Mei C., Zhou J., Peng X. Simulation and Optimization of Furnaces and Kilns for Nonferrous Metallurgical Engineering

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2 Modeling of the Thermophysical Processes in FKNME 

practice. If the dissipation term is ignored and the turbulent viscosity hypothesis

⎛⎞

⎜⎟

⎝⎠

= is adopted to Eq.2.81, the sensible enthalpy equation of the mixture

can be obtained in the form

rad R

HHQQ

tc t

ρλ∂

σ∂

⎧⎫

⎛⎞

∂

⎪⎪

+∇ − + ∇ = + +

⎜⎟

⎨⎬

⎜⎟

∂

⎪⎪

⎝⎠

⎩⎭

• U

(2.83)

where

is the turbulent diffusivity of enthalpy,

is the turbulent viscosityˈ

the turbulent Prandtl numbers. The sensible enthalpy is defined as

ref

()d ()d

HcTT cTT k

′′ ′′

=− ++

∫∫

where c

is specific heat weighed on mass fraction, c

is specific heat of species

before reaction, T

ref

is the reference temperature of enthalpy.

The source term from chemical reaction is

()

Δ 0

QRH=−

∑

(2.84)

The reaction heat

ΔH

˄J/kmol˅can be calculated based on species specific

enthalpy of formation at the given reference temperature. For n

kilo-molar

products or reactants, it is

()

Δ d

jijififip

HT nMhT cTT

⎧⎫

⎪⎪

′′

⎨⎬

⎪⎪

⎩⎭

∑

∫

(2.85)

A set of basic equations of chemical fluid dynamics in combustion are got in

term of the above species conservation, enthalpy, momentum and continuity

equations:

a) Continuity equation:

()

∂

+∇ =

∂

• U

b) Species conservation equation:

()

ii ii

YYS

ρΓ

⎛⎞

∂

+∇ − + ∇ =

⎜⎟

∂

⎝⎠

• U

c) Momentum equation:

()

∂

+∇ =−∇ + +∇ +

∂

••

UU F g

where

refers to Reynolds stress tensor

()

eff eff

μμρ

=∇+∇− ∇−•Π UU UI I

where p is pressure;

is the volume force;

eff

is the effective fluid viscosity

(equal to the sum of molecule viscosity and the turbulent viscosity, i.e.

), I is

the unit tensor. The dyad (a two-order of tensor) is defined by

  Ping Zhou, Feng Mei and Hui Cai

()

B="#

d) The enthalpy equation refers to Eq. 2.83.

2.3.2



Gaseous combustion models

There are very strong interactions between the turbulence and chemical reaction in

the combustion. The chemical reaction has an effect on the density and viscosity

due to heat release, which further influences turbulence. On the other hand, the

turbulence influences the combustion by intensively mixing reactants and products.

From the former section, in order to solve the basic equations of chemical fluid

dynamics in the combustion, it is needed to solve the second order nonlinear

partial differential equation with the source term including average chemical

reaction rate. Therefore, the key of the turbulent reaction model is how to model

the average chemical reaction rate. It is difficult to develop a general model

because it is simultaneously influenced by turbulent mix, molecule transport and

chemical reaction. So far, of the models mentioned above (Fan and Wang, 1992;

Fan et al., 1987; Zhou, 1994; Carol, 1987; Zhao et al., 1994; Zhen and Zhou, 1996;

CFX-4.2 Solver, 1997), the mixed-is-burnt and eddy-break-up models are most

widely applied.

To study combustion phenomena in the combustion devices, the heat

effects caused by combustion (such as the distribution of temperature and

heat flux) are mainly considered. Moreover, the influence of chemical

reaction on flow is also caused by its heat effect. Therefore, “a simple

chemical reaction system” is usually used to simulate complicated reaction

dynamics processes.

In the simple chemical reaction system, it is assumed:

a) The turbulent transport coefficient of all species are all the same at each point

of the flow field, i.e.

˄subscripts F, O and P stand for fuel, oxidant and

product respectively

˅.

b) Fuel and oxidant are combined in a fixed ratio i, the stoichiometric ratio,

such that:

()

fuel + k

oxidant 1+ k

productii⎯⎯→ (2.86)

Obviously, if any two of the three species concentrations are available in this

system, the third one can also be solved.

The mixture fraction f for the reaction can be defined by

χχ

−

(2.87)

where

=−

(2.88)

2 Modeling of the Thermophysical Processes in FKNME 

where Y is mass fraction. So,

o and

in Eq. 2.87 refer to

value of oxidant and

fuel respectively. Furthermore,

=−

, ǂ

=1.

In definition

ˈf is always positive, attaining its stoichiometric value

when

= 0. Thus:

(2.89)

The mean value of the mixture fraction f satisfies the following conservative

transport equation without source term and is a scalar with conservation feature.

()

ρμ

σσ

⎛⎞

∂

+∇ −∇ + ∇ =

⎜⎟

∂

⎝⎠

••U (2.90)

where

ρ is the fluid density, U is the mean fluid velocity,

and

are the

equivalent laminar and turbulent Prandtl numbers respectively and both equal

to 0.9.

2.3.2.1



Mixed-is-burnt model

The mixed-is-burnt model is mostly used to simulate the turbulent diffusion flame,

which feature much higher chemical reaction rates comparing to the mixing rates

between fuel and oxidant.

The mixed-is-burnt model assumes that:

a) The chemical reaction rate is infinite.

b) Fuel and oxidant cannot coexist instantaneously.

For the above diffusion flame, the instantaneous mass fraction can be calculated

based on instantaneous mixture fraction by the following relationship.

If f

˚f

, the mixture is made of fuel and product. There are

FOPF

, 0 , 1

YYYY

−

===−

−

(2.91)

If f

˘f

, the mixture is made of oxidant and product. There are

FO PO

0 , 1 , 1

YY YY

==− =− (2.92)

Although most combustions in real installations are taking place in turbulent

flow, we generally pay attention to the distribution of the mean value of various

variables, instead of their instantaneous value. However, we haven’t yet known the

relationship between the mean mass fraction or the mean temperature and the

mean mixture fraction. So, in order to obtain the mean mass fraction and the mean

temperature, the concept of the probability density function (Fan and Wang, 1992;

Fan et al., 1987; Zhou, 1994; Zhen and Zhou, 1996; Meng, 1997) has been

proposed.

For random mixture fraction “

ā fluctuating with time from 0 to 1, its

  Ping Zhou, Feng Mei and Hui Cai

probability existing in the region [f , fˇdf] can be defined as p(f)df, and p(f) is

called the probability density function.

()d 1pf f=

∫

(2.93)

The mean mixture fraction

and variance of the mixture fraction

′

˄for

convenience,

′

is expressed as g

˅are determined by

()d

∫

(2.94)

222

( ) ()d ()d ()

gff

ff f

=− = −

∫∫

(2.95)

For any function

(f), its mean and variance are:

() ( ) ( ,) dx

ϕϕ

∫

(2.96)

22 2

() ( ) ( ,) d ()xfpfxf

ϕϕ ϕ

′

=−

∫

(2.97)

In order to determine p (f), apart from calculating

, it is necessary to get g.

The modeled form of g equation used is

() ()

gT g

ρμ

ρμρ

σσ

⎛⎞

∂

+∇ −∇ + ∇ = ∇ −

⎜⎟

∂

⎝⎠

••U (2.98)

where k is the turbulent kinetic energy;

ε is the turbulent dissipation rate; C

and

are the empirical constants and their values are respectively 2.8 and 2.0.

If the expression of p (f) is given,

and g are got from the corresponding

equations, then the mean mass fraction of species can be obtained. This model is

called as k-

ε-

-g model. The mean mass fraction of fuel, oxidant and product are

respectively:

()

max ,0 d

⎛⎞

−

⎜⎟

−

⎝⎠

∫

(2.99)

max 1 ,0 ( )d

⎛⎞

=−

⎜⎟

⎝⎠

∫

(2.100)

and

PFO

1YYY=− − (2.101)

Therefore, the crucial problem is how to determine the probability density

function (PDF) of the mixture fraction. At present, there are mainly three kinds of

methods to determine PDF:

a) p(f) is specified on the knowledges of the turbulent fluctuation (Fan and

Wang, 1992; Fan et al., 1987; Zhou, 1994; Carol, 1987).

b) The control equations of p(f) are constructed and solved. (Zhou, 1994; Zhen

and Zhou, 1996) Moreover, the PDF transport equation, which includes the joint

probability density function of velocity and chemical thermo-dynamics parameters,

2 Modeling of the Thermophysical Processes in FKNME 

can be use to predict accurately any complicated chemical reaction mechanism.

But this method needs lots of computing time, hence is limited in the engineering

application.

c) The ESCIMO (engulfment-stretching-coherence-interdiffusion-interaction-

moving observer), which was proposed by Spalding, can analyze quantitatively

the influence of the factors such as turbulence, molecule transport and chemical

dynamics of turbulent combustion (Fan and Wang, 1992; Carol, 1987), in which

the complicated molecule transport and chemical dynamics model is able to be

used. However, when this method is used to analyze more complicated practical

problem, there are a lot of problems to be solved, for example, back flow and

unsteady process.

The first method is more extensively applied in the engineering problems. In

addition to the simple form of p(f), the velocities and temperatures predicted by

this method are in better agreement with the practical result. Here, two simple

PDF models are introduced.

Double delta function

Presented by Spalding, it assumes that: f takes only two values: f

and f

; if the

time fraction is

when f = f

, then the time fraction must be (1

) when f = f

. So,

the mixture fraction varies with time in a rectangular wave way, which indicates

that fuel and oxidant cannot simultaneously present at the same point. A double

delta function has the following form

() ( ) ( )

(1 )pf f f

αδ α δ

−+

=+− (2.102)

In the practical application,

is usually set to be 0.5, then

ffg

=+ ˈ

ffg

−

=−

Beta function

Compared to p (f) in a double delta function, the predicted result by adopting p(f)

in a beta function is closer to the experimental data (Dong,1998). This PDF model

has the following form:

()

fff

−

∫

(2.103)

(1 )

⎡⎤

−

=−

⎢⎥

⎣⎦

(2.104)

(1 )

(1 ) 1

⎡⎤

−

=− −

⎢⎥

⎣⎦

(2.105)

2.3.2.2



Eddy break-up model (EBU)

The turbulent premix flames can be considered as many micro gaseous eddy

groups which includes already burned eddies and going to be burned eddies in

  Ping Zhou, Feng Mei and Hui Cai

the different extent. The chemical reactions take place on the interface of the two

eddies. It is assumed that: the chemical reaction rate depends on the rate of the

micro groups of gaseous fuel breaking into smaller micro groups in the

turbulence; the break up rate is directly proportional to the dissipation rate of the

turbulent fluctuant kinetic energy. Therefore, the source term (that is the mean

chemical reaction rate) in the species conservation equation can be compoted in

terms of k and

(Fan and Wang, 1992; Fan et al., 1987; Zhou, 1994; Carol,

1987), this is called as eddy break-up model (EBU). In the eddy break-up model,

an explicit equation is solved for the mass fraction of the fuel:

()

FFRAlim

YYCCM

ρμ

με

ρρ

σσ

⎛⎞

∂

+∇ −∇ + ∇ =−

⎜⎟

∂

⎝⎠

••U (2.106)

with

23.6 , viscous mixin

model

4.0 , collision mixing model

με

⎧

⎛⎞

⎪

⎜⎟

⎨

⎝⎠

⎪

⎩

Aie

infinite rate chemistr

finite rate chemistr

1.0,

1.0, ,

0.0, < ,

CDaD

Da D

⎧

⎪

=≥

⎨

⎪

⎩

lim

can be determined by solving transport equation or algebra expression. Here,

it is modeled as

()

lim

min , , without a product term

min , , , with a product term

⎧

⎛⎞

⎪

⎜⎟

⎝⎠

⎪

⎨

⎛⎞

⎪

⎜⎟

⎪

⎝⎠

⎩

Damkohler number, Da, is defined by

≡

where

≡ , it has the dimensions of time and is called as the turbulent

diffusion time.

is the chemical reaction time:

()

CH CH F O

AYY

τρρ

= (2.107)

where A

is rate constant; T

is activation temperature; a and b are exponents

for the fuel and oxidant density respectively; D

is the ignition/extinction value

of D.

If Da>>1, the combustion is the finite reaction flow controlled by diffusion; if

Da<<1, the combustion is the finite reaction flow controlled by kinetics.

The mass fraction of oxidant and product are given by

2 Modeling of the Thermophysical Processes in FKNME 

−

=− −

(2.108)

PFO

1YYY=− − (2.109)

2.3.2.3



Magnussen soot model

Soot is a kind of black solid particles taking formation when carbon fuel is

burned in a poor oxygen cnciornment. The soot in gaseous flames can result in

significantly enhancement of radiative heat transfer, but soot emission cause

serious environmental pollution. Detail chemical kinetic models of soot

formation are very complex, and there are many factors influencing formation

and space distribution of soot, namely turbulent scale, chemical stoichiometric

ratio and mixture fraction. There are a lot of models of soot formation, all of

which include some empirical parameters (Shadman, 1989; Magnussen, 1989;

Tesener et al., 1971). Here, we only introduce the mathematics model of soot

formation in gaseous flame, which was suggested by Magnussen.

The formation of soot particles occurs in two steps. In the first step radical

nuclei are created. Radical nuclei are defined to be the active sites on particles

from which the soot deposits will grow. The rate of formation of the radical nuclei

is given by

() ()

nnn

ρμ

σσ

⎛⎞

∂

+∇ −∇ + ∇ = + − −

⎜⎟

∂

⎝⎠

••U (2.110)

where n is the concentration of radical nuclei , mol/kg; N is the soot particle

concentration , kg/kg; n

is the rate of spontaneous formation of radical nuclei;

n −

) and g

are constants which in the case of an acetylene flame take the

values:

()

fg−= (2.111)

10g

−

= (2.112)

Radical nuclei are spontaneously formed at the rate

0Cfu

nAYC

−

= (2.113)

where Y

is the mass fraction of carbon in the fuel and C

is the mean

concentration of fuel (kg/m

). In the case of an acetylene flame the constants A

and E take the values:

13.5 10A =× (2.114)

910

=× (2.115)

The soot particle diameter d can be related to A, and there is the following

form:

constantAd =•

(2.116)

  Ping Zhou, Feng Mei and Hui Cai

The rate of soot particle formation is assumed to depend on the interaction

between the active radical nuclei and the carbon radicals, which combined

with the fact that the radical nuclei are destroyed on the surface of the soot

particles:

() ()

NNabNn

ρμ

σσ

⎛⎞

∂

+∇ −∇ + ∇ = −

⎜⎟

∂

⎝⎠

••U (2.117)

where a and b are constants.For the case of an acetylene flame, take the values

10a = ,

810b

−

=× .

2.3.3



Droplet and particle combustion models

In this section, combustion of coal and oil is to be taken as examples respectively

for introduction of particle and droplet combustion models.

2.3.3.1



Coal combustion in gas phase

The combustion of a coal particle is a two-stage process: the devolatilization of

raw coal particle and the oxidation of residual char. At the first stage, the coal

particles give off volatile, which is burned around char particle and the gaseous

flame in space is formed. At the second stage, the gas-solid two-phase combustion

takes place between the chars and oxidants. So, two separate gases are given off

by the particles, the volatiles and the char products, the latter come from the

burning of carbon within the particle.

The simulation of coal particles combustion is to solve jointly the gas-particle

two-phase flow model and the volatile-char oxidizing reaction model. The

coupling of the particle phase, the gaseous pahse and the concentration filed can

be handled by the PSIC method (Zhou,1994;Zhen and Zhou,1996). In the gas

phase, the description of the mixed-is-burnt ( k-

¦-f-g) and the eddy break-up

(EBU) models previously addressed are all applicable to the combustion of

volatile as fuel, apart from the volatiles, oxidant and products form only the part of

the gas phase that is not char products.

If the mixed-is-burnt model is being used, the instantaneous mass fractions are

given by:

()

ST PC

1ff Y−ı

()

ST PC

, 0

ff Y

−−

−

(2.118)

()

ST PC

1ff Y<−:

FOPC

0 , 1

YYY

==−−

(2.119)

If the eddy break-up model is used, and the equation for mass fraction of fuel is

2 Modeling of the Thermophysical Processes in FKNME 

unchanged, the mass fraction of oxidant is given by:

OF PC

YY Y

−

=− − −

(2.120)

For both models, the amount of products is given by:

PFOPC

1YYYY=− − − (2.121)

where Y

is the mass fraction of char products, the definition of mixture fraction

and stoichiometric mixture fraction can be seen in Section 2.3.2.

2.3.3.2



Coal pyrolysis and devolatilization

There are single-reaction model (Badzioch and Hawksley,1970) and two-reaction

model (Ubhayakar et al., 1976) to describe the devolatilization of the coal. In the

single-reaction model (Field et al., 1967), the coal is considered to have fixed

fractions of volatiles, char and ash. The rate of production of the volatile gases is

given by the first order reaction (Gibb,1985):

()

Vfv

km m

=− (2.122)

where m refers to the mass of volatiles which have escaped from unit mass of raw coal;

refers to the total yield of volatiles, the rate constant k

follows Arrhenius law:

exp

⎛⎞

=−

⎜⎟

⎝⎠

(2.123)

where T

is the temperature of coal particle (assumed uniform); A

and E

preexponential factor and activation energy respectively, and they are empirical

constants and vary with the type of coal.

Integration of Eq. 2.122 gives the fractional volatile yield as a function of time:

(

)

1exp d

=− −

∫

(2.124)

Since the coal particles are heated by the furnace gases, T

and k

varies with

time. The single-reaction model is only appropriate for the devolatilization at the

medium temperature. E

, A

and m

are functions of temperature, and their

values at the higher temperature are very different from that at the lower

temperature or at the medium temperature. Therefore, the two-reaction model is

more generally used. It assumes two-reaction with different rate parameters

compete to pyrolyse the raw coal. Considering the raw coal (dry and ash-free)

with the mass m

, then

volatiles˄m

˅+ residual char˄m

ch1

raw coal m

volatiles˄m

˅+ residual char˄m

ch2

(1ˉ

)

(1ˉ

)

  Ping Zhou, Feng Mei and Hui Cai

and

are the equivalent percentages of the volatiles for the corresponding

reaction. The first reaction dominates at lower particle temperatures and the

second reaction dominates at higher temperatures. Generally, the volatile

is less

than

during the devolatilization of coal. As a result, the total yield of volatiles

will depend on the temperature history of the particle, and will increase with

temperature. k

and k

is the rate constant following Arrhenius law:

exp

⎛⎞

=−

⎜⎟

⎝⎠

(n=1,2)

At time t, assume that a coal particle originally with unit mass consists of mass

of raw coal, mass m

of residual char after devolatilization has occurred, and

mass m

of ash. The reaction rate constants k

and k

determine the rate of

conversion of the raw coal

()

12c

kkm

=− + (2.125)

The rate of volatiles production is given by:

()

11 2 2 c

kkm

αα

=+ (2.126)

and the rate of char formation is:

()( )

()

11 2 2 c

kkm

αα

=− +− (2.127)

The initial value of m

is equal to (1 − m

); The values of

, A

and E

can

be got from approximate analysis of the coal.

, A

and E

can be got from

the pyrolysis analysis. Some references suggest: A

=3.7×10



=1.46×10



, E

=7.41×10

kJ/mol, E

=2.525×10

kJ/mol (Smoot and Pratt,

1983;Zhou,1986).

2.3.3.3



Char oxidation (heterogeneous reaction)

Although there are many models for describing char oxidation, Field model (Field

et al., 1967) and Gibb model (Gibb, 1985) are usually used in the simulation. The

former is a simple reaction model and the latter takes into account the diffusion of

oxidant within the pores of the chars. Since the combustion of the chars is much

slower than the coal devolatilization, it will decide the time of coal particle burnt

out.

In the Field (Fu and Wei, 1984), a char particle is considered to be a spherical

particle surrounded by a stagnant boundary layer (called stagnation film) through

which oxygen must diffuse before it reacts with the char. The oxidation rate of the

char is limited by the diffusion of oxygen to the external surface of the char

particle and by the effective char reactive rate. The diffusion rate of oxygen is

given by k

ˉp

), where p

is the partial pressure of oxygen in the furnace

gases far from the particle boundary layer and p

is the oxygen pressure at the