Chapter 2
but they are extremely complex and their solution, even with the aid of large computers, requires simplifications and assumptions to be made about a number of aspects, e.g. choice of a turbulence
approximation, treatment of boundaries and numerical methods of solution. The simplifications and assumptions chosen, together with the fundamental equations, make up a mathematical model. It is obvious that such mathematical models need to be validated against experimental measurements.
2.2.2.1. Mathematical model.
Mathematical models are primarily confined to heat transfer and solidification phenomena where the fluid flow is usually treated rather simply. Such models were presented by a number of authors,
f441 [451
among them are Savage and Pritchard, J Adenis et al., J Hills, Donaldson and H e s s , ^ ^ Mizikar, F a h i d y ^ ^ and Szekely and Stanek. Later, as the development of the turbulence t h e o r y a n d the improvement of numerical methods for solving multidimensional turbulent flow problems have progressed,
mathematical models of fluid flow in steelmaking processes have been developed by several investigators. t-^-61] Recently, several
mathematical models have been developed to analyse fluid flow in continuous casting tundishes. Up to now, however, little
work has been published on mathematical models of fluid flow in continuous casting moulds. The following section, therefore, reviews work dealing with the mathematical models of fluid flow in continuous casting tundishes.
r 631
The mathematical model developed by Debroy and Sychterz is a two-
Chapter 2
dimensional one, which describes isothermal, incompressible, steady state and turbulent fluid flow in tundishes. The fundamental
equations used for flow predictions are the equation of continuity and the Navier-Stokes equation in two dimensions. For the computation of the turbulent viscosity, the prandtl's mixing length
r 681 hypothesis is used, which can be written as :-
- Pi2 du
By
where: p: density of the medium; p^\ turbulent viscosity; I: prandtl mixing length;
|3u/3y|: absolute value of the velocity gradient along a direction perpendicular to the direction of flow.
The mixing length is defined as:-
1 = 0.4y.
where: y: distance to the nearest wall. The effective viscosity is expressed as:-
“off
" " t +
where: Peff' effective viscosity;
p: molecular viscosity of the medium.
f 621
Tanaka et al. introduced a three-dimensional mathematical model
Chapter 2
of tundish systems. The equation of continuity and the Navier-Stokes equation in three dimensions were used as the fundamental equations. They employed the k-e model of Jones and Launder to calculate the turbulent viscosity. According to Jones and Launder, the turbulent viscosity is determined as:-
H t - K 1pk*/e
The governing equations for k and e are respectively:-
3 /j __ 3/c
(puLk - ) = G - pe dx. 1 a. dx.k i
and
3 /i ff de
(pu e - ) = (K2G - K3pe) e/k dx. l a e dx.l
du . du . du . where: G = p — + — -)
dx. dx. d x .
i i J
k: turbulence kinetic energy;
e: rate of dissipation of turbulence kinetic energy; A*eff: effective viscosity;
turbulent viscosity;
K x, K2 , K3 , a empirical constants.
They solved the above equations together with boundary conditions. The results were found to be in a good agreement with the
Chapter 2
experimental results they obtained from the physical models.
Similar mathematical models were developed by He and Sahai, Szekely and El-Kaddah^^ and Ilegbusi and Szekely.
Szekely et al. used mathematical models to study fluid flow in the tundishes when flow control devices, such as dams and wires, were employed. The PHOENICS computational package was used for the solution of the equations. And the removal of inclusions in different tundish designs was also assessed.
2.2.2.2. Methods of solving mathematical models
The solution procedures of the mathematical models of fluid flow in continuous casting system are mainly through the solution of the finite difference equations which are derived from the governing differential equations. Thus the methods involve the derivation of finite difference formulations from the differential equations and boundary conditions as well as methods for solving the resulting set of simultaneous non-linear algebraic equations.
The following methods have been used to derive finite difference formulations from the differential equations and the boundary conditions: - ^
a) from Taylor-series expansions;
b) through the integration over a finite volume.
There are many kinds of schemes which may be used in deriving finite difference equations. They are, mainly:-
a) central difference scheme;
Chapter 2
b) upwind difference scheme.
For the solution of a set of simultaneous non-linear algebraic
equations, successive-substitution techniques must be employed. When using this method, initial guesses for the values of the variables are substituted into successive-substitution formulae which have been derived from the algebraic equations, and new values are calculated, then these values are used as new guesses to the solution and so on. Such procedures are commonly referred as iterative. Iterative methods of solving simultaneous algebraic equations may be divided into two
[70]