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

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9 Multiple Modeling of the Single-ended Radiant Tubes 

might also drive faster temperature increase due to better diffusion effect.

However, the temperature would eventually slide down once all fuel had been

consumed. On the other hand, combusting under lower excess air ratio showed

flatter temperature profile. The model failed to predict the temperature fluctuation

along the tubes that we had observed in experiment under high excess air ratio,

which could be caused by the 3D to 2D simplification process. Note that there

were recirculation zones vertical to the axial flow as shown in Fig. 9.4(c) , (d).

These recirculations brought a part of oxidant from the annular air stream down to

the central axial fuel streams, leading to better mixing and quicker reaction. In the

reduced 2D model, however, the radial jets acted like a barrier between the

annular air stream and the central fuel stream, leading to slower mixing process.

The impact of total fuel supply to the temperature was primarily the overall

temperature level. Fig. 9.13 shows the good agreement between the measurements

and the prediction.

The model was validated by the 6 experimental cases. The results were quite

satisfactory, which made it possible for us to continue the dimension reduction

process to 1D modeling of the SER tube.

Fig. 9.13



A Comparison of predicted tube wall temperatures against the measurements at

different fuel supply(the numbers of curves are related in Table 9.2)

(a)For inner tube; (b) For outer tube

9.4



One-dimensional Modeling of the SER Tube

The slim geometry of the SER tube, especially the annular channel enclosed

between the inner tube and the outer tube, physically justified the 1D treatment of

the flow in a remarkable part of the computational domain. However, the

following assumptions were still needed:

a) The 3D-structured burner head could be simplified into a single entry.

b) The gases could be treated as perfectly mixed from the very beginning of the

  Feng Mei

process.

c) The chemical reactions could be considered as “fuel + air = products”.

d) The flow and heat transfer processes could be considered as one-dimensional.

The 1D model of combustion is shown in Fig.9.14.

Fig. 9.14



1D model of combustion

The governing equations set under above assumptions can be written as:

Energy conservation within the inner tubes:

1111

()

pg g

mc P T T

ωωω

−

•=−

QPq

ωω

−

−− (9.4)

Heat transfer across the inner tube wall:

()

11 1 1 11

gg g

RTTq

ωωω

−−

⎡⎤

−+

⎣⎦

()

21 1 2 12 21

gg g

RTTqq

ωω ωω ω

δα

−−−

⎡⎤

=+ − + −

⎣⎦

(9.5)

Energy conservation within the annular channel:

qPqP

2212

ωω

−

−− (9.6)

Note that chemical reactions had been finished before entering into this region.

Therefore there is no more heat release term in Eq.9.6.

Heat transfer across the outer tube wall:

()

22 2 2 22 12 2

rr r

gg s

RTTqqRq

ωωωωω ω

αδ

−−−−

⎡⎤

−+ + =+

⎣⎦

(9.7)

The radiation of the gases to the inside wall of the inner tube:

()()

11 11

11 1

gg g

TaT

ωω

εσ ε

−

⎡⎤

−

⎣⎦

−− −

(9.8)

The radiation of the gases to the annular enclosure made by the inner and outer tube:

2222 222

gggg ggg

qEaE TaT

ωω

εσε

⎡⎤

=−= −

⎣⎦

(9.9)

The radiation of the inner tube to the outer tube:

ωω

εε

−

⎛⎞

−+

⎜⎟

⎝⎠

(9.10)

() ( )

21 1 2 22 2 2

pg gg g

mc P T T P T T

ωω ω ω

αα

−−

=−+ −

9 Multiple Modeling of the Single-ended Radiant Tubes 

The radiation of the outer tube to the environment (here referring to the larger

cylindrical enclosure of the experimental setup):

εε

⎟

⎠

⎞

⎜

⎝

⎛

−

q (9.11)

where E

is the blackbody emissivity.

In the equations above,

denotes the convective heat transfer coefficient,

W/(m

• K); P denotes the perimeter of the tube, m; Q

denotes heat release,

kW/kg ; q

denotes specific radiation heat flux , W/m

; R

is inner tube diameter ,

m ; R

is outer tube diameter , m ;¥ is tube thickness , m; T is temperature, K ; m

is mass flow rate , kg/s ; c

is specific heat capacity; σ is Stefan-Boltzmann

constant,

σ = 5.67h10

W/(m

• K

The emissivities of the gases as function of partial pressure, total pressure and

temperature were determined using separate subroutine. The heat transfer

coefficients were calculated by empirical equations and were fixed to 20 W/(m

gK)

for locations where Reynolds number is below 2,300.

The radial and convective heat transfer estimated by the above equations were

found lower than measurements, which was largely considered as an accumulated

error due to the continuous dimension reduction process. Corrections were

therefore introduced based on general experience and the 2D simulation results.

The corrections were either formulated as constants or expression with values

varying from 1.0~1.3.

On top of the above equations, we still need two equations for the continuity

and pressure drop:

()

= (9.12)

dd 2

vp v

xx D

++ = (9.13)

where v denotes the velocity;p denotes pressure; f denotes the coefficient of flow

resistance ; D denotes the equivalent diameter of the flow channel.

So far all necessary equations had been ready except a governing equation for

the fuel consuming rate in Eq. 9.4. This was the crucial information we were going

to obtain from the 2D modeling results. By running the 2D model we managed to

map out the fuel consuming rate curves at various operation conditions as shown

in Fig.9.15 and Table 9.2, where the cases 1 to 6 were simulations to the actual

experiments with temperature measurements whereas the case 1

to 4

were results

of numerical extrapolation to extreme operational conditions. Higher resolution

mapping of fuel consuming rates could be obtained by running simulations at

more operation points.

  Feng Mei

Fig. 9.15 The fuel consuming rate simulated by 2D model

(The curves were numbers as in Table 9.2)

The fuel consumption rates could be structured either by algebraic expressions

or by a lookup table. Both lead to a correlation between the fuel consume rate and

operation parameters. This correlation could be generally expressed as:

()

,,,m

ϕψ

(9.14)

where

φ,

, P respectively denote premix ratio, excess air ratio and quantity of

fuel supply. Given these parameters, the fuel consuming rate could be solely

determined by the physical location x defined as the distance to the burner head

(entrance). The combustion heat release would be then computed by substituting

Eq.9.14 into Eq.9.4. The thermophysical properties of the combustion gases were

calculated with the help of a component ratio index k(x):

()

=−

(9.15)

where

denotes the initial fuel flow rate. The local air flow rate

()m

and the

combustion products flow rate (

) were respectively:

()

1mm kx

=−

⎡⎤

⎣⎦

(9.16)

afaf

mm mmm=+−−

&& &&&

(9.17)

The equations set from Eq.9.4 to Eq.9.17 makes up the 1D SER tube model. The

initial temperatures of the supplying fuel and air were determined by separated

subroutine computing the performance of heat recuperator.

9 Multiple Modeling of the Single-ended Radiant Tubes 

The predictions of this 1D model are shown in Fig. 9.16 with good agreement to

the measurements. Note that the better predictions at the dead end were resulted

from the introduction of the experimental correction factors into the heat transfer

equations.

Fig. 9.16



A comparison of the 1D simulation with measurements

References

Bilger R W (1987) Turbulent jet diffusion flames. Prog. Energy Combustion Sciences, Jan.

Borghi R (1998) Turbulent combustion modeling. Prog. Energy Combust. Sci., 14:

245~292

Bowman C T, Cohen L S (1975) Influence of aerodynamic phenomena on pollutant formation in

combustion ĉ: Experimental results. EPA-650/2-75-061a, U Environmental Protection

Agency

Chan S H, Kumar K (1990) Analytical investigation of SER recuperator performance. 13th

annual energy-sources technology conference and exhibition, New Orleans: Jan., 14~18

Harder R F, Viskanta R, Ramadhyani S (1987) Gas-Fired Radiant Tubes: a review of

literature. Topical Report

Kreinin E V, Kafiren Yu P (1980) Combustion of Gases in Radiant Tubes. Nerda,

Leningrad

  Feng Mei

Libby P A, Williams F A eds. (1980) Turbulent Reacting Flows, Springer-Verlag, Berlin

Lisienko V G, Sklyar F R, Kryuchenkov Yu V, Toritsyn L N, Volkov V V, Tikhotskii A I

(1986) Combined solution of the problem of external heat transfer and heat transfer

inside a gas radiantion tube for thermal furnaces with a protective atmosphere. Journal

of Eng. Physics, 50

Mei F (1999) Experimental and numerical investigation of a single-ended radiant tube.

Thesis(ph. D), Facult Polytechnique de Mons, Belgium

O’Brien E E (1980) The probability density function approach to reacting turbulent flows,

Turbulent reacting flows. P. A . Libby and F. A. Williams (Eds.), Topics in Applied

Physics 44, Springer-Verlag, Heidelberg, 184~207

Ramaurthy H, Ramadhyani S, Viskanta R (1994) A two-dimensional axisymmetric model

for combusting. Reacting and radiating flows in radiant tubes, J. of the Institute of

Energy, Sept.

Ramamurthy H, Ramadhyani S, Viskanta R (1995) A thermal system model for a

radiant-tube continuous reheating furnace. J. of Materials Engineering and performance,

Oct., 14(5)

Ramamurthy H, Ramadhyani S, Viskanta R (1997) Development of fuel burn-up and wall

heat transfer correlations for flows in radiant tube. Numerical Heat Transfer, 31(6)

Spalding D B (1997) Combustion Theory applied to engineering. Imperial College Report

HTS/77/1



Multi-objective Systematic

Optimization of FKNME

Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

As most engineers may agree, it is not very difficult to improve the performance

of an operating furnace or kiln by some means. The difficulties lie in finding the

most effective strategy to optimize the system with a well-developed practical

approach and bringing the ideas into practice. There fore, theoretically sound

methodologies supported by necessary analyzing tools are needed. In this chapter,

the optimization methodology and the objective functions are discussed for the

FKNME optimization.

10.1



Introduction

The optimization of engineering processes and production operations is to improve

technical level and economic benefit through refining and adjusting system

structures and operating conditions. The needs of optimization never end as the

requirement to performance is always rising. Due to this nature, optimization

process in general is regarded as an important source of technological developments

and innovations in industries.

10.1.1



A historic review

The methodology of optimization has been remarkably evolved. At the preliminary

stage, only empirical optimization could be carried out by intuitions and common

senses. This was later replaced by a more efficient analytical optimization approach

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

based on experimental inductions, which was usually single-objective oriented. It is

not until the recent decades that a large variety of researching tools have been

developed so that the analyzing, implementing and result validating in the

optimization process can be carried out in a more systematic and efficient way. This

has made it possible for multi-objective optimization, which is also called the

“systematic optimization” (Liu, 1994; Liu and Bao, 1999).

As new theories keep emerging over time, more and more new tools have been

developed and used for optimization. As a result, qualitative analysis was firstly

replaced by quantitative analysis tools such as mathematic programming and

computer simulation. After that, more complicated artificial intelligence theories

(such as expert system, fussy analysis, neural network, and genetic algorithm etc.)

introduced, which enable us to carry out optimization in a broader and deeper

range jointly using quantitative and qualitative analyses, accurate and fuzzy

methods and algorithm-based mathematic modeling and searching techniques

based on database of recognizable patterns.

Optimizations were originally more passive-natured because they were usually

actions consequently taken when the system was sufficiently understood after long

time practice. Nevertheless, it has been more like an action taken aiming at proactively

improving performance of the system nowadays. The latest trend is to launch

optimization process as early as the prototyping stage for new system development in

order to maximize the benefits of system operation at the best performing conditions.

Before the environmental issues had aroused global concerns, the objective

function set for optimization were usually those measuring economic interests or

technical performance. Today, optimization may target objective functions that

reflect the interests of the society, environment or the balanced development

between technology, society, economy and environment (Hu, 1990).

10.1.2



The three principles for the FKNME systematic optimization

Generically speaking, the idea of systematic optimization is reflected by the

following three principles (Mei et al., 1996):

a) Attention should be paid equally to the optimizations of the FKNME

structure and its operational processing. Here “structure” means the main body of

the FKNME (including their geometries, liner materials and technical

configurations) and their thermal systems (including heat/electricity supplying,

ventilation and gas exhausting, materials loading and unloading mechanisms). The

“operational processing” includes the compositions of the loading materials, the

combustion conditions and the procedures for material feeding/ discharging, air

blasting and heat/electricity supplying institutions.

b) The optimization should integrate needs at different levels, namely the

working mechanisms ˄at the micro level˅, the FKNME structures ˄at the middle

10 Multi-objective Systematic Optimization of FKNME 

level˅ and the FKNME performances ˄at the macro level˅. The working

mechanisms include the spatiotemporal characteristics of the fields of

temperatures, component concentrations, heat-releasing rates and magnetic force

etc. This information not only reveals the microdynamics of various transfer and

physical chemistry processes in the FKNME, but also reflects the coupling effects

between these processes. The optimization of macro performances should be

realized through changing the microdynamics, which can only be observed by

measuring the spatiotemporal characteristics of the relevant vector/scalar fields.

On the other hand, the change of the microdynamics has to be done by modifying

the FKNME structures or the operational processes. Fig.10.1 illustrates the

inter-dependency and/or determining relationship among the three levels.

Fig. 10.1 Full-scale optimization of furnaces and kilns

c) A balance must be made among technological advancement, economical

profitability and social interests. Pursuing economical profitability and technical

excellence but ignoring its impacts to the environment is shortsighted and

local-minded. From the sustainable growth point of view, we should encourage a

systematic optimization that harmonizes the interests from all of the social,

economical and technological aspects.

10.2



Objectives of the FKNME Systematic Optimization

Even though the objectives of optimization are widely variable depending on

different furnaces and kilns, there are a few aspects in common, such as

maximizing the output rates, the product quality and the furnace/kiln’s service

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

lifespan. Particularly in the metallurgical engineering area, the furnaces and kilns

are mostly well known for their remarkable energy consumption and pollution.

Therefore, the objectives of optimization for these installations should also include

minimizing the energy consumption and pollutive emissions in addition to the

three aforementioned objectives.

Mathematically, the multi-objective optimization problem can be expressed as:

[]

,,,

xx x= Kx

To solve:

min ( )

or :

max ( )

(10.1)

with constraints:

(x)İ

For most of the FKNME, f

(x), f

(x), …, f

(x) or

(), (), , ()

ϕϕ ϕ

xx must

cover functions as addressed in the following sections.

10.2.1



Unit output functions

The unit output rate (or bed capability) is defined as the producing capability per

unit bed area or per unit volume of the furnace (t/(m

gh) or t/(m

gh)). The rising

of unit output rate is the ultimate way to lower down the consumption and increase

the profitability over unit cost:

The definition by unit bed area is:

(t /(m h))

•

= ǂǂ

(10.2)

The definition by unit volume is:

(t /(m h))

•

= ǂǂ

(10.3)

where G is the mass processed or produced in

hours by the furnace; F and V

are respectively the bed area and the working volume of the furnace.

The forms of a and G, although vary with different furnaces and kilns, should

generally fall into one of the following:

a) Output rate function defined by the heat transfer capability inside the furnace:

()

mgimmimm

TTTH

∑

−+−

ασεε

(t/h) (10.4)

where

and T

are respectively the surface emissivity and surface temperature

(in Kelvin) of the furnace loads;

is the total incident radiation on the mth