- Email: [email protected]
Simulation study of dynamic heating process of molybdenum for sheet suited for rigid transport components design
Available online at www.sciencedirect.com Available online at www.sciencedirect.com
ScienceDirect ScienceDirect
Transportation Research Procedia 00 (2019) 000–000 Available online at www.sciencedirect.com Transportation Research Procedia 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Transportation Research Procedia 40 (2019) 442–448 www.elsevier.com/locate/procedia
13th International Scientific Conference on Sustainable, Modern and Safe Transport 13th International 2019), Scientific Conference on Sustainable, and Safe Transport (TRANSCOM High Tatras, Novy Smokovec –Modern Grand Hotel Bellevue, (TRANSCOM 2019),Slovak High Tatras, Novy Smokovec – Grand Hotel Bellevue, Republic, May 29-31, 2019 Slovak Republic, May 29-31, 2019
Simulation study of dynamic heating process of molybdenum for Simulation study of dynamic heating process of molybdenum for sheet suited for rigid transport components design sheet suited for rigid transport components design a b Miroslav Paveleka*, Tibor Donicb Miroslav Pavelek *, Tibor Donic
Department of mechatronics and electronics, University of Zilina, Zilina 01026, Slovakia b Research Centreand of University Zilina. Zilina 01026,Zilina Slovakia Department of mechatronics electronics,ofUniversity of Zilina, 01026, Slovakia b Research Centre of University of Zilina. Zilina 01026, Slovakia
a a
Abstract Abstract This paper deals with the development of the simulation model of the moving molybdenum sheet, while the molybdenum sheet is This paper deals resistive with the heating. development of the simulationenvironments model of the moving sheet, while theStatic molybdenum sheet in is heated by direct COMSOL/MATLAB are usedmolybdenum to create mentioned model. model created COMSOL is compared measurement and based onenvironments this comparison the dynamic (COMSOL/MATLAB) adjusted. heated by direct resistivewith heating. COMSOL/MATLAB are used to createmodel mentioned model. Static modeliscreated in COMSOL is compared with measurement and basedfor on different this comparison the dynamic model (COMSOL/MATLAB) is adjusted. The dynamic model is created as in-loop simulation initial conditions of heating elements position and temperature The dynamicwithin modelmolybdenum is created assheet. in-loop for different conditionsdistribution of heating within elements position and temperature distribution Thesimulation result of dynamic modelinitial is temperature molybdenum sheet for given heating elements and speed.sheet. For that reason, proposedmodel modeliscan be used fordistribution optimization of direct resistive heating distribution withinshift molybdenum The resultthe of dynamic temperature within molybdenum sheet forsystem given heating elementssheet. shift and speed. For that reason, the proposed model can be used for optimization of direct resistive heating system of molybdenum of molybdenum sheet. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published byof Elsevier B.V. committee of the 13th International Scientific Conference on Sustainable, Peer-review under responsibility the scientific Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Peer-review under responsibility of the scientific Modern (TRANSCOM 2019). Modern and and Safe Safe Transport Transport (TRANSCOM 2019).committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Keywords: molybdenum sheet; direct resistive heating; finite element method. Keywords: molybdenum sheet; direct resistive heating; finite element method.
1. Introduction 1. Introduction This article discusses the usage of finite element method for the design of a precise simulation model of resistive This article discusses the usage finite element method theconsider design of a precise simulation of resistive heating process. Main issue was toofdevelop a model, whichfor will dynamic properties of model a heating system, heatinga process. Main sheet issue was to develop model, which will consider dynamic properties where molybdenum is being moved abetween two heating electrodes. Proposed model of hasa heating to be ofsystem, a high where a molybdenum sheet requirements is being moved twoofheating electrodes.ofProposed model has to be of a high accuracy in order to model andbetween the needs the development the equipment for the heating and accuracy in order to model requirements and the needs of the development of the equipment for the heating and * Corresponding author. address:author. [email protected] * E-mail Corresponding E-mail address: [email protected] 2352-1465 © 2018 The Authors. Published by Elsevier B.V. Peer-review©under responsibility of the scientific committee 2352-1465 2018 The Authors. Published by Elsevier B.V. of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). 2352-1465 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). 10.1016/j.trpro.2019.07.064
2
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
443
shaping the molybdenum sheets into molds. Molybdenum is one of the few materials suitable for this purpose in a view of the long-term high-temperature load of the mold. The mold is exposed to the temperature gradient of up to 100 °C/mm at the maximum temperature of 2150 °C within the longitudinal direction of the container, where the melt is located whereby at adjacent part the corundum as an already growth product is located. Based on the extreme conditions during the process, it is crucial to not decrease the mechanical abilities during the shaping process of the mold. For that purpose, the structural changes and possibilities of avoiding them by properly set pre-heating respectively heating/cooling system of Mo sheets need to be predicted. Currently, the molds are made of molybdenum sheet having a thickness of 0,5 mm, made by powder metallurgy technology, i.e. by plastic deformation and sintering procedure. Specifically, the molybdenum sheet Grade M1 corresponding to the American standard ASTM B386 or GB 3877, where the chemical composition is approximately the same. As heating elements, standard copper electrodes are used. All of the mentioned material properties are precisely defined within proposed simulation model, which consider dynamic change of the molybdenum sheet location within the heating system. [1-3] Nomenclature a b c d h x xe xe1 ye NEP NXP NYP NZP xp yp zp SOE SHIFT tstop NOS TNOS IT POE I σ ρ ε k Cp i
width of molybdenum sheet length of molybdenum sheet thickness of molybdenum sheet electrode diameter electrode height distance between electrodes electrode position in x direction starting position of electrode in x direction electrode position in y direction number of electrode partitions number of molybdenum sheet partitions in x direction number of molybdenum sheet partitions in y direction number of molybdenum sheet partitions in z direction length of molybdenum sheet part width of molybdenum sheet part height of molybdenum sheet part speed of electrode movement final electrode shift partial simulation time number of simulation steps ratio of motion speed to partial simulation time initial temperature electrode pressure electrode current electrical conductivity density relative permittivity thermal conductivity specific heat capacity at constant pressure simulation step
2. Simulation model and material definitions The simulation model is developed in COMSOL environment as a 3D model with the options for reconfiguration the mutual position of electrodes, the electrode material and the pressure on the electrodes. The model is composed
444
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
3
of the molybdenum sheet and two electrodes. The molybdenum sheet is modeled as a block (domain) with a wanted width(a), depth(b) and thickness(c) and electrodes are modeled as cylinders with a wanted diameter(d), height(h) and distance(x) between them. The geometry (Fig.1.) is then complemented by physic setup. The Multiphysics “Joule heating (JH)” is used for the model. This Multiphysics is formed by the connection of “Electrical circuit (EC)” physic and “Heat transfer in solid (HTS)” physic. The physic “Electrical circuit” is used to set up the value of input current trough top boundary of one electrode and grounding through the top boundary of the other electrode. Within the EC module, the required contact pressure applied to the electrodes and the roughness of the contact between the electrode and Mo sheet can be adjusted. The HTS is used to set up parameters as initial temperature of the system and the method in which the temperature is transferred to the surrounding area. The proper function of the model is achieved by setting the spherical domain with a diameter of one meter, which is filled with air. All simulations are the set as time domain simulation, so we can determine the influence of studied materials on the speed of the heating of the Mo sheet. [4-6]
Fig. 1. Model of direct resistive Mo sheet heating – geometry.
Fig. 2. Model of direct resistive Mo sheet heating divided to subdomains.
Advanced model of molybdenum sheet in motion is created as a script in MATLAB environment. To be able to simulate the heating process of the molybdenum in motion, the simulation had to be divided into partial simulations. The partial simulation is basically time-limited step of simulation with constant position of the electrodes. As the simulation itself the geometrical model (Fig.2.) needed to be also divided to parts, which serve as elements to obtain temperature distribution (final conditions) from one step of the simulation and as input elements to define initial conditions of next simulation step, in which the position of the electrodes is changed. The number of partial simulation is given by (1), while the time of partial simulation is given by (2). 𝑁𝑁𝑁𝑁𝑁𝑁 = 0.1 ∗ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆/𝑆𝑆𝑆𝑆𝑆𝑆
𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆/𝑆𝑆𝑆𝑆𝑆𝑆/𝑁𝑁𝑁𝑁𝑁𝑁
(1) (2)
The position of electrodes given in each simulation step is determine by (3) in x-direction and by (4) in y-direction.
4
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
445
(3)
𝑥𝑥𝑥𝑥 = 𝑥𝑥𝑥𝑥1 + (𝑆𝑆𝑆𝑆𝑆𝑆/𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇) ∗ (𝑖𝑖 − 1)
𝑦𝑦𝑦𝑦 = 𝑎𝑎/2 − 𝑥𝑥/2
(4)
𝑥𝑥𝑥𝑥 = 𝑏𝑏/𝑁𝑁𝑁𝑁𝑁𝑁
(5)
𝑧𝑧𝑧𝑧 = 𝑐𝑐/𝑁𝑁𝑁𝑁𝑁𝑁
(7)
Equations (5), (6) and (7) determines the dimensions of the subdomains of the molybdenum sheet.
(6)
𝑦𝑦𝑦𝑦 = 𝑎𝑎/𝑁𝑁𝑁𝑁𝑁𝑁
The mentioned JH module of the COMSOL environment uses well-known equations for modeling a heat transfer in solid materials (8). 𝜌𝜌𝐶𝐶𝑃𝑃
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
(8)
− 𝛻𝛻 ⋅ (𝑘𝑘𝛻𝛻𝑇𝑇) = 𝑄𝑄
In this case, the heat source “Q” is determined by Joule-Lenz law of resistive loses in material structure due to the current flow through this structure. The material parameters and other model settings as final conditions settings of domains can be seen in Table 1 [5]. Table 1. Model settings material and final conditions Parameter/Domain
Molybdenum sheet
Electrodes
Material
Molybdenum
Copper
Electrical conductivity [S/m]
17.9e6
5.5e7
Density [kg/m^3]
10280
8700
Relative permittivity [-]
1
1
Thermal conductivity [W/(°C*m)]
142
400
Specific heat capacity at constant pressure [J/(°C*kg)]
254
385
Current conservation
YES
YES
Electric insulation
YES
YES
Thermal insulation
YES
YES
Heat flux
Sphere (1 m diameter with initial 20 °C)
3. Simulation results Simulation results are obtained for model with parameters defined in Table 2. Table 2. Parameters of simulation model. Parameter
a [mm]
b [mm]
c [mm]
d [mm]
h [mm]
x [mm]
xe [mm]
xe1 [mm]
Value
150
200
1
20
50
75
(xe1,i)
30
Parameter
ye [mm]
NXP [-]
NYP [-]
NZP [-]
xp [mm]
yp [mm]
zp [mm]
SOE [mm/s]
Value
52.5
40
30
2
4
5
0.5
1
Parameter
SHIFT [-]
tstop [s]
NOS [-]
TNOS [-]
IT [°C]
POE [MPa]
I [A]
i [-]
Value
140
10
14
0.1
20
0.3
1500
(1,NOS)
446
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
5
Fig.3 and Fig.4. show basic concept of simulation results of the proposed model. The first part of the results Fig.3a shows initial temperature distribution before the current flow trough the electrodes is aplied. Fig.3b is temperature distribution after 10 s this temperature distribution is than transferd to next simulation step with slightly adjusted electrode position as can be seen on Fig.3c and the 10 s simulation interval starts again. The result from secound simulation step can be seen at Fig.3d.
Fig. 3. Simulation results of molybdenum sheet with electrode shift.
Fig. 4. Simulation results of molybdenum sheet with electrode shift.
Fig.4. is showing the final temperature distribution of all remaing simulation steps, while the minimal final temperature (last step) of the molybdenum sheet is at value 343 °C and the maximal temperature at value 599 °C.
6
Miroslav Pavelek et al. / Transportation Research00Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia (2019) 000–000
447
Fig. 5. Simulation model of molybdenum sheet without electrode shift.
In comparison the simulation with static (parameters that does’t include electrode shift are the same as in first model) electrodes was made (Fig.5.) with simulation time 140 s position of electrodes in the midlle of the plate. In this case the minimal final temperature is at value of 335 °C and the maximal temperature at value 462 °C. 4. Conclusion In this paper, the design of thermal finite element simulation model of Mo sheet was described. The main aim was to target exact modeling of thermal field in the structure of Mo sheet, while dynamic motion of the investigated system was considered. This motion is required, when more precise and uniform thermal distribution within the heated sheet is required. The main problem during the investigation is the proper definition of the material properties of Mo sheets. Each manufacturer has different material composition. Therefore, the simulation model has to have the possibility of reconfiguration in order to meet experimental results. Second approach of presented paper was focused on the design of the model, which will consider a dynamic change of the heated place within Mo sheet, i.e. translation move was applied. Various software products have limitations regarding the dynamic movement of the investigated sample, therefore special script in MATLAB environment was developed. With the use of this approach, each iteration of the simulation accepts previous results, which are repeatedly implemented within the computation solution. Given proposal of the solution was required due to future works, which will be focused on the design of multi-physics simulation model of Mo sheet heating and molding system, where high-validity simulation models are expected to be used namely for further investigations of mechanical parameters and other molybdenum restrictions. Also experimental comparisons will be further realized and compared with simulations. Acknowledgements The authors would like to thank to the national grant agency APVV for project support No. APVV-0396-15 and No. APVV 14-0284 – Study of useful properties of molded molybdenum sheets applicable for horizontal crystallisation of sapphire monocrystals References [1] Karban, P., Mach, F., Dolezel, I., 2012. Hard-couple model of local direct resistance heating of thin sheets. In: Journal of Computational and Applied Mathematics 236(18), 4725-4731, 2012. https://doi.org/10.1016/j.cam.2012.02.036 [2] Plansee Group, https://www.plansee.com/en/index.html [15.12.2018] [3] Mori, K., Maki, S., Tanaka, Y., 2005. Warm and Hot Stamping of ultra-high tensile strenght sheets using resistance heating”. In: CIRP Annals-Manufacturing Technology, 54(), 209-212, 2005. [4] Frivaldsky, M., Drgona, P., Spanik, P., 2013. Experimental analysis and optimization of key parameters of ZVS mode and its application in the proposed LLC converter designed for distributed power system application, In: International Journal of Electrical Power and Energy Systems, vol: 47, pp. 448-456, DOI: 10.1016/j.ijepes.2012.11.016 [5] Dughiero, F., Forzan, M., Pozza, C., Sieni, E., 2012. A translational couple electromagnetic and thermal innovative model for induction welding of tubes. In: IEEE Transactions on Magnetics 48(2), 483-486, 2012. https://doi.org/10.1109/TMAG.2011.2174972
448
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
[6] Frivaldsky, M., Cuntala, J., Spanik, P., 2014. Simple and accurate thermal simulation model of supercapacitor for development of module solutions, In: International Journal of thermal Sciences, vol. 84, pp. 34-47, Oct. 2014, DOI: 10.1016/j.ijthermalsci.2014.04.005
7
ScienceDirect ScienceDirect
Transportation Research Procedia 00 (2019) 000–000 Available online at www.sciencedirect.com Transportation Research Procedia 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Transportation Research Procedia 40 (2019) 442–448 www.elsevier.com/locate/procedia
13th International Scientific Conference on Sustainable, Modern and Safe Transport 13th International 2019), Scientific Conference on Sustainable, and Safe Transport (TRANSCOM High Tatras, Novy Smokovec –Modern Grand Hotel Bellevue, (TRANSCOM 2019),Slovak High Tatras, Novy Smokovec – Grand Hotel Bellevue, Republic, May 29-31, 2019 Slovak Republic, May 29-31, 2019
Simulation study of dynamic heating process of molybdenum for Simulation study of dynamic heating process of molybdenum for sheet suited for rigid transport components design sheet suited for rigid transport components design a b Miroslav Paveleka*, Tibor Donicb Miroslav Pavelek *, Tibor Donic
Department of mechatronics and electronics, University of Zilina, Zilina 01026, Slovakia b Research Centreand of University Zilina. Zilina 01026,Zilina Slovakia Department of mechatronics electronics,ofUniversity of Zilina, 01026, Slovakia b Research Centre of University of Zilina. Zilina 01026, Slovakia
a a
Abstract Abstract This paper deals with the development of the simulation model of the moving molybdenum sheet, while the molybdenum sheet is This paper deals resistive with the heating. development of the simulationenvironments model of the moving sheet, while theStatic molybdenum sheet in is heated by direct COMSOL/MATLAB are usedmolybdenum to create mentioned model. model created COMSOL is compared measurement and based onenvironments this comparison the dynamic (COMSOL/MATLAB) adjusted. heated by direct resistivewith heating. COMSOL/MATLAB are used to createmodel mentioned model. Static modeliscreated in COMSOL is compared with measurement and basedfor on different this comparison the dynamic model (COMSOL/MATLAB) is adjusted. The dynamic model is created as in-loop simulation initial conditions of heating elements position and temperature The dynamicwithin modelmolybdenum is created assheet. in-loop for different conditionsdistribution of heating within elements position and temperature distribution Thesimulation result of dynamic modelinitial is temperature molybdenum sheet for given heating elements and speed.sheet. For that reason, proposedmodel modeliscan be used fordistribution optimization of direct resistive heating distribution withinshift molybdenum The resultthe of dynamic temperature within molybdenum sheet forsystem given heating elementssheet. shift and speed. For that reason, the proposed model can be used for optimization of direct resistive heating system of molybdenum of molybdenum sheet. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published byof Elsevier B.V. committee of the 13th International Scientific Conference on Sustainable, Peer-review under responsibility the scientific Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Peer-review under responsibility of the scientific Modern (TRANSCOM 2019). Modern and and Safe Safe Transport Transport (TRANSCOM 2019).committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Keywords: molybdenum sheet; direct resistive heating; finite element method. Keywords: molybdenum sheet; direct resistive heating; finite element method.
1. Introduction 1. Introduction This article discusses the usage of finite element method for the design of a precise simulation model of resistive This article discusses the usage finite element method theconsider design of a precise simulation of resistive heating process. Main issue was toofdevelop a model, whichfor will dynamic properties of model a heating system, heatinga process. Main sheet issue was to develop model, which will consider dynamic properties where molybdenum is being moved abetween two heating electrodes. Proposed model of hasa heating to be ofsystem, a high where a molybdenum sheet requirements is being moved twoofheating electrodes.ofProposed model has to be of a high accuracy in order to model andbetween the needs the development the equipment for the heating and accuracy in order to model requirements and the needs of the development of the equipment for the heating and * Corresponding author. address:author. [email protected] * E-mail Corresponding E-mail address: [email protected] 2352-1465 © 2018 The Authors. Published by Elsevier B.V. Peer-review©under responsibility of the scientific committee 2352-1465 2018 The Authors. Published by Elsevier B.V. of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). 2352-1465 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 13th International Scientific Conference on Sustainable, Modern and Safe Transport (TRANSCOM 2019). 10.1016/j.trpro.2019.07.064
2
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
443
shaping the molybdenum sheets into molds. Molybdenum is one of the few materials suitable for this purpose in a view of the long-term high-temperature load of the mold. The mold is exposed to the temperature gradient of up to 100 °C/mm at the maximum temperature of 2150 °C within the longitudinal direction of the container, where the melt is located whereby at adjacent part the corundum as an already growth product is located. Based on the extreme conditions during the process, it is crucial to not decrease the mechanical abilities during the shaping process of the mold. For that purpose, the structural changes and possibilities of avoiding them by properly set pre-heating respectively heating/cooling system of Mo sheets need to be predicted. Currently, the molds are made of molybdenum sheet having a thickness of 0,5 mm, made by powder metallurgy technology, i.e. by plastic deformation and sintering procedure. Specifically, the molybdenum sheet Grade M1 corresponding to the American standard ASTM B386 or GB 3877, where the chemical composition is approximately the same. As heating elements, standard copper electrodes are used. All of the mentioned material properties are precisely defined within proposed simulation model, which consider dynamic change of the molybdenum sheet location within the heating system. [1-3] Nomenclature a b c d h x xe xe1 ye NEP NXP NYP NZP xp yp zp SOE SHIFT tstop NOS TNOS IT POE I σ ρ ε k Cp i
width of molybdenum sheet length of molybdenum sheet thickness of molybdenum sheet electrode diameter electrode height distance between electrodes electrode position in x direction starting position of electrode in x direction electrode position in y direction number of electrode partitions number of molybdenum sheet partitions in x direction number of molybdenum sheet partitions in y direction number of molybdenum sheet partitions in z direction length of molybdenum sheet part width of molybdenum sheet part height of molybdenum sheet part speed of electrode movement final electrode shift partial simulation time number of simulation steps ratio of motion speed to partial simulation time initial temperature electrode pressure electrode current electrical conductivity density relative permittivity thermal conductivity specific heat capacity at constant pressure simulation step
2. Simulation model and material definitions The simulation model is developed in COMSOL environment as a 3D model with the options for reconfiguration the mutual position of electrodes, the electrode material and the pressure on the electrodes. The model is composed
444
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
3
of the molybdenum sheet and two electrodes. The molybdenum sheet is modeled as a block (domain) with a wanted width(a), depth(b) and thickness(c) and electrodes are modeled as cylinders with a wanted diameter(d), height(h) and distance(x) between them. The geometry (Fig.1.) is then complemented by physic setup. The Multiphysics “Joule heating (JH)” is used for the model. This Multiphysics is formed by the connection of “Electrical circuit (EC)” physic and “Heat transfer in solid (HTS)” physic. The physic “Electrical circuit” is used to set up the value of input current trough top boundary of one electrode and grounding through the top boundary of the other electrode. Within the EC module, the required contact pressure applied to the electrodes and the roughness of the contact between the electrode and Mo sheet can be adjusted. The HTS is used to set up parameters as initial temperature of the system and the method in which the temperature is transferred to the surrounding area. The proper function of the model is achieved by setting the spherical domain with a diameter of one meter, which is filled with air. All simulations are the set as time domain simulation, so we can determine the influence of studied materials on the speed of the heating of the Mo sheet. [4-6]
Fig. 1. Model of direct resistive Mo sheet heating – geometry.
Fig. 2. Model of direct resistive Mo sheet heating divided to subdomains.
Advanced model of molybdenum sheet in motion is created as a script in MATLAB environment. To be able to simulate the heating process of the molybdenum in motion, the simulation had to be divided into partial simulations. The partial simulation is basically time-limited step of simulation with constant position of the electrodes. As the simulation itself the geometrical model (Fig.2.) needed to be also divided to parts, which serve as elements to obtain temperature distribution (final conditions) from one step of the simulation and as input elements to define initial conditions of next simulation step, in which the position of the electrodes is changed. The number of partial simulation is given by (1), while the time of partial simulation is given by (2). 𝑁𝑁𝑁𝑁𝑁𝑁 = 0.1 ∗ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆/𝑆𝑆𝑆𝑆𝑆𝑆
𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆/𝑆𝑆𝑆𝑆𝑆𝑆/𝑁𝑁𝑁𝑁𝑁𝑁
(1) (2)
The position of electrodes given in each simulation step is determine by (3) in x-direction and by (4) in y-direction.
4
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
445
(3)
𝑥𝑥𝑥𝑥 = 𝑥𝑥𝑥𝑥1 + (𝑆𝑆𝑆𝑆𝑆𝑆/𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇) ∗ (𝑖𝑖 − 1)
𝑦𝑦𝑦𝑦 = 𝑎𝑎/2 − 𝑥𝑥/2
(4)
𝑥𝑥𝑥𝑥 = 𝑏𝑏/𝑁𝑁𝑁𝑁𝑁𝑁
(5)
𝑧𝑧𝑧𝑧 = 𝑐𝑐/𝑁𝑁𝑁𝑁𝑁𝑁
(7)
Equations (5), (6) and (7) determines the dimensions of the subdomains of the molybdenum sheet.
(6)
𝑦𝑦𝑦𝑦 = 𝑎𝑎/𝑁𝑁𝑁𝑁𝑁𝑁
The mentioned JH module of the COMSOL environment uses well-known equations for modeling a heat transfer in solid materials (8). 𝜌𝜌𝐶𝐶𝑃𝑃
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
(8)
− 𝛻𝛻 ⋅ (𝑘𝑘𝛻𝛻𝑇𝑇) = 𝑄𝑄
In this case, the heat source “Q” is determined by Joule-Lenz law of resistive loses in material structure due to the current flow through this structure. The material parameters and other model settings as final conditions settings of domains can be seen in Table 1 [5]. Table 1. Model settings material and final conditions Parameter/Domain
Molybdenum sheet
Electrodes
Material
Molybdenum
Copper
Electrical conductivity [S/m]
17.9e6
5.5e7
Density [kg/m^3]
10280
8700
Relative permittivity [-]
1
1
Thermal conductivity [W/(°C*m)]
142
400
Specific heat capacity at constant pressure [J/(°C*kg)]
254
385
Current conservation
YES
YES
Electric insulation
YES
YES
Thermal insulation
YES
YES
Heat flux
Sphere (1 m diameter with initial 20 °C)
3. Simulation results Simulation results are obtained for model with parameters defined in Table 2. Table 2. Parameters of simulation model. Parameter
a [mm]
b [mm]
c [mm]
d [mm]
h [mm]
x [mm]
xe [mm]
xe1 [mm]
Value
150
200
1
20
50
75
(xe1,i)
30
Parameter
ye [mm]
NXP [-]
NYP [-]
NZP [-]
xp [mm]
yp [mm]
zp [mm]
SOE [mm/s]
Value
52.5
40
30
2
4
5
0.5
1
Parameter
SHIFT [-]
tstop [s]
NOS [-]
TNOS [-]
IT [°C]
POE [MPa]
I [A]
i [-]
Value
140
10
14
0.1
20
0.3
1500
(1,NOS)
446
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
5
Fig.3 and Fig.4. show basic concept of simulation results of the proposed model. The first part of the results Fig.3a shows initial temperature distribution before the current flow trough the electrodes is aplied. Fig.3b is temperature distribution after 10 s this temperature distribution is than transferd to next simulation step with slightly adjusted electrode position as can be seen on Fig.3c and the 10 s simulation interval starts again. The result from secound simulation step can be seen at Fig.3d.
Fig. 3. Simulation results of molybdenum sheet with electrode shift.
Fig. 4. Simulation results of molybdenum sheet with electrode shift.
Fig.4. is showing the final temperature distribution of all remaing simulation steps, while the minimal final temperature (last step) of the molybdenum sheet is at value 343 °C and the maximal temperature at value 599 °C.
6
Miroslav Pavelek et al. / Transportation Research00Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia (2019) 000–000
447
Fig. 5. Simulation model of molybdenum sheet without electrode shift.
In comparison the simulation with static (parameters that does’t include electrode shift are the same as in first model) electrodes was made (Fig.5.) with simulation time 140 s position of electrodes in the midlle of the plate. In this case the minimal final temperature is at value of 335 °C and the maximal temperature at value 462 °C. 4. Conclusion In this paper, the design of thermal finite element simulation model of Mo sheet was described. The main aim was to target exact modeling of thermal field in the structure of Mo sheet, while dynamic motion of the investigated system was considered. This motion is required, when more precise and uniform thermal distribution within the heated sheet is required. The main problem during the investigation is the proper definition of the material properties of Mo sheets. Each manufacturer has different material composition. Therefore, the simulation model has to have the possibility of reconfiguration in order to meet experimental results. Second approach of presented paper was focused on the design of the model, which will consider a dynamic change of the heated place within Mo sheet, i.e. translation move was applied. Various software products have limitations regarding the dynamic movement of the investigated sample, therefore special script in MATLAB environment was developed. With the use of this approach, each iteration of the simulation accepts previous results, which are repeatedly implemented within the computation solution. Given proposal of the solution was required due to future works, which will be focused on the design of multi-physics simulation model of Mo sheet heating and molding system, where high-validity simulation models are expected to be used namely for further investigations of mechanical parameters and other molybdenum restrictions. Also experimental comparisons will be further realized and compared with simulations. Acknowledgements The authors would like to thank to the national grant agency APVV for project support No. APVV-0396-15 and No. APVV 14-0284 – Study of useful properties of molded molybdenum sheets applicable for horizontal crystallisation of sapphire monocrystals References [1] Karban, P., Mach, F., Dolezel, I., 2012. Hard-couple model of local direct resistance heating of thin sheets. In: Journal of Computational and Applied Mathematics 236(18), 4725-4731, 2012. https://doi.org/10.1016/j.cam.2012.02.036 [2] Plansee Group, https://www.plansee.com/en/index.html [15.12.2018] [3] Mori, K., Maki, S., Tanaka, Y., 2005. Warm and Hot Stamping of ultra-high tensile strenght sheets using resistance heating”. In: CIRP Annals-Manufacturing Technology, 54(), 209-212, 2005. [4] Frivaldsky, M., Drgona, P., Spanik, P., 2013. Experimental analysis and optimization of key parameters of ZVS mode and its application in the proposed LLC converter designed for distributed power system application, In: International Journal of Electrical Power and Energy Systems, vol: 47, pp. 448-456, DOI: 10.1016/j.ijepes.2012.11.016 [5] Dughiero, F., Forzan, M., Pozza, C., Sieni, E., 2012. A translational couple electromagnetic and thermal innovative model for induction welding of tubes. In: IEEE Transactions on Magnetics 48(2), 483-486, 2012. https://doi.org/10.1109/TMAG.2011.2174972
448
Miroslav Pavelek et al. / Transportation Research Procedia 40 (2019) 442–448 Pavelek, Donic / Transportation Research Procedia 00 (2019) 000–000
[6] Frivaldsky, M., Cuntala, J., Spanik, P., 2014. Simple and accurate thermal simulation model of supercapacitor for development of module solutions, In: International Journal of thermal Sciences, vol. 84, pp. 34-47, Oct. 2014, DOI: 10.1016/j.ijthermalsci.2014.04.005
7