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

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

is assumed that particles are perfectly mixed, that is

()

−

(8.128)

where the average residence time

t is

= (s) (8.129)

where

is the mass flow rate of the solid, kg/s; If the mass of the particles has

largely diminished in the dense phase, the mass flow rate can also be estimated by

averaging the feeding flow rate and the discharging flow rate;

M is the total

mass of the solids in the dense phase, kg.

For N dense phase bed layers in series, the age distribution function at the outlet

() e

(1)!

Ntt

−

⎛⎞

⎜⎟

−

⎝⎠

(8.130)

where

Ntt

/= .

Usually Eq. 8. 127, Eq. 8. 128, and Eq. 8. 130 should be solved simultaneously

from N=1 to N=N. The computation procedure is quite complex. In reference

(G.S.1993) the correlation diagram of

X against dimensionless residence time

(

/tt) is listed based on the analysis results (referr to Fig. 8.12), which will be

useful in engineering calculation.

Fig. 8.12 Relationship between solid average reaction rate

X and dimensionless time

The various dense phase models discussed above, to some extent, enable us to

understand the physics and the reaction processes in the fluidized bed. Researches

have also been reported on using chaos theory to analyze and predict the

instantaneous temperature, pressure and heat transfer coefficient at various points

in the dense phase (Karamavruc and Clark, 1996; Yong et al., 1999; Himmelblau,

2000). This opens new doors for the investigations in this area.

  Chi Mei and Shaoduan Ou

References

Cen K F, Fan J R(1990) Theory and Computation of Gas-Solid Multi-Phase Flow in

Engineering (in Chinese). Zhejiang University Press, Hangzhou

Cen K F, Ni M J et al (1997) Theory, Design and Operation of Circulating Fluidized Bed.

(in Chinese). China Electric Power Press, Beijing

Davison J F (1961) Symposium on Fluidization-Discussion. Trans. Inst. Chem. Eng. 39:

230~232

Davison J F, Harrison D (1963) Fluidized Particles. Cambridge: Cambridge Univ. Press

Darton R C, Lanauze R D et al (1977) Bubble growth due to coalescence in fluidized beds.

Trans. Inst. Chem. Eng., 55: 274~280

Grace J R, Clift R (1974) On the two-phase theory of fluidization. Chemical Engineering

Science, 29: 327

G S (1993) Handbook of Multi-phase Flow and Heat Transfer (in Chinese). Mechanical

Industry Press,Beijng

Himmelblau D M (2000) Applications of artificial neural networks in chemical engineering.

Korean Journal of Chemical Engineering, 17(4).

Karamavruc A I, Clark N N (1996) Application of deterministic chaos theory to local

instantaneous temperature, pressure and heat transfer coefficients in a gas fluidized bed.

J. of Energy Resources Technology. (Transactions of the ASME), Sept, 118: 214~219

Kim Jimin , Han Guiyoung (2007). Simulation of bubbling fluidized bed of fine particles

using CFD. Korean Journal of Chemical Engineering, 24(3).

Kunii D, Levenspiel O (1969) Fluidization Engineering. New York: Wiley

Levenspiel O (1962) Chemical Reaction Engineering. New York: Willey

Movi S, Wen C Y (1976) Simulation of Fluidized Bed Reactor Performance by Modified

Bubble Assemblage Model, in Fluidization Technology. Washington: Hemisphere, 1:

189~203

Murray J D (1965) On the mathematics of fluidization: steady motion of fully developed

bubbles. J. Fluid Mech, 21: 465~493

Rowe P N, Partridge B A (1965) An X-ray study of bubbles in fluidized beds. Trans. Inst.

Chem. Eng., 43: 157~175

Smoot L D, Partt D T (1979) Pulverized Coal Combustion and Gasification. Plenum, New

York

Sohn H Y, Wadsworth M E (1979) Rate Processes of Extractive Metallurgy. New York:

Plenum

The Editorial Committee of “Chemical Engineering Handbook”(1989). Chemical

Engineering Handbook (6) (in Chinese). Chemical Industry Press, Beijing

Wen C Y (1968) Noncatalytic Heterogeneous Solid-Fluid Reaction Models. Ind.

Eng.Chem.,(9):34~54

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

Xiao X G, Xie Y G (1997) Metallurgical Reaction Engineering Series: Fundamental of

Metallurgical Reaction Engineering (in Chinese). Metallurgical Industry Press,Beijing

Yong Kang et al (1999) Particle flow behavior in three-phase fluidized beds. Korean

Journal of Chemical Engineering, 16(6)

Yoshida K, Kunii D (1986) Stimulus and response of gas concentration in bubbling

fluidized beds. J. Chem. Eng. Jpn. 1: 11~16

Zhou L X (1994) Theory and Numerical Simulation of Gas-Particle Two-Phase Flow and

Combustion in Turbulence (in Chinese). Science Press, Beijing

 

Multiple Modeling of the Single-

ended Radiant Tubes

Feng Mei

In this chapter the modeling of a gas-fired single-ended radiant (SER) tube is

presented in full detail. The physical processes and models widely include

turbulence, flow relaminarization, partially premixed combustion, radiation heat

transfer, near-wall phenomena, dimension reduction modeling etc. The focus of

this chapter, however, is not the SER tube modeling but the demonstration of the

methodology, modeling effect assessment and problem-solving techniques that can

be widely applied in numerical simulation. This chapter also demonstrates a few

possible means to simplify or decompose a complex modeling problem into a

number of smaller but easier modeling tasks as well as possible means to make the

best use of the available computational resources for achieving optimized

modeling results.

9.1



Introduction

In the foregoing chapters a lot of numerical modeling techniques have been

discussed for analyzing and controlling the FKNME and their thermal engineering

processes. However, simulation results themselves can also be the objects of

simulation. In this chapter we are going to discuss such a case, which is the

modeling of the single-ended radiant (SER) tubes with a three dimensional burner.

The objective of this study is twofold to optimize the operation condition of the

SER tubes and to develop a 1D model so that quick online simulation and control

can be realized. Given that the involved fluid dynamic and thermal physical

  Feng Mei

processes are very complex and three-dimensional; the major effort for this study

is to find a numerical solution to simplify the complexity and reduce the

dimensions before the 1D model can be properly established. Baring all these in

mind we focus no longer on the accuracy of any individual model employed but

the overall effectiveness of the simulation and the compatibility among all

employed models and numerical techniques.

9.1.1



The SER tubes and the investigation of SER tubes

In applications such as heating, melting, sintering and heat treating, the workload

often has to be stay free from contamination. For this purpose we can either use

electrical heating elements, which are easily controllable for the heating power, or

use radiation heat from tubes that are internally heated by gas-fired burners. This

kind of tubes is called gas-fired radiant tubes. The single-ended radiant (SER) tube

is one of the radiant tube family members with double tube structure and more

uniform temperature distribution over the outer tube. The SER tube also shows

advantages such as easy installation and high efficient heat recovery(Harder,1987).

The working mechanism of the SER tube is as follows (Mei, 1999; Lisienko et al.,

1986; Harder, 1987): as illustrated in Fig. 9.1, the partially premixed fuel (nature

gas + air) is introduced in from inlet 1 while the rest of air flows in from inlet 2.

The supplying air is preheated in the heat recuperator before it meets the partially

premixed fuel at the burner outlet where combustion takes place. The flames are

confined within the inner tube but the combustion products flow through down to

the opening end and turn 180° into the annular channel enclosed by the inner tube

and outer tube. The outer tube is heated by the combustion products (by

convection) and the inner tube (by radiation) on the one side and radiates heat to

the workload in the furnace on the other side. To ensure a uniform temperature

distribution along the moving direction of the workload or to optimize the heat

flux distribution, a number of radiant tubes are usually installed in rows in the

furnace.

Fig. 9.1 A scheme of SER tube

1üInlet of nature gas and premixed air; 2üInlet of air; 3üHeat exchanger; 4üBurner;

5üOuter tube; 6üInner tube; 7üOutlet of combustion products

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

The combustion and fluid flow in the SER tubes are three-dimensional (see

Fig.9.2 and Fig. 9.3). As a matter of fact, 60% of total heat is released within the

area close to the burner head where the fluid flow and combustion show strong 3D

pattern. As the gases flow further downstream, 2D pattern gradually dominates

until the gases enter into the annular channel. Due to the narrow space and high

temperature the flow turns to be 1D plug flow in the annular channel. The

continuous change of flow pattern brings us a problem: how many dimensions

should we model the flow? A 3D model seems appropriate from the viewpoint of

precisely simulating the combustion processes but apparently consumes too much

computation resources and may lead to serious convergence difficulty. A 2D or 1D

model seems more practical in terms of numerical approach but has to

compromise remarkably in terms of prediction accuracy.

One more difficulty comes from the so-called “relaminarization” phenomenon:

once the gases are heated up the viscosity substantially rises and the turbulent flow

will be transforming back into laminar flow. This is particularly true in the annular

channel because of the substantial reduced characteristic length. These changes

lead to the rapid falling of the Reynolds number, which causes the reverse

transition from turbulent flow to laminar flow. Together with many other concerns,

the above two problems bring up serious challenge to the simulation work.

Fig. 9.2



The exterior of burner in SER tube

Lisienko et al. (Lisienko et al., 1986) proposed a solution to overcome this

situation: in their study of straight-through radiant tube, they suggested first

simulating the fluid flow and combustion in a single radiant tube and then apply a

simple numerical fitting to the computed “fuel consuming rate curve”, which was

a indication of the intensity of heat release along the flow direction. By applying

this consuming rate curve to represent the complex combustion process they

managed to establish a new one-dimensional model for the fluid flow and heat

transfer processes. This approach was approved successful because the fuel

consumption rate had been found very efficient reflecting the major characteristics

  Feng Mei

of the complicated combustion processes, which largely determined the heat

transfer and fluid flow processes. Lisienko et al. published their study of 2D to 1D

reduction based on this approach in 1986. They applied a laminar flow model in

the 2D simulation and introduced a correction for the turbulent combustion. A

correlation between the fuel consuming rate and the Reynolds number had been

developed and then applied in the 1D model. This correlation was also found

effective in predicting structure as simple as the straight-through radiant tube.

Viskanta and his students (Harder et al.,1987; Ramaurthy,1994) proposed using

low Reynolds number turbulent 2D model based on their wide review on the then

published studies. Their fuel consuming rate correlation has taken the burner

structure into account. A one-dimensional model has developed by applying this

fuel consuming rate correlation.

The above investigations focused on the simple-structured straight through

radiant tube whereas in this chapter we present more complicated single-ended

radiant (SER) tube. The extra difficulties for SER tube include:1. 3D burner

structure, which can no longer be simulated by 2D model; 2. Partially premix

combustion, which represents characteristics of both premix combustion and

diffusion combustion; 3. Double channels, which force a U-turn of the flow. This

U-turn is a challenge to many low Reynolds number (LRN) turbulent models

because most of these models were not developed to consider sudden change of

flow direction.

9.1.2



The overall modeling strategy

The two major issues in this modeling were to reduce dimension and to select

appropriate turbulence and combustion models.

The burner head shown in Fig. 9.2 implies the flow will be three-dimensional.

The modeling of three-dimensional require tremendous turbulent flow was

therefore indispensable but would computational effort if a full-scale simulation

has to be carried out. This direct approach is sometimes not feasible if the required

computation goes beyond available computer capacity.

A better solution would be to start with a study of the cold flow to understand

the turbulence in the neighborhood of the burner head before we develop an

appropriate full scale 2D model in using the information obtained from the 3D

model as boundary conditions (see Fig.9.2). The objective of the 2D modeling is

to determine the influence of the operational conditions, i.e. premix ratio, excess

air ratio and mass flow rate, to the combustion process so that the correlation

between the fuel consuming rate and the operational conditions can be

established. So long as this correlation can be made, the 1D modeling becomes

possible.

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

Selecting appropriate models were the second key issue. Note that in most

practical cases the optimization of the overall system is superior to that of any

part of the system. As discussed previously, the modeling of the turbulence was

found of the major technical difficulty for modeling of the SER tubes. We

therefore ranked the turbulence model as first priority, followed by the

combustion model as the second priority. We need a reliable combustion model

because our two major interests in this application, namely the temperature

distribution along the tubes and the fuel consuming rate, are both closely

associated with the combustion process. The radiation model ranksproirity No.3

on top of the wall functions due to the fact that most heat is transferred by

radiation at high temperature whereas convection (estimated by the wall

functions) is relatively less important.

The priorities have been ranked and now we may move on to select specific

model for each process. To do so it is necessary to have a good understanding

about the model performance and the physical processes. In the upcoming sections

we are going to discuss how the models are accessed and selected against the

physical phenomenon. Apart from the pure theoretical concerns in this selecting

process, we also have to make necessary compromise with the available CFD code.

Generically speaking, self-development of CFD code was less advised because it

often consumes a very large part of research resources but not necessarily

guarantee better overall results. It is actually more advisable to use commercial

CFD packages as much as possible even though adaptation has to be made from

time to time.

The CFD package used for this study was Fluent 4.0. All the employed models,

except the turbulence model and the combustion model, are the built-in standard

models of Fluent 4.0.

9.2



3D Cold State Simulation of the SER Tube

Fig. 9.3 illustrates the burner head structure of the SER tube. As stated in the

foregoing section, the purpose of this modeling was to understand the separating

and mixing patterns of the turbulent flow within the burner head structure. Note

that the tube is much longer than 10 times tube diameter that we may reasonably

assume the dead end of the SER tube has no influence to the flow field from the

entry side where the burner head is located. Therefore we remove the outer tube from

the computation domain in order to reduce computation load. The standard RNG k-ε

model is applied (see next section for the reasoning of this selection) with a finite

difference mesh grid of 17h27h150 and the SIMPLE scheme. The computation

domain is 1/4 of the flow field.

  Feng Mei

Fig. 9.3



A 3D model of nature gas burner

(a) Radial distribution of meshes;(b) Axial distribution of meshes

Fig. 9.4 shows the 3D flow field structure at a characteristic Reynolds number

(based on the inner tube diameter) of 9658. The physical positions of the cross

sections are marked in Fig.9.5. Four recirculation zones are identified in Fig.9.4.

One of these recirculation zones is located inside the burner head where the

incoming fuel flow is split into an axial flow and four radial flows. The other three

recirculation zones are all in the area where the central flow (partially premixed

fuel) and the annular flow (air) are mixing. The sizes of the recirculation zone I

and III are found different in Fig.9.4(a), (b), which strongly indicate their

three-dimensional characteristics. By contrast, the recirculation zone II is basically

independent of the sections, implying that the flow has been gradually reduced

into 2D structure. Fig.9.4(c), (d) illustrate the recirculation of the radial injections.

No strong entrainment is identified before the radial injections meet the annular

injection. Fig. 9.6 compares the radial flow velocities in a range of 90 degrees

along the circumference direction at J=13 and J=21. About 95% of the momentum

is retained within 1/2 of the area even though the peak velocity has reduced 60%

from section J=13 to J=21. This analysis can be used to measure how much the

flow is “three-dimensional” because the velocities should be identical along the

circumference direction in 2D flow. This means that under the condition of

identical total flow rate and same J level a 3D injection flow should be much more

concentrated than a 2D injection. The mixing effects in these two situations are