DE10333869A1 - Laying out installation for melting and/or refining glass melt comprises using target value for dimensions and/or operating parameters of installation - Google Patents

Abstract

Laying out an installation for melting and/or refining a glass melt comprises determining a target value for the dimensions and/or operating parameters of the installation, determining a first set of installation dimensions, determining a first set of operating parameters and material parameters of the glass melt, simulating the melting and/or refining process with simulation of behavior of the bubbles in the melt, comparing the simulation results with the corresponding target value, determining a further set of dimensions of the installation based on the comparison results, determining a further set of operating parameters of the installation and material parameters based on the comparison results, repeating the previous 4 steps until the simulation result lies in the framework of a prescribed confidence interval around a prescribed value. Independent claims are also included for the following: (1) installation melting and/or refining a glass melt; and (2) a glass product produced by the installation.

Description

The The invention relates to the design of a plant for melting and / or Purify of glass according to claim 1.

In the manufacture of glass, a melt is usually produced, which is refined before being fed to further processing. Lautering is understood to mean the removal of bubbles and dissolved gases from molten glass, which are continuous in devices such as those in the application DE 199 39 779 A1 Applicant described installation can be performed.

Usually will the purification made of glass melts by adding substances that at a temperature increase Release gases that diffuse into existing bubbles and enlarge them. Thereby can the bubbles rise faster from the melt and out of this escape.

In continuously operated plants, the refining can be done by applying reduced pressure. This method and its embodiments are, for example, in EP 0 989 099 A1 . EP 0 967 180 B1 . EP 0 231 518 B1 and JP 11255519 A described.

In order to remove foam, refining gas can also be produced by refining agents in the molten glass, for example according to the Applicant's application DE 199 39 771 A1 ,

Further, when refining, the high frequency explanation method developed by the Applicant can be used without the use of toxic refining agents such as As ₂ O ₃ or Sb ₂ O ₃ . This method is based on that the melt is passed through an alternating electric field of a coil, wherein the field is coupled directly into the glass. The aggregate walls are designed as skulls, whereby the glass melt reaches a temperature of more than 1900 ° C. The high-frequency explanation method is in the following patents DE 199 39 773 A1 and DE 199 39 786 A1 described in detail.

When refining, furthermore, the high temperature refining method developed by the applicant can be used. In the application DE 102 56 657.7 describes a heating device and a melting unit, which allow efficient heating of the melt due to improved cooling. In DE 102 56 594.5 describe a method and apparatus for heating melts so that the walls of the melter can be cooled sufficiently and at the same time more energy is supplied to the melt than are withdrawn through the cooled walls.

conditioned due to the reactions occurring during the melting process contains a Glass melt Gases in chemically and / or physically dissolved form as well as in the form of bubbles.

The goal of any refining process is to remove, as far as possible, the bubbles contained in the molten glass and at least part of the dissolved gases. These gases include, for example, H ₂ O, CO ₂ , SO ₂ , N ₂ , O ₂ and Ar.

Fundamentals of the processes occurring during refining are described in the German patent application of the applicant DE 103 04 973.8-45 whose content is incorporated by reference in its entirety in the present application.

Around already in the design phase of the plant statements for example about the Operating behavior of the production plant under different conditions To win, the operating behavior of the plant can melt and / or lautering of glass determined by simulation calculations and / or simulated become. This requires the behavior of the melt dispersed bubbles, which forms the continuum phase, if possible realistically to be able to describe.

In her contribution "Behavior of bubble files in glass melting space ", Glass Sci. Technol. 76 (2003), pp. 71-80 J. Matyáš and L. Nimec is a parameter study that examines the influence of start and boundary conditions on the bubble size distribution in a model smelting room and on the glass quality. Of Further is the ability estimated from the model, the main sources for To identify bubbles in the melting chamber.

there The authors describe the behavior of a single bladder as Basis of a model for the Bubble concentration. There will be a limited number of bubbles their growth and traveled path in the melting chamber is pursued. Each tracked bubble then represents all the bubbles in a class, which is defined by start position and output size.

Of the Disadvantage of this approach, however, is that only individual bubbles to be viewed as. The behavior of a single bubble in ambient Continuum phase, however, is not comparable to behavior from bubbles in an environment dispersed throughout the continuum phase further bubbles, because these influence the behavior of the dispersion made of melt and bubbles lasting.

T. Roi, O. Seidel, G. Nölle and D. Höhne choose Therefore, in their publication "Modeling of the bubble population in glass melts ", Glass Sci. Technol. 67 (1994), p. 263-271 the approach of the population balance equation to the total bladder balance while the purification to describe. It is not just a bubble, but a large number uniform Bubbles considered, for the movement of the bubbles, their growth and the formation of new bubbles is calculated. However, only the behavior of the Gas bubbles are considered in the surrounding continuum phase and the effects neglecting the bubbles dispersed in the molten glass. Especially becomes the spatial Distribution of flow parameters the glass dispersion except Eight left.

The corresponding calculations were made in a two-dimensional computer model carried out, which, however, due to strong simplifications not suitable is to simulate concrete melting tanks, but only fundamental relationships can represent. For example, the coalescence - that is, the confluence of Bubbles to form a new, bigger bubble - not considered.

It Therefore, it is an object of the invention, the predictive quality of the simulation the behavior of gas bubbles in high-viscosity liquids such as in particular To improve glass melting and thus the quality of the design of the plant to increase.

A Another object of the invention is the formation of new bubbles as well as the retroactive effect the bubble movement, in particular their buoyancy, on the flow of Consider melt to be able to. moreover It is an object of the invention to coalesce the bubbles, in particular due to different buoyancy speeds in the design included.

The Inventors have realized that the aerosol technique of the approach of the moment method is known with which for the formation of particles in the gaseous environment that develops Particle size distribution can be modeled.

So describe A. Bensberg, P. Roth, R. Brink and H. Lange in their Post "Modeling of particle evolution in aerosol reactors with coflowing gaseous reactants ", AIChE Journal 45 (1999), pp. 2097-2106, the modeling of simultaneous formation and coagulation - that is, aggregation from at least two particles to a cluster, with the difference for coalescence is that the solid particles generally do not flow together - from particles at superimposed convective and diffusive transport.

With this approach can for particles an aerosol in the surrounding gas phase for a considered location temporal change the particle size distribution considering convective particle transport, the formation of clusters more critical Size as well the increase or loss of particles in a size interval be calculated by coagulation. Through a diffusion approach In addition to the coagulation, the influence of those dispersed in the gas continuum is also evident Particles on the local particle size distribution considered.

Surprisingly it has been shown that the Moment method not only on gas-dispersed particles, but also on in liquid dispersed gas bubbles can be applied.

The Tasks are therefore already easily by a procedure according to claim 1 solved.

• a) des Festlegens eines Zielwertes für zumindest eine Abmessung und/oder zumindest einen Betriebsparameter der Anlage,
• b) des Festlegens eines ersten Satzes von Abmessungen der Anlage,
• c) des Festlegens eines ersten Satzes der Betriebsparameter der Anlage und der Stoffparameter der zu läuternden Schmelze,
• d) des Simulierens des Schmelz- und/oder Läuterprozesses mit Simulation des Verhaltens der Blasen in der Schmelze,
• e) des Vergleichens des Simulationsergebnisses mit den entsprechenden Zielwert,
• f) des Festlegens eines aufgrund des Vergleichsergebnisses angepaßten weiteren Satzes von Abmessungen der Anlage,
• g) des Festlegens eines aufgrund des Vergleichsergebnisses angepaßten weiteren Satzes der Betriebsparameter der Anlage und der Stoffparameter der zu läuternden Schmelze,
• h) des Wiederholens der Schritte d) bis g) solange, bis das Simulationsergebnis im Rahmen eines vorgebbaren Vertrauensbereiches um den entsprechenden Zielwert liegt, und ist dadurch gekennzeichnet, daß das Simulieren des Verhaltens der Blasen in der Schmelze basierend auf der Momentenmethode durchgeführt wird.

The inventive method for designing a system for melting and / or refining glass comprises the steps

• a) setting a target value for at least one dimension and / or at least one operating parameter of the plant,
• b) setting a first set of dimensions of the installation,
• c) determining a first set of the operating parameters of the plant and the substance parameters of the melt to be purified,
• d) simulating the melting and / or refining process with simulation of the behavior of the bubbles in the melt,
• e) comparing the simulation result with the corresponding target value,
• (f) the determination of a further set of dimensions of the installation, adapted to the result of the comparison,
• g) determining a further set of operating parameters of the plant adapted to the result of the comparison and the substance parameters of the melt to be purified,
• h) repeating steps d) to g) until the simulation result lies within a predefinable confidence interval around the corresponding target value, and is characterized in that the simulation of the behavior of the bubbles in the melt is carried out based on the moment method.

On In this way, the invention provides for the first time a method for laying out a plant for melting and / or refining glass available at which by applying the moment method, the formation of new bubbles as well as the retroactive effect the bubble movement, in particular their buoyancy, on the flow of Melt considered becomes. Furthermore, the chemical reaction of the bubbles on the Considered melt, by completely containing the gas exchange between the liquid and the bubble phase is. A resulting degassing and a shift of the equilibrium of redox reactions in the liquid phase are described. In addition, for the first time coagulation of the bubbles, especially due to different buoyancy rates included in the design become. This allows the predictive quality of the simulation and thus the goodness the design of the system advantageously increased significantly become.

Especially is dimensioning possible, which the dimensions so on the positioning and performance the devices with which energy is supplied to the system, votes that one preferably small design with correspondingly low, but sufficient residence time ensures. In this way, the inventive method allows a particularly efficient operation of the system in terms of energy consumption and the time required to produce a refined melt with the desired Properties.

The Inventive solution provides In addition, a plant for melting and / or refining glass available, which Has dimensions according to the above mentioned methods with respect to at least one target size of the dimensions and / or the operating parameters are matched.

The Invention will now be described with reference to an embodiment with reference on the attached Figure described.

It shows

1 : A schematic representation of a melting and refining trough with enlarged sections of a melt containing bubbles.

The in the following Quotes are listed at the end of the description.

A glass melt consists not only of liquid glass but of a suspension of liquid and dispersed gas bubbles. In particular, in a melting tank, the order of magnitude of the bubble diameter ranges from a few nanometers to a few millimeters and thus covers at least six orders of magnitude. Figure 1 outlines this condition. Gases dissolved in the liquid glass diffuse into the gas bubbles or vice versa. At high supersaturation of the dissolved gases can. also form new bubbles. The bubbles continue to grow through coagulation and grow faster as they grow larger. All these phenomena can be described by means of the moment method, which is already successfully used in aerosol physics, ie in the formation of particles in gas flows, see [1], [2], [3], [4]. The models used so far can neither regenerate bubbles describe the retroactive effect of the bubble movement on the flow of the melt. Also, the coagulation of bubbles by unterhiedlich fast buoyancy speeds is not yet detected. With the moment method these restrictions do not exist. It only comes to its validity limit if the volume fraction of the bubble phase clearly exceeds 10%.

Basic Adoption of the method is that itself always a bubble size spectrum according to a sets unimodal logarithmic normal distribution. In case of Aerosols agrees with this assumption very well with experimental findings, as the above cited works show. Also for bubble-containing liquid flows could this assumption has already been confirmed experimentally, see [5]. This Approach is only inadequate where in an area with on average huge Bladder relies on strong new formation of blisters. This happens in the way that the Number of newly formed very small bubbles of the order of magnitude the already existing much larger bubbles is, see rather a bimodal bubble size distribution to adjust. Occurs in the processes to be described one such Case up, check whether the moment model is a bimodal logarithmic normal distribution for the Bubble size spectrum can be extended. in principle But it is also possible in a unimodal distribution function that constantly new Bubbles form, even if the mean bubble diameter clear greater than that is the newly formed bubbles.

betrachtet. Eine solche Kugel ist oben links, in Abbildung 1 skizziert. r_K muß so gewählt werden können, daß für eine systemtypische makroskopische Abmessung L und die Zahl der Blasen n_K in K gilt:

In a fluid F, a bubble phase B is suspended. For averaging, a sphere K of radius r _K and volume

considered. Such a sphere is sketched at the top left, in Figure 1. r _K must be able to be chosen so that for a system-typical macroscopic dimension L and the number of bubbles n _K in K:

ist. α bezechnet darin die Zahl der Moleküle in der Blase.The bubbles are assumed to be spherical so that their volume is equal

is. α is the number of molecules in the bubble.

eingeführt. Damit definieren wir den Volumenbruch der Blasen mit α Molekülen als

q_αβ ist der Ortsvektor des Mittelpunkts einer mit α Molekülen. Der zweite Index β ist einzuführen, da es in der Kugel K mehrere Blasen mit α Molekülen an unterschiedlichen Orten und mit unterschiedlichen Gechwindigkeiten geben kann.For further calculations, the function becomes

introduced. This is how we define the volume fraction of bubbles with α molecules as

q _αβ is the position vector of the center of one with α molecules. The second index β is to be introduced because in the sphere K there can be several bubbles with α molecules at different locations and at different speeds.

The volume fraction of the entire bubble phase is then

It can easily be shown that

• a)
  
  V_K,F ist der ganze vom Fluid in der Kugel K eingenommene Raum. Für diesen Mittelwert können die Materialgesetze des reinen Fluids angesetzt werden.
• b) 〈Ψ〉s(x,t) = (1 – ε(x,t))〈Ψ〉(x,t) (9)〈Ψ〉_s ist der auf das Schmelzevolumen bezogene Mittelwert. Es handelt sich somit um eine Partialgröße des Fluids in der Schmelze, etwa der Partialdichte.
• c)
  
  Für diesen Mittelwert gelten wieder die Marerialgesetze des Blaseninhalts.
• d)
  
  Dieser Mittelwert kann wiederum als eine Partialgröße der Blasenphase angesehen werden.
• e)
  
  Auch Für diesen Mittelwert gelten die Marerialgesetze des Blaseninhalts.
• f)
  
  Hier handelt es sich wieder um eine Partialgröße der Blasenphase. Auch hier kann leicht gezeigt werden, daß alt
• 
  Bei der Formulieriung von Bilanzgleichungen sind noch zwei weitere Mittelwerte nützlich, nämlich
• g)
  
• h)
  

ρ_F und ρ_B sind dabei die Materialdichten des reinen Fluids und der reinen Blase.For the different phases and reference volumes, different averages for a be sweet size Ψ defines:

• a)
  
  V _{K, F} is the entire space occupied by the fluid in the sphere K. For this mean, the material laws of the pure fluid can be applied.
• b) <Ψ> s (x, t) = (1 - ε (x, t)) <Ψ> (x, t) (9) <Ψ> _s is the mean value related to the melt volume. It is thus a partial size of the fluid in the melt, such as the partial density.
• c)
  
  For this average, the Marerialgesetze of bladder content apply again.
• d)
  
  This mean can again be considered as a partial size of the bubble phase.
• e)
  
  Also for this average, the Marerialgesetze the bubble content apply.
• f)
  
  This is again a partial size of the bubble phase. Again, it can be easily shown that old
• 
  When formulating balance sheet equations, two other averages are useful, namely
• G)
  
• H)
  

ρ _F and ρ _B are the material densities of the pure fluid and the pure bubble.

n_K ist der Normaleneinheitsvektor auf der Oberfläche der Kugel K. Φ stellt die Zufuhr über die Systemoberfläe dar (z.B. Spannungstensor bei Ψ = ν_F). z ist die Zufuhr ins Innere von K (z.B. Schwerkraft bei Ψ = ν_F). ζ schließlich ist die Produktion aufgrund der Wechselwirkung zwischen den beiden Phasen.The ball K is an open system with fixed surface. A balance equation for a general size Ψ of the fluid phase then has the general form

n _K is the normal unit vector on the surface of the sphere K. Φ represents the input over the system surface (eg stress tensor at Ψ = ν _F ). z is the inflow into the interior of K (eg gravity at Ψ = ν _F ). ζ finally, the production is due to the interaction between the two phases.

To transform the interal form of the balance equation into the differential form, we first establish a relationship between the δ function and the Π function. The δ function is defined to hold

We define another Π function as

A comparison of equations (18) and (19) with equation (4) shows that the identity Π 1 (| x - x∫ |) - Π (x - x∫) (20) applies. For the Cartesian coordinates of the gradient of Π ₁ then applies

This can also be formulated in vector notation as V K ∇ x Π 1 (| x - x∫ |) = δ (| x - x '| - r K n) K (22)

This allows the general balance equation (17) to be transformed

The mean value definition in equation (8) is the general balance equation derived from it for regular points in differential form

In preparation for the derivation of a general balance equation for the bubble phase we form first

q ._αβ ist die Geschwindikeit dcr Blase αβ. Damit können wir bilden

That can be reworded too

q. _αβ is the velocity of the bubble αβ. We can form that

Where m is. _α defined as

ρ_B ist die Dichte und q . die Geschwindigkeit der Blasenphase. Die beiden Terme auf der linken Seite der Gleichung stellen die lokale zeitliche Änderung und den konvektiven Transport von Ψ. Der erste Term auf der rechten Seite stellt den Austausch zwischen den beiden Phasen dar. Der zweite Term auf der rechten Seite stellt Fluß und Produktion von Ψ in da Blasenphase dar. Er ist zu ermitteln aus der Betrachtung des einzelnen Blse.With the help of the mean value definition in equation (10), this equation can be transformed into the form of a balance equation

ρ _B is the density and q. the speed of the bubble phase. The two terms on the left side of the equation represent the local time change and the convective transport of Ψ. The first term on the right represents the exchange between the two phases. The second term on the right represents flow and production of Ψ in the bubble phase. It is to be determined from the consideration of the single bubble.

Equations (24) and (27) formulate balance equations for any size Ψ for both phases. In this form, all sizes depend on both location and time. In the following considerations, this dependency is no longer explicitly stated. Only in case of deviations from it is pointed out. The balance equations are formulated in such a way that the individual terms for both phases are based on the constant volume V _K. For the conservation quantities mass, momentum and energy then the production terms have to be negative.

For the mass balance, the general parameters of the general balance equation for the fluid phases are interpreted as

The mass balance for the fluid phase is then according to equation (24)

Equation (27) gives the mass balance for the bubble phase phase in the mold

As already noted above for the general case, the production terms of the two phases must be negatively equal, so that

The summation of the two mass balances then provides

dann erhalten wir zum einen nach Gleichung ((31) eine Massenbilanz der üblichen Form

Define density and velocity of the melt as

then, on the one hand, according to equation ((31), we obtain a mass balance of the usual form

The fact that ρ _{S is} the density of the melt results from the fact that ρ _S V _{K is} actually the total mass in the sphere K.

g ist die spezifische Schwerkraft. Detailliertere Kenntnisse über den Produktionsterm ζ_I erhalten wir aus der Betrachtung der Blasenphase.For the momentum balance of the fluid phase, the general parameters of the general balance equation are interpreted as

g is the specific gravity. More detailed knowledge about the production term ζ _{I is} obtained from the consideration of the bubble phase.

With the help of equation (24) we obtain the momentum balance first in the form

The general size Ψ of the general balance equation for the bubble phase becomes Ψ = q in the case of the momentum balance. interpreted. From equation (27) we obtain for the balance equation

auszuwerten. m_αq .._αβ ist gleich Masse × Beschleunigung einer Blase mit Radius r_α. Dann können wir schreiben mαq ..αβ = mαg + Fαβ (37) For further evaluation, the term is now

evaluate. m _α q .. _αβ is equal to mass × acceleration of a bubble with radius r _α . Then we can write m α q .. αβ = m α g + F αβ (37)

F _αβ is the force exerted by the fluid on the bladder surface. Multiplication by Π (x - q _αβ ) and summation over β

die auf das Blasenvoumen bezogene Oberflächenkraft. Nach Gleichung (10) erhalten wir daraus für die Mittelwerte

That's it

the surface force related to bladder volume. According to equation (10) we obtain from this for the mean values

According to equation (14) we obtain after summation over α

We can finally transform equation (36)

Since the momentum is also a conserved quantity, the second and the third term on the right side, the production terms, must be negative equal to the production term of the momentum balance of the fluid phase. By equation (35) follows

Summation of the two momentum balances (35) and (42) yields first with the equations (32) and (33)

The convective terms require detailed consideration. According to the mean definitions (12) and (13), we first approximate

The relative velocity between the two phases is defined as

According to the definitions (12), (13), (32) and (33) can be derived therefrom

In it, Y _{B is} the mass fraction of the bubble phase in the melt

With the help of approximation (45) the canvective terms can be transformed

We can transform the momentum balance (44) with it

Let's define the stress tensor of the melt as t S = (1 - ε) <t F > - ρ S Y B (1 - Y B ) ωω (51) then the momentum balance gets the usual shape

μ_F ist die Viskasität des flüssigen Glases und I der Einheitstensor. Damit erhält der Spannungstensor für die Schmelze insgesamt die Gestalt

A momentum balance of this form is released by Fluent. It must therefore be ensured that the equation of matter for the stress tensor is of the form of the equation (51). For <t _F > the matarial equation of a compressible Newtonian fluid can be used, ie

μ _F is the viscosity of the liquid glass and I the unit tensor. This gives the stress tensor for the melt as a whole the shape

To simplify this expression, let's consider the size estimates of the quantities that occur. The characteristic velocity U, length L, viscosity μ _{F, o} , gas density ρ _{B, o} , density of the melt ρ _{F, o} and volume fraction of the gas phase ε ₀ have the values in a typical glass flow

A key size for simplifications is here mass breakage of the bubble phase Y _B. For its magnitude applies

For typical glass melts, it is of the same order of magnitude as u _S , so that O (ω) = O (ν S ) = U (57)

Then we can write to equation (47) ₁ in a good approximation ν F ≈ ν S (58)

In the friction angle τ _{S of} the stress tensor, which is the branch term on the right side of Equation (54), we can replace ν _F by ν _S , and we write

Let us now estimate the magnitude order of the last term on the right side of equation (54) versus the magnitude of the friction component τ _S.

We first form this

So is O (ρ S Y B (1 - Y B ) ωω) «O (τ S (62) so that we can neglect the last term in equation (54). The material equation for the stress tensor of the melt then reduces

gelöst. μ_S ist die Viskosität der Schmelze, für die wir nach obigen Überlegungen setzen μS = (1 – ε)μF (65) In detail, in Fluent, a momentum balance of the shape becomes

solved. μ _S is the viscosity of the melt, for which we set according to the above considerations μ S = (1 - ε) μ F (65)

F is in fluent at freely programmable source term, for which we set equations (63) and (64) F = ∇ (εp) (66)

With equations (64), (65) and (66) provide all the information with Fluent the momentum balance for to dissolve a melt, contains the bubbles of a volume fraction ε.

φ_F ist der Wärmeflußvektor des flüssigen Glases. Mit Hilfe von Gleichung (24) erhalten wir die Energiebilanz zunächst in der Form

For the energy balance of the fluid phase, the general parameters of the general balance equation are interpreted as

φ _F is the heat flow vector of the liquid glass. With the help of equation (24) we first get the energy balance in the form

λ_F ist die Wärmeleitfähigkat des flüssigen Glases, J_ϑ der Diffusionsfluß der im flüssigen Glas gelösten Gasspezies ϑ und h_ϑ die zugehörige spezifische Enthalpie. Die generelle Größe Ψ der allgemeinen Bilanzgleichung für die Blasenphase wird im Fall der Energiebilanz als Ψ = e_B interpretiert. Aus Gleichung (27) erhalten wir damit für die Bilanzgleichung

Taking into account the gas diffusion in the liquid glass can be used as a material law for the heat flux vector

λ _F is the heat conductivity of the liquid glass, J _{θ is} the diffusion flux of the gas species dissolved in the liquid glass θ and h _θ the associated specific enthalpy. The general variable Ψ of the general balance equation for the bubble phase is interpreted as Ψ = e _B in the case of the energy balance. From equation (27) we obtain for the balance equation

auszuwerten. Vernachlässigen wir den Einfluß der Reibungskräfte an der Blasenoberfläche auf den Energiehaushalt der Blasen, dann können wir die Energiebilanz für die einzelne Blase formulieren als

For further evaluation, the term is now

evaluate. If we ignore the influence of frictional forces on the bubble surface on the energy balance of the bubbles, then we can formulate the energy balance for the individual bubble as

der dortige Wärmeflußvektor und

die dortige Geschwindigkeit der Blasenoberfläche, für die gilt

Where n _{α, β are} the normal unit vector on the bubble surface,

the local heat flux vector and

the local speed of the bubble surface, for which applies

λ_α ist die Wärmeleitfähigkeit des Gasgemischs in der Blase und κ_α die Wärmeübergangszahl zwische Blaseninnerem un umgebenden flüssigem Glas. Kann sowohl (ΔT)_αβ auf der Blasenoberfläche als auch ∇T im Blasenvolumen als konstant angesehen werden, läßt sich Gleichung (70) umformen zu

We now approximately assume that the temperature field in the bubble of radius τ _α to one (ΔT) _{αβ is} identical to that of the surrounding fluid. As a material equation for the heat flux vector at the bladder surface we therefore use

λ _α is the thermal conductivity of the gas mixture in the bubble and κ _{α is} the heat transfer coefficient between the bubbles inside and surrounding liquid glass. If both (ΔT) _αβ on the bubble surface and ∇T in the bubble volume can be considered as constant, equation (70) can be transformed

ν_α ist dabei definiert als

wobei V_ref ein beliebiges Referenzvolume ist. Nach Summation über α erhalten wir weiter

Now according to equation (14) we form the mean

ν _α is defined as

where V _{ref is} any reference volume. After summation over α we get further

ist der Mittelwert der Divergenz des Wärmeflüsktors in den Blasen. Er ist im wesentlichen bestimmt durch den Verlauf des Temperaturgradienten, der – wie obenbereits angemerkt – in Blasen und Fluid als gleich angenommen wird. Dann können wir setzen

The term

is the mean of the divergence of the heat flow in the bubbles. It is essentially determined by the course of the temperature gradient, which, as noted above, is assumed to be the same in bubbles and fluid. Then we can bet

The first term on the right represents the Flussterm, the second another source term. For the energy balance of the bubble phases we obtain the equations (69), (75) and (76)

The last term on the right represents the supply to the interior, the other terms therefore the production. We can identify the production term of the fluid phase as

wenn die folgenden Materialgesetze eingehalten werden.By summing the energy balances of the two phases, we obtain a balance equation of the shape

if the following material laws are complied with.

eine spezifische Wärmekapazität der Blase

eine spezifische Wärmekapazitätdifferenz zwischen Blase und flüssigem Glas

eine Wärmeleitfähigkeit des Schmelze λ_S,0, ein Diffusionskoeffizient D_ϑ,o, eine Gasspezieskonzentration im flüssigen Glas

eine Temperatur T₀ und einen Druck p₀ ein, die die Werte haben

We can simplify material laws even by size-scale comparisons. For this purpose we introduce a specific heat capacity of the liquid glass as further characteristic quantities

a specific heat capacity of the bladder

a specific heat capacity difference between bubble and liquid glass

a thermal conductivity of the melt λ _{S, 0} , a diffusion coefficient D _{θ, o} , a gas species concentration in the liquid glass

a temperature T ₀ and a pressure p ₀ , which have the values

während der zweite Term nach den Gleichungen (55) und (56) von der Größenordnung

ist. Wir können daher die Materialgleichung (80) annähern durch

The first two terms in equation (80) are of the order of magnitude

during the second term according to equations (55) and (56) of the order of magnitude

is. We can therefore approximate the material equation (80)

The specific heat capacity of the melt is then

Here, c _{p, F} and c _{p, B are} the specific heat capacities of liquid glass and bubble. The three terms of the heat flux vector on the right side of equation (81) are of order of magnitude

The leading term is therefore the first of the three. The other two are so much smaller that they can be neglected. The material equation (81) then simplifies φ S = -Λ S ∇T (89)

For the species balance of gases dissolved in liquid glass, the general parameters of the general balance equation for bathing phases are interpreted as

Y_F,ϑ ist des Massenbruch der Gasspezies ϑ innerhalb des flüssigen Glases. Ihr Massenbruch in der Schmelze ist dann YS,ϑ = (1 – YB)ŶF,ϑ (91) The species balance is then according to equation (24)

Y _{F, θ} is the mass fraction of the gas species θ within the liquid glass. Your mass melt break is then Y S, θ = (1 - Y B ) Y? F, θ (91)

With equations (4), (15) and (47) ₁ we obtain the balance equation for this mass fraction

The diffusion flux _vector J _θ is formally given the shape

Since, according to equation (54), Y _{B is} on the order of 10 ^-5 , the second term is also smaller by about this order of magnitude. We therefore approach the material law for the diffusion flux vector J θ = -Ρ S D θ ∇y S, θ (94)

The second term on the second page of equation (92) is actually a term of flux and should be added to the diffusion flux vector. However, the material law used by Fluent as standard for the diffusion flux vector is in the form of Eq. (94). It then only has to be entered according to the material law for D _θ . The second term on the right hand side is then treated as a source term in terms of programming.

The actual source is composed of a contribution of the chemical reaction within the liquid glass and a contribution of the mass exchange between the phases. We therefore start formally

sind übliche Ansätze zu wählen, etwa dem Ratengesetz nach Arrhenius. Eine explizite Form des Phasenaustauschterms

wird bei der Betrachtung des Blasenwachstums bestimmt.For the chemical reaction term

are usual approaches to choose, such as the installment law by Arrhenius. An explicit form of the phase change term

is determined by considering the bubble growth.

The basis for the treatment of the bubble phase is the aerosol-dynamic equation in the farm

die mittlere Ge schwindigkeit von Blasen mit α Molekülen,

der Diffusionskoeffizient der Blasen mit α Molekülen, J die Massenproduktionsdichte von Blasenkeimen, α_* die Zahl der Moleküle in einem Blasenkeim,

seine Masse, β_αγ die Stoßhäufigkeit von Blasen mit α Molekülen und Blasen mit γ Molekülen, η die Koagulationswahrscheinlichkeit zweier stoßender Blasen,

die Übergangshäufigkeit von Molekülen aus dem flüssigen Glas in eine die Blese mit α Molekülen und

die Übergangshäufigkeit für den umgekehrten Vorgang.It describes the behavior of a phase dispersed in a fluid. The dispersed particles can be of different sizes, collide with each other and coagulate and exchange pale with the surrounding fluid. N _α denote the number density of bubbles with α molecules,

the mean velocity of bubbles with α molecules,

the diffusion coefficient of the bubbles with α molecules, J the mass production density of bubble nuclei, α _* the number of molecules in a bubble germ,

its mass, β _αγ the collision frequency of bubbles with α molecules and bubbles with γ molecules, η the coagulation probability of two colliding bubbles,

the transition frequency of molecules from the liquid glass into a bubble with α molecules and

the transition frequency for the reverse process.

Für die mittlere Geschwinigkeit von Blasen mit Radius r_α können wir daher ansetzen

The bases move relative to the liquid glass at a buoyancy rate ν _{A, α} and a diffusion flux

For the mean velocity of bubbles with radius r _α , we can therefore begin

With the help of equation (47) ₂ and a diffusion approach we obtain from this

die Auftriebsgeschwindigkeit einer Blase mit α Molekülen. Die aerosoldynamische Gleichung kann damit geschrieben werden als

It is

the buoyancy rate of a bubble with α molecules. The aerosol dynamic equation can be written as

In the calculation of the buoyancy speed, we simplistically assume that buoyancy and friction forces are always in equilibrium. The buoyancy force on a bubble with radius r _α

Therein σ is the surface tension, R the general gas constant and M _B is the spatially and time variable molecular weight of the gas mixture in the bubble. A dependency of the molar mass on the bubble radius is not taken into account in the model developed here. The expression corresponding to the Stokes friction for a round bubble with radius r _α

In mechanical equilibrium, the sum of these two forces is zero. From this, the buoyancy rate relative to the fluid phase can then be calculated

A ist die Avogadro-Zahl und D_ϑ* der Diffusionskoeffizient derjenigen Gasspezies im flüssigen Glas, aus der die Blasenkeime gebildet werden. Das Materialgesetz für den Diffusionskoefizienten ist so konstruiert, daß es für größere Radien den Diffusionskoeffizienten von Brownschen Teilchen liefert. Im Grenzwert zu einer hypothetischen einmolekularen Blase, also für sehr kleine Blasen, liefert es den Gasdiffusionskoeffizienten D_ϑ*. r₁ ist der Radius einer solchen hypothetischen einmolekularen Blase. Wenn man berücksichtigt, daß bei sehr kleinen Blasen der Innendruck praktisch ausschließlich durch die Oberflächenspannung bestimmt wird, kann man aus der thermischen Zustandsgleichung für ideale Gase die Gleichung

ableiten.The diffusion behavior of bubbles usually corresponds to that of Brownian particles. We call this area a continuum area. In the case of very small bubbles, for example in the nucleation phase, the diffusion behavior corresponds more closely to that of the gases dissolved in the liquid glass. We are talking about the molecular field here. As a material equation for the radii-dependent diffusion coefficient, which is valid for both ranges, we therefore assume

A is the Avogadro number and D _{θ * is} the diffusion coefficient of those gas species in the liquid glass from which the bubble nuclei are formed. The material law for the diffusion coefficient is designed to provide the diffusion coefficient of Brownian particles for larger radii. At the limit of a hypothetical single-molecular bubble, ie for very small bubbles, it provides the gas diffusion coefficient D _{θ *} . r ₁ is the radius of such a hypothetical single-molecular bubble. Taking into account that for very small bubbles, the internal pressure is determined almost exclusively by the surface tension, one can from the equation of thermal equation for ideal gases, the equation

derived.

Collisions between two bubbles are caused, on the one hand, by the Brownian motion of the bubbles, which also underlie the diffusion behavior. On the other hand, bubbles collide due to different lift speeds. For the shock frequency, we therefore start first

ein Materialgesetz, das in der Aerosolphysik dem Kontiuumsbereich entspricht, siehe [4], und schreiben

In analogy to the above diffusion approach choose for us

a material law that corresponds to the contraction region in aerosol physics, see [4], and write

bewegt, durchfegt während eines Zeitintervalls Δt ein Kollisionsvolumen von der GrößeA bubble with radius r _α - short α-bubble -, due to different buoyancy velocities relative to a bubble with radius r _γ - short γ-bubble - with the speed

moves, sweeps a collision volume of the size during a time interval .DELTA.t

If the center of a γ-bubble is within this volume, a collision of the two bubbles takes place. The expected value for the collision of a given α-bubble with a γ-bubble is then

Then the number density of such bursts is equal to Δt Ż α, γ Δt = N α N γ V. α, γ Δt (109)

und wir können identifizierenOn the other hand, this shock number density is the same

and we can identify

ableiten. Damit erhalten wir schließlich für die zugehörige Stoßhäufigkeit

η ist die Wahrscheinlichkeit, daß zwei Blasen, die kollidieren, auch koagulieren, also sich vereinigen. Diese Koagulationswahrscheinlichkeit hängt im wesentlichen von von der Kontaktdauer der beiden stoßenden Blasen, damit von ihrer Relativgeschwindigkeit und damit unter anderem von ihren Radien, wie Gleichung (112) zeigt. Aufgrund der hohen Viskosität von Glasschmelzen ist die Relativgeschwindigkeit maximal in ener Größenordnung von 10^–2 m/s. Es wird angenommen, daß sich daraus eine hinreichend lange Kontaktdauer für stoßende Blasen ergibt und die Koagulationswahrscheinlichkeit für alle Radien nahe eins liegt. Insbesondere wird die Koagulationswahrscheinlichkeit als unabhängig von den Radien der stoßenden Blasen angenommen.With the help of equation (102) we can use the expression for the relative velocity

derived. This finally gives us the associated shock frequency

η is the probability that two bubbles that collide also coagulate, thus unite. This coagulation probability depends essentially on the contact duration of the two blasting bubbles, and thus on their relative velocity and thus, inter alia, on their radii, as shown in equation (112). Due to the high viscosity of glass melts, the relative speed is maximum in the order of 10 ^-2 m / s. It is assumed that this results in a sufficiently long contact duration for bursting bubbles and that the coagulation probability for all radii is close to one. In particular, the coagulation probability is assumed to be independent of the radii of the puffing bubbles.

gehen wir davon aus, daß alle Gasmoleküle des Blase, die auf die Blasenoberfläche auftreffen, in die flüssige Phasen übergehen. Die Zahl der pro Flächen- und Zeiteinheit auftreffenden Moleküle der Spezies ϑ ist nach einfachen Argumenten der kinetischen Gastheorie gleich

To calculate the transition frequency

Let us assume that all gas molecules of the bladder, which hit the bladder surface, pass into the liquid phases. The number of molecules of the species θ impinging per unit area and time is the same according to simple arguments of the kinetic theory of gases

Where X _{B, θ is} the mole fraction of the gas species θ in the bubble and M _{θ is} the species molar mass. The transition probability then results from multiplication with the bubble surface and summation across all species, and we get

dann sind die Übergangswahrscheinlichkeiten in beiden Richtungen gleich, wenn gilt

Let us denote the saturation vapor pressure of the species θ in the glass with p _θ (T) and the molar fraction of the gas species θ in the liquid glass at the bubble surface

then the transition probabilities are the same in both directions, if true

Then follows for the transition probability of the reverse process

ist die Konzentration in Mol/m³ der Gasspezies ϑ im flüssigen Glas an der Blasenoberfläche. Der Ansatz für die Übergangswahrscheinlichkeit lautet dann

For example, as shown in [6], we can substitute the saturation vapor pressure p _θ (T) by the solubility L _θ according to

is the concentration in mol / m ^{3 of} the gas species θ in the liquid glass at the bubble surface. The approach to the transition probability is then

aus dem Inneren des flüssigen Glases weit weg von einer Blase ersetzt werden, da diese bei kleiner Blasenkonzentration ungefähr gleich der mittleren Konzentration der Gasspezies ϑ im flüssigen Glas ist. Der Stofftransport

in Mol/s eines gelösten Gases auf dem Inneren des flüssigen Glases an die Blasenoberfläche wird näherungsweise beschrieben durch den Ansatz

Sh_α,ϑ ist die Sherwood-Zahl und Pe_α,ϑ die Peclet-Zahl. Der damit beschriebene Stoffstrom zur Blasenoberfläche durchdringe diese auch, so daß gilt

However, the concentrations at the bladder surface are unknown and still have to be determined by the concentration

be replaced from the inside of the liquid glass far away from a bubble, since this is at a small bubble concentration approximately equal to the average concentration of the gas species θ in the liquid glass. The substance transport

in mol / s of a dissolved gas on the interior of the liquid glass to the bubble surface is approximately described by the approach

Sh _{α, θ} is the Sherwood number and Pe _{α, θ is} the Peclet number. The flow of material to the bubble surface thus described also penetrate this, so that applies

The above considerations apply not only to the sum of the gas species but also to each species. In the equations above, only summation over all species may need to be omitted. With equations (115), (119), (120) and (121), we then obtain for the concentration since gas species θ at the bubble surface

auf, die wir nun in guter Näherung bestimmen können zu

In the later analysis, the difference emerges

which we can now determine to a good approximation

τ_g ist der geometrische Mittelwert des Blasenradius. In dieser Form können wir nachfolgende Integrale nicht explizit berechnen. Für ein Vereinfachung nutzen wir aus, daß gilt

Using Equation (102), we can approximate the radius dependence of the Sherwood number

τ _g is the geometric mean of the bubble radius. In this form, we can not explicitly calculate subsequent integrals. For simplicity, we take advantage of that

ab:

From this we derive as approximate equation for the Sherwood number for

from:

N(r_α)dr_α ist die Zahldichte von Blasen mit Radien zwischen r_α und r_α + dr_α. Die Verteilungsfunktion enthält drei Parameter, nämlich

• • die Gesamtzahldiehte der Blasen N_B,
• • den mittleren Blasenradius r_g und
• • die Standardabweichung ω.

The basis of the moment model is the assumption of a logarithmic normal distribution for the bubble size spectrum according to

N (r _α ) dr _α is the number density of bubbles with radii between r _α and r _α + dr _α . The distribution function contains three parameters, namely

• The total number of bubbles N _B ,
• • the mean bubble radius r _g and
• • the standard deviation ω.

All parameters are functions of place and time. Both the total density of the bubbles as well as their average size and the width of their size distribution are thus temporally and spatially variable. In determining these three parameters, the procedure is to derive balance equations for three moments of the distribution function from the aerosol dynamic equation. Such moment is defined as

According to the equation of thermal equation for ideal gases, there is a relationship between the number of molecules α and the radius r _α

By equations (127), (128), and (129), we obtain for the 0th, 1st, and 2nd moments

zu lösen. Dabei bedeuten

The parameter N _B is identical to the 0th moment. To determine the other two parameters r _g and ω is the non-linear system of equations

to solve. Mean

With the specification of such a distribution function and the introduction of Moments can now various sizes and Drawings in their explicit dependency on the parameters and moments of the distribution function are determined.

i) volume fraction of the bubble phase

Using the distribution function, the volume fraction of bubbles with radius r _α can be expressed as ε α = V α N α

und kann nun mit Gleichung (127) berechnet werden zu

The volume fraction of the entire bubble phase is then the same

and can now be calculated with equation (127)

ii) badgers and mass fracture the bubble phase

an, dann kann diese Größe als der in Gleichung (11) definierte Mittelwert betrachtet werden. Nach den Gleichungen (13) und (14) kann die Partialdichte der gesamten Blasenphase in der Schmelze berechnet werden als

Give the density of a bubble with radius r _α according to the equation of thermal equation for ideal gases

then this quantity may be considered as the mean defined in equation (11). According to equations (13) and (14), the partial density of the entire bubble phase in the melt can be calculated as

Then

The mass fraction of the bubble phase is the same

ii) density of the melt

According to Equation (32) we obtain from this as a material equation for the density of the melt ρs = (1 - ε) ⟨ρF⟩ + M B M 1 (138)

For ⟨ρ _F⟩ a material equation is to be selected from the liquid glass of each variety.

iii) relative speed between the two phases

die mittlere Geschwindigkeit der gesamten Blasenphase

berechnet werden gemäß

According to equations (11) and (13) can be calculated from the mean velocity of the bubbles with radius

the mean velocity of the entire bubble phase

be calculated according to

According to equation (98), it follows for the mean bubble velocity

und ν_S eliminieren und erhalten als Bestimmungsgleichung für die Relativgeschwindigkeit zwichen den beiden Phasen

Using Equation (47) ₂ , we can derive the velocities from them

and ν _S eliminate and obtain as a determination equation for the relative velocity between the two phases

Equations (102), (103), and (127) allow us to approximate the intergals

there is

iv) moment balance equations

ν_A,l ist der Auftriebstransportterm, für den gilt

By multiplying the aerodynamic equation (99) by (α / A) ^l , summation by α and transition to the continuous spectrum we obtain

ν _{A, l} is the lift transport term for which applies

The productio terms for l ≥ 1 mean

These Equations also apply to the case l = 0 with the exception of the last product iosterm, which then disappears. there that is, it is to the balance equation for the number density of the bubbles. The last term only affects the size of the bubbles, but not on their numbers.

Equations (102), (127), and 129) allow us to calculate the lift transport term

bis zum linearen Glied in α um α_g = α(r_g), also um den Wert von α, bei dem die Verteilungsfunktion Ihr Maximum hat. Wir nähern also

To transform the diffusion term into an expression explicitly in M _l , we develop the diffusion coefficient

to the linear term in α around α _g = α (r _g ), ie around the value of α, at which the distribution function has its maximum. So we are approaching

there means

With The help of equations (143) and (129) is given in more explicit terms shape

With this we can bring the moment balance into the form:

ist, kann dann ebenfalls eindeutig bestimmt werden.However, the balance equation for the lth moment now also contains the (l + 1) th moment. In the balance equation for the second moment, therefore, the third moment appears, for which again no balance equation is to be solved. If the first three moments M _0, M ₁ and M _{2 are} known, the three parameters of the distribution function can be calculated unambiguously, as has been shown. The 3rd moment equals the equations (128) and (129)

is, can then also be clearly determined.

The production terms can be calculated with the help of some approximations

und

sind ebenfalls identisch mit dem jeweiligen Summand in den früheren Ausdrücken, wo über ϑ summiert wurde. Aus der Annahme der Radienunabhängigkeit der Gaszusammemsetzung der Blasen folgt: αϑ = XB,ϑα ⇒ Nαϑ = Nα ⇒ NBϑ = NB, rgϑ = rg, ωϑ = ω (172) Diffusion coefficient and shock frequency are identical to the previously introduced terms, since the underlying phenomena depend only on the bubble size, but not on their composition. The transitional probabilities

and

are also identical to the respective summand in the earlier expressions where θ was summed. From the assumption of the radial independence of the gas composition of the bubbles follows: α θ = X B, θ α ⇒ N αθ = N α ⇒ N Bθ = N B , r gθ = r G , ω θ = ω (172)

For the 1st Moment applies

The associated balance equation for the 1-th moment can then be determined in an analogous manner to above as

sind bereits in den Gleichungen (162) bis (169) angegeben. Auf der rechten Seite der Momentenbilanzgleichung taucht auch das 2-te Moment auf. Dafür analog zum 1-ten

Formulas for calculating the production term

are already given in equations (162) to (169). On the right side of the moment balance equation also the second moment appears. For analogous to the 1-th

Since we can determine the molar fractions of the gas species in the bubble phase by means of Equation (173) ₂ , we obtain the bubbles for the mean molar mass

vii) production term of Species balance equation of the liquid phase

die Stoffmenge der Gasspezies ϑ, die pro Zeit- und Volumeneinheit von der Flüssigkeits- in die Blasenphase übergeht. Dieser Term ist dann negativ gleich dem Phasenaustauschterm

der Speziesbilanzgleichung der flüssigen Phase. Für den Quellterm der Speziesbilanzgleichung in Gleichung (95) können wir dann schreiben

M _{1, θ} is equal to the amount of substance of the gas species θ contained in the bubbles relative to the melt volume. Then

the amount of substance of the gas species θ, which passes from the liquid to the bubble phase per unit time and volume. This term is then negatively equal to the phase change term

the species balance equation of the liquid phase. For the source term of the species balance equation in Equation (95) we can then write

Since the complete balance system of equations and the corresponding material equations are each numbered the same as in the first occurrence in the previous text:

While the Description of the refining process with the previously available has standing models, the invention overcomes the application of the moment model these limitations. With previously available Models can not take into account the formation of new bubbles In addition, there is no feedback the bubble movement, in particular their buoyancy, on the flow of Melt, and the coagulation or coalescence of the bubble, for example, due to different buoyancy rates, is not considered.

On the other hand can with the invention except overcoming the above-mentioned disadvantages also considered further effects become. Thus, the diffusive gas transport from the liquid glass into the gas bubbles and vice versa, incorporating the buoyancy rate considered become. In addition, the size distribution the bubbles are included in the model as a logarithmic normal distribution.

Of Further, the gas composition in the bubble phase becomes timed and locally disbanded calculated. The gas density in the bubble phase is dependent of bubble size, position - ie in particular the depth - the Bubble as well as from the gas composition considered.

In addition, will the relative movement of the bubble phase to the liquid glass due to diffusive Transport in the form of Brownian motion and lift taken into account.

Further can by the invention when applying the moment model changes the fabric sizes, in particular the density, the thermal conductivity and the specific heat capacity of the melt considered the bubble phase become.

in the With regard to the coagulation or coalescence of the bubbles can also influence the diffusion movement and different buoyancy rates within the bubble phase considered become.

When using the method according to the invention, the following points should also be noted:
In one place and at a time, all bubbles have the same gas composition regardless of their size. This then applies in particular to the bladder germs, although these contain predominantly the gas species which is present in a highly supersaturated concentration and thus causes the formation of new nuclei. This simplification is justified if we proceed from the following idea:
A gas species is so oversaturated that spontaneously forms a bubble nuclei. In the immediate vicinity of the bubble nuclei, the concentration of these dissolved gas species is initially greatly reduced. The remaining gas species can now diffuse into the bubble nuclei more intensely. In this way, a middle gas composition quickly sets in even in very small bubbles.

The coagulant probability lies essentially on the contact duration of the two blasting bubbles, thus on their relative velocity and thus among other things on their radii. However, the present model assumes a constant coagulation probability. Due to the high viscosity of glass melts, the relative speed is on the order of 10 ^-2 m / s. It is assumed that this results in a sufficiently long contact data for congested blisters and the coagulation probability for all radii is close to one. In deriving the expression used here for the collision frequency by Brownian motion, this assumption was implicitly assumed. Because of the small diffusion coefficients of the dissolved gases and the high viscosity of the melt, it can be assumed that coagulation by Brownian motion does not play a major role anyway.

Of Further may depend on the assumption of a logarithmically normal distribution Bubble size distribution to deviate. In this case, the term is for the bubble size distribution adjust accordingly. This can be the case, for example. if existing big Bubbles use strong nucleation, giving a bimodal bubble size distribution with a. Maximum at the mean size of the newly formed small Bubbles and another one at the medium size of the big bubbles sets. This Effect is particularly with regard to the size-dependent coagulation and coalescence processes to take into account.

There for the Bubble phase no own equation of motion is solved, accelerating effects not considered in the bubble movement. Due to the high viscosity However, this effect can be considered negligible in glass melts be accepted small.

• • die Neubildung von Blasenkeimen J
• • die Masse
  
  und die Stoffmenge
  
  von Blasenkeimen
• • Koagulationswahrscheinlichkeit η zweier stoßender Blasen
• • Randbedingungen für die Momentenbilanzgleichungen an Blasdüsen

For the specific application, therefore, only the following values must be determined:

• • the regeneration of blister nuclei J
• • the mass
  
  and the amount of substance
  
  of bladder germs
• • Coagulation probability η of two colliding bubbles
• • boundary conditions for the moment balance equations on tuyeres

thereupon can the invention for the design of the plant will be used, the predictive quality of the simulation Behavior of gas bubbles in high-viscosity molten glass clearly improves and thus the goodness the design increased becomes.

In summary let yourself so say that a bubble continuum model is provided with that in a melt the interaction between gas bubbles and liquid Glass completely can be described.

• • Auftrieb durch Verringerung der Dichte der Schmelze als Ganze,
• • Massenaustausch zwischen den beiden Phasen und
• • Veränderung von Materialparametern (z.B. Viskosität). Grundannahmen des Modells sind, dass
• • die Blasenphase durch Mittelwertbildung als Kontinuumsphase behandelt und
• • das Blasengrößenspektrum stets durch eine logarithmische Normalverteilung beschrieben werden kann

The possible interactions are

• Buoyancy by reducing the density of the melt as a whole,
• • Mass exchange between the two phases and
• • Change of material parameters (eg viscosity). Basic assumptions of the model are that
• • the bubble phase treated by averaging as a continuum phase and
• • The bubble size spectrum can always be described by a logarithmic normal distribution

Experimental investigations confirm such a size distribution especially for glass melts. The size distribution of the bubbles then obeys the expression

• • Gesamtzahldichte der Blasen N_B,
• • mittlerer Blasenradius r_g und
• • Breite der Größenverteilung (Standardabweichung) ω,

die sich zeitlich und räumlich in einer Glasströmung ändern können. Um die Änderungen dieser drei Parameter beschreiben zu können, wurden Bilanzgleichungen für drei Momente der Größenverteilung hergeleitet, die mit Potenzen der in den Blasen enthaltenen Stoffmenge gebildet werden. Ein solches Moment wird somit definiert als

It contains the three parameters

• Total number density of bubbles N _B ,
• • average bubble radius r _g and
• Width of the size distribution (standard deviation) ω,

which can change in time and space in a glass flow. To be able to describe the changes of these three parameters, balance equations were derived for three moments of size distribution, which are formed with powers of the amount of substance contained in the bubbles. Such a moment is thus defined as

v →_s ist der mittlere Geschwindigkeitsvektor der Schmelze, V →_Auf,l der Auftriebsgeschwindig-keitsvektor des Moments l und

der Diffusionskoffizient der Blasen aufgrund ihrer Brownschen Bewegung. P_l ist ein Quellterm, in dem Effekte berücksichtigt werden können wie Massenaustausch zwischen den beiden Phasen. und Koagulation etwa aufgrund unterschiedlicher Auftriebsgeschwindigkeiten unterschiedlich großer Blasen. Bei bekannten Werten der Momente M_l könne nicht nur die Parameter N_B, r_g und ω bestimmt werden, sondern auch der Volumenanteil der Blasenphase e. Diese geht dann in die Dichte ρ_s, Viskosität μ_s und Wärmeleitfähigkeit λ_s der Schmelze ein über die Beziehungen

ρ_F, μ_F und λ_F sind die entsprechenden Werte des blasenfreien flüssigen Glases. Die Gleichungen sind so formuliert, dass sie mit konventionellen Strömungsprogrammen gekoppelt werden können. Insbesondere die CFD-Software Fluent bietet mit Programmierschnittstellen die Möglichkeit, ein solches Modell zu implementieren.Where A is the Avogadro number and a is the number of molecules. The balance equations have the form

v → _s is the average velocity vector of the melt, V → U _{, l} the buoyancy velocity vector of the moment l and

the diffusion coefficient of the bubbles due to their Brownian motion. P _l is a source term in which effects can be considered, such as mass exchange between the two phases. and coagulation due to different buoyancy rates of different sized bubbles. With known values of the moments M ₁ not only the parameters N _B , r _g and ω can be determined, but also the volume fraction of the bubble phase e. This then enters into the density ρ _s , viscosity μ _s and thermal conductivity λ _{s of} the melt via the relationships

ρ _F , μ _F and λ _F are the corresponding values of the bubble-free liquid glass. The equations are formulated so that they can be coupled with conventional flow programs. In particular, the CFD software Fluent offers with programming interfaces the possibility to implement such a model.

List of symbols and formula symbols

literature

• [1] K.W. Lee, H. Chen, and J.A. Gieseke. Log-normally preserving size distribution for brownian coagulation in the free-molecule regime. Aerosol Sci. and Technol., 3: 53-62, 1984.
• [2] E.R. Whitby, P.H. McMurry, U. Shankar, and F.S. Binkowski. Modal aerosol dynamics modeling. EPO reports, pages 600 / 3-91 / 020, 1,991th
• [3] K.W. Lee; Y. J. LEE. and D.S. Han. The log-normal size distribution theory for brownian coagulation in the low knudsen number regime. J. Colloid Interface Sci., 188: 486-492, 1997.
• [4] A. Bensberg, P. Roth, R. Brink, and L. Lange. Modeling of Particle evolution in aerosal reactors with coflowing gaseous reactants. AlChE Journal, 45 (14): 2097-2106, 1999th
• [5] Cs. Sisak and Z. Ormós. Investigation of two-phase fluidised system iii. characterization of size distributions of bubbles produced by venturi-type gas distributor. Hung. J. Ind. Chem., 13: 503, 1985th
• [6] A. Bensberg, C. Berndhäuser, K. Jochem, D. Köpsel, F.-T. Lentes, W. Muschik, T. Pfeifer, J. Piroth, and G. Weidmann. Modeling of Chemical Purification with the TNO model LN 72/99.

Claims (3)

Method for designing a plant for melting and / or lautering a liquid, in particular a glass melt containing the steps a) of Set a target value for at least one dimension and / or at least one operating parameter the plant, b) setting a first set of dimensions the plant, c) setting a first set of operating parameters the plant and the substance parameter of the melt to be purified, d) simulating the melting and / or refining process with simulation the behavior of bubbles in the melt, e) of comparison the simulation result with the corresponding target value, f) the setting of a further adapted based on the comparison result Set of dimensions of the plant, g) setting a based on the comparison result adapted further set of operating parameters the plant and the substance parameter of the melt to be purified, H) repeating steps d) to g) until the simulation result within the scope of a predefined confidence interval around the corresponding one Goal is to simulate the behavior of the bubbles in the melt is carried out based on the moment method. Plant for melting and / or refining glass with dimensions, which according to one Method according to claim 1 with respect to each other with regard to at least a target size of the dimensions and / or the operating parameters are matched. Product, in particular glass product, manufactured in a system according to claim Second