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 

particle surface. The value of k

is given by.

0.75

2.53 10

−

⎛⎞

⎜⎟

⎝⎠

(2.128)

where R

is the particle radius; T

is the particle temperature; T

is the far-field

temperature

˗p is the local pressure and p

is atmospheric pressure. The char

oxidation rate per unit area of particle surface is given by k

. The chemical

reaction rate coefficient k

is given by

ccp

exp

kAT

⎛⎞

=−

⎜⎟

⎝⎠

(2.129)

where A

and T

depend on the type of coal and their values can refer to the

relative references (Smoot and Pratt,1983;Zhou,1986;Wall,1986). For this model,

and k

are in units of kg/(m

• atm • s). The overall reaction rate of a particle is

given by:

()

dc g

4π

−

−−

kk pR



(2.130)

and is controlled by the smaller rates of the k

and k

Gibb model (Gibb, 1985) takes into account the void fraction

of the char

particle, the particle volume /internal surface ratio a, the effective internal

diffusion coefficient D

of oxygen within the pores, and the molar ratio

carbon atoms/oxygen molecules.

The oxidation mechanism of carbon can be characterized by the parameter

then the oxides are produced according to the equation:

() ( )

CO 2 1CO 2 CO

φφφ

+⎯⎯→− +−

(2.131)

The value of

is assumed to depend on the particle temperature T

()

exp

⎛⎞

−

=−

⎜⎟

−

⎝⎠

(2.132)

where A

and T

are constants. Gibb suggests A

=2500K and T

=6240K.

By solving the oxygen diffusion equation analytically, the following equation is

obtained for the rate of decrease in the char mass m

()

123

kkk

ερ

−

∞

=− + +

−



(2.133)

where the far field oxygen concentration C

∞

is taken to be the time-averaged value

obtained from the gas phase calculation;

is the density of the char; k

is the rate

of external diffusion; k

is the surface reaction rate; k

is the rate of internal

diffusion and surface reaction. These are defined as follows:

(2.134)

  Ping Zhou, Feng Mei and Hui Cai

where D represents the external diffusion coefficient of oxygen in the surrounding

gas.

()

=−

(2.135)

where k

is the carbon oxidation rate, defined by the modified Arrhenius equation:

ccp

exp

kAT

⎛⎞

=−

⎜⎟

⎝⎠

(2.136)

where A

and T

are constants. Gibb recommends A

=14m/s and T

=21580K.

Further

()

coth 1 /kk a

ββ β

=− (2.137)

0.5

⎛⎞

⎜⎟

⎝⎠

(2.138)

Field model may predict sufficiently accurate when the temperature is not too

high but correction on the reduction and combustion of CO

must be considered if

the temperature is high (Zhou, 1982).

2.3.3.4



Oil combustion

The simulation on oil combustion is similar to that on coal combustion. The

mass fractions of fuel, oxidant and product in gaseous phase are predicted

through computing mixture fraction, which is identical to the aforementioned

gaseous combustion model (Guo, 1997). The difference is that both coal

devolatilization and carbon oxidation need to be considered in coal

combustion, which are the chemical reactions in solid and gaseous phase.

However, for oil combustion, the evaporation of the drops needs only to be

considered. Generally, there are no chemical reactions at the surface of oil

droplets and the combustions only carry out in volatiles given off by the

evaporation of the drops.

Spherical liquid droplets are assumed to be heated up to their boiling point

temperature and then to evaporate at a rate determined by the heat transfer to the

droplet and the latent heat of the liquid. The decrease in droplet diameter is

computed from the evaporative mass loss based on the assumption that the liquid

density remains constant.

In 1993, Barreiros suggested a mathematic model for oil combustion

(Barreiros et al.,1993). At any instant t, the droplet is assumed to have

diameter d

, a uniform temperature T

, while liquid density

and specific

heat capacity c

are assumed to be constant. According to the film theory for

the evaporation of liquid droplets, once the gas temperature T

exceeds the

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

boiling point temperature T

, the evaporation equation for the droplets is

given by:

()

d( ) 4

ln 1

dNu

=− + (2.139)

that is

()

pp p

ln 1

dNu

tcd

=− + (2.140)

where

is thermal conductivity; B is the mass transfer number defined by:

()

BTT

=− (2.141)

where L is the latent heat of vaporization of the liquid and c

is specific heat of gas,

is the modified Nusselt number and is given by:

Nu C Nu

= •

where

0.5 0.3333

20.55Nu Re Pr=+

(2.142)

0.5

0.5 0.3333

1.3333

1.237

1 0.278 1CRePr

RePr

−

⎛⎞

=+ +

⎜⎟

⎝⎠

(2.143)

where Re is the Reynolds number of the liquid droplet, that is:

Re v v

=−

(2.144)

where

is gas density, v

and v

are gas velocity and droplet velocity respectively;

Pr is the Prandtl number:

(2.145)

Hence, the decrease rate of droplet diameter is given by:

()

pp p

ln 1

tcd

=− +

(2.146)

Heat transfer to the droplet is modeled by the equation:

()

pg p

gp p 0p

Pp pPp

pp p

d6 d

TNu d

TT I T

tcdtcd

=−++− (2.147)

where I

is the radiative heat flux through environment to droplets˗

is emissivity

of droplet˗

is the Stefan-Boltzmann constant.

  Ping Zhou, Feng Mei and Hui Cai

2.3.4



models

given off from combustion consist of mostly nitric oxide NO and nitrogen

dioxide NO

. Besides, there is also minor nitrous oxide N

O. During the

combustion, the concentration of NO is higher than that of NO

, and NO

produced from NO. Thus, the key to the reaction dynamic model of NO

is to

research into the formation of NO. NO

is classified as thermal NO

, prompt

and fuel NO

, based on three distinct chemical kinetic processes that form

Thermal NO

is formed by oxidation of atmospheric molecular nitrogen N

high temperature. According to Zeldovich mechanism, it is generally described by

ˇO NOˇN (2.148)

ˇO

NOˇO (2.149)

Prompt NO

is formed by a series of reactions and many possible intermediate

species between hydrocarbon and N

. Based on Fenimore mechanism, it is

generally simplified as:

ˇN

HCNˇN (2.150)

ˇN

HCNˇNH (2.151)

ˇO

NOˇO (2.152)

ˇOH NOˇH (2.153)

ˇO

NOˇCO (2.154)

Fuel NO

is produced by oxidation of nitrogenous compound in devolatilized

fuel. Although the reaction mechanism for the formation and reduction of fuel

is unclear, the following observations can be made based on the researches

reported in the past a few years (Mao et al., 1998):

a) In the normal combustion conditions, nitrogenous organic compound in fuel

are heated and decomposed into HCN, NH

and intermediate product CN, etc,

which are emitted from fuel with volatiles. They are called volatile-N.

Nitrogenous compound remained in char are called char-N.

b) The percentages of HCN and NH

among volatile-N depend not only on the

types of fuel and the character of volatiles, but also on the chemical character such

as combination state between N and hydrocarbon. Besides, they are related to the

combustion conditions such as temperature.

c) In oxidizing atmosphere, HCN is oxidized into NO; In reducing atmosphere,

NO and HCN are reduced into N

The simplified model representing the forming and reducing mechanism of fuel

is showed in Fig. 2.5.

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

Fig. 2.5 Main path for fuel NO

formation and reduction

For thermal and prompt NO

, only the transport equation for NO species is

needed:

()

ONONO

YYS

σσ

⎛⎞

∂

+∇ −∇ + ∇ =

⎜⎟

∂

⎝⎠

••U (2.155)

For coal and oil combustion, in addition to the above equation, the transport

equations for HCN and NH

species are needed:

()

HCN

HCN HCN HCN

YYS

σσ

⎛⎞

∂

+∇ −∇ + ∇ =

⎜⎟

∂

⎝⎠

••U (2.156)

()

HNHNH

333

YYS

σσ

∂

⎛⎞

+∇ −∇ + ∇ =

⎜⎟

∂

⎝⎠

••U (2.157)

where Y

, Y

HCN

and

Y are respectively mass fractions of NO, HCN and NH

;

, S

HCN

and

S are respectively source or sink term in the mass fraction

transport equations for NO, HCN and NH

Since the mass fractions of those pollutants are generally small (<10

−3

), they are

calculated as passive combustion scalars, that is their influence on flow velocity

and other scalars, such as temperature, pressure and the concentration of other

species, can be neglected.

2.3.4.1 Thermal NO

formation

According to Zeldovich mechanism, the net rate of formation of thermal NO is

  Ping Zhou, Feng Mei and Hui Cai

given by

[

]

[][] [][] [][] [][]

12221 2

dNO

ON NO NON NOO

kkk k

−−

=+− −

(2.158)

In order to calculate the formation rate of NO, the steady state assumption for

[N] is used (Mao, 1998), thus we obtain:

[

]

[

]

[

]

[

]

[

]

[] [ ]

12 2 2 1 2

22 1

dNO 2O( O N NO )

dONO

kk k k

tkk

−−

−

(2.159)

where [NO], [O

], [N

], [O] and [N] refer to the molar concentration of

corresponding species respectively, and their units are mol/m

. Their reaction rate

coefficients are respectively given by

(Dong, 1998):

=1.8×10

exp(−38370/T) (m

molgV

-1

=3.8×10

exp(−425/T) (m

molgV

=1.8×10

Texp(−4680/T) (m

molgV

-2

=3.8×10

Texp(−20820/T) (m

molgV

Based on the assumption that the dissociation reactions of [O] reach a partial

equilibrium, we have:

[] [ ]

0.5

O 36.64 O exp( 27123/ )TT=− (mol/m

) (2.160)

In the transport equation for mass fraction of NO, the NO source term due to

thermal NO

mechanism is:

[

]

thermal,NO NO

dNO

= (2.161)

where M

represents the molecule weight of NO.

2.3.4.2 Prompt NO

formation

Prompt NO

mainly result from the combustion circumstance with poor

oxygen and rich fuel in which there are more CH

atomic groups. In

hydrocarbon flame, the atomic groups of CH and CH

, via the reactions

Eq.2.150 and Eq.2.151, form the intermediate products, and they are further

oxidized into prompt NO

. Prompt NO

formation is proportional to the

number of carbon atoms present per unit volume. Soete’s research results show

that, for most hydrocarbon fuel, the control equation of prompt NO

formation

rate can be described as (Soete,1975):

[

]

[][][ ]

spr 2 2

dNO

ONFuelexp

tRT

⎛⎞

−

⎜⎟

⎝⎠

(2.162)

where both

k and

are experimental constants and their values can refer to

reference (Dupont and Porkashnian,1993).

is the order of oxidation reaction

which depends on combustion conditions, as

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

1.0, <4.1 10

3.95 0.9ln , 4.1 10 <1.1 10

0.35 0.1ln , 1.1 10 <0.03

0 , 0.03

−

−−

−

⎧

⎪

−− ×< ×

⎪

⎨

−− ×<

⎪

⎩

(2.163)

where

x is oxygen molar fraction; f

is Soete model’s correction factor which

incorporates the fuel type, as

4.75 0.0819 23.2 32 12.2fn

φφ φ

=+ − + − (2.164)

where n is the number of carbon atoms per molar hydrocarbon fuel;

is the

equivalence ratio.

Therefore, in the transport equation for mass fraction of NO, the NO source

term due to prompt NO

mechanism is:

[

]

prompt,NO NO

dNO

= (2.165)

In the common combustion, prompt NO

accounts for a very small part, namely,

for coal combustion, prompt NO

formation is less than 5% of the total.

2.3.4.3



Fuel NO

formation

HCN path

The main reaction paths of fuel NO

by the intermediate products HCN are shown

as Fig. 2.6.

Fig. 2.6 The main reaction paths of fuel NO

by the intermediate products HCN

In terms of the NO reaction mechanism suggested by Lockwood and the

assumption that all char-N can be directly converted into NO (Lockwood,1992),

the source terms in the equations for mass fraction can be described as:

HCN pf,HCN HCN 1 HCN 2

SS S S

−−

=++ (kg/(m

• s)) (2.166)

OC,NONO1NO2NO3

SS S S S

−− −

=++ + (kg/(m

• s)) (2.167)

The rate of HCN production is equivalent to the rate of liquid fuel released into

  Ping Zhou, Feng Mei and Hui Cai

the gas phase through evaporation or the rate of solid fuel released into the volatile

through volatilization. That is:

pf,HCN pf NF HCN N

//SSmMMV= (2.168)

where S

is the rate of fuel release into the gas phase through evaporation or

volatilization; m

is the mass fraction of nitrogen in fuel; M

HCN

and M

are

respectively the molecule weight of HCN and N; V is the cell volume.

The mass consumption rates of HCN which appear in Eq.2.168 are computed as:

HCN 1 1 HCN

/SRMpRT

−

=− (2.169)

HCN 2 2 HCN

/SRMpRT

−

=− (2.170)

where R

and R

are conversion rates of HCN in reactions (ĉ) and (Ċ)

respectively, s

-1

; p is pressure, Pa, T is the mean temperature, K; R is the general

gas constant.

According to Soete’s research result, we have:

211HCNO 1

exp( /( ))RAxx ERT

=− (2.171)

22HCNNO 2

exp( /( ))RAxx ERT=− (2.172)

where

is the order of oxidation reaction and is the same as Eq.2.163, T is the

instantaneous temperature; x is the molar fraction and

=3.5×10

˄s



=3.0×10

˄s



=2.805×10

˄J/mol˅

=2.512×10

˄J/mol˅

is produced in reaction (ĉ) but destroyed in reaction (Ċ), thus the source

terms for Eq.2.167, i.e. S

NO−1

and S

NO−2

, are evaluated as follows:

NO 1 HCN 1 NO HCN 1 NO

//()SSMMRMpRT

−−

=− = (2.173)

NO 2 HCN 2 NO HCN 2 NO

//()SSMM RMpRT

−−

==− (2.174)

The mass consumption rates of NO in reaction (

ċ), S

NO-3

(kg/(m

• s)), can be

expressed as:

NO 3 BET s NO 3

/1000SACMR

−

= (2.175)

where A

BET

is BET surface area

Ზ

/kg); C

is the concentration of particles, kg/m

;

ᲖBET surface area is a method for measuring specific surface area, which is suggested by Brunauer,

Emmet and Teller and based on multi-layer absorption theory for the gas absorbed by the surface of solid

particles (refer to “Chemical Engineering Handbook”(in Chinese),1989). It is called as BET method and the

specific surface area tested by means of this method is called as BET surface.

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

and the heterogeneous reaction rate of NO reduction on the char surface R

can be

modeled

by (Dong,1998):

33NO 3

exp( /( ))RAx ERTp=− (2.176)

where A

=230mol/ (atm • m

BET

• s); E

=1.428×10

J/mol; T is the mean

temperature, K; p is pressure, atm,latm=101325Pa; x

is the molar fraction of

NO.

The formation rate of NO from char-N is given by:

C,NO C NC NO N

//SSmMMV= (2.177)

where S

C,NO

is the char burnout rate, kg/s; and m

is the mass fraction of nitrogen

in the char.

For the liquid fuel, both S

C,NO

and S

NO−3

are equal to zero.

path

The main reaction paths of fuel NO

by the intermediate products NH

are

shown as Fig.2.7. The overall reaction are:

Fig. 2.7 The main reaction paths of fuel NO

by the intermediate products NH

ˇO

⎯

⎯→ NOˇH

Oˇ0.5H

(2.178)

ˇNO

⎯

⎯→ N

ˇH

Oˇ0.5H

(2.179)

According to Soete’s research result, the reaction rates are respectively given

by:

1NHO

4.0 10 exp( 32000 /( ))kxx RT

=× − (2.180)

2NHNO

1.8 10 exp( 27000 /( ))kxx RT=× − (2.181)

where

is the oxidizing reaction order just as Eq.2.163.

The source terms in the transport equations can be described as follows:

NH pf,NH NH NH

333132

SS S S

−−

=++ (kg/(m

• s)) (2.182)

123

OC,NONO NO NO

SS S S S

−−−

=+++ (kg/(m

• s)) (2.183)

  Ping Zhou, Feng Mei and Hui Cai

The calculating method of S

pf, NH3

is similar to that in HCN paths, and there is:

pf,NH pf NF NH N

//SSmMMV= (2.184)

where

M is the molecule weight of NH

31 3

NH 1 NH

/SkMpRT

−

=− (2.185)

32 3

NH 2 NH

/SkMpRT

−

=− (2.186)

NO is produced in reaction (

ĉ) but destroyed in reaction (Ċ) and (ċ), we

have

131

NO NH NO 1 NO

//()SSMMkMpRT

−−

=− = (2.187)

232 3

NO NH NO NH 2 NO

//()SSMM kMpRT

−−

==− (2.188)

The calculating method of S

C,NO

and S

NO-3

are similar to that in HCN paths.

2.4



Simulation of Magnetic Field

Electric field and magnetic field widely exist in the power equipments in

FKNME. The interaction between the magnetic field and the current flow in the

melt generates electromagnetic forces causes the melt flow and interface wave

thus effecting the production directly. So, to simulate the melt flow and its flow

field, the current field must be first computed before the magnetic field can be

determined.

2.4.1



Physical models

The magnetic field in FKNME is a joint effect of the currents in the

electrode (anode, cathode, bus bar), the electric heating elements and the

melt. Meanwhile, because of the ferromagnetic materials such as the jack

and steel structure of the furnace, influences of ferromagnetic shields

should also be considered in order to simulate precisely the magnetic field

in the furnace.

Simplification has to be made if the electric field and the magnetic field are to

be simulated in a complex FKNME. Besides simplifying the furnace structure

and the shape of electrical conductor, the following simplification still has to be

done.

The time-averaged value is used to represent the current flow in the melt to

eliminate the current fluctuation with the operation process. The fluctuation is

actually the outcome of many factors and it is difficult to be described in a

mathematical way.

The current flow in the electric conduction elements is simplified into the