ISIJ International, Vol. 49 (2009), No. 4, pp. 495–504

1. Introduction

Direct study of full RH degasser is difficult to be com-puted as it involves two phase flow which is conceptuallydifficult and extremely time consuming to solve on a com-puter (typical computer time for an actual RH process toachieve steady circulation flow rate is of the order of 20 d).In order to understand the RH degasser operation experi-ments were conducted with 1/5 to 1/10 water models.1–5) Inwater models, attention was focused on the different param-eters such as snorkel diameter, gas injection rate, number ofnozzles etc., affecting the liquid circulation rate which pro-vides only flow field and mass flow rates of water. The cir-culation rate obtained by water models cannot be adoptedfor the real RH systems as thermal interactions are nor-mally neglected in the model studies.

Mixing time measurements in the real RH degasser wasalso carried out by few researchers.1) But carrying the ex-periments in plant is difficult and involves lots of practicaldifficulties. In order to understand the fluid flow behavior inthe ladle, two dimensional and three dimensional modelswere developed.1,4,6–10) In a two dimensional model,1,6) shortcircuiting of flow between the legs was observed whichdoes not represent the real situation, but the three dimen-sional models7–10) never showed short circuiting. In all theseladle studies, the presence of the vacuum vessel was ig-nored (RH operation was not taken in to picture) because itwas too difficult to simulate the flow in both the ladle andthe RH vessel. This made the above studies limited to onlyladle portion where the two phase flow simulations wereabsent.

The gas–liquid two phase flow in the up leg snorkel playsa decisive role in creating the circulation in the RH de-gasser. As the flow of molten steel in the ladle can not beseparated from liquid flows in the vacuum vessel andsnorkels, results obtained by the above studies are only

qualitative.Later, some researchers11–13) used pseudo single fluid

(homogeneous fluid) model to study the full RH systemwhich includes ladle, snorkels and vacuum vessel. In thesemodels the shape and position of gas bubbles/plumes andgas hold up were considered and the buoyancy force wasincluded in the momentum equations as source terms to ob-tain the circulation flow rate. However, these models couldnot be generalized in a CFD code to predict the circulationflow rate, because of presence of lot of empiricism in themodel.

In order to properly analyze the full RH degasser opera-tion, two phase analysis is needed. The shortcomings aris-ing due to the above methods can be overcome by usingmultiphase models. There are two main methods used formodeling two-phase flows namely the Eulerian–Eulerianand the Eulerian–Lagrangian approaches. The Eulerian–Eulerian approach is based on the concept of interpenetrat-ing continua, for which all the phases are treated as contin-uous media with properties analogous to those of a fluid.Both phases are coupled by inter phase forces. From themathematical modeling point of view both fluids are de-scribed by similar equations of conservation. In the Eu-lerian–Lagrangian approach the particles, bubbles anddroplets are considered as discrete phase and the trajectoryof each individual particle is tracked in space using theequations of motion based on Newton’s second law. Thismethod is physically realistic than the Eulerian–Eulerianapproach for dilute flows.

Recently an attempt was made to analyze the full RH de-gasser using the Eulerian–Eulerian model14,15) and the pres-ent authors have made a contribution to it. The thermal andpressure variations on the gas bubbles were neglected inthese studies. The flow predicted by these models is similarto those obtained in previous studies and the circulationflow rate predicted for the cold cases could match reason-

Prediction of Circulation Flow Rate in the RH Degasser UsingDiscrete Phase Particle Modeling

P. Anil KISHAN and Sukanta K. DASH

Department of Mechanical Engineering, IIT, Kharagpur, India, 721 302. E-mail: sdash@mech.iitkgp.ernet.in

(Received on October 2, 2008; accepted on January 15, 2009)

Conservation equations for mass and momentum with a two equation k–e model are solved for the con-tinuous phase along with a discrete phase particle modeling (representing gas bubbles) in the RH degasserto predict the circulation flow rate of water in a scaled down model and then the numerical solution hasbeen extended to the real plant case for the prediction of steel circulation flow rate in the actual RH vessel.The prediction of the circulation flow rate of water from the present numerical solution matches reasonablywell with that of the experimental observation, taking into account various uncertainties those have beenimbedded in the numerical model. RH operation for multi up legs and single down leg for a water modelshows that the circulation flow rate falls with the number of up legs and there is an optimum number ofdown legs for which the circulation flow rate is the maximum for the case of a single up leg. For the actualRH operation in plant it was seen that the circulation flow rate increases with the increase in snorkel diame-ter and snorkel immersion depth (SID). However, it is apparent that there is existence of optimum SID formaximum circulation flow rate. For different down leg immersion depth the circulation flow rate in the RHdepends heavily on the up leg immersion depth. The actual RH operation of the plant for the multi up legand down leg cases was found to be exactly similar in nature to that of the water model cases.

KEY WORDS: circulation flow rate; multi leg RH; discrete particle modeling.

495 © 2009 ISIJ

ably well with that of the experimental observations. Butfor the actual RH degasser, this model predicts the circula-tion rate far away from the actual circulation rate becausethe thermal effects of the gas expansion could not be takeninto account in the model. The reason for this will be dis-cussed later in the text and can be read from the article ofthe present authors.15)

All the above studies have so far considered two leg RHsystems which consist of one up leg and one down leg. Liand Tsukihashi13) studied the effect of multi leg RH systemon circulation rate. In that study, one down leg was kept atthe centre, and three up legs were placed surrounding thedown leg at equal angles. The circulation flow rate of multileg RH system was found to be more compared to conven-tional RH system.

In the present study, the recirculation in the RH degasserwith gas injection was solved using Eulerian–Lagrangianapproach which is well suited for dilute flows. In thismethod, liquid phase is considered to be a continuum andgas phase is treated as dispersed phase in the form of bub-bles. Each computational cell contains the continuous anddispersed phase to some fraction. Emphasis was made onthe effect of different snorkel sizes, snorkel immersiondepths along with multi up legs and down legs for the cir-culation flow rate of the liquid in the RH system.

2. Governing Equations with the Assumptions

The unsteady mass and momentum transfer of rising gasbubbles to liquid is considered using the following assump-tions.1. The flow is assumed to be isothermal. So the energy

equation need not be solved. In reality, the argon gaswhich is injected into steel attains the temperature likethat of steel within a much shorter time compared to itsaverage residence time in the up leg and in the RH ves-sel.

2. The presence of slag on top surface of the liquid steelin the ladle was neglected in the study.

3. The flow is essentially Newtonian and incompressiblewith constant properties.

4. Momentum transport between the continuous phaseand discrete phase can be represented by the interphase coupling terms.

5. Bubble–bubble interactions were assumed to be negli-gible and hence, the drag coefficient correlation, appli-cable to a single bubble–fluid system, was applied forthe appropriate range of Reynolds number.

6. Turbulence is limited to be a property of the liquidphase, and it may be described using the modified twoequation k–e model.

7. Discrete mono size spherical bubbles were assumed toform at the nozzle tip. Thus the size of the bubblesforming at the nozzle or orifice was assumed to beknown a priori from the available correlation reportedin the literature16)

...........................(1)

8. The size of the bubble was estimated on the basis ofthe gas flow rate and was considered to remain invari-ant during its rise through the liquid.

9. Collapse of gas bubbles near the free surface and thesplashing due to them is neglected. The free surface ofthe liquid changes little bit due to splashing and thisdoes not affect the circulation flow rate.

With the above assumptions, the governing equations forcontinuous phase are written as follows.

2.1. Continuous Phase EquationsThe governing equations for the continuous phase of

multiphase flows can be derived based on the Navier–Stokes equations for single-phase flows. Considering theexistence of dispersed particles, a volume-averaging tech-nique17) was used to develop a set of partial differentialequations to describe the mass and momentum conserva-tion. The volume-averaged forms of incompressible, un-steady conservation equations can be written as16–18)

..................(2)

.......(3)

The velocity fluctuations are related to the mean stress fieldthrough the Boussinesq hypothesis:

...........(4)

Based on Newton’s third law of motion, the forces acting onthe particles by liquid is equal to the reaction force on theliquid. Therefore, the momentum transfer from particles toliquid is taken into account by adding the particle–fluid in-teraction force ∑F which can be obtained by summing theforces that are acting on all the particles in the cell by thefluid. This force includes drag force, virtual mass force andother forces acting on the particles by the fluid which is ex-plained in the inter phase coupling.

The 3-D model of unsteady, liquid phase equations mustbe supplemented with additional relationships to achievethe closure. The closure relationships used in the presentmodel refer to two-phase bubbly flow, with continuous liq-uid phase and dispersed gas phase.

2.2. Dispersed Phase Equations: Equations of Motionfor the Bubbles

The analysis of the bubbles has been carried out via aLagrangian approach, in which the trajectories of individualbubbles are determined in space and time. The time rate ofchange of momentum (proportional to mass*acceleration)of a discrete bubble is the result of various forces acting onthe bubble. Forces acting on a particle include inter phaseforces between fluid and particle and forces imposed by ex-ternal fields. The appropriate form of Newton’s second lawof motion for the bubble, is represented as

..........(5)

Right hand side of Eq. (5) represents the forces acting onthe particle/bubble which include gravity force, buoyancyforce, drag force, virtual mass force, and other fluid–parti-cle interaction forces which include Basset force, Liftforces, Brownian force etc., those were neglected in thepresent study. Drag force, virtual mass force and other in-teraction forces are the inter phase forces between the liq-uid and gas phase. The calculations of different forces areshown below.

The gravity force and buoyancy force on the particle isgiven as follows.

md

dtF F F F Fi

i i i i ib G B D VM Int

v� � � � �, , , , ,

ρ ρ δ μl l l′ ′∂∂

∂∂

⎛

⎝⎜⎞

⎠⎟u u k

U

x

U

xi j iji

j

j

i

� � �2

3

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

⎛

⎝⎜⎞

⎠⎟

∂∂

′ ′ ∑

tU

xU U

p

x x

U

x

U

x

xu u

VF

ij

i j

i j

i

j

j

i

ji j i i

( ) ( )α ρ α ρ

α α μ

α ρ α ρ

l l l l

l l l

l l l l

�

�� � �

� � �1

g

∂∂

∂∂t x

Uj

i( ) ( )α ρ α ρl l l l� �0

dQ

b �0 352

0 2

.

.

g

⎛

⎝⎜

⎞

⎠⎟

ISIJ International, Vol. 49 (2009), No. 4

496© 2009 ISIJ

FG,i�FB,i�Vgi(rg�r l) .........................(6)

The position equation for the trajectories of the bubbles aredefined as

...................................(7)

2.3. Inter Phase CouplingThe interaction between fluid and particle is known as

phase coupling or phase exchange. The coupling of the dis-crete Lagrangian and continuous Eulerian phases is themost difficult part of the modeling. There are three kinds ofphase couplings namely, mass coupling, momentum cou-pling and energy coupling. Various models have been pro-posed for these couplings. If the particle concentration issufficiently low, then the so-called one-way coupling isenough. One-way coupling means that particles are affectedby fluid but fluid is not affected by particles. In this case,the theoretical treatment is simple where the equations offluid motion are same as single phase flows.

As the concentration increases, two-way coupling isneeded. In the two-way coupling, both phases are affectedby each other. The fluid phase is affected by particles andparticles are affected by fluid. As the concentration of dis-crete phase increases further, in addition to the two-waycoupling, we have to consider the particle–particle interac-tion. In this work, mass and energy coupling are not de-scribed anymore because there is no phase change takingplace. In the present case, we are using two way couplingwhere both phases are coupled with each other, because theconcentration of bubbles is not insignificant in the medium.

2.4. Types of Coupling2.4.1. Volume Fraction Coupling

In two phase bubbly flow, the dispersed phase bubblescan become quite large due to coalescence, merging toform extended cylindrical bubbles which can have trans-verse diameters approaching the pipe diameter. For this rea-son, the volume occupied by the dispersed particles cannotbe neglected. In the present simulations, the diameters ofbubbles are not negligible compared to the size of the cell.Therefore volume fraction of the discrete phase bubbles inthe computation cells has been considered. The volumefraction in the continuous phase equations results fromthese spatially distributed discrete bubbles. However, in thecontinuous phase equations, the volume fraction is a con-tinuous field variable with a spatially smooth distribution,as in the classical two-fluid model.

The volume fraction of dispersed phase is calculated as

..................................(8)

assuming all the bubbles are of uniform size.And the volume fraction of the liquid phase is calculated

from

ag�a l�1 (valid for each computational cell) .....(9)

2.4.2. Momentum Transfer CouplingThe momentum exchange term in the liquid phase equa-

tion of motion can be evaluated from the concept that theinter phase exchange forces experienced by the particles actwith equal magnitude but in opposite directions in the liq-uid phase. The drag force acting on a suspended particle isproportional to the relative velocity between the phases andis given as

...............(10)

where A is the exposed frontal area of the particle to the di-rection of the incoming flow, CD is the drag coefficient,which is a function of the particle Reynolds number, Re.For rigid spherical particles, the drag coefficient, CD, can beestimated by the following equations

.......................(11)

Where Re is relative Reynolds number

.........................(12)

where a1, a2, and a3 are constants applicable for smoothspherical particles over several ranges of Re given by Morsiand Alexander.20)

The forces due to acceleration of the relative velocity canbe divided into two parts: virtual mass effect and the bassetforce. The virtual mass effect relates to the force requiredto accelerate the surrounding fluid. This force is some timescalled the apparent mass force because it is equivalent toadding a mass to the sphere. The added mass force ac-counts for the resistance of the fluid mass that is moving atthe same acceleration as the particle. For a spherical parti-cle, the volume of the added mass is equal to one-half ofthe particle volume, so that

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

The Basset term describes the force due to the laggingboundary layer development with changing relative veloc-ity. But such forces are normally not present in a bubblyflow, so we do not describe them here. So the inter phasecoupling force calls for drag force and virtual mass whichhas been used through Eqs. (10), (13). The bubble rotationand bubble coalescence during the bubble motion are alsoignored.

2.5. Turbulence ModelingSeveral alternatives have been proposed to estimate the

effective viscosity of turbulent liquid phase in gas–liquidflows. Modified form of standard k–e model performs satis-factorily. The governing transport equations for turbulentkinetic energy, k and its dissipation rate, e is represented asfollows.16,18,19)

.........................................(14)

....(15)

The distribution of turbulent viscosity is obtained from

∂∂

∂∂

∂∂

∂∂

⎛

⎝⎜⎞

⎠⎟

′ ′∂∂

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

t xU

x x

C u uU

xP

kC

k

jj

j j

i ji

j

( ) ( )α ρ ε α ρ ε

αμσ

ε

α ρε

ρε

ε

ε ε

l l l l

lt

l l b l

�

�

� � � �1 2

2

∂∂

∂∂

∂∂

∂∂

⎛

⎝⎜⎞

⎠⎟′ ′

∂∂

⎛

⎝⎜⎞

⎠⎟

tk

xU k

x

k

xu u

U

xP

jj

j k ji j

j

i

( ) ( )α ρ α ρ

αμσ

α ρ ρ ε

l l l l

lt

l l l b

�

� � � � �

F Vdu

dt

d

dti i

VM l� �1

2ρ

v⎛

⎝⎜⎞

⎠⎟

Re��ρ

μl b

l

d ui iv

C aa a

D � � �12 3

2Re Re

F C A u ui i iD D l, ( )� � �1

2ρ v vi i

αgcell

�NV

V

dx

dti

i� v

ISIJ International, Vol. 49 (2009), No. 4

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m t�r lCmk2/e ..............................(16)

Where Cm, Ce1, Ce2, s k and se are constants and their val-ues are as follows. Cm�0.09; Ce1�1.44; Ce2�1.92; s k�1.0and se�1.3 according to Launder and Spalding.21)

It is to be noted that the mass conservation (Eq. (2)), mo-mentum conservation (Eq. (3)) and the turbulent k and eequations (Eqs. (14), (15)) are the most general in form. Ifag is zero or too small (means no gas bubble being injectedto the system or too less gas being injected) then theseequations become the governing equations for a single fluidflow in any system.

Pb is the production rate of turbulence energy which isdue to the interaction between the mean flow field and thebubble motion. In literature18) it was reported that one mayneglect the contribution of extra turbulence generation dueto the interaction between fluid flow and bubble motion.Therefore, in the present work, Pb was set to zero. As a re-sult, the modified k–e model reduces to standard k–emodel. In the present study, the effect of fluid turbulence onbubble motion is not considered. As a result, average liquidvelocity is used in the calculation of source terms instead ofinstantaneous velocities of the liquid. If the fluid turbulenceon particle trajectories is to be considered, then we have touse the instantaneous velocity instead of average velocity inthe calculation of the source terms. The instantaneous ve-locity consists of average velocity and fluctuating compo-nent. Calculation of fluctuating components is available inliterature16,18) but is not required in the present numericalsimulation.

3. Boundary Conditions

No slip boundary condition was used at all the imperme-able walls for continuous phase while bubbles were allowedto reflect at these impermeable walls. Wall functions wereconsidered at the wall for computations of near wall k and evalues. The gas bubbles enter the solution domain from thenozzles at a rate determined by the gas flow rate. The gasbubbles are allowed to escape from the degasser at the rateat which it reaches the free surface. The free surface of theliquid in the RH was considered to be flat with zero shearstress condition (the free surface being the domain bound-ary) and the bubbles were allowed to escape from this sur-face. It is to be mentioned here that the exact shape of thefree surface in the RH vessel is immaterial if one wishes todetermine the circulation flow rate in the RH system. At theinlet of the nozzle the gas velocity is prescribed because itis known from the experiment and actual operation and theturbulence intensity was set to 2% with a turbulent viscos-ity ratio of 10 to start with.

4. Numerical Modeling

The equations to be solved are the mass and momentumconservation equations for the liquid (Eqs. (2) and (3)) andthe equations which describe the transport of turbulencequantities k and e (eqs. (14) and (15)). These partial differ-ential equations are coupled with equations of motion ofbubbles which are solved to obtain bubble trajectories andvelocities, volume fraction and momentum exchange be-tween phases. The momentum equation of liquid phasealong with turbulence closure equations are integrated overa control volume and then discretized to yield algebraicequations by using finite volume method. The discretizedequations thus obtained were solved by the algebraic multigrid solver of Fluent 6.3 CFD code by incorporating theabove boundary conditions. In discretization process, firstorder upwind scheme was used for convective variables andcentral differencing for diffusion terms. The combination of

discrete particle simulation and continuous phase simula-tions was achieved in the following way.

First, velocity and pressure fields are assumed through-out the computational domain. The discrete phase bubblevelocities for each particle are obtained by integrating theparticles equation using Trapezoidal integration. Fromthese the volume fraction and fluid–particle interactionforces in each computational cell are calculated. The con-tinuous phase velocity distribution is calculated by usingthe momentum source terms (which are the sum of fluid-particle interactions of all particles in each computationalcell). The continuity equation is used to correct the pressurefield in the domain. The pressure and the velocities of con-tinuous phase are then updated for the next iteration to con-tinue. The solutions of continuous phase equations involveseveral iterations before convergence for each time step.Once the solution is converged, the positions of particlesare determined using the position equation (Eq. (7)) for thenext time step. The computation returns to the beginning ofthe loop where the particle motion equations are integrated.In this way, both dispersed phase and continuous phases arecoupled via inter phase coupling. The simulations were car-ried till steady state conditions for mass circulation in theRH were achieved.

In this way continuous and discrete phases are coupledby inter phase coupling forces while satisfying momentumconservation equations. The convergence of the equationswas monitored by whole field residual of each variable andwhen this variable fell below 10�3 the solution was as-sumed to have converged for that time step. Steady state so-lution for the circulation flow rate could be achieved by per-forming the simulation for more than 10 s which takesabout a computer time of more than 20 d.

Near the nozzle inlet, grids were tetrahedral and awayfrom the nozzles hexahedral grids could be placed to havebetter accuracy. Due to the connection problems of the noz-zles, with up leg tetrahedral cells were placed near the noz-zles although these type of cells reduce the accuracy of so-lution to some extent. The nozzle arrangements for differ-ent cases are shown in the corresponding figures (Fig. 1,and Figs. 3 and 4). The time step used in the simulation was0.0001 s during the entire period of simulation. The totalnumber of cells for a particular case was always more than70 000 with finer cells near the nozzle and in the up leg.Grid independent test was carried for almost all the cases ofthe water model since the simulation time was limited toonly 4 d. But for the case of actual plant RH system it wasimpossible to carry out grid independent test for all thecases as because each run was taking more than 20 d. Onlyone grid independent test for the plant RH system was done90 000 cells and the mass circulation rate was almost samelike the case where 70 000 cells were used. The propertiesof the fluids used in the simulation can be read from Table1.

5. Results and Discussions

At the beginning of a typical calculation, bubbles ofknown diameter (from Eq. (1)) were introduced into the do-main through 10 nozzles and their position tracked as afunction of time till the gas bubbles reached the free surfacewhere these bubbles were allowed to escape from the sys-tem. In the present calculations, the bubble size created atthe nozzle tip is around 3–6 mm for water–air model and8–16 mm for steel-argon system depending on the flow rateat ambient conditions. The bubbles drag the liquid alongwith it inside the up leg. Thus the liquid from the ladle getssucked in to the RH vessel through the up leg and then re-turns to the ladle again through the down leg. This way theliquid movement creates mixing in the ladle as well as in

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the RH vessel. The flow in the ladle percolates to the pe-ripheral wall along the bottom and ascends along the ladleside wall, forming a number of small loops in the ladle.Subsequently, the bulk of the stream flows circuitously to-wards the entrance of the up snorkel, and hence the circula-tory flow of the molten steel being treated is complete. Theliquid free surface in the ladle is relatively motionless: theliquid in these regions do not mix very vigorously.

5.1. One Up Leg and One Down Leg for the WaterModel

In order to get a numerical confidence we first started acase for a water model experiment. The schematic diagramand details of geometry of the water model are given in Fig.1(a) and Fig. 1(b). Figure 1(c) shows the nozzle arrange-ment in the up leg. There are two rows of nozzles in the upleg with 5 nozzles in each row spaced at 72° to each otherin a symmetric manner. The circulation flow rate of waterwas numerically computed from the present CFD computa-tion and has been plotted against the air injection rate inFig. 2 (triangles are numerically computed and line is acurve fit) where a comparison with the experimental obser-vation can be seen. The experimental measurement wasdone by adding 50 g of KCl solution to the setup when thesetup was almost in steady state. Then the concentration of the tracer (C ) (KCl solution) was measured with time,and plotted as C/C∞ against time where C∞ was the finalconcentration in the vessel. The peak area method (essen-tial idea of Nakanishi et al.1)) was used to compute the cir-culation flow rate, details of which can be seen from Ap-pendix-A. We did not use any velocity measurement in thedown leg to compute the circulation flow rate because thatwould have been a very expensive setup. The height ofwater in the vacuum vessel was controlled by the amount ofvacuum being produced by the pump in the vessel. The ex-periment was not directly done by us, rather was done in anindustrial R&D of Tata Steel. Almost all the data are classi-fied, so we can not show the details of the raw readings.Only the end result was given to us for comparison pur-poses. The number and size distribution of bubbles in theexperiment could not be measured because any such meas-urement will call for very elaborate setup which again willincrease the cost of the experiment tremendously. The com-parison with the experimental observation is not very bad ifwe think that the governing equations describing the physi-cal process is not all that simple. It can be seen from Fig. 2that the circulation flow rate has not reached a state of satu-

ration. After reaching a state of saturation the circulationflow rate will decrease with the gas flow rate because athigh gas flow rate there will be too many bubbles present inthe up leg which will choke the up leg and this will notallow the liquid to rise in to the up leg for which the circu-lation flow rate will decrease. The present study agrees withthe results of Park et al.12) where there is a saturation valuefor circulation rate.

5.2. Multiple Up Legs and Multiple Down Legs for theWater Model

Figure 3 shows the diagram for a two up legs and onedown leg case with the nozzle arrangement while Fig. 4shows the diagram for three up legs and one down leg for ascaled down RH system. From Figs. 3 and 4 it can beclearly seen that there is only one row of nozzles used foreach up leg instead of two which was used for the one upleg and one down leg RH system. The diameters of the uplegs and down legs in multi leg RH system is chosen on thebasis of equal cross sectional area for up leg and down legcompared to a two leg RH system (one up leg and onedown leg). The same RH system can be interchanged to getnew cases like one up leg and two down legs along withone up leg and three down legs. Figure 4(d) shows the ve-locity vector at a cross section AA passing through the upleg and one down leg. Figure 4(e) shows the vector at across section of BB. From the field vector it is clear thatthere is enough mixing in the ladle as well as in the upperRH vessel. It must be noted here that the RH vessel heightmay be anything (as seen from Fig. 4) but the field vectorswill be only seen up to the height where the liquid is pres-ent in the vessel. The height of the liquid in the vessel willdepend on the amount of suction being applied to the RHvessel. Figures 5(a) and 5(b) show the field vectors at crosssection AA and BB when the RH system has one up legand three down legs. In any situation the mixing in the ladleis very clearly visible due to the RH operation.

Figure 6 shows (symbols are CFD computation and linesare curve fitted) the circulation flow rate in the water modelwith multiple up legs and only one down leg. It can be seenfrom the Figure that with the increase of up legs the circula-tion flow rate in the system has decreased. The amount of

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499 © 2009 ISIJ

Fig. 1. (a) Schematic diagram of experimental water model RH degasser. (b) 3-D viewof RH degasser (water model). (c) Plan view of nozzle arrangement in up leg.

Table 1. Properties of materialsused in the simulation.

Fig. 2. Water circulation rate in the RH de-gasser as a function of air injectionrate in the up leg: A comparison be-tween experimental measurementsand present computations (up legsize�80 mm down leg size�80 mmSID�170 mm).

gas being injected for one up leg system is the same as thatfor three up leg system in our present computation. Butfrom practical operation point of view the diameter of theup legs has to be reduced. This introduces extra resistanceto the flow of gas in the up leg and hence the velocity of gasfalls in the up leg as a result and hence the drag force im-parted to the fluid on turn also decreases. This is the reasonfor which the circulation flow rate falls in the RH systemwith the increase with the up legs. Li and Tsukihashi13)

found more circulation rate for three up leg and one downleg RH system in their computational model because theirmodel had incorporated lots of empirical formula from theexperiments. With such empiricism the model can not be in

general applied to a CFD study with a general CFD pack-age which has been done in the present case.

Figure 7 shows (symbols are CFD computation and linesare curve fitted) the circulation flow rate in the water modelRH system with one up leg and many down legs. To accom-modate the down leg in the fixed ladle we have to reducethe size of the down leg when the number of down legs in-creases (according to the above equivalent cross sectionalarea concept). When the gas flow rate in to the up leg isfixed the driving force for the circulatory motion in the RHsystem is fixed. But the circulation flow rate in the RH sys-tem is seen to vary with the number of down legs. Whenthe down legs are two, the circulation flow rate is the maxi-

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Fig. 3. (a) Water model RH degasser with two uplegs and one down leg. (SID�170 mm)(geometry scalable). (b) Plan view of RHdegasser. (c) Plan view of nozzle arrange-ment in up legs.

Fig. 5. Flow field in the water model RH degasser with three down legs and oneup leg (up leg size�80 mm, down leg size�50 mm and SID�170 mm).(a) At section A–A, y�0 mm plane (as shown in Fig. 5(c)). (b) At sectionB–B, x�55 mm plane (as shown in Fig. 5(c)).

Fig. 6. Water circulation rate in the RH de-gasser with multiple up legs and onedown leg (up leg size (1)�80 mm,(2)�60 mm, (3)�50 mm, down legsize�80 mm, SID�170 mm).

Fig. 4. (a) Water model RH degasser with three up legs and one down leg (geometryscalable). (b) Plan view of RH degasser. (c) Plan view of nozzle arrangement inup leg. (d) Flow field at section A–A, y�0 mm plane (SID�170 mm) (as shownin Fig. 4(f)). (e) Flow field at section B–B, x�55 mm plane (SID�170 mm) (asshown in Fig. 4(f)).

mum and it again falls when the number of down legs in-creases to three. This can be explained in the followingmanner. Through the up leg gas gets pushed into the vac-uum vessel and thereby creates a head for the fluid to movein the system. The fluid flow will be highest if the systemencounters less resistance. This happens in the case of twodown leg system compared to a three down leg systemwhere the system resistance rises due to small diameterdown legs being present more in number. This phenomenoncan also be thought this way. If we want to discharge waterfrom a dam (on the other side of which there is a constanthead of water available) through a pipe or through manynumbers of pipes but smaller in diameter then in whichcase we get more flow rate? Surely when the system resist-ance is the lowest we get highest flow rate. The system re-sistance depends on the pipe diameter (length of each pipeis equal here and also in the RH system) and also on theflow rate through it, via the friction factor being influencedby the Reynolds number. So there are always an optimumnumber of pipes with specified diameters which gives thehighest flow (unless someone totally opens the dam whichis not the case we are considering). Our present RH systemis exactly a replica of the dam problem where we get high-est circulation rate with only two down legs.

5.3. Actual RH System Used in the PlantAfter having a water model study done, we extended the

numerical simulation to a real plant case. The RH systemused in the plant is shown in Fig. 8(a) with all its dimen-sions and Fig. 8(b) shows the three dimensional view of thesystem. The objective is to compute the steel circulationflow rate in the RH system. The circulation of steel isshown in Fig. 9 as a function of argon injection rate andsnorkel size. It can be seen from Fig. 9 that as the snorkelsize increases the circulation flow rate in the RH system in-creases. When the diameter of the up leg or down leg(snorkel size) increases it offers less flow resistance so thecirculation flow increases. But in any case this numericalsimulation will not be able to predict the actual circulationflow rate that is encountered in the plant, because argonwhen injected into the up leg will expand tremendouslywhich will cause more drag force on the fluid and hence thecirculation flow rate will be much more in real practice. Butthe formulation we did here is capable of doing it providedwe use the energy equation to be solved and then predictthe circulation flow rate. But for the time being this type ofcomputation seems to be almost impossible. The reason forthis will be discussed little later from now. But the presentmodel can predict the circulation flow rate for a cold case

very well as we have already seen in the case of the watermodel.

5.4. Effect of Snorkel Immersion DepthFigure 10 shows the circulation flow rate when the SID

is changing from 500 to 1 500 mm. When the SID is1 500 mm, the liquid column above the nozzle level to thefree surface in the RH vessel is higher than that for the casewhen the SID is 500 mm. For a particular argon blowingrate the driving force created by the argon bubble to dragthe liquid along with it will depend on the height of the liq-uid column that is available in the up leg (from the nozzleto the free surface in the RH vessel). If the liquid columnheight is higher, then the bubbles will impart more kineticenergy to the liquid column before they come out at the freesurface. This is because the bubbles get enough length toimpart the energy to the fluid because they take more timeto travel up the liquid column. Although the mass of theliquid column for 1 500 mm SID is higher than that for thecase of 500 mm SID but it has acquired lot of kinetic en-ergy so it gets more upward velocity which helps it to moveup and hence the circulation flow rate becomes higher forthe case of 1 500 mm SID compared to that for the case of500 mm SID. For the case of 500 mm SID the length of the liquid column is much shorter compared to that of1 500 mm SID and hence the bubbles get very short time toescape out at the free surface and they can not impart allthe kinetic energy to the fluid because their surface escapevelocity is much higher here compared to the case of1 500 mm SID. So the circulation flow rate for 500 mm SIDremains lower compared to the case of 1 500 mm SID forlower argon flow rate. When the argon flow rate increasesthen the kinetic energy that is imparted by the argon bub-bles to the liquid column also increases but this time theliquid in the up leg of the 500 mm SID will acquire highervelocity because of its lower mass (lower column length),so the circulation flow rate at 1 600 lpm of argon flow rate ishigher for the case of 500 mm SID compared to the case of1 500 mm SID. The circulation flow rate for the 1 000 mmSID remains in between the 500 mm and 1 500 mm SID tillthe argon flow rate of 1 200 lpm and falls after an argonflow rate of 1400lpm compared to the case of 500 mm SID.

So far we discussed about the drag force being impartedby the argon bubbles to the liquid column in order to give ita circulation flow rate in the RH system. However, the dragforce imparted by the bubbles normally depends on the slipvelocity of the bubbles compared to the liquid mass sur-rounding it. So the slip velocity can be zero when the bub-ble velocity and the liquid velocity in the up leg are about

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Fig. 7. Water circulation rate in the RH de-gasser with multiple down legs andone up leg (down leg size (1)�80 mm,(2)�60 mm, (3)�50 mm, up legsize�80 mm, SID�170 mm).

Fig. 8. (a) Schematic diagram of RH degasser used in the plant. (b) 3-D viewof RH degasser (geometry scalable).

the same. Then the circulation flow rate can fall in the sys-tem which can be seen to be happening for the case of1 000 mm SID at 1 600 lpm of argon flow rate.

5.5. Variable Immersion Depth for up Leg and DownLeg

If the down leg immersion depth is varied along with theup leg immersion depth then the circulation flow rate ofsteel in the RH system can be very peculiar. Such a plot ofcirculation flow rate at an argon injection of 1 600 lpm isshown in Fig. 11 where the down leg and up leg immersiondepth varies. At low down leg SID of 100 to 500 mm, an upleg SID of 500 mm produces highest circulation flow ratecompared to the case of 1 000 mm and 1 500 mm of up legSID. But at 1 000 mm of down leg SID the case just re-verses completely and again at a down leg SID of 2 000 mmthe case again reverses completely. The up leg immersiondepth has very less role to play in the circulation flow rate.It is the down leg which injects the fluid in to the ladle.When the down leg depth is low the jet coming out of itwill hit the bottom of the ladle with less strength which willcreate lesser circulation in the ladle but the jet will get

enough suction length in the ladle to suck more fluid withit. So the strength of the recirculation in the ladle will be acombined effect of the impact of the jet and the length ofthe plume suction created by the jet. So it is expected thatthe recirculation strength will vary with the immersiondepth of the down leg and one can expect maxima and min-ima on the curve in such type of a combined variation ofthe circulation strength. On top of this effect, the immersiondepth of the up leg will also add some minor effect andhence the curve can be a wave shape curve which we haveobtained in Fig. 11.

5.6. Effect of Multiple Up Legs and Down Legs forPlant Size RH Operation

Figure 12(a) shows the plant size RH system with twoup legs and one down leg with SID of 500 mm. Figure12(c) shows the velocity field at a cross section of AA andFig. 12(d) at a cross section of BB. Both the vector plotsshow strong circulation in the ladle. Figure 13 shows thevelocity field in the RH system with one up leg and twodown legs with a SID of 500 mm (this has been obtained byjust interchanging the down leg and up legs in Fig. 12). Thevector plots at cross sections of AA and BB are shown forthis case in Fig. 13(a) and Fig. 13(b).

Figure 14 shows the circulation flow rate of steel versusthe argon flow rate when there are two up legs one down legand one up leg and two down legs. We have discussed ear-lier for the water model that the circulation flow rate will beless if we increase the number of up legs (argon flow re-maining same and up leg size decreasing). Exactly the samephenomenon is repeated here for the circulation flow rate ofsteel in the plant size RH system which can be seen fromFig.14. We have also seen earlier for the water model that

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Fig. 9. Steel circulation rate in the RH degasser as afunction of air injection for different snorkelsizes (SID�500 mm).

Fig. 10. Steel circulation rate in the actual RHdegasser as a function of air injectionrate for different snorkel immersiondepth (snorkel size�450 mm).

Fig. 11. Steel circulation rate in the RH degasseras a function of DSID and different USID(snorkel size�450 mm, argon injectionrate�1 600 lpm).

Fig. 12. (a) RH degasser for the plant, with two up legs and one down leg. (b) Plan view ofRH degasser. (c) Flow field at section A–A, y�0 mm plane (SID�500 mm) (asshown in Fig. 12(e)). (d) Flow field at section B–B, x�400 mm plane(SID�500 mm) (as shown in Fig. 12(e)).

the circulation flow rate becomes the highest for two downlegs and one up leg and again here for the steel circulationrate we see the same phenomenon exactly repeating (Fig.14).

Figure 15 shows the circulation flow rate of steel forplant size RH when there are two down legs and one up leg.When the down leg size is higher the circulation flow rateof steel is higher. For the higher down leg size with a SIDof 1 500 mm the circulation flow rate is again higher com-pared to lower SID of 500 mm and 1 000 mm.

6. Difficulty to Reproduce Plant Result

In the actual plant practice the circulation flow rate in theRH vessel is much higher compared to what has been pre-sented here in Fig. 14. In the present CFD model the expan-sion of the argon gas due to sudden heating in the liquidsteel is not taken into account due to several difficulties.Had that been taken into account then the circulation flowrate would have been much higher. In order to take the ex-pansion of the gas in the liquid steel one has to solve theenergy equation along with the equation of state for the gaswhich considerably adds to computer time, and was foundalmost impossible with our computers. The gas within avery short time attains the temperature of liquid steel (about1 823 K) inside the bath and hence expands to about 15times inside the liquid bath. So in order to take into accountthis effect one has to use a very fine grid at the nozzle inletand use extremely small time step, which considerably in-creases the computational time and also the computationbecomes unstable for even 800 Lt/min of argon flow whichis the bare minimum flow for an industrial setup. So it wasalmost impossible to get a feasible solution with the energyequation being activated along with all other equations inplace. Otherwise also one could try to put higher nozzleinlet velocity for argon to take into account higher volumeflow of argon inside the bath. This process also calls for in-stability because the inlet velocities become as large as 250to 300 m/s at the inlet of each nozzle for the minimumargon flow of 800 Lt/min in a real plant case. So a practicalsolution of the plant case with or without the energy equa-tion being solved looks very difficult for the time being.However, the present method is a step towards that whichcan predict at least the circulation flow rate in a RH vesselfrom first principles with the use of CFD with very littleempiricism in the CFD model (only empirical formula

being used is the drag law).

7. Conclusions

The conservation of mass and momentum equations(with suitable source terms to represent interfacial drag, liftand virtual mass forces) have been solved numerically withthe k–e model; using an unstructured grid; to predict theflow field and the distribution of the liquid and gas bubblesinside the experimental set up as well as for the RH ladleand vessel. By using the flow field the circulation flow in-side the devices could be computed for various gas flowrates. It was found from the present numerical simulationthat multiple up legs do not help in augmenting the circula-tion flow rate in a RH vessel where as one up leg and twodown legs help to increase the circulation flow rate. At lowargon injection rate the lower SID will produce lower circu-lation rate but at higher injection rate the lower SID willproduce higher circulation rate only for two leg RH system.However, for the multi leg RH system, lower SID may notproduce higher circulation flow rate at higher argon flowrate because the circulation flow rate becomes too sensitiveto the diameters of the down legs. Circulation flow rate in-creases with the increase in the size of the down leg and upleg. Although the cold model study produces circulationflow rate closer to the experimental observation but thepresent model could not produce circulation rate of steel inthe RH vessel to the actual observation of the plant due toseveral difficulties, one of those is the inability to takeargon expansion in the RH system. But the present compu-tation is a realistic step towards the exact computation ofthe plant case because the present simulation is a computa-tion from the first principle without much empiricismimbedded in it.

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Fig. 13. Flow field in the RH degasser with two down legs and one up leg (up legsize�450 mm, down leg size�320 mm, SID�500 mm). (a) At section A–A,y�0 mm plane (as shown in Fig. 13(c)). (b) At section B–B, x�400 mm plane (asshown in Fig. 13(c)).

Fig. 14. Comparison of steel circulation rate formulti-leg RH degasser (up leg size(1)�450 mm, (2)�320 mm, (3)�450 mm,down leg size (1)�450 mm, (2)�450 mm,(3)�320 mm, SID�500 mm).

Fig. 15. Comparison of steel circulation rate for multidown leg RH degasser as a function of differentSID (down leg size (1)�320 mm, (2)�400 mm,up leg size (1), (2)�450 mm).

NomenclatureA : Exposed frontal area of the bubble in the direction

of incoming flow (m2)CD : Drag coefficientdb : Diameter of bubble or particle (m)F : Interaction forces between liquid and gas phases (N)g� : Acceleration due to gravity (m/s2)k : Turbulent kinetic energy (m2/s2)

mb : Mass of the bubble (kg)N : Number of bubbles in the computational cellp : Pressure (Pa)

Pb : Production of turbulence due to particle drag(kg/m s3)

Q : Gas injection rate (m3/s)Re : Relative Reynolds numberUj : Average velocity of liquid (m/s)ui� : Fluctuating velocity component of liquid (m/s)V : Volume of the bubble (m3)

Vcell : Volume of the computational cell (m3)vi : Velocity of bubble (m/s)xi : Position vector of bubble (m)

ag : Volume fraction of dispersed phasea l : Volume fraction of liquid phasee : Turbulent energy dissipation rate (m2/s3)

m l : Viscosity of the liquid phase (Pa s)m t : Turbulent viscosity (Pa s)rg : Density of bubble (kg/m3)r l : Density of liquid (kg/m3)

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C. A. Perim and G. A. Vargas Filho: Ironmaking Steelmaking, 31(2004), 37.

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(1974), 269.

Appendix A

50 g solution of KCL is introduced to the system by a sy-ringe and the tracer concentration is measured with time bythe probe which is inserted to the down leg. The probe actu-ally measures the conductivity of the solution at that point,but the computer converts the conductivity to the concen-tration by reading a predefined conductivity versus concen-tration chart which is normally calibrated very carefully. A“C” versus “t” curve can be obtained this way (Fig. A-1).When the system comes to steady state the final concentra-tion in the ladle will be C∞ which can be straight read fromthe “C” versus “t” plot. Another plot for C/C∞ versus “t”has to be drawn which will look exactly the same like thatof “C” versus “t” but will have different scale on the y axisbecause the y axis is now divided by C∞ (see Fig. A-1)

.........................(A-1)

The net amount of KCL solution added to the system isknown and can be found out from Eq. (A-1) where the inte-gration limit for time t1 and the circulation flow rate m areunknown. The C versus t curve is generated experimentallyand known. The net mass of the water in the RH degassersetup at the beginning of the experiment is MSol, which isknown. The final concentration in the system will be as perEq. (A-2).

.............................(A-2)

If we divide C∞ to both sides of Eq. (A-1) then we get (A-3)where the circulation flow rate m and the limit of integra-tion t1 are again unknown but this gives us a relation withMsol which is desired so that circulation flow rate can befound out. It is to be noted that Eqs. (A-1) and (A-3) are notseparate independent equations, rather (A-3) is a scaledequation of (A-1).

...............(A-3)

So, the task of determining t1 remains on the physics of thesituation. When the tracer is added to the up le, it comes tothe down leg slowly and its concentration increase withtime. But after some time the concentration reaches a low-est value, this means all the tracer have passed the downleg. Then the concentration slowly rises, which means thetracer after passing one loop in the system is again appear-ing at the down leg. So the point of lowest concentration inthe C or (C/C∞) versus t plot will signify a time of one cir-culation in the system. Therefore, t1 is the time when thesystem concentration is the lowest.

After knowing t1 the circulation flow rate m can be com-puted from Eq. (A-3) which will be accurate to a large ex-tent. It must be noted that without ever measuring the ve-locity the circulation flow could be computed.

«mC

Cdt

m

CM

t

∞ ∞∫0

1

� �KClSol

Cm

M∞ � KCl

Sol

«m Cdt mt

0

1

∫ � KCl

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Fig. A-1. A typical Concentration and normalized concentration plot for a tracer experiment.