Numerical solutions and application of computational dynamics in Chemical Engineering

Computation Fluid Dynamics(CFD): A Numerical Solution to Fluid Flow Problems

N. Chakraborti² and Nishant K. Singh¹

¹ Nishant K. Singh, department of Chemical and Bioengineering,

National Institute of Technology, Jalandhar – 144011, Punjab, India

²Prof. Nirupam Chakraborti, Department of Metallurgical & Materials Engineering,

Indian Institute of Technology, Kharagpur - 721302, WB, India

Tel: 91-3222-283286, E mail: nchakrab@metal.iitkgp.ernet.in

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.

NOTATION

U X-direction velocity

V Y-direction velocity

W Z-direction velocity

t Temperature

k Turbulent kinetic energy

ε Kinematic rate of dissipation

C Mass fraction of tracer species

Cp Specific heat (liquid steel) (J/kg K)

D_eff Effective diffusivity (m²/s)

D₀ Binary diffusivity

E_max Maximum dimensionless concentration in C-curve

g Gravity acceleration

h Enthalpy (J/kg)

H Height dimensions of the tundish

K_eff Effective thermal conductivity (W/m k)

K ₀ Laminar thermal conductivity (W/m k)

P Static pressure (N/s²)

Pr_t Turbulent prandtl number = 0.85

Q Total volumetric flow through tundish

Re Reynolds number

Sc _t Turbulent Schmidt number = 0.7

α Volumetric thermal expansion co-efficient (m³/m² )

ρ Liquid steel density (kg/m³)

ρ_c Density of tracer fluid (kg/m³)

μ _eff Liquid effective viscosity

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.

Figure (1): Tundish with weirs and dams.

Governing Differential Equations

The governing equations provided below are adequate for obtaining a numerical solution for both the flow and temperature fields. Various symbols used in them are explained in the nomenclature.

(a) Continuity equation

(2.1)

(b) conservation of momentum

(2.2)

(2.3)

(d) Governing Equations for turbulent kinetic energy and its dissipation

(2.4)

(2.5)

(e) Mass Conservation equation for tracer particle

(2.6)

In the above set of coupled equations, m_t is defined as:

(2.7)

Further, G, the turbulence generation term, is defined as:

(2.8)

In the above equations, various turbulence parameters are taken as per standard turbulence model [12] as:

and (2.9)

The two driving buoyancy forces in the equation of motion and, are due to thermal gradient and steel flow respectively. Here represents a body force due to thermal gradients existing inside the vessel, and can be expressed as:

(2.10)

It is important to recognize here that in the above equation ρ represents a volume-averaged continuum density, and the momentum source term ( ), which takes into account the momentum transfer due to purging.

Initial and boundary conditions

Correct initial and boundary conditions are required to realistically simulate the flow. At t= 0 we have assumed that the vessel is empty at a specified temperature and with increase in time vessel is filled up with molten steel. At t = 0 derivatives of temperature, concentration of tracer are set to zero. No-slip conditions of zero velocity are introduced at the static walls. At the immediate vicinity of the tundish wall, logarithmic wall functions are used for calculating the shear stresses, turbulent kinetic energy, energy dissipation, and the velocity components parallel to the tundish wall. Details regarding the wall functions are described elsewhere. Heat losses from the side walls of the container are estimated by prescribing appropriate convective heat transfer coefficients at those locations, and writing a convective-conductive flux balance at the fluid-wall interface. The bottom of the vessel is treated as a horizontal wall and the boundary conditions imposed are the same as those for the tundish wall. For simplicity it is assumed that the top free surface of the liquid is at and coincides with the computational domain. No fluid flow is allowed across side boundary. The velocity at surface is taken as zero, as well as the concentration gradient at the top surface. Boundary conditions for tundish are shown in the figure 2 below:

Line Callout 1 (Accent Bar): V=0,
∂u/∂y =0,
∂w/∂y =0

Line Callout 1 (Accent Bar): ∂u/∂y =0,∂v/∂y=0,
∂w/∂y =0,∂T/∂y=0,
∂C/∂y=0,∂k/∂y=0,
∂ε/∂y=0

Figure 2: Boundary condition for fluid flow

SOLUTION STRATEGY

Accurate computations of momentum, heat and species transport fields need to be performed. To achieve this goal, coupled governing differential equations for fluid flow, heat transfer and mass transfer are discretized using a fully time-implicit finite volume technique, utilized in earlier works. The governing differential equations were discretized and the power-law differencing scheme was used to evaluate the finite-volume coefficients. The coupled fluid flow equations are solved using a pressure based primitive variable formulation according to the SIMPLER algorithm [13, 1]. Convection-diffusion terms in the discretized governing equations are modeled by employing a power-law differencing scheme .The resultant system of algebraic equations is iteratively solved by a line-by-line tridiagonal matrix algorithm (TDMA) solver, widely used in many transport calculations [14]. With this method, the boundary information is quickly transmitted to the interior of the domain by direction alteration of 'sweeping', resulting in a quick convergence. Also, as an aid for handling non-linearities, controlled convergence is achieved by the introduction of suitable under-relaxation parameters in the iterative scheme. Numerical simulations are performed with molten steel as the constituent material initially present in the problem domain. The computational domain is taken to be a non-axi-symmetric in Cartesian coordinate system. The maximum height of the vessel, H, has been taken as 0.65 m which is quite typical for the vessels used in many integrated plants. The thermo-physical properties used for simulating the problem are listed in Table 1 and 2.

Choice of grid size and time step

For the present simulation the choice of grid size is determined by two criteria. Those two criteria are as follows:

Criterion-1: In order to obtain an estimate of boundary layer thickness (δ) at the rigid side-walls, an appropriate scaling analysis is performed, following a procedure outlined in Bejan[13]. Accordingly, in order to resolve the thermal and flow behavior within the boundary layer, at least a few (typically five) grid points are accommodated inside it.

Criterion -2: In order to specify the correct eddy diffusivity values near the wall, it is necessary to control the grid size in its vicinity. If the grid points immediately adjacent to the wall fall beyond the 'near wall' regime but the diffusion coefficients are still evaluated by log-law, the simulation may lead to an erroneous estimation of the eddy diffusion coefficients. Therefore, it is necessary to have a grid distribution such that the grid adjacent to the wall always falls within the buffer layer. This condition is mathematically given as:

(3.1)

One can satisfy the above requirement by ensuring that some grids remains within the above bound, even for the maximum k value obtained in the solution. The value of k is typically low as the wall is approached, the appropriate grid size, which meets the above requirement, is found by trial and error method using limiting values of k.

Satisfying the two requirements discussed above, a non-uniform grid size of 65 is employed along the x-direction and a non-uniform grid size of 35x35 is employed along the y-z plane perpendicular to the x- direction. We have used denser griding scheme for the the area near the wall as well as near the dams and weirs, so that the above two criteria will not be violated. However, grid independence study confirms that a finer grid system does not alter the results appreciably. Transient simulations are performed using a time step size of 0.001 s. Although in the implicit time-discretization scheme employed for the present formulation, a coarser time-step could be used, it is observed that a time step size which is coarser than that chosen for the present simulation disturbs accuracy of the solution considerably.

Convergence Criteria

Convergence in the computed values is checked by monitoring the magnitude of all the primitive variables (u, T, k, ε) within inner iterations. Specifically, the following convergence criterion has been used based on the method of relative error:

Where Φ is the value of any primitive variable at a particular grid point at the current iteration level, Φ old is the value of the same variable at the same grid point at the previous iteration. Φ max is the maximum absolute value of the same over the entire domain.

MODEL DESCRIPTION

The flow profile of the tundish was calculated using standard k-ε turbulence model with standard wall functions. This model solves for incompressible mass conservation equation and three momentum balance equations for the velocities u_i
,and pressure P, throughout the domain of tundish:

= 0 (4.1)

= + + (4.2)

where = (4.3)

The effective viscosity, μ_eff, depends mainly on the turbulence variables k and ε, found by solving transport equations described in governing differential equation section (b).

The effect of thermal convection on flow is accounted by last term of the 2^nd equation (known as Boussinesq approximation) which account for thermal buoyancy force. The heat transfer model solves a 3-D energy transport equation:

= (4.4 )

Where (4.5)

A typical tundish configuration is shown in figure 1. At the inlet, velocity is fixed at 0.89 m/s in the negative y- direction which corresponds to casting speed of 0.55 m/min for a section 150x 250 mm.

The boundary condition over the outlet plane is the standard condition of zero gradience for velocity, pressure and the temperature. The top surface is a zero shear stress wall plane. Standard wall functions, based on the high reynolds no turbulence model of Launder and spalding [3] are implied in the other faces.

Table 1: Material property of liquid steel

Property

Unit

Value

Density

Molecular viscosity (μ)

Specific heat capacity (C_p)

Co-eff of expansion (α)

Thermal conductivity (K)

Molecular weight

Kg/m³

Kg/m.s

J/kg K

K^-1

W/m K

Kg/kg mole

7010-(T- T_liq)x 0.883

where T= operating temperature (⁰K) and

Tliq = liquidus temperature (1808 ⁰K)

5.55. 10^-3

680

1.10^-4

2473.24-1.211 T ( ⁰K)

55.85

Table 2: Casting Variables

Casting Variable

Unit

Value

Casting speed

Volumetric flow rate

Inlet Reynolds number

Inlet turbulence intensity

Inlet turbulence energy

Inlet turbulence dissipation rate

outlet turbulence intensity

Operating temperature

m/min

m³/s

m2/s2

m2/s3

⁰K

0.55

0.0034375

81114

3.89

0.0018154

0.0025938

3.73

1823

RESULTS AND DISCUSSION

Tundish is characterized on the basis of its flow profile under different conditions. Flow profile for different tundish configuration is evaluated at the following planes:

Vertical longitudinal xy-planes at inlet, outlet and mid plane.
Vertical transverse yz-planes at inlet and outlet plane.
Horizontal xz-plane.

It has been observed that the flow is truly three-dimensional as the flow field in various parallel planes differed significantly. It is also observed that the plugging inlet of liquid goes down to the bottom of the tundish (Figures 3(c), 4(c), 5(c)) where it spreads radially (Figures 3(b), 4(b), 5(b)). The stream then proceeds towards the longitudinal as well as transverse side walls. It then ascends upwards towards the top free surface and travels towards the middle of the tundish thereby creating a recirculating flow in the longitudinal and transverse direction.

In the existing tundish configuration (without dams and weirs), the recirculating zone in the transverse plane is symmetric as the metal inlet is located in the symmetric plane. In the central region of the tundish, rapid mixing was observed (see figure 3(a)). A short circuit stream was observed. This would lead to the very low residence time for the steel, thus giving inclusion very little time to float out. Finally, the streams from the two central nozzles were found to flare excessively, in a cyclic pattern. The cause for the flaring was the turbulence generated by the impact of the ladle stream.

We also made analysis of effect of weirs and dams in the existing tundish configuration. As shown in the figure, there is an obvious increase in mixing of components in some of the regions of tundish, when we use weirs and dams both at a time (Figure 5(a), 5(b), 5(c)). However no significant increase in turbulence is observed, when only dams are used (Figure 4(a), 4(b), 4(c)). The regions of better mixing are the regions where the turbulence is very high. The turbulence is created by the weir and dams in this case (Figure 5).

The insertion of a set of dams resulted in the elimination of short circuited streams (Figure 4). We can observe the significant amount of improvement in flaring of the two central streams from the tundish, when we introduce weirs in the arrangement (Figure 5).