- Email: [email protected]
An electromagnetic and thermal analysis of a cold crucible melting
International Communications in Heat and Mass Transfer 32 (2005) 1325 – 1336 www.elsevier.com/locate/ichmt
An electromagnetic and thermal analysis of a cold crucible meltingB J.H. Song a,T, B.T. Min a, J.H. Kim a, H.W. Kim a, S.W. Hong a, S.H. Chung b a b
KAERI, 150 Dukjindong, Yusunggu, Daejeon, 305-353, South Korea KIST, 39-1 Hawolgokdong, Seongbuk-gu, Seoul, 136-791, South Korea Available online 22 August 2005
Abstract To investigate the performance of a cold crucible employed for the melting of corium, which is a mixture of UO2 and ZrO2, a computational analysis of the coupled electro-magnetic field, heat transfer, and fluid flow is carried out. Governing differential equations, basic numerical methods, computational model, and the results of the simulation of cold crucible melting are discussed. By comparing the analysis results with the experimental data, it is shown that the proposed computational model reasonably predicts the fundamental characteristics of a cold crucible melting. D 2005 Elsevier Ltd. All rights reserved. Keywords: Cold crucible melting; Phase change; Electro-magnetic field; Corium
1. Introduction One of the promising techniques for the melting of refractory material is the cold crucible melting [1–3]. The cold crucible meting is basically an inductive heating of electrically conducting melt by an alternating electromagnetic field. Heating is accomplished by ohmic losses caused by eddy currents induced in the melt. The skull formed between the very high-temperature melt and the cold crucible acts as both a self-container and an effective insulator. B
Communicated by K. Suzuki and S. Nishio. T Corresponding author. E-mail address: [email protected] (J.H. Song).
0735-1933/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2005.07.015
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While significant demands exist in the industry, the implementation of a cold crucible technique for a specific application encounters many engineering problems as it deals with very high temperature and complicated processes. And the experimental data for the performance of the cold crucible are rare. Therefore, a computational tool, which can predict the performance of cold crucible melting would be very helpful in solving engineering problems. In this paper a computational model for the cold crucible melting of corium, which is a mixture of UO2 and ZrO2 and represents the nuclear reactor material, is suggested. The results of the analysis are discussed in comparison with the experimental data. In the experiment, which simulate the molten fuel and coolant interaction phenomenon of the severe accidents in nuclear reactor [4,5], the cold crucible technique was adapted to make molten liquid from a mixture of UO2 and ZrO2, which has a melting temperature near 3000 K.
2. Analytical models and numerical analysis method The analysis of cold crucible melting involves both the analysis of electromagnetic field around the components of the cold crucible and the fluid flow and heat transfer in the melt. By Ampere’s law, the current flowing through an induction coil placed around the cold crucible generates a magnetic field. A voltage will be induced in a conductor proportional to the rate of change of magnetic flux opposite in sign such that a current generated by this induced potential resists the initiating magnetic field by Faraday’s law of induction. From these laws, the governing differential equations for the electromagnetic field can be written as below j ð1=lm j AÞ þ ixrA ep x2 A ¼ Jo
ð1Þ
where A is a magnetic vector potential defined as j A= B, B is magnetic flux density, Jo is electric source density, l m is magnetic permeability, e p is permittivity, x is the angular frequency of the electric field, and r is electric conductivity. The governing differential equations for the fluid motion and heat transfer for the melt in the cold crucible consist of continuity, momentum, and energy equation [6]. The continuity equation for the liquid phase, the momentum equation for the liquid motion modeled by an incompressible turbulent flow equation, where electromagnetic force, buoyancy force, and drag force between the liquid and solute are explicitly modeled, and energy equation are as below Bq=Bt þ jdðquÞ ¼ 0 ð2Þ BðquÞ=Bt þ jd ðquuÞ ¼ Fem þ Fb þ Fd jp þ jdðl þ lt Þju
ð3Þ
Fb ¼ qbgðT To Þ;
ð4Þ
Fd ¼ C ð1 fl Þ2 =fl3 ðu u s Þ
ð5Þ
B qCp T =Bt þ jd qCp uT ¼ jdkT jT þ Qem þ Lðdfs =dT ÞBT =Bt
ð6Þ
where q is the density of the fluid, l is the viscosity, l t is the turbulent viscosity, Fem is electromagnetic force, Fb is buoyancy force, Fd is the drag force in the solid liquid mixture [7], f l is the liquid fraction, C is a constant accounting for the mushy-region morphology, us is the velocity of the solid. C p is the
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specific heat of the liquid, k T is the thermal conductivity, L is the latent heat, f s is the solid fraction. The turbulence is modeled by k–e model [8] below Bðqk Þ=Bt þ jdðquk Þ ¼ jd ðl þ lt =rk Þjk þ G qe þ Sk;drag
ð7Þ
BðqeÞ=Bt þ jd ðqueÞ ¼ jd ðl þ lt =re Þje þ e=k ðC1e G C2e qeÞ þ Se;drag
ð8Þ
lt ¼ qC l k 2 =e
ð9Þ
G ¼ lt Bui =Buj þ Bui =Buj d Buj =Bui þ Buj =Bui
ð10Þ
Sk;drag ¼ 106 1 f12 =f12 k;
ð11Þ
Se;drag ¼ 106 1 f12 =f12 e
ð12Þ
where k is kinetic energy, e is dissipation rate, C l = 0.09, r k = 1.0, r e = 1.30, C 1e = 1.44, C 2e = 1.92, S k,drag and S e,drag are damping source terms. To analyze the electromagnetic field around a cold crucible, which is made of segmented bottom chamber and water-cooled copper tubes connected to it, the finite element method is applied. It solves the full three-dimensional electromagnetic field. The calculated joule heat generation and electromagnetic force on the fluid are used as source terms for driving the fluid flow and heat transfer. Finite element formulation starts by multiplying a trial function % on both side of the magnetic vector potential equation in (1) and taking the volume integral over a control volume V over a segment. Then the volume integral is converted to a new form by using vector identities and the Gauss’s divergence theorem and taking the magnetic vector potential A as a trial function. Z Z Z 2 %d ixrA ex A dV ¼ %dJo dV ð13Þ %d j ð1=lm j AÞdV þ Z
jd ½A ð1=lm j AÞdV ¼
Z
ð1=lm j AÞd ðn AÞdS þ
Z
Ad ½n ð1=lm j AÞdS: ð14Þ
At planes of symmetry and electrically conducting wall, the tangential component of electric field or magnetic vector potential is zero and the normal component of the magnetic field is zero. At the boundary of the magnetically conducting wall, the tangential component of magnetic field and the normal component of electric field is zero. At the infinite boundary, A becomes zero. Then Eq. (14) can be transformed as below, which is subject to a finite element approximation [6]. Z Z Z 2 ðj AÞdð1=lm j AÞdV þ Ad ixrA ex A dV ¼ AdJo dV : ð15Þ As the magnetic potential within each element can be interpolated by using the value defined on the edge of hexahedral element, we can obtain a linear system of simultaneous equations as below Z 1 ð16Þ Kij ¼ R ðj Ni Þd 1=lm j Nj dV
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Kij2
¼R
Bi ¼
Z
ixr ex2 Ni dNj dV
ð17Þ
Z
ð18Þ
Ni dJo dV ;
R Kij1 þ Kij2 Aj ¼ Bi
ð19Þ
where Ni is the vector shape function. From this equation we determine the magnetic vector potential in each element. Then we can determine the electromagnetic force and joule heating E ¼ ixA; Fem ¼ Je rE;
Qem ¼ Je dJe =r
ð20Þ
where E is electric field strength, Je is induced electric current density, and Qem is joule heating by induced current. A finite volume method is used to numerically solve the governing differential equation for the fluid flow and heat transfer in Eqs. (2)–(12). As the procedure is well established [6,7], it is not repeated in this paper. It is noted that the same hexahedral control volume is used in the finite volume method for consistency.
3. Characteristics of the cold crucible melting and input model A cold crucible used in the experiment [2,4] is shown in Fig. 1. It consists of water-cooled copper tubes, bottom chamber, and a plug at the bottom center. The material model for the numerical analysis and size is shown in Fig. 2, where a vertical section of the crucible is shown. The lower region consists of a plug at the center and a bottom chamber. The vertical wall on the right hand side represents the fingers. The rings outside the crucible represent the induction coil. A mixture of UO2 pellets and ZrO2 powder at the weight ratio of 78/22 is charged in the cold crucible. The total weight is about 17 kg. The
Fig. 1. A typical shape of a cold crucible.
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Fig. 2. Material model for the crucible.
first layer at the bottom represents the ZrO2 powder. The second layer represents the mixture of the UO2 pellets and ZrO2 powder. The third layer represents the ZrO2 powder layer with Zr ring as an initiator. The fourth layer represents a mixture of the UO2 pellets and ZrO2 powder. The fifth layer is the ZrO2 powder, which is used to minimize the loss due to radiation. As each segment of the cold crucible, which is defined for a finger, has the same initial and boundary condition, a periodic boundary condition can be applied for the azimuthal faces. For each segment, there are nodes in radial, vertical, and azimuthal direction, which define the infinitesimal control volumes for the numerical analysis. Nodal scheme in axial and radial direction is shown in Fig. 3. There are 52 nodes in radial direction, 96 nodes in the axial direction, and 8 nodes in azimuthal direction. Fine meshes are
Fig. 3. Node scheme for a vertical plane of cold crucible.
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used in the lower part of the charge near the bottom chamber, near the boundary of the finger, and initiator. As the temperature of the charged material changes significantly during the melting process, temperature dependent thermo-physical and electromagnetic properties [9] are incorporated in the analysis by utilizing experimental data [9]. The permeability of 4p 10 7 Henry/m and permittivity of 10 9/(36p) Farad/m are used. As the temperature at the cold crucible inlet is maintained constant by the chiller, the temperature of the cold crucible surface is assumed to be constant at 25 8C in the analysis. There is a solid-to-solid contact between the melt and crucible at the bottom portion. So, the heat transfer to the plug and bottom portion of the crucible is by conduction through the crust between the melt and the crucible. In the finger region, there is a solid–solid contact between the charged material and fingers at first. As the melting progresses, the thickness of the crust region decreases. In addition, net electromagnetic force pushes away the melt into the center region. So, it is quite possible there can be an air gap between the molten material and finger region. An air gap is explicitly modeled in the numerical analysis as a boundary condition. Fig. 4 shows input current, input voltage, and Q factor, which is the ratio of stored energy in the condenser to the input power, during the actual experiment. The power distribution among components of the cold crucible during the melting is shown in Fig. 5. It is shown that input power is used to heat up the melt and compensate the heat loss through the crucible and coil. The electric conductivity plays a major role in inducing eddy currents in the charged material. The electrical conductivity of corium rapidly increases when there is a phase change. So, the absorption capacity of the cold crucible increases when the charged material changes phases from solid to liquid. That is indicated by a sudden decrease in Q factor and a sudden increase in the current in Fig. 4. We can notice that majority of charged material reached the molten state near 1600 s. After 1600 s, the input power was maintained constant and the melting zone expands slowly. While the melting zone expands and the thickness of unmelted zone decreases, the heat loss through the cold crucible increases. Then the input voltage needs to be increased further to expand the melting zone, which is indicated by an increase in power after 3500 s in Fig. 5. After the charged material melted completely around 6500 s, the input power was reduced to cool the melt. When it solidified, the Q factor increases due to changes in the 30
Q
500
25
Current
20 400 15
300
10
200 Voltage
5
100
0
0 0
2000
4000
6000
Time(sec)
Fig. 4. Voltage, current, and coupling factor.
8000
Q
DC Current(A)/Voltage(V)
600
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80
Power(kW)
60
Total input power Power absorbed to melt
40
Crucible loss
20
Coil loss
0 0
2000
4000
6000
8000
Time(sec)
Fig. 5. Power consumption in the crucible.
electric conductivity from liquid to solid. The Q factor at time zero is bigger than that of the solid phase, as the crucible is filled with powder and pellet initially. The heat losses are both by electromagnetic dissipation and heat transfer. The skin effect [1] is developed on the surface of the coil and the crucible made of copper. The current induced on the surface results in ohmic losses. There exists additional heat loss due to heat transfer from the melt in the cold crucible. After a certain amount of charged material melted, since the current increased due to effective coupling, the ohmic loss through the coil and crucible decrease. During this period the power absorbed in the melt increases. This phase is shown in Fig. 5 during 1300–2000 s. Fig. 5 indicates that the input power was carefully controlled during the operation to take into account the characteristic of cold crucible melting discussed above. The power control consisted of four phases of initial heat up phase, constant power phase, crust control phase, and cool down phase.
4. Analysis results The analysis of the cold crucible melting was performed by the computer program, which was constructed from the numerical method discussed above. As the electromagnetic field, fluid flow, and heat transfer are coupled, the calculations were carried out in two steps. In the first step, the power distribution among the crucible, charge, and coil was determined from the analysis of the electromagnetic field around the crucible. As a second step, the heat transfer and fluid flow analysis was performed using the calculated joule heat generation and electro-magnetic force, to get the melt temperature, velocity of the convective motion, and the heat losses around the cold crucible. The newly calculated state of melt is used for the electromagnetic analysis in the next time step. The analysis results indicate that there are four distinct phases: initiation of melting, initiation of close coupling between the melt and magnetic flux, close coupling, and cooling phase. These phases were also identified in the experimental data in Figs. 4 and 5 as discussed above. For the analysis, the input power was maintained at 90 kW. Though the power was maintained constant in the numerical analysis while the power was carefully controlled for operational efficiency in the experiment, the fundamental characteristics of cold crucible melting were easily identified and the
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qualitative comparison of the numerical analysis results and experimental data was possible. The results of the analysis and the discussions are provided for each phase below. 4.1. Initiation of Melting This period corresponds to the initial part of the melting. The calculated flow pattern and isothermal lines at 300 s are shown in the Fig. 6(a), while the joule heat generation is shown in Fig. 6(b). The isothermal lines are at 200 8C interval and the outermost line indicates 200 8C. It is shown that the zirconium ring almost melted and became a heat source for the zirconia powder nearby. Still the power consumption in the charge is very small compared to the total input power. The power consumed in the crucible and coil is 60.9 kW and 22.6 kW. They are dissipated as heat loss. The input power absorbed in the charge is only 6.4 kW. So, the coupling between the charge and the magnetic flux is weak. Melt volume is 0.028S . Though the heat generation has very large difference, that is, maximum 1490 W/cm3 at the center of initiator, 100 W/ cm3 at the surface, it is limited in a narrow regime around the initiator as shown Fig. 6(b). 4.2. Initiation of close coupling between the melt and magnetic flux This period is identified by a rapid decrease in Q factor shown in Fig. 4 and increase in power absorption in the melt shown in Fig 5. Fig. 7(a) and (b) indicate the fluid flow and isothermal lines at 1260 s. The isothermal lines are at 100 8C interval. The lowest isothermal line is the line near the bottom portion
(a)
(b)
200 C 400 C
100 W/cm3
600 C
Tmax=2344 C
Hmax= 1490 W/cm3
Tinterval=200 C
Fig. 6. Melting performance at 300 s.
J.H. Song et al. / Int. Commun. Heat and Mass Transf. 32 (2005) 1325–1336
(b)
40 W/cm 3 80 W/cm 3
(a)
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Hmax= 285 W/cm3
Fig. 7. Melting performance at 1260 s.
of the crucible, which is 100 8C. The maximum temperature is 2643 8C. The power absorbed in the charge is increased to 41.6 kW. The heat loss at the bottom, the finger, and the vent hole is 0.42 kW, 2.87 kW, and 0.015 kW, respectively, the radiation from the crust is small as 0.02 kW. The remainder, 38.3 kW, is used to heat up the charged material. The joule heat generation is shown in Fig. 7(b). The melt volume is 0.827S and melt average temperature is 2643 8C. It is shown that the upper surface depressed due to initial porosity of the charged material, while the melted zone occupies the crucible substantially. 4.3. Close coupling This phase corresponds to the quasi-steady state shown after 1800 s in Figs. 4 and 5. Fig. 8(a) and (b) show the fluid flow, isothermal lines, and joule heat generation at 2000 s. The isothermal lines are at every 100 8C starting from 2500 8C. The maximum temperature is 3042 8C. It is shown that the charged material is fully melted and there exists a thin layer at the bottom. There is a good mixing in the liquid region due to the electromagnetic force and buoyancy force. The maximum liquid velocity is 33 cm/s. The flow is in a turbulent regime. The absorbed power in the melt is 68.8 kW (76.4%), while the heat loss to the finger, bottom, vent hole, and upper crust is 25.3 kW, 38.0 kW, 2.8 kW, and 8.2 kW, respectively. So, the balance is slightly negative. Then, the thickness of the lower crust would decrease a little bit to make the balance. It is known that the meniscus height is usually higher at the center than at the margin in the cold crucible melting even when the crust covers the melt surface. In our analysis, we have not considered this phenomenon, because its variation is estimated
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(b)
3000 C
3
m 20 W /c 3 m 40 W /c 3 /c m 10 0 W
(a)
2900 C
2800 C
Tmax=3040 C Tinterval=100 C
Hmax= 674 W/cm3
Fig. 8. Melting performance at 2000 s.
small in this experimental condition and this analysis mainly focuses on the overall evaluation for the performance of the cold crucible melting for UO2/ZrO2 mixtures. Fig. 8(b) shows the joule heat generation. The lines indicate the contours of the same volumetric heat flux. The lines are at every 20 W/cm3 starting from 20.0 kW/cm3. It is shown that the heat generation is concentrated near the wall region, which is quite typical of an induction heating of an oxide material. In the experiment, the temperature was measured using a two-color pyrometer in gray body condition through cavity or a tungsten thermo-well. After the oxidation of an initiator zirconium metal, the charged material temperature increased from initiator melting temperature (about 1700 8C) to corium melting temperature (2560 8C) for about 200 s. And then it increased about 3200 8C within 600 s. Though the melt temperature is measured on the melt surface, it is similar to the calculated value, as shown Figs. 7(a) and 8(a). 4.4. Power distribution and steady state heat balance Fig. 9 shows the calculated power absorption among cold crucible components during first 2000 s. Initially the induction power is mainly dissipated by the cooling water supplied to the crucible and induction coil, as there is only a weak coupling. As the melting progresses, the coupling between the melt and the electromagnetic field increases. When the amount of molten material reaches a critical volume, close coupling starts and the induction heating became effective. Comparison of Figs. 5 and 8 indicates that the analysis result is consistent with the experimental result in terms of heat distribution among various components of the crucible. In the analysis result, the power
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80 Charge Crucible Coil
Power (kW)
60
40
20
0
0
500
1000
1500
2000
Time (Sec)
Fig. 9. Power distribution in the cold crucible.
absorption is very low as 7% initially and it increases to 76% at 2000 s as shown in Fig. 8. Meanwhile, about 55% of the total input power is absorbed into the melt in experiment as shown in Fig. 5. The 25% and 20% of the total power is lost into the crucible and coil in experiment, while 20% and 10% of the total power is lost into the crucible and coil in the calculation. So, there is not much difference between the experiment and calculation. The steady state heat balance is of interest in terms of determining the input power to compensate for the heat losses. Initially, the melting zone starts near the crucible wall and the initiator and expands while the power is absorbed in the melt. The heat loss through the crust increases as the melt temperature increases. When the input power is bigger than the heat loss, the remainder of the absorbed power is used to erode the crust layer. As the thickness of the crust layer decrease, the heat loss increases. This will lead to a new steady state. This process is repeated during the close coupling period. The experimentally measured temperature indicated that the melt temperature is well above the melting point [1]. The melt temperature was expected to be not much different from the melting temperature, as the temperature of the inner surface of the crust, where the phase change occurs, should be melting temperature and there would be strong convective motion in the molten pool of corium due to buoyancy force and electromagnetic force. This melt super heat could be explained by an air gap between the melt and the crucible in the vertical interface. It is supported by the fact that when the mixture fully melted, the electromagnetic force pushes the molten liquid towards the center. The analysis results indicate that the melt super heat could be maintained by the assumption of air gap between the melt and the crucible and support this argument.
5. Conclusion A computational tool, which analyzes a coupled electromagnetic and thermal field around the cold crucible, for the evaluation of the performance of the cold crucible melting is suggested. The comparison of the computational results and experimental data demonstrates that the fundamental characteristics of the cold crucible melting can be reliably predicted by the computational method proposed in the present
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paper. The model will be very useful for the design of a cold crucible and the analyses of cold crucible melting experiments. Nomenclature E Electric field strength r Electric conductivity q Density Turbulent viscosity lt Drag force in the mixture Fd B Magnetic flux density Magnetic permeability lm u Velocity vector Electromagnetic force Fem % Trial function Applied electric current density Jo e Permittivity l Viscosity Buoyancy force Fb A Magnetic vector potential
Acknowledgement The study has been performed under the Long- and Mid-Term Nuclear R&D Program supported by MOST, Republic of Korea.
References [1] V.I. Aleksandrov, et al., in: E. Kaldis (Eds.), Current Topics in Material Science, vol. 1, 1978, p. 421, Chapter 6. [2] S.W. Hong, B.T. Min, J.H. Song, H.D. Kim, Materials Science & Engineering. A, Structural Materials: Properties, Microstructure and Processing 357 (2003) 297. [3] D. Gombert, J.R. Richardson, Nuclear Technology 141 (3) (2003) 301. [4] J.H. Song, et al., Journal of the Korean Nuclear Society 33 (4) (2002) 344. [5] D. Magallon, I. Huhtiniemi, Nuclear Engineering and Design 204 (2001) 369. [6] Y.W. Cho, et al., Modelling and Simulation in Materials Science and Engineering 4 (1996) 11. [7] M.R. Aboutalebi, M. Hasai, R.I.L. Guthrie, Metallurgical and Materials Transactions. B 26B (3) (1995) 731. [8] A.D. Brent, V.R. Voller, K.J. Reid, Numerical Heat Transfer 13 (1988) 297. [9] RASPLAV Project, RP-TR-5 and RP-TR-7 (1995).
An electromagnetic and thermal analysis of a cold crucible meltingB J.H. Song a,T, B.T. Min a, J.H. Kim a, H.W. Kim a, S.W. Hong a, S.H. Chung b a b
KAERI, 150 Dukjindong, Yusunggu, Daejeon, 305-353, South Korea KIST, 39-1 Hawolgokdong, Seongbuk-gu, Seoul, 136-791, South Korea Available online 22 August 2005
Abstract To investigate the performance of a cold crucible employed for the melting of corium, which is a mixture of UO2 and ZrO2, a computational analysis of the coupled electro-magnetic field, heat transfer, and fluid flow is carried out. Governing differential equations, basic numerical methods, computational model, and the results of the simulation of cold crucible melting are discussed. By comparing the analysis results with the experimental data, it is shown that the proposed computational model reasonably predicts the fundamental characteristics of a cold crucible melting. D 2005 Elsevier Ltd. All rights reserved. Keywords: Cold crucible melting; Phase change; Electro-magnetic field; Corium
1. Introduction One of the promising techniques for the melting of refractory material is the cold crucible melting [1–3]. The cold crucible meting is basically an inductive heating of electrically conducting melt by an alternating electromagnetic field. Heating is accomplished by ohmic losses caused by eddy currents induced in the melt. The skull formed between the very high-temperature melt and the cold crucible acts as both a self-container and an effective insulator. B
Communicated by K. Suzuki and S. Nishio. T Corresponding author. E-mail address: [email protected] (J.H. Song).
0735-1933/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2005.07.015
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While significant demands exist in the industry, the implementation of a cold crucible technique for a specific application encounters many engineering problems as it deals with very high temperature and complicated processes. And the experimental data for the performance of the cold crucible are rare. Therefore, a computational tool, which can predict the performance of cold crucible melting would be very helpful in solving engineering problems. In this paper a computational model for the cold crucible melting of corium, which is a mixture of UO2 and ZrO2 and represents the nuclear reactor material, is suggested. The results of the analysis are discussed in comparison with the experimental data. In the experiment, which simulate the molten fuel and coolant interaction phenomenon of the severe accidents in nuclear reactor [4,5], the cold crucible technique was adapted to make molten liquid from a mixture of UO2 and ZrO2, which has a melting temperature near 3000 K.
2. Analytical models and numerical analysis method The analysis of cold crucible melting involves both the analysis of electromagnetic field around the components of the cold crucible and the fluid flow and heat transfer in the melt. By Ampere’s law, the current flowing through an induction coil placed around the cold crucible generates a magnetic field. A voltage will be induced in a conductor proportional to the rate of change of magnetic flux opposite in sign such that a current generated by this induced potential resists the initiating magnetic field by Faraday’s law of induction. From these laws, the governing differential equations for the electromagnetic field can be written as below j ð1=lm j AÞ þ ixrA ep x2 A ¼ Jo
ð1Þ
where A is a magnetic vector potential defined as j A= B, B is magnetic flux density, Jo is electric source density, l m is magnetic permeability, e p is permittivity, x is the angular frequency of the electric field, and r is electric conductivity. The governing differential equations for the fluid motion and heat transfer for the melt in the cold crucible consist of continuity, momentum, and energy equation [6]. The continuity equation for the liquid phase, the momentum equation for the liquid motion modeled by an incompressible turbulent flow equation, where electromagnetic force, buoyancy force, and drag force between the liquid and solute are explicitly modeled, and energy equation are as below Bq=Bt þ jdðquÞ ¼ 0 ð2Þ BðquÞ=Bt þ jd ðquuÞ ¼ Fem þ Fb þ Fd jp þ jdðl þ lt Þju
ð3Þ
Fb ¼ qbgðT To Þ;
ð4Þ
Fd ¼ C ð1 fl Þ2 =fl3 ðu u s Þ
ð5Þ
B qCp T =Bt þ jd qCp uT ¼ jdkT jT þ Qem þ Lðdfs =dT ÞBT =Bt
ð6Þ
where q is the density of the fluid, l is the viscosity, l t is the turbulent viscosity, Fem is electromagnetic force, Fb is buoyancy force, Fd is the drag force in the solid liquid mixture [7], f l is the liquid fraction, C is a constant accounting for the mushy-region morphology, us is the velocity of the solid. C p is the
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specific heat of the liquid, k T is the thermal conductivity, L is the latent heat, f s is the solid fraction. The turbulence is modeled by k–e model [8] below Bðqk Þ=Bt þ jdðquk Þ ¼ jd ðl þ lt =rk Þjk þ G qe þ Sk;drag
ð7Þ
BðqeÞ=Bt þ jd ðqueÞ ¼ jd ðl þ lt =re Þje þ e=k ðC1e G C2e qeÞ þ Se;drag
ð8Þ
lt ¼ qC l k 2 =e
ð9Þ
G ¼ lt Bui =Buj þ Bui =Buj d Buj =Bui þ Buj =Bui
ð10Þ
Sk;drag ¼ 106 1 f12 =f12 k;
ð11Þ
Se;drag ¼ 106 1 f12 =f12 e
ð12Þ
where k is kinetic energy, e is dissipation rate, C l = 0.09, r k = 1.0, r e = 1.30, C 1e = 1.44, C 2e = 1.92, S k,drag and S e,drag are damping source terms. To analyze the electromagnetic field around a cold crucible, which is made of segmented bottom chamber and water-cooled copper tubes connected to it, the finite element method is applied. It solves the full three-dimensional electromagnetic field. The calculated joule heat generation and electromagnetic force on the fluid are used as source terms for driving the fluid flow and heat transfer. Finite element formulation starts by multiplying a trial function % on both side of the magnetic vector potential equation in (1) and taking the volume integral over a control volume V over a segment. Then the volume integral is converted to a new form by using vector identities and the Gauss’s divergence theorem and taking the magnetic vector potential A as a trial function. Z Z Z 2 %d ixrA ex A dV ¼ %dJo dV ð13Þ %d j ð1=lm j AÞdV þ Z
jd ½A ð1=lm j AÞdV ¼
Z
ð1=lm j AÞd ðn AÞdS þ
Z
Ad ½n ð1=lm j AÞdS: ð14Þ
At planes of symmetry and electrically conducting wall, the tangential component of electric field or magnetic vector potential is zero and the normal component of the magnetic field is zero. At the boundary of the magnetically conducting wall, the tangential component of magnetic field and the normal component of electric field is zero. At the infinite boundary, A becomes zero. Then Eq. (14) can be transformed as below, which is subject to a finite element approximation [6]. Z Z Z 2 ðj AÞdð1=lm j AÞdV þ Ad ixrA ex A dV ¼ AdJo dV : ð15Þ As the magnetic potential within each element can be interpolated by using the value defined on the edge of hexahedral element, we can obtain a linear system of simultaneous equations as below Z 1 ð16Þ Kij ¼ R ðj Ni Þd 1=lm j Nj dV
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Kij2
¼R
Bi ¼
Z
ixr ex2 Ni dNj dV
ð17Þ
Z
ð18Þ
Ni dJo dV ;
R Kij1 þ Kij2 Aj ¼ Bi
ð19Þ
where Ni is the vector shape function. From this equation we determine the magnetic vector potential in each element. Then we can determine the electromagnetic force and joule heating E ¼ ixA; Fem ¼ Je rE;
Qem ¼ Je dJe =r
ð20Þ
where E is electric field strength, Je is induced electric current density, and Qem is joule heating by induced current. A finite volume method is used to numerically solve the governing differential equation for the fluid flow and heat transfer in Eqs. (2)–(12). As the procedure is well established [6,7], it is not repeated in this paper. It is noted that the same hexahedral control volume is used in the finite volume method for consistency.
3. Characteristics of the cold crucible melting and input model A cold crucible used in the experiment [2,4] is shown in Fig. 1. It consists of water-cooled copper tubes, bottom chamber, and a plug at the bottom center. The material model for the numerical analysis and size is shown in Fig. 2, where a vertical section of the crucible is shown. The lower region consists of a plug at the center and a bottom chamber. The vertical wall on the right hand side represents the fingers. The rings outside the crucible represent the induction coil. A mixture of UO2 pellets and ZrO2 powder at the weight ratio of 78/22 is charged in the cold crucible. The total weight is about 17 kg. The
Fig. 1. A typical shape of a cold crucible.
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Fig. 2. Material model for the crucible.
first layer at the bottom represents the ZrO2 powder. The second layer represents the mixture of the UO2 pellets and ZrO2 powder. The third layer represents the ZrO2 powder layer with Zr ring as an initiator. The fourth layer represents a mixture of the UO2 pellets and ZrO2 powder. The fifth layer is the ZrO2 powder, which is used to minimize the loss due to radiation. As each segment of the cold crucible, which is defined for a finger, has the same initial and boundary condition, a periodic boundary condition can be applied for the azimuthal faces. For each segment, there are nodes in radial, vertical, and azimuthal direction, which define the infinitesimal control volumes for the numerical analysis. Nodal scheme in axial and radial direction is shown in Fig. 3. There are 52 nodes in radial direction, 96 nodes in the axial direction, and 8 nodes in azimuthal direction. Fine meshes are
Fig. 3. Node scheme for a vertical plane of cold crucible.
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used in the lower part of the charge near the bottom chamber, near the boundary of the finger, and initiator. As the temperature of the charged material changes significantly during the melting process, temperature dependent thermo-physical and electromagnetic properties [9] are incorporated in the analysis by utilizing experimental data [9]. The permeability of 4p 10 7 Henry/m and permittivity of 10 9/(36p) Farad/m are used. As the temperature at the cold crucible inlet is maintained constant by the chiller, the temperature of the cold crucible surface is assumed to be constant at 25 8C in the analysis. There is a solid-to-solid contact between the melt and crucible at the bottom portion. So, the heat transfer to the plug and bottom portion of the crucible is by conduction through the crust between the melt and the crucible. In the finger region, there is a solid–solid contact between the charged material and fingers at first. As the melting progresses, the thickness of the crust region decreases. In addition, net electromagnetic force pushes away the melt into the center region. So, it is quite possible there can be an air gap between the molten material and finger region. An air gap is explicitly modeled in the numerical analysis as a boundary condition. Fig. 4 shows input current, input voltage, and Q factor, which is the ratio of stored energy in the condenser to the input power, during the actual experiment. The power distribution among components of the cold crucible during the melting is shown in Fig. 5. It is shown that input power is used to heat up the melt and compensate the heat loss through the crucible and coil. The electric conductivity plays a major role in inducing eddy currents in the charged material. The electrical conductivity of corium rapidly increases when there is a phase change. So, the absorption capacity of the cold crucible increases when the charged material changes phases from solid to liquid. That is indicated by a sudden decrease in Q factor and a sudden increase in the current in Fig. 4. We can notice that majority of charged material reached the molten state near 1600 s. After 1600 s, the input power was maintained constant and the melting zone expands slowly. While the melting zone expands and the thickness of unmelted zone decreases, the heat loss through the cold crucible increases. Then the input voltage needs to be increased further to expand the melting zone, which is indicated by an increase in power after 3500 s in Fig. 5. After the charged material melted completely around 6500 s, the input power was reduced to cool the melt. When it solidified, the Q factor increases due to changes in the 30
Q
500
25
Current
20 400 15
300
10
200 Voltage
5
100
0
0 0
2000
4000
6000
Time(sec)
Fig. 4. Voltage, current, and coupling factor.
8000
Q
DC Current(A)/Voltage(V)
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80
Power(kW)
60
Total input power Power absorbed to melt
40
Crucible loss
20
Coil loss
0 0
2000
4000
6000
8000
Time(sec)
Fig. 5. Power consumption in the crucible.
electric conductivity from liquid to solid. The Q factor at time zero is bigger than that of the solid phase, as the crucible is filled with powder and pellet initially. The heat losses are both by electromagnetic dissipation and heat transfer. The skin effect [1] is developed on the surface of the coil and the crucible made of copper. The current induced on the surface results in ohmic losses. There exists additional heat loss due to heat transfer from the melt in the cold crucible. After a certain amount of charged material melted, since the current increased due to effective coupling, the ohmic loss through the coil and crucible decrease. During this period the power absorbed in the melt increases. This phase is shown in Fig. 5 during 1300–2000 s. Fig. 5 indicates that the input power was carefully controlled during the operation to take into account the characteristic of cold crucible melting discussed above. The power control consisted of four phases of initial heat up phase, constant power phase, crust control phase, and cool down phase.
4. Analysis results The analysis of the cold crucible melting was performed by the computer program, which was constructed from the numerical method discussed above. As the electromagnetic field, fluid flow, and heat transfer are coupled, the calculations were carried out in two steps. In the first step, the power distribution among the crucible, charge, and coil was determined from the analysis of the electromagnetic field around the crucible. As a second step, the heat transfer and fluid flow analysis was performed using the calculated joule heat generation and electro-magnetic force, to get the melt temperature, velocity of the convective motion, and the heat losses around the cold crucible. The newly calculated state of melt is used for the electromagnetic analysis in the next time step. The analysis results indicate that there are four distinct phases: initiation of melting, initiation of close coupling between the melt and magnetic flux, close coupling, and cooling phase. These phases were also identified in the experimental data in Figs. 4 and 5 as discussed above. For the analysis, the input power was maintained at 90 kW. Though the power was maintained constant in the numerical analysis while the power was carefully controlled for operational efficiency in the experiment, the fundamental characteristics of cold crucible melting were easily identified and the
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qualitative comparison of the numerical analysis results and experimental data was possible. The results of the analysis and the discussions are provided for each phase below. 4.1. Initiation of Melting This period corresponds to the initial part of the melting. The calculated flow pattern and isothermal lines at 300 s are shown in the Fig. 6(a), while the joule heat generation is shown in Fig. 6(b). The isothermal lines are at 200 8C interval and the outermost line indicates 200 8C. It is shown that the zirconium ring almost melted and became a heat source for the zirconia powder nearby. Still the power consumption in the charge is very small compared to the total input power. The power consumed in the crucible and coil is 60.9 kW and 22.6 kW. They are dissipated as heat loss. The input power absorbed in the charge is only 6.4 kW. So, the coupling between the charge and the magnetic flux is weak. Melt volume is 0.028S . Though the heat generation has very large difference, that is, maximum 1490 W/cm3 at the center of initiator, 100 W/ cm3 at the surface, it is limited in a narrow regime around the initiator as shown Fig. 6(b). 4.2. Initiation of close coupling between the melt and magnetic flux This period is identified by a rapid decrease in Q factor shown in Fig. 4 and increase in power absorption in the melt shown in Fig 5. Fig. 7(a) and (b) indicate the fluid flow and isothermal lines at 1260 s. The isothermal lines are at 100 8C interval. The lowest isothermal line is the line near the bottom portion
(a)
(b)
200 C 400 C
100 W/cm3
600 C
Tmax=2344 C
Hmax= 1490 W/cm3
Tinterval=200 C
Fig. 6. Melting performance at 300 s.
J.H. Song et al. / Int. Commun. Heat and Mass Transf. 32 (2005) 1325–1336
(b)
40 W/cm 3 80 W/cm 3
(a)
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Hmax= 285 W/cm3
Fig. 7. Melting performance at 1260 s.
of the crucible, which is 100 8C. The maximum temperature is 2643 8C. The power absorbed in the charge is increased to 41.6 kW. The heat loss at the bottom, the finger, and the vent hole is 0.42 kW, 2.87 kW, and 0.015 kW, respectively, the radiation from the crust is small as 0.02 kW. The remainder, 38.3 kW, is used to heat up the charged material. The joule heat generation is shown in Fig. 7(b). The melt volume is 0.827S and melt average temperature is 2643 8C. It is shown that the upper surface depressed due to initial porosity of the charged material, while the melted zone occupies the crucible substantially. 4.3. Close coupling This phase corresponds to the quasi-steady state shown after 1800 s in Figs. 4 and 5. Fig. 8(a) and (b) show the fluid flow, isothermal lines, and joule heat generation at 2000 s. The isothermal lines are at every 100 8C starting from 2500 8C. The maximum temperature is 3042 8C. It is shown that the charged material is fully melted and there exists a thin layer at the bottom. There is a good mixing in the liquid region due to the electromagnetic force and buoyancy force. The maximum liquid velocity is 33 cm/s. The flow is in a turbulent regime. The absorbed power in the melt is 68.8 kW (76.4%), while the heat loss to the finger, bottom, vent hole, and upper crust is 25.3 kW, 38.0 kW, 2.8 kW, and 8.2 kW, respectively. So, the balance is slightly negative. Then, the thickness of the lower crust would decrease a little bit to make the balance. It is known that the meniscus height is usually higher at the center than at the margin in the cold crucible melting even when the crust covers the melt surface. In our analysis, we have not considered this phenomenon, because its variation is estimated
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(b)
3000 C
3
m 20 W /c 3 m 40 W /c 3 /c m 10 0 W
(a)
2900 C
2800 C
Tmax=3040 C Tinterval=100 C
Hmax= 674 W/cm3
Fig. 8. Melting performance at 2000 s.
small in this experimental condition and this analysis mainly focuses on the overall evaluation for the performance of the cold crucible melting for UO2/ZrO2 mixtures. Fig. 8(b) shows the joule heat generation. The lines indicate the contours of the same volumetric heat flux. The lines are at every 20 W/cm3 starting from 20.0 kW/cm3. It is shown that the heat generation is concentrated near the wall region, which is quite typical of an induction heating of an oxide material. In the experiment, the temperature was measured using a two-color pyrometer in gray body condition through cavity or a tungsten thermo-well. After the oxidation of an initiator zirconium metal, the charged material temperature increased from initiator melting temperature (about 1700 8C) to corium melting temperature (2560 8C) for about 200 s. And then it increased about 3200 8C within 600 s. Though the melt temperature is measured on the melt surface, it is similar to the calculated value, as shown Figs. 7(a) and 8(a). 4.4. Power distribution and steady state heat balance Fig. 9 shows the calculated power absorption among cold crucible components during first 2000 s. Initially the induction power is mainly dissipated by the cooling water supplied to the crucible and induction coil, as there is only a weak coupling. As the melting progresses, the coupling between the melt and the electromagnetic field increases. When the amount of molten material reaches a critical volume, close coupling starts and the induction heating became effective. Comparison of Figs. 5 and 8 indicates that the analysis result is consistent with the experimental result in terms of heat distribution among various components of the crucible. In the analysis result, the power
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80 Charge Crucible Coil
Power (kW)
60
40
20
0
0
500
1000
1500
2000
Time (Sec)
Fig. 9. Power distribution in the cold crucible.
absorption is very low as 7% initially and it increases to 76% at 2000 s as shown in Fig. 8. Meanwhile, about 55% of the total input power is absorbed into the melt in experiment as shown in Fig. 5. The 25% and 20% of the total power is lost into the crucible and coil in experiment, while 20% and 10% of the total power is lost into the crucible and coil in the calculation. So, there is not much difference between the experiment and calculation. The steady state heat balance is of interest in terms of determining the input power to compensate for the heat losses. Initially, the melting zone starts near the crucible wall and the initiator and expands while the power is absorbed in the melt. The heat loss through the crust increases as the melt temperature increases. When the input power is bigger than the heat loss, the remainder of the absorbed power is used to erode the crust layer. As the thickness of the crust layer decrease, the heat loss increases. This will lead to a new steady state. This process is repeated during the close coupling period. The experimentally measured temperature indicated that the melt temperature is well above the melting point [1]. The melt temperature was expected to be not much different from the melting temperature, as the temperature of the inner surface of the crust, where the phase change occurs, should be melting temperature and there would be strong convective motion in the molten pool of corium due to buoyancy force and electromagnetic force. This melt super heat could be explained by an air gap between the melt and the crucible in the vertical interface. It is supported by the fact that when the mixture fully melted, the electromagnetic force pushes the molten liquid towards the center. The analysis results indicate that the melt super heat could be maintained by the assumption of air gap between the melt and the crucible and support this argument.
5. Conclusion A computational tool, which analyzes a coupled electromagnetic and thermal field around the cold crucible, for the evaluation of the performance of the cold crucible melting is suggested. The comparison of the computational results and experimental data demonstrates that the fundamental characteristics of the cold crucible melting can be reliably predicted by the computational method proposed in the present
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paper. The model will be very useful for the design of a cold crucible and the analyses of cold crucible melting experiments. Nomenclature E Electric field strength r Electric conductivity q Density Turbulent viscosity lt Drag force in the mixture Fd B Magnetic flux density Magnetic permeability lm u Velocity vector Electromagnetic force Fem % Trial function Applied electric current density Jo e Permittivity l Viscosity Buoyancy force Fb A Magnetic vector potential
Acknowledgement The study has been performed under the Long- and Mid-Term Nuclear R&D Program supported by MOST, Republic of Korea.
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