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Source publication
An unconditionally stable fully explicit finite difference method for solution of conduction dominated phase-change problems
is presented. This method is based on an asymmetric stable finite difference scheme for approximation of diffusion terms and
application of the temperature recovery method as a phase-change modeling method. The computational...
Context in source publication
Context 1
... fixed temperature boundary condition of 1150 °C was applied at the ingot/mold boundary. The model problem, with the thermophysical properties and initial and boundary conditions, is summarized in Figure 9 and Table V, respectively. The numerical analysis was done with a uniform 81 · 81 and 161 · 161 grids with various values of Fourier number (0.25, 0.5, 1.0, and 1.5). ...
Similar publications
A truly meshless method for the analysis of phase-change heat conduction problems is presented. The method is based on the global Galerkin weak formulation, and the RBFs are used for the construction of the shape functions. A modified version of the Cartesian transformation method (CTM) is developed for the efficient and accurate evaluation of the...
Citations
... There are many mathematical models and numerical approaches to solve phase change problems in alloying systems [cf. 26,[27][28][29][30]. The optimal determination of boundary heat flux to maintain the desired temperature distribution in a binary alloy solidification problem has been considered in [31]. ...
Thermal optimization of the vertical continuous casting process is considered in the present study. The goal is to find the optimal distribution of the temperature and interfacial heat transfer coefficients corresponding to the primary and secondary cooling systems, in addition to the pulling speed, such that the solidification along the main axis of strand approaches to the unidirectional solidification mode. Unlike many thermal optimization of phase change problems in which the desirable (target) temperature, temperature gradient, or interface position are assumed to be a priori known, a desirable shape feature of the freezing interface (not its explicit position) is assumed to be known in the present study. In fact, the goal is equivalent to attain a nearly flat solid-liquid interface that is featured by its zero mean curvature. The objective functional is defined as a function of the temperature and mean curvature of the freezing interface. The solidus and liquidus iso-contours of the temperature field are used to implicitly determine the solid-liquid mushy region, i.e., the freezing interface. The temperature field is computed by solving a quasi steady-state nonlinear heat conduction equation. A smoothing method is employed to ensure the differentiability of heat equation. The explicit form of first-order necessary optimality conditions is derived. A gradient descent algorithm is introduced to find local solutions of the corresponding optimization problem. Numerical results support the feasibility and success of the presented method.
... The implicit interface tracking methods itself are divided into direct and indirect (latent heat evolution) approaches (c.f. [14]). In the former approach, the position of solidliquid interface is interpolated by an implicit function on a fixed spatial grid using either the sharp interface [15][16][17][18] or diffuse interface [5] models. ...
The regularity problem of heat conduction equation corresponding to modeling of solidification based on the latent heat evolution approach is considered in the present study. It is shown that the corresponding PDE is actually semi-smooth, an issue that has not been taken into account explicitly in the related literature. A general smoothing (regularization) strategy is introduced to solve this problem. More specifically, the smoothed version of the effective heat capacity method is presented in this work. The presented approach is applied to model the quasi steady-state heat transfer problem in the continuous casting process. Numerical experiments demonstrate the success of the presented regularization strategy.
... On the other hand, this kind of methods involves iterative calculations when dealing with non-linear equations. Tavakoli and Davami (2007) showed that the product of the number of time-steps by the number of iterations when using an implicit method might be higher than the number of time-steps when using an explicit method. They took cases presented by Swaminathan and Voller (1992) and compared their results generated with an implicit method to results generated with a fully explicit method. ...
Building thermal mass is a key parameter defining the ability of a building to mitigate inside temperature variations and to maintain a better thermal comfort. Increasing the thermal mass of a lightweight building can be achieved by using Phase Change Materials (PCMs). These materials offer a high energy storage capacity (using latent energy) and a nearly constant temperature phase change. They can be integrated conveniently in net-zero energy buildings. The current interest for these buildings and for better power demand management strategies requires accurate transient simulation of heavy and highly insulated slabs or walls with short time-steps (lower than or equal to 5 minutes). This represents a challenge for codes that were mainly developed for yearly energy load calculations with a time-step of 1 hour. It is the case of the TRNSYS building model (called Type 56) which presents limitations when modeling heavy and highly insulated slabs with short time-steps. These limitations come from the method used by TRNSYS for modeling conduction heat transfer through walls which is known as the Conduction Transfer Function (CTF) method. In particular, problems have been identified in the generation of CTF coefficients used to model the walls thermal response. This method is also unable to define layers with variable thermophysical properties, as displayed by PCMs. PCM modeling is further hindered by the limited information provided by manufacturers: physical properties are often incomplete or incorrect. Finally, current models are unable to represent the whole complexity of PCM thermal behavior: they rarely include different properties for melting and solidification (hysteresis); they sometimes take into account variable thermal conductivity; but they never model subcooling effects. All these challenges are tackled in this thesis and solutions are proposed. The first part (chapter 4) focuses on improving the CTF method in TRNSYS through state-space modeling, significantly decreasing the achievable time-steps. A complete solution to this issue can be reached by implementing in TRNSYS a wall model using a finite-difference method and coupling it with the CTF method in Type 56 (chapter 5). The second part (chapter 6) proposes an in-depth characterization of thermophysical properties of a PCM used in different test-benches, i.e. the density (Appendix A), the thermal conductivity (Appendix B) and the thermal capacity.
... It seems that in spite of large time increment, the computational efficiency of implicit methods is essentially not better than explicit methods for solidification modeling. Tavakoli and Davami [27] summarize the advantages and the drawbacks of implicit and explicit schemes according to the following properties: efficiency, robustness, adjustability, accuracy, simplicity of implementation, and scalability. From the revue, it is obvious that in this case (heat transfer with phase change/solidification), the performance of the implicit method is not better than that of the explicit ones because the problem is nonlinear. ...
... The best results were reached by the alternating direction explicit scheme (ADE) proposed by Larkin, [28] which is modification of Saul'yev's method. Unlike the classic explicit scheme, the ADE is more stable (Tavakoli and Davami [27] shows that in some cases unconditionally), which allows larger time steps. The ADE scheme is asymmetric and the governing equation is solved independently in the downward (À) and upward (þ) direction. ...
A supervision algorithm for controlling of continuous casting (CC) process is presented. The control strategy is based on the observation of temperature distribution through the casting strand. The algorithm is composed of two parts, an original 3D transient numerical model of the temperature field and the fuzzy-regulation model. The numerical model calculates and predicts the temperature distribution while the fuzzy-regulation model tracks the temperature in specific areas and tunes the casting parameters such as the casting speed, the cooling intensities in the secondary cooling, etc. The main goal is to keep surface and core temperatures in the specific ranges corresponding with the hot ductility of steel and adequately reacts on the variable casting conditions. The results show good and robust control behavior, fast response to dynamic system changes and general applicability for any CC process.
... Prior to closing this section we would like to point out to an elegant discussion on the shortcomings of the Niyama criterion by Spittle et al. [22] which is partly in agreement with our discussion. is governed by heat conduction [23,24]: ...
The goal of the present study is to predict the
formation of solidification induced defects in castings by
thermal criteria functions. In a criterion function method,
the heat transfer equation is firstly solved, and then the susceptibility
of defect formation at every point in the casting
is predicted by computing a local function, criterion function,
using results of the thermal analysis. In the first part
of the paper, some famous criteria functions, in particular,
the Pellini and Niyama criteria, are analyzed and their
shortcomings are discussed in details. Then, a new criterion
function is suggested to decrease the shape-dependency
issue of the former criteria. The feasibility of the new
method is studied by comparing numerical results against
some archived experimental data.
... Degree of superchilling can be predicted using either analytical or numerical methods-finite difference (FDM), finite elements (FEM), or finite volume method (FVM)-of the heat transfer equations. Analytical methods offer an exact solution and are mathematically elegant; however, due to their limitations, analytical solutions are mainly for one-dimensional cases with simple initial and boundary conditions and constant thermal properties (Abbas et al., 2004;Pham, 2008;Tavakoli and Davami, 2007). The advantage of numerical methods over analytical is that effects of the phase change over a range of temperatures, changing thermal properties, the step change in thermal conductivity over the same range, and that the heterogeneity of food products can be analyzed (Delgado and Sun, 2001;Pham, 2006;Resende et al., 2007;Zuritz and Singh, 1985;Tavakoli and Davami, 2007;Wang et al., 2007). ...
... Analytical methods offer an exact solution and are mathematically elegant; however, due to their limitations, analytical solutions are mainly for one-dimensional cases with simple initial and boundary conditions and constant thermal properties (Abbas et al., 2004;Pham, 2008;Tavakoli and Davami, 2007). The advantage of numerical methods over analytical is that effects of the phase change over a range of temperatures, changing thermal properties, the step change in thermal conductivity over the same range, and that the heterogeneity of food products can be analyzed (Delgado and Sun, 2001;Pham, 2006;Resende et al., 2007;Zuritz and Singh, 1985;Tavakoli and Davami, 2007;Wang et al., 2007). ...
... Studies on phase change problems have been carried out using the finite difference method (Clavier et al., 1994;Pham, 2006Pham, , 2008Tavakoli and Davami, 2007;Idelsohn et al., 1994;Voller, 1984Voller, , 1987Kim and Kaviany, 1990;Muhieddine et al., 2009;Wang et al., 2007;Wilson and Singh, 1987;Mannapperuma and Singh, 1989;Abbas et al., 2004). These studies have applied the finite difference method by using enthalpy or temperature formulation of heat conduction and have concluded that the method is successful to study the phase change problems. ...
The food superchilling process is of increasing importance because of its benefit in achieving food quality and extending shelf life of food products. The rate of the superchilling process is critical to the products’ quality and to the productivity of the process, and therefore the superchilling dynamics are of extreme importance. The objective of this work was to develop a one-dimensional implicit finite difference numerical model for predicting partial freezing time necessary to achieve an optimal degree of superchilling in foods and to validate the model experimentally. The evaluation of degree of superchilling was determined using finite slab and measured by using a calorimetry method. There is a good level of agreement between numerical simulation and laboratory experimental results.
... In this study, we propose the non-uniform Cartesian and cylindrical grids as the spatial domain discretization and the finite difference method for the numerical solution of governing equation (1). The literature [11] summarizes the advantages and the drawbacks of implicit and explicit schemes in solidification problems according to following properties: efficiency, robustness, adjustable, accuracy, simplicity of implementation, and scalability. There is shown that the performance of the implicit method is not better than that of the explicit method in the category of phase-change problems. ...
... There is shown that the performance of the implicit method is not better than that of the explicit method in the category of phase-change problems. The numerical model is discretized by the stable explicit scheme according to the Saul'yev asymmetric scheme [11] with the alternating direction explicit method (ADE) [12]. ...
Nowadays, continuous casting (CC) is used for providing almost one hundred percent of steel world production. The control of quality in CC products cannot be achieved without the knowledge of heat transfer and solidification of cast slabs. The solidifying slab passing through the caster is subjected to variable thermal conditions and mechanical stresses which can cause many serious defects in final structure of slabs. These defects might be eliminated by the optimal control of casting process. This paper describes an algorithm used to obtain time dependant casting parameters for stabilization of the casting process where high productivity is preserved. The presented algorithm can be used in two regimes. The first regime (off-line) can be used to find the control parameters such the casting speed and cooling intensities in the secondary cooling zone in order to get slab surface and core temperatures in specific ranges. The second regime allows the on-line regulation, stabilization of CC process and an immediate reaction to non-standard casting conditions such as a casting speed variation, or a breakdown of nozzle or cooling circuit in the secondary cooling. The algorithm is created as coupling between our original three-dimensional numerical model of temperature field and the regulation algorithm based on the fuzzy logic approach. The simulations indicate a good regulation efficiency and applicability for the real casting process.
... Since (1) and (2) should be solved within every topology evolution iteration, efficient numerical solution of these equations is a key to develop a practical optimization tool. In Tavakoli and Davami (2007a) a fast solidification solver has been introduced to solve conduction dominated solidification problems. Later, authors (Tavakoli and Davami 2007b) extend application of this solver to simulate real world sand casting process. ...
The optimal design of a casting feeding system is considered. The riser topology is systematically modified to minimize the
riser volume, while simultaneously ensuring that no defect appears in the product. In this approach, we combine finite-difference
analysis of the solidification process with evolutionary topology optimization to systematically improve the feeding system
design. The outstanding features of the presented method are: its efficiency, ease of implementation and simplifying definition
of the initial design. The efficiency and capability of the presented method are supported by illustrative examples.
... As the numerical optimization is an iterative procedure, efficient numerical method with reasonable accuracy is more desirable. In [28] an stable explicit finite difference method for solution of conduction dominated phase change problems was presented. The computational cost of this method is same as an explicit method (per time step) while it is free from the time step limitation due to stability criteria. ...
... For brevity of presentation we refer interested readers to Refs. [27,28] for more details about mathematical and numerical modeling of casting solidification used in the present study. ...
A method for automatic optimal feeder design in steel casting processes is presented. The initial design is the casting part (without feeders) which is placed in a suitable mold box. Design of each feeder contains the following steps: determination of the feeder-neck connection point on the casting surface, initial feeder design, feeder shape optimization and feeder topology optimization. Completing designing the first feeder, the method attends to designing the next one, if it is required, and the same procedure will be repeated. In the presented method, feeders are designed in a descending order of their sizes. The feasibility of the presented method is supported with an illustrative example.
... It seems that in spite of large time increment, the computational efficiency of implicit methods is not essentially better than explicit methods for solidification modeling. In [5] this issue has been discussed in detail. In [5] we present an unconditionally stable fully explicit finite difference method for the simulation of 2D solidification problems on simple geometries. ...
... In [5] this issue has been discussed in detail. In [5] we present an unconditionally stable fully explicit finite difference method for the simulation of 2D solidification problems on simple geometries. The computational cost of this method is the same as an explicit method per time step, while it is free from the time step limitation due to the stability criterion. ...
... There are several methods to incorporate the phase change effect in the heat conduction equation (for good survey, see [8,9]). Following [5], the temperature recovery method [10,11] is used in this study. This method consists of transforming the latent heat into an equivalent number of degrees by division of latent heat by the specific heat. ...
An efficient numerical method for simulation of casting solidification is presented. The presented method is based on a stable explicit finite difference solution of the heat equation in conjunction with the temperature recovery method to incorporate latent heat effect. The computational cost of the presented method is approximately the same as an explicit method (per time step), while it is free from the time step limitation due to the stability criterion. A simple domain decomposition method is included to improve the computational performance of the presented method. The efficiency, stability and accuracy of the presented method are supported with illustrative examples. Copyright © 2007 John Wiley & Sons, Ltd.
