Author's personal copy

Transcription

1 International Journal of Thermal Sciences 60 (2012) 142e152 Contents lists available at SciVerse ScienceDirect International Journal of Thermal Sciences journal homepage: A simplified mathematical model of glass melt convection in a cold crucible induction melter Sugilal Gopalakrishnan a,b, André Thess a, * a Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology, P.O. Box , Ilmenau, Germany b Nuclear Recycle Group, Bhabha Atomic Research Centre, Mumbai , India article info abstract Article history: Received 5 July 2011 Received in revised form 16 March 2012 Accepted 5 June 2012 Available online 13 July 2012 Keywords: Convection Vitrification Glass melting Glass melt homogenization Cold crucible induction melting Cold crucible induction melting is emerging as a promising technology for immobilizing nuclear waste in glass matrices. Since the transport properties such as viscosity and electrical conductivity of molten glass exhibit strong temperature dependencies, performance of the cold crucible induction melter is highly sensitive to the thermal field prevailing in the molten glass pool. A simplified mathematical model was developed to numerically investigate the impact of molten glass properties such as viscosity, thermal conductivity and electrical conductivity on the performance of a cold crucible induction melter meant for high level radioactive waste vitrification. The present investigation of the thermal convection using the simplified model confirms the findings of previous studies. Numerical simulation of thermal convection shows that low electrical conductivity and high viscosity existing in the cooler parts of the molten glass bath can lead to poor electromagnetic induction and localized heating in the present melter-inductor configuration. The stable thermal stratification due to bottom cooling leads to a relatively stagnant fluid layer in the lower part of the glass melt. These features of the thermal convection can limit the heat transfer and mixing in the glass melt which in turn can affect both the melting capacity and product homogeneity adversely. Mechanical stirring using a water-cooled stirrer can overcome these limitations. The present study confirms that mechanical stirring of the glass melt can enhance the electromagnetic induction through thermal homogenization. Mechanical mixing eliminates the relatively stagnant fluid layer observed in thermal convection. Distribution of induced power, temperature and velocity predicted by the simplified model exhibit matching characteristics of the results obtained by other investigators. The simplified model reduces the computational load substantially as it eliminates complex electromagnetic computations. Ó 2012 Elsevier Masson SAS. All rights reserved. 1. Introduction Cold crucible induction melting technology (CCIM) for nuclear waste vitrification is under active development in several countries like France [1], Russia [2], United States [3], South Korea [4] and India [5]. With the new melting technology, more waste can be loaded into glass matrix; more aluminium can be contained in high melting glass; waste feed can be melted faster and the melter can have a longer service life [6]. The CCIM technology can substantially reduce vitrified waste volume and thereby, cut down the cost of interim storage and final disposal significantly. In cold crucible induction glass melting, glass is directly heated by electromagnetic induction employing a segmented crucible which is manufactured from contiguous segments forming a cylindrical * Corresponding author. Tel.: þ ; fax: þ address: thess@tu-ilmenau.de (A. Thess). volume, but separated by a thin layer of electrically insulating material [7]. An inductor surrounding the segmented crucible induces eddy currents in each segment creating an electromagnetic field inside the crucible. Such an arrangement is equivalent to two aircored transformers e one formed between the induction coil and the crucible segments and the second formed between the segments and the charge. Thus, the segmented crucible acts as a field concentrator, indirectly reducing the gap between the induction coil and the charge [8]. In order to avoid high temperature glass corrosion of the metallic crucible, internal cooling of the segments is provided. This cooling produces a solidified glass layer, which acts as a protection against glass corrosion along the melter inner wall (see Fig. 1). High temperature availability without substantial corrosion makes the CCIM a promising technology for nuclear waste vitrification. Physical phenomena taking place inside the cold crucible induction melter are extremely complex and strongly interrelated. Apart from other electrical parameters, the direct induction heating of the glass melt is strongly governed by its physical properties /$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved.

2 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e Nomenclature B magnetic flux density, T c P specific heat, J kg -1 K -1 g acceleration due to gravity, m s -2 h heat transfer coefficient, W m -2 K -1 J electric current density, A m 2 k thermal conductivity, W m -1 K -1 L characteristic length, m p pressure, N m -2 Q volumetric heat source, J m -3 r radius given by x and y coordinates, m t time, s T temperature, K V velocity in stationary frame, m s -1 V r relative velocity in rotating frame, m s -1 x, y, z Cartesian coordinates, m Greek symbols b coefficient of volumetric expansion, K 1 g surface tension coefficient, N m -1 K -1 ε emissivity h viscosity, kg m 1 s e1 m magnetic permeability, H m -1 r density, kg m -3 s electrical conductivity, S m -1 u angular frequency of current, rad s -1 U angular velocity, rad s -1 V gradient operator, m 1 D difference of a quantity Subscript 0 reference value Transport properties such as dynamic viscosity and electrical conductivity of molten glass exhibit strong temperature dependencies. Since a wide temperature range prevails during the cold crucible induction melting, these properties vary in the glass melt by several orders of magnitude [9,10]. Since the performance of the cold crucible induction melter is highly sensitive to the glass properties, a thorough understanding of the behavior of the glass melt is essential for optimizing the melter design with regard to its industrial implementation. The scope of an experimental investigation to understand the thermal convection in the molten glass is limited due to the hostile environment inside the melter. On the contrary, a numerical investigation permits to analyze the effect of various transport properties on the convective behavior of the molten glass. Fig. 1. Schematic representation of cold crucible induction glass melting: (a) top view (b) sectional view. Until now, only a few studies have been carried out to investigate thermal convection of molten glass in a cold crucible induction melter. In 1996, Schiff et al. carried out a numerical simulation of thermal convection of glass melt in a cylindrical induction furnace with water-cooled walls and bottom [11]. They modeled electromagnetic, hydrodynamic and thermal aspects but the electromagnetic modeling neglected the effect of water cooling at the boundaries. An analytical solution of the Bessel equation was used to obtain the magnetic field intensity in the glass melt. Later Hawkes used FIDAP software to simulate melting of glass in a cold crucible induction heated melter [12]. The electromagnetic modeling was carried out using the electromagnetic vector potential. A power controller was implemented in the model to maintain a desired power in the glass melt. In their studies, both Schiff and Hawkes employed axisymmetric models. Jacoutot et al. established a multi-physics coupling procedure to numerically simulate thermal convection in molten glass heated by induction [13]. They employed a 2-D, axisymmetric power source, which was obtained by solving the electromagnetic induction equations using the integral method. The thermal and hydrodynamic aspects were solved using the FLUENT software. They obtained a steady, 3-D, laminar flow for glass melt with Raz10 5 and Prz10 3. Roach et al. used ANSYS-based finite element model to predict operational behavior for specific materials (borosilicate and other glass melts) and system configurations [10]. Sauvage et al. numerically investigated the thermoconvective flow of molten glass heated by direct induction in a cold crucible [14]. Their results showed a steady, axisymmetrical flow for a power of 45 kw in the glass bath with a height of m and radius of 0.25 m. However, loss of axisymmetry was observed for a power of 55 kw owing to thermoconvective instabilities. These instabilities appeared as convection cells formed by the rising hot glass similar to Benard- Marangoni cells. The number and size of the convection cells varied with time. As a continuation of this study, Burn et al. carried out a full 3D strong iterative coupling between induction, thermal and hydrodynamic phenomena using the two commercial software FLUX and FLUENT [15]. The commercial software FLUX is used to solve induction equations and FLUENT software to solve the NaviereStokes and thermal equations. Two different meshes are used in such simulations as the mesh refinement requirements are different for the induction and hydrodynamic phenomena. Interpolations between the two meshes are required to couple the finite element (FLUX) and the finite volume (FLUENT) based software.

3 144 The aim of the present study is to develop a simplified mathematical model for the calculation of induction heating of glass melt in a cold crucible induction melter and use this model to investigate the convective behavior of glass melts for different melter configurations. Two different glasses are used for this purpose. Two melter configurations are also considered: the first deals with natural convection in an un-stirred glass melt and the second deals with the forced convection in a mechanically stirred glass melt. The simplified mathematical model is expected to reduce the computational load significantly. 2. Modeling approach The electromagnetic induction and induced power density distribution in the glass melt in a cold crucible induction melter strongly depend on configuration of the melter-inductor system and physical properties of the glass melt. The segmented crucible considered for the present studycomprises of 56 water-cooled stainless steel tubes (1 inch, BWG 12) arranged in a circular array to hold a molten glass pool of 0.50 m diameter [8].Thefloor of the melter is made of a 0.01 m thick stainless steel plate and has a diameter of 0.6 m. A transistor-based induction heating power supply system with an operating frequency of 200 khz is used for melting the glass inside the segmented crucible. A single-turn copper coil with an inside diameter of 0.6 m and height of 0.15 m is located 0.05 m above the water-cooled floor to reduce the induction heating of the metallic floor. A simplified electromagnetic model is developed for the calculation of induction heating in the glass melt inside the segmented crucible. This simplified model is used for studying both free and forced convection of glass melt. In both problems, the main goal is to predict the spatial distribution of the velocity and temperature fields for different glasses Electromagnetic model The magnetic Reynolds number (Re m ¼ m 0 s 0 V 0 L,wherem 0 is the magnetic permeability, s 0 is the electrical conductivity at the reference temperature T 0, V 0, is the characteristic velocity scale and L is the characteristic length [16]) and the ratio of the length scale of the furnace and wave length of the oscillating field (N f ¼ um 0 s 0 L 2,whereu is the frequency of the electric field) are typically small in glass melting operations, for which the electromagnetic field can be decoupled from the flow field [17]. In this case, the induction equations describing the electromagnetic quantities in the glass melt can be expressed in terms of the electric scalar potential and the magnetic vector potential. Using the scalar and vector potentials, the induced current density and the corresponding induction heat in the glass melt can be calculated. Since the main goal of this study is to develop a simplified mathematical model for the calculation of induction heating of glass melt in a cold crucible induction melter, an approximated volumetric heat source can be used to represent the induction heat in the glass melt. Based on the general solution of the field equation for an infinitely long cylinder excited by a close-coupled axial coil (solenoid), the current density induced in the azimuthal direction can be expressed in its dimensionless form as Jðx; yþ ¼ J 1ðk rþ (1) J 1 ðkr me Þ p where r ð¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx 2 þ y 2 ÞÞ is the radius; k (¼(1 i)/d me ) is the complex wave number; and J 1 is the 1st order Bessel function of the first kind; r me and d me are the radius and skin depth of the glass melt, respectively [18]. For a given glass, the skin depth is governed by its electrical conductivity and the frequency of the coil current. The skin depth, d me,isdefined by the Eq. (2), S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e152 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d me ¼ 1 ; pm 0 s 0 f (2) where m 0 is the relative magnetic permeability (m 0 ¼ 1 for glass melt), s 0 is the electrical conductivity at T 0 and f is the frequency. When r me [d me, the induced current decreases exponentially with depth. Researchers have determined empirically that for cylindrical melters the diameter should be approximately a factor of 3.8 times the skin depth at the normal frequency of operation [19]. Assuming d me ¼ r me /2 for the present CCIM application, induced current density can be approximated using Eq. (1) as Jðx; yþ ¼ J 1ðð2 2iÞðr=r me ÞÞ : (3) J 1 ð2 2iÞ For a finite cylinder excited by a short coil, the end effects lead to a distortion of the electromagnetic field and induced current distribution becomes sensitive to the melter-inductor configuration. The induced current is also influenced by the the electrical conductivity of the medium. In normal operation of cold crucible induction melting, the total power induced in the glass melt is regulated at a required level. Based on the above aspects, a nondimensional, volumetric induction heat source in the glass melt can be approximated as - Q J ðx; y; zþ ¼jJðx; yþj 2 f z f T f c ; (4) where J(x,y) is the dimensionless current density given by the Eq. (3). The factors f z, f T and f c are to account for end effects in a given melter-inductor configuration, effect of temperature dependent electrical conductivity of the glass melt and power control by induction heating system, respectively. The factor f z in Eq. (4) is selected as a function of z-coordinate (m) to account for the end effects on the induced heat in the glass melt for the given melter-inductor configuration. A general expression for the f z factor is intuitively chosen for the given coil configuration and can be represented by the equation: f z ¼ 1 a þ sin 1 þ a bpz L c : (5) In Eq. (5), parameters a, b and c can be selected to suit a given melter-inductor configuration. For example, a ¼ b ¼ c ¼ 0 represents an infinitely long cylinder excited by a close-coupled axial coil. The parameter a is used to account for the presence of the metallic floor which drastically reduces the induction heat in the glass melt near the floor. The parameters b and c of the sinusoidal function are used to reflect the impact of the relative vertical position of the inductor with respect to the glass melt. For the present melter-coil configuration, a ¼ 0.1, b ¼ 0.8 and c ¼ 0.25 are used. Eq. (4) takes electrical conductivity of the glass melt into account through the factor f T which is expressed in terms of the ratio of the electrical conductivity s(t) at local temperature T and the electrical conductivity s(t 0 ) at a reference temperature T 0 as given by Eq. (6). f T ¼ sðtþ sðt 0 Þ The factor f c used to maintain a constant power in the glass melt is given by the equation: f c ¼ Q 0 Z ; (7) jjðx; yþj 2 f z f T dv V (6)

4 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e where Q 0 is the power required to be maintained in the glass melt with a volume V. The quantity in the denominator of the right hand side of the Eq. (7) represents the total heat induced in the glass melt at a given point of time. The factor f c is computed at the end of every iteration and it is used in the next iteration to scale the local heat generation such that the total power in the melt is maintained at the desired level, Q Hydrodynamic and thermal models Hydrodynamic and thermal calculations carried out under the present study are based on the following assumptions: The flow is assumed to be laminar, 3D and unsteady. The fluid is assumed to be incompressible and Newtonian. The Boussinesq assumption is valid to model buoyancy. Lorentz forces in the glass melt are neglected. Viscous heating is negligible compared to induction heating. Two different melter configurations were considered for the present study. The first case was chosen to study natural convection in an un-stirred glass melt and the second to study the forced convection in a mechanically stirred glass melt. Hydrodynamic and thermal models used for these cases are presented here Natural convection The computational domain considered for the natural convection in an un-stirred glass melt is a cylindrical molten glass bath with a radius of 0.25 m and a height of 0.2 m (see Fig. 2(a)). The complete model domain is selected for the numerical simulation of the natural convection case as non-axisymmetric thermoconvective instabilities can occur under certain operating conditions. The molten glass can be considered as a Newtonian fluid [17]. Using the Boussinesq assumption, laminar thermal convection in the glass melt can be described by the unsteady NaviereStokes equation, r vv h vt þrv$vv ¼ VpþV$ hðtþ VV þvv Ti þr 0 gbðt T 0 Þ; (8) together with the incompressibility condition, V$V ¼ 0; (9) where V is the velocity vector; p is the pressure; g is the acceleration due to gravity; r, b, h(t) are the density, coefficient of thermal expansion and viscosity, respectively. The symbolic notations VV and VV T represent vv i /vx j and vv j /vx i, respectively [20]. A ratio of the electromagnetic body force to the gravitational body force, (J 0 B 0 /r 0 gb(t T 0 ), where J 0 and B 0 are the characteristic electric current density and magnetic flux density, respectively), can be used to compare the relative strength of the two body forces in the glass melt. Since J 0 B 0 =r 0 gbðt T 0 Þ1 in poorly electricallyconducting, nonmagnetic and highly viscous glass melts, the electromagnetic body forces can be neglected. The hydrodynamic boundary condition at the free surface can be defined by equating the viscous strain and the thermal strain as given by vu vt h ¼ g : (10) vz surface vr surface In Eq. (10), g ¼ vs s /vt, where s s is the surface tension. For molten glass,g ¼ 10 4 Nm 1 K 1 can be used [14]. This boundary condition describes surface-tension-driven (Marangoni) Fig. 2. Computational domains used for numerical simulations: (a) natural convection in molten glass with radius ¼ 250 mm and height ¼ 200 mm (full model with cells); (b) forced convection in molten glass with paddle mixer (1/4th model with cells). convection. It can be implemented in FLUENT by specifying the g value at the free surface using the Marangoni stress option of the wall boundary condition [21]. A stationary wall with no-slip boundary condition is used for rest of the boundaries. The temperature T in the glass melt can be obtained by solving the energy conservation equation, vt rc P vt þ rc PV$VT ¼ V$ðkðTÞVTÞþQ J ; (11) where the term Q J is the volumetric heat source in the glass melt on account of induction heating derived in the previous section (Eq. (4)). The viscous heating is very small compared to the Joule heat generation in the glass and is neglected as the Brinkman number, Br ¼ h 0 V 2 0 /k 0 DT, satisfies the condition Br << 1. In Eq. (11), heat transfer by internal radiation is accounted using an equivalent thermal conductivity k, which includes both thermal conductivity and internal radiation effects.

5 146 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e152 At the free surface, a mixed thermal boundary condition described by Eq. (12) is used with an emissivity ε ¼ 0:9, surface heat transfer coefficient h s ¼ 20 W/m 2 K and surrounding temperatures T a ¼ 303 K and T ar ¼ 423 K. qðr; z me Þ¼εs sb T s 4 T4 ar þ h s ðt s T a Þ (12) Identical boundary conditions were used in similar investigations carried out by previous researchers [13e15,22]. The radiation term in the Eq. (12) is valid for the present case as the boundary surface is flat (shape factor F 1e1 ¼ 0) and surrounded by a relatively large enclosure (A 1 /A 2 ¼ 0) [23]. The value of the glass emissivity depends on the specific glass under consideration. Additionally, this parameter can also dependent on the glass temperature. Hawkes used a constant value for glass emissivity ðε ¼ 0:85Þ in his simulation [12] and Burn et al. used ε ¼ 0:9 in their study [15]. The combined convection and external radiation boundary conditions can be implemented in FLUENT using the mixed option of the wall thermal boundary condition [21]. An isothermal boundary condition with T ¼ 800 K is used to represent the water-cooled walls and floor. This condition assumes that the thermal gradient on account of the wall cooling is essentially confined to a thin layer of solidified glass [24] Forced convection Flow instabilities and symmetry breaking bifurcations are generally observed in fluids under stirring at high Reynolds numbers. For the present forced convection problem, the Reynolds number (rl 2 U/h) is O(1) on account of high viscosity and low angular velocity. In forced convection of glass melt in a cold crucible induction melter, flow instabilities and symmetry breaking bifurcations are not reported by previous researchers who used complete flow domain for their computations [22]. Therefore, an 1/4 th model was used in the present study to reduce the computational efforts. The computational domain comprising of 1/4 th of the molten glass bath (having a radius of 0.25 m and a height of 0.2 m) with a paddle type stirrer is used for the numerical simulation of forced convection (see Fig. 2(b)). The diameter, width and thickness of the paddle are 0.3 m, 0.05 m and 0.01 m, respectively. The shaft has a diameter of 0.05 m and is positioned m above the melter floor. Since there are no stators or baffles, it is possible to simulate forced convection in the glass bath using a coordinate system that moves with the rotating equipment. The equations of motion can be solved in a rotating frame, and the acceleration of the fluid can be augmented by additional terms in the momentum equation [22]. The incompressibility condition, NaviereStokes, and thermal equations can be written for the rotating frame as V$V r ¼ 0; (13) r vv r vt þ rv$ðv rv r Þþrð2UV r þ U U rþ ¼ Vp þ V$½hðTÞðVV r þ VV T r ÞŠ þ r 0gbðT T 0 Þ; (14) rc P vt vt þ rc PV$ðV r TÞ¼V$ðkðTÞVTÞþQ J ; (15) where V r is the relative velocity in the rotating frame and U is the angular velocity of the rotating frame relative to the stationary reference frame. The fluid velocities can be transformed from the stationary frame to the rotating frame using the relation V r ¼ V þ U r, where r is the position vector of an arbitrary point in the computational domain from the origin of the rotating frame. The momentum equation contains two additional acceleration terms: the Coriolis acceleration ð2u V r Þ and the centripetal Table 1 Physical properties of the Glass A (temperature T in Kelvin). Property Unit Test glass Density (r) kg/m Coefficient of expansion (b) 1/K Surface tension coefficient (g) N/m K Specific heat (c P ) J/kg K 1535 Viscosity (h) Pa s 0.82 þ e ( T/41.194) Electrical conductivity (s) S/m T Thermal conductivity (k) W/m K T 2 þ 0.013T 5.25 acceleration ðu U rþ. The present forced convection problem is simulated with FLUENT using single rotating frame method. In this method, a separate fluid region containing the impeller and moving with the given angular velocity is specified [21]. The hydrodynamic boundary conditions for the free surface of the glass bath and the crucible walls are implemented using stationary wall with zero tangential shear stress and stationary wall with no-slip, respectively. A moving wall boundary condition with a given absolute speed (in rpm) is used for the impeller. The symmetry planes are specified by periodic boundary condition of rotational type and can be implemented by setting the flux at the outlet boundary equal to the flux entering the inlet boundary. An isothermal boundary condition with T ¼ 800 K is used to represent the water-cooled mechanical stirrer. Rest of the thermal boundaries are treated similar to those employed for the natural convection study Numerical solution The model equations were solved using the commercial computational fluid dynamics software FLUENT. The Cartesian coordinate system was used for the FLUENT simulations. The model equations were solved using a second order accurate finite volume discretization scheme. The SIMPLE algorithm was employed for the pressure-velocity coupling. Two different glass compositions were used to investigate the effect of glass properties on the convective behavior of glass melt in a cold crucible induction melter. Glass A is characterized by low viscosity and high electrical conductivity whereas Glass B has relatively higher viscosity and lower electrical conductivity. The physical properties of the Glass A and the Glass B are given in Table 1 and Table 2, respectively. The physical properties of the Glass B given in Table 2 are based on the values used by Jacoutot et al. [22]. Temperature-fitted dependence functions are used for viscosity, electrical conductivity and thermal conductivity. The specific heat is assumed to be constant, which is valid for the temperature range 800e1800 K [22]. In order to investigate the effect of viscosity and electrical conductivity on the convective behavior of the molten glass, all other properties were kept the same for both the glasses. Table 2 Physical properties of the Glass B (temperature T in Kelvin). Property Unit Waste glass Density (r) kg/m Coefficient of expansion (b) 1/K Surface tension coefficient (g) N/m K Specific heat (c P ) J/kg K 1535 Viscosity (h) Pa s e ( T/67.295) Electrical conductivity (s) S/m T Thermal conductivity (k) (natural convection) W/ mk T 2 þ 0.013T 5.25 Thermal conductivity (k) (forced convection) W/ mk T 2 þ 0.011T 5.187

6 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e The commercial, finite element based software COMSOL Multiphysics was used to validate the simplified mathematical model developed in the present study. Using COMSOL Multiphysics, an axisymmetric model with strong coupling between induction, thermal and hydrodynamic phenomena was solved for the present melter-inductor configuration and Glass B. The COMSOL Multiphysics predicted a temperature range of Ke K in the glass melt for a total induction power of 25 kw. The corresponding temperature range obtained using FLUENT with the simplified mathematical model is 773 Ke K, which is in very good agreement with the COMSOL Multiphysics solution. Three-dimensional simulations were carried out with a total power of 45 kw induced in the glass bath to compare the effect of glass melt properties on thermal convection; and numerical investigations were conducted with a paddle type stirrer rotating at 10 rpm in the glass melt with a total power of 70 kw to study the effect of mechanical stirring. Both Glass A and Glass B were used for this purpose. In the forced convection study, Glass B was assigned a relatively lower thermal conductivity to compare the effect of thermal conductivity on convective behavior of stirred bath (see Table 2). 3. Results Natural convection in glass melt is expected to be strongly governed by the physical properties of the molten glass as well as the Joule heating power induced in the melt. Natural convection in both Glass A and Glass B were numerically investigated. These results are presented in section 3.1. In order to compare the effectiveness of natural convection, complimentary investigations were carried out for a mechanically stirred glass bath with a paddle type stirrer. The forced convection results are presented in section Natural convection In order to compare the effect of glass melt properties on the thermal convection, numerical simulations were carried out for Glass A and Glass B with a total power of 45 kw induced in the glass bath. Both Glass A and Glass B exhibit steady, axisymmetrical solutions for this power. Fig. 3(a) and (b) show the volumetric power induced in the vertical section of the glass bath for Glass A and Glass B, respectively. Fig. 3(a) and (b) illustrate that strong induction heating occurs towards the upper periphery of the glass melt where the magnetic flux density and electrical conductivity are relatively higher. Peak values of the power density obtained for Glass A and Glass B are W/m 3 and W/m 3, respectively. Temperature distribution and flow field in the vertical section of the glass bath corresponding to Fig. 3(a) and (b) are presented in Figs. 4 and 5, respectively. In Fig. 5, the left halves show velocity vector plots and the right halves show the streamline plots. The computed flow fields consist of a primary circulation with a downward motion at the pool center and a secondary circulation with a downward motion at the melter wall (see Fig. 5). The hot glass near the melter wall moves upward and bulk of it is driven towards the center of the free surface of the glass melt because of the buoyancy and thermocapillary forces. Remaining part of the flow is driven towards the melter wall where it sinks due to relatively higher density on account of water-cooling of melter walls. The stable thermal stratification in the lower portion due to water cooling of melter floor results in a relatively stagnant fluid layer towards the bottom of the glass pool. The simulation results showed a temperature range of 800e1392 K and velocity range of 0e m/s in Glass A and 798e1643 K and 0e m/s in Glass B. The reduction of the flow field strength in Glass B is due to its relatively higher viscosity. The weak flow field in turn has resulted in a higher Fig. 3. Joule heating power density (W/m 3 ) distribution in the vertical section of the glass bath for a total induced power of 45 kw. Panels shown are for (a) Glass A and (b) Glass B.

7 148 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e152 Fig. 4. Temperature (K) distribution in the vertical section of the glass bath for a total induced power of 45 kw. Panels shown are for (a) Glass A and (b) Glass B. Fig. 5. Flow field in the vertical section of the glass bath for a total induced power of 45 kw. Panels shown are for (a) Glass A and (b) Glass B. [left half: velocity (m/s) vector plot; right half: streamline plot].

8 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e peak melt temperature 1643 K in Glass B as against 1392 K in Glass A (see Fig. 4). The volumetric heat distribution, thermal and flow fields predicted by the simplified model exhibit matching characteristics of the results obtained with strong coupling between induction, thermal and hydrodynamic phenomena under identical operating conditions [15]. The power control factor f c for maintaining the power at 45 kw level is 1.08 for Glass A and 2.25 for Glass B. This is an indication of poor electromagnetic induction in Glass B. Thus, low electrical conductivity and high viscosity of molten glass can lead to poor electromagnetic induction and localized heating in the present melter-inductor configuration. The stable thermal stratification due to bottom cooling leads to a relatively stagnant fluid layer in the lower part of the glass melt. This numerical simulation of the thermal convection provided results consistent with findings of prior studies [13e15]. Induction heating is maximum in the upper part of the annular region near the walls of cylindrical glass melt. Water cooling of the melter walls and floor lead to cooler zones adjacent to the walls and bottom. Induction heating of the areas of cooler glass is reduced due to the lower electrical conductivity and higher viscosity of the cooler glass. These features of the localized heating and cooling, and limited thermal convection in the glass pool, can limit the heat transfer and mixing which in turn can affect both the melting capacity and product homogeneity adversely Forced convection Numerical simulation of the thermal convection confirms that low electrical conductivity and high viscosity of the glass melt can lead to poor electromagnetic induction and localized heating. Computed flow fields show the presence of a stagnant fluid layer in the lower part of the glass melt due to stable thermal stratification caused by bottom cooling of the cold crucible induction melter. Weak flow field and poor mixing under thermal convection can limit the melting capacity and lead to poor product quality. Mechanical stirring using a water-cooled stirrer can overcome these limitations [22]. In order to investigate the effect of mechanical stirring, numerical investigations were carried out with a paddle type stirrer rotating at 10 rpm in the glass melt with an induced power of 70 kw. The higher operating power is on account of the enhanced cooling requirements. Both Glass A and Glass B were used for investigating the effect of mechanical stirring. In order to assess the effect of thermal conductivity on convective behavior of stirred bath, Glass B was Fig. 6. Joule heating power density (W/m 3 ) distribution in the vertical mid-plane of the paddle in the stirred glass bath for a total induced power of 70 kw and paddle angular velocity of 10 rpm. Panels shown are: (a) Glass A; (b) Glass B. Fig. 7. Temperature (K) distribution in the vertical mid-plane of the paddle in the stirred glass bath for a total induced power of 70 kw and paddle angular velocity of 10 rpm. Panels shown are: (a) Glass A; (b) Glass B.

9 150 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e152 Fig. 8. Temperature (K) distribution in different horizontal planes in the stirred Glass B for a total induced power of 70 kw and paddle angular velocity of 10 rpm. The panels given are: (a) z ¼ m; (b) z ¼ 0.1 m; (c) z ¼ m; (d) z ¼ 0.2 m. Fig. 9. Particle trajectories of 20 points uniformly distributed in the mid-plane of the paddle between (x ¼ 0.24, y ¼ 0, z ¼ 0.005) and (x ¼ 0.24, y ¼ 0, z ¼ 0.195) in the mechanically stirred Glass A for a total induced power of 70 kw and paddle angular velocity of 10 rpm. assigned a relatively lower thermal conductivity (see Table 2). Fig. 6 compares the Joule heating power density induced in a vertical plane in the stirred bath of Glass A and Glass B. A comparison of Figs. 3 and 6 clearly indicates the effect of mechanical stirring on the induction heating of the glass melt. Fig. 6 confirms that forced convection employing a mechanical stirrer leads to relatively more uniform induction heating in comparison with thermal convection. Eqs. (4) and (6) describe the influence of local temperature on the volumetric heat source caused by electromagnetic induction. Thus, the effect of mechanical stirring on the Joule heating power density can be explained in terms of the temperature distribution in the glass melt. Fig. 7 compares the temperature distribution in the vertical mid-plane of the paddle in the stirred glass bath. Temperature distributions in different horizontal planes in the stirred bath of Glass B are presented in Fig. 8. Relatively higher viscosity and lower thermal conductivity of Glass B lead to higher maximum melt temperature compared to Glass A for the same induced power. A comparison of Figs. 4 and 7 clearly indicates that the mechanical stirring eliminates thermal stratification to a greater extent and improves the thermal homogeneity in the glass melt significantly. The volumetric heat distribution and thermal field predicted by the simplified model exhibit similar characteristics of the results obtained by Jacoutot et al. [22].

10 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e Table 3 Comparison of various quantities in Glass A and Glass B obtained for thermal convection in glass melt with a melt power of 45 kw. Physical quantity Unit Glass A Glass B Maximum temperature K Average temperature K Maximum velocity m/s Maximum power density W/m Average viscosity Pa s Average thermal conductivity W/m K Average electrical conductivity S/m Power control factor e Cooling losses % In order to visualize the flow pattern in the mechanically stirred glass bath, particle trajectories of 20 points uniformly distributed in the mid-plane of the paddle between (x ¼ 0.24, y ¼ 0, z ¼ 0.005) and (x ¼ 0.24, y ¼ 0, z ¼ 0.195) are shown in Fig. 9. The particle trajectories indicate that hot glass from pool surface is continuously pulled into the bottom interior by the stirrer. This stirring action is responsible for better heating at the lower part of the glass bath (compare Figs. (4) and (7)). The rotating paddle also eliminates stagnant fluid layer and thereby, enhances thermal and chemical homogenization of the glass melt. Various quantities numerically predicted for Glass A and Glass B under natural convection and mechanical stirring are compared in Table 3 and Table 4, respectively. Table 3 shows a substantially higher maximum melt temperature in Glass B for the same power although thermal conductivities of both the glasses are very close. This is on account of the weak thermal convection in Glass B due to its relatively higher viscosity and highly localized heating. A substantially high power control factor (f c ) shows a relatively poor electromagnetic induction and localized heating for Glass B. Unlike in thermal convection, mechanical stirring of the glass melt improves the electromagnetic induction through improved thermal homogenization. This can be confirmed by comparing the power control factors obtained for Glass A and Glass B. The power control factors for Glass A and Glass B are, respectively, 1.08 and 2.25 for natural convection and 1.2 and 1.36 for forced convection. These values indicate that the difference in power control factors for the two different glasses is reduced substantially as the natural convection is replaced by forced convection. The mechanical stirring substantially reduces the peak melt temperature keeping the bulk glass temperature unaffected, in Glass A. A deviation from this trend exhibited by Glass B is expected to be on account of the lower thermal conductivity used in the case of mechanically stirring (compare Tables 3 and 4). The thermal homogenization and product quality enhancement by mechanical stirring is possible only at the cost of additional cooling losses. The cooling losses under thermal convection for Glass A and Glass B are 58.41% and 52.88%, respectively (refer Table 4 Comparison of various quantities in Glass A and Glass B obtained for mechanically stirred glass melt with a melt power of 70 kw. Physical quantity Unit Glass A Glass B Maximum temperature K Average temperature K Maximum velocity m/s Maximum power density W/m Average viscosity Pa s Average thermal conductivity W/m K Average electrical conductivity S/m Power control factor e Cooling losses % Table 3). Corresponding cooling losses with mechanical stirring are 80.10% and 82.36%, respectively (see Table 4). 4. Conclusions A simplified mathematical model was developed to numerically investigate the impact of molten glass properties such as viscosity, thermal conductivity and electrical conductivity on the performance of a cold crucible induction melter meant for high level waste vitrification. The volumetric heat distribution, thermal and flow fields predicted by the simplified model exhibit matching characteristics of the results obtained by other researchers employing strong coupling between induction, thermal and hydrodynamic phenomena under identical operating conditions. Although complex electromagnetic computations can give more accurate results, the simplified model reduces the computational load substantially as it avoids numerical solution of electromagnetic induction equations. Two different glass compositions were used for the numerical simulations. A detailed study was carried out to characterize thermal convection of glass melt with total induced power of 45 kw. Present numerical simulation of the thermal convection employing a simplified mathematical model provided results consistent with findings of prior studies. Induction heating is maximum in the upper part of the annular region near the walls of cylindrical glass melt. Water cooling of the melter walls and floor lead too cooler zones adjacent to the walls and bottom. Bottom cooling in the cold crucible induction melter leads to a relatively stagnant fluid layer in the lower part of the glass melt due to stable thermal stratification. Induction heating of the areas of cooler glass is reduced due to the lower electrical conductivity and higher viscosity of the cooler glass. These features of the localized heating and cooling, and limited thermal convection in the glass pool, can limit the heat transfer and mixing which in turn can affect both the melting capacity and product homogeneity adversely. Mechanical stirring using a water-cooled stirrer can overcome the above mentioned limitations. Numerical investigations were carried out to simulate the glass melt conditions in presence of a paddle type stirrer rotating at 10 rpm in molten glass with a total induced power of 70 kw. The rotating paddle eliminates stagnant fluid layer and thereby, enhances thermal and chemical homogenization of the glass melt. The simplified model also confirms that forced convection in molten glass enhances the electromagnetic induction through improved thermal homogenization. Acknowledgements The authors gratefully acknowledge financial support for SG s stay at Ilmenau University of Technology provided by the German Research Foundation (Deutsche Forschungsgemeinschaft) in the framework of the Research Training Group (Graduiertenkolleg) Lorentz force velocimetry and Lorentz force eddy current testing. Part of the present work was prepared during this stay. References [1] R. Do. Quang, A. Jensen, A. Prod homme, R. Fatoux, J. Lacombe, Integrated pilot plant for a large cold crucible induction melter, in: Proceedings of the Waste Management 2002 Conference, Tucson, USA, [2] F.A. Lifanov, M.I. Ojovan, S.V. Stefanovsky, R. Burcl, Feasibility and expedience to vitrify NPP operational waste, in: Proceedings of the Waste Management 2003 Conference, Tucson, USA, [3] D. Gombert, J.R. Richardson, J.A. Roach, N.R. Soelberg, Cold crucible induction melter prototype at the Idaho national engineering and environmental laboratory, in: Proceedings of the Waste Management 2004 Conference, Tucson, USA, 2004.

11 152 S. Gopalakrishnan, A. Thess / International Journal of Thermal Sciences 60 (2012) 142e152 [4] C.W. Kim, J.K. Park, S.W. Shin, T.W. Hwang, J.H. Ha, M.J. Song, Vitrification of simulated LILW using induction cold crucible melter technology, in: Proceedings of the Waste Management 2006 Conference, Tucson, AZ, USA, [5] G. Sugilal, P.B.S. Sengar, Cold crucible induction melting technology for vitrification of high level waste: development and status in India, in: Proceedings of the Waste Management 2008 Conference, Phoenix, USA, [6] A Year of Successful Investments - Office of Engineering and Technology 2009 Annual Report, US Department of Energy, Washington DC, USA, [7] A. Jouan, R. Boen, S. Merlin, P. Roux, A warm heart in a cold body e melter technology for tomorrow, in: Proceedings of the Spectrum 96, International Topical Meeting on nuclear and Hazardous waste Management, Seattle, USA, [8] G. Sugilal, Experimental study of natural convection in a glass pool inside a cold crucible induction melter, International Journal of Thermal Sciences 47 (2008) 918e925. [9] J.A. Roach, J.G. Richardson, Technical development of new concepts for operation and control of cold crucible induction melters for vitrification of radioactive wastes, in: Proceedings of the Waste Management 2006 Conference, Tucson, AZ, USA, [10] J.A. Roach, D.B. Lopukh, A.P. Martynov, B.S. Polevodov, S.I. Chepluk, Advanced modeling of cold crucible induction melting for process control and optimization, in: Proceedings of the Waste Management 2008 Conference, Phoenix, USA, [11] V.K. Schiff, A.N. Zamyatin, A.A. Zhilin, Numerical simulation of thermal convection of a glass melt in a cylindrical induction furnace, Glass Science and Technology 69 (1996) 379e386. [12] G. Hawkes, Modelling a cold crucible induction heated melter, in: Proceedings of 2003 FIDAP/POLYFLOW User Group Meeting, Evanston Illionois, USA, [13] L. Jacoutot, P. Brun, A. Gagnoud, Y. Fautrelle, Numerical modelling of natural convection in molten glass heated by induction, Chemical Engineering and Processing 47 (2008) 449e455. [14] E. Sauvage, A. Gagnoud, Y. Fautrelle, P. Brun, J. Lacombe, Thermoconvective flow of molten glass heated by direct induction in a cold crucible, Magnetohydrodynamics 45 (2009) 535e542. [15] P. Burn, E. Sauvage, E. Chauvin, 3-D thermal, hydrodynamic and magnetic modelling of elaboration of glass by induction in cold crucible, in: Proceedings of the Waste Management 2009 Conference, Phoenix, USA, [16] P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, [17] M.K. Choudhary, A three dimensional mathematical model for flow and heat transfer in electrical glass furnaces, IEEE Transactions on Industry Applications IA 22 (1986) 912e921. [18] E.J. Davies, Conduction and Induction Heating, Peregrinus, London, [19] D. Venable, T.P. Kim, Radio Frequency Heating Fundamentals and Applications, Westinghouse Electric Corporation, [20] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley and Sons, [21] User s Guide, FLUENT 6.3 Documentation Fluent, Fluent, Lebanon, NH, [22] L. Jacoutot, Y. Fautrelle, A. Gagnoud, P. Brun, J. Lacombe, Numerical modeling of coupled phenomena in a mechanically stirred molten-glass bath heated by induction, Chemical Engineering Science 63 (2008) 2391e2401. [23] J.P. Holman, Heat Transfer, Mc Graw Hill, New York, [24] E. Sauvage, A. Gagnoud, Y. Fautrelle, P. Brun, J. Lacombe, Numerical simulation of glass flow heated by direct induction in a cold crucible, in: Proceedings of the 6th International Conference on Electromagnetic Processing of Materials, Dresden, Germany, 2009.