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Abstract:
The equations governing the fluid flow problems are the continuity (conservation of mass), the Navier-Stokes (conservation of momentum) and the energy equations. These equations form a system of coupled non-linear partial differential equations. In general, closed form analytical solutions are possible only if these partial differential equations can be made linear, either because of non-linear terms naturally drop out or because of non-linear terms are small compared to other terms so that they can be neglected. If the non-linearity in the governing partial differential equations can not be neglected, which is the situations in most engineering flows, then numerical methods are needed to obtain solutions. The well known discretization methods used in CFD are Finite Difference Method (FDM), Finite Volume Method (FVM) and Finite Element Method (FEM).
In the present study, the features of FEM and FVM have been discussed and detailed flow analysis of a metallurgical tundish is done using FVM (SIMPLER algorithm based). A mathematical model has been prepared for tundish geometry and its boundary conditions have also been defined. A robust solution strategy is also prepared and proper choice of grid, time space and convergence criteria have been ascertained. Results of the tundish modeling are shown and different types of approximations of first and second order derivative used in our work are also discussed. Analysis includes flow visualization of tundish with dams, with both dams & weirs and without dams & weirs.
INTRODUCTION
Computational Fluid Dynamics (CFD) is concerned with obtaining numerical solution to fluid flow problems by computational methods. The advent of high-speed and large-memory computers has enabled CFD to obtain solutions to many flow problems including those that are compressible or incompressible, laminar or turbulent, chemically reacting or non-reacting. The equations governing the fluid flow problem are the continuity (conservation of mass), the Navier-Stoke [1, 2, 3] (conservation of momentum), and the energy equations. These equations form a system of coupled non-linear partial differential equations (PDEs). In general, closed form analytical solutions are possible only if these PDEs can be made linear, either because non-linear terms naturally drop out (e.g., fully developed flows in ducts and flows that are inviscid and irrotational everywhere) or because nonlinear terms are small compared to other terms so that they can be neglected (e.g., creeping flows, small amplitude sloshing of liquid etc.). If the non-linearities in the governing PDEs cannot be neglected, which is the situation for most engineering flows; the numerical methods are needed to obtain solutions. To solve these equations, we need to have appropriate boundary conditions and a suitable initial condition for non-permanent flow. By adjusting the source-term [1, 2] of the problem-considered, model can be extended for many applications such as heat transfer and transportation of fluid. The most accurate discretization technique used for fluid flow is Finite volume method. This method can be used in complex geometries and often leads to robust and cheap schemes. It also allows local conservativity of the numerical flux, which is quite attractive for the modeling of problems for which the flux is of importance.
The problem associated with the non-linearity can be solved by various methods. One way to deal this problem is to consider the corresponding time-dependent problem and use operator splitting method such as Peaceman-Rachford type or the θ method of time discretization, (Saramito and Piau [4] and Suresh kumar et al. [5]) to decouple the effect associated with non-linearity and incompressibility respectively. For the Newtonian flow Bristeau et al. [6] found much better results. Others tried to somehow linearize the non-linear term. An impressive result for this kind of try is achieved by Baranger and Sandri [7].
First time in early sixties, Harlow and Welch [8] uses incompressible viscous flow solver based on finite volume discretization. Due to the lack of a time-evolution equation for the pressure and the occurrence of the continuity equation, straight forward discretization of the incompressible flow equations leads to an unphysical odd–even coupling of the pressure. The most suitable approach to avoid this problem is to use different grid points for the pressure and the velocity (such grids are usually referred as "staggered grids"). Harlow and Welch, in their model put the pressure at the cell centers and the normal components of the velocity vector at the middle of the faces (this staggering of the variables on Cartesian grids has been described in details by Patankar). Recently a model proposed by Hwang [9] developed staggered grids for unstructured meshes where the pressure is stored at the centroids and the velocity vector at the cell faces of triangular meshes. Koshizuka et al. [10] have proposed a finite volume scheme for 2D grids where the velocity is stored at the centroids of the cells and the pressure on the vertices. For 2D meshes, Perot has proposed a finite volume scheme where the normal velocity components are stored at the face mid-points and the pressure at the circumcenters of the triangles.
However extending staggered unstructured meshes to general 3D unstructured grids is not straightforward but mostly pressure weighted interpolation based approach is used. Recently a more effective pressure-weighted interpolation (PWI) has been proposed by Miller and Griffiths [11]. After that, other researchers also found satisfactory results on unstructured collocated meshes using the PWI method. These flow-solvers mostly use temporal discretizations based on pressure-correction methods where a time step is divided into sub-steps. Firstly prediction of a starred velocity field is made that do not satisfy the continuity equation. Then a correction involving the pressure is computed such that the continuity equation is satisfied and this way iteration follows. The temporal accuracy of solver for the velocity is of the same order as the order of accuracy of the time stepping method (first order for Euler, second order for Cranck–Nickolson and so on), but the accuracy for the pressure evolution is only first order.
We have used a cell-centered finite volume scheme for our work. The time discretization of Navier-stokes equation is based upon the fractional time step method. The spatial discretization is based upon the recent published results of Patankar.
The tundish in a continuous casting is an intermediate vessel between the ladle, a batch vessel and the mould with a continuous operation. It serves as a metal reservoir and distributor. Continuous casting is one of the prominent methods of production of casts. Effective design and operation of continuous casting machines needs complete analysis of the continuous casting process. Recent works (P. Gardin a, M. Brunet) in this field has shown tundish a major component in the continuous casting process. It is used to enhance oxide inclusion separation and provides a continuous flow of liquid steel to the mould during ladle exchanges. With the development of sequential continuous casting, the increase in casting speed and the demand for improved quality, there has been a move towards tundish of larger capacity. It is also used to enhance the mixing of the different components of cast in the tundish.
Mathematical model is a useful tool for characterization of flow profile of liquid in tundish. A large number of works have been carried out on flow modeling in tundish. Rajeev K. Singh [12] has reported some decent flow visualizations and made RTD analysis of tundish using Fluent Inc. software. They characterize tundish of different configurations using a 3-D steady state mathematical simulation. In our present work, apart from generation of flow profiles of steel inside the tundish, the effect of weirs and dams have been predicted for flow behavior inside the tundish.
MATHEMATICAL MODELING
Physical Problem
The problem considered here is a metallurgical tundish. This is illustrated in figure 1. In previous development in the field of tundish metallurgy and the fact that the tundish is the last vessel that steel is in contact with prior to solidification , research programs were initiated to optimize the fluid dynamics in the tundish. In addition some operational problem had to be solved, which were:
a) Skull formation at the extremities of the tundish, which resulted in a gradual decrease in the working volume of the tundish.
b) Flaring of the two central streams from the tundish.
Flow visualization helps in making vital decisions about the designing and modeling of the process. The procedure of flow visualization, in practice consisted of achieving steady state in the tundish, injection of a tracer (methyl blue dye) in the ladle stream, observation of the tracer and recording through still photography. In order to modify the flow characteristic in the tundish, a variety of flow control devices (dams, weirs and slotted baffles) were inserted at various locations. A schematic diagram is shown below in figure 1. In the absence of flow control devices, which is the current situation in the most of the industries in India, large dead zones were observed at the two extremities of the tundish. These zones could lead to the skull formation in tundish, especially at low superheats, because of heat losses by radiation from the bath surface. The whole experimental procedure is complicated, and difficult to implement economically therefore computat-ional techniques are more preferred.
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