ISIJ International, Vol. 33 (1993). No. 6, pp. 61 9-639Review

Mathematical Modeling of the Flow of Four Fluids in a PackedBed

Jun-ichiro YAGl

Chairperson. Committeeon Transport Phenomenain Gas-Solid-Liquid Packed Beds. Society of Basic Research on SpecificSubjects, The lron and Steel Institute of Japan; Institute for AdvancedMaterials Processing, Tohoku University. Katahira,Aoba-ku, Sendai, Miyagi-ken. 980 Japan.

(Received on December24. l992; accepted in final form on March l9. 1993)

Macroscopic flow phenomenaplay important roles not only for improving productivity and energyefficiency but also for achieving the stable operation in metallurgical and chemical reactors. This reviewpaper deals with flow phenomenaof four fluids which are single-phase or multi-phase flow of gas, fineparticles, Iiquid and packed particles. In someprevious researches on the multi-phase flow, fundamentalequations were derived for a continuous fluid phaseand dispersed phaseswith different types of modeling.However, in this paper, continuous flow wasassumedfor each phase in the derivation of the equation ofmotion for obtaining the numerical solution. The model has been applied to the simulation of not only fourphase flows but also one to three phaseflows. Typical examplesfor application will be described for severalprocesses.

KEYWORDS:mathematical modeling; four fluids; packed bed; Navier-Stokes equation; equation of con-tinuity; equation of motion; energy balance; interaction parameter; probability modelj continuous flowmodel.

1. Introduction

Packedbed processes are widely applied to chemicalreaction and heat transfer operations in variousindustries. Typical examples of such processes are blastfurnace, shaft furnace, coke oven. Merz furnace forcalcination of limestone, melting furnace of sludge,exhaust heat recovery process, dust collector, heat andpowder storage processes and so on. In addition, afluidized bed reactor which is a typical example oftwo-phase flow is often used in the chemical engineeringfield.

Theflow phenomenaof packedparticles, powders, gasand liquid play important roles not only for achievingthe stable operation of these processes but also for im-proving the operating efficiency. For this reason, manyattempts are being madeto understand fluid flow in manykinds of processes. However,they are not yet summarizedsystematically up to the present. Previous researches onfluid fiow in these processes have been mainly done forone-phaseor two-phase fiow of fiuids; gas-solid systemssuch as moving bed reactor, fluidized bed reactor,pneumatic transportation of powders and gas1iquidsystems such as slag foaming of metallurgical processes,gas blowing into degassing processes of metal, water flowfor cooling the nuclear reactor and spray cooling.Furthermore, for liquid-solid systems the researches arenowbeing conducted on CWM(Coal Water Mixture)and COM(Coal Oil Mixture).

In addition, Iiquid flow and movementof solids in

packedbeds are also being studied in order to clarify the

61 9

fiow phenomenain the blast furnace and in the sludgemelting furnace.

The four fluids in a packed bed are multi-phase flowsof gas, Iiquid, fine powdersand packed particles whichare movedby gravity. Thesefour fluids which affect eachothers through the exchangesof momentum,massandheat flow in the packed bed. Examples of four fiuids

processes are blast furnace and moving bed type sludgemelting furnace. However, manyprocesses of one- tothree-fluids are used in industry as mentionedpreviously.Furthermore, three-phase fiow is observed in the racewayof a blast furnace, and gas-solid two phase flow withvaporization and combustion of coal powdersoccurs inthe blow pipe during injection of pulverized coal.

In this paper, continuous flow is assumedfor eachphase in the derivation of the equation of motion and ageneralized mathematical modelfor the flow of four fluidswill be proposed for obtaining the numerical solution.

Themodelcan be applied to the simulation of not onlyfour-phase flow but also one- to three-phase flow. Typicalexamplesof the application will be described for several

processes together with the equations for the estimationof interaction andsource terms in the equations of motionand of continuity.

2. Formulation of Four Fluid Flows

For estimating the macroscopic flow phenomenoninthe packed bed, fundamental equations describing theflow phenomenonare derived. In the packed bed inwhich four fluids coexist as shownin Fig. l, occupation

C 1993 ISIJ

Y

Liquid Solid

~ ~

ISIJ Internationai, Vol.

L^oooooooooooooooooooooooooooooooooooooooooooooooo

Packedbed

t ~Gas Fine particle

Fig. 1. Fluid flows in the packed bed.

Ls ~ Ep + Ef(ps) + 81(5} + Lf(1,sJ

8p jG sj

8g

F L8r L1(d) j

Lfti,s)

81(s)

Ef(p,s)

Lj el(d) + sr(Ld)

Fig. 2. Four fluids in the packed bed.

Lf(i,d)

33 (1993), No. 6

Table 1. Classification of the state of fluid flows.

No. Code GAS LIQUID POWDERExample

1 100 c shaft furnaGe, Maerzfurnace

2 ol o cl~quid filtration through packedbed,liquid movementin meta, bath

3 120 c SemiC absorption

4 103 c Ddust colleGtion, fluidized bed, turbulent PC[combustion

5 130 c D spray cooling

6 ~23 c SemiC D gas-powdermixed flow with liquid flow

7 124 c Sernj C C liquid-pGWder mixed flow with gas tlaw

8 12314 c SemiC DlC two phase flows of liquid-powder andsolid-powder

9 134 c D C gas ftow and dispersed liquid dropletwith powder

10 13314 c o DlC gas*pQWdermixed fiQw and liquid dro~letwith powder

@,:~Ff9

e p + e f(p, 8) +'~,r.' E I (s) + e f(1, s)

~J~pl!

ratio of each fluid is expressed by the volume fraction 8as shownin Fig. 2. This is explained as follows:

(1) Gasphase fills the voidage of the packed bed. (8g)

(2) Flowing liquid and powder phase in the liquid

are regarded as liquid phase. The static hold-up ac-companiedby packed particles is not included in theliquid phase. (Eq. (1))

(3) Flowing powdersare regarded as powderphase.

Thepowdersaccompaniedby packedparticles and liquid

phase are not included in the powderphase. (ef)

(4) Packedparticles go downdue to gravity in the

movingbed. Thepowderand liquid phases, which exist

as static hold-up, are treated as packedparticles. (Eq. (2))

81 =81(d) +ef(1, d) ..........................(1)

e.= 8p+ef(p, s) +81(s) +8f(1, s) .......,,.....(2)

Fromthese assumptions, the following equation cande derived.

8g+ef+el+8*= I ..............(3)

Concerning the state of eachphase, both one fiuid flowof gas or liquid, and two fluid flows of liquid or gasphase with packed particles maybe expressed by the

following fundamental equations given by Eqs. (4) to(22). However, in the system in which gas phasebecomesdiscontinuous due to high flow rate of liquid in the twophase flow of gas and liquid, the gas phase is assumedto be completely accompaniedby liquid phase and the

system is expressed by the same equation describedbelow, although the liquid phase have different flowcharacteristics and physical properties from normalliquid phase. Aswith the above, whenthe powderphaseis continuous at higher powderconcentration as seen in

C 1993 ISIJ 620

O (D

ef Pfl e l(d) + 8 f(1 d)

Fig. 3. Interactron parameters amongfour fluids

pneumatic transportation, both phasesof gas andpowderare assumedas a single phase. In the other words, in this

paper, gas phase is always continuous phase whenthethree phases coexist. According to the assumptions, theflow systems are classified as listed in Table I togetherwith the code and someexamples. Here, C, Dand Cmeancontinuous phase, discontinuous phaseandwetteddiscontinuous phaserespectively which correspond to the

code numbers.In addition, packedparticles are always considered as

continuous phase, howeveremptybeds are also includedin this study.

Generalized equations of continuity and motion offour fluids defined above can be described as follows.

For the process analyses, the Navier-Stokes equationwhich is based on the continuous model is used as the

equation of motion. Therefore, there probably are somediscrepancies in the discontinuous models and also in

actual physical phenomenafrom the model proposedhere. Someparameters in the equations which must bedetermined from experimental results were introduced.

Figure 3 shows the relationships of the interaction

parameters amongfluids for the derivation of funda-

mental equations.

Equation of continuity;(1) for the gas:

a (SgPg)+V (egpguH'g) ..........(4)=~~Sg,i..

at i k

The term ~i ~k Sgk,i desrgnates a source term mcluding

ISIJ International. Vol.

chemical reactions. The term expresses the amount of

gas phaseincreased due to phasechangesandabsorption

or desorption of gaseous componentsat the surface ofliquid, powderand packed particles.

(2) for the fine particles:

a (8fpf)+v'(efpfuH'f)=~~S;,i ••••••••••••(5)at i k

The term ~i~kS~,i meanspowder generation bychemical reactions and physical destruction of packedparticles. S' and S} meanspowdercollection by liquid

flow and by accumulation of powder in the packed bedrespectively.

(3) for the liquid:

a (8lpl)+V'(elptuH>1)=~~S~i ••••••••••••••(6)

et i k

The term ~i~k Slk'iexpresses liquid formation due to

chemical reactions. S~ and Sf meansmelting of packedparticles and powder collection by liquid flow respec-tively.

(4) for the packed particles:

a (esPs)+v'(8sPsuH's)=~~S~i ..............(7)

et i k

The term ~i~kS~,i corresponds to solid deposition

on packed particles due to chemical reactions anddisappearance of packed particles by the physicaldestruction. S: and S! meanmelting of packedparticles

and powderaccumulation on packed particles.

Equation of motion:(1) for the gas:

a(egpgu~'g) -~ H,at +v ' (egPgugug)

=V'

(8g/lgVuH'g)~eg grad Pg+FH'~+F~;

+Flf

......(8)

On the gas fiow in the packed bed, the interaction

parameter betweenpacked particles and gas is given byErgun's relationl) expressed by Eq. (9). Usually the first

term in the right hand side of the equation is negligiblein the range of high Reynolds number.

g~ ags H' ~~ ~ ~ H* H'Fsg

368 g s g gs g s g s= - 150,4 ' (u -u )+1.75p a (u -u )Iu -u l.(9)

Here, contact area ags betweengas and solid is calculated

from Eq. (lO).2)

ags=68s/csds~ais""" """""(10)

r _)o,i

xplul

als = (ags +als)L I - exp l .45(ags +als)Pl

2 ~ 2 ).( .) }Jo.05 o.75

plal(ags+ais)

o 2

l

(ags +ais)(Plul) ~(plul )gpl 2 ~

The interaction parameter between gas and liquid is

obtained from Eq. (1 1)3) by using drag coefficient.

H>__ l3-agI ~, ~, ~, H*Fl Cdpgl ug-ul l(ug-ul )/dlJ ""'(1 1)

agt +asl 4

621

33 (1993), No. 6

agl O34/d(Frl)~1/2(We)2/3

The interaction parameter between powder and gasis obtained by using drag coefficient Cd4) and voidagefunction expressed by Eq. (13) which wasproposed byRichardson and Zaki.5)

F~- -3C~pgl uH'g-u~'f l(uH'g- uH'f

)8f...........(12)

H>

4cfdfC~=e~4.7Cd

.............(13)

If Rep I .O, then Cd=24/RepIf 1.0 then Cd=(24/Rep)(1 +0.15Re0.69)If 103 then Cd=0.44

.(14)

(2) for the fine particles:

a(Bfpfu~'f ) H>~at +v ' (8f pfuf uf )

-~ ~' H'= - ef grad Pf +F~+F, +F~+8f pf~ . , . . . (15)

~The interaction parameter F} rs given by Shibata etal.6) as Eq. (16).

= *

-> I H. ~.~.H,F~ - 2DPf8fluf~u,1(uf~u.)Fk

..(16)

Here, D* designates hydrodynamic equivalent diam-eter. Thecoefficient Fk is given by Eq. (17) as a functionof Froude number.

12.89/Frl'61 (vertical direction)Fk-

14.98/Frl'33 (horizontaldirection)"""""(17)

H> ~,As F~ rs equal to -F~, the value ofFlf

is given byEq. (12). Howevert~he interaction parameter betweenpowder and liquid F~ have not yet been fully studiedtherefore the equation for estimating the value has notbeen reported.(3) for the liquid:

a(81 plu~'I ) ~'~'at +v ' (81 plul ul )

~ ~ ~ ~= ~81 grad Pl +F~+Ff +F'~'1g

+F'~'11

+elplH'9

=[3

HF 180{8s+el(s)}2,ll81-~ ~FF~ - (ul~us)

d~{el +81 (s)}

.(18)

~* *~/' I ~ ~F~Fg_

. (19)

.(20)

Becausemanydifferent types of liquid flow have beenobserved, it is necessary to examine the effect of flow

-~ H*condition on the values of Flg, Ff and~(4) for the packed particles:

8(e*psu~'* ) H>H,+v'(8 pu u )at * s * *

= - 8, grad Ps+F~':

+Flf

+F~9

+;:~+

8,ps~••••••(21)

~,Theinteraction term betweensolid particles F~ is givenby the following equation.

C 1993 ISIJ

ISIJ International, Vol. 33 (1993), No, 6

F~'ss

=8susV2u~s

" " - - "(22)

A few papers have reported the value of ~s Which is

the interaction coefficient amongparticles.

3. Examplesof Application

3.1. GasFlow in a PackedBedMathematical modeling and numerical analysis on the

packedbed or movingbed reactors have beenconductedbecause they usually shownon-uniform flow as secn in

the shaft furnace and blast furnace. Stream lines of gasand pressure distribution in the reactors can be obtainedas the simultaneous solution of both equations ofcontinuity (Eq. (4)) and motion (Eq. (8)) on gas. Ergun'sequation (Eq. (9)) is widely adopted for the evaluationof the interaction prameter between gas and solid in

the equation of motion. In this case, the equation ofcontinuity is obtained from Eq. (4) by considering the

source term at steady state. Furthermore, Ergun'sequation expressed by Eq. (23) which is the equation of

H, H,motion, is derived from Eq. (8) by estimating F~, F~andthe viscous term at steady state, and by giving zero to

u, and to al' in Eq. (10) for obtaining the value of ~H*-eggradPg+F~=0

.......,..........(23)

In the numerical analysis of non-uniform flow field,

the following methodshave beenapplied. Theyare finite

difference approximation7) using stream function, finite

element method,8,9) characteristic curve methodlo) andSIMPLEmethodll,i2) for direct solution on pressureand velocity distributions.

The effectiveness of Eq. (23) was examined bycomparing the calculated curves with the measuredvalues. With three dimensional modelexperiments, Ohnoand Kond08) observed the pressure loss in the packed

R~=~J':D g/R=0,4

Packed materials

A c [:~ Glass sphere lOmmc

~~ Glass sphere 2mm#

l Rlng plate

A B c 2000

meas. calc

AoBO

1500 caDe

~E:~

Iooo~~!

_bJ9_ Ya-

r500

Total flow rate

Air400Nm'/H

~48 ~ oo-480~-

bed having uniform andnon-uniform packedstructures.The results obtained were comparedwith the calculated

ones. Anexampleof the results is shownin Fig. 4. Wanget al.1 2) measuredthe pressure distribution in the packedbed and the distribution of gas velocity at the outlet of

an two-dimensional packedbedhaving free-board abovethe bed. Figure 5 shows the comparison of calculatedvalues andmeasuredones. Goodagreementwasobtainedin both Figs. 4 and 5. In the numerical calculation,

although Ohnoet al, neglected the convective term,(ug ' V)ug as described in Eq. (23), the term wasconsideredin the modelproposedbyWanget al. as given by Eq. (24).

It waspointed out by Wanget al.12) and Choudharyet al.13) that the convective term exerted influence oncalculated results for the case where stream lines wereextremely curved.

8gPg(u~'g

'

grad)u~'g H'. .

.(24)= eggradPg+F~••••••-

Decreaseof particle size and increase of voidage occurdue to combustion of coke and, softening and meltingof ores in the sintering process. Thesephenomenawereconsidered in the following equations by Kasai etal.14)

aF~/at K~ (Tb-Tld)'(1-F~) .....,,......(25)

a8g/at=K.•aF~lat+ CR~ ••••••••,••••(26)

._u'l~

>039 O 39

R tMM]

(C)

p (pa)-Calc'

e EXD'

O

I 5O

l iOO

: t50E8'o

~~{~;~~~~~LtBO

~90

[e'S~

I 220

1'~0e

Ti90

100 20.0 300 400 500

Pressure (mmH,O)

Fig. 4. Comparisonofpressure distribution at wall measuredand calculated for cylinder shapemodel with eccentric

ring plate arrangement.B)

-Calco ExD,

320

280

240

200

~ 160

N

l 20

80

40

o

P (Pa)

tuyere

o

, 50

Fig. 5.

: iOO

tuyere 162.0 ! 150

27 \~~\~rs2'o~e:O

~\~~~j~ tSO

tuyere 2al' 220-1'o o olJ2 'o 2 7-0

~Q~:

i90

39 o 39a39 39

R (Mnl A (MMl(a) (b)

(a) Isobars of gas at A-A vertical plane(b) Isobars of gas at B-B vertical plane(c) Velocity distribution at the top of freeboardObserveddata and calculated results by 3dimensionalmodel. 12)

C 1993 ISIJ 622

ISIJ International, Vol.

Adp=Kd•AF~....

..........(27)

Here, F~ designates melting ratio of the sinter feed,

R~is combustion rate of coke. Tb and Tl are respectivelythe bed temperature and the liquidus temperature in thephase diagram. K~, K, and Kd are constant.

In addition, Kasai et a!, carried out the research oniron ore sintering. A mixture of coke and mini-pellets

which were prepared from a spherical alumina particle

(2mmin diameter) as a central core and a mixture ofpulverized hematite ore and limestone as the shell layer

wasused for the sintering experiment by using a sintering

pot. Pressure loss in the sintering pot wascalculated fromthe Ergun's equation with the values of voidage andparticle size obtained from Eqs. (25) to (27). Becausethe

one dimensional non-steady state model was used in

this calculation, gas flow rate was easily calculated byconsidering the gas volume change due to chemicalreactions. Comparison of the calculated and the ob-served results showedgood agreement as depicted in

Fig. 6.

Yoshinagaand Kub01~)also constructed a simulationmodelof the sintering process andconducted simulationcalculation for a sinter pot and a DL-type continuoussintering machine. In this study, gas velocity and pres-sure drop in the sinter bed wascalculated based on theErgun's equation. Calculated results on the flow rate of

gas, temperature and gas concentration coincided well

with experimental data. In the calculation, different

values of drag coefficient were given to the alreadysintered, sintering and not yet sintered parts of thesintering bed.

There was a cohesive zone having extremely lowpermeability in the lower part of a blast furnace. Evenfor such layers of mutually combinedparticles formedafter melting, Ergun's equation could be applied to theevaluation of pressure drop by using I.75+22Srl'4instead of I .75 which is the coefficient in the inertial term

225

200

175

150

- 125

~100

a 75o

a'* 200,u,

Q* 175IL

150

125

IOO

75

o75

o50-

0.25 E

33 (1993), No, 6

in the Ergun's equation. 16) Furthermore, permeability ofthe cohesive zonecould also be evaluated by consideringthe decrease of voidage due to cohesion.17)

For the flow phenomenonin a blast furnace, someresearches were carried out as follows. They are thenumerical analysis of model experiments on the forma-tion of cohesive zone,i8) the change of permeability in

a blast furnace during the operation23) with moistureaddition, the estimation of radial distribution of gas flowrate in the packed bed of layered structure by a twodimensional mathematical model. Figure 79) illustrates

layered structure of burdenmaterial, existence of racewayand cohesive zone. In this figure rather uniform radial

pressure distribution and nonuniform mass velocitydistribution were observed.

Figure 8showsan examplel9) of gas flow distributionin a coke dry quencher which recovers sensible heat ofhot coke by inert gas.

3.2. Liquid Flow in a PackedBedFor the liquid fiow in the packedbed completely filled

35

30

25

20

~ 15

iO

S

O

- 5[KO/M SJORE-COKE-

COHESIVEZONE

RACEWAY

DERDZONEFORSOLIO

S SlO O lOR (M)

35

30

25

20

N 15

10

5

C

OtKO/M27

a5 2s~/)~i~//2~

i7.

is

i2

~~i5a

~l5o'B

IO O lO5SR CM)

Fig. 7. Layered structure and gas flow vector, isobars andcontour lines of averaged massvelocity of gas com-puted by a mathematical model of blast furnace.9)

Case1rl

c._____~c(P)

- Cornputed-- --- Measured

J 11'

..

~'~~T~~~

C(,se 2rso

---'~csc~

-- oc

'__i r 7.J .~.- ,,J

5 10 2015

Time [minl

o _ 300'5o1; 200

v1" o0g EE

075ls o ~

0.5o Q u',,,

:.1" ~

o25

uo 300 ::v,

200

OO

5 o

Fig. 6. Comparisonof measuredand computedchanges in

pressure drop of bed and superficial gas velocity

through bed at two different cases.14)

(Cases I and 2are for alumina core pellets with shell

layer containing 10 and 20wt'/, of CaO.)

i

ho, 2

hQ' 4

h -0'5

,.

!o, 8

!!l [ kg /rrFs li

Fig, 8, Distribution of mass flux of the quenching gas in

CDQ'I9)

623 C 1993 ISIJ

ISIJ International, Vol. 33 (1 993). No. 6

with liquid, the relation between liquid pressure andliquid velocity can be described by Darcy's equation.20)

Acontinuous fiow model21) which is anextended Darcy'sequation wasproposed to the two-phase fiow of gas andliquid in the packedbed. Onthe other hand, a probabilitymodel29) wasdeveloped in which flow behavior of liquid

through a packed bed was regarded as the probability

process.In this section, both cases will be explained below. In

the continuous model, Iiquid flow is thought as con-tinuous phase. Darcy's and Fanning's equations were

-~adopted for the evaluation of the fiow resistance F~betweeH,n solid particles and liquid and of the friction

force F~betweenliquid and gas phases respectively. Theequation of motion was described by the followingequation.

H* -~-elgradPl+F~+F~+8lpl~9 =0 ..........(28)

Liquid hold up, ht Wasobtained from Eq. (31)22) byadding the static and the dynamichold-up given by Eqs.(29) and (30).

81(s) =0.0194Ga~•0254Cp~o.0044 .....,,(29)

el =8. 1220Re~•581Ga~0.482 Cp~•298. . . . . .

.(30)

ht=el(s)+81"""'

-""""(31)

Re~=plplipds/pl"'

"""""(32)

Ga~=p~9(ip,d.)3/(,~2e2)....

..........(33)Is

Cp~= (1/cos el){pl9(csds)2/(al8~)} •••••••••••(34)

Furthermore, the diameter of the liquid droplet wasdefined by Eq. (35) regarding the diameter as the largest

diameter of the liquid droplets in the densest packedbedof the spheres having the diameter, d*.

dl= {(2f•- 3)/3}d. ......................(35)

In the other study,24) Ergun's equation modified onthe basis of the experimental data in the gas-liquid

countercurrent packedbed wasalso proposed instead ofFanning's e_~quation for the evaluation of interaction

parameter Fl'

Figure 9 shows a comparison of measuredand cal-

culated results according to the above theory for thechangeof liquid flow rate distribution. In the experiment,water wassupplied as a point source into the fixed bed.

The result obtained has proven the effectiveness of thetheory. Figure 102i) showsthe simulated results of liquid

flow in an actual blast furnace which were computedbya two-dimensional modelbasedon the continuous model.

The flow behavior of pig iron was regarded asincompressive fluid flow and was described by thefollowing two equations. The first one was the equationof continuity which was obtained by neglecting the

unsteady state and the source terms. Thesecondone wasthe equation of motion given by Eq. (36) which wasderived from Eq. (18) by assuming steady-staH,te and byconsidering gravity and the interaction force F~betweenliquid and particles as external force terms.

E

~

~:~

Vof ,$

2.0 Obs. H20-Glycerin (50-50)

Cal. VI = 434cm3/min

IJ) H=5cm Dp= 6.02 mmo , = 0.00654 kg/m•s

o

1.O

H=10cmo

o

1~ o o H*15cmo o o o

1.Ocro o H=20cm

o o o o o o

2.0

H=SOcm1,o

o o o

2,0

H=40cm1.O

aQo o e a

1 2 3 4 5Distance trom center r lcm]

Frg. 9. Distribution of liquid massvelocity.21)

Slag Metal

TD= 1450 C

DpD= 0.02 m

Slag Metal

To = 1,350'C

slag Metal slag Metal

Fig. 10. Effect ofcoke diameter anddeadmantemperature onslag and pig iron flow in the lower part of blast

furnace. 21)

81pl(uH>1

'

grad)uH'I ~ H'~81grad Pl +F; +F~+el

pl~~g ...(36)

FH':=el/41V2uH>1"""""'(37)

H*lgnoring the term F~, Eq. (36) can be applied directly

for the coke free zone in the blast furnace hearth. How-~'

~;er, the viscous force of liquid F: becamesmaller than

F~ for the flow in the coke packed bed. Therefore, the

interaction term betweensolid and liquid wassubstituted

by Carman's pressure loss equation for the packed bedin laminar fiow region.25) The interaction term is given

by Eqs. (38) and (39).

~ ~*F~=- CBul""

"""""(38)

C 1993 ISIJ 624

ISIJ lnternational, Vol.

CB=1808~pl/81(c'd.)2....

.........,(39)

Whenpressure loss wasevaluated by Ergun's equation,

Eq. (40) could be used instead of Eq. (39). In this caseonly laminar fiow term26) wasnecessary becauseparticle

Reynolds numberfor the flow of pig iron in the hearth

wasestimated as less than lO.

CB=1508~,~l/81(ip,d,)2....

....,.....(40)

Fig. 11.

l I \\\~~~\* \_:~~~~~~i

(a) A-l

(b) A-2

_

_'/'=~~~d~

--~ ~'i//#i':!J_j

,tt t~ '. NN~~~\\\\\\~.:I~ \\tt*~\ \**\\ *--~~~~r___*-dl//

-//

(c) B-l

/

.\

(d) B~~

Calculated velocity distribution.27) (solid lines:

lines obtained by the model experiment)flow

33 (1993), No. 6

Figure 11 illustrates the computedresults on the flowbehavior of pig iron by using Eq. (36), and Eqs. (38) to(40). The computed results coincided well with the

experimental results of the modelexperiment using wateras shownin Fig. 12.

Figure 13 showsa comparison of the computedandexperimental results for different size distribution ofpacked particles. The two sets of results fitted well.27)

3.3. Movementof PackedParticles

Theoretical analysis for the movementof packedparticles due to gravity was started quite recently andthe mathematical description is still not well established.

Regarding the moving granular particles as the

continuous flow in passive state, the plasticity theory

model in which the velocity distribution appeared dueto the internal stress distribution wasapplied to analysethe movementof the deadman.As the result, the shapeof the deadmanwasestimated approximately.

According to the model experiments for the descentof particles in the shaft part of ablast furnace, the streamlines of the particles was found to be estimated as thelines from the point of intersection which wasobtainedby extending the shaft wall upwards.

The movementof particles is expressed by the fol-

10wing equations when the potential flow is assumedand considered the friction force for the motion24) asEq. (42).

H,-e*gradP*+F~=0........

..........(41)

~* H*F~=Ku* .............(42)

The equation of continuity given by Eq. (43) wasderived from Eq. (7) by assuming steady state andnonexisting of liquid and powderphases.

drv 8,p,u~,

=~S,gi.........

...........(43)

50

(a) A-1

(b) A-2

(c) B-1

Fig. 12. Visualized

hearths.

27)

flow

(d) B~3

lines in the 2-dimensional model

625

o : (a)

iA : (b) ExperimeRtal

40c3 : (c)

IJ '

___

calculatcd

:~,

30

~~,, A ,H I L20 !llf'

10

oo

Al):l

'~12

e( rad )

dp=lOmm dp=5mm

dp* Q tlet dp= dp=iomm5mm 10mm

c277mm c277mm

,e400mm '400mm ,400mm

(a) (b) (G)

Fig. 13. Relationship between traveling time and e,27)

(e : angle between outlet and positions of tracer in-

jection)

C 1993 ISIJ

(D)Shaft angle 83'o

(a)

EE Strea~lline

Timeline

$J~::

ao*rfaJ

~::

J

Strea~l

line

ISIJ International. Vol.

3S ~ 333•103(m/s l

30

2S

20

Is

EN lO

5

O

tlll{1 (b)tll:lllll

lll llll

IllIIll ltIlt

IIIIltlII I IIlllll,llllll I ItIll}:llijilt:t Illll

lIIII ll t

:

~1}

l

O 10Sr (rrll

Fig. 14. Stream line and time line for solid flow observed in

a model experiment (a) and solid flow vector com-puted from the potential fiow model (b).30)

Source terms were calculated to each zone from theoverall material balance for packed particles whichshowedshrinkage by the softening and melting of iron

ores in the cohesive zone, by the gasification around thecohesive zone and by the combustion of coke in themelt-down zone and in the deadman.

Figure 14 shows both the stream and time lines30)

obtained by a model experiment, and solid flow vectorcomputedfrom the potential flow assumption using givenshape of the deadman.Both results showsqualitativelyfavourable comparisonalthough the operating conditionis different. Onthe other hand, in the lower part of theshaft, moreaccurate velocity field32) wasobtained by anequation of motion derived considering viscous term anddrag force based on Ergun's equation as the externalforce in Navier-Stokes equation. As seen from Eq. (45),

the viscosity ps for the objective system wasnecessary todetermine the interaction force between solid particles.

8,p.(uH>.'

grad)uH'. ~> ~~-e*grad P* +F~+F~+e*p*H'g ....(44)

F'~..

= e,p.V2u~'s

" " " " "(45)

From the theory mentioned above, Chen et al.33)

obtained the optimal value of ps in the simulation of theexperimental results for the descent of sand particles in

a two dimensional blast furnace model. Anexample ofthe computedresults is shownin Fig. 15. In this figure0.07 Pa' s was found as the value of ~* to give the bestsimulation.

Basedon the theory for determining velocity field fromthe stress distribution, the following equation of motionwas derived and the interaction force between solidparticles wasexpressed by using the stress.

D(esP,uH~.)/Dt

=FH'ss

+8, ps~• • • • • • • • • • •

(46)

~~__8 div~Fs~

* s """"""' """""(47)

Onthe basis of the so-called kinematic model, whichdescribes the movementof the particle bed on the basisof the probability theory focusing the nonuniform

C 1993 ISIJ 626

33 (1993), No. 6

Mesh33 2

E:

o

d(m) p(kg/m ) c() c() E G(kg/s)

0.0019 2600,0 4s 25 0.51 5.97•10~3

cQmputtd '*oo"'structure (a)u5=0.15Pa' s (b)us=0.07Pa'

Fig.

r(cm)

l 5.

~~~

Fig. 16.

33~;

~

O

33 2

ElO -

15

182022

24

g4 98 84 98o*(,~)

Simulation results ofdescent by gravity.33)

Kinematic

o

s

observld

(c)u5=0.005Pa ' s33.2

lO

15

18

2022

24

~

oo

,(em)o 84 9884 98

*('~)

imelines for packed particles

Hodel

t-OsGO-t2V

180)

2,O,

30035a42~

fcj

Potential Flot,

Movementof solid observed (a), computed bykinematic model (b) and potential flow mode(c) for

two cases.36) (bottom discharge and sidewall dis-

charge)

movementof particles, following equation wasobtainedas the relation between velocities in the longitudinaldirection u*. and in the radial direction u*..

us'= ~Beu*./ar.....

...........(48)

The equation of massconservation was described byusing ~i~k S.k,i

as the source term.

au../ez + (1lr)a(rusr)/er =~~S,ki ...........(49)

ikSubstituting Eq. (48) for Eq. (49), Eq. (50) which is

similar to the diffusion equation is derived.

ISIJ International, Vol. 33

(a] (b)f lll

'

I ~~T /~lrJ'l'~l~fi;:l

f-~-1'Hri~ (~) (b)

il

}111

IIlr~ifltlti;!ii

i}1I)Iiill

li~!~~Htri l~~'

f;~1~H~H~7'_~FiE}1~rti {11111

Il}

!H+Hr~l' -rrrr

l}1

1

ll}l_f _}~j~~t:~~~c

l

Jri_~~~~:'~=TEr~{

l-

j:} tj l~~Fi~' ~J~ICl

HI~'~~ ~~~l~'rl I

Illlttjt:jll tl{lii'IllLl~rph~h~ ~;' :'

l_Lr~~~~_~'~~h+~'~~1~~~~:b"_

LI I~L~tlL~1-T7H~~}'~~~~t Il

t:

I

Ijltil~l! ~hl;!1)

tl

tl[tljl

f t

1((1

Ilf!:

lx~:'

_t ~th~~:i:if-

{}:!-

{~:li

l{, t lr+*~~: i ~---'i

'l[IL~lllll:}'~~iL;h:H

I

LtlT't'l!ll{

~'$

ti]iv!l~~:~::L~~~~~~~~:~~~t~(_~~~~L~:f:J~ ~}] l

tiii:"~t :"~isL'

raceway raeeway(1} Tlme lines for the cases of (2) stream lines for the cases of

{a) B=0 5 and (b) B=1'o (m) (a) B=0'5 and (b) B ~1 o (Iu)

Figl 17' sohd flow pattern in a biast furnace computedbykinematic model' 38)

au*. B au e2u

az r er +B +~~S,, .........(50)- *' *'

ar2 i kApplicability of the kinematic modelwasexaminedby

two dimensional model experiments. The results areillustrated in Fig. 16.36) In the figure solid movementinthe packed bed having rather low bed height wasexplained well by the kinematic model. But the potentialflow model is difficult to describe such solid flowspecifically around the particles discharge hole. Theval-

ue of B in the kinematic model has to be obtainedempirically. Figure 1733) showsa computedexampleforthe effect of B-value on the descending behavior ofparticles. Distribution of the radial velocity becomesfiat

with the increase of B-value.

The analytical methodbased on the continuous flowtheory is successful for describing simply the movementof granular particles, but does not explain principally thephenomenaof hanging and slip which are difficult

problems in the blast furnace operation. It should benoted that the kinematic model can not consider theeffect of the gas flow on the solid flow. Currently a newmethod for analyzing macroscopic behavior of thepackedbed is introduced effectively. In this method, theequation of motion for each particle is solved withappropriate interaction parameters between particles.

However, this methodconsumeslong computing time.

3.4. Two-phaseFlow of PackedParticles and GasIn the shaft furnace for direct reduction and in the

blast furnace stack, effect of gas fiow on the descentbehavior of particles has been considered to be not sosignificant, therefore, the analysis of a single phaseflow30,39) has beenapplied by neglecting the effect of gasflow on the movementof the particles up to the present.However,Shimizu et al. 39) found from their experimentsthat the voidage in the moving bed where the particles

movedownwardswashigher than that in the fixed bedwhere the particles were stationary as shownin Fig. 18.

Furthermore, gas flow field in the moving bed wasobtained correctly by introducing the modification onthe voidage to the equation of motion for the fixed bed.

627

(1 993). No. 6

l

'A)

Fig. 18.

Fig.

~

a,

u

o~6 D mmo~4

o~2o.60

hlov ~~~~" O 450'\~~Cg be(! X 5.33

~~~ ___ e 6.50

~ A 7.40

__o~~~~~

o. 54 \0.52

__,c'!'teor

beef

~~Lte~~~~

0.5

a4o. 5 0.60 065 070 7075 080

cs ( - )Relationship between e and ip. of fixed and movingbeds,30)

Free surface of pe] Iets ~Q,

U: : i ;~~_f~.i.~

(kPa)ro

i rll lc: l /L

1 l[I :,Lh Z~~(:oE~ 3 Calculated area

~' 4'1Ttjj's~~4fQ'\J

;FO sLaJ TrriTIT. : 5f'c:

a'l 7u dr~ 8

i~ cas Inlet g Shaft furnace

J~i;:::;s~

*:~i~: i;

19.

calculated area

Gasvelocity Gaspressure

Calculated gas fiow and gas pressure distribution.40)

$h I.8

uoa'

>v)

oEacT'

cr]c:

oJO

voo'

v'v'

oEo

1:'

oo:

o 02 o.4 0.6 0,8 1.0

Radia[ distance r(-)

Fig. 20. Longitudinal andradial distribution ofmassvelocities

of G. and G, for H2 reduction.41)

Minakami and Toyama40)analyzed numerically thedistributions of the gas velocity andpressure in the shaftfurnace applying the potential flow model to solid fiowand Ergun's equation to gas flow. Figure 19 showsanexample of computedresults indicating an useful tech-

nique to prevent the sticking of particles. Yagi andOhmori.41)presented anexampleof numerical simulationfor the shaft furnace producing sponge iron as shownin Fig. 20 on the nonuniform gas flow due to the non-

C 1993 ISIJ

la)

1.6 r:O

T 1.4

N 0.2e)

1.2 0.4 0.6

vl

a0.8

cr, i ,o1•O

o ~8O 1,0 2,0 3,0 4.C

Longitudinal distance z{-)o~

[bJ z :3,96

I o 04

o - o.1 02

v,

~-0•2o04

o _0,3

ISIJ International, Vol. 33

600 6Oo

500 soo

400 400

500 300

200 200

IOO lOO

o ~140 100 IOO140 4

(1 993), No. 6

O100 IOO140

[!R ( - )

Excer iment ) ca Iculat ion>

Fig. 21. Time lines obtained through the cold model ex-periment of the descend of burden and through thecalculation using kinematic model,43)

uniform packed structure in which void fraction andparticle size increases from the wall to the central axis.

Satoh et al.42) madea process model for evaluating

heat recovery in the coke dry quenching process. Inthis analysis, numerical computation was conductedby using the heat balance equation together with thepotential flow model for solid flow and Ergun's equa-tion for gas flow resulting in good agreement betweencomputed temperatures of gas and coke and actual

operating data. Yamaokaet al.43) studied mathemati-cally a shaft type prereduction furnace for the develop-

mentof a newironmaking process (SC-method). In this

study, the kinematic model mentioned in the previoussection wasadopted for the solid flow. Figure 21 shows

a comparison between the measuredand the computedresults. The comparison shows qualitatively goodagreement.

Kuwabaraet al.32) conducted the model experiments

on the descending of the burden materials in the blast

furnace using dry ice and glass beads. In this study,

preheated nitrogen gas wasintroduced to melt downthe

dry ice, therefore in this experiment upwardflow of gasand downwardflow of burden materials were occurred.

For the simulation of the descent behavior of burdenmaterials in the modelexperiment, the potential flow in

the shaft and Naviel~Stokes equation in the contracting

part in the cold modelwere selected for the mathematicalmodel.

H> au*, au*.divu*= ..........(51)=0 .....ez ar

-~-> 8VP+e.,4.V ~:+8,p, g-VP ....

8,p.(uH'.' V)u. = - , g

(52)

The value for the VPgin Eq. (52) was obtained fromErgun's equation given by Eq. (9).

Figure 22 showsthe computedstream lines and timelines. Deadmanwasassumedto be the stationary domainof solid particles determined before the computing bystress analysis. The computed results showedqualita-

tively similar to the experimental results. However,particles around the deadmanandnear the wall behavedas aviscous fluid. For the simulation basedon the viscousflow approximation as mentioned above, viscosity of the

,c

Ee,

EO1ca,

>fO

H

C~(t,

EO1C

12 Q'>

!Q14 hL;

1.O 0.0 1.O

r/R(-)(a)Potential flow approxiunation(b)Potential flow (in shaft)-viscous flow

(below belly) approximation

Fig. 22. Calculated profiles of streamlines and timelines ofparticles

.

32)a) without consideration of voidage functionb) with consideration of voidage function

solid flow has to be given. But the value has not beenstudied well. In the above computation,32) constantparticle Reynolds number25 wasassumed.

3.5. Gas, Fine Particles Two-phaseFlow in PackedBedsRegarding the high injection rate of pulverized coal

into a blast furnace, researches on the accumulation andthe residence behaviors of powders in packed beds wererecently carried out. Shibata et al.6,44) proposed thefollowing equations considering two interaction param-eters between powders and gas, and between powdersand packed particles.

Gasphase;Equation of continuity div(egPgu~'g)=0

..............(53)

Equation of motion~* ~ H, ~div(egPgugug)= -8 gradP +(F +Ff) ........(54)

Powderphase;

Equatron of contmurty div(efpfu~~f )=0 .. ..........(55)

~H* H* ~'Equation of motion div(efpfufuf)=F~ +F~ ••(56)

The in~teraction forces between g~~ and packedparticles F~, betweengas and ~.Powders F~, and betweenpowdersandpackedparticles F! were given by Eqs. (9),

(12) and (16) respectively.

The voidage function proposed by Richardson andZaki wasused for drag coefficient C~in

F~'grFigure 236)

showsthe effect of the function representing substantial

improvement. Thecoefficient of additional pressure loss

Fk Wasdetermined as Eq. (17) on the basis of the

experiments as shownin Fig. 24.44) By using this model,accumulating behavior of powders in a nonuniformpackedbed wasexaminedand the computedresults wascomparedfavorably with experimental results as shownin Fig. 25.44) Yamaoka45,46)studied the bahavior ofpowdersin the packedbed in the similar wayas Shibataet al. However, the modelshavesomedifferences because

Yamaokaet al.'s model considers the non-steady state

C 1993 ISIJ 628

ISIJ lnternationa], Vol. 33 (1993), No. 6

l:

O.~:-

O=OO,!,a,

,:

l'O

5~o~1-C:

u,

o,=,a,

~~S,:~

20

10

o

Fig.

a)

~II

Allen's law~~~~ \f~e~i

o

23.

Fig. 24.

u,ro

~~~'!'O'5 ,~

~~~o~O '~o u' 10r:ooa'~:,:oG,,QL,P~~o~~

o

b)

\I

Allenls law~,

~--\~l~~~_~TEr~e~)'QICID

C:

1o 20 30 40 o lo 20 30

Reynolds number(- ) Reynetdsnumber(- )Relation between Reynolds numberand interactioncoefficient of gas-powder. 6)

l~-LL,,,

,O

O,D

:,,O,,,

,D

C~,O

CO1:1::,1,

OC(,O~E,DOO

104

103

102

1ol

\~'~;~~~~/o

Fk =14.98

Horizontal direction

Frl.33

+' C]L1 \-perpendicular direction

t~:Z~Fk =

12.89JLC~]\r

Frl.61i~.~~~)o

Powderfeed d~~rate (kg/m2s) ~\:~~*

0.1 Ib:~~*

ooAA[]l~c**

PackedbedGlassSinter

CokeSinter

Coke

PowderGlass

Coal dust

1Oo1O'2 10-1 1Oo 1O1

Froude numberUp/Vi~~g(- )Relation between Froude numberand coefficient ofadditional pressure loss Fk.6)

('/.)

0.51.O

1.52.0

RC 12

Rds (mm)

R8.0

R5,0

C3,.o

34

ef (el.)

: 0,40.6

: 0,8

1,o

RC

Rd~mm)

2 4

(Observed) (Computed)

Fig. 25. Comparison between observed and computed dis-

tributions of powderhold up in a two-dimensionalnonuniform packed bed.44)

40

term in the fundamental equations. In addition, the in-

teraction force betweengas andpowdersFlf

is estimatedby Ergun's equation using relative velocity betweengasandpowder, andEq. (57) which expressed the interactionforce betweenpowdersandpackedparticles wasderivedfrom the momentumchange due to collision betweenpacked particles and powders.

~, 3 1+efs 8sP.'efpf(d.+df)2 H, ~,F~-2 1-ef' p,d~+pfd~ Iufluf ""'(57)

Kusakabeet al.47,48) directly measured the powdervelocity in the packed bed by a photometrical method,andthe measureddata wereexplainable by Eq. (58) whichwas derived by considering gravity, the interaction

parameter betweenpowdersand gas, and the collison ofpowders with packed particles. Because of one-di-mensional analysis, the following equation wasadoptedas a fundamental equation in this study.

Equation of motion:

H,duf 4 H* ~. H* -~pg lug~ufl(ug-uf)dt ~TC~

pfdf~>

g~ ~ b,-~ u2

1 p- g-2Fk ..........(58)

DpfFor the mathematical analysis on the combustion of

pulverized coal, petroleum cokeparticles, heavyoil, waterpitch slurry and water coal slurry in the combustionfurnace, two or three dimensional turbulent flow analysis

has been applied. The equations of continuity and mo-tion for gas phase, and the equations of motion for pow-ders, for droplets of slurry and heavy oil are expressedby the following commonformula written as Eqs. (59)

to (60).

V'

(egP9u~'gc)

=V(FcVip) +Sc+Sd... .... ..... .

(59)

where, designation of ip. Fc and Sc are listed in Table2. The equations of motion for powders and liquiddroplets are given by Eq. (60) using a cylindrical

coordinate system.

duf'/dt =G(ug. - uf')

duf'/dt =G(ug. - uf') +ufolrf

duf/dt =G(ugo~ufo)+uf'ufo/rf """"""(60)H> -~where, G=3pegCdRef/4pfd~, Ref =pf Iug- uf Idflpg.

The friction of a single particle moving in fluid flow,that is, the amount of momentumexchangewas con-sidered in the Particle-Source-In-Cell model. In addition,heat andmasstransfer dueto chemical reactions occurredas the exchanges between different phases. The mathe-matical description on this phenomenonfor gas phase

wasthe sameas the continuous modelgiven by Eqs. (4)

and (8). But the motion of particles was described byEq. (60) which was in the Lagrange type and the locusof each particle was followed.

For the pulverized coal combustion, a two competingreaction model expressed by Eq. (61) was used for therelease of volatile matter.69)

Rl (Volatiles) + (Char)

(Rawcoal)ocl (1 -

ocl)(61)~~~'R2(Volatiles) + (Char) """

a2 (1 -oc2)

Themixedcontrol theory on the chemical reaction andthe diffusion of oxygen was applied to the combustionreaction of char expressed by Eq. (62).

629 C 1993 ISIJ

ISIJ International, Vol. 33 (1 993), No. 6

Table 2. Turbulent diffusron coeffic~ent rc and source term Sc.

Conservation of c Fc Sc

Mass

Axial momentum

Radial momentum

Kinetic energy

l

uge

ugr

k

o'tef r

uef f

uef f

ak

,l eff

o

ee ~a au I a eu epgz gr+az

'4 rr az r ar Prrrez az

ec ~ T~a augz I a augr ap

+ 2pugr

az'lrf

ar r erPrrr

ar arerr

Dissipation rate g

G-p8

ac k(CIG Cpe)

Cl

l .44

C2

1.92

crk

0.90

(TC

l .22

{(+

""'-' )}( )]e~ ' a~ ' e~ a~ 'G=~ 2 "' "' "' + "'+ +.*'

e. a, , a, a.

(o) sarajJ V M. ::~9. 5z

(b) Witbank v M=34.3z

(e)

Fig. 26.

To ihe fyo v. M. =44 .6;~

1500K - --- 2000K 2500K

Computedtemperature distributions in blow pipe ofa blast furnace.49)

k.1 k.2

C+11202 CO, C0+11202 C02 ""(62)

Equation (63) was used as an overall reaction rateconstant.

1lk,f 1lk.1 + 1/kf"""'

"""""(63)

The locus and temperature distribution of fine coalpowders were estimated in the blowing pipe of a blastfurnace by the mathematical analysis on fluid flow, heattransfer and chemical reaction between gas and fine

powders. Figure 2649) showsthe computedresults of thechange in temperature distribution according to thechangeof volatile matter content of the fine coal powders.

For the combustion of petroleum coke and the residueof heavy oil droplets, almost the samerate equation asthe combustion of char mentioned abovewasapplied.50)

As the vaporization of water proceeded in the initial

period of the combustion of heavy oil droplets, heattransfer control was assumedin the analysis of this

process, and the heat transfer coefficient was evaluatedby Ranz-Marshall's equation considering the burstingof vapor.

dmf/dt = - I~(d~/CFf) In( I+A)h..(64)

A~Cpf(Tf ~Tst)/LH

P O101Mpa

40 cm

Free Board (Air)

J

-T- 5'54 cml~i~

Eoco

--' 1'1

*-'~~', e)

-..yi

1~L. X iT

lS"~

Fig. 27.

fftfft

vg = 25cm/,;1.27 cm

,~ =1P = 0.106MpavgJet' = 578cmls

Two-dimensional bed with conditionsperiment and in computation.53)

used in ex-

Combustionof the slurry consists of the vaporizationof water included in the droplet and of the subsequentrelease of volatile matter. For the vaporization process,the rate equation for the heavy oil droplet mentionedabove was applied. Howeyer, Ist-order reaction modelproposed by badzioch et a/.51) wasused for the release

of volatile matter.

dV/dt=K(V* - V).........

..........(65)

In the mathematical modeling of a bubbling typefluidized bed, two models considering (model-A) andneglecting (model-B) the pressure loss in powderphasehave been proposed previously. Boillard et al. 52) mea-sured the voidage distribution in the experimentillustrated in Fig. 27 in which the jet flow towards an

C 1993 ISIJ 630

ISIJ International. Vol. 33 (1993), No. 6

a

1.O

O.9

0.8

O.7

0.6

O.5

0.4

0.3

0.2

+ Exper]ment8f Points, 2.5 cmfrQm Center

Predicted Average Poroslty

,dodel A. Stress 1Model B Stress 1- ,

'

~

l':;/++++++++++++++

++ ~++ ll

j~

1.2

1.0

>* 0.8Hc,,

opE~0

0.6~

o.4

OIL

0.2

Porosity at 10 cmfroln center

- ComputaLiomwith kinetic theory model(16 x 22 node5)

-.- Co't]puLatioll with kinetic theory model.'~ d ~~*(16 x 42 nodes) / q

/9~---- Colnputation of Bolliuard (1986)o Experimor]t ,~'

-~~'(f.

C'a'rl~~~~

oa o

1.O

O.9

0.8

O.7

0.6

0.5

0.4

0.3

0.2

o 1O 20 30 40 50 6OBed Height (cm)

+ Experimenta~ Points, 10 cm from Center

Predicted Average Porosity

Model A, Stress 1

- Model B, Stress I /L+

p++

~rF

r

f+_,~i~~~~~~~+.

0.00

Fig. 29.

~

a

lo 20 30 40 50BEDHEIGHT(cm)

Comparisonof computedtime-averaged porosity asfunction of bed height at 10cm from centerwith experiment by Ding andGidaspowandpreviouscomputeddata of Bouillard and Gidaspow.53)

1.O

O.9

0.8

O.T

0.6

0.5

0.4

0.3

0.2

10 20 30 40 50Bed Height (cm)

+ Experlmental Points, 17 cm from Center

Predicted Average Porosity

Model A, Stress 1

- Model B, Stress I :fr~+ll

60

Fig. 28.

o 1o 20 30 40 50 60Bed Height (cm)

Experimental and theoretical vertical time-averagedporosity profiles for two hydrodynamic models.52)

obstacle settled in the uniformly fluidized bed wasintroduced.

Nosignificant difference wasfound betweenmodel-Aand model-B from the comparison of the observed dataand the calculated results as shownin Fig. 28. However,for further advancementof the mathematical model,Ding andGidaspow53) proposed the following equationsadding a gravity term for gas and viscous terms for gasand solid. In this equation, the viscosity of powderphasewhich has been neglected or regarded as constantpreviously was estimated on the basis of the moleculardynamics

.

Equation of continuity for gas phase.

a(pgeg)/at +V•

(pgegu~'g)=O.. .... ..........(66)

Equation of motion for gas phase.

e( pgegu~'g)/at +V•

(pgeguH'gu~,g)=

-egVPg+egPg9H'+v'l:~>g+p(uH'f~u~'g) .......(67)

631

Equation of continuity for powderphase.

e(pf8f)/at +v '

(pf8fuH'f )=O ..............(68)

Equation of motion for powderphase.

a(pfefu~~f )/at +v ' (pf8fu~'f u~'f )=

~8fVPf+8fpfg~'+v.T~'f+p(u~'g-u~'f) ""'(69)

Where, the stress term for powderphase T~f includedthe pressure of powderphase, the shear stress and theforce due to volume viscosity are expressed by thefollowing equation as a function of voidage, density ofpowder, diameter of powder and vibration energy ofpowderphase.

Stress term:T~'g

=2Egktg[1/2{V '

u~>g

+(V '

u~g)T}

- (1/3)V '

uH>gl]....(70-a)

->Tf = ~Pf +ef~f(V uH'f

)I

~28f'lf [1/2{V '

uH>f

+(V '

u~f )T} ~(1/3)V '

uH'fl]

....(70-b)

Pressure of powderphase:

Pf=8fpf{1 +2(1 +e)8fg.)T ...............(71)

Where, go3 [1-( 8f )1!31J

~~ 8f ~**

Volumeviscosity:

0.5~f = (4/3)efpf dfg.(1 +e)(T/I~:)

. . . . . . . . . . .

.(72)

Shearing viscosity:

o.5~f = (4/5)ef pfdfg.(1 +e)(TllL). . . . . . . . . . .

.(73)

The drag coefficient Pappeared in Eq. (69) wasexpressed by different equations for the range of 8g asdescribed below.

If 8g is less than 0.8, then Pis given by Eq. (74) accordingto Ergun's equation.

~ -~

p=1508f'lg

+ 1.75Pg8fl ug-ufl

egd~ df

If eg is larger than 0.8, then pis given by Eq. (75)

according to the drag coefficient for a single sphere.

C 1993 ISIJ

ISIJ International. Vol. 33 (1993), No. 6

3 ~* H,

= CggPgPfeflu -ufl

_2.65p 4 dg eg ..........(75)

df cf24(1+0,15Re~•68?)Rep (Rep

Where, Cd(Rep~1000)~ 0.44

Figure 29 shows the computed results from theabove-mentioned theory. The results computedshowedbetter agreement with experimental data than thosecomputedby models-A and -B.

3.6. Ga~HLiquidTwo-phaseFlow in PackedBedsFor analyzing gas-1iquid two-phase flow in packed

beds, estimation of the liquid fiowing region is quiteimportant. This is similar to the flow of packedparticlesin two-phase flow of gas and particles explained in theprevious section. On the other hand, gas fiow can bedescribed by Ergun's equation considering the friction

forces in both boundaries of gas-packed particles andgas-1iquid.

Szekely and Kajiwara54) analyzed the counter-currentflow of gas and liquid in a packed bed as follows.

Gasflow: V•(egPgu~~g)=0....,.

...,......(76)

H>8gVPg=F~•••••

••••••••••(77)

Llquid fiow V (8lpluH'I)=0........

..........(78)

81VPl =f3uH'I

+el pl ~.. . . . . . ... ,

....(79)

f3 = 180e~pl/d~81""""'

where,. . . . . . . . . .

(80)

Liquid hold-up was estimated by the followingequation.

ht =h. +hd=0.03 +0. 147[{8t+ 81 (s)}ul]o.63 (ul [cm/s]).(81)

In comparing Eq. (77) with Eq. (8), this analysisconsidered only the decrease of volume ratio occupiedby gas due to liquid in the packedbed for evaluating theeffect of liquid on gas flow, but did not consider thechange of friction force due to the change of contactsurface area between gas and solid by liquid hold-~up

and the interaction force between gas and liquid Fl.

Furthermore, ~was found by comparing Eq. (79)witgh

Eq. (18) that F~ wasnot considered.Figures 30 and 31 show the stream lines of gas and

liquid which were computedfrom the theory merrtionedabove. Dotted lines depicted the results assuming the

constant hold up (ht =0.06) and solid lines were obtainedfor the case of ht computedby Eq. (81). Thestream lines

of gas were not so affected by liquid hold-up, how-ever, those of liquid were considerably affected. Thiscomputation was carried out for the two-dimensionalpacked bed to which the liquid was uniformly pouredfrom the top. Results showedthe formation of a dry

zone only near the inlet of gas.Recently liquid fiow in a packed bed was examined

for the case where the liquid wassupplied from a point

source. The results computed by the continuous flowmodel was reported55) not to simulate well the exper-imental results as shownin Fig. 32. Fromthe results, it

wasfound that the continuous flow modelcan compute

I. 4,.5

12

h l. lc-

w 1.0x

O.9

ttJ

J O. eZo,!, O.7~:UJ

2 O. 65O.5

ga$injeclio,, 0.4

pQintO.~;

O.20.I

O,O

1,,1g

o.o o,i 0.2 0.3 o,~ 0.5 0.6 0.7 oe o9 Io

, lf I , l

,,' , ,1 '

,l I / , ll I

,1 ! l! / I I l/ / / / // . / / /r // / /rr/ / / / /./ ./ / /

/ // /_/ /

,ll/

///,l

C 1993 ISIJ

o,o o.1 0.2 o~; 0.4 0,5 0.6 0.7 0.8 0,9 ro

DIMENSIONLESSOISTANCE

with ht =0.06 in two phase region

- - -: computedresults by Szekely and KajiwaraFig. 30. Gas stream function for Q,=900cm3/min, Qg=

0.28Nm3/min, H=60cm,I=15cm and dp=3.07

mm.

s4)

H:::t-

UJ::

Cl,

tlJ

JZOZtJ,

2,:

gasinJection

paint

"II

0.0 0.1 02 0,3 0,4 0.5 0.6 0.7 0.8 OS I.OI,4

l.3

l,2

l,l

I.o

O.9

O,e0.7

o.e

0,5

0.4

0,5

O.2

O. I0,0

t L1 \t L t L

lt \ I t I I1 11 ILt ill

\ \ \ I I itlt

tl lil '\ L it : II I

\\ tlll I

llt I ltll t I :jtt't l

Illll I :l ll ll,llIlt I l

Illl , IIt:1:t l lf,1,t , l

' 1l lltl : 1 1III I

,l

lIlIIlI

632

O.O 0,1 0.2 OS 0,4 05 0.6 07 0.8 0.9 lDI~IENSIONLESSDISTANCE

o

with h* =0.06 in two phase region

- - : computedresults by Szekely and KajiwaraFrg 31 Llquid stream functron for Q=900cm3/min, Qg=

0.28 Nm3/min, H=60cm,I= 15cm(position of in-

jection nozzle) and dp=3.07 mm.s4)

velocity vector only for the specified region andwaseasyto extend to three-dimensional analysis, but it cannotdetermine the flow region of liquid and the boundarybetweendry and wet zones.

To deal with these flows, a probability model whichconsidered discontinuous fiow of liquid, Iiquid flowingregion, dispersion of liquid flow due to packed particles

and nonuniform liquid flow caused by gas flow wasdeveloped

.

In the probability model, the liquid flow distribution

at the nearest two points in the next level wascalculated

by the probability of transfer which wasdetermiend by

20 - ixiO-~ [m/s)

ISIJ

J~\~}\~\\\j\~~~~j~\~\~~~}T:~~~~T~~\~\\~~~~~~\~~~~}\\\~~~

\\\~~~~~\~(\~;\\~~

~~: t :~\\~~\~\~\\\\~ )i ; t l~\\\\\\\\L~tt~\\\\\\1L~~11l\\\1~Lllltl t I l' i I t I I I

l i I'i I t ' I I

lnternationa l, Vol.33 (1993), No. 6

20

~Jt~lO

*

o30 24 i8 612

X (cm)

O9=3x I 0-3Nms/s

O

1~1~,-10~h

.*{~ O,E:

e- "80

~':.* 60

"IL''* 'L' 40

~' 'L'2a

Er o,~,:'

Oll3. 33x I O-'k glsBound8ry of liquid

flo~

_EXp.

Exp.

Calc.

18 12 6x (cm)

,iguid flow and liguidat tht reeeivers

a

Fig. 32.

Comparison of measured liquid flow

distribution with computed one by the

continuous model.55)

(a) Liquid flole vector

30 24

(b)Boundary ofdistribution

20

~~

~]o~

O

- ixiO~' (m/s)

~l\.

30 18 6 O12

Og=3x

X (cm)

1o-a NmIs

O

20

~~_101~)

h

IFSe~; a5' h8a

t ': 60

'* 't' 40~$h~.,.

20*.:~r

:~':' O

ol:F3. 3 3X 1o-lkg/s

Boundary of liquidf low

al .

-EXp.

Exp.

Cal .

Fig. 33.

Comparison of measured liquid flowdistribution with computed one by the

probability-continuous model.55)

(a) Li quid flol~ veclor

30 24

(b) Boundary ofdistribution

the drag force from gas flow to the droplets and gravity

whendroplets descend through a packed bed havingnetwork structure of void. Andalso it wasassumedthat

someof liquid droplets distributed to the two points weresimultaneously re-distributed to the neighboring fourpoints at the samelevel by the local disturbance in gasvelocity. The two-dimensional movementof the liquid

droplets wasdescribed as follows:

1ld I ~= d)(~~d2x I~-d

pl pgug') cos~Cdt26 24

~ pl(~ d

~)9=

pl(~d d2ydt2

-)(

gC(!1

d I~ 2 sin~2pud4 g'

.

(82)

.(83)

In order to compensate the shortcomings of the

continuous and the probability models, it was found tobe effective to combine both models. In the combinedmodel, Iiquid flow region was firstly obtained by the

probability model. After that, Iiquid flow vector wascomputedby the continuous flow model in the regionobtained. Figure 33 shows a simulated result for the

counter-current flow of gas and liquid in the packedbed

633

18 12 6x (cm)

liquid flole and liguidat the reeeivers

O

computed in the two-dimensional general coordinatesystem, proving the applicability of this numericalanalysis. This methodwas applied to the analysis of asludge melting furnace having a coke packed bed.

For analyzing liquid flow phenomenawith the meltingof solid particles in a packed bed, semi-hot modelexperiments were conducted. The procedure was thatparaffin chips were melted by introducing the preheatednitrogen gas from the lower part of the side wall into

the packed bed using glass beads and paraffin chips aspackedparticles. Acorresponding mathematical simula-tion wascarried out.56) In this simulation, the combinedmodel for the liquid flow mentioned abovewasused to

computethe flow region and the liquid flow vector. Asshownin Fig. 34, computedand observed longitudinaldistributions of pressure and temperature showedcon-siderably good agreement.

Asfor the fundamental equations usedfor this analysis,

the equations of continuity for gas and liquid are given

by Eqs. (76) and (78), and the equations of motion for

gas and liquid are represented by Eqs. (84) and (85)

res pectively.

~~~~ H>div[8gPgugugJ= - 8g grad Pg+F~••••

••••••••••(84)

H>H* H* ~ ~'= 81pl9 +F~+F~- 81 grad P (85)div[ez Plul ul I =

C 1993 ISIJ

240

Pressure drop (Pa)

O 500 1OOC240

ISIJ

lg2

~]44~

~ 96~

48

o

International. Vol. 33

Pressure drop (Pa)

O 500 iOOO240

(1 993),

Pressure

O 500

No. 6

Exp' ca[ci;:Lio -A -.- :

Al ' o

IA

IA

IA\\\

(a)

\

o

o

o

~

O 7S ISOTemp, ('C)

emp.

1g2

~~ 144

~~ 96

48

O

~1IA

~IA

jA

}Al(b) t

drop (Pa)

1OOO

EXpA,Cald~~es.

o -rernp

O.

75 150Telnp. ('C)

lg2

~i]44

="F~i 96

48

O(c)

Exp. Cctl_c.1=pres.

-2:pres.

O

'•O

lle

.,,

'

Fig. 34.

Comparisonbetween observed data andcomputed curves by a model for thelongitudinal distribution of temperatureand pressure.56)

: model predictions

- - - : experimental results

o 500Pressure

1OOOdrop (Fa)

a) LowFlux

VerticalPosition,

m

Liquid Streamlines1.6

1~

0.6

O

l fl ll Il If fI

IIfII/

O 0.6 O.g t20.3O 0.6 O.g t20.3

Horizontal Positlon, mb) High Flux

VerticalPosition,

m

0,3 o.6 0.9 1.2

Hortzontal Positton, mFig. 35. Liquid streamlines in trickle bed with restricted

outlet.57)

For the parallel flow of gas and liquid, an exampleofanalysis wasreported5 7) wherein the equation of motion,the unsteady state term and the convective term were{}~glected_~and ErgH,un's equation was applied to obtain

F~ and F~, and F~ was ignored. Figure 35 shows acomparison between measured and computed liquidstreamlines. Favorable comparison proved that thefundamental equations were also able to be applied tothe parallel fiow operation.

3.7. Four Fluid Flows Including Three Fluid Flows in

PackedBedsYamaoka58)analyzed hanging behaviors causedby the

accumulation of dust in the moving bed by expandingthe two phase flow theory of gas and solid in the packedbed. Fundamentalequations usedweredescribed below:

C 1993 ISIJ 634

Equations of continuity and motion for gas.

a (8gpg)+div(8gPguH'g) O .....,....(86)at

a (8gPgu~g)+e grad P ~ ~'

at~-Ffg=0

....,(87)

Equations of material balance and motion for dust.

a 8fpf+divefpfu~~f R R ..........(88)at

a -~ ~' ~*

atEfpfuf~F~-F~=0 ...

..........(89)

~'Th_>e term F~ wascalculated from Eq. (16) and the

terms F~ and F~ were obtained by Eqs. (9) and (12),

res pectively.

Sticking rate R, of dust on the surface of packedparticles and detaching rate Rdof dust from the surfaceof packed particles were expressed as follows:

R.=8gPfluH'f

l(1 -(1 -x)f)......

...........(90)

Rd=(8~-8g)P} I PgluH'gl2

(91)(-1- exp 2 Ed

If sticking probability of a dust particle to packedparticles and collision frequency of the dust particles in

the movementof unit bed length was defined as x andf, respectively, the sticking probability of the dust particle

to packed particles in the movementof unit bed length

was expressed by the term (1 - (1 -x)f). The flow rateof dust particles coming to the unit volume in unit time

wasexpressed by 8fpflu~'f l• Thecollision frequency f was '

determined by the theory of mean free path in themolecular dynamics.

3 (d. +df)2f=1 (eg +8f)

d~ ""'"""""'(92)

Thedetaching rate of the dust from the packedparticles

given by Eq. (91) was derived on the basis of the as-sumptions that the rate wasproportional to the amount

ISIJ International, Vol.

of the dust on the surface of the packed particles anddecreased with the increase of the kinetic energy of theflowing gas. In the equations Ed, p~ and e~ were the

constant with the dimension of energy, volume densityof the dust sticking to the packed particles and thevoidage of the packed bed before any dust sticks

respectively.

The voidage of the packed bed wasgiven by the fol-

lowing equation which considered the sticking and thedetaching of the dust particles.

e 8.+div(e,u~'.)= (R R) ..........(93)at p~

The kinematic model wasadopted for the estimationof the descending rate of packedparticles. This equationdid not consider the interactions of the packed particles

with gas and with dust.

In the experiments58) gas pressure was confirmed toaffect the occurrence of hanging therefore the pressureloss level (PL) defined by the following equation wasintroduced as the index of hanging.

PL=Wp/Ap........

..........(94)

Here, Apwas the gas pressure difference between the

top and the specified level in the bed. WpWasthe vertical

load on burden materials obtained from Janssen'sequation59) at the specified level. It was defined thathanging occurs at PL>1.0. In the case of x=0.05 andEd=2OOOkgm/s2, computedresults coincided well withexperimental data as shownin Fig. 36.

Asludge melting furnace with a coke packed bed is anewindustrial process constructed recently as the processfor environmental protection. In this process, the reactionof packed coke with air introduced from the tuyeresgenerates heat which is used not only for heating of gasand coke but also for dehydration, drying and meltingof the sludge charged as illustrated in Fig. 37. Thesludgereleased thermal energy in its combustion process.Therefore, this is a typical high temperature process withthe three-phase flows of packed coke, gas and moltensludge. For evaluating the performance of this process,amathematical simulation modelwasproposedby Wanget al.60) In the model, the gas flow was estimated byErgun's equation with the convective term and the liquid

fiow was computed by the combined model of theprobability andthe continuous flow modelsas mentionedin the previous section. Thekinematic modelwasadoptedfor the estimation of solid flow. Consequently, thefundamental equations given by Eqs. (95), (97) and (99)

werederived from Eqs. (4), (6) and (7) by assumingsteadystate andby neglecting the transfer amongphases. Theseequations are the equations of continuity for each phase,Interaction forces to the other two-phases were con-sidered in the equation of motions for gas and liquid.

But the fiow of packed particles was solved indepen-dently.

Gasflow: div(8gpgu~~g)=~~Sg,i......

.........,(95)

ikH' H' H> ~ldiv(8gpgugug)= -eg grad Pg+F~+Fg .......(96)

33 (1993). No.

llii

Us

UstlIIIlltlllltllItlil

lltllllf]lIIIIIIIILIillllIII]Ill]]ILI11tlfll lIII

J'fll liltll'tl

Fig. 36.

Fig. 37.

U~

U

6

U/ Pf ef e~

:~~~~.1- /

045040o300.20

o. 5

P

20mmA4

40

60

eO

OO120

CaseA: Hangingoccurred

u; pf ef e~

o450.40o~oo20

P

20mmAa

40

60

80

IOO

oo(\1~

8f

oclr

e,

o~'~oroc,

PL

0.25

o~50

o 75oo

PL

0,25

o~o

0.75\c~!.

CaseB: Stable operation

Movementsof packed bed, gas and powders cal-

culated by the two dimensional mathematical sim-ulation model.58)

Air

SIQg~Ges

Coke v Studge

CombustibleGost

I

Cbke' ' ~)•'O~'tSj ludge

o~a~O~~~ia~~hiSiag

,'*....=

Schematic representation of a packedbed in a sludgemelting furnace.59)

Liquid flow: div(ElpluH'I) =~~Slk'i

""""""""'(97)

ik-> ~' -~~~'div(8l pl ul ul )= ~8l pl9 +F~+F~- 81 grad Pl "'(98)

Flow of packedparticles: div(espsuH>s)=~~S~,i ....(99)

ikusr

" " " " " (IOO)= =B(auszlar),

B=2.5ds"'

drv(aspespsus) ~~S~f (p=coke, sludge) .....(lO1)

ik

635 C 1993 ISIJ

l .30

:::

- 0.65N

0.00

(a)

l (m/ sl

I

ISIJ International. Vol.

l.O 0.5R CM)

(b)

~\~~--~~_~ i\~' j

II

33 (1993), No. 6

~\

0.0 1.0 0.5R CM)

P(Pa)9-1700S-lSOO7-ISOO6-iIOO5- 9004- 100i- SOO2- 300l- 100

O.O

(a) flow vector(b) isobars

Fig. 38.

Computedresults of gas flow.s9)

l .30

Z:

- O.S5N

O* Oo

- : I 0~4(ml•)

l.O O.S 0.0R (M)

(b)

l.O

R0.5

(M)0.0

(c)

_:~~~~/ jUQTv') 4'_~1~_=__'_// iI-s' oxiolI 2-2' sxio 4:3-2' ox l0~4

I 4-1' sx'o-s

ls-1' oxl0~4;ry/I

s-s' oxl0~51-3' oxl0~5t-1' oxi0-5

I

l

l .O 0.5• 0.0R CM)

(a) fiow vector(b) streamlines(c) contourlinesFig. 39.

Solid fiow computedby kinematic model,s9)

l .30

::

- 0.65N

0.00

: ' o~e(rt/1)

nteltfnl l~'1'

//i///)//)!:1 '*'

dry ' '

son' '

(a) _~

*~

(b)

0.00 1.OO O. 50 OR (M]

~~

(c)

l OO 0.50R (M)

OOI .OO

a*,*1+a* .= I .....,,...,...(l02)

Becausethe packed particles phase consisted of cokeand sludge layers, the volume fractions of the two layers

were estimated by Eqs. (lOl) and (l02).

In the above equations, the effect of gas and liquid

phases on descet behavior of the packed particles wasnot considered. Figures 37 to 40 show an example ofcomputedresults. In these figures preferential fiow of gasand higher pressure were found in peripheral region anddown-flow of gas appeareddue to the exhaustion of someof the gas from the sludge hole in the lower part of the

bed. Higher descent velocity of the packed particles wascaused by the strong consumption of coke in the

combustion zone near the gas inlet. The velocity vectorof the molten sludge was computedby the continuousflow model after determination of the dry zone by the

probability model. The measurementon the flow phe-

nomenaof three-phases in an industrial sludge melt-ing furnace wasquite difficult and wasnot reported yet.

However, the computed results shown in this paperseemto be reasonable from the practical data on the

consumption rate of coke, the processing rate of sludge

and the distribution of gas temperature.

Anumberof one- and two-dimensional mathematicalmodels of the blast furnace constructed on the basis of

O. 50R CM)

0.0D

(a) flow vector(b) streamlines(c) grid generationFig. 40.

Computedresults of liquid flow.59)

the transport phenomenatheory have been reported.

However, attention wasfocused on the flow phenomenaof only oneor two phases in the mostmodels. Themodelincluding the flows of four phaseshas not been reported.

As a general blast furnace model, one- and two-dimensional models have been developed up to the

present and have been applied to the analysis of actual

processes. In the one-dimensional models,61 ~64) chemi-cal reactions and heat transfer were mainly investigated,

however, in the two dimensional models,23,65 ~68) radial

non-uniform flow played a moresignificant role.

In the generalized blast furnace model, heat and masstransfer and chemical reactions have also to be consider-

ed as well as fiuid flow to solve simultaneously the dis-

tributions of teniperature, concentration and velocity.

Consequently, the model is quite complex, therefore, thefiuid flow equations included in the model is not alwaysthe best expression. In this review, the fiow equationsincluded in the blast furnace model proposed bySugiyamaet al,63) which seemsone of the best modelsto explain the inner state of the blast furnace at presentwill be described below. This model also includes the

fundamental equations for heat and masstransfer.

Gasflow; equation of continuity

C 1993 ISIJ 636

ISIJ International. Vol, 33 (1993). No. 6

l

PressI~

_~ -

Solid Gas

RR1=0.1

RR2-0.1.. O. 7Rh]--'-_1rO

RR\.3!~~0.3

'\.//'\ / O5

r'R 2~10 o. 7~~L/e;"~/91O 0.9

~oo

4

800

I~oo

eo600

800

Stepwise Total

\SC~

1400~2(b

1200 1600

Solid

1600

1800

Gasa) Stream c) Temperatureb) Reduction

lines distributiondegree

Fig. 41. Numerical simulation of a blast furnace process.31)

1200

drv(egPguH>g)=~~Sg,i...

..........(103)

ikequation of motion

~*8ggradPg=F~•,••••

••••••••••(104)

Solid flow; equation of continuity

div(8* psu~s)= " " " " " (I05)~~S~,i•-

ikequation of motion

8* grad P* =Ku~'*

. ... . . ... .(l06)

Liquid flow; equation of continuity

div(8l plu~~1)= " " " " " (I07)~~S~i••-

ikequation of motion

H* H,elgradPl=F~+F~+8lplH'9""

"""""(l08)

Manyreaction rates were considered in the model.These63)are for the reduction reactions of iron ores with

COandH2as gas-solid reactions, the reduction reactionof molten FeOwith carbon as solid-liquid reaction, theshrinkage of ore in the cohesive zone, the combustionandgasification of coke in lower part of the blast furnace.

From these, source and sink terms in the equation ofcontinuity for each phase were obtained based on therates of the reactions and the rates of the masstransferbetween the different phases. However, this model hasbeen considered as three fluid flows by neglectingpowder flow. Figure 41 showsan example of computedresults illustrating the stream lines of gas, packedparticles

and liquid (slag and hot metal), and the isobars of gas,which are the flow characteristics of three phases as well

as the distributions of reduction degree of iron ores, gasand solid temperatures. These three fluid flows areaffected mutually. In addition, the temperature and thereduction degree distributions affect significantly the flowbehavior of the three fluids through the change of thepacked structure as the changeof the shape of cohesivelayers. Therefore, gas flow showssharp changearoundthe cohesive zone.

637

4. Concluding Remarks

Simulation of flow phenomenagives useful informa-tion for the process design and improvement of theoperation results.

In order to achieve high efficiency operation of packedbed processes like blast furnace, each flow in multi-fluidphases should be solved based on the equation ofcontinuity and the equation of motion in the form ofNavier-Stokes equation. This paper proposed a gen-eralized mathematical model on the flow phenomenainthe packed bed processes including four different fiuid

phases of gas, Iiquid, powdersandpackedparticles. Thismodel can also be applied to the process analysis ofone- to three-phase fiows. However,mixedfiow processesof gas-1iquid phases and of powder-liquid phases arealso used as actual industrial processes. For these

processes, previous studies were conducted by applyingthe methodfor a single-phase flow and the method fordiscontinuous phase of liquid droplets and powderparticles. The method proposed in this paper has apossibility to be extended to the analysis of flowphenomenaof such mixed flow.

Coauthors

This report was prepared by all membersof TheCommittee on Transport Phenomenain Gas, SolidLiquid in PackedBeds, TheJoint ResearchSociety, Thelron and Steel Institute of Japan. The names andaffiliations of the members are listed as follows:Tomohiro Akiyama (Tohoku Univ.), Kuniyoshi Ishii

(Hokkaido Univ.), Kunihiko Ishii (NKKCorp.). Kazuolchifuji (Kawasaki Steel Corp.). Takanobu Inada(Sumitomo Metal Industries, Ltd.), Shin-ichi Inaba(Kobe Steel. Ltd.), Yuji lwanaga (Sumitomo MetalIndustries, Ltd.). Tateo Usui (Osaka Univ.), KazutomoOhtake (Toyohashi Univ. of Tech.), Yoshio Okuno(Secretary General, Nippon Steel Corp.). KatsumiKusakabe (Kyushu Univ.). Jokichi Shinoda (NisshinSteel Co., Ltd.), Kunio Shinohara (Hokkaido Univ.),MasatakaShimizu (Kobe Steel. Ltd.). MasayasuSugata(Post-Secretary General, Nippon Steel Corp.). TakashiSugiyama (Nippon Steel Corp.). Hiroshi Takahashi(Muroran Inst. of Tech.), Reijiro Takahashi (TohokuUniv.), Kanji Takeda (Kawasaki Steel Corp.). Seiji

Taguchi (Kawasaki Steel Corp.). Katsuhiro Tanaka(Nisshin Steel Co., Ltd.), Takeshi Furukawa (NKKCorp.), MasayukiHorio (Tokyo Univ. ofTech. &Agri.),

Takatoshi Miura (Tohoku Univ.). Seiji Morooka(Kyushu Univ.), Jun-ichiro Yagi (Chairperson, TohokuUniv.) and MotozoYasuno(Kawasaki Steel Corp.)

Nomenclature

aij : Contact surface area of i and j phases in

unit volume of bed (m2/m3(bed))

B: Constant for the kinematic model (m)Cd: Drag coefficient (-)

Cp~: Modified capillary number(-)Cpf : Constant pressure specific heat of fuel oil

(J/kg K)D* : Hydrodynamicequivalent diameter (m)

C 1993 ISIJ

DT:

dd'p' f'

ds:

dl:

e:

efs '

Fm:

Frl :

Frf :

Fk:

Fj'

F; :

Ga :m

g:l"ht' hs' nd ,

hc :

I:

k:1.

.kd, "c2 '

v'K8' Km'Ad '

L'H.mf :Pi :Ap:

r:

R~:Rem

'

Rep:

D.Rd' l\s '

rf :

~~ :Sj,i

jk

S S-c' d '

T.Tb, I ld '

T T '

f' sf '

C 1993 ISIJ

ISIJ International, Vol.

Equivalent diameter of gas flow channel in

packed bed (m)Diameters of packedparticle and powders,respectively (m)Diameter of the particle including accom-panying powderand/or liquid (m)Diameter of liquid droplet (m)Coefficient of restitution (-)Repulsion coefficient between powderandpacked particle (-)Melting fraction of powder layer in outer-shell (-)Froude numberfor liquid flow (-)

(=Iu~'I l/~D*g)

Froude numberfor powderflow (-)

(=Iu~fII ~D*g)

Coefficient given by Eq. (17) (-)Interaction force of i phase on j pha~e(N/m3)Term of drag force due to viscosity in iphase (N/m3)Modified Galliler number(-)

[={p~9(ip,d.)3}1(,x 282)]l*

Acceleration of gravity (m/s2)

Total, static and dynamic hold-up ofliquid, respectively (-)Convective heat transfer coefficient

(W/m2K)Unit tensor (-)Kinetic energy of turbulence (m2/s2)

Reaction rate constant of Eq. (63) (1/s)

Constant of Eqs. (25), (26) and (27) re-spectively (1/s), (1/s K), (m)Latent heat for evaporation (J/kg)

Massof powder (kg/particle)

Pressure of i phase (Pa)Pressure loss (kg/m2)Radial coordinate in cylindrical coordinate

system (m)combustion rate of coke (kg(C)/m3(bed) s)

Modified Reynolds number(-)

{= (pl Iu~'I lc,d.)/~l}

Particle Reynolds number(-)

{= (ip.d. IuH'g IPg)1,t.}

Detaching and Sticking rates of powderfrom/to packed particles, respectively(kg/m3(bed) s)

Radius of particle (m)

Masstransfer rate from kphase to j phasedue to the changeof i (k=g, s, e, f, where,

k~j) (kg/m3(bed) s)

Source term shownin Table 2Bed temperature and temperature on theliquidus line corresponding to chemicalcomposition of melt, respectively (K)Temperature of fuel oil droplet and satu-ration temperature, respectively (K)

638

33 (1993), No. 6

(3/2)T: Vibration energy (m2/s2)

t: Time (s)

ui : Real velocity of i phase (i=g, s,l, f) in the

bed (m/s)V, V* : Volatile matter included in coal and finally

released volatile matter, respectively (kg)

We: Webernumber{=(dpPll u~>ll2)lal} (~)Wp: Load (kg/m2)

z : Longitudinal coordinate (m)

Greek letters

8., 61: Critical surface tension and surface ten-sion, respectively (N/m)

8: Dissipation rate ofturbulent energy (m2/s2)

8i : Volume fraction of i phase (i=g, s,l, f)

(-)81(d), el(s) : Dynamicand static hold-up of liquid (-)

8f(/, d) : Volume fraction of powder accompaniedby dynamichold-up of liquid (-)

8f(p, s), ef(1, s) : Volume fractions of powder accom-panied by packed particles and static

hold-up of liquid, respectively (-)e: Circumferential coordinate in cylindrical

coordinate system (rad)Ol : Contact angle ofliquid on the solid surface

(rad)

Pi : Viscosity of i phase (i=g, s, e) (Pa ' s)

Pi : Density of i phase (i=g, s,l, f) (kg/m3)

Tg' Tf' T* : Stresses ofgas, powderandpackedparticles

phases, respectively (N/m2)c*, ipf : Shape factors of packed particle and

powder, respectively (-)

~: Angle betweenx-axis andgas flow direction(rad)

l)

2)

3)

4)

5)

6)

7)

8)

9)

lO)

l l)

l2)

13)

l4)

l5)

16)

17)

18)

REFERENCESS. Ergun: Ind. Eng. Chem., 45, (1953), 447.

K. Onda, H. Takeuchi and Y. Koyama:KagakuKogaku, 31(1967), 126.

J. Umada,H. Shinohara and M. Tsubakihara: KagakuKogaku,27 (1963), 978.

Kagaku Kogaku Binran (5th ed.), ed, by Soc. Chem. Eng.,Japan, Maruzen, Tokyo, (1988), 232.J. F. Richardson and W.N. Zaki: Trans. Chem.Eng,, 32 (1954),

35.

K. Shibata, M. Shimizu, S. Inaba, R. Takahashi and J. Yagi:

Tetsu-to-Hagan~, 77 (1991), 236.

M. Kuwabara, S. Takane, K. Sekido and I. Muchi: Tetsu-to-

Hagan~,77 (1991), 1593.

Y. Ohnoand K, Kondo: Tetsu-to-Hagan~, 73 (1987), 2088 and2036.J. Kudoand J. Yagi: Tetsu-to-Hagan~, 73 (1987), 2020.

M. Hatano and K. Kurita: Tetsu-to-Hagan~, 66 (1980), 1898.

S. V. Patankar: Numerical Heat Transfer and Fluid Flow,McGraw-Hill, NewYork, (1980).J. Wang,R. Takahashi and J, Yagi: KagakuKogakuRonbunshu,16 (1990), 723.

M. Choudhary, M. Propster and J. Szekely: AIChEJ., 22 (1976),

600,

E. Kasai, J. Yagi and Y. Ohmori: Tetsu-to-Hagan~, 70 (1984),

I567.

M, Yoshinaga and T. Kubo: SumitomoMet., 29 (1977), 383.

T. Sugiyama, Y. Satoh, M. Nakamuraand Y. Hara: Tetsu-to-

Hagan~,66 (1990), 1908.

K. Takatani and Y. Iwanaga: Tetsu-to-Hagan~, 73 (1987), 980.

T. Irita, T. Isoyama, Y. Hara, Y. Okuno, Y. Kanayamaand K.

l9)

20)

21)

22)

23)

24)

25)

26)

27)

28)

29)

30)

31)

32)

33)

34)

35)

36)

37)

38)

39)

40)

41)

42)

43)

ISIJ International. Vol. 33

Tashiro: Tetsu-to-Hagan~, 68 (1982), 2295. 44)

H. Satoh, T. Miura and N. Iwakiri: J. Fuel Soc. Jpn.. 70 (1991),

l 136. 45)

A. E. Scheidegger: The Physics of Flow Through Porous Media 46)(3rd ed.), Univ. Toronto Press, Toronto, (1974), 74. 47)T. Sugiyama, A. Nakagawa.S. Shibaike and Y. Oda: Tetsu-to-

Hagan~,73 (1987), 2044. 48)T. Fukutake and V. Rajakumar: Tetsu-to-Hagan~, 66 (1980),

1937. 49)

K. Kurita: Dr. Thesis, Study on the Elucidation of the Phe-

nomenain Blast Furnace by Mathematical Model, TokyoUniv., 50)(1986).

T. SugiyamaandM. Sugata: Seitetsu Kenkyu, (1987), No. 325, 34. 5l)

T. Fukutake and K. Okabe: Tetsu-to-Hagan~, 60 (1974), 507,

K. Tachimori, J. Ohno, M. Nakamuraand Y. Hara: Tetsu-to-

Hagan~,70 (1984), 2224. 52)

Y. Tomita, H, Ohkusuand T. Fukuda: Nisshin Steel Tech. Rep.,

56 (1987), l. 53)a) K. Shibata, Y. Kimura, M. Shimizu and S. Inaba: JOURNEES 54)

SIDERURGIQUESATS1989, Paris, (1989); b) S. Inaba, K.Shibata, Y. Kimura and M. Shimizu: The 54th Committee 55)

(Ironmaking), TheJapanScociety for the Promotion of Science

(JSPS). Rep, No. 1864 (July, 1988). 56)

Y. Ohnoand M. Schneider: Tetsu-to-Hagan~, 74 (1988), 1923.

M. Shimizu, A. Yamaguchi, S. Inaba and K, Narita: Tetsu-to- 57)

Hagan~,68 (1982), 936. 58)T. Sugiyama,Y. Hayashi, M. Sugata, H. Shibaike andN. Suzuki: 59)

CAMP-ISIJ, I (1988), 22.

M. Kuwabara,K. Isobe, K. Mio and I. Muchi: Tetsu-to-Hagan~, 60)

74 (1988), 1734. 61)J. Chen, T. Akiyama, H. Nogami, J. Yagi and H. Takahashi:

ISIJ Int,, 33 (1992), 664. 62)S. C. Cowin: PowderTech., 9(1974), 61, 63)

R. M. Neddermanand U. Tuzun: PowderTech., 22 (1979), 243.J. Wang,R. Takahashi and J. Yagi: KagakuKogakuRonbunshu, 64)

17 (1991), 179.

T, Tanaka,Y. Kajiwara andT. Inada: Tetsu-to-Hagan~, 74 (1988), 65)

2262.

T. Inada: Private communication. 66)S. Toyama,H. Hirosue and K. Uchida: J. Soc. PowderTechnol.(Jpn.), 10 (1973), 146. 67)S. Minakami and S. Toyama: KagakuKogakuRonbunshu, 18(1992), 205. 68)J. Yagi andY. Ohmori: Res, Inst. Min. Dress. Met.. TohokuUniv.,

35 (1979), 115. 69)

H. Satoh, H. Nogami, T. Miura and H. Iwakiri: Preprint for

23rd AutumnConf., Soc. Chem.Eng. Jpn., (1990), Oct. 70)

H. Yamaoka,T. Miyazaki andY. Kamei: CAMP-ISIJ, I (1988),

14.

(1 993), No. 6

K. Shibata, M. Shimizu, S. Inaba, R, Takahashi and J. Yagi:Tetsu-to-Hagan~, 77 (1991), 1267.

H. Yamaoka:Tetsu-to-Hagan~, 72 (1986), 403.

H. Yamaoka:Tetsu-to-Hagan~, 72 (1986), 2194.

K. Kusakabe, T, Yamaki, S. Morooka and H. Matsuyama:Tetsu-to-Hagan~, 77 (1991), 1407.

K. Kusakabe, T. Yamakiand S. Morooka: Tetsu-to-Haganb, 77(1991), 1413.

H. Nogami, T. Miura and T. Furukawa: Tetsu-to-Hagan~, 78(1992), 1222.

H. Aoki, S. Tanno, T. Miura and M. Ohnishi: Trans. Jpn. Soc.Mech. Engrs. B, 27 (1991), No. 538, 2152.

H, Aoki, A. Furuhata, T. Amagasa,S. Tanno, T. Miura, S.

Ohtani andM. Ohguro: J. Soc. PowderTechnol. (Jpn.), 68 (1989),

l053.J. X. Boillard, R. W. Luczkowski and D. Didaspow: AIChEJ.,

35 (1989), 908.J. Ding and D. Gidaspow: AIChEJ., 36 (1990), 523.J. Szekely andY. Kajiwara: Trans. Iron Steel Inst. Jpn., 19 (1979),

76.

J. Wang,R. Takahashi and J. Yagi: Tetsu-to-Hagan~, 77 (1991),

1585.

J. Wang,R. Takahashi and J. Yagi: Tetsu-to-Hagan~, 78 (1992),

to be submitted.

D. H. Andersonand A. V. Sapre: AIChEJ., 37 (1991), 377.

H. Yamaoka:Tetsu-to-Hagan~, 77 (1991), 1633.J. Wang,R. Takahashi and J. Yagi: KagakuKogakuRonbunshu,18 (1992), 659.J. Yagi and I. Muchi: Trans. Iron Steel Inst. Jpn., 10 (1970), 392.S. Taguchi, H. Kubo, N. Tsuchiya and K. Okabe: Tetsu-to-

Hagan~,68 (1982), 2303.C. J. Fielden and B. I. Wood:JISI, 206 (1968), 650.

M. Kuwabaraand I. Muchi: Tetsu-to-Hagan~, 61 (1975), 301 and787.J. Yagi, K. Takedaand Y. Ohmori: Trans. Iron Steel Inst. Jpn.,

22 (1982), 884,

M. Hatano, K. Kurita, H. Yamaokaand T. Yokoi: Tetsu-to-

Hagan~,68 (1982), 2369.

M, Kuwabara, S. Takane, K. Sekido and I. Muchi: Tetsu-to-

Hagan~,77 (1991), 1593.

M, Kuwabara, K. Isobe, K. Mio, K. Nakanishi and I. Muchi:Proc. 2nd Japan-Australia Symp.. ISIJ, Tokyo, (1983), 193.

T. Sugiyama:94, 95th NishiyamaMemorialSeminar, ISIJ, Tokyo,(1983), 131.

H. Kobayashi, J. B. Howardand A. F. Sarofim: 16th Int. Symp.on Combustion, (1976), 411.

H. Kokubu, A. Sasaki. S. Taguchi and N. Tsuchiya: Tetsu-to-

Hagan~,68 (1982), 2338,

639 C 1993 ISIJ