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

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7 Coupling Simulation of Four-field in Flame Furnace 

Fig. 7.10 A sketch of flow field of swirling jet-flow

Fig. 7.11 computed particle traces in reaction shaft in the case of employing swirling nozzle

b) To achieve particle dispersion in the air, the swirl inducing design is superior

to the old design having parallel distribution on air inlets. Some characteristics of

the swirling design are:

ü The irregularity of particle motion has decreased, which is helpful in

reducing the dust generation ratio.

üRegulating the swirl intensity of particles is relatively easy to achieve in practice.

  Ping Zhou, Zhuo Chen and Kai Xie

üThe incoming swirl velocity of particles is greatly affects the swirling of

particles in the reaction shaft.

c) The axial velocity of the process air is an important factor for controlling

and sustaining the cone shape of the suspension mixture. Increasing the axial

velocity of the process air will decrease the trend of centrifugal motion of the

falling particles. Moreover, the velocity of the process air at the burner exit has a

great impact on the average axial velocity of particles.

d) The irregular motion of particles and their erosion effect on the lining are

influenced by the recirculation of fluids in the reaction shaft. Most particles

having irregular motion return from the bottom to the top of the reaction shaft.

In-depth research of the recirculation and its size is necessary for reduction of the

irregular motion of particles.

e) It is found that an increase of swirl number of the process air results in an

increase of dust generation ratio. The reasons are:

üAfter leaving the central swirling region, the particles are free from the

constraints of the process air and drift in the reaction shaft.

üSome particles, especially those of medium sizes, move along the periphery

of the suspension mixture. When the swirl number increases, these particles can

easily enter the stagnant region at the interface of the shaft and the settler, and

flow foward the uptake shaft with the gas.

f) It is proposed that the burner structure for intensified smelting can be

modified in the following aspects:

üThe pneumatic conveying method can be used to transport the concentrate

particles into the shaft. The conveyed air flow should be controlled to meet the

needs of burning decomposed sulfur vapor while avoiding backfiring. The swirl

number of the conveyed air should be in the range of 1.0~1.5.

üThe central oxygen pipe may continue to be used, or be replaced with a

high-speed jet nozzle which can also double as a combustion stabilizer.

üThe distribution air should be removed to simplify the structure of the

concentrate burner.

üThe two-ring structure of the burner is suggested to be replaced by a single

ring. Oxygen enrichment of the process air is allowed to be adjusted and its

velocity is limited around 150m/s. The swirl number should be within the range

of 0~0.5 tangential velocity of the incoming process air.

üThe volume of flue gas can be reduced by increasing oxygen enrichment of

the process air, which can also decrease the average flue velocity.

References

Bao Yunran (1997) Strategy of intensive economy incensement and energy development

7 Coupling Simulation of Four-field in Flame Furnace 

(in Chinese). Energy Research and Information, 13(1): 1

Chen Hongrong, Mei Chi, Xie Kai et al (2004) Operation optimization of concentrate

burner in copper flash smelting furnace (in Chinese). Transactions of Nonferrous

Metals Society of China, 14(3):631̚636

Li Xinfeng, Mei Chi, Zhou Ping et al (2003) Mathematical model of multistage and

multiphase chemical reactions in flash furnace (in Chinese). Transactions of Nonferrous

Metals Society of China, 13(1):204̚207

Lu Zhongwu (1995) Flame Furnaces (in Chinese). Metallurgical Industry Press, Beijing

Ma Aichun, Zhou Jiemin, Ou Jianping et al (2006) CFD prediction of physical field for

multi-air channel pulverized coal burner in rotary kiln (in Chinese), 13(1):75̚79

Mao Jianxiong et al (1998) Clean combustion of Coal (in Chinese). Science Press, Beijing

Mei Chi et al (1999) Hologram simulation of modern furnaces and kilns (in Chinese).

Journal of Central South University of Technology, 30(6): 592̚596

Mei Chi et al (2003) Form and application of high effective reaction zone in copper flash

smelting (in Chinese). Nonferrous Metallurgy, 55(4):85̚88

Ren Hongjiu (2001) Bath Smelting of Non-ferrous Metallurgy (in Chinese). Metallurgical

Industry Press, Beijing

Ou JianpingˈMa Aichun, Zhan Shuhua et al (2007) Dynamic simulation on effect of flame

arrangement on thermal process of regenerative reheating furnace (in Chinese). Journal

of Central South University of Technology, 14(2):243̚247

Xie Kai (2005) Several Theory and Operation Optimization Challenges in the Development

of Modern Copper Flash Smelters (in Chinese). Thesis (PhD), Central South University

Xie Kai, Mei Chi, Zhang Quan et al (2003) Research of uneven temperature field in the

zinc fractionating tower furnace chamber (in Chinese). Journal of Central South

University of Technology(Natural Science), 34(1):32̚35



Modeling of Dilute and Dense

Phase in Generalized Fluidization

Chi Mei and Shaoduan Ou

The gas-particle two-phase flow has been widely and intensively applied and

investigated in recent decades. It is very difficult to describe such a non-linear and

stochastic industrial system using accurate mathematical language. In this chapter,

the principles and its applications are introduced and discussed regarding the

modeling of the gas-particle two-phase flow.

8.1



Introduction

In this book, the term “gas-particle generalized fluidization” refers to:

a) Not only the processes of gas passing through bulk layer of powders to form

a fluidized bed, but also the dilute phase fluidization of the gas-particle system and

the suspended motion of the gas-particle two-phase flow.

b) Not only the conventional fluidization of motion in the vertical direction, but

also the two-phase flows in the horizontal or an arbitrary direction. Many FKNME

such as fluidized roaster, air-flow drying system, pulverized coal combustion

system, flash smelting and flash converting furnaces of sulfide concentrate, etc.,

all belong to the category of the generalized fluidization.

Table 8.1 shows the classification of the generalized fluidization systems based

on the gas-particle interactions and the thermal engineering characteristics of

FKNME. The range of the characteristic parameters for each class can be found

from Fig. 8.1 to Fig. 8.3.

  Chi Mei and Shaoduan Ou

Table 8.1



Operating conditions of the gas-particle generalized fluidization systems

Item Dense phase Dilute phase

Fluidized

condition

Fixed or

moving

bed

Bubbling

fluidized bed

Turbulent

fluidized bed

Circulating

fluidized bed

Suspension

fluidized bed

Linear

velocity

of gas flow

/ mgs

−1

Porosity of

the mixed

system/%

<45 45~70 70~80 80~95 >95

Engineering

applications

Solid fuel;

Stratified

burning;

vertical

shaft

furnace;

blast smelting

furnace

Fluidized

roaster;

fluidized hea

treatment

furnace;

powder dense

phase

transport;

fluidized

burning boiler

Fluidized

roaster;

fluidized

burning boiler

Dilute Phase

fluidized roaster;

Rapid fluidized

bed

Pneumatic

conveying

system;

Carrier Drying

system;

pulverized

coal or

heav

oil

burner;

flash

smelting

and converting

furnace

ķ u

is the critical fluidization velocity of the bulk layer. For materials with wide size distributions,

()

2~700 10Ar =×

, characteristic size is taken as

∑

is sphericity, with the following experimental correlation (Cen et al.,1997)

0.528

0.584

ppg

0.056

0.294

ρρ

−

⎛⎞

⎜⎟

⎝⎠

(8.1)

where v

is kinematic viscosity of fluidized gas, m

/s.

ĸ u

is the velocity at the chocking point.If the flow rate is lower than u

the gas flow can no longer

carry particles (average characteristic size) out from the bed layer; meanwhile, the pressure drop in the dense

phase bed layer will increase dramatically. The formula for estimating u

()

0.288 0.69

0.2

ch b p p t

1.463 /

ugDGD Re

ρρ

•

−

⎡⎤

⎛⎞⎛⎞

⎢⎥

⎜⎟⎜⎟

⎢⎥

⎝⎠⎝⎠

⎣⎦

(8.2)

where D

is the equivalent size of the bed layer, m; G

is the solid phase mass flow, kg/s; Re

the Reynolds

number corresponds to the terminal velocity.

Ĺ u

is the converting velocity of the turbulent fluidized bed to the dilute phase. The characteristics (such

as larger bubble disappearing, linguiform gas flow, particles dispersing, upward ejection along the bed height

with zigzag form) of turbulent fluidized bed are gradually disappearing, the interface between the dense and

dilute phase starts becoming fuzzy, the fluctuation of pressure drop in the bed layer becomes smaller, and the

8 Modeling of Dilute and Dense Phase in Generalized Fluidization 

frequency becomes higher. The estimation value of

u is (G.S.,1993)

()

0.5

ck s p

7.0 0.77ud

•≈−

(8.3)

where

•

is the product of particle material density and average size.

ĺ u

is the velocity at the converting point when the interface between dense and dilute phase

starts disappearing. When the gas flow rate exceeds u

, the system becomes a fairly uniform dilute

phase. The formula for estimating u

is as follows:

()( )()

0.422

0.96

0.344

FD p p pgg bp t

0.684 / / /uGD DdRe

μρρρ

−

••

⎡⎤

≈

⎣⎦

(8.4)

The physical meanings of symbols in this equation are the same as those in ĸ.

Fig. 8.1 Relation between the flow condition and porosit

of bed la

ers and the

velocity of the gas flow (Cen et al.,1997)

Fig. 8.2



Particle concentration of the fluidized system (Cen et al.,1997)

  Chi Mei and Shaoduan Ou

8.2



Particle Size Distribution Models

Most bulks used for industrial purposes are composed of particles (powders) of

different sizes. There are great differences between particles with different sizes in

the fluidized devices, in terms of dynamics, diffusion behavior and interface

reaction rate, etc. Therefore, in order to develop a model for a generalized

fluidization system, first of all, it is necessary to model accurately the size

distribution of the particle groups.

For any given particle out of a group of particles (

= ), it is defined that

)(

is the probability distribution density (or distribution density, in short) of

Fig. 8.3 Particle distribution characteristics of the vertical fluidization

system (Cen et al.,1997)

8 Modeling of Dilute and Dense Phase in Generalized Fluidization 

d at

()

() lim lim

iii

fpd

δδ

δδδ

δδ

Δ→ Δ→

′

<< +Δ

ΔΔ

(8.5)

The generalized integration of distribution density in the range of zero to

the “distribution function” of the random variable d .

δδδδ

d )()0()(

∫

=<<=

iii

fdPF (8.6)

The physical interpretation of the distribution function is the existence frequency

of all particles with sizes smaller than

. It is also called the cumulative

distribution passing sieve:

iii

δδ

d )(

∫

= (8.7)

The existence frequency of particles with diameter larger than

is defined as

the cumulative distribution on sieve:

δδ

d )(

∫

∞

= (8.8)

Obviously, 1d )(

==+

∫

∞

iiii

fRD

δδ

(8.9)

In addition,

δδ

)(

−== (8.10)

Several common distribution functions will be introduced in the following(G.S.,1993;The

editorial committee of “Chemical Engineering Handbook”,1989).

8.2.1



Normal distribution model

Normal distribution model is expressed as follows:

()

⎥

⎦

⎤

⎢

⎣

⎡

−

−=

exp

π2

)(

(8.11)

where

a is a model constant, which is the particle size of maximal distribution

density, or the mathematical expectation (

M ) of a random variable d .

() ()

dii

δδδδ

∞

+∞

−∞

∑

∫

The arithmetic average of the measured values of

d usually oscillates around its

mathematical expectation

M , and therefore

Ma (i.e., the particle size when 5.0=D )

where

is the standard deviation

()

δδδδσ

∫

∞+

∞−

−=

  Chi Mei and Shaoduan Ou

The particle size distribution will be wider as

increases; and the particle

size distribution will be more focused around

a if

decreases.

The cumulative distribution passing sieve, as

() d exp d

2π

δδ

δδ δ

⎡⎤

−

== −

⎢⎥

⎣⎦

∫∫

(8.12)

or by approximation

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

−

+≈

δδ

erf1

D (8.13)

()

xerf is the error function of

, which can be found in many mathematical or

statistical handbooks.

8.2.2



Logarithmic probability distribution model

Many industrial bulks are close to this type of distribution with probably some

deviations at the two extremes.

In the equal sectional coordinates, )(

is asymmetrically distributed (as

shown in Fig. 8.4). A coordinates transformation must be carried out if normal

distribution transform is needed.

Assuming

lg= , then

()

⎥

⎦

⎤

⎢

⎣

⎡

−

−=

exp

π2

)(

ζζ

f (8.14)

()

( ) d 1 erf

ζζ

ζζζ

⎡⎤

⎛⎞

−

⎢⎥

⎜⎟

==+

⎜⎟

⎢⎥

⎝⎠

⎣⎦

∫

(8.15)

As the logarithmic probability distribution functions, the particle number, surface

area and weight distribution of the particle group follow the same rule and have

the same standard deviation

Fig. 8.4 Logarithmic probability distribution of particle group A and B