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Energy distribution and melting efficiency in glass melting channel: Diagram of melt flow types and effect of melt input temperature
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
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Energy distribution and melting efficiency in glass melting channel: Diagram of melt flow types and effect of melt input temperature ⁎
Lukáš Hrbek, Marcela Jebavá , 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
A B S T R A C T
Keywords: Glass melting Melt flow Mathematical modelling Energetic model Space utilization Melting performance
The synergy between glass melting and glass flow character was investigated by mathematical modelling of a Joule heated glass melting channel with a central longitudinal row of vertical heating electrodes. The glass melt containing sand particles and bubbles entered the channel where different types of the melt flow were set up. The sand particle dissolution and the bubble removal were monitored up to the achievement of phenomena completion. The increasing fraction of energy supplied to the input region of the channel proved to be crucial for setting up the beneficial character of the melt flow and consequent increase of the melting performance. When the temperature of the inputting melt decreased, the melting performance also decreased. The relations of the “energetic model” were derived and applied to predict the beneficial type and conditions of the melt flow. The best results have been obtained for the flow character near the uniform flow. The beneficial state was achieved when each region of the channel was supplied by the energy needed there for glass heating and heat losses. The impact of the temperature of the inputting melt on the detailed character of the melt flow was discussed with the help of the energetic model.
1. Introduction In the horizontal glass melting spaces, > 80% of the energy should be delivered to the input part of the melting space to fulfil the energy demands for batch conversion to glass melt. The high energy demand is satisfied by the energy supplied to the batch and molten glass from both the combustion and the melting space. The heat is partially supplied to the glass by electric boosting (below the batch blanket) and by the massive transport of the hot melt from the region of maximal temperatures (hot spot) towards the batch area by natural convection of the melt. The pumping of the heat by natural convection is currently used in most industrial facilities. Nevertheless, a high tax has to be paid for this opportunity, because longitudinal circulations with low melting effect occupy most of the melting space, reduce the melt residence time in the space, and drastically restrict the melting performance when the homogenization phenomena should be fully accomplished. Several technological studies dealt with the problem of melt residence time in the space, as was presented e.g. in [1] or by Beerkens in [2]. Beerkens declared that the minimum residence time was typically only 15–20% of the average residence time. Therefore, spaces, whether horiziontal or vertical, without circulation patterns appeared to be the most advantageous. Hence, the further effort of glass researchers and
⁎
technologists had to be focused on restricting the longitudinal melt circulations either by higher delivery of energy to the batch region or by the segmentation of the glass melting process. The former approach, however, faces technical and technological problems with the positioning and operation of the heat sources (burners, electrodes) and scarce knowledge of the resulting melt flow characters and their mutual transitions. The latter approach leading to segmented melting systems prevents the macroscopic back-flows and increases the overall melting space. Here, separate melting phenomena, such as the conversion of batch to glass, sand dissolution, and bubble removal, take place. Therefore, the optimal conditions for each individual phenomenon can be set up in the relevant melting segment. The facilities with separated the space for batch conversion [3] and segmented melters containing three or four parts were proposed [4–6]. However, it is necessary to mention the problems of segmented melters that result from their relatively complex design and exacting operation. The relatively high heat losses can be expected as well owing to the separated melter structure. Nevertheless, to segment the melting system into only two spaces seems to simplify the problem: the space for batch conversion from the solid to liquid state with powerful conversion technology [7–9], combined with the homogenization space to terminate both dissolution and fining phenomena under tuned conditions. Thus, the
Corresponding author at: University of Chemistry and Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic E-mail addresses: [email protected] (L. Hrbek), [email protected] (M. Jebavá), [email protected] (L. Němec).
https://doi.org/10.1016/j.jnoncrysol.2017.12.009 Received 15 September 2017; Received in revised form 1 December 2017; Accepted 3 December 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Hrbek, L., Journal of Non-Crystalline Solids (2017), https://doi.org/10.1016/j.jnoncrysol.2017.12.009
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The relation for quantity τFref is given in ([15], Eq. (5)). The former expressions in Eqs. (2a, b) and (3a, b) are valid for the case of sand dissolution being the controlling phenomenon, and the latter ones for bubble removal as the controlling phenomenon.
need arose for the relevant theoretical solution of the homogenization spaces prior to their construction. The authors of this article solved the simultaneous sand particle dissolution and bubble removal at a constant average temperature of the glass melt of 1420 °C in the electrically heated horizontal melting channel by mathematical modelling [10]. The melt flow approaching a uniform character and possibly a helical flow were preferentially set up in the space using a longitudinal row of heating electrodes. The results proved that both melting phenomena can efficiently be accomplished in one space under the condition of a controlled melt flow. The melting performance of the channel increased with the increasing input temperature of the melt and with the increasing portion of energy located in the input zone of the space. The effect was indicated by the increasing space utilization. As the space utilization and melting performance increased, the transition of the flow character from the flow with longitudinal melt circulations to an almost uniform flow was observed. In this article, an “energetic model” will be introduced as an efficient tool for the prediction of the character of the longitudinal melt flow when the energy distribution in the space, structural heat losses, theoretical heat, or the kinetics of homogenization processes will change. The results of the mathematical modelling of cases with different melt input temperatures and detailed inspection of the melt flow character in the melting channel will be discussed in the light of the energetic model. The melt flow diagram will be presented and the conditions of high melting performance at respective melt input temperatures will be determined as a function of the longitudinal energy distribution in the space.
2.1. Relations of the energetic model The resulting character of the melt flow particularly depends on the horizontal distribution of the supplied energy to the space, as was already mentioned in [10]. Allow us to imagine a horizontal space with an input of either batch or cool glass melt and characterized by heating the input material and by heat losses through boundaries. Most of the heat needs to be fed at the space input where the batch undergoes chemical transformation and/or the melt should be heated to the average temperature; the structural heat losses also have to be compensated proportionally. If the local needs of energy are not balanced, the natural longitudinal circulations occur in the space. The characteristic velocity of the longitudinal circulations vlc can be introduced [15]; the velocity represents the overall intensity of the longitudinal circulations. The space can then be divided into the input region where the energy is consumed for batch conversion (in an industrial furnace approximately the batch region) and/or melt heating and covering the relevant heat losses, and the following region where only the heat losses should be covered. The latter is called the “free level” region. The driving force of longitudinal circulations can be described using the difference between the energy actually delivered to either the input or free level region and the energy actually needed in the relevant region to maintain the energetically balanced state. The balanced state means the situation when each region of the melting space is supplied by the right amount of energy needed for batch conversion and/or heating of the melt and covering the heat losses. The characteristic velocity of the longitudinal circulation vlc was conventionally placed in the longitudinal axis at the level of the free level region, approximately in the middle of the length of longitudinal circulations; the characteristic velocity expresses the circulation intensity by its value and the flow direction by its sign. Its value is considered linearly proportional to the driving force:
2. Theoretical part The quantity “utilization of the melting space u” has been used for the evaluation of the character of the melt flow with respect to sand dissolution and bubble removal [10–15]. The utilization of continual space for sand dissolution or bubble removal (fining) expresses the relation between the time for accomplishment of the controlling melting phenomenon τHref and the mean residence time of the melt in the space τG under the critical conditions and at the average temperature in the channel. The critical state then describes the situation when the sand particle of maximum size or the bubble of the initially minimal radius is removed (dissolved or refined) right at the output from the space:
uF , D =
τHref τG
, τG =
V , u∈ V˙
0; 1
T ̇ vlc = −C1 [(1 − k1 )(HM M + Ḣ L) − (1 − ξ ) Ḣ L]
where k1 is the fraction of energy actually delivered to the input region of the channel, HMT is the specific energy for batch reactions, phase and modification transitions, and for heating of the contents to the exit temperature of the channel (theoretical heat) in J/kg, Ṁ is the mass L flow rate (also the critical melting performance Ṁ crit ) in kg/s, Ḣ is the total heat flux through channel boundaries (the flux of heat losses) in W, ξ is the fraction of heat losses belonging to the input region, and C1 is the constant of the proportionality in the space under the given average temperature. Thus, the first term in the brackets signifies the energy actually delivered to the free level region of the channel and the second one the energy actually needed to cover the relevant heat losses. As already mentioned, longitudinal melt circulations develop in the energetically unbalanced state between both regions. If the batch (cool melt) input is situated left, anticlockwise longitudinal circulations arise when the actual heat flux delivered to the free level region is higher L than the balanced one, given by the term (1 − ξ ) Ḣ in Eq. (4). In the opposite case when the actual heat flux is lower than the balanced term, the clockwise circulations set up. The flow situation is schematically presented in Fig. 1 of reference [16]. If vlc = 0, then Ṁ = Ṁ bal in Eq. (4), where Ṁ bal is the flow rate through the space under balanced energy distribution. No longitudinal circulation ideally exists in this case and the flow in the channel should be uniform, described only by the throughput velocity of the melt vfr. The value of Ṁ bal is then derived from Eq. (4) at vlc = 0:
(1)
where the index by u designates the controlling melting phenomenon (F – fining, D – sand dissolution). τHref is either the reference time of fining τFref, the time the minimum bubble needs to ascend the distance equivalent to the height of the glass level in a quiescent liquid and at the average temperature in the channel, or the average time of sand dissolution τDave, V is the volume of the space (m3), and V̇ is the volume flow rate (m3/s). For the plug flow, τHref = τG is valid, so uF,D = 1 [13]. 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 HML decrease, and the critical performance of the melting process Ṁ crit increases with space utilization according to:
HML =
Ḣ LτFref 1 Ḣ LτDave 1 or HML = , ρV uD ρV uF
Vϱ Vρ uF , Ṁ crit = uD or Ṁ crit = τFref τDave
(4)
(2a, b)
(3a, b)
Ḣ L (k − ξ ) Ṁ bal = T 1 HM (1 − k1 )
where HM are the specific heat losses (J/kg), HL̇ is the total heat flux across the space boundaries (J/s), and ρ is the glass density (kg/m3). L
2
(5)
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As is apparent for the variable k1: for k1 = ξ, Ṁ bal = 0 and for k1 → 1, Ṁ bal → ∞. If k1 < ξ, no real value of Ṁ bal occurs. The function Ṁ bal (k1) is depicted in Fig. 3 as the red curve. The curve Ṁ bal (k1) forms the boundary line between the anticlockwise longitudinal melt circulations on the left side of the curve where the value of vlc is negative and the clockwise circulations on the right side of the curve characterized by positive values of vlc (Eq. (4)). The vicinity of the Ṁ bal (k1) curve represents the region of approximately uniform flow. The area of uniform-like flow is characterized by high utilization values and, consequently, by high melting performances and low specific heat losses. Such favourable melt flow area is conventionally defined with the assistance of vlc and vfr velocities. The characteristic throughput velocity of the melt vfr is here defined in the same point as the velocity vlc, i.e. at the longitudinal axis of the glass level. The left boundary curve of the uniform-like region is described by the convention that the absolute value of the characteristic velocity of longitudinal circulations | vlc | is equivalent to the characteristic velocity of the through flow vfr in the definition point:
vfr = |vlc|,
Fig. 1. The axonometric view of the melting channel with the level input of the glass melt (the vertical arrow). The inner length is 6.77 m (6.225 m to the transversal refractory barrier), the width is 2 m, and the height of the glass level is 1 m.
first six electrodes formed the input region and the remaining ten electrodes the free level region of the channel. The Glass Furnace Model [17] was used for mathematical modelling which provided the temperature, velocity, and electric power fields in the space. The solution was coupled with submodels of sand dissolution and bubble removal, prepared in cooperation with the author's workplace. The melt, sand particles, and bubbles trajectories were traced along their pathways. The float glass was chosen as the model glass. The temperature dependence of the glass density and kinematic viscosity in the temperature range of 900–1800 °C are presented in [16]. The average sand dissolution rate and average bubble growth rates obtained by laboratory experiments were used to follow the history of the particles. The temperature dependences of both quantities are also presented in [16]. The semiempirical model of the bubble behavior was applied [18]. The history of around 104 bubbles (having the initial size of 5 × 10− 5 m) and sand particles (the initial size of 5 × 10− 4 m) with regularly located starting points at the given input were traced. Bubble nucleation inside the space was not considered. The trajectories and sand radii were calculated with the time step mostly at 1 s and the time step for bubble radii calculation at 0.1 s. A calculation grid was used with a distance of 2.5 cm between grid points. The controlling phenomenon was given by the critical particle (sand or bubble) first reaching the transversal refractory barrier. This state was characterized by: the critical mass melting performance Ṁ crit , the mean residence time of the melt in the space τG, the fining time of the critical bubble τFcrit or the average sand dissolution time τDave. The final value of the space utilization of the controlling phenomenon was calculated from Eq. (1), the specific energy consumption from Eq. (2a, b). To validate Eqs. (7)–(8), the character of the melt flow was inspected in another set of simulations under conditions of different melt flow rates. The melt flow rate and the relevant value of k1 were determined under which the character of the melt flow changed from the existing longitudinal circulation flow to the unifom flow and vice versa. The picture of the longitudinal melt flow in this case was characterized by the spring point located approximately at the halfway point of the melting channel. The melt input temperature varied between 1120 and 1320 °C and the constant average temperature of 1420 °C was maintained in the space. Finally, the amount of energy supplied to the input region of the space (to the first six electrodes) varied between 45 and 100% (the value of k1 varied between 0.45 and 1) in each set of calculations.
(6)
where vfr = C2 Ṁ is always positive and C2 is a constant calculated from the melt flow rate in cross section of the space. The conditions given by eE. (6) and by definitions of both velocities determine the border values of the uniform-like area on the left of the Ṁ bal (k1) curve:
Ṁ eq left =
C1 Ḣ L (k1 − ξ ) T − C2 C1 (1 − k1 ) HM
(7)
Here, Ṁ eq left is the melt flow rate under conditions given by Eq. (6) and demarcates the boundary line of the still favourable character of the melt flow. If k1 shifts to the right and behind the curve Ṁ bal (k1) , both vlc and vfr are positive and the resulting flow velocity on the melt level is the sum of both velocity components. The condition vfr = vlc is entered and the border line of the favourable flow region right of the curve of state Ṁ bal (k1) is defined as:
Ṁ eq right =
C1 Ḣ L (k1 − ξ ) T + C2 C1 (1 − k1 ) HM
(8)
Thus, the Ṁ eq right curve is found right of the Ṁ bal (k1) curve. Both curves are depicted in Fig. 3 (the blue and green curves). Eqs. (4)–(8) describe the melt flow picture from the point of view of the longitudinal energy distribution. The verification of the distribution of flow types according to Eq. (5) by the mathematical model and the impact of the input temperature of the melt (represented here by the quantity HMT which grows if the input temperature decreases) on the critical melting performance, will be treated in this work. 3. Calculation procedure and conditions The simple melting channel, the same as in [10], was chosen for mathematical simulations corresponding to the small horizontal melting furnace, as is clear from the view in Fig. 1. The melting channel was an orthogonal space with the inner length being 6.77 m, the width 2 m and the height of the glass level 1 m. Three layers of refractory materials surrounded the inner space. The glass level was insulated owing to the necessary boundary condition and hence the upper structure was not considered. The transversal refractory barrier on the right side of the space served as the reference point where the sand dissolution or bubble removal should be terminated – and signalled the achievement of the critical state. The glass melt flowed into the channel by a strip of level at the front wall (the width of 2 m, the length of 0.2 m, see the arrow and the strip in Fig. 1). Sixteen vertical bottom electrodes were located regularly in the longitudinal axis of the space. The standard height of electrodes in the case with the level input of the melt was 0.3 m. The
4. Results of calculation Two sets of calculations were executed in this work to elucidate the impact of the melt flow character on the melting efficiency. In the first round of calculations, the boundary lines demarcating the beneficial 3
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Fig. 2. a–e The character of the longitudinal 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 corresponds to ¼ or ¾ of the space width (XZ plane). The arrows → and ← highlight the main directions of the melt flow, the arrows ↓ point at the input and the spring point positions. The melt flow character develops at the growing melt flow rate Ṁ and at constant values of k1. 2a–c: k1 = 0.8, Ṁ = 1, 2, and 6 kg/s. 2d–e: k1 = 0.6, Ṁ = 1, and 5.8 kg/s. tinput = 1320 °C.
4
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(5)) was crossed, the uniform-like flow set in the free level region of the space, as Fig. 2c shows. The further increase of Ṁ was followed at k1 = 0.6 (for the position look at Fig. 3). The growth of Ṁ above the Ṁ bal (k1) curve led to crossing the second boundary line Ṁ eq left (Eq. (7)); the spring point shifted to the output and the anticlockwise longitudinal circulations started in the space (Fig. 2d). At even higher values of Ṁ , the spring point shifted more to the output and the anticlockwise circulations strengthened, which is demonstrated by Fig. 2e. So, Fig. 2a–e demonstrate the impact of the intensity of flow rate (increasing Ṁ ) on the flow character in the free level region of the space (clockwise, uniform, anticlockwise) and also on the spring point position under the condition of identical longitudinal energy distribution in the channel (two values of k1 had to be actually used). Fig. 3 clarifies these changes in the character of the melt flow graphically: when the flow rate Ṁ increases at constant energy distribution k1 (starting at low Ṁ ) then the flow character changes from clockwise type (under Ṁ bal curve) through the uniform type (around Ṁ bal curve) to the anticlockwise type (above Ṁ bal curve). Fig. 3 provides the curves of Ṁ eq left (energ ), Ṁ eq right (energ ), and ̇ Mbal (k1) according to Eqs. (7)–(8) and (5), as well as the values of Ṁ eq left (mat ) and Ṁ eqright (mat ) ascertained by mathematical modelling. The following parameters were used: Ḣ L = 3.27 × 105 W (J/s), ξ = 0.36, HMT = 1.269 × 105 J/kg, C2 = 4.9 × 10− 4 m/kg, and C1 = 1.72 × 10− 8 m/J. The average value of C1 was calculated using three different modelled cases when the longitudinal melt velocity on the melt level was read (at X = 4000 mm and at ¼ of the space width). To obtain the relevant value of vlc, the average value of the longitudinal velocity was recalculated to the level axis and the relevant throughput velocity was subtracted. As Fig. 3 shows, the area of the unifom-like flow demarcated by the values of Ṁ eq left (mat ) and Ṁ eq right (mat ) appears slightly narrower and shifted to the lower values of Ṁ at lower values of k1 comparing with the area demarcated by the curves Ṁ eq left (energ ) and Ṁ eq right (energ ) . The difference is caused by a slightly deformed structure of the flow sets in the real case with heating by electrodes. Still, both solutions agree
Fig. 3. The curves Ṁ eq right (energ )(k1), Ṁ eq left (energ )(k1), and Ṁ bal (k1) calculated according to Eqs. (7)–(8) and (5), as well as the values of Ṁ eq right (mat ) and Ṁ eq left (mat ) ascertained by mathematical modelling (blue and green points). tinput = 1320 °C. The oval pictograms designate the orientation of the longitudinal circulations of the melt and the pictogram at the Ṁ bal (k1) curve indicates the uniform flow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
area of uniform-like flow, described by Eqs. (7) and (8), were verified. Fig. 2a–e illustrates the development of the longitudinal character of the melt flow at k1 > ξ. If the values of the melt flow rate Ṁ at constant k1 = 0.8 were sufficiently low (situated below the Ṁ bal curve, see Fig. 3), the clockwise longitudinal circulations of the melt dominated and the spring point appeared near the input, as Fig. 2a shows. When Ṁ increased at constant k1, the melt flow character was half way to the uniform-like flow (Eq. (8)). The weak forward flow near the bottom arose right behind the spring point (Fig. 2b) and then gradually strengthened with increasing flow rate. When the Ṁ bal (k1) curve (Eq.
Table 1 The values of the modelled cases with the longitudinal row of electrodes and different input temperatures of the melt. k1 is the fraction of total energy delivered to the input region of the channel, uF and uD are the relevant utilizations of the channel, τDave and τFref are the relevant reference times of sand dissolution and bubble removal, τG is the mean residence time of the melt in the channel, Ṁ crit is the critical melting performance, and HMLare the specific heat losses. The cases presented already in [10] are labelled by asterisks. Case No.
tinput (°C)
k1
u (controlling)
τDave, τFref (s)
τG (s)
HML (kJ/kg)
Ṁ crit (kg/s) (t/day)
1* 2* 3* 4* 5* 6* 7* 8* 9* 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
1320 1320 1320 1320 1320 1320 1320 1320 1320 1220 1220 1220 1220 1220 1220 1220 1220 1120 1120 1120 1120 1120 1120 1120 1120 1120 1120
0.45 0.60 0.70 0.75 0.80 0.825 0.85 0.90 1.00 0.45 0.60 0.70 0.80 0.83 0.85 0.90 1.00 0.45 0.60 0.70 0.80 0.825 0.85 0.87 0.89 0.91 1.0
uF = 0.34 uF = 0.38 uF = 0.42 uF = 0.45 uF = 0.47 uF = 0.46 uF = 0.40 uD = 0.27 uD = 0.41 uF = 0.21 uF = 0.25 uF = 0.28 uF = 0.34 uF = 0.37 uF = 0.35 uD = 0.27 uD = 0.30 uF = 0.17 uF = 0.19 uF = 0.23 uF = 0.28 uF = 0.29 uF = 0.28 uF = 0.26 uF = 0.25 uF = 0.23 uD = 0.23
τFref = 1878 τFref = 1877 τFref = 1879 τFref = 1878 τFref = 1878 τFref = 1877 τDave = 2059 τDave = 2078 τDave = 1903 τFref = 1920 τFref = 1877 τFref = 1877 τFref = 1879 τFref = 2048 τFref = 1877 τDave = 2141 τDave = 1952 τFref = 1877 τFref = 1875 τFref = 1923 τFref = 2048 τFref = 1889 τFref = 1890 τFref = 1879 τFref = 1972 τFref = 1887 τDave = 2153
5
5560 4985 4448 4130 3988 4072 5163 7608 4663 9029 7603 6723 5560 5558 5452 7814 6567 11232 9734 8499 7225 6881 6719 7223 8027 8255 9320
5.2 5.8 6.5 7.0 7.25 7.1 5.6 3.8 6.2 3.2 3.8 4.3 5.2 5.2 5.3 3.7 4.4 2.6 3.0 3.4 4.0 4.2 4.3 4.0 3.6 3.5 3.1
449.3 501.1 561.6 604.8 626.4 613.4 483.4 328.3 535.7 276.5 328.3 371.5 449.3 449.3 457.9 319.7 380.2 224.6 259.2 293.8 345.6 362.9 371.5 345.6 311.0 302.4 267.8
62.9 56.4 50.3 46.7 45.9 46.1 58.3 85.9 52.9 101.3 85.2 75.3 62.2 61.9 61.1 87.5 73.8 123.8 107.1 94.2 79.5 80.0 74.6 80.2 88.9 91.7 103.3
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overall. The curves Ṁ eq left (energ )(k1), Ṁ eq right (energ )(k1) and Ṁ bal (k1) thus provide fundamental prerequisite information about the beneficial areas of glass melting in terms of the melt flow character. The other set of mathematical simulations was focussed on the role of the melt input temperature tinput. The melt input temperature, being the melt output temperature of the previous batch conversion space, reflects the conditions of the conversion space and, therefore, is an important technological factor. tinput is involved in the value of HMT in Eq. (4) where HMT represents the theoretical heat – here the heat needed for glass heating from the input to the exit temperature of the melt in the channel. The main values significant for evaluation of melting efficiency are presented in Table 1. Table 1 shows that, at the melt input temperatures 1220 and 1120 °C, the maximum melting performance and, consequently, the minimum specific heat losses were achieved very close to the maximum value of the space utilization. This was in agreement with the values achieved at 1320 °C in [10], labelled here in Table 1 by asterisks. The simultaneous development of technological quantities with the space utilization provides evidence about the dominating role of the melt flow character under constant average temperature in the space.
Fig. 4. The critical melting performance Ṁ crit as a function of the fraction of energy located in the input part of the space k1 at three melt input temperatures 1120, 1220 and 1320 °C, and the relevant Ṁ bal (k1) curves. The curve Ṁ crit – 1320 °C, 3:1: the energy delivered to the input part (k1 = 0.8) was further partitioned in the ratio of 3:1 in favour of the first pair of electrodes.
5. Discussion of results anticlockwise longitudinal circulations. Note that the values of Ṁ crit for the longitudinal row of electrodes are high even at lower values of k1, as was already reported in [10]. The maximum critical melting performances are achieved slightly left of the Ṁ bal (k1) curve. The respective longitudinal flow established at these optimum conditions represents an almost uniform flow and weak anticlockwise longitudinal circulations. The effect of the slow anticlockwise longitudinal circulations lies in reducing the maximal forward throughoutput velocity near the melt level. This effect enhances the space utilization. Similar optimum conditions were repeatedly found in [11–15] where the optimum flow conditions were examined in the space with preset temperature distributions; the results indicate the most probable universal picture of the optimum flow character. The values of Ṁ crit substantionaly decrease with the declining melt input temperature at the constant k1 and in the area left from the relevant Ṁ bal (k1) curves. The explanation comes from the definition of vlc by Eq. (4); higher values of HMT (i.e. the lower melt input temperatures) indicate higher absolute values of vlc. Higher absolute values of vlc lead to lower values of the space utilization at the same flow rate and to the occurrence of either undissolved sand particles or bubbles in the outputting melt. Graphically, the Ṁ bal (k1) dependence becomes more curved at higher values of HMT (see Fig. 4) and shows lower values of Ṁ bal at the given k1. The new values of Ṁ crit should therefore decrease to approach the lower Ṁ bal (k1) curve and not to increase the new value of vlc too much. Thus, the cases with the melt input temperatures of 1120 and 1220 °C show the lower positions of their Ṁ bal (k1) curves and the relevant values of Ṁ crit should be also lower, as compared with the values at the melt input temperature of 1320 °C. Fig. 5 then shows the practically parallel developments of Ṁ crit and u as a function of k1 at 1220 °C. It confirms the dominating role of the melt flow character (represented by the value of the space utilization) for the melting efficiency. The role of the intensity of the longitudinal melt circulations can be more clearly ilustrated by the 3D representation of the characteristic values of the velocity component of the longitudinal circulations vlc. Fig. 6 presents the values of vlc as a function of the flow rate Ṁ and the fraction of energy delivered to the input region k1. Eq. (4) and the values of HMT, Ḣ L , and ξ were applied, as was presented in the section Results of calculation. The absolute values of vlc are plotted in Fig. 6 to be clearer; in fact, the values are negative left of the Ṁ bal (k1) curve and positive to the right of it. The Ṁ bal (k1) curve in the base of the 3D graph (in the XY plane) designates the areas of the anticlockwise and clockwise longitudinal circulations. The zero values of vlc exist at the Ṁ bal (k1) curve and the
5.1. The diagram of melt flow types The diagram of the longitudinal melt flow types drawn in coordinates of the flow rate Ṁ against the the fraction of energy delivered to the input part of the space k1 is crucial to be able to understand the impact of the melt flow character on the melting efficiency in continuous glass melting spaces. The diagram valid for the melting channel is presented in Fig. 3 but a similar picture can be acquired for the classic melting facility with batch blanket. The Ṁ bal (k1) curve splits the plane [Ṁ , k1] into the left part lying at lower values of k1 and characterized by anticlockwise longitudinal circulations of the melt, and the right part at higher k1 values where the clockwise longitudinal circulations are produced. The Ṁ bal (k1) curve itself indicates the zero value of the intensity of the longitudinal circulations and only uniform flow dominates here. Since the presence of longitudinal circulations reduces the utilization of the space [10], the utilization should be high along the Ṁ bal (k1) curve and should decrease with the increasing distance from the curve. The precise information about the intensity of the longitudinal circulations vlc is then provided by Eq. (4). The conventionally defined area between Ṁ eq left (k1) and Ṁ eq right (k1) curves in Fig. 3, surrounding the Ṁ bal (k1) function, depicts the area of Ṁ and k1 suitable for the application of the beneficial melt flow. The following results of the mathematical modelling prove the beneficial applicability of this area. The sufficient agreement between courses of both Ṁ eq (energ ) curves and points of Ṁ eq (mat ) in Fig. 3 justifies the assumption of a linear relation between the values of vlc and the driving force of longitudinal circulations (Eq. (4)) in the examined area of flow rates. If the values of HMT, Ḣ L , and ξ are approximately known (i.e. the curve Ṁ bal (k1) can be drawn), the profitable area from the point of view of the melt flow can be assessed in advance. 5.2. The influence of the melt input temperature tinput (represented by the theoretical heat HMT) To express the significance of the melt input temperature (the energy necessary for heating the melt being inputted), three sets of calculations at the melt input temperatures 1320 [10], 1220, and 1120 °C have been executed and plotted in Fig. 4 in the form of Ṁ crit values versus k1. Different melt input temperatures imply different values of HMT and, consequently, different curves of Ṁ bal (k1) ; therefore, three relevant Ṁ bal (k1) curves are plotted along with the values of Ṁ crit . The calculated Ṁ crit values grow in the area left from the Ṁ bal (k1) curve with increasing k1 in all three cases owing to the decreasing intensity of the 6
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Fig. 5. The values of the critical melting performance Ṁ crit and the space utilization of the controlling phenomenon u as a function of the fraction of energy located in the input region of the space k1. tinput = 1220 °C.
Fig. 7. The absolute values of the characteristic velocity of longitudinal circulations |vlc | valid for the standard case with the longitudinal row of electrodes as a function of the fraction of energy delivered to the input region of the space k1. tinput = 1320 °C. energ – the values coming from the energetic model (Eq. (4)), mat – the values coming from the numerical solution by the mathematical model.
pictures of mathematical modelling can also influence the accuracy of the values. Despite the difference, the values coming from the energetic model appear suitable for semi-quantitative predictions. Both sets of | vlc | values achieve the zero at k1 = 0.82; this point corresponds to the intersection of Ṁ crit (k1) and Ṁ bal (k1) curves (Fig. 4 also shows this). The small values of | vlc |, which should exist at k1 = 0.8, then correspond to the intensity of the anticlockwise circulations when Ṁ crit reaches its maximal value. Figs. 6 and 7 based on the values of | vlc | thus make the beneficial area of high melting performances easily available thanks merely to knowledge of a few integral quantities. Nevertheless, some unexpected behavior can still be observed for Ṁ crit (k1) dependences in Fig. 4. If the values of Ṁ crit are situated just at the Ṁ bal (k1) curve or very close to it, the values of the space utilization should roughly correspond to those found for the uniform or helical flows and should be essentially independent of the melt input temperature. The values of the maximum critical melting performance close to or at the Ṁ bal (k1) curve should then be similar. However, the maximum values of the critical melting performance obviously decrease with decreasing melt input temperature (see Fig. 4). The values of the space utilization in the case of the isothermal uniform flow were found to be 0.445 for the sand dissolution and 0.67 for bubble removal [13], and for the helical flow were found between 0.6 and 0.8 [11–12,14–15]. On the other hand, the interpolated values of the space utilization at the respective Ṁ bal (k1) curves were uF = 0.45 at the melt input temperature 1320 °C, 0.32 at 1220 °C and only 0.25 at 1120 °C. Hence, the real utilization values were substantially lower than expected. More precisely, the maximum critical melting performance decreased as the melt input temperature decreased. The reason for this behavior should be sought in the detailed character of the melt flow. From the flow pictures, it can be seen that the cold melt entering the space through the level naturally falls to the bottom and forms the longitudinal circulation flow near the space input. The lower the melt input temperature is, the higher the support of the original londitudinal circulations – caused by the energetically unbalanced state – can be expected. The arising powerful melt flow then affects the melt flow character in the entire space and over the entire interval of k1, including the area close to the Ṁ bal (k1) curve. When k1 values occur in the area of the Ṁ bal curve, the melt flow shows the character of the uniform-like flow (see Fig. 2c). However, at low melt input temperature, the powerful circulations that arise near the input considerably support the following forward flow near the bottom. In order to keep the constant value of the flow rate, a backflow in the centre of the height arises as a consequence of both massive forward flows; therefore, the space
Fig. 6. The characteristic (absolute) values of the velocity component of longitudinal circulations | vlc | as a function of the flow rate Ṁ and the fraction of energy delivered to the input region k1. tinput = 1320 °C. Red curve: the values of | vlc | at Ṁ crit . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
region of low values of | vlc | along the curve defines the area as beneficial for glass melting. The expressions in the edges represent the border values of vlc at k1 = 0 and 1. The areas with high values of | vlc | and, consequently, low space utilization, particularly exist at low values of k1 and high values of Ṁ . At k1 = 1, the vlc values are independent of the flow rate. The red curve in the graph represents the spatial development of |vlc | values for the critical melting performances Ṁ crit and relevant values of k1 (standard case 1*–9* in Table 1). The values of | vlc | decrease with the growing k1 left of the Ṁ bal (k1) curve but they grow to the right of it. When doing a section through the Ṁ crit points and relevant k1 values, the decrease and increase of | vlc | values is clearer as Fig. 7 shows. Both values of |vlc | coming from Eq. (4) (energetic model) and those read from the pictures of the melt flow (mathematical model) are plotted in Fig. 7 for comparison. The values of | vlc |mat are larger than those calculated from the model of energy distribution according to Eq. (4); nevertheless, they show equivalent tendencies. The difference between the mentioned values is primarily caused by deviation of the real melt flow (showed by the mathematical model) from the concept of the smooth distribution of energy represented by Eq. (4) of the energetic model. Furthermore, the real reading of the melt velocities from the 7
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Fig. 8. The character of the longitudinal 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 corresponds to ¼ or ¾ of the space width (XZ plane). The arrows → and ← highlight the main directions of the melt flow, the arrow ↓points to the input and the spring point positions. tinput = 1120 °C, k1 = 0.89 (case 25 in Table 1).
As Fig. 9a shows at k1 = 0.8, the character of the longitudinal melt flow in the second region of the space is approximately uniform and the space utilization, as well as the critical melting performance, achieve their maximum values (see Table 1, case 5). In Fig. 9b, the clockwise longitudinal circulations arise, and both the space utilization and the critical melting performance obviously decrease (see Table 1, case 8). However, the decrease stops or even the critical melting performance increases again between k1 = 0.9 and 1 at 1220 °C and particularly at 1320 °C (although the intensity of the clockwise longitudinal circulations further increases). This fact was already discussed in [10], and the increase was attributed to the decrease of the average time of sand dissolution. However, it seems that this factor plays only a partial role. The detailed inspection of the melt flow at k1 = 1 has revealed the ultimate shift of the spring point to the space input, even before the end
utilization should decrease owing to newly arising longitudinal circulations. If the melt input temperature approaches the average temperature in the space, the backflow disappears and the values of both the space utilization and the melting performance correspond approximately to the uniform flow. The typical picture of the melt with two forward flows is shown in Fig. 8. As Fig. 4 demonstrates, the values of the critical melting performance steeply decrease behind the maximum values at all three melt input temperatures. The principal decrease signalizes the arising clockwise longitudinal circulations of the melt right of the relevant Ṁ bal (k1) curves. The values of vlc grow with growing k1 and the space utilization, as well as the critical melting performance, should decrease. Fig. 9a–b show how the flow character typically changes at the melt input temperature of 1320 °C.
Fig. 9. a–bThe character of the longitudinal melt flow patterns marked by the streamlines of the melt when the melt is heated along the central longitudinal axis of the space. The arrows → and ← highlight the main directions of the melt flow, the arrow ↓points at the input and the spring point positions. tinput = 1320 °C. a: k1 = 0.8, b: k1 = 0.9.
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the conditions of the melt flow character approaching a uniform flow and explained the impact of the melt input temperature on the melting efficiency. At present, authors cooperate with specialists in the field of electric boosting installation in melting spaces to solve the practical problem of massive heating in the input region. The theoretical investigation of further factors of the process – such as the finer distribution of energy and heating means in the space and the impact of heat losses by both the mathematical and energetic model – is planned for the near future, as well as the cooperative solution of batch conversion problems.
of the intensively heated input region. The critical particle trajectories passing through the spring point then had a better chance to be transported to the space sides by the strong transversal circulations. Consequently, sand particles or bubbles moved there with lower forward velocities and had enough time to be dissolved or removed; this effect enabled higher flow rates. Nevertheless, the practical installation of all the delivered energy to the input appears not to be realistic. The impact of the melt input temperature (or the theoretical heat in cases involving the presence of the batch) is in the energetic model expressed by the quantity HMT (Eqs. (4)–(5)). Summarizing the role of the melt input temperature, the growth of HMT increases the intensity of the overall longitudinal circulations according to Eq. (4), and thus decreases the space utilization along with the melting performance in the interval of k1 left of the Ṁ bal (k1) curve. Growing HMT causes the Ṁ bal (k1) curve to become more curved, and a larger value of k1 is needed to achieve the maximum utilization and performance values (situated close to the Ṁ bal (k1) curve). The high values of HMT also evoke strong local circulations in the input region of the space, strengthen the overall longitudinal circulations, and reduce the values of the utilization and the performance, even in the area of the Ṁ bal (k1) curve. More effort should therefore be focused on the detailed distribution of energy and its improvement in the input region of the melting space. The preliminary results of the calculations characterized by highest values of critical melting performances confirm this intention for the case with finer distribution of the delivered energy in the input part of the channel: Ṁ crit – 1320 °C, 3:1 in Fig. 4. The interval of k1 right of the Ṁ bal (k1) curve shows the rapid decrease of both the space utilization and the critical melting performance, but a repeated increase can occur close to k1 = 1 owing to the shift of the spring point to the input. The impact of the melt input temperature was thus elucidated and the optimal conditions of the longitudinal energy distribution in the channel were determined by the energetic model. The results have practical significance for implementation owing to the very high specific melting performance that is achievable. The specific heat losses are correspondingly reduced. The technical solution of this approach mostly assumes high energy supply to the input region of the space and to the central longitudinal section through the space. The electric heating is more flexible and preferred to heating by burners. The installation of powerful heating in the input region of the space should be solved by the proper amount and arrangement of electrodes to work without their overloading and without overheating both the melt and refractory material. If the results have to be applied to melting spaces with a batch blanket, the question of suitable energy distribution in the input region appears even more urgent. The improved approaches to the rapid batch-to-glass conversion should be examined and applied to adjust the batch conversion capacity to the high sand dissolution and fining capacity of the melt. In a favourable case, the melting space could be miniaturized without a decrease of the melting performance.
Acknowledgement This work has been supported by the project of the Technology Agency of the Czech Republic, No. TH02020316, “Advanced technologies of glass production”. References [1] F. Simonis, H. de Waal, R. Beerkens, Influence of furnace design and operation parameters on the residence time distribution of glass tanks, predicted by computer simulation, Collected Papers XIVth International Congress on Glass. New Delhi, India, 1986, pp. 118–127. [2] R. Beerkens, Analysis of elementary process steps in industrial melting tanks – some ideas on innovation in industrial glass melting, Ceramics-Silikáty 52 (2008) 206–217. [3] S. Wiltzsch, Evaluation principle in respect of the batch-melting zone of a glass melt furnace, Glass Technol.: Eur. J. Glass Sci. Technol. A 58 (2017) 105–115, http://dx. doi.org/10.13036/17533546.58.4.008. [4] C.P. Ross, G.L. Tincher, Glass Melting Technology: A Technical and Economic Assessment, Glass Manufacturing Industry Council, Westerville, OH, 2004, pp. 59–87. [5] E.D. Spinosa, Modular glassmelting, Am. Cer. Soc. Bull. 83 (2004) 33–35. [6] R. Beerkens, Modular melting, part 2 industrial glassmelting process requirement, Am. Cer. Soc. Bull. 83 (2004) 35–38. [7] D. Rue, W. Kunc, G. Aronchi, Charles DrummondIII (Ed.), Operation of a Pilot-Scale Submerged Combustion Melter, Proceeding 68th Conference on Glass Problems, 2007, pp. 125–135 (Columbus, OH). [8] D.J. Bender, J.G. Hnat, A.F. Litka, L.W. Donaldson Jr., G.K. Ridderbusch, D.J. Tessari, J.R. Sacks, Advanced glass melter research continues to make progress, Glass Ind. 3 (1991) 10–37. [9] O. Sakamoto, Innovative Energy Saving Glass Melting Technology, 59 Res. Reports Asahi Glass Co. Ltd., 2009, pp. 55–60. [10] J. Hrbek, P. Kocourková, M. Jebavá, P. Cincibusová, L. Němec, Bubble removal and sand dissolution in an electrically heated glass melting channel with defined melt flow examined by mathematical modelling, J. Non-Cryst. Solids 456 (2017) 101–113, http://dx.doi.org/10.1016/j.jnoncrysol.2016.11.013. [11] M. Polák, L. Němec, Glass melting and its innovation potentials: the combination of transversal and longitudinal circulations and its influence on space utilization, J.Non-Cryst. Solids 357 (2011) 3108–3116, http://dx.doi.org/10.1016/j. jnoncrysol.2011.04.020. [12] P. Cincibusová, L. Němec, Sand dissolution and bubble removal in a model glassmelting channel with melt circulation, Glass Technol.: Eur. J. Glass Sci. Technol. A 53 (2012) 150–157. [13] L. Němec, P. Cincibusová, Sand dissolution and bubble removal in a model glass melting channel with a uniform melt flow, Glass Technol.: Eur. J. Glass Sci. Technol. A 53 (2012) 279–286. [14] M. Polák, L. Němec, Mathematical modelling of sand dissolution in a glass melting channel with controlled melt flow, J. Non-Cryst. Solids 358 (2012) 1210–1216, http://dx.doi.org/10.1016/j.jnoncrysol.2012.02.021. [15] P. Cincibusová, L. Němec, Mathematical modelling of bubble removal from a glass melting channel with defined melt flow and the relation between the optimal melting conditions of bubble removal and sand dissolution, Glass Technol.: Eur. J. Glass Sci. Technol. A 56 (2015) 52–62. [16] M. Jebavá, P. Dyrčíková, L. Němec, Modelling of the controlled melt flow in a glass melting space – its melting performance and heat losses, J. Non-Cryst. Solids 430 (2015) 52–63, http://dx.doi.org/10.1016/j.jnoncrysol.2015.08.039. [17] Glass Furnace Model 4.18, Glass Service, Inc., Vsetín, Czech Republic, (2016). [18] L. Němec, M. Vernerová, P. Cicibusová, M. Jebavá, J. Kloužek, The semiempirical model of the multicomponent bubble behavior in glass melts, Ceramics-Silikáty 56 (2012) 367–373.
6. Conclusion In the investigated melting channel as the second part of the twospace segment furnace, two crucial homogenization phenomena in the melt, sand dissolution and bubble removal, require partially different conditions to be met. This work indicates that both phenomena may effectively be realized in one melting space, provided the uniform flow (possibly transformed to the helical flow) is established in the module. The best results obtained by mathematical modelling were attained for the cases with energy concentrated in the input region of the space (longitudinal energy distribution) and with the central longitudinal energy barrier realized by the longitudinal row of electrodes. The growing value of the theoretical heat showed a significant decrease of the space utilization and the melting performance. The presented and applied energetic model introduced an idea of the idealized longitudinal melt flow under the smooth distribution of energy. It predicted
Notation list k1: the fraction of energy delivered to the input region of the channel (1) vcl: the characteristic velocity of the longitudinal circulations of the melt (m/s) vfr: the maximal forward (throughput) velocity of the melt (m/s) uH: the utilization of the space for the controlling homogenization phenomenon (1) uD: the utilization of the space for sand particle dissolution (1)
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V: the inner volume of the melting channel (m3) V̇ : the volume flow rate of the melt through the channel (m3/s) ξ: the fraction of heat losses belonging to the input region of the channel (1) ρ: the glass density (kg/m3) τDave: the average time of sand particle dissolution in the channel (s) τFref: the time the bubble needs to ascend the distance equivalent to the height of the glass melt in quiescent state (no melt flow) (s) τG: the mean residence time of the melt in the channel (τG = V / V̇ ) (s) τHref: the time for completion of the controlling melting phenomenon (s) Balanced energy distribution (balanced state): the state in which each region of the melting channel (space) is supplied with the right amount of energy needed for batch conversion and/or melt heating and for compensation of heat losses Critical state: the state when the sand particle of the maximum size or the bubble of initially minimum size is removed (dissolved or refined) right at the output from the melting space (when the particle achieves the transversal refractory barrier near the output) Input region: the longitudinal region of the melting channel where the energy is supplied for batch conversion and/or melt heating and for compensation of heat losses Free level region: the longitudinal region of the melting channel where the energy is supplied only for compensation of heat losses
uF: the utilization of the space for bubble removal (1) tinput: the temperature of the melt being inputted into the channel by the strip of the glass level at the front wall (°C) C1: the constant of proportionality between the characteristic velocity of the longitudinal circulations and the driving force of longitudinal circulations (Eq. (4)) (m/J) C2: the constant of proportionality between the maximal forward (throughput) melt velocity and the mass flow rate of the melt (m/kg) Ḣ L : the total heat flux across the channel boundaries (J/s) L HM : the specific heat losses (J/kg) HMT: the specific energy for batch conversion and heating of the batch and/or glass melt to the exit temperature of the space (the specific theoretical heat) (J/kg) Ṁ : the mass flow rate of the melt (melting performance) through the channel (kg/s) Ṁ bal : the mass flow rate of the melt under balanced energy distribution (kg/s) Ṁ bal (k1) : the curve of the mass flow rate under balanced energy distribution as a function of the fraction of energy delivered to the input region of the channel k1 (kg/s) Ṁ crit : the critical melting performance (the melting performance in the critical state) (kg/ s) Ṁ eq left : the melt flow rate under condition vfr = | vlc | demarcating the left boundary of the area of the uniform melt flow (kg/s) Ṁ eq right : the melt flow rate under condition vfr = vlc demarcating the right boundary of the area of the uniform melt flow (kg/s)
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Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Energy distribution and melting efficiency in glass melting channel: Diagram of melt flow types and effect of melt input temperature ⁎
Lukáš Hrbek, Marcela Jebavá , 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
A B S T R A C T
Keywords: Glass melting Melt flow Mathematical modelling Energetic model Space utilization Melting performance
The synergy between glass melting and glass flow character was investigated by mathematical modelling of a Joule heated glass melting channel with a central longitudinal row of vertical heating electrodes. The glass melt containing sand particles and bubbles entered the channel where different types of the melt flow were set up. The sand particle dissolution and the bubble removal were monitored up to the achievement of phenomena completion. The increasing fraction of energy supplied to the input region of the channel proved to be crucial for setting up the beneficial character of the melt flow and consequent increase of the melting performance. When the temperature of the inputting melt decreased, the melting performance also decreased. The relations of the “energetic model” were derived and applied to predict the beneficial type and conditions of the melt flow. The best results have been obtained for the flow character near the uniform flow. The beneficial state was achieved when each region of the channel was supplied by the energy needed there for glass heating and heat losses. The impact of the temperature of the inputting melt on the detailed character of the melt flow was discussed with the help of the energetic model.
1. Introduction In the horizontal glass melting spaces, > 80% of the energy should be delivered to the input part of the melting space to fulfil the energy demands for batch conversion to glass melt. The high energy demand is satisfied by the energy supplied to the batch and molten glass from both the combustion and the melting space. The heat is partially supplied to the glass by electric boosting (below the batch blanket) and by the massive transport of the hot melt from the region of maximal temperatures (hot spot) towards the batch area by natural convection of the melt. The pumping of the heat by natural convection is currently used in most industrial facilities. Nevertheless, a high tax has to be paid for this opportunity, because longitudinal circulations with low melting effect occupy most of the melting space, reduce the melt residence time in the space, and drastically restrict the melting performance when the homogenization phenomena should be fully accomplished. Several technological studies dealt with the problem of melt residence time in the space, as was presented e.g. in [1] or by Beerkens in [2]. Beerkens declared that the minimum residence time was typically only 15–20% of the average residence time. Therefore, spaces, whether horiziontal or vertical, without circulation patterns appeared to be the most advantageous. Hence, the further effort of glass researchers and
⁎
technologists had to be focused on restricting the longitudinal melt circulations either by higher delivery of energy to the batch region or by the segmentation of the glass melting process. The former approach, however, faces technical and technological problems with the positioning and operation of the heat sources (burners, electrodes) and scarce knowledge of the resulting melt flow characters and their mutual transitions. The latter approach leading to segmented melting systems prevents the macroscopic back-flows and increases the overall melting space. Here, separate melting phenomena, such as the conversion of batch to glass, sand dissolution, and bubble removal, take place. Therefore, the optimal conditions for each individual phenomenon can be set up in the relevant melting segment. The facilities with separated the space for batch conversion [3] and segmented melters containing three or four parts were proposed [4–6]. However, it is necessary to mention the problems of segmented melters that result from their relatively complex design and exacting operation. The relatively high heat losses can be expected as well owing to the separated melter structure. Nevertheless, to segment the melting system into only two spaces seems to simplify the problem: the space for batch conversion from the solid to liquid state with powerful conversion technology [7–9], combined with the homogenization space to terminate both dissolution and fining phenomena under tuned conditions. Thus, the
Corresponding author at: University of Chemistry and Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic E-mail addresses: [email protected] (L. Hrbek), [email protected] (M. Jebavá), [email protected] (L. Němec).
https://doi.org/10.1016/j.jnoncrysol.2017.12.009 Received 15 September 2017; Received in revised form 1 December 2017; Accepted 3 December 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Hrbek, L., Journal of Non-Crystalline Solids (2017), https://doi.org/10.1016/j.jnoncrysol.2017.12.009
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The relation for quantity τFref is given in ([15], Eq. (5)). The former expressions in Eqs. (2a, b) and (3a, b) are valid for the case of sand dissolution being the controlling phenomenon, and the latter ones for bubble removal as the controlling phenomenon.
need arose for the relevant theoretical solution of the homogenization spaces prior to their construction. The authors of this article solved the simultaneous sand particle dissolution and bubble removal at a constant average temperature of the glass melt of 1420 °C in the electrically heated horizontal melting channel by mathematical modelling [10]. The melt flow approaching a uniform character and possibly a helical flow were preferentially set up in the space using a longitudinal row of heating electrodes. The results proved that both melting phenomena can efficiently be accomplished in one space under the condition of a controlled melt flow. The melting performance of the channel increased with the increasing input temperature of the melt and with the increasing portion of energy located in the input zone of the space. The effect was indicated by the increasing space utilization. As the space utilization and melting performance increased, the transition of the flow character from the flow with longitudinal melt circulations to an almost uniform flow was observed. In this article, an “energetic model” will be introduced as an efficient tool for the prediction of the character of the longitudinal melt flow when the energy distribution in the space, structural heat losses, theoretical heat, or the kinetics of homogenization processes will change. The results of the mathematical modelling of cases with different melt input temperatures and detailed inspection of the melt flow character in the melting channel will be discussed in the light of the energetic model. The melt flow diagram will be presented and the conditions of high melting performance at respective melt input temperatures will be determined as a function of the longitudinal energy distribution in the space.
2.1. Relations of the energetic model The resulting character of the melt flow particularly depends on the horizontal distribution of the supplied energy to the space, as was already mentioned in [10]. Allow us to imagine a horizontal space with an input of either batch or cool glass melt and characterized by heating the input material and by heat losses through boundaries. Most of the heat needs to be fed at the space input where the batch undergoes chemical transformation and/or the melt should be heated to the average temperature; the structural heat losses also have to be compensated proportionally. If the local needs of energy are not balanced, the natural longitudinal circulations occur in the space. The characteristic velocity of the longitudinal circulations vlc can be introduced [15]; the velocity represents the overall intensity of the longitudinal circulations. The space can then be divided into the input region where the energy is consumed for batch conversion (in an industrial furnace approximately the batch region) and/or melt heating and covering the relevant heat losses, and the following region where only the heat losses should be covered. The latter is called the “free level” region. The driving force of longitudinal circulations can be described using the difference between the energy actually delivered to either the input or free level region and the energy actually needed in the relevant region to maintain the energetically balanced state. The balanced state means the situation when each region of the melting space is supplied by the right amount of energy needed for batch conversion and/or heating of the melt and covering the heat losses. The characteristic velocity of the longitudinal circulation vlc was conventionally placed in the longitudinal axis at the level of the free level region, approximately in the middle of the length of longitudinal circulations; the characteristic velocity expresses the circulation intensity by its value and the flow direction by its sign. Its value is considered linearly proportional to the driving force:
2. Theoretical part The quantity “utilization of the melting space u” has been used for the evaluation of the character of the melt flow with respect to sand dissolution and bubble removal [10–15]. The utilization of continual space for sand dissolution or bubble removal (fining) expresses the relation between the time for accomplishment of the controlling melting phenomenon τHref and the mean residence time of the melt in the space τG under the critical conditions and at the average temperature in the channel. The critical state then describes the situation when the sand particle of maximum size or the bubble of the initially minimal radius is removed (dissolved or refined) right at the output from the space:
uF , D =
τHref τG
, τG =
V , u∈ V˙
0; 1
T ̇ vlc = −C1 [(1 − k1 )(HM M + Ḣ L) − (1 − ξ ) Ḣ L]
where k1 is the fraction of energy actually delivered to the input region of the channel, HMT is the specific energy for batch reactions, phase and modification transitions, and for heating of the contents to the exit temperature of the channel (theoretical heat) in J/kg, Ṁ is the mass L flow rate (also the critical melting performance Ṁ crit ) in kg/s, Ḣ is the total heat flux through channel boundaries (the flux of heat losses) in W, ξ is the fraction of heat losses belonging to the input region, and C1 is the constant of the proportionality in the space under the given average temperature. Thus, the first term in the brackets signifies the energy actually delivered to the free level region of the channel and the second one the energy actually needed to cover the relevant heat losses. As already mentioned, longitudinal melt circulations develop in the energetically unbalanced state between both regions. If the batch (cool melt) input is situated left, anticlockwise longitudinal circulations arise when the actual heat flux delivered to the free level region is higher L than the balanced one, given by the term (1 − ξ ) Ḣ in Eq. (4). In the opposite case when the actual heat flux is lower than the balanced term, the clockwise circulations set up. The flow situation is schematically presented in Fig. 1 of reference [16]. If vlc = 0, then Ṁ = Ṁ bal in Eq. (4), where Ṁ bal is the flow rate through the space under balanced energy distribution. No longitudinal circulation ideally exists in this case and the flow in the channel should be uniform, described only by the throughput velocity of the melt vfr. The value of Ṁ bal is then derived from Eq. (4) at vlc = 0:
(1)
where the index by u designates the controlling melting phenomenon (F – fining, D – sand dissolution). τHref is either the reference time of fining τFref, the time the minimum bubble needs to ascend the distance equivalent to the height of the glass level in a quiescent liquid and at the average temperature in the channel, or the average time of sand dissolution τDave, V is the volume of the space (m3), and V̇ is the volume flow rate (m3/s). For the plug flow, τHref = τG is valid, so uF,D = 1 [13]. 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 HML decrease, and the critical performance of the melting process Ṁ crit increases with space utilization according to:
HML =
Ḣ LτFref 1 Ḣ LτDave 1 or HML = , ρV uD ρV uF
Vϱ Vρ uF , Ṁ crit = uD or Ṁ crit = τFref τDave
(4)
(2a, b)
(3a, b)
Ḣ L (k − ξ ) Ṁ bal = T 1 HM (1 − k1 )
where HM are the specific heat losses (J/kg), HL̇ is the total heat flux across the space boundaries (J/s), and ρ is the glass density (kg/m3). L
2
(5)
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As is apparent for the variable k1: for k1 = ξ, Ṁ bal = 0 and for k1 → 1, Ṁ bal → ∞. If k1 < ξ, no real value of Ṁ bal occurs. The function Ṁ bal (k1) is depicted in Fig. 3 as the red curve. The curve Ṁ bal (k1) forms the boundary line between the anticlockwise longitudinal melt circulations on the left side of the curve where the value of vlc is negative and the clockwise circulations on the right side of the curve characterized by positive values of vlc (Eq. (4)). The vicinity of the Ṁ bal (k1) curve represents the region of approximately uniform flow. The area of uniform-like flow is characterized by high utilization values and, consequently, by high melting performances and low specific heat losses. Such favourable melt flow area is conventionally defined with the assistance of vlc and vfr velocities. The characteristic throughput velocity of the melt vfr is here defined in the same point as the velocity vlc, i.e. at the longitudinal axis of the glass level. The left boundary curve of the uniform-like region is described by the convention that the absolute value of the characteristic velocity of longitudinal circulations | vlc | is equivalent to the characteristic velocity of the through flow vfr in the definition point:
vfr = |vlc|,
Fig. 1. The axonometric view of the melting channel with the level input of the glass melt (the vertical arrow). The inner length is 6.77 m (6.225 m to the transversal refractory barrier), the width is 2 m, and the height of the glass level is 1 m.
first six electrodes formed the input region and the remaining ten electrodes the free level region of the channel. The Glass Furnace Model [17] was used for mathematical modelling which provided the temperature, velocity, and electric power fields in the space. The solution was coupled with submodels of sand dissolution and bubble removal, prepared in cooperation with the author's workplace. The melt, sand particles, and bubbles trajectories were traced along their pathways. The float glass was chosen as the model glass. The temperature dependence of the glass density and kinematic viscosity in the temperature range of 900–1800 °C are presented in [16]. The average sand dissolution rate and average bubble growth rates obtained by laboratory experiments were used to follow the history of the particles. The temperature dependences of both quantities are also presented in [16]. The semiempirical model of the bubble behavior was applied [18]. The history of around 104 bubbles (having the initial size of 5 × 10− 5 m) and sand particles (the initial size of 5 × 10− 4 m) with regularly located starting points at the given input were traced. Bubble nucleation inside the space was not considered. The trajectories and sand radii were calculated with the time step mostly at 1 s and the time step for bubble radii calculation at 0.1 s. A calculation grid was used with a distance of 2.5 cm between grid points. The controlling phenomenon was given by the critical particle (sand or bubble) first reaching the transversal refractory barrier. This state was characterized by: the critical mass melting performance Ṁ crit , the mean residence time of the melt in the space τG, the fining time of the critical bubble τFcrit or the average sand dissolution time τDave. The final value of the space utilization of the controlling phenomenon was calculated from Eq. (1), the specific energy consumption from Eq. (2a, b). To validate Eqs. (7)–(8), the character of the melt flow was inspected in another set of simulations under conditions of different melt flow rates. The melt flow rate and the relevant value of k1 were determined under which the character of the melt flow changed from the existing longitudinal circulation flow to the unifom flow and vice versa. The picture of the longitudinal melt flow in this case was characterized by the spring point located approximately at the halfway point of the melting channel. The melt input temperature varied between 1120 and 1320 °C and the constant average temperature of 1420 °C was maintained in the space. Finally, the amount of energy supplied to the input region of the space (to the first six electrodes) varied between 45 and 100% (the value of k1 varied between 0.45 and 1) in each set of calculations.
(6)
where vfr = C2 Ṁ is always positive and C2 is a constant calculated from the melt flow rate in cross section of the space. The conditions given by eE. (6) and by definitions of both velocities determine the border values of the uniform-like area on the left of the Ṁ bal (k1) curve:
Ṁ eq left =
C1 Ḣ L (k1 − ξ ) T − C2 C1 (1 − k1 ) HM
(7)
Here, Ṁ eq left is the melt flow rate under conditions given by Eq. (6) and demarcates the boundary line of the still favourable character of the melt flow. If k1 shifts to the right and behind the curve Ṁ bal (k1) , both vlc and vfr are positive and the resulting flow velocity on the melt level is the sum of both velocity components. The condition vfr = vlc is entered and the border line of the favourable flow region right of the curve of state Ṁ bal (k1) is defined as:
Ṁ eq right =
C1 Ḣ L (k1 − ξ ) T + C2 C1 (1 − k1 ) HM
(8)
Thus, the Ṁ eq right curve is found right of the Ṁ bal (k1) curve. Both curves are depicted in Fig. 3 (the blue and green curves). Eqs. (4)–(8) describe the melt flow picture from the point of view of the longitudinal energy distribution. The verification of the distribution of flow types according to Eq. (5) by the mathematical model and the impact of the input temperature of the melt (represented here by the quantity HMT which grows if the input temperature decreases) on the critical melting performance, will be treated in this work. 3. Calculation procedure and conditions The simple melting channel, the same as in [10], was chosen for mathematical simulations corresponding to the small horizontal melting furnace, as is clear from the view in Fig. 1. The melting channel was an orthogonal space with the inner length being 6.77 m, the width 2 m and the height of the glass level 1 m. Three layers of refractory materials surrounded the inner space. The glass level was insulated owing to the necessary boundary condition and hence the upper structure was not considered. The transversal refractory barrier on the right side of the space served as the reference point where the sand dissolution or bubble removal should be terminated – and signalled the achievement of the critical state. The glass melt flowed into the channel by a strip of level at the front wall (the width of 2 m, the length of 0.2 m, see the arrow and the strip in Fig. 1). Sixteen vertical bottom electrodes were located regularly in the longitudinal axis of the space. The standard height of electrodes in the case with the level input of the melt was 0.3 m. The
4. Results of calculation Two sets of calculations were executed in this work to elucidate the impact of the melt flow character on the melting efficiency. In the first round of calculations, the boundary lines demarcating the beneficial 3
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Fig. 2. a–e The character of the longitudinal 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 corresponds to ¼ or ¾ of the space width (XZ plane). The arrows → and ← highlight the main directions of the melt flow, the arrows ↓ point at the input and the spring point positions. The melt flow character develops at the growing melt flow rate Ṁ and at constant values of k1. 2a–c: k1 = 0.8, Ṁ = 1, 2, and 6 kg/s. 2d–e: k1 = 0.6, Ṁ = 1, and 5.8 kg/s. tinput = 1320 °C.
4
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(5)) was crossed, the uniform-like flow set in the free level region of the space, as Fig. 2c shows. The further increase of Ṁ was followed at k1 = 0.6 (for the position look at Fig. 3). The growth of Ṁ above the Ṁ bal (k1) curve led to crossing the second boundary line Ṁ eq left (Eq. (7)); the spring point shifted to the output and the anticlockwise longitudinal circulations started in the space (Fig. 2d). At even higher values of Ṁ , the spring point shifted more to the output and the anticlockwise circulations strengthened, which is demonstrated by Fig. 2e. So, Fig. 2a–e demonstrate the impact of the intensity of flow rate (increasing Ṁ ) on the flow character in the free level region of the space (clockwise, uniform, anticlockwise) and also on the spring point position under the condition of identical longitudinal energy distribution in the channel (two values of k1 had to be actually used). Fig. 3 clarifies these changes in the character of the melt flow graphically: when the flow rate Ṁ increases at constant energy distribution k1 (starting at low Ṁ ) then the flow character changes from clockwise type (under Ṁ bal curve) through the uniform type (around Ṁ bal curve) to the anticlockwise type (above Ṁ bal curve). Fig. 3 provides the curves of Ṁ eq left (energ ), Ṁ eq right (energ ), and ̇ Mbal (k1) according to Eqs. (7)–(8) and (5), as well as the values of Ṁ eq left (mat ) and Ṁ eqright (mat ) ascertained by mathematical modelling. The following parameters were used: Ḣ L = 3.27 × 105 W (J/s), ξ = 0.36, HMT = 1.269 × 105 J/kg, C2 = 4.9 × 10− 4 m/kg, and C1 = 1.72 × 10− 8 m/J. The average value of C1 was calculated using three different modelled cases when the longitudinal melt velocity on the melt level was read (at X = 4000 mm and at ¼ of the space width). To obtain the relevant value of vlc, the average value of the longitudinal velocity was recalculated to the level axis and the relevant throughput velocity was subtracted. As Fig. 3 shows, the area of the unifom-like flow demarcated by the values of Ṁ eq left (mat ) and Ṁ eq right (mat ) appears slightly narrower and shifted to the lower values of Ṁ at lower values of k1 comparing with the area demarcated by the curves Ṁ eq left (energ ) and Ṁ eq right (energ ) . The difference is caused by a slightly deformed structure of the flow sets in the real case with heating by electrodes. Still, both solutions agree
Fig. 3. The curves Ṁ eq right (energ )(k1), Ṁ eq left (energ )(k1), and Ṁ bal (k1) calculated according to Eqs. (7)–(8) and (5), as well as the values of Ṁ eq right (mat ) and Ṁ eq left (mat ) ascertained by mathematical modelling (blue and green points). tinput = 1320 °C. The oval pictograms designate the orientation of the longitudinal circulations of the melt and the pictogram at the Ṁ bal (k1) curve indicates the uniform flow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
area of uniform-like flow, described by Eqs. (7) and (8), were verified. Fig. 2a–e illustrates the development of the longitudinal character of the melt flow at k1 > ξ. If the values of the melt flow rate Ṁ at constant k1 = 0.8 were sufficiently low (situated below the Ṁ bal curve, see Fig. 3), the clockwise longitudinal circulations of the melt dominated and the spring point appeared near the input, as Fig. 2a shows. When Ṁ increased at constant k1, the melt flow character was half way to the uniform-like flow (Eq. (8)). The weak forward flow near the bottom arose right behind the spring point (Fig. 2b) and then gradually strengthened with increasing flow rate. When the Ṁ bal (k1) curve (Eq.
Table 1 The values of the modelled cases with the longitudinal row of electrodes and different input temperatures of the melt. k1 is the fraction of total energy delivered to the input region of the channel, uF and uD are the relevant utilizations of the channel, τDave and τFref are the relevant reference times of sand dissolution and bubble removal, τG is the mean residence time of the melt in the channel, Ṁ crit is the critical melting performance, and HMLare the specific heat losses. The cases presented already in [10] are labelled by asterisks. Case No.
tinput (°C)
k1
u (controlling)
τDave, τFref (s)
τG (s)
HML (kJ/kg)
Ṁ crit (kg/s) (t/day)
1* 2* 3* 4* 5* 6* 7* 8* 9* 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
1320 1320 1320 1320 1320 1320 1320 1320 1320 1220 1220 1220 1220 1220 1220 1220 1220 1120 1120 1120 1120 1120 1120 1120 1120 1120 1120
0.45 0.60 0.70 0.75 0.80 0.825 0.85 0.90 1.00 0.45 0.60 0.70 0.80 0.83 0.85 0.90 1.00 0.45 0.60 0.70 0.80 0.825 0.85 0.87 0.89 0.91 1.0
uF = 0.34 uF = 0.38 uF = 0.42 uF = 0.45 uF = 0.47 uF = 0.46 uF = 0.40 uD = 0.27 uD = 0.41 uF = 0.21 uF = 0.25 uF = 0.28 uF = 0.34 uF = 0.37 uF = 0.35 uD = 0.27 uD = 0.30 uF = 0.17 uF = 0.19 uF = 0.23 uF = 0.28 uF = 0.29 uF = 0.28 uF = 0.26 uF = 0.25 uF = 0.23 uD = 0.23
τFref = 1878 τFref = 1877 τFref = 1879 τFref = 1878 τFref = 1878 τFref = 1877 τDave = 2059 τDave = 2078 τDave = 1903 τFref = 1920 τFref = 1877 τFref = 1877 τFref = 1879 τFref = 2048 τFref = 1877 τDave = 2141 τDave = 1952 τFref = 1877 τFref = 1875 τFref = 1923 τFref = 2048 τFref = 1889 τFref = 1890 τFref = 1879 τFref = 1972 τFref = 1887 τDave = 2153
5
5560 4985 4448 4130 3988 4072 5163 7608 4663 9029 7603 6723 5560 5558 5452 7814 6567 11232 9734 8499 7225 6881 6719 7223 8027 8255 9320
5.2 5.8 6.5 7.0 7.25 7.1 5.6 3.8 6.2 3.2 3.8 4.3 5.2 5.2 5.3 3.7 4.4 2.6 3.0 3.4 4.0 4.2 4.3 4.0 3.6 3.5 3.1
449.3 501.1 561.6 604.8 626.4 613.4 483.4 328.3 535.7 276.5 328.3 371.5 449.3 449.3 457.9 319.7 380.2 224.6 259.2 293.8 345.6 362.9 371.5 345.6 311.0 302.4 267.8
62.9 56.4 50.3 46.7 45.9 46.1 58.3 85.9 52.9 101.3 85.2 75.3 62.2 61.9 61.1 87.5 73.8 123.8 107.1 94.2 79.5 80.0 74.6 80.2 88.9 91.7 103.3
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overall. The curves Ṁ eq left (energ )(k1), Ṁ eq right (energ )(k1) and Ṁ bal (k1) thus provide fundamental prerequisite information about the beneficial areas of glass melting in terms of the melt flow character. The other set of mathematical simulations was focussed on the role of the melt input temperature tinput. The melt input temperature, being the melt output temperature of the previous batch conversion space, reflects the conditions of the conversion space and, therefore, is an important technological factor. tinput is involved in the value of HMT in Eq. (4) where HMT represents the theoretical heat – here the heat needed for glass heating from the input to the exit temperature of the melt in the channel. The main values significant for evaluation of melting efficiency are presented in Table 1. Table 1 shows that, at the melt input temperatures 1220 and 1120 °C, the maximum melting performance and, consequently, the minimum specific heat losses were achieved very close to the maximum value of the space utilization. This was in agreement with the values achieved at 1320 °C in [10], labelled here in Table 1 by asterisks. The simultaneous development of technological quantities with the space utilization provides evidence about the dominating role of the melt flow character under constant average temperature in the space.
Fig. 4. The critical melting performance Ṁ crit as a function of the fraction of energy located in the input part of the space k1 at three melt input temperatures 1120, 1220 and 1320 °C, and the relevant Ṁ bal (k1) curves. The curve Ṁ crit – 1320 °C, 3:1: the energy delivered to the input part (k1 = 0.8) was further partitioned in the ratio of 3:1 in favour of the first pair of electrodes.
5. Discussion of results anticlockwise longitudinal circulations. Note that the values of Ṁ crit for the longitudinal row of electrodes are high even at lower values of k1, as was already reported in [10]. The maximum critical melting performances are achieved slightly left of the Ṁ bal (k1) curve. The respective longitudinal flow established at these optimum conditions represents an almost uniform flow and weak anticlockwise longitudinal circulations. The effect of the slow anticlockwise longitudinal circulations lies in reducing the maximal forward throughoutput velocity near the melt level. This effect enhances the space utilization. Similar optimum conditions were repeatedly found in [11–15] where the optimum flow conditions were examined in the space with preset temperature distributions; the results indicate the most probable universal picture of the optimum flow character. The values of Ṁ crit substantionaly decrease with the declining melt input temperature at the constant k1 and in the area left from the relevant Ṁ bal (k1) curves. The explanation comes from the definition of vlc by Eq. (4); higher values of HMT (i.e. the lower melt input temperatures) indicate higher absolute values of vlc. Higher absolute values of vlc lead to lower values of the space utilization at the same flow rate and to the occurrence of either undissolved sand particles or bubbles in the outputting melt. Graphically, the Ṁ bal (k1) dependence becomes more curved at higher values of HMT (see Fig. 4) and shows lower values of Ṁ bal at the given k1. The new values of Ṁ crit should therefore decrease to approach the lower Ṁ bal (k1) curve and not to increase the new value of vlc too much. Thus, the cases with the melt input temperatures of 1120 and 1220 °C show the lower positions of their Ṁ bal (k1) curves and the relevant values of Ṁ crit should be also lower, as compared with the values at the melt input temperature of 1320 °C. Fig. 5 then shows the practically parallel developments of Ṁ crit and u as a function of k1 at 1220 °C. It confirms the dominating role of the melt flow character (represented by the value of the space utilization) for the melting efficiency. The role of the intensity of the longitudinal melt circulations can be more clearly ilustrated by the 3D representation of the characteristic values of the velocity component of the longitudinal circulations vlc. Fig. 6 presents the values of vlc as a function of the flow rate Ṁ and the fraction of energy delivered to the input region k1. Eq. (4) and the values of HMT, Ḣ L , and ξ were applied, as was presented in the section Results of calculation. The absolute values of vlc are plotted in Fig. 6 to be clearer; in fact, the values are negative left of the Ṁ bal (k1) curve and positive to the right of it. The Ṁ bal (k1) curve in the base of the 3D graph (in the XY plane) designates the areas of the anticlockwise and clockwise longitudinal circulations. The zero values of vlc exist at the Ṁ bal (k1) curve and the
5.1. The diagram of melt flow types The diagram of the longitudinal melt flow types drawn in coordinates of the flow rate Ṁ against the the fraction of energy delivered to the input part of the space k1 is crucial to be able to understand the impact of the melt flow character on the melting efficiency in continuous glass melting spaces. The diagram valid for the melting channel is presented in Fig. 3 but a similar picture can be acquired for the classic melting facility with batch blanket. The Ṁ bal (k1) curve splits the plane [Ṁ , k1] into the left part lying at lower values of k1 and characterized by anticlockwise longitudinal circulations of the melt, and the right part at higher k1 values where the clockwise longitudinal circulations are produced. The Ṁ bal (k1) curve itself indicates the zero value of the intensity of the longitudinal circulations and only uniform flow dominates here. Since the presence of longitudinal circulations reduces the utilization of the space [10], the utilization should be high along the Ṁ bal (k1) curve and should decrease with the increasing distance from the curve. The precise information about the intensity of the longitudinal circulations vlc is then provided by Eq. (4). The conventionally defined area between Ṁ eq left (k1) and Ṁ eq right (k1) curves in Fig. 3, surrounding the Ṁ bal (k1) function, depicts the area of Ṁ and k1 suitable for the application of the beneficial melt flow. The following results of the mathematical modelling prove the beneficial applicability of this area. The sufficient agreement between courses of both Ṁ eq (energ ) curves and points of Ṁ eq (mat ) in Fig. 3 justifies the assumption of a linear relation between the values of vlc and the driving force of longitudinal circulations (Eq. (4)) in the examined area of flow rates. If the values of HMT, Ḣ L , and ξ are approximately known (i.e. the curve Ṁ bal (k1) can be drawn), the profitable area from the point of view of the melt flow can be assessed in advance. 5.2. The influence of the melt input temperature tinput (represented by the theoretical heat HMT) To express the significance of the melt input temperature (the energy necessary for heating the melt being inputted), three sets of calculations at the melt input temperatures 1320 [10], 1220, and 1120 °C have been executed and plotted in Fig. 4 in the form of Ṁ crit values versus k1. Different melt input temperatures imply different values of HMT and, consequently, different curves of Ṁ bal (k1) ; therefore, three relevant Ṁ bal (k1) curves are plotted along with the values of Ṁ crit . The calculated Ṁ crit values grow in the area left from the Ṁ bal (k1) curve with increasing k1 in all three cases owing to the decreasing intensity of the 6
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Fig. 5. The values of the critical melting performance Ṁ crit and the space utilization of the controlling phenomenon u as a function of the fraction of energy located in the input region of the space k1. tinput = 1220 °C.
Fig. 7. The absolute values of the characteristic velocity of longitudinal circulations |vlc | valid for the standard case with the longitudinal row of electrodes as a function of the fraction of energy delivered to the input region of the space k1. tinput = 1320 °C. energ – the values coming from the energetic model (Eq. (4)), mat – the values coming from the numerical solution by the mathematical model.
pictures of mathematical modelling can also influence the accuracy of the values. Despite the difference, the values coming from the energetic model appear suitable for semi-quantitative predictions. Both sets of | vlc | values achieve the zero at k1 = 0.82; this point corresponds to the intersection of Ṁ crit (k1) and Ṁ bal (k1) curves (Fig. 4 also shows this). The small values of | vlc |, which should exist at k1 = 0.8, then correspond to the intensity of the anticlockwise circulations when Ṁ crit reaches its maximal value. Figs. 6 and 7 based on the values of | vlc | thus make the beneficial area of high melting performances easily available thanks merely to knowledge of a few integral quantities. Nevertheless, some unexpected behavior can still be observed for Ṁ crit (k1) dependences in Fig. 4. If the values of Ṁ crit are situated just at the Ṁ bal (k1) curve or very close to it, the values of the space utilization should roughly correspond to those found for the uniform or helical flows and should be essentially independent of the melt input temperature. The values of the maximum critical melting performance close to or at the Ṁ bal (k1) curve should then be similar. However, the maximum values of the critical melting performance obviously decrease with decreasing melt input temperature (see Fig. 4). The values of the space utilization in the case of the isothermal uniform flow were found to be 0.445 for the sand dissolution and 0.67 for bubble removal [13], and for the helical flow were found between 0.6 and 0.8 [11–12,14–15]. On the other hand, the interpolated values of the space utilization at the respective Ṁ bal (k1) curves were uF = 0.45 at the melt input temperature 1320 °C, 0.32 at 1220 °C and only 0.25 at 1120 °C. Hence, the real utilization values were substantially lower than expected. More precisely, the maximum critical melting performance decreased as the melt input temperature decreased. The reason for this behavior should be sought in the detailed character of the melt flow. From the flow pictures, it can be seen that the cold melt entering the space through the level naturally falls to the bottom and forms the longitudinal circulation flow near the space input. The lower the melt input temperature is, the higher the support of the original londitudinal circulations – caused by the energetically unbalanced state – can be expected. The arising powerful melt flow then affects the melt flow character in the entire space and over the entire interval of k1, including the area close to the Ṁ bal (k1) curve. When k1 values occur in the area of the Ṁ bal curve, the melt flow shows the character of the uniform-like flow (see Fig. 2c). However, at low melt input temperature, the powerful circulations that arise near the input considerably support the following forward flow near the bottom. In order to keep the constant value of the flow rate, a backflow in the centre of the height arises as a consequence of both massive forward flows; therefore, the space
Fig. 6. The characteristic (absolute) values of the velocity component of longitudinal circulations | vlc | as a function of the flow rate Ṁ and the fraction of energy delivered to the input region k1. tinput = 1320 °C. Red curve: the values of | vlc | at Ṁ crit . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
region of low values of | vlc | along the curve defines the area as beneficial for glass melting. The expressions in the edges represent the border values of vlc at k1 = 0 and 1. The areas with high values of | vlc | and, consequently, low space utilization, particularly exist at low values of k1 and high values of Ṁ . At k1 = 1, the vlc values are independent of the flow rate. The red curve in the graph represents the spatial development of |vlc | values for the critical melting performances Ṁ crit and relevant values of k1 (standard case 1*–9* in Table 1). The values of | vlc | decrease with the growing k1 left of the Ṁ bal (k1) curve but they grow to the right of it. When doing a section through the Ṁ crit points and relevant k1 values, the decrease and increase of | vlc | values is clearer as Fig. 7 shows. Both values of |vlc | coming from Eq. (4) (energetic model) and those read from the pictures of the melt flow (mathematical model) are plotted in Fig. 7 for comparison. The values of | vlc |mat are larger than those calculated from the model of energy distribution according to Eq. (4); nevertheless, they show equivalent tendencies. The difference between the mentioned values is primarily caused by deviation of the real melt flow (showed by the mathematical model) from the concept of the smooth distribution of energy represented by Eq. (4) of the energetic model. Furthermore, the real reading of the melt velocities from the 7
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Fig. 8. The character of the longitudinal 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 corresponds to ¼ or ¾ of the space width (XZ plane). The arrows → and ← highlight the main directions of the melt flow, the arrow ↓points to the input and the spring point positions. tinput = 1120 °C, k1 = 0.89 (case 25 in Table 1).
As Fig. 9a shows at k1 = 0.8, the character of the longitudinal melt flow in the second region of the space is approximately uniform and the space utilization, as well as the critical melting performance, achieve their maximum values (see Table 1, case 5). In Fig. 9b, the clockwise longitudinal circulations arise, and both the space utilization and the critical melting performance obviously decrease (see Table 1, case 8). However, the decrease stops or even the critical melting performance increases again between k1 = 0.9 and 1 at 1220 °C and particularly at 1320 °C (although the intensity of the clockwise longitudinal circulations further increases). This fact was already discussed in [10], and the increase was attributed to the decrease of the average time of sand dissolution. However, it seems that this factor plays only a partial role. The detailed inspection of the melt flow at k1 = 1 has revealed the ultimate shift of the spring point to the space input, even before the end
utilization should decrease owing to newly arising longitudinal circulations. If the melt input temperature approaches the average temperature in the space, the backflow disappears and the values of both the space utilization and the melting performance correspond approximately to the uniform flow. The typical picture of the melt with two forward flows is shown in Fig. 8. As Fig. 4 demonstrates, the values of the critical melting performance steeply decrease behind the maximum values at all three melt input temperatures. The principal decrease signalizes the arising clockwise longitudinal circulations of the melt right of the relevant Ṁ bal (k1) curves. The values of vlc grow with growing k1 and the space utilization, as well as the critical melting performance, should decrease. Fig. 9a–b show how the flow character typically changes at the melt input temperature of 1320 °C.
Fig. 9. a–bThe character of the longitudinal melt flow patterns marked by the streamlines of the melt when the melt is heated along the central longitudinal axis of the space. The arrows → and ← highlight the main directions of the melt flow, the arrow ↓points at the input and the spring point positions. tinput = 1320 °C. a: k1 = 0.8, b: k1 = 0.9.
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the conditions of the melt flow character approaching a uniform flow and explained the impact of the melt input temperature on the melting efficiency. At present, authors cooperate with specialists in the field of electric boosting installation in melting spaces to solve the practical problem of massive heating in the input region. The theoretical investigation of further factors of the process – such as the finer distribution of energy and heating means in the space and the impact of heat losses by both the mathematical and energetic model – is planned for the near future, as well as the cooperative solution of batch conversion problems.
of the intensively heated input region. The critical particle trajectories passing through the spring point then had a better chance to be transported to the space sides by the strong transversal circulations. Consequently, sand particles or bubbles moved there with lower forward velocities and had enough time to be dissolved or removed; this effect enabled higher flow rates. Nevertheless, the practical installation of all the delivered energy to the input appears not to be realistic. The impact of the melt input temperature (or the theoretical heat in cases involving the presence of the batch) is in the energetic model expressed by the quantity HMT (Eqs. (4)–(5)). Summarizing the role of the melt input temperature, the growth of HMT increases the intensity of the overall longitudinal circulations according to Eq. (4), and thus decreases the space utilization along with the melting performance in the interval of k1 left of the Ṁ bal (k1) curve. Growing HMT causes the Ṁ bal (k1) curve to become more curved, and a larger value of k1 is needed to achieve the maximum utilization and performance values (situated close to the Ṁ bal (k1) curve). The high values of HMT also evoke strong local circulations in the input region of the space, strengthen the overall longitudinal circulations, and reduce the values of the utilization and the performance, even in the area of the Ṁ bal (k1) curve. More effort should therefore be focused on the detailed distribution of energy and its improvement in the input region of the melting space. The preliminary results of the calculations characterized by highest values of critical melting performances confirm this intention for the case with finer distribution of the delivered energy in the input part of the channel: Ṁ crit – 1320 °C, 3:1 in Fig. 4. The interval of k1 right of the Ṁ bal (k1) curve shows the rapid decrease of both the space utilization and the critical melting performance, but a repeated increase can occur close to k1 = 1 owing to the shift of the spring point to the input. The impact of the melt input temperature was thus elucidated and the optimal conditions of the longitudinal energy distribution in the channel were determined by the energetic model. The results have practical significance for implementation owing to the very high specific melting performance that is achievable. The specific heat losses are correspondingly reduced. The technical solution of this approach mostly assumes high energy supply to the input region of the space and to the central longitudinal section through the space. The electric heating is more flexible and preferred to heating by burners. The installation of powerful heating in the input region of the space should be solved by the proper amount and arrangement of electrodes to work without their overloading and without overheating both the melt and refractory material. If the results have to be applied to melting spaces with a batch blanket, the question of suitable energy distribution in the input region appears even more urgent. The improved approaches to the rapid batch-to-glass conversion should be examined and applied to adjust the batch conversion capacity to the high sand dissolution and fining capacity of the melt. In a favourable case, the melting space could be miniaturized without a decrease of the melting performance.
Acknowledgement This work has been supported by the project of the Technology Agency of the Czech Republic, No. TH02020316, “Advanced technologies of glass production”. References [1] F. Simonis, H. de Waal, R. Beerkens, Influence of furnace design and operation parameters on the residence time distribution of glass tanks, predicted by computer simulation, Collected Papers XIVth International Congress on Glass. New Delhi, India, 1986, pp. 118–127. [2] R. Beerkens, Analysis of elementary process steps in industrial melting tanks – some ideas on innovation in industrial glass melting, Ceramics-Silikáty 52 (2008) 206–217. [3] S. Wiltzsch, Evaluation principle in respect of the batch-melting zone of a glass melt furnace, Glass Technol.: Eur. J. Glass Sci. Technol. A 58 (2017) 105–115, http://dx. doi.org/10.13036/17533546.58.4.008. [4] C.P. Ross, G.L. Tincher, Glass Melting Technology: A Technical and Economic Assessment, Glass Manufacturing Industry Council, Westerville, OH, 2004, pp. 59–87. [5] E.D. Spinosa, Modular glassmelting, Am. Cer. Soc. Bull. 83 (2004) 33–35. [6] R. Beerkens, Modular melting, part 2 industrial glassmelting process requirement, Am. Cer. Soc. Bull. 83 (2004) 35–38. [7] D. Rue, W. Kunc, G. Aronchi, Charles DrummondIII (Ed.), Operation of a Pilot-Scale Submerged Combustion Melter, Proceeding 68th Conference on Glass Problems, 2007, pp. 125–135 (Columbus, OH). [8] D.J. Bender, J.G. Hnat, A.F. Litka, L.W. Donaldson Jr., G.K. Ridderbusch, D.J. Tessari, J.R. Sacks, Advanced glass melter research continues to make progress, Glass Ind. 3 (1991) 10–37. [9] O. Sakamoto, Innovative Energy Saving Glass Melting Technology, 59 Res. Reports Asahi Glass Co. Ltd., 2009, pp. 55–60. [10] J. Hrbek, P. Kocourková, M. Jebavá, P. Cincibusová, L. Němec, Bubble removal and sand dissolution in an electrically heated glass melting channel with defined melt flow examined by mathematical modelling, J. Non-Cryst. Solids 456 (2017) 101–113, http://dx.doi.org/10.1016/j.jnoncrysol.2016.11.013. [11] M. Polák, L. Němec, Glass melting and its innovation potentials: the combination of transversal and longitudinal circulations and its influence on space utilization, J.Non-Cryst. Solids 357 (2011) 3108–3116, http://dx.doi.org/10.1016/j. jnoncrysol.2011.04.020. [12] P. Cincibusová, L. Němec, Sand dissolution and bubble removal in a model glassmelting channel with melt circulation, Glass Technol.: Eur. J. Glass Sci. Technol. A 53 (2012) 150–157. [13] L. Němec, P. Cincibusová, Sand dissolution and bubble removal in a model glass melting channel with a uniform melt flow, Glass Technol.: Eur. J. Glass Sci. Technol. A 53 (2012) 279–286. [14] M. Polák, L. Němec, Mathematical modelling of sand dissolution in a glass melting channel with controlled melt flow, J. Non-Cryst. Solids 358 (2012) 1210–1216, http://dx.doi.org/10.1016/j.jnoncrysol.2012.02.021. [15] P. Cincibusová, L. Němec, Mathematical modelling of bubble removal from a glass melting channel with defined melt flow and the relation between the optimal melting conditions of bubble removal and sand dissolution, Glass Technol.: Eur. J. Glass Sci. Technol. A 56 (2015) 52–62. [16] M. Jebavá, P. Dyrčíková, L. Němec, Modelling of the controlled melt flow in a glass melting space – its melting performance and heat losses, J. Non-Cryst. Solids 430 (2015) 52–63, http://dx.doi.org/10.1016/j.jnoncrysol.2015.08.039. [17] Glass Furnace Model 4.18, Glass Service, Inc., Vsetín, Czech Republic, (2016). [18] L. Němec, M. Vernerová, P. Cicibusová, M. Jebavá, J. Kloužek, The semiempirical model of the multicomponent bubble behavior in glass melts, Ceramics-Silikáty 56 (2012) 367–373.
6. Conclusion In the investigated melting channel as the second part of the twospace segment furnace, two crucial homogenization phenomena in the melt, sand dissolution and bubble removal, require partially different conditions to be met. This work indicates that both phenomena may effectively be realized in one melting space, provided the uniform flow (possibly transformed to the helical flow) is established in the module. The best results obtained by mathematical modelling were attained for the cases with energy concentrated in the input region of the space (longitudinal energy distribution) and with the central longitudinal energy barrier realized by the longitudinal row of electrodes. The growing value of the theoretical heat showed a significant decrease of the space utilization and the melting performance. The presented and applied energetic model introduced an idea of the idealized longitudinal melt flow under the smooth distribution of energy. It predicted
Notation list k1: the fraction of energy delivered to the input region of the channel (1) vcl: the characteristic velocity of the longitudinal circulations of the melt (m/s) vfr: the maximal forward (throughput) velocity of the melt (m/s) uH: the utilization of the space for the controlling homogenization phenomenon (1) uD: the utilization of the space for sand particle dissolution (1)
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V: the inner volume of the melting channel (m3) V̇ : the volume flow rate of the melt through the channel (m3/s) ξ: the fraction of heat losses belonging to the input region of the channel (1) ρ: the glass density (kg/m3) τDave: the average time of sand particle dissolution in the channel (s) τFref: the time the bubble needs to ascend the distance equivalent to the height of the glass melt in quiescent state (no melt flow) (s) τG: the mean residence time of the melt in the channel (τG = V / V̇ ) (s) τHref: the time for completion of the controlling melting phenomenon (s) Balanced energy distribution (balanced state): the state in which each region of the melting channel (space) is supplied with the right amount of energy needed for batch conversion and/or melt heating and for compensation of heat losses Critical state: the state when the sand particle of the maximum size or the bubble of initially minimum size is removed (dissolved or refined) right at the output from the melting space (when the particle achieves the transversal refractory barrier near the output) Input region: the longitudinal region of the melting channel where the energy is supplied for batch conversion and/or melt heating and for compensation of heat losses Free level region: the longitudinal region of the melting channel where the energy is supplied only for compensation of heat losses
uF: the utilization of the space for bubble removal (1) tinput: the temperature of the melt being inputted into the channel by the strip of the glass level at the front wall (°C) C1: the constant of proportionality between the characteristic velocity of the longitudinal circulations and the driving force of longitudinal circulations (Eq. (4)) (m/J) C2: the constant of proportionality between the maximal forward (throughput) melt velocity and the mass flow rate of the melt (m/kg) Ḣ L : the total heat flux across the channel boundaries (J/s) L HM : the specific heat losses (J/kg) HMT: the specific energy for batch conversion and heating of the batch and/or glass melt to the exit temperature of the space (the specific theoretical heat) (J/kg) Ṁ : the mass flow rate of the melt (melting performance) through the channel (kg/s) Ṁ bal : the mass flow rate of the melt under balanced energy distribution (kg/s) Ṁ bal (k1) : the curve of the mass flow rate under balanced energy distribution as a function of the fraction of energy delivered to the input region of the channel k1 (kg/s) Ṁ crit : the critical melting performance (the melting performance in the critical state) (kg/ s) Ṁ eq left : the melt flow rate under condition vfr = | vlc | demarcating the left boundary of the area of the uniform melt flow (kg/s) Ṁ eq right : the melt flow rate under condition vfr = vlc demarcating the right boundary of the area of the uniform melt flow (kg/s)
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