A Finite Element Model for Numerical Analysis of Sintering

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1 A Finite Element Model for Numerical Analysis of Sintering DANIELA CÂRSTEA High-School Group of Railways, Craiova ION CÂRSTEA Department of Computer Engineering and Communication University of Craiova Str. Doljului nr. 14, bl. C8c, sc.1, apt.7, Craiova ALEXANDRU ADRIAN CÂRSTEA University of Craiova Abstract: A finite element model is presented to simulate coupling of thermal, electrical and mechanical behaviour of electric current-activated sintering. In many real systems there are natural couplings of the physical fields that interact so that this interaction can not be ignored. The material properties are dependent on electrical field and temperature in the system and displacement and stress distributions depend on material properties. These aspects are included in our work with emphasis on the development of numerical models using the finite element method. Key-Words: - Coupled fields; Finite element method; Sintering. 1 Introduction Current-activated sintering methods are presented in the professional literature as efficient methods to produce materials with performant properties. In these methods the large electric currents are used, and external loads are applied during processing. Joule- Lenz effect of the electric current produces high heating rates and can enhance diffusion or/and reaction processes. The external loads generate a stress that plays a significant role in the densification process. The influence of some relevant parameters on the sintering is of great importance for the engineer so that a good model must include it must be analysed. The temperature inhomogenities can affect the materials produced. More, we can control the sintering by stress distributions so that the mechanical aspects can not be neglected. The processing pressure is another important parameter for the materials quality. In our work the focus is on studying the influence of various material and control parameters using a finite element model for coupled fields. Analytical solutions for the electrical engineering problems are limited to some simple applications and ignore some physical phenomena. For complex problems the accurate models are necessary and the numerical solutions are efficient approaches for an optimal design and operation. With the advent of modern digital computers, many numerical models were developed and they become widely used in the scientific computing. We use the old algorithms and transform them for the new architectures but we must invent new algorithms having in our mind the computational power of the new computers. The efficient design of the electromagnetic devices has resulted in more stringent specifications and a demand for optimal operation, which is very important in high-performance electrical power systems. More exacting specifications have demanded during the design stage the development of accurate methods of predicting the performance characteristics of these devices. 1.2.Coupled models Many areas of engineering require the solution of problem in which the electromagnetic field equations are coupled to other partial differential equations, such as those describing thermal field, fluid flow or stress behaviour. These phenomena are described by equations that are coupled. The coupling between the fields is a natural phenomenon and only in a simplified ISSN: ISBN:

2 approach the field analysis can be treated as independent problem. In several cases, it is possible a decoupling and a cascade solution of the coupled equations. Another attractive and efficient approach of solving coupled differential equations is to consider the set as a single system. In this way a single linear algebraic system for the whole set of differential equations is obtained after discretization, and is solved to a single step. If one or more equations are non-linear, non-linear iterations of the whole system are required. The equations of the electromagnetic fields and heat dissipation in electrical engineering are coupled because the most of the material properties are temperature dependent and the heat sources represent the effects of the electromagnetic field. Thus, the electrical conductivity depends on the temperature and Joule s effect of the electrical current represents the main source for heating. Temperature has a significant effect of the stress distribution through thermal expansion. Fig. 1 Schematic principle The thermal effects of the electromagnetic field are both desirable and undesirable phenomenon. Thus, in conducting parts of some electromagnetic devices (coils of the large-power transformers, current bars, cables conductors, conductors of the electric machines etc) the heating is an undesirable phenomenon. The heat is generated by ohmic losses of the driving currents and eddy currents induced in conducting materials. But in induction heating devices for welding the heating is a desirable phenomenon. The thermal effect of the electromagnetic field is the treatment base for many electric materials in industry. As target example we consider the device from the Fig. 1 presented in [2]. The device has an axisymmetric configuration so that we use the cylindrical co-ordinate system Orz. The significances of the elements from the figure are 1 copper electrodes, 2 spacers, 3 plungers, 4 die, and 5 the sample. The sample can be a copper or alumina. The material for spacer, plunger and die is graphite. 2 Mathematical modelling of the electrical field The mathematical model for the sintering process is based on a set of governing equations for a three-way dynamic coupling of the fields. These equations are presented in the professional literature and are known as equations of the mathematical physics so that we shall not present in detail. The three fields have not the same time constant. Thus, the dynamic elastic behaviour and electrical potential reach the steady state in a much shorter period compared with the heat transfer so that we can use quasistatic models for these fields. The immediate consequences consist in reducing the complexity of modelling and simulation. The mathematical model for the electromagnetic field is based on Maxwell s equations for some particular cases. A complete physical description of electromagnetic field is given by Maxwell s equations in terms of five field vectors: the magnetic field H, the magnetic flux density B, the electric field E, the electric field density D, and the current density J. In low-frequency formulations, the quantities satisfy Maxwell s equations [3]: H = J (1) B E= t (2) div B = 0 (3) div D = ρc (4) with ρ c the charge density, σ the electric conductivity, and µ the magnetic permeability. For simplicity we give up to the bold notations for vectors. The second set of relationships, called the constitutive relations, is for linear materials: B = µ H; D = εe; J = σe The formulation with vector and scalar potentials has the mathematical advantage that boundary ISSN: ISBN:

3 conditions are more often easily formed in potentials than in the fields themselves. The magnetic vector potential is a vector A such that the flux density B is derivable from it by the curl ( ) operation The complexity of the mathematical model for electromagnetic field was one of the main reasons to find and develop new computation methods. All methods can be included in one of the following classes: Manipulation of the equations so that some unknowns are eliminated Definition of some potential functions from where the field unknowns can be obtained by simple processing Finding of some assumptions that simplifies the computation for practical problems The potential formulations seem attractive because of their computational advantages. One of these consists in the fact the boundary conditions are easily framed in the potentials than in the field themselves. In our work we consider the charge conservation law for quasistatic case: J = 0 ; J = σ E (5) with: ρ - the material resistivity, E - the electric strength and J the current density. A 2D-field model was developed for a resistive distribution of the electric field. A scalar electric potential φ is introduced by the relation [3]: E = ϕ (6) Laplace s equation describes the field distribution (for anisotropic materials): ( σ ϕ) = 0 (7) where σ is the electrical conductivity. This model is based on the real assumption that the electrical potential reaches the steady state in a much shorter period compared with the time constant of the heat transfer. 3 Mathematical modelling of the thermal field The thermal field is described by the heat conduction equation: [(cγ )( T ) T ] + [ k( T ) T ] = q (8) t T ( x,0) = T ( x) x Ω (9) 0 where: T (x, t) is the temperature in the spatial point x at the time t; k is the tensor of thermal conductivity; γ is mass density; c is the specific heat that depends on T; q is the density of the heat sources that depends on T, and T 0 (x) is the initial temperature. In the coupled problems we use the formula: q = ρ ( T ) J 2 (10) with ρ the electrical resistivity of the material. Equation (10) is solved with boundary and initial conditions. The boundary conditions can be of different types: Dirichlet condition for a prescribed temperature on the boundary; convection condition; radiation condition, and mixed condition. Radiation can be regarded as a simple surface loss subtracting from the surface power input. The Stefan- Boltzmann law gives the radiation loss. If the body is radiating to a surface at absolute temperature T Kelvin, the radiation loss is defined by [3]: P ( 4 4 r = ε r C 0 T T ) (11) where ε r is the emissivity coefficient of the surface (dimensionless) and T is the absolute surface temperature in Kelvin (K). The constant C 0 is W/m 2 K 4. For low temperatures the radiation loss is negligible but in our target example it must be considered. Consequently, it is convenient to use coupled models and accurate methods for computation of the heat penetration in the conductors, especially in some electromagnetic devices as the induction heating devices or sintering apparatus. 4 Mathematical modelling of the mechanical system In a stress analysis problem the displacement, strain and stress are of great importance. The physical quantities for stress analysis are: Displacement vector δ Strain vector ε and its principal values Stress vector σ and its principal values Some relevant criteria (Tresca criterion, Drucker-Prager criterion, Mohr-Coulomb criterion, Von Mises stress) For axisymmetric problems, the displacement field is assumed to be defined by the two components of the displacement vector in direction Or and Oz. Only three components of strain and stress tensors are independent in both plane stress and plane strain cases and four components for the axisymmetric problems due to the radial deformation. The equilibrium equations for axisymmetric problems are: 1 ( rσ r ) τ + r r z rz = f r (12) ISSN: ISBN:

4 1 ( rτ rz ) σ z + = f z (13) r r z where σ r, σ z τ rz are the stress components, and f r, f z are components of the volume force vector. Temperature strain is determined by the coefficients of thermal expansion and temperature difference between strained and strainless states. Components of the thermal strain for axisymmetric problem and orthotropic material are defined by the following equation: α z α r ε 0 = T (14) α θ 0 where α z, α r, α θ are the coefficients of thermal expansion along the corresponding axes for orthotropic material, and T is the temperature difference between strained and strainless states. For linear elasticity, the stresses are related to the strains by the constitutive law (Hooke's law): { σ} = [ D ]({ ε} { ε 0}) (15) where [D] is a matrix of elastic constants (Young's modulus, Poisson's ratio, shear modulus), and {ε 0 } is the column vector for the initial thermal strain. the software software products of type CAD offer a lot of program packages based on the FEM so that a specialist can use these packages in his interest area. In our research we used the product Quickfield of the Tera Analysis company [5]. Fig. 3 Final temperature on Oz versus space In Fig. 2 the analysis domain is shown. A half of the whole domain is used because of the symmetry. The mesh is built with triangles and linear interpolation functions are used. 5 Finite element model The numerical models for coupled analysis are obtained by discretizing in time and space the mathematical models presented in previous sections. The time dependent case require considerable more computing than the steady state since the time adds an extra dimension. The problem is if we do a time discretization firstly and then a space discretization, or firstly we do a space discretization and then the time is discretized. The second approach has an essential advantage: for the lumped-parameter system obtained by time discretization we have a very large number of methods because the classic theory of the lumped parameter was developed very much. Fig. 2 - The field domain The finite element method (FEM) is presented in a rich literature so that it is not necessary to present in this paper [1]. More, many companies in the area of Fig. 4 Temperature versus time in the sample center In our numerical simulation the sample material is alumina and the material of spacers, plungers and die is graphite. The time for simulation was 500 seconds. In Fig. 3 the final temperature along the axis Oz is plotted. The maximum value is in the sample. In Fig. 4 the temperature evolution in time at the point from the sample centre is shown. The temperature and the external load on the plunger generate the forces that appear in apparatus. From the stress analysis a deformation appear in the electrode. ISSN: ISBN:

5 4 Conclusion The problem of coupled fields in electrical engineering is a complex problem in terms of computing resources. In practice the coupled fields are treated independently in some simplified assumptions. The accuracy of the numerical computation is poor. With the new architectures, a multidisciplinary research is possible. Some computational aspects were presented with emphasis on the coupled problems. In coupled problems a hierarchy of decomposition can be defined with a substantial reduction of the computation complexity. References: [1]. Segerlind.L.J., Applied Element Analysis, John Wiley and Sons, 1984, USA. [2]. Wang, S., Casolco, S.R., Xu, G., Garay, J.R., Finite element modeling of electric currentactivated sintering: the effect of coupled electric potential temperature and stress. In: Acta Materialia 55(2007) [3]. Cârstea, D., Cârstea, I., CAD in electrical engineering. The finite element method. Editor SITECH Craiova, Romania. [4]. Cârstea, I., Advanced Algorithms for Coupled Problems in Electrical Engineering. In: Mathematical Methods and Computational Techniques in Research and Education. Published by WSEAS Press, ISSN: ; ISBN: Pg [5]. *** QuickField program, version 5.4. Page web: Company: Tera analysis. ISSN: ISBN: