Scope of Present Study

This study established a hot-blast stove model to simulate on-gas and on-blast cycles of the hot-blast stove and found out the best mixing proportion between BFG and COG to promote the hot-blast stove efficiency and minimize the use of COG. Due to the limitation of simulation software and the pore distribution in checkers of hot-blast

stove being uniform, this study used porous media approximation to simulate the checkers.

In order to achieve the goal mentioned above, it is important to establish the optimization operation technique of hot-blast stove. During this technology development process, it is necessary to determine the mixing and burning characteristics of BFG and COG, heat transfer of solids and gases between regenerative bricks, cold blast and waste gas.

Then it can establish the internal flow field and thermal field to identify the control mechanism of every parameter in the hot blast generation process completely. Basically, the main purpose of this technical research and development is to provide the theoretical fundament of optimization operation technique and reduce the experimental times in the scene of hot-blast stove operation. In addition, it can use this model to appraise the advanced combustion technology in the future, for example, the feasibility of rich oxygen combustion or bio-fuel application to the hot-blast stove.

CHAPTER 2

MATHEMATICAL MODEL

2.1 Domain Description

The hot-blast stove of the China Steel Cooperation is illustrated in Fig. 2-1. The hot-blast stove is a heat exchange equipment constructed from two cylindrical refractory shells filled with checker bricks and refractory materials. Its architecture can be divided into four parts (see Fig.

2-2): combustion chamber, checker chamber, dome and burner. Inlet1 is the inlet of mixed fuel gas of BFG and COG, inlet2 is the one of combustion air, inlet3 is the inlet of cold blast, outlet1 and outlet2 are the exits of waste gases and outlet3 is the exit of hot blast.

On gas cycle, the mixed fuel gas of BFG and COG burns with the combustion air. The combustion products, or waste gases, ascend through the dome, and then descends down through the checker bricks, transferring energy to them. Finally, the waste gas arrives at the bottom of checker chamber. This on-gas cycle is about fifty minutes and enables the checker bricks to reach the required temperature, and then the hot-blast stove stops providing the mixed fuel gas and the combustion air. Then the on-gas stage changes to the stage of stove change, whose transition is about ten minutes. Subsequently, the hot-blast stove changes to the on-blast cycle.

The cold blast starts to enter the bottom of hot-blast stove, and it is heated as passing up through the checker bricks into the dome, and then proceeds into the combustion chamber before exiting the hot-blast stove through the hot blast valve. Eventually, the hot blast leaves the hot-blast

stove and is mixed with the cold blast to achieve the required hot blast temperature of blast furnace. The on-blast cycle is about sixty minutes.

When this hot-blast stove changes to the on-gas cycle, another stove provides the prepared hot blast to the blast furnace.

This study simulates the condition of mixed fuel gas of BFG and COG to burn with combustion air in the hot-blast stove and the heat transfer between gas and solid. From the simulation, this study intends to find out the best proportion between BFG and COG in the mixed fuel gas to increase the thermal efficiency of hot-blast stove.

2.2 Governing Equations

In order to make the physical problem more tractable and simplified, some assumptions are made as follows:

1. All gaseous mixtures are regarded as the ideal gases.

2. Specific heat capacity for each checker is constant.

3. Conductivity for each checker is piece-wise linear.

4. Neglect the radiation heat transfer.

5. One-step global reactions of hydrogen, carbon monoxide, methane and ethylene with air are adopted to represent the chemical reaction (combustion) of mixed fuel gas of BFG and COG.

6. Use porous media approximation to simulate the checkers.

Based on the assumptions mentioned above, the governing equations are given in the following:

2.2.1 The Continuity and Momentum Equation

Turbulent flows are characterized by fluctuating velocity field. In Reynolds averaging, the solution variables in the instantaneous (exact) Navier-Stokes equations are decomposed into the mean (ensemble-averaged or time-averaged) and fluctuating components. For the velocity components:

Likewise, for pressure and other scalar quantities:

φ =φ +φ′ (2-2) where ϕ denotes a scalar such as pressure, energy, or species concentration. Substituting expressions of this form for the flow variables into the instantaneous continuity and momentum equations and taking a time (or ensemble) average (and dropping the overbar on the mean velocity, u) yields the ensemble-averaged momentum equations. They can be written in Cartesian tensor form as:

( )

=0

Equations (2-3) and (2-4) are called Reynolds-averaged Navier-Stokes (RANS) equations. They have the same general form as the instantaneous Navier-Stokes equations, with the velocities and other solution variables now representing ensemble-averaged (or time-averaged)

values. Additional terms now appear that represent the effects of turbulence. These Reynolds stresses, −ρu_i^′u_j^′ , must be modeled in order to close Equation (2-4).

For variable-density flows, Equations (2-3) and (2-4) can be interpreted as Favre-averaged Navier-Stokes equations, with the velocities representing mass-averaged values. As such, Equations 2-3 and 2-4 can be applied to density-varying flows.

The Reynolds-averaged approach to turbulence modeling requires that the Reynolds stresses in Equation 2-4 be appropriately modeled. A common method employs the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients:

ij

2.2.2 The Energy Conservation Equation

In FLUENT, turbulent heat transport is modeled using the concept of Reynolds' analogy to turbulent momentum transfer. The “modeled”

energy equation is thus given by the following:

[ ] i ij eff h

The term involving (τ_ij)_eff represents the viscous heating, and is always computed in the coupled solvers. It is not computed by default in the segregated solver, but it can be enabled in the Viscous Model panel.

For the standard k−ε models, the effective thermal conductivity is given by where k, in this case, is the thermal conductivity. The default value of the turbulent Prandtl number is 0.85. You can change the value of the turbulent Prandtl number in the Viscous Model panel. S_h includes the heat of chemical reaction, and any other volumetric heat sources you have defined. where sensible enthalpy h is defined for ideal gases as

j energy due to chemical reaction:

⁼⁻

∑

volumetric rate of creation of species j.

2.2.3 Momentum Equation in Porous Media

Porous media are modeled by an addition of a momentum source term (Si) to the standard fluid flow equations. The source term is composed of two parts: a viscous loss term (Darcy law, the first term on the right-hand side of Equation 2-13) and an inertial loss term (the second term on the right-hand side of Equation 2-13).

⎟⎟⎠ D and C are prescribed matrices. This momentum sink contributes to the pressure gradient in the porous cell, creating a pressure drop that is proportional to the fluid velocity (or velocity squared) in the cell.

2.2.4 Energy Equation in Porous Media

FLUENT solves the standard energy transport equation in porous media regions with modifications to the conduction flux and the transient terms only. In the porous medium, the conduction flux uses an effective conductivity and the transient term includes the thermal inertia of the solid region on the medium described as follows:

( ) ( )

^hf

E_f = total fluid energy computed by FLUENT as the volumetric average of the fluid conductivity and the solid conductivity, which is in form of

s f

eff k k

k =γ +(1−γ) (2-15) where γ is the porosity of the medium, k_f is the fluid phase thermal conductivity and k_s is the solid medium thermal conductivity.

2.2.5 The Species Transport Equation

When choosing to solve conservation equations for chemical species, FLUENT predicts the local mass fraction of each species, Yi, through the solution of a convection-diffusion equation for the ith species. This conservation equation takes the following general form:

Y_i vY_i J_i R_i S_i t +∇⋅ =−∇⋅ + +

∂

∂ (ρ ) (ρr ) r (2-16)

where Ri is the net rate of production of species i by chemical reaction (described later in the Section 2.2.6) and Si is the rate of creation by addition from the dispersed phase plus any user-defined sources. An

equation of this form will be solved for N - 1 species where N is the total number of fluid phase chemical species present in the system. Since the mass fraction of the species must sum to unity, the Nth mass fraction is determined as one minus the sum of the N - 1 solved mass fractions. To minimize numerical error, the Nth species should be selected as that species with the overall largest mass fraction, such as N2 when the oxidizer is air.

In turbulent flows, FLUENT computes the mass diffusion in the following form:

turbulent viscosity and Dt is the turbulent diffusivity). The default Sct

is 0.7. Note that turbulent diffusion generally overwhelms laminar diffusion, and the specification of detailed laminar diffusion properties in turbulent flows is generally not warranted.

2.2.6 The Finite-Rate/Eddy-Dissipation Model

The reaction rates that appear as source terms in the species transport equation (2-15) are computed as follows. To consider the turbulent chemistry model, based on the work of Magnussen and Hjertager (1976), called the Finite-Rate/Eddy-Dissipation model. The finite-rate model computes the chemical source terms using Arrhenius expressions, ignoring the effects of turbulent fluctuations. The net source of the chemical species i due to reaction R_i, is computed as the sum of the

Arrhenius reaction sources over the Nr reactions in which the species Arrhenius molar rate of creation/destruction of species i in reaction r is given by:

Γ represents the net effect of third bodies on the reaction rate. This term is given by

By default, FLUENT does not include third-body effects in the reaction

rate calculation.

Here one step global reactions of hydrogen, carbon monoxide, methane and ethylene are adopted, the rate exponent, pre-exponential factor, temperature exponent and activation energy is described as follows: with reaction rate expression:

[ ] [ ][ ]2 2 (2)Carbon monoxide combustion reaction:

2

5 2

.

0 O CO

CO+ → (2-24) with reaction rate expression:

[ ] (2.239 10¹²)exp( 1.7 10⁸/RT)[ ][ ] [CO O₂ ^0.25 H₂O]^0.5

dt CO

d =− × − × (2-25)

(3)Methane combustion reaction

CH₄ +2O₂ →CO₂ +2H₂O (2-26) with reaction rate expression:

[ ] [ ] [ ]2 ^1.3 with reaction rate expression:

[ ] [ ] [ ]2 ^1.65

The standard k−ε model is a semi-empirical model based on model transport equations for the turbulence kinetic energy (k) and its dissipation rate (ε). The model transport equation for k is derived from the exact equation, while the model transport equation for ε was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart.

In the derivation of the k−ε model, it was assumed that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k−ε model is therefore valid only for fully turbulent flows.

The standard k−ε model in FLUENT falls within this class of turbulence model and has become the workhorse of practical engineering flow calculations in the time since it was proposed by Launder and Spalding (1972). Robustness, economy, and reasonable accuracy for a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. It is a semi-empirical model, and the derivation

of the model equations relies on phenomenological considerations and empiricism.

Standard Wall Functions

The standard wall functions in FLUENT are based on the proposal of Launder and Spalding (1974), and have been most widely used for industrial flows.

Momentum

The law-of-the-wall for mean velocity yields * 1ln( *)

Reynolds' analogy between momentum and energy transport gives a similar logarithmic law for mean temperature. As in the law-of-the-wall for mean velocity, the law-of-the-wall for temperature employed in FLUENT comprises the following two different laws:

„ Linear law for the thermal conduction sublayer where conduction is important.

„ Logarithmic law for the turbulent region where effects of turbulence dominate conduction.

The thickness of the thermal conduction layer is, in general, different from the thickness of the (momentum) viscous sublayer, and changes from fluid to fluid. For example, the thickness of the thermal sublayer for a high-Prandtl-number fluid (e.g., oil) is much less than its momentum sublayer thickness. For fluids of low Prandtl numbers (e.g., liquid metal), on the contrary, it is much larger than the momentum sublayer thickness.

In highly compressible flows, the temperature distribution in the near-wall region can be significantly different from that of low subsonic flows, due to the heating by viscous dissipation. In FLUENT, the temperature wall functions include the contribution from the viscous heating.

The law-of-the-wall implemented in FLUENT has the following composite form:

^{* 14 12}

{

^{Pr 2 (Pr Pr) 2}

}

where P is computed by using the formula given by Jayatilleke (1969):

^P

[

^{e t}

]

Note that, for the segregated solver, the terms

2

will be included in Equation 2-33 only for compressible flow calculations.

The non-dimensional thermal sublayer thickness, yT^* , in Equation 2-33 is computed as the y^* value at which the linear law and the logarithmic law intersect, given the molecular Prandtl number of the fluid being modeled.

The procedure of applying the law-of-the-wall for temperature is as follows. Once the physical properties of the fluid being modeled are specified, its molecular Prandtl number is computed. Then, given the molecular Prandtl number, the thermal sublayer thickness, y_T^* , is computed from the intersection of the linear and logarithmic profiles, and stored.

During the iteration, depending on the y_T^* value at the near-wall cell, either the linear or the logarithmic profile in Equation 2-33 is applied to compute the wall temperature T_w or heat flux q& (depending on the type of the thermal boundary conditions).

The function for P given by Equation 2-34 is relevant for the smooth walls. For the rough walls, however, this function is modified as follows:

P

where E′ is wall function constant modified for the rough walls.

Species

When using wall functions for species transport, FLUENT assumes that species transport behaves analogously to heat transfer. Similarly to Equation 2-33, the law-of-the-wall for species can be expressed for constant property flow with no viscous dissipation as

w and turbulent Schmidt numbers, and J_i,w is the diffusion flux of species i at the wall. Note that P_c and y_c^* are calculated in a similar way as P and

*

yc, with the difference being that the Prandtl numbers are always replaced by the corresponding Schmidt numbers.

Turbulence

In the k−ε models, the k equation is solved in the whole domain including the wall-adjacent cells. The boundary condition for k imposed at the wall is the wall-adjacent cells, which are the source terms in the k equation, are computed on the basis of the local equilibrium hypothesis. Under this assumption, the production of k and its dissipation rate are assumed to be equal in the wall-adjacent control volume.

Thus, the production of k is computed from

The ε equation is not solved at the wall-adjacent cells, but instead is computed using Equation 2-39.

2.2.7.1 Transport Equations for the Standard ^k−ε Model

The turbulence kinetic energy, k , and its rate of dissipation, ε , are obtained from the following transport equations:

_{k b M k} In these equations, Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated as described in Section 2.2.7.1.1: Modeling Turbulent Production in the k−ε Models.

Gb is the generation of turbulence kinetic energy due to buoyancy, calculated as described in Section 2.2.7.1.2: Effects of Buoyancy on Turbulence in the k−ε Models. YM represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, calculated as described in Section 2.2.7.1.3: Effects of

Compressibility on Turbulence in the k−ε Models. C_1ε, C_2εand C_3ε are constants. σ_k and σ_ε are the turbulent Prandtl numbers for k and

ε, respectively. S_k and S_ε are user-defined source terms.

Modeling the Turbulent Viscosity

The turbulent (or eddy) viscosity, μ_t , is computed by combining k

and ε as follows:

ρ ε

μ_t = C_μ ^k2 (2-42) where C_μ is a constant.

Model Constants

The model constants C1_ε ; C2_ε ; C_μ ; σk , and σ_ε have the following default values:

ε

C1 =1.44, C_2ε=1.92, C_μ=0.09, σ_k=1.0, σ_ε=1.3

These default values have been determined from experiments with air and water for fundamental turbulent shear flows including homogeneous shear flows and decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows.

Although the default values of the model constants are the standard ones most widely accepted, you can change them (if needed) in the Viscous Model panel.

2.2.7.1.1 Modeling Turbulent Production in the ^k−ε Models

The term Gk, representing the production of turbulence kinetic energy, is modeled identically for the standard k−ε models. From the exact equation for the transport of k, this term may be defined as

To evaluate Gk in a manner consistent with the Boussinesq hypothesis,

G_k =μ_tS² (2-44) where S is the modulus of the mean rate-of-strain tensor, defined as

S≡ 2SijSij (2-45)

2.2.7.1.2 Effects of Buoyancy on Turbulence in the ^k−ε Models When a non-zero gravity field and temperature gradient are present simultaneously, the k−ε models in FLUENT account for the generation of k due to buoyancy(Gb in Equation 2-40), and the corresponding contribution to the production of ε in Equation 2-41.

The generation of turbulence due to buoyancy is given by component of the gravitational vector in the ith direction. For the standard k−ε models, the default value of Pr_t is 0.85. The coefficient of thermal expansion, β , is defined as

_p For ideal gases, Equation (2-46) reduces to

It can be seen from the transport equation for k (Equation 2-40) that turbulence kinetic energy tends to be augmented (G_b >0) in unstable stratification. For stable stratification, buoyancy tends to suppress the turbulence (G_b <0) . In FLUENT, the effects of buoyancy on the generation of k are always included when you have both a non-zero gravity field and a non-zero temperature (or density) gradient.

While the buoyancy effects on the generation of k are relatively well understood, the effect on ε is less clear. In FLUENT, by default, the buoyancy effects on ε are neglected simply by setting Gb to zero in the transport equation for ε (Equation 2-41).

However, it can include the buoyancy effects on ε in the Viscous Model panel. In this case, the value of Gb given by Equation 2-48 is used in the transport equation for ε (Equation 2-41).

The degree to which ε is affected by the buoyancy is determined by the constant C3_ε . In FLUENT, C3_ε is not specified, but is instead calculated according to the following relation:

u

C_3ε =tanh v (2-49) where v is the component of the flow velocity parallel to the gravitational vector and u is the component of the flow velocity perpendicular to the gravitational vector. In this way, C_3ε will become 1

for buoyant shear layers for which the main flow direction is aligned with the direction of gravity. For buoyant shear layers that are perpendicular to the gravitational vector, C_3ε will become zero.

2.2.7.1.3 Effects of Compressibility on Turbulence in the ^k−ε Models

For high-Mach-number flows, compressibility affects turbulence through so-called dilatation dissipation, which is normally neglected in the modeling of incompressible flows. Neglecting the dilatation dissipation fails to predict the observed decrease in spreading rate with increasing Mach number for compressible mixing and other free shear layers. To account for these effects in the k−ε models in FLUENT, the dilatation dissipation term, Y_M , is included in the k equation. This

For high-Mach-number flows, compressibility affects turbulence through so-called dilatation dissipation, which is normally neglected in the modeling of incompressible flows. Neglecting the dilatation dissipation fails to predict the observed decrease in spreading rate with increasing Mach number for compressible mixing and other free shear layers. To account for these effects in the k−ε models in FLUENT, the dilatation dissipation term, Y_M , is included in the k equation. This

在文檔中 熱風爐之燃燒、流場與熱傳數值模擬分析 (頁 24-0)