CHAPTER 2 MATHEMATICAL MODEL
2.4 Initial Conditions and Boundary Conditions
2.4.2 Boundary Conditions
The establishment of boundary conditions is based on the data provided by the China Steel Cooperation.
In the model domain, the boundary conditions are specified as the following: three inlets, three outlets, solid, gas, and wall boundary conditions.
An example involved with the calculation methods and steps for the setting of boundary conditions are given in detail as following.
Operation Conditions:
The operating pressure of hot-blast stove is set to be 101325 Pascal
and the gravity is set to be -9.8 m/s2 in y-direction.
The Inlet Boundary Conditions
There are three inlet boundaries. Inlet1 supplies the mixed fuel gas, inlet2 supplies the combustion air, and inlet3 supplies the cold blast.
1. Mixed Fuel Gas Inlet Boundary Conditions of Hot-Blast Stove The boundary conditions here demand for velocity, temperature, and species mass fraction.
Step1. Decide velocity and temperature:
The boundary conditions of velocity and temperature in the inlet of mixed fuel gas are given from the data provided by the China Steel Cooperation.
Step2. Set specie compositions and mass fraction of mixed fuel gas:
The boundary conditions of species compositions and mass fraction in the inlet of mixed fuel gas are given from the data provided by the China Steel Cooperation.
2. Combustion Air Inlet Boundary Conditions of Hot-Blast Stove The boundary conditions here demand for velocity, temperature, and species mass fraction.
Step1. Decide velocity and temperature:
The boundary conditions of velocity and temperature in the inlet of combustion air are given from the data provided by the China Steel Cooperation.
Step2. Set species compositions and mass fraction of mixed fuel gas:
The boundary conditions of specie compositions and mass fraction in the inlet of combustion air are given from the data provided by the China Steel Cooperation.
3. Cold Blast Inlet Boundary Conditions of Hot-Blast Stove
The boundary conditions here demand for velocity, temperature, and species mass fraction.
Step1. Decide velocity and temperature:
The boundary conditions of velocity and temperature in the inlet of cold blast are given from the data provided by the China Steel Cooperation.
Step2. Set specie compositions and mass fraction of mixed fuel gas:
The boundary conditions of specie compositions and mass fraction in the inlet of cold blast are given from the data provided by the China Steel Cooperation.
.
The Outlet Boundary Conditions
There are three outlet boundaries. Outlet1 and outlet2 discharge the waste gas, outlet3 discharges the hot blast.
The boundary conditions here demand for pressure only, and the pressure here is set to be 101325 Pascal.
Wall Boundary Conditions
Excepting the inlet and outlet boundaries, the remaining geometry is all wall boundaries, which are the no-slip boundary conditions (u, v, w=0) for velocity and adiabatic.
Porous Media
The top view of real checkers in the hot-blast stove is shown in Fig.
2-3. The porosity of every checkers is 0.225, obtained by following formula.
Porosity = wetted volume of fluid / total volume of checker region = 4296 ×π ×(55/2)2 /π ×(7600 /2)2 = 0.225
In Fluent, the porous media model can be used for a wide variety of problems, including flows through packed beds, filter papers, perforated plates, flow distributors, and tube banks. Due to the limitation of simulation software and the uniform pore distribution requirement of checkers in hot-blast stove, this study applied porous media approximation to simulate the checkers.
The materials and heights for each layer of checkers in the hot-blast stove are shown in Fig. 2-4 schematically, and their corresponding thermal properties used for simulation are listed in Table 2-1. The specific heats are maintained constant for each layer of checkers. As to the conductivities, the linear interpolations are applied for the temperature range in the simulation.
Table 2-1 Thermal properties of checkers for simulation Thermal property
Material
Specific heat capacity (J/kg-K)
Thermal conductivity (W/m-K)
273K 973K 2000K S21 1,180
1.40 1.57 2.00 273K 973K 2000K CRN130 1,180
2.21 2.04 1.98 273K 973K 2000K SF125 1,050
1.28 1.35 1.40 273K 973K 2000K SF120 1,050
1.28 1.35 1.40 273K 973K 2000K SF115 1,050
1.28 1.35 1.40
CHAPTER 3
INTRODUCTION TO NUMERICAL ALGORITHM
3.1 Introduction to FLUENT Software
FLUENT is a state-of-the-art computer program for modeling fluid flow and heat transfer in complex geometries. It provides complete mesh flexibility, including the ability to solve the flow problems using unstructured meshes that can be generated about complex geometries with relative ease. Supported mesh types include 2D triangular/quadrilateral, 3D tetrahedral/hexahedral/pyramid, and mixed (hybrid) meshes. FLUENT also allows to refine or coarsen grid based on the flow solution.
FLUENT is written in the C computer language and makes full use of the flexibility and power offered by the language. Consequently, true dynamic memory allocation, efficient data structures, and flexible solver control are all possible. In addition, FLUENT uses a client/server architecture, which allows it to run as separate simultaneous processes on client desktop workstations and powerful compute servers. This architecture allows for efficient execution, interactive control, and complete flexibility between different types of machines or operating systems.
All functions required to compute a solution and display the results are accessible in FLUENT through an interactive, menu-driven interface.
3.2 Numerical Method for FLUENT
FLUENT uses Segregated Solver method to solve the governing integral equations for the conservation of mass and momentum, and (when appropriate) for energy and other scalars such as turbulence and chemical species. In case a control-volume-based technique is used that consists of:
z Division of the domain into discrete control volumes using a computational grid.
z Integration of the governing equations on the individual control volumes to construct algebraic equations for the discrete dependent variables such as velocities, pressure, temperature, and conserved scalars.
z Linearization of the discretized equations and solution of the resultant linear equation system to yield updated values of the dependent variables.
3.2.1 Segregated Solution Method
Using this approach, the governing equations are solved sequentially (i.e., segregated from one another). Because the governing equations are non-linear (and coupled), several iterations of the solution loop must be performed before a converged solution is obtained. Each iteration consists of the steps illustrated in Fig. 3-1 and outlined below:
1. Fluid properties are updated, based on the current solution. (If the calculation has just begun, the fluid properties will be updated based on the initialized solution.)
2. The u, v, and w momentum equations are each solved in turn using
current values for pressure and face mass fluxes, in order to update the velocity field.
3. Since the velocities obtained in Step 2 may not satisfy the continuity equation locally, a Poisson-type equation for the pressure correction is derived from the continuity equation and the linearized momentum equations. This pressure correction equation is then solved to obtain the necessary corrections to the pressure and velocity fields and the face mass fluxes such that continuity is satisfied.
4. Where appropriate equations for scalars such as turbulence, energy, species, and radiation are solved using the previously updated values of the other variables.
5. When interphase coupling is to be included, the source terms in the appropriate continuous phase equations may be updated with a discrete phase trajectory calculation.
6. A check for convergence of the equation set is made.
These steps are continued until the convergence criteria are met.
3.2.2 Linearization: Implicit
In the segregated solution method the discrete, non-linear governing equations are linearized to produce a system of equations for the dependent variables in every computational cell. The resultant linear system is then solved to yield an updated flow-field solution.
The manner in which the governing equations are linearized may take an implicit form with respect to the dependent variable (or set of variables) of interest.
The implicit form is described in the following:
z Implicit: For a given variable, the unknown value in each cell is computed using a relation that includes both existing and unknown values from neighboring cells. Therefore each unknown will appear in more than one equation in the system, and these equations must be solved simultaneously to give the unknown quantities.
In the segregated solution method each discrete governing equation is linearized implicitly with respect to that equation's dependent variable.
This will result in a system of linear equations with one equation for each cell in the domain. Because there is only one equation per cell, this is sometimes called a scalar system of equations. A point implicit (Gauss-Seidel) linear equation solver is used in conjunction with an algebraic multigrid (AMG) method to solve the resultant scalar system of equations for the dependent variable in each cell. For example, the x-momentum equation is linearized to produce a system of equations in which u velocity is the unknown. Simultaneous solution of this equation system (using the scalar AMG solver) yields an updated u-velocity field.
In summary, the segregated approach solves for a single variable field (e.g., p) by considering all cells at the same time. It then solves for the next variable field by again considering all cells at the same time, and so on.
There is no explicit option for the segregated solver.
3.2.3 Discretization
FLUENT uses a control-volume-based technique to convert the governing equations to algebraic equations that can be solved numerically.
This control volume technique consists of integrating the governing equations about each control volume, yielding discrete equations that
conserve each quantity on a control-volume basis.
Discretization of the governing equations can be illustrated most easily by considering the steady-state conservation equation for transport of a scalar quantity φ. This is demonstrated by the following equation written in integral form for an arbitrary control volume V as follows:
∫
⋅ =∫
Γ∇ ⋅ +∫
computational domain. The two-dimension, triangular cell shown in Fig.3-2 is an example of such a control volume. Discretization of Equation 3-1on a given cell yields
v A A S V
Arf
= area of face f
(∇φ)n = magnitude of ∇φ normal to face f V = cell volume
The equations solved by FLUENT take the same general form as the one given above and apply readily to multi-dimension, unstructured meshes composed of arbitrary polyhedral.
By default, FLUENT stores discrete values of the scalar φ at the cell center (c0 and c1 in Fig. 3-2). However, face values φf are required for the convection terms in Equation 3-2 and must be interpolated from the cell center values. This is accomplished using an upwind scheme.
3.2.3.1 First-Order Upwind Scheme
When first-order accuracy is desired, quantities at cell faces are determined by assuming that the cell-center values of any field variable represent a cell-average value and hold throughout the entire cell; the face quantities are identical to the cell quantities. Thus when first-order upwind is selected, the face value ϕf is set equal to the cell-center value of ϕ in the upstream cell.
3.2.4 SIMPLE Algorithm
The SIMPLE algorithm uses a relationship between velocity and pressure corrections to enforce mass conservation and to obtain the pressure field.
If the momentum equation is solved with a guessed pressure field p*,
the resulting face flux J*f , computed from Equation 3-3 two cells on either side of the face, and Jˆf contains the influence of velocities in these cell. The term df is a function of ap, the average of the momentum equation ap coefficients for the cells on either side of face
f .) does not satisfy the continuity equation. Consequently, a correction J ′f is added to the face flux J*f so that the corrected face flux, Jf
Jf =J*f +J′f (3-5) satisfies the continuity equation. The SIMPLE algorithm postulates that
J ′f be written as
J′f =df(pco′ − pc′1) (3-6) where p′ is the cell pressure correction.
The SIMPLE algorithm substitutes the flux correction equations (Equations 3-5 and 3-6) into the discrete continuity equation (N
∑
faces =f f
f A
J 0)
to obtain a discrete equation for the pressure correction p′ in the cell:
′=
∑
′ +=N
∑
faces The pressure-correction equation (Equation 3-7) may be solved using the algebraic multigrid (AMG) method. Once a solution is obtained, the cell pressure and the face flux are used correctly.p= p* +αpp′ (3-9) Jf = J*f +df(pco′ − pc′1) (3-10) Here αp is the under-relaxation factor for pressure. The corrected face flux Jf, satisfies the discrete continuity equation identically during each iteration.
3.3 Computational Procedure of Simulation
The complete operating procedure for using FLUENT package software is carried out through the following processes sequentially.
3.3.1 Model Geometry
For FLUENT calculations, it is necessary to build a model firstly.
This study used the pre-processor software Solid Works to build the hot-blast stove model as shown in Fig. 3-3. It has to divide the hot-blast stove into finite volumes in this step in order to generate grids conveniently.
The inlets of mixed fuel gas and combustion air are built as cylinders with a diameter of 1.4 m, and the inlet of cold blast is built as a cylinder with a diameter of 1.6 m. The outlets of waste gas are built as cylinders
with a diameter of 1.6 m, and the outlet of hot blast is built as a cylinder with a diameter of 1.54 m.
3.3.2 Grid Generation
After building the hot-blast stove model, it has to use the pre-processor Gambit to generate grids as shown in Fig. 3-4. It defines the different grid sizes in different volumes in this step. Defining the smaller grid size for the smaller volume will increase the accuracy of the simulation, but it must consider the applicability of the grid size. If it adopts too small grid size in this step, the simulation time will be influenced. Besides, if the largest grid size is different from the smallest one too much, it will influence the FLUENT calculation.
3.3.3 FLUENT Calculation
Once determine the important features of the problem that one wants to solve, it will follow the basic procedural steps shown below.
1. Create the model geometry and grid.
2. Start the appropriate solver for 2D or 3D modeling.
3. Import the grid.
4. Check the grid.
5. Select the solver formulation.
6. Choose the basic equations to be solved: laminar or turbulent (or inviscid), chemical species or reaction, heat transfer models, etc.
Identify additional models needed: fans, heat exchangers, porous media, etc.
7. Specify material properties.
8. Specify the boundary conditions.
9. Adjust the solution control parameters.
10. Initialize the flow field.
11. Calculate a solution.
12. Examine the results.
13. Save the results.
14. If necessary, refine the grid or consider revisions to the numerical or physical model.
3.4 Grid and Time Step Tests
In order to obtain the acceptable numerical solution, this study applies the structure and unstructured grids produced from geometry models to carry out the grid and time step tests. The grid and time step tests all include on-gas and on-blast cycles.
This study uses the root mean square percentage error method [15]
to appraise the accuracy of simulation. The root mean square percentage error is calculated from the equation that is written as follows:
[
( )
] 121. On-Gas Cycle:
The boundary conditions of inlet1 and inlet2 (see Fig. 2-2 for the locations) are described as follows: Velocity and temperature in the inlet of mixed fuel gas are 27.2 m/s and 491 K respectively, and the mass fraction is tabulated in Table 3-1. Velocity and temperature in the inlet of combustion air are 12.9 m/s and 319 K respectively.
Table 3-1 Mass fractions in the inlet of mixed fuel gas BFG+COG
Species Mass fraction
CO2 0.327968
CO 0.207077
H2 0.003213
N2 0.458232
O2 0.000023
CH4 0.002933
C2H4 0.000554
There are two waste gas outlets (outlet1 and outlet2) (see Fig. 2-2 for the locations) in the hot-blast stove. The waste gas temperature listed in the experimental data is the temperature that is measured when the waste gases at outlet1 and outlet2 mix. Therefore, the waste gas temperature of simulation mentioned below is using the average waste gas temperature at oulet1 and outlet2.
The purpose for simulating the on-gas cycle is to obtain the waste
gas temperature.
Four different grid distributions (densities) are tested: they are 1027246, 1088989, 1324338 and 1834332 respectively. And three different time steps are tested: 60-, 30- and 15-second respectively. The test results are given in Table 3-2 to Table 3-3 and Fig. 3-5 to Fig. 3-6, in which the locations of inlet1, inlet2, outlet1 and outlet2 are illustrated schematically in Fig. 2-2. To consider the computational time and accuracy, the grid number of 1088989 and time step of 30 second are selected here.
Table 3-2 Grid test results of different grid densities for on-gas cycle Waste gas temperature (Time step= 60 sec)
Grid Number RMSE (%)
1027246 2.1 1088989 2.0 1324338 2.0 1834332 2.4
Table 3-3 Time step test results for on-gas cycle Waste gas temperature (Grid number= 1088989)
Time step (sec) RMSE (%)
60 2.0 30 2.0 15 2.1
2. On-Blast Cycle:
The boundary conditions of inlet3 (see Fig. 2-2 for the location) are described as follows: Velocity and temperature in the inlet of cold blast (air) are 39.3 m/s and 493 K respectively.
The purpose for simulating the on-blast cycle is to obtain the average hot blast temperature at outlet3 (see Fig. 2-2 for the location).
Four different grid distributions (densities) are tested: they are 1027246, 1088989, 1324338 and 1834332, respectively. And three different time steps are tested: 60-, the 30- and 15-second respectively.
The test results are given in Table 3-4 to Table 3-5 and Fig. 3-7 to Fig. 3-8, in which the locations of inlet3 and outlet3 are illustrated schematically in Fig. 2-2. To consider the computational time and accuracy, the grid number of 1088989 and time step of 30 second are selected here.
Table 3-4 Grid test results of different grid densities for on-blast cycle Hot blast temperature (Time step= 60 sec)
Grid Number RMSE (%)
1027246 1.3 1088989 1.3 1324338 1.4 1834332 1.5
Table 3-5 Time step test results for on-blast cycle Hot blast temperature (Grid number= 1088989)
Time step (sec) RMSE (%)
60 1.3 30 1.2 15 1.3
PC of Intel Core 2 with CPU 2.40 GHz, 2.93 GB RAM is applied to carry out the computation, and the absolute convergence criteria for x-velocity, y-velocity, z-velocity, k and epsilon are selected as 0.001, and the one for energy is 0.00001. Then the computational time for a typical simulation in the on-gas cycle needs about 10 hours, whereas it spends about 2 hours for the on-blast cycle simulation.
CHAPTER 4
RESULTS AND DISCUSSION
The main gaseous fuels used in hot-blast stove are the carbon monoxide and hydrogen, which are contained in the gas coming from the blast furnace (BFG). Using BFG as the fuel gas sometimes cannot reach the required hot blast temperature, so it must be enriched by the other fuel gas, such as coke oven gas (COG), with a higher net calorific value. The mixture of BFG and COG is constituted of CH4, CO and other usual molecules found in hydrocarbon flames. The compositions and corresponding volume fraction of BFG and COG provided by China Steel Cooperation (CSC) are summarized in Table 4-1.
Table 4-1 Respective Compositions (% in volume) of BFG and COG
BFG COG
CO2 23.02 2.13
CO 22.73 7.12
H2 3.74 56.08
N2 50.51 6
O2 0 0.1
CH4 0 25.78
C2H4 0 2.79
From the information provided by CSC, the price of BFG (0.38 NT$/Nm3) is about 10 times that of COG (3.94 NT$/Nm3), and the production of COG is much less than BFG. Therefore, if CSC can minimize the usage of COG, it will be able to reduce the operation cost of hot-blast stove greatly that meets the requirement for saving energy.
Accordingly, this study conducted a parametric study to change the respective mixing volume flow rates, i.e. heating values, of BFG and COG under a specified fuel mixture supply rate and the excess air ratio. The purpose is to find the optimal mixing proportion between BFG and COG that can achieve the best hot-blast stove efficiency effectively.
Before the parametric study was carried out, a reference case was described in details in Section 4.1 first. The operation process of hot-blast stove can be divided into on-gas and on-blast cycles; hence, the discussion of reference case includes both cycles.
4.1 Reference Case
This study used a commercial package software FLUENT to simulate the on-gas and on-blast cycles of hot-blast stove. The experimental data on April 9th, 2008 provided by CSC are set to be the boundary conditions for FLUENT calculation. The boundary conditions of on-gas and on-blast cycles can be referred in Section 3.4, and they are not repeated here. To
This study used a commercial package software FLUENT to simulate the on-gas and on-blast cycles of hot-blast stove. The experimental data on April 9th, 2008 provided by CSC are set to be the boundary conditions for FLUENT calculation. The boundary conditions of on-gas and on-blast cycles can be referred in Section 3.4, and they are not repeated here. To