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Inductive heating with a stepped diameter crucible
Applied Thermal Engineering 102 (2016) 149–157
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Inductive heating with a stepped diameter crucible Yoav Hadad, Eytan Kochavi, Avi Levy ⇑ Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8410501, Israel
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
We present a novel solution for
Stepped diameter crucible with SHC. On the right, a section of the SHC is shown with current flow.
vacuum inductive heating of a gradual crucible. A numerical model was built to estimate the temperature gradient in the crucible. The method was validated with an experimental inductive furnace setup. The solution was shown enable to use in gradual diameter crucible.
a r t i c l e
i n f o
Article history: Received 29 September 2015 Accepted 28 March 2016 Available online 31 March 2016 Keywords: Inductive heating Stepped diameter crucible Variable diameter crucible Secondary coil
a b s t r a c t Induced heating in a vacuum environment is the most common method for high precision and purity casting. A significant disadvantage in this process is that it is designed for heating in a crucible of uniform diameter. However, it is not suited for the heating of a crucible with stepped diameter. The use of a variable diameter crucible enables to place a larger amount of raw material into the wider diameter section of the crucible. After the material melts, the liquid fills the narrow part of the crucible and enables better control of liquid flow into the casting mold. Hence, in this research we present a novel solution for vacuum inductive heating in a stepped diameter crucible by using a secondary heating coil (SHC). In order to examine the SHC solution, numerical models were developed to describe the temperature distribution and the heat generation in the crucible. Experiments on a vacuum induction furnace were conducted to validate the numerical models of the SHC. A good match was acquired between the numerical and experimental results. The results of the simulations and experiments have shown great improvement in the capability of heating a stepped diameter crucible with an SHC. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Induction heating process is widely applied in industrial operations such as thermal treatment [1], precision hardening [2] for parts with complex geometry and metal casting [3]. The dynamics
⇑ Corresponding author. Tel.: +972 8 6477092; fax: +972 8 6477130. E-mail address: [email protected] (A. Levy). http://dx.doi.org/10.1016/j.applthermaleng.2016.03.151 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
of induction heating is described by Maxwell’s equations. An alternating current in a coil produces a time-varying magnetic field in its surroundings with equal frequency to the coil current. This magnetic field causes eddy currents on the surface of the workpiece, located inside the coil. The resultant eddy currents are opposite in direction to the coil current, as due to the Lenz’s law. The eddy currents heat the conductor according to the Joule effect. Induction heating is a complex combination of electromagnetic and heat transfer phenomena. The design of an induction heating
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system has to be optimized for each application and requires a complicated analysis. Most of the analytical models were obtained for simple form [4,5]. The first of the numerical techniques to be widely used for electro-heating problems was finite difference, and the method is still used today in certain applications [6–8]. Induced heating in a vacuum environment has become more popular for high precision and purity casting. In this method a crucible is made of material with high melting temperature, such as Molybdenum and Graphite. The induced electric field generates vortex currents in the crucible and heats it to melt the alloy inside. The alloys in the casting industry come in the form of bars and plates such that the dimensions of the crucible limit the amount of raw material able to be inserted into the crucible. Therefore, there is a need to reshape the raw material to fit into the crucible (Fig. 1). The use of a stepped diameter crucible enables placing a larger amount of material into the wider region part of the crucible. After the material melts, the liquid fills the narrow part of the crucible and enables better control of liquid flow into the casting mold. A significant disadvantage of vacuum induction heating is that it is suitable for heating uniform diameter crucibles, and not stepped diameter ones. This disadvantage is most significant when the induction coil is placed outside of the heating zone. In this configuration the current flow is in the opposite direction of the coil current by Faraday’s law, therefore the eddy current will be concentrated in the large diameter. Hence, the heating mechanisms of the narrow diameter are radiation and conduction from the large diameter and not by Joule effect. As a result, the temperature in the crucible’s bottom is low. To prevent solidification, it is required to increase the temperature of the crucible in the large diameter [9]. Higher temperature requires an expensive crucible material and may cause increase in the amount of pollutants in the casting material. Dhakal et al. [10] presented an induction furnace cavity for heat treatment of high purity niobium in this furnace, the niobium susceptor can is inductively heated and heat is transferred to the Niobium cavity by radiation. Another approach is to change the magnetic field density by changing the coil geometry [11]. Tudbury [12] offered the combination of two coils; in this case the current in the narrow diameter coil will be greater than in the wide diameter coil, such that the difference in magnetic flux compensates for the change in geometry. A similar idea is presented in Zinn and Semiatin [13]. It is recommended to change the density of the coil to compensate the change in the geometry of the model. These solutions are especially suitable for gradual changes in geometry and not for sharp drops in diameter as described in Fig. 1. Another method uses a cold crucible [14–16]. A water-cooled crucible is made from copper segments. An induction coil is
Stepped crucible
Thermocouple sleeve
wrapped around the crucible. The induced currents produced in each of the sectors make the cold crucible act as SHC while the induction coil acts as primary coil. This method is effective but a leak of water into the oven can cause steam explosion, and is therefore not preferred. A solution for heating of small parts in large induction coil by means of an ‘‘insert” is presented in [17]. The operating principle is an alternating current within the induction coil, which induces the eddy current flowing within the insert in the opposite direction. However, the slot within the insert breaks the eddy current flow, forcing it to complete a loop on the internal surface of the insert. Current flow on the inside surface of the insert creates a magnetic field of its own, which, in turn induces a current within the workpiece. The present study uses a SHC for heating a stepped crucible, as illustrate in Fig. 1. Experimental and numerical study is conducted. The prediction of the developed model is validated with the experimental data.
2. Experimental setup An induction furnace experimental setup was designed and built to validate the numerical model. The system shown in Fig. 2 is designed for high vacuum and a temperature of 1100 °C. A 304 L stainless steel stepped crucible with 36 g of tin in its bottom was placed in the furnace. An additional thermal insulation is obtained with an alumina cylinder surrounding the crucible. The vacuum chamber is a quartz tube sealed with a water cooled flange. Radiation shields in the top of the furnace decrease heat losses. A rotor pump is used to evacuate the inside chamber to a pressure of 103 mbar. Temperature measurements are conducted with K-type thermocouples. Accuracy of the K-type thermocouples was approximately 0.2%. One thermocouple is placed within the tin material and the second is mounted on the side of the crucible. The experiments were conducted both with and without the secondary coil. The induction power in the furnace was 1.1–1.2 kW with AC frequency of 8800 Hz. The temperature of the cooling water was 25 °C.
3. Governing equations 3.1. Electromagnetic fields The Maxwell’s equations in differential form which describe electromagnetic field is written as [18]:
Quartz tube
Secondary coil
Main coil
Fig. 1. Stepped diameter crucible with SHC. On the right, a section of the SHC is shown with current flow.
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! ! D ¼ er e0 E ! ! B ¼ lr l0 H ! ! ! J ¼ rE þ J e
Top flange
ð5Þ ð6Þ
where the parameters e; l and r denote, respectively, the permit! tivity, magnetic and electrical conduction of the material. J e is the externally generated current density. The magneto-quasi-static approximation consists in neglecting the displacement currents ! @ D =@t in Eq. (1). This approximation therefore leads to neglecting the propagation phenomena, as presents in Bay et al. [19]. This approximation is realistic also here, for the reason that in this works we used, with a commercial program COMSOL Multiphysics, Eq. (1) while accounting for displacement currents. It can be helpful to formulate the problems in terms of magnetic ! vector potential A and the electric scalar potential u. They are given by:
Quartz tube
Crucible Alumina insulator
ð4Þ
Cooling water pipes
Coil
! ! B ¼r A ! ! @A E ¼ ru @t
Lower flange
ð7Þ ð8Þ
Using the definitions of magnetic potential, the constitutive relationship in Eq. (1) now has the form:
1
Fig. 2. Picture of induction furnace.
l ! @D @t ! ! @B r E ¼ @t ! r B ¼0 !
!
r H ¼ J þ
ð1Þ ð2Þ ð3Þ
!
! @D ! þ Je @t
ð9Þ
Thus a time-harmonic electromagnetic field can be introduced:
! ! J e ¼ ðiwr w2 er e0 Þ A þ r
1
lr l0
!
r A
þ ðr þ iwer e0 Þru ð10Þ
where w is the angular frequency. The electric and magnetic potentials are not uniquely defined from the electric and magnetic fields. The variable transformation of the potentials is called a gauge transformation [20]. To obtain a unique solution, a constraint variable, W is used to impose a condition on the derivatives of the magnetic vector potential. Therefore, its absolute value does not have particular significance while only its gradient enters the equations.
error magnetic flux density (%)
! ! where H is magnetic field intensity, J is conduction current den! ! sity, D is electric flux density, E is electric field intensity, B is magnetic field density. These equations include constitutive relations that describe macroscopic properties and can be given for linear material as:
!
rr A ¼ rE þ
freq (Hz) Fig. 3. The error of the two-dimensional magnetic field model as a function of frequency.
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Fig. 4. Hollow tubes in magnetic field at a frequency of 10 (kHz): (a) two-dimensional model and (b) three-dimensional model.
e¼! A A þ rW @W ~ ¼u u @t Choosing W ¼ iu=w results in
! e þr J e ¼ ðiwr w2 er e0 Þ A
ð12Þ
In order to solve the above set of equation, a set of boundary conditions is required. The impedance boundary condition [21] provides a condition that is useful at boundaries where the skin depth is small, or alternatively, if the conductor is thick in comparison to the skin depth.
ð13Þ
! n H þ
ð11Þ
1
lr l0
e rA
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! e0 er ir=w n ð E nÞ ¼ 0
lr l0
ð14Þ
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153
3.2. Heat transfer For the heat transfer analysis, the following equation has been used [22]:
qcp ðTÞ
@T ¼ r ðkrTÞ þ Q @t
ð15Þ
where q is density and cp ðTÞ is specific heat capacity of the workpiece and Q is the heat source term of the heat. In this case, this heat is generated by the induced currents and can be described as:
Q¼
! 1 Reð J E Þ 2
ð16Þ
where Re is the real part of the production. As the material heats up in the crucible, it is melted. During the phase transition, a significant amount of latent heat is absorbed. The total amount of heat enters per unit mass of alloy during the transition is given by the enthalpy change, DH. In addition the specific heat capacity, cp ðTÞ also changes considerably during this transition. As opposed to pure metals, an alloy generally undergoes a broad temperature transition zone, over several degrees, in which a mixture of both solid and molten material co-exists in a ‘‘mushy” zone. To account for the latent heat related to the phase transition, the specific heat capacity was calculated by [23].
cp ðTÞ ¼ cp ðTÞ þ dDHðTÞ
ð17Þ
where DH(T) is the latent heat of the transition and d is a Gaussian function given by Fig. 5. Overall flowcharts for the numerical simulation procedure.
The boundary condition approximates this penetration to avoid the need to include an additional domain in the model. The material properties that appear in the equation are those for the conductive material excluded from the model.
2 mÞ exp ðTT 2 ðDTÞ pffiffiffiffi d¼ DT p
ð18Þ
All boundary conditions are radiation in vacuum conditions for ideal gray bodies. Thus q, the surface heat flux, is given by:
q ¼ fðG rr T 4 Þ where f is emissivity, G is irradiation and constant.
Fig. 6. Magnetic field and temperature distributions in the furnace without SHC.
ð19Þ
rr is Stefan–Boltzman
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Fig. 7. Current density SHC.
(3D)
(2Daxis-symmetric)
Fig. 8. Overall flowcharts for the numerical simulation procedure with SHC.
4. Numerical solution The governing equations introduced above were solved numerically by the finite elements method, using COMSOL Multiphysics version 4.3 with the ‘‘induction heating” model. The magnetic field Eq. (13) was solved in the frequency domain and the heat transfer Eq. (16) was solved in the time domain. The current was intro-
duced with ‘‘coil group domain” in 2D model and ‘‘surface current density” in the 3D model. 4.1. Verification problems The verification of the models utilized in the numerical solution is by means of two test cases: namely; comparing the magnetic
Y. Hadad et al. / Applied Thermal Engineering 102 (2016) 149–157
field at the center of the empty short coil and heat induction of sleeve with analytical models. 4.1.1. The magnetic field in the center of a short coil The magnetic field in the center of a helical coil, of length l ¼ 0:12 m and radius R ¼ 0:06 m with N ¼ 15 tightly wound turns, can be obtained by the following expression [13]:
l0 N I ffi ¼ 0:022T B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 R2 þ l
ð20Þ
This case was examined in 2D and 3D numerical models. Simulations were made for the frequency range of 0–10 kHz. The error in the 2D model is sensitive to the frequency simulated. The error as a function of the frequency is given in Fig. 3. It can be seen that up to 2 kHz the results fit the analytical solution. Above this range, the absolute value of the error grows with frequency up to 2.6%. The cause for this error is the assumption that the coil current has homogeneous distributions, which in fact deviates due to the proximity effect. The geometry of the 3D model is equivalent to that of the 2D model. To save computation time, only one half of the coil was modeled exploiting the symmetry of the problem. The geometry of the coil is described as half of the rings, and the boundary condition on the coil was ‘‘surface current density”. This definition satisfies the assumption of the analytical Eq. (21), a homogeneous distribution of the current. The error obtained was 4% and does not dependent of frequency. 4.1.2. Heat induction of a hollow tube In this section, a comparison is made between the analytical solution [12] for the power dissipation of a hollow tube, with the concept of equivalent resistance to the numerical results. The solution is of an inductor-coil with a cross section of quadrangular and the length of the coil and the hollow tube are equal. The calculation was performed for a hollow tube with a 60 mm outer diameter, 1.8 mm thickness and 148 mm length, electric resistivity
prevent heating of the upper flange. A spiral water cooled copper induction six turns coil is placed around the vacuum chamber. In the numerical model the Air domain was large enough to emulate semi-infinite environment. The induction coil was described by a circular cross-section made of copper with the fixed temperature. The boundary outside the air domain was ambient temperature and magnetic insulation, also were defined fix temperature in the coil and the cooling zones on the flanges of 298 K. Radiation heat transfer was defined between the interior boundary in the furnace by Eq. (19) and the exteriors boundary was defined with radiation and natural convection. The latent heat of tin was be calculated by Eq. (18).
4.2.2. Numerical solution for experimental furnace with SHC In the case that includes the SHC the geometry becomes more complex. Therefore, a 3D model is required to describe the magnetic field and the SHC non-symmetrical geometry and currents. Note that only the critical parts of the magnetic field had to be represented. The rest of the furnace geometry, which is cylindrical, is described with a 2D axisymmetric model to further compute the heat transfer. The models’ solutions were obtained in two stages. The first stage is a 3D model that computes the magnetic field in order to obtain the heat source in the crucible due to the addition of the SHC, as a function of temperature. Components of the furnace which have no effect on the magnetic field were neglected, as shown in Fig. 7. The furnace operates at 8.8 kHz. At this frequency the current flows very close to the surface of the conducting regions. The skin depth is less than 0.7 mm in the copper. Estimation of the eddy current requires at least three linear elements per skin depth, which allows capturing the variation of the field. To save computer resources we used in surface impedance boundary conditions Eq. (14) in the SHC, similar to the procedure used in the work of Hayt [24]. In the second stage, the physics of heat transfer was solved using a 2D axisymmetric
7:24 107 X m, and the electric current is 141 A RMS on a frequency of 10 kHz. The power dissipation predicted by the analytical model is:
P ¼ 1214 W
ð21Þ
This case was examined by 2D and 3D models. The deviations is of the numerical simulations from the analytical solution was found to be 1.67% and 1.75% respectively. Fig. 4 shows the models and the calculated magnetic flux density. 4.2. The numerical model of the induction furnace The numerical model was examined with and without the presence of a SHC. A 2D axisymmetric model was used for the simulations without a SHC. For the case with SHC, A 3D model was used to solve the electromagnetic heating within the crucible and a 2D axisymmetric model solved the heat transfer. The use of 3D and 2D models had to be made in order to save computing resources. 4.2.1. Numerical solution for experimental furnace without SHC The simulation was carried out using the software COMSOL Multiphysics version 4.3 with ‘‘induction heating” physics. The electromagnetic field is solved in the frequency domain (14), while temperatures are solved in the time domain (16). A detailed flow chart of the solution algorithm is shown in Fig. 5. The geometry of the model is presented in Fig. 6. A stepped crucible, made of 304 L stainless steel, contains tin in the bottom. The crucible is placed in an alumina tube for thermal insulation purposes. A quartz tube is used as the vacuum chamber and is sealed between two 304 L stainless steel flanges. Molybdenum radiation shields
155
Fig. 9. Temperature distributions in the furnace with SHC.
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temperature in the tin continues to increase until it reaches the temperature in the large diameter of the crucible due to the use of the SHC. Fig. 10b presents a comparison between the experimental result and the numerical result of the process of heating and melting tin with the SHC. Although the predictions of the simulation for the case without a SHC are much better than for the
model. This model uses the heat source as computed by the first 3D model in the first stage. A detailed flow chart of the numerical simulation procedure is shown in Fig. 8. Fig. 7 shows the path and the current density in the SHC and crucible. As presented in Fig. 7 the SHC concentrated the current in the narrow diameter. Fig. 9 shows the temperature distribution in the experimental furnace. It may be seen, that the use of the SHC changes the temperature profile in the crucible such that the temperature obtained in the narrow diameter is higher than the temperature in the wide diameter. This is a more desirable result that ensures proper flow of melt. 4.3. Comparison between calculated and experimental results The numerical predictions are compared with the experimental result. The thermo-physical and electromagnetic properties of the furnace were taken from Comsol library material. Thermocouples located in the large diameter of the crucible and in the tin measured the temperatures during the heating. The temperature vs. time response without the SHC is seen in Fig. 10a. As shown, the temperature measured in the tin is lower than the temperature in the wall. Moreover the calculated temperature in the experimental furnace is compatible with the experimental measurements. The same experiment was conducted with SHC. The results can be seen in Fig. 10b. The temperature of the tin in the melting phase is lower than the one in the crucible. There is a trend reversal around the melting temperature. While the tin melts, the temperature in the crucible rises. In the end of the melting, the
a
Fig. 11. The geometry of a secondary coil.
700 600
Temp [°C]
500 Experimental wall crucible temperature Experimental n temperature Calculaon n temperature
400 300 200
Calculaon wall crucible temperature
100 0 0
200
400
600
800
1000
1200
1400
me [sec]
b
700 600 Experimental wall crucible temperature Experimental n temperature
Temp [°C]
500 400 300
Calculaon n temperature
200
Calculaon wall crucible temperature
100 0 0
200
400
600
800
1000
1200
1400
me [Sec] Fig. 10. Comparison between the experimental results to numeric results: (a) heating and melting tin without SHC and (b) with SHC.
Y. Hadad et al. / Applied Thermal Engineering 102 (2016) 149–157
References
32.5
Heang power in the narrow diameter [W]
157
32 31.5 Efficient ratio
31 30.5 30 29.5 1
1.5
2
2.5
3
3.5
4
Relave heights (a/b) Fig. 12. Heating power in the narrow diameter vs. relative heights.
case with a SHC, the comparison is able to demonstrate the efficiency of the SHC. 4.4. Parametric analysis In order to achieve maximum heating efficiency with a SHC, it is necessary to get an optimal design. Let a and b be the heights of the outer and inner surfaces of the SHC, respectively, as shown in Fig. 11. By keeping a larger than b, it is possible to increase the current density in the inside diameter, and by that increasing the magnetic field on the crucible. On the other hand, it increases the magnetic losses. Here, a parametric analysis for the second coil geometry is presented in Fig. 11. The purpose is to find an optimal ratio a=b which maximizes the heating power of the in the narrow diameter part of the crucible. The computations were performed for a current of 141 A and at a frequency of 8800 Hz. The results are seen in Fig. 12. The optimal heights’ ratio is between 1.2 and 1.7 which is provides a maximum heating power of the SHC. 5. Summary In this work, a novel approach in induction heating of a stepped diameter crucible was proposed and studied. The benefits of using a SHC are demonstrated. This suggested approach can be used as a low cost and effective solution for melting or heat treatment applications. The suggested method was examined experimentally and simulated numerically. The obtained results show that the SHC could be successfully used for heating a stepped diameter crucible. The numerical model derived for this case can correctly predict the efficiency of the SHC and can be used for furnace design. In this case, the SHC is made of copper in order to decrease electrical losses. For cases where copper cannot be used, it is recommended to use for SHC the material of the crucible.
[1] H. Jiang, T.H. Nguyen, M. Prud’homme, Optimal control of induction heating for semi-solid aluminum alloy forming, J. Mater. Process. Technol. 189 (1) (2007) 182–191. [2] H. Kristoffersen, P. Vomacka, Influence of process parameters for induction hardening on residual stresses, Mater. Des. 22 (8) (2001) 637–644. [3] S. Ray, Synthesis of cast metal matrix particulate composites, J. Mater. Sci. 28 (20) (1993) 5397–5413. [4] M.W. Kennedy, H. Akhtar, J.A. Bakken, R.E. Aune, Analytical and experimental validation of electromagnetic simulations using COMSOLÒ, reinductance, induction heating and magnetic fields, in: Proceedings of 2011 COMSOL Conference Stuttgart, Germany, 2011. [5] J. Davies, P. Simpson, Induction Heating Handbook, McGraw-Hill Companies, 1979. [6] J.D. Lavers, Numerical solution methods for electroheat problems, Magn., IEEE Trans. 19 (6) (1983) 2566–2572. [7] O. Ben-David et al., Magnetohydrodynamic flow excited by rotating permanent magnets in an orthogonal container, Phys. Fluids (1994-Present) 26 (9) (2014) 097–104. [8] O. Ben-David, A. Levy, B. Mikhailovich, A. Azulay, Impact of rotating permanent magnets on gallium melting in an orthogonal container, Int. J. Heat Mass Transf. 81 (2015) 373–382. [9] A. Barka, J.J. Bernard, B. Benyoucef, Thermal behavior of a conductor submitted to skin effect, Appl. Therm. Eng. 23 (10) (2003) 1261–1274. [10] P. Dhakal, G. Ciovati, W. Rigby, J. Wallace, G.R. Myneni, Design and performance of a new induction furnace for heat treatment of superconducting radiofrequency niobium cavities, Rev. Sci. Instrum. 83 (6) (2012) 065–105. [11] V.L. Rudnev, Joining components by induction heating, Part I, Heat Treat. Prog. 5 (2) (2005) 19–22. [12] C.A. Tudbury, Basics of Induction Heating, John F. Rider. Inc., New Rochelle, NY, 1960. [13] S. Zinn, S.L. Semiatin, Elements of Induction Heating: Design, Control, and Applications, ASM International, 1988. [14] J.-R. Yang, J.-R. Chen, H.S. Ding, J-J. mGuo, J-C. Han, H.-Z. Fu, Thermal characteristics of induction heating in cold crucible used for directional solidification, Appl. Therm. Eng. 59 (1) (2013) 69–76. [15] I. Quintana, Z. Azpilgain, D. Pardo, I. Hurtado, Numerical modeling of cold crucible induction melting, in: Proceedings of 2011 COMSOL Conference Stuttgart, Germany, 2011. [16] A. Morita, H. Fukuib, H. Tadanoa, S. Hayashia, J. Hasegawab, M. Niinomic, Alloying titanium and tantalum by cold crucible levitation melting (CCLM) furnace, Mater. Sci. Eng., A 280 (1) (2000) 208–213. [17] V. Rudnev, D. Loveless, R.L. Cook, M. Black, Handbook of Induction Heating, CRC Press, 2002. [18] Ruth Chabav, Bruce Sherwood, John E. Gastineau, Post-use review. Electric and magnetic interactions, Am. J. Phys. 63 (7) (1995) 669. [19] F. Bay, V. Labbe, Y. Favennec, J.L. Chenot, A numerical model for induction heating processes coupling electromagnetism and thermomechanics, Int. J. Numer. Meth. Eng. 58 (6) (2003) 839–867. [20] A. Kovetz, The Principles of Electromagnetic Theory, CUP Archive, 1990. ISBN: 9780521399975. [21] I. Nathan, L. Di-Rienzo, S. Yuferev, Surface impedance boundary conditions– application to computer-aided electromagnetic design, in: Proc. 11th International Conference on Development and Application Systems, Suceava, Romania, May 17–19, 2012. [22] J.P. Holman, Heat Transfer, 9th, McGraw-Hill Companies, 2002. [23] T. Basak, K.G. Ayappa, Analysis of microwave thawing of slabs with effective heat capacity method, AIChE J. 43 (7) (1997) 1662–1674. [24] W.H. Hayt, Engineering Electromagnetics, McGrawY Hill. Inc., New York, NY, 1981.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Inductive heating with a stepped diameter crucible Yoav Hadad, Eytan Kochavi, Avi Levy ⇑ Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8410501, Israel
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
We present a novel solution for
Stepped diameter crucible with SHC. On the right, a section of the SHC is shown with current flow.
vacuum inductive heating of a gradual crucible. A numerical model was built to estimate the temperature gradient in the crucible. The method was validated with an experimental inductive furnace setup. The solution was shown enable to use in gradual diameter crucible.
a r t i c l e
i n f o
Article history: Received 29 September 2015 Accepted 28 March 2016 Available online 31 March 2016 Keywords: Inductive heating Stepped diameter crucible Variable diameter crucible Secondary coil
a b s t r a c t Induced heating in a vacuum environment is the most common method for high precision and purity casting. A significant disadvantage in this process is that it is designed for heating in a crucible of uniform diameter. However, it is not suited for the heating of a crucible with stepped diameter. The use of a variable diameter crucible enables to place a larger amount of raw material into the wider diameter section of the crucible. After the material melts, the liquid fills the narrow part of the crucible and enables better control of liquid flow into the casting mold. Hence, in this research we present a novel solution for vacuum inductive heating in a stepped diameter crucible by using a secondary heating coil (SHC). In order to examine the SHC solution, numerical models were developed to describe the temperature distribution and the heat generation in the crucible. Experiments on a vacuum induction furnace were conducted to validate the numerical models of the SHC. A good match was acquired between the numerical and experimental results. The results of the simulations and experiments have shown great improvement in the capability of heating a stepped diameter crucible with an SHC. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Induction heating process is widely applied in industrial operations such as thermal treatment [1], precision hardening [2] for parts with complex geometry and metal casting [3]. The dynamics
⇑ Corresponding author. Tel.: +972 8 6477092; fax: +972 8 6477130. E-mail address: [email protected] (A. Levy). http://dx.doi.org/10.1016/j.applthermaleng.2016.03.151 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
of induction heating is described by Maxwell’s equations. An alternating current in a coil produces a time-varying magnetic field in its surroundings with equal frequency to the coil current. This magnetic field causes eddy currents on the surface of the workpiece, located inside the coil. The resultant eddy currents are opposite in direction to the coil current, as due to the Lenz’s law. The eddy currents heat the conductor according to the Joule effect. Induction heating is a complex combination of electromagnetic and heat transfer phenomena. The design of an induction heating
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system has to be optimized for each application and requires a complicated analysis. Most of the analytical models were obtained for simple form [4,5]. The first of the numerical techniques to be widely used for electro-heating problems was finite difference, and the method is still used today in certain applications [6–8]. Induced heating in a vacuum environment has become more popular for high precision and purity casting. In this method a crucible is made of material with high melting temperature, such as Molybdenum and Graphite. The induced electric field generates vortex currents in the crucible and heats it to melt the alloy inside. The alloys in the casting industry come in the form of bars and plates such that the dimensions of the crucible limit the amount of raw material able to be inserted into the crucible. Therefore, there is a need to reshape the raw material to fit into the crucible (Fig. 1). The use of a stepped diameter crucible enables placing a larger amount of material into the wider region part of the crucible. After the material melts, the liquid fills the narrow part of the crucible and enables better control of liquid flow into the casting mold. A significant disadvantage of vacuum induction heating is that it is suitable for heating uniform diameter crucibles, and not stepped diameter ones. This disadvantage is most significant when the induction coil is placed outside of the heating zone. In this configuration the current flow is in the opposite direction of the coil current by Faraday’s law, therefore the eddy current will be concentrated in the large diameter. Hence, the heating mechanisms of the narrow diameter are radiation and conduction from the large diameter and not by Joule effect. As a result, the temperature in the crucible’s bottom is low. To prevent solidification, it is required to increase the temperature of the crucible in the large diameter [9]. Higher temperature requires an expensive crucible material and may cause increase in the amount of pollutants in the casting material. Dhakal et al. [10] presented an induction furnace cavity for heat treatment of high purity niobium in this furnace, the niobium susceptor can is inductively heated and heat is transferred to the Niobium cavity by radiation. Another approach is to change the magnetic field density by changing the coil geometry [11]. Tudbury [12] offered the combination of two coils; in this case the current in the narrow diameter coil will be greater than in the wide diameter coil, such that the difference in magnetic flux compensates for the change in geometry. A similar idea is presented in Zinn and Semiatin [13]. It is recommended to change the density of the coil to compensate the change in the geometry of the model. These solutions are especially suitable for gradual changes in geometry and not for sharp drops in diameter as described in Fig. 1. Another method uses a cold crucible [14–16]. A water-cooled crucible is made from copper segments. An induction coil is
Stepped crucible
Thermocouple sleeve
wrapped around the crucible. The induced currents produced in each of the sectors make the cold crucible act as SHC while the induction coil acts as primary coil. This method is effective but a leak of water into the oven can cause steam explosion, and is therefore not preferred. A solution for heating of small parts in large induction coil by means of an ‘‘insert” is presented in [17]. The operating principle is an alternating current within the induction coil, which induces the eddy current flowing within the insert in the opposite direction. However, the slot within the insert breaks the eddy current flow, forcing it to complete a loop on the internal surface of the insert. Current flow on the inside surface of the insert creates a magnetic field of its own, which, in turn induces a current within the workpiece. The present study uses a SHC for heating a stepped crucible, as illustrate in Fig. 1. Experimental and numerical study is conducted. The prediction of the developed model is validated with the experimental data.
2. Experimental setup An induction furnace experimental setup was designed and built to validate the numerical model. The system shown in Fig. 2 is designed for high vacuum and a temperature of 1100 °C. A 304 L stainless steel stepped crucible with 36 g of tin in its bottom was placed in the furnace. An additional thermal insulation is obtained with an alumina cylinder surrounding the crucible. The vacuum chamber is a quartz tube sealed with a water cooled flange. Radiation shields in the top of the furnace decrease heat losses. A rotor pump is used to evacuate the inside chamber to a pressure of 103 mbar. Temperature measurements are conducted with K-type thermocouples. Accuracy of the K-type thermocouples was approximately 0.2%. One thermocouple is placed within the tin material and the second is mounted on the side of the crucible. The experiments were conducted both with and without the secondary coil. The induction power in the furnace was 1.1–1.2 kW with AC frequency of 8800 Hz. The temperature of the cooling water was 25 °C.
3. Governing equations 3.1. Electromagnetic fields The Maxwell’s equations in differential form which describe electromagnetic field is written as [18]:
Quartz tube
Secondary coil
Main coil
Fig. 1. Stepped diameter crucible with SHC. On the right, a section of the SHC is shown with current flow.
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! ! D ¼ er e0 E ! ! B ¼ lr l0 H ! ! ! J ¼ rE þ J e
Top flange
ð5Þ ð6Þ
where the parameters e; l and r denote, respectively, the permit! tivity, magnetic and electrical conduction of the material. J e is the externally generated current density. The magneto-quasi-static approximation consists in neglecting the displacement currents ! @ D =@t in Eq. (1). This approximation therefore leads to neglecting the propagation phenomena, as presents in Bay et al. [19]. This approximation is realistic also here, for the reason that in this works we used, with a commercial program COMSOL Multiphysics, Eq. (1) while accounting for displacement currents. It can be helpful to formulate the problems in terms of magnetic ! vector potential A and the electric scalar potential u. They are given by:
Quartz tube
Crucible Alumina insulator
ð4Þ
Cooling water pipes
Coil
! ! B ¼r A ! ! @A E ¼ ru @t
Lower flange
ð7Þ ð8Þ
Using the definitions of magnetic potential, the constitutive relationship in Eq. (1) now has the form:
1
Fig. 2. Picture of induction furnace.
l ! @D @t ! ! @B r E ¼ @t ! r B ¼0 !
!
r H ¼ J þ
ð1Þ ð2Þ ð3Þ
!
! @D ! þ Je @t
ð9Þ
Thus a time-harmonic electromagnetic field can be introduced:
! ! J e ¼ ðiwr w2 er e0 Þ A þ r
1
lr l0
!
r A
þ ðr þ iwer e0 Þru ð10Þ
where w is the angular frequency. The electric and magnetic potentials are not uniquely defined from the electric and magnetic fields. The variable transformation of the potentials is called a gauge transformation [20]. To obtain a unique solution, a constraint variable, W is used to impose a condition on the derivatives of the magnetic vector potential. Therefore, its absolute value does not have particular significance while only its gradient enters the equations.
error magnetic flux density (%)
! ! where H is magnetic field intensity, J is conduction current den! ! sity, D is electric flux density, E is electric field intensity, B is magnetic field density. These equations include constitutive relations that describe macroscopic properties and can be given for linear material as:
!
rr A ¼ rE þ
freq (Hz) Fig. 3. The error of the two-dimensional magnetic field model as a function of frequency.
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Fig. 4. Hollow tubes in magnetic field at a frequency of 10 (kHz): (a) two-dimensional model and (b) three-dimensional model.
e¼! A A þ rW @W ~ ¼u u @t Choosing W ¼ iu=w results in
! e þr J e ¼ ðiwr w2 er e0 Þ A
ð12Þ
In order to solve the above set of equation, a set of boundary conditions is required. The impedance boundary condition [21] provides a condition that is useful at boundaries where the skin depth is small, or alternatively, if the conductor is thick in comparison to the skin depth.
ð13Þ
! n H þ
ð11Þ
1
lr l0
e rA
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! e0 er ir=w n ð E nÞ ¼ 0
lr l0
ð14Þ
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3.2. Heat transfer For the heat transfer analysis, the following equation has been used [22]:
qcp ðTÞ
@T ¼ r ðkrTÞ þ Q @t
ð15Þ
where q is density and cp ðTÞ is specific heat capacity of the workpiece and Q is the heat source term of the heat. In this case, this heat is generated by the induced currents and can be described as:
Q¼
! 1 Reð J E Þ 2
ð16Þ
where Re is the real part of the production. As the material heats up in the crucible, it is melted. During the phase transition, a significant amount of latent heat is absorbed. The total amount of heat enters per unit mass of alloy during the transition is given by the enthalpy change, DH. In addition the specific heat capacity, cp ðTÞ also changes considerably during this transition. As opposed to pure metals, an alloy generally undergoes a broad temperature transition zone, over several degrees, in which a mixture of both solid and molten material co-exists in a ‘‘mushy” zone. To account for the latent heat related to the phase transition, the specific heat capacity was calculated by [23].
cp ðTÞ ¼ cp ðTÞ þ dDHðTÞ
ð17Þ
where DH(T) is the latent heat of the transition and d is a Gaussian function given by Fig. 5. Overall flowcharts for the numerical simulation procedure.
The boundary condition approximates this penetration to avoid the need to include an additional domain in the model. The material properties that appear in the equation are those for the conductive material excluded from the model.
2 mÞ exp ðTT 2 ðDTÞ pffiffiffiffi d¼ DT p
ð18Þ
All boundary conditions are radiation in vacuum conditions for ideal gray bodies. Thus q, the surface heat flux, is given by:
q ¼ fðG rr T 4 Þ where f is emissivity, G is irradiation and constant.
Fig. 6. Magnetic field and temperature distributions in the furnace without SHC.
ð19Þ
rr is Stefan–Boltzman
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Fig. 7. Current density SHC.
(3D)
(2Daxis-symmetric)
Fig. 8. Overall flowcharts for the numerical simulation procedure with SHC.
4. Numerical solution The governing equations introduced above were solved numerically by the finite elements method, using COMSOL Multiphysics version 4.3 with the ‘‘induction heating” model. The magnetic field Eq. (13) was solved in the frequency domain and the heat transfer Eq. (16) was solved in the time domain. The current was intro-
duced with ‘‘coil group domain” in 2D model and ‘‘surface current density” in the 3D model. 4.1. Verification problems The verification of the models utilized in the numerical solution is by means of two test cases: namely; comparing the magnetic
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field at the center of the empty short coil and heat induction of sleeve with analytical models. 4.1.1. The magnetic field in the center of a short coil The magnetic field in the center of a helical coil, of length l ¼ 0:12 m and radius R ¼ 0:06 m with N ¼ 15 tightly wound turns, can be obtained by the following expression [13]:
l0 N I ffi ¼ 0:022T B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 R2 þ l
ð20Þ
This case was examined in 2D and 3D numerical models. Simulations were made for the frequency range of 0–10 kHz. The error in the 2D model is sensitive to the frequency simulated. The error as a function of the frequency is given in Fig. 3. It can be seen that up to 2 kHz the results fit the analytical solution. Above this range, the absolute value of the error grows with frequency up to 2.6%. The cause for this error is the assumption that the coil current has homogeneous distributions, which in fact deviates due to the proximity effect. The geometry of the 3D model is equivalent to that of the 2D model. To save computation time, only one half of the coil was modeled exploiting the symmetry of the problem. The geometry of the coil is described as half of the rings, and the boundary condition on the coil was ‘‘surface current density”. This definition satisfies the assumption of the analytical Eq. (21), a homogeneous distribution of the current. The error obtained was 4% and does not dependent of frequency. 4.1.2. Heat induction of a hollow tube In this section, a comparison is made between the analytical solution [12] for the power dissipation of a hollow tube, with the concept of equivalent resistance to the numerical results. The solution is of an inductor-coil with a cross section of quadrangular and the length of the coil and the hollow tube are equal. The calculation was performed for a hollow tube with a 60 mm outer diameter, 1.8 mm thickness and 148 mm length, electric resistivity
prevent heating of the upper flange. A spiral water cooled copper induction six turns coil is placed around the vacuum chamber. In the numerical model the Air domain was large enough to emulate semi-infinite environment. The induction coil was described by a circular cross-section made of copper with the fixed temperature. The boundary outside the air domain was ambient temperature and magnetic insulation, also were defined fix temperature in the coil and the cooling zones on the flanges of 298 K. Radiation heat transfer was defined between the interior boundary in the furnace by Eq. (19) and the exteriors boundary was defined with radiation and natural convection. The latent heat of tin was be calculated by Eq. (18).
4.2.2. Numerical solution for experimental furnace with SHC In the case that includes the SHC the geometry becomes more complex. Therefore, a 3D model is required to describe the magnetic field and the SHC non-symmetrical geometry and currents. Note that only the critical parts of the magnetic field had to be represented. The rest of the furnace geometry, which is cylindrical, is described with a 2D axisymmetric model to further compute the heat transfer. The models’ solutions were obtained in two stages. The first stage is a 3D model that computes the magnetic field in order to obtain the heat source in the crucible due to the addition of the SHC, as a function of temperature. Components of the furnace which have no effect on the magnetic field were neglected, as shown in Fig. 7. The furnace operates at 8.8 kHz. At this frequency the current flows very close to the surface of the conducting regions. The skin depth is less than 0.7 mm in the copper. Estimation of the eddy current requires at least three linear elements per skin depth, which allows capturing the variation of the field. To save computer resources we used in surface impedance boundary conditions Eq. (14) in the SHC, similar to the procedure used in the work of Hayt [24]. In the second stage, the physics of heat transfer was solved using a 2D axisymmetric
7:24 107 X m, and the electric current is 141 A RMS on a frequency of 10 kHz. The power dissipation predicted by the analytical model is:
P ¼ 1214 W
ð21Þ
This case was examined by 2D and 3D models. The deviations is of the numerical simulations from the analytical solution was found to be 1.67% and 1.75% respectively. Fig. 4 shows the models and the calculated magnetic flux density. 4.2. The numerical model of the induction furnace The numerical model was examined with and without the presence of a SHC. A 2D axisymmetric model was used for the simulations without a SHC. For the case with SHC, A 3D model was used to solve the electromagnetic heating within the crucible and a 2D axisymmetric model solved the heat transfer. The use of 3D and 2D models had to be made in order to save computing resources. 4.2.1. Numerical solution for experimental furnace without SHC The simulation was carried out using the software COMSOL Multiphysics version 4.3 with ‘‘induction heating” physics. The electromagnetic field is solved in the frequency domain (14), while temperatures are solved in the time domain (16). A detailed flow chart of the solution algorithm is shown in Fig. 5. The geometry of the model is presented in Fig. 6. A stepped crucible, made of 304 L stainless steel, contains tin in the bottom. The crucible is placed in an alumina tube for thermal insulation purposes. A quartz tube is used as the vacuum chamber and is sealed between two 304 L stainless steel flanges. Molybdenum radiation shields
155
Fig. 9. Temperature distributions in the furnace with SHC.
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temperature in the tin continues to increase until it reaches the temperature in the large diameter of the crucible due to the use of the SHC. Fig. 10b presents a comparison between the experimental result and the numerical result of the process of heating and melting tin with the SHC. Although the predictions of the simulation for the case without a SHC are much better than for the
model. This model uses the heat source as computed by the first 3D model in the first stage. A detailed flow chart of the numerical simulation procedure is shown in Fig. 8. Fig. 7 shows the path and the current density in the SHC and crucible. As presented in Fig. 7 the SHC concentrated the current in the narrow diameter. Fig. 9 shows the temperature distribution in the experimental furnace. It may be seen, that the use of the SHC changes the temperature profile in the crucible such that the temperature obtained in the narrow diameter is higher than the temperature in the wide diameter. This is a more desirable result that ensures proper flow of melt. 4.3. Comparison between calculated and experimental results The numerical predictions are compared with the experimental result. The thermo-physical and electromagnetic properties of the furnace were taken from Comsol library material. Thermocouples located in the large diameter of the crucible and in the tin measured the temperatures during the heating. The temperature vs. time response without the SHC is seen in Fig. 10a. As shown, the temperature measured in the tin is lower than the temperature in the wall. Moreover the calculated temperature in the experimental furnace is compatible with the experimental measurements. The same experiment was conducted with SHC. The results can be seen in Fig. 10b. The temperature of the tin in the melting phase is lower than the one in the crucible. There is a trend reversal around the melting temperature. While the tin melts, the temperature in the crucible rises. In the end of the melting, the
a
Fig. 11. The geometry of a secondary coil.
700 600
Temp [°C]
500 Experimental wall crucible temperature Experimental n temperature Calculaon n temperature
400 300 200
Calculaon wall crucible temperature
100 0 0
200
400
600
800
1000
1200
1400
me [sec]
b
700 600 Experimental wall crucible temperature Experimental n temperature
Temp [°C]
500 400 300
Calculaon n temperature
200
Calculaon wall crucible temperature
100 0 0
200
400
600
800
1000
1200
1400
me [Sec] Fig. 10. Comparison between the experimental results to numeric results: (a) heating and melting tin without SHC and (b) with SHC.
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References
32.5
Heang power in the narrow diameter [W]
157
32 31.5 Efficient ratio
31 30.5 30 29.5 1
1.5
2
2.5
3
3.5
4
Relave heights (a/b) Fig. 12. Heating power in the narrow diameter vs. relative heights.
case with a SHC, the comparison is able to demonstrate the efficiency of the SHC. 4.4. Parametric analysis In order to achieve maximum heating efficiency with a SHC, it is necessary to get an optimal design. Let a and b be the heights of the outer and inner surfaces of the SHC, respectively, as shown in Fig. 11. By keeping a larger than b, it is possible to increase the current density in the inside diameter, and by that increasing the magnetic field on the crucible. On the other hand, it increases the magnetic losses. Here, a parametric analysis for the second coil geometry is presented in Fig. 11. The purpose is to find an optimal ratio a=b which maximizes the heating power of the in the narrow diameter part of the crucible. The computations were performed for a current of 141 A and at a frequency of 8800 Hz. The results are seen in Fig. 12. The optimal heights’ ratio is between 1.2 and 1.7 which is provides a maximum heating power of the SHC. 5. Summary In this work, a novel approach in induction heating of a stepped diameter crucible was proposed and studied. The benefits of using a SHC are demonstrated. This suggested approach can be used as a low cost and effective solution for melting or heat treatment applications. The suggested method was examined experimentally and simulated numerically. The obtained results show that the SHC could be successfully used for heating a stepped diameter crucible. The numerical model derived for this case can correctly predict the efficiency of the SHC and can be used for furnace design. In this case, the SHC is made of copper in order to decrease electrical losses. For cases where copper cannot be used, it is recommended to use for SHC the material of the crucible.
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