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 

application, what we need to know are only time-averaged velocity, time-averaged

enthalpy and time-averaged turbulent features, and it is unnecessary to master the

details about production and development of the turbulence. Therefore, RAS method

has been widely applied in solving the practical engineering problems.

In the Reynolds-averaged simulation, according to different Reynolds number

(Re) , there are different models for closing time-averaged equations. For the

turbulence with high Re, standard k-

two-equation model and RNG model are

often used. For turbulence with low Re or the flow in near-wall regions, low Re

model is often used. In addition, RNG model is also used in model turbulence

with low Re (refer to Section 2.1).

In the case of the molten metal with high temperature, the fluid flowability

varies greatly with the type of the metal. For example, the viscosity of liquid steel

is 8.7×10



/s, the viscosity of nickel matte slag is 9.38×10

-5

/s, the viscosity of

nickel matte is 1.11×10



/s (Kaiser and Downing,1977) . At the same time, the

melt velocity greatly varies with the different equipments. For example, the

velocity of liquid aluminum in the nozzles of high speed roll-casting is about 5m/s,

while the velocity of high temperature melt in bath pool smelting is only of the

order of 10



m/s. Therefore, when the flow of high temperature melt is simulated,

it is necessary to choose the suitable turbulent model according to the

characteristics of the melt or equipments.

2.5.2



Electromagnetic flow

For most practical fluid motion, body force F

often appears in the form of gravity

and buoyancy. But, when the electrical current goes through the molten metal or

the external of furnace, it will induce an electromagnetic field and an additional

body force called as electromagnetic force, such as in aluminum reduction cell,

high-frequency smelting, electroslag remelting and ladle refining with

electromagnetic stirring.

Fig.2.13 is a sketch of equipment used in industrial production. In the

equipment, electromagnetic stirring has an important influence on the dynamic

conditions in the metal smelting process. For an induction furnace, when current

goes through coil (Fig.2.13 (a)), induced current is produced in the crucible melt

and further forms an electromagnetic field, which drives the melt to move. For an

electric arc furnace (Fig.2.13 (b)) there is also induced current and melt circulation

motion.

When theoretically analysis is carried out to study the fluid motion driven by

the electromagnetic force, it is necessary to couple the electromagnetic force

governing equations (including Maxwell equations and Ohm’s law) and the fluid

flow governing equation (Navier-Stokes equations).

  Ping Zhou, Feng Mei and Hui Cai

In the area of electromagnetic fuid dynamics, Maxwell equation is expressed by

(Szekely, 1985; Li et al., 1985; Deng et al., 1997; Zhou and Mei; 1997a, b):

∂

∇× = −

∂

˄Faraday’s law˅ (2.207)

∇× =BJ ˄Ampere’s law˅ (2.208)

0∇=

• B ˄Continuity condition of magnetic field˅ (2.209)

0∇=• J ˄Law of current continuity˅ (2.210)

Fig. 2.13 The sketch of equipment with fluid motion driven by electromagnetic force

If there is a fluid motion, Ohm’s law can be written as

()

σ=+×JEUB (2.211)

where E is electric field intensity,V/m; B is magnetic flux density,Wb/m

; J is

current density, A/m

; U is melt velocity, m/s;

is magnetic permeability, H/m;

is electric conductivity, S/m.

The second term at the right of Eq.2.211 stands for current induced by the

convective motion of the melt. In most practical application related to induction

stirring, the term is often neglected because of low melt velocity (Szekely,1985).

Then Eq. 2.211 can be approximately expressed by

σ=JE

(2.212)

Current density

J and magnetic flux density B can be got by solving Eq. 2.207 ~Eq.

2.210. Therefore electromagnetic body force (Lorentz Force) can be obtained

based on the following equation.

=×FJB

(2.213)

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

The major difficulty to solve Lorentz Force is that the boundary conditions of

the electromagnetic field (or the value of

B, E and J) are not easily obtainable

unless through complex and repeated field computations.

In practical application, above equations are often simplified according to the

specific conditions of the problem. By eliminating variables

E and J among Eq.

2.207

̚Eq. 2.210, the governing equation of the magnetic flux density can be given

∂

()

=∇× × + ∇UB B (2.214)

where

= . It is called as diffusion coefficient of magnetic field, m

/s.

The first term on the right hand side of the above equation represents the

convection transport of

B and the second term represents the diffusion transport of

B. For most electromagnetic flow phenomena in smelting process, it is possible to

assume that diffusion transport caused by magnetic flux density is far more than

convection, that is Magnetic Reynolds Number Rm is quite small

(Li et al., 1985).

Then above equation can be simplified as

∂

η=∇B (2.215)

In this way, transport equation of magnetic flux density is independent of

momentum equation, which simplifies the computation. Here, magnetic Reynolds

number Rm is equal to the ratio of magnetic convection and magnetic diffusion,

and there is

00 m

/Rm U L

= (2.216)

where U

and L

are the characteristic velocity and characteristic length

respectively.

For static magnetic field, there is no eddy current, and for steady liquid

metal with high electric conductivity, there is no free electric charge. In this

case, it is convenient to introduce magnetic vector potential

A which is

defined as

=∇×BA (2.217)

and

 



∂

−

∂

 (2.218)

as well as

0•∇=A

(2.219)

When above definition is combined with Maxwell equations, there is

∇=A

∂

(2.220)

  Ping Zhou, Feng Mei and Hui Cai

or 

∇=−AJ  (2.221)

The computation of electromagnetic force field becomes simpler if

B at the

boundary is given. Otherwise, these boundary conditions must be computed by

means of the current distribution in coils. There fore the determination of the

additional body force (mainly refer to Lorentz Force) is always the key process for

solving any magnetic fluid dynamic problem.

ASEA-SKF is a kind of ladle refining equipment and used in deoxidation and

composition last adjustment of special steel. To improve the effect, moving wave and

low frequency coil current are adopted. Fig. 2.14 shows the simulated velocity field in

ASEA-SKF, and major parameters in computation are listed in Table 2.7

(Szekely, 1985).

Table 2.7 The major parameters used in simulating melt velocity field in ASEA-SKF

Parameter

Value

Melt density

7200 kg/m

Melt viscosity 6×10

−3

Pa·s

Depth of bath pool 1.7 m

Radius of bath pool 1.13 m

Electrical resistivity

4×10

−6

Ω·m

Magnetic permeability

1.26×10

−6

H/m

Permittivity

8.85×10

−2

F/m

Frequency 1.4 Hz

The maximum magnetic intensity in z

direction measured at inner wall

3.69×10

A/m

Fig. 2.14



Sketch of fluid motion driven by electromagnetic force

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

2.5.3



The melt motion resulting from jet-flow

Gas jet is a very popularly applied technique in modern metal extraction and

refining, such as top-blowing BOF used in steel smelting, converter, Noranda

furnace and Vanukov furnace used in smelting or refining copper, lead, zinc and

nickel. When gas is jetted into these equipments, the processes, such as bubble

formation, separation, rising and pumping up, result in the melt circulation, which

intensify a series of metallurgical physical processes including mass transfer and

chemical reaction at interface, engulfment of slag-metal interface, and removal of

inclusion (Chen and Mei,1993).

According to the contact mode between the gas phase and the liquid phase in

the reaction, external gas jet can be divided into two types:

a) Impinging gas jet. There is a certain distance from gas nozzle to solid or

liquid surface and gas is only ejected to solid or liquid surface, such as LD

converter.

b) Immersed gas jet. A gas nozzles or orifice is immersed into liquid, as copper

blow converter (Fig. 2.15).

Fig. 2.15 Sketch of copper converting furnace

Per installing location of gas nozzle, immersed gas jet can also be divided into

bottom-blowing, horizon-blowing (side-blowing) and top-blowing. In extraction

and refining of nonferrous metals, the former two are broadly applied.

The influence of gas jet technique on the intensifying metallurgical physical

chemistry and service life directly depends on the flow pattern and turbulent

characteristics of fluid in bath pool. However, the flow pattern and turbulent

characteristics of fluid depend on many factors, such as the bubble behavior,

jetting mode, geometry of bath pool, physical properties of melt and gas volume

  Ping Zhou, Feng Mei and Hui Cai

jetted, etc. To understand better the heat and mass transfers in gas-liquid

metallurgical reactors, metallurgical industries have carried out a lot of researches

in the recent a few decades including the following areas:

a) The formation, motion, mass transfer and reaction behavior of bubble in

metallurgical melt.

b) Bubbling, jetting and their conversion conditions for immersed nozzles in the

melt.

c) The behaviors of high pressure air jetting into free space and impinging

liquid surface.

d) The stirring effect of gas on the melt.

e) The melt flow driven by gas stirring.

f) Simulation on operation process in metallurgical gas-liquid reactor.

When gas is jetted into bath pool by nozzle or immersed spray gun,

continuous gas phase moves in the liquid and there is a gas-liquid interface.

Because of the stability of the interface, jet-flow forms entrainment from

ambient medium, which makes jet-flow expand and its velocity reduce. The gas

flow behavior at the section of nozzle outlet has an influence on jet penetration,

melt motion, nozzle erosion, solid particles penetration and the gas-liquid-solid

inter-phase reaction.

2.5.3.1



Free jet characteristic in gas-gas jet system

The structure of a free jet flow is shown in Fig.2.16. In gas-gas jet system, gas is

jetted at velocity

from the nozzle with diameter d

. It can be assumed that gas

on section

AA′ is of the uniform velocity. As jet-flow moves forward in x axis

direction, ambient fluid is engulfed which makes jet-flow expand, but central

gas still keeps the initial velocity, this region is called the potential core zone.

Because of engulfment, the diameter of potential core zone gradually reduces

until zero as the increase of

x. The distance, from the nozzle outlet to the

location where the velocity on the jet-flow axis starts to reduce, is called the

length of potential core zone. The periphery of potential core zone is mixing

zone where the velocity gradually decreases until the velocity at peripheral

boundary is equal to zero. The width of mixing zone gradually increases with

This area is a developing zone of jet-flow, and the gas flow is called the starting

stage and is about (0

̚6.4)d

. In this zone, expansion velocity of jet-flow can be

described by the expansion angle, which is usually obtained from measurements.

For gas-gas system, expansion angle is 20

°̚26°. After jet-flow pass through

starting stage, gas velocity on the axis begins to decrease and jet-flow goes into

the second stage namely the transition stage. In this stage, gas velocity tends to

be identical and the whole transition zone belongs to mixing zone, which is

located in (6.4

̚ 8.0)d

. Jet-flow has fully developed from section CC′,

time-averaged velocities in axis direction on different Sections have the similar

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

distribution. When the central velocity of jet-flow is reduced to zero, the energy

of jet-flow is completely consumed and the jet-flow vanishes into environment.

Fig. 2.16 The sketch of free jet configuration

(I is starting stage; II is transition stage; III is basic stage)

2.5.3.2



Limited jet characteristic in gas-liquid system

Smelting process finds often jets of gas to high temperature melt. Fig.2.17 shows

the experimental results of gas jet-flow in gas-liquid two-phase system. The

results indicate that, the axis velocity starts to decline much earlier than that in

the gas to gas jet. Measurements showed that the axis velocity started to decline

at distance of

to 2d

from nozzle. The pattern of velocity declines within the

first (0.8 ~ 5)

is quite different from the pattern with distance >5d

. In the

latter case, the declining accelerate remarkably and the axis velocity (

20d0

) has

dropped to 1/10 of the initial velocity. In contrast, in the case of gas to gas jet it

takes about 60

distance for the axis velocity to reduce to 1/10. The buoyancy

of gas bubbles in the liquid phase is the major factor affecting the behavior of

jet-flow.

Fig. 2.17 The decline of axis relative velocity of gas jet in gas and liquid

2.5.3.3



Penetration depth of gas jet

The depth of penetration is an important characteristic parameter of gas jet in bath

pool. Many researches have been reported on this topic (Hu et al., 1999; Devia,

1998).

  Ping Zhou, Feng Mei and Hui Cai

Kozlovszky and Rhys proposed an empirical correlation for vertical jetting of

air or helium into water (Kozlowski, 1986):

/ d

=1.92FrĄ

0.384

˄air-water˅ (2.222)

0.367

/2.23

′

=ld Fr ˄helium-water˅ (2.223)

An empirical correlation of penetration depth in horizon-blowing jet-flow has

been proposed by Hofield and Brimakeheim (Hoefele, 1979):

0.46

0gl

/10.7 (/)

ρρ

′

=ld Fr (2.224)

where l

is penetration depth; d

is nozzle diameter;

and

are gas and liquid

density, respectively; Fr′ is modified Froude number

lg 0

()

ρρ

′

−

2.5.3.4



Track of immerged horizontal gas jet

By experiments of immerged horizontal jet-flow it has been found that jet-flow is

dispersed at a range of distance from the nozzle. This is because the engulfment

effect greatly slows down the jet velocity. Once the horizontal velocity of the jet

flow reduces to a level smaller than the rising velocity of the bubbles in the liquid

phase, the vertical component velocity of the jet starts to dominate and the jet flow

will be broken into bubbles (Szekely, 1985).

Sehmreis and Szekely studied the influence of buoyancy and liquid

engulfment in jet flow via mass and momentum conservation analysis. By

assuming that both the gas volume fraction and the expansion of the jet flow are

related with the horizontal distance from the nozzle, they worked out an

equation describing the trajectory of the jet flow (refer to Fig. 2.18) (Szekely,

1985) as follows:

1/2

12 2

4tan 1

−

⎡⎤

⎛⎞

′

⎢⎥

⎜⎟

⎢⎥

⎝⎠

⎣⎦

rxc

(2.225)

where

and x

stand for vertical and horizontal dimensionless distance of the

jet-flow from the cone angle origin, which can be calculated by taking nozzle

diameter as characteristic size;

is the expansion angle where jet-flow pass

the origin;

and d are respectively jet-flow diameter on the section of nozzle

outlet and of horizontal distance

x; c is gas volume fraction on the section of

horizontal distance

x, which is given by the following formula (Szekely,

1985).

1/2

(1 )

cc c

⎛⎞

⎟

⎜

⎟

⎜

=+−

⎟

⎜

⎟

⎜

⎟

⎝⎠

(2.226)

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

Fig. 2.18 Schematic of horizontal jet in liquid phase with buoyancy effect

Fig.2.19 shows the trajectories of the axis of a jet flow in water computed by

Eq.2.225 and Eq.2.226 under the condition of

=20eand different modified

Froude numbers. Suggestions of modification of this model can be found in

literature (Zhu et al., 1998).

Fig. 2.19 Calculations of the axis trajectory of a horizontal

jet flow in water at different modified Froude numbers

2.5.3.5



Simulation on fluid flow in bath pool with immerged gas jet

Eq.2.204 and Eq.2.205 can also be used to describe the governing differential

equations for fluid flow in gas stirring system. But here the body force F

represents the buoyancy of gas bubbles, or c

where c stands for gas volume

fraction in gas-liquid two-phase region. Since the buoyancy resulting from

temperature difference is much less than that resulted from bubbles, the movement

of the melt is mainly driven by the gas buoyancy and therefore the thermal

buoyancy can be basically ignored when the fluid flow is to be studied. Obviously,

properly determining F

is the key in studying the flow phenomena of the melt in

jetting bath pool.

Generally, the rising region of immerged jet-flow

˄or called two-phase region˅

the distribution of the gas phase in the radial direction can be described by

Gaussian distribution. Compared to bottom-blowing jet side-blowing jet shows

  Ping Zhou, Feng Mei and Hui Cai

larger radius but lower maximal void fraction on the axis of the two-phase region.

Different researchers reported their studies and empirical expressions on the void

fraction in two-phase region (Szekely, 1985; Zhu et al., 1998; Castillejos, 1987).

Fig.2.20 shows the predicted flows in a steel ladle with side-blowing and

bottom-blowing. The side-blowing is found superior to bottom-blowing from the

viewpoint of stirring and mixing effects.

Interested readers are referred to the literature (Szekely, 1985; Zhan et al., 2003;

Zhu et al., 2006; Pan and Zhu, 2007; Zhan and Wang, 2006) for fluid flows in bath

pool with gas jets.

Fig. 2.20 Simulations on the melt flow in a steel ladle with

side-blowing and bottom-blowing jet

(a) Side-blowing; (b) Bottom-blowing

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