite only several percent of the total energy being available for the transversal circulations. When · · the ratio of Hinput =Htot approaches 1, no energy is available for transversal circulations and the melt flow in the region under the free level partially simulates the uniform character (see Fig. 11). Generally, the uniform flow shows high values of the space utilization [27], so the relatively · · high utilization value is found at Hinput =Htot = 1 as well. The sequence · · of melt flow patterns developing at an increasing value of Hinput =Htot is in agreement with Fig. 1; the system is only augmented with the case involving two forward flows.

· Fig. 14. The dependence of the space utilization uD, mass melting performance M (t/day), and the specific heat losses HLM (kJ/kg) on the fraction of energy supplied into the melt inflow region (the region of the simulated glass batch).

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Fig. 15. The average values of the maximal longitudinal component of the velocity of the forward flow near the bottom and the average longitudinal melt velocity on the level in the longitudinal section through the space corresponding to ¼ or ¾ of the space width (XZ plane) and measured at X = 2100, 3100, 4100 and 5100 mm, as well as the positions · · of the spring point Xsp as a function of Hinput =Htot . (X is the distance from the inner part of the front wall).

The following Fig. 15 shows the average values of the maximal longitudinal velocities of the forward flow near the bottom, the average values of the longitudinal velocities on the level, and the positions of the · · spring point Xsp as a function of Hinput =Htot . Fig. 15 demonstrates that the absolute values of both melt velocities — expressing the intensity · · of the longitudinal circulations — decrease with increasing Hinput =Htot , although the flow rate through the space (melting performance, see Fig. 14) grows. The decrease of the bottom velocity slightly slows down in the region of steeply increasing melting performance at · · Hinput =Htot = 0.93–0.96, then continues to a zero velocity value. Here, the working forward flow switches to the level; hence, the values of vlevel become positive and the value of Xsp shifts to the space input. The character of the velocity courses thus coincide with the shift of · · the spring point towards the space input at Hinput =Htot = 0.95. The · · critical trajectory runs near the level at Hinput =Htot →1 , so the fast increase of the level velocity explains the slight decrease of the space · · utilization at the final value of Hinput =Htot ¼ 1. The circularity of the melt flow was assessed, defined as the ratio of   v circ =v long where both velocities are the average values of the relevant velocity components, read in four points of the transversal section through the space at X = 3600 mm. The values of circularity in Fig. 16 show the same tendency as the utilization values and confirm the beneficial role of the helical-like flow. The circularity of the melt flow · · grows up to the value Hinput =Htot = 0.93, although the amount of energy supplied to the region of the free level and, consequently, the intensity of the transversal circulations permanently decrease. The increase is caused by the fact that the value of the average longitudinal velocity decreases faster than the relevant value of the circulation velocity. This is demonstrated by both curves in Fig. 16. The high values of the space · · utilization are preserved even at higher values of Hinput =Htot which can be explained by the emerging tendency of the melt to set up a uniform flow, characterized as well by high utilization values. When the character of the melt flow is changed, the time-temperature history of the melting phenomena also varies and contributes to the resulting value of the melting performance and specific heat losses (see Eqs. (4a) and (5a)). The values of the average sand dissolution times (the sand dissolution is the predominantly controlling phenomenon in these calculations) change nevertheless only slightly, as Table 1 shows. The percentage contributions of the time–temperature history

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 Fig. 16. The value of the space utilization and the value of  vcirc =v long as a function of · · Hinput =Htot .

(represented by the value of τDave) and space utilization (represented by the value of uD) are provided in Fig. 17. As is manifest from the figure, the principal beneficial effect should be attributed to the increase of the space utilization, consequently to the establishment of the helical-like flow of the melt. The strengthening of the role of the time–temperature · · history, at Hinput =Htot N 0.9, is caused by the elevation of the working forward flow to the level where higher temperatures prevail. · · According the presented results, the very high values of Hinput =Htot characterize an efficient melt flow. Therefore, it is assumed that the horizontal distribution of energy has to be far from the balanced state in most calculated cases. The following Fig. 18 demonstrates the distance between the percentages of energy supplied to the batch region in the balanced state and in the presented results. The glass batch with 50% · · of cullet is used for the calculations of the ratio of Hinput =Htot in the balanced state when considering Eq. (2): · · · Hinput H TM MþξHL ¼ · · · · Htot H TM MþξHL þ ð1−ξÞHL :

ð14Þ

The comparison of both curves in Fig. 18 informs that the actual energetic state of the modelled cases is far from the balanced one

Fig. 17. The percentage contribution of the time-temperature history (represented by the value of τDave) and space utilization (represented by the value of uD) to the percentage · · · growth of the melting performance M. The reference values are taken at Hinput =Htot = 0.2.

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heating elements 1 and 2 (93% of energy supplied to the input region), namely case 7 in Table 1 and cases 2 and 10–11 in Table 2, do not prefer unambiguously the type of heating closely below the input level. Yet this type of heating could be preferred owing to the anticipated increase of temperature in the batch blanket (when applied instead of the inflowing melt) and the acceleration of the batch conversion. The assumed effect should, however, be verified. 6. Conclusions

Fig. 18. The percentage fraction of the energy supplied to the melt inflow region as a function of the specific mass melting performance. ▼: the fraction of energy in the case of the balanced · · state 100Hinputbal =Htot , the batch of the glass float was applied with 50% of cullet. ■: the · · fraction of energy supplied in the present calculations: 100Hinput =Htot .

in most calculations. Strong longitudinal circulations set up in the calculated cases, and hence prevent the efficient installation of the · · helical flow. The values of Hinput =Htot around 0.9 and higher are needed to approach the balanced state and satisfactory values of the space utilization. This can be both a technological and an economical problem for application. However, the high melting performance achievable by the simulation of the helical flow provides an opportunity to work at a reasonably unbalanced energetic state which provides only a partial effect of the controlled melt flow. The other factors enhancing the intensity of transversal circulations should therefore be examined — reduced sidewall insulation or higher average melting temperatures. The effect of the vertical energy distribution can satisfactorily be assessed only in the region of the free level. The shift of the energy to the bottom increases the space utilization by less than about 10% (see Table 1, case 7 and Table 2, case 2, the compared cases have different controlling phenomena), but the mass melting performance increases by about 20%. The increase of the space utilization and the further increase of the mass melting performance can be explained by the temperature increase by about 6 °C in the working forward flow near the bottom. The lower melt viscosity enhances the transversal circulations near the bottom (their intensity is inversely proportional to the viscosity of the melt [17]) and the higher temperature slightly accelerates the melting phenomena (see the acceptable values in Tables 1 and 2). Both small effects explain the approximately 20% increase of the mass melting performance. The effect of the vertical energy distribution in the melt inflow region provides only indications. The planar-like heating with the entire energy supplied below the melt inflow (cases 6–8 in Table 2) provides much lower average sand dissolution times and slightly higher space utilization in case 8, characterized by the heating near the level. The picture of the melt flow shows forward flows — near the bottom and along the level (Fig. 13). The critical trajectory is located in the upper high temperature forward flow which explains the lower value of τDave = 1456 s in Table 2 and the increase of the space utilization from 0.085 to 0.12 between cases 6 and 8. The comparison with cases 8–9 in Table 1, characterized by the same character of the longitudinal flow, makes it clear that this type of the melt flow beneficial for the melting process. As Fig. 16 shows, the high values of the space utilization in cases 8–9 (Table 1) are caused by the gradual installation of the uniform type of flow with low values of circularity. The same change of the melt character probably holds for the planar-like heating near the level presented in Fig. 13 and characterized by the increase of the space utilization from 0.085 to 0.12 between cases 6 and 8 in Table 2. The application of different vertical combinations of the central longitudinal heating by

The presented results were achieved by using the model space, still different from the industrial glass melting furnace. However, the results show that the opportune character of the melt flow should be installed in the melting space and, consequently, the melting performance can be increased and the specific heat losses lowered even several times owing to the increase of the space utilization. The application of the helical-like flow, derived from theoretical considerations and from the previous modelling results, appears to be the suitable type of the controlled flow. The helical-like flow arises as a superposition of the imposed transversal circulations to the natural longitudinal flow patterns. The character and intensity of the longitudinal flow substantially affects the quality of the resulting flow. The uniform flow (without longitudinal circulations) assembled with the imposed transversal circulations promises the best results. The uniform flow — applied by itself — also provides high utilization values, but it requires an almost balanced energy distribution in the space. Such flow character seems to be hardly realizable; yet the conditions of its installation should be further studied. The combination of the longitudinal flow patterns with the transversal circulations is but real. The successful realization of the helical-like flow also demands the location of a substantial part of energy in the input region where the highest energetic needs exist; otherwise, a helical-like flow is insufficient. The gradual transition of the energy towards the input region primarily slows down the intensive longitudinal melt circulations with the backflow along the level, facilitates the onset of a helical flow, and increases the space utilization. The amount of energy supplied to the input region should practically be at least 80% of the total energy amount in order to acquire good results. With the further transfer of energy, the longitudinal melt flow passes the stage of the balanced energy distribution which is characterized by the uniform-like flow or flow with forward flows along the level and bottom. These cases show the maximal values of the space utilization and occur at more than 90% of the total energy located in the input region. If all the energy is located in the input region, the longitudinal circulations develop with the forward flow near the level and the space utilization slightly decreases. The presented principles of the melt flow character are valid also for real melting facilities, and hence they are at least partially applicable. The character of heating applied in this work corresponds to the inner energy source as is Joulean heat with the use of electrodes; consequently, the presented efficient heating arrangements can be satisfactorily simulated. However, the practical chances of heating from above, represented by the combustion process and use of burners, have not yet been clarified. Heating by burners can restrict the helical flow, but it stabilizes the uniform flow if balanced energy distribution is ensured. The opportunities of the application of both types of efficient melt flow under the use of burners should therefore be studied. However, the supply and distribution of so much energy in the space input region can be a technological and an economical problem. Another problem arises from the fact that the batch conversion and melt homogenization phenomena are practically serial processes, so the increase of the melting performance owing to higher space utilization should be balanced by the increase of the batch conversion capacity. The practical effective melt flow control aspires therefore for new furnace constructions with a substantially reduced region of the free level, possibly extended batch region and with a modified heating system. At present, the partial achievement

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of the helical-like flow in the existing melting facilities (container furnace) seems to be the realizable variant. In order to improve this variant, the increase of intensity of transversal circulations by further factors mentioned in this work and the increase of the batch conversion capacity should be also studied. Acknowledgements This work has been supported by the long-term conceptual development research organization RVO 67985891 and by Specific University Research (MSMT No. 20/2015). References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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