A strategy for an efficient simulation of countcorrent flows in the iron blast furnace

A strategy for an efficient simulation of countercurrent flows in the iron blast furnace

D. A. Bennett

Department of Mathematics and Statistics, Newcastle upon Tyne Polytechnic, Newcastle upon Tyne, UK

R. Bradley

Department of Aerospace Engineering, University of Glasgow, Glasgow, UK

A recently developed anisotropic Ergun equation for layered porous media is embedded into equations characterizing a chemically inert, axially symmetric model of the iron blast furnace. A novel transformation is used to deal with the fitrnace geometry, and additional modifications caterfor convection, heat transfer, and the softening and melting of the iron ore. The equations ure identified as examples of a generic convection-anisotropic diffusion problem, and a modular solution method is proposed for both single- and double-phase cases. The result is an algorithm that is significantly more efficient than existing techniques. It is used to demonstrate some features relating the gas distribution to the shape, structure, and location of the cohesive zone.

Keywords: Ergun, blast furnace, porous media, anisotropic, simulation, countercurrent

Introduction

The production of iron by an operating blast furnace is based on a simple principle. The top of the furnace is charged with alternate layers of coke and flux (mainly iron ore), and hot air is blown through the furnace from tuyeres close to its base. The hot gases generated by the burning of the coke are carried up through the furnace stack, melting and then reducing the iron ore. The molten iron runs to the base of the furnace, from where it can be tapped. There are two major flows in this process-one of gas from the base to the top of the furnace and a much slower flow of solids descending in the opposite direction.

Early attempts to model the production process used a combination of uniform gas and solids flows with stagewise mass and enthalpy balances. Engineers and chemists were able to use these models to improve furnace design and to increase efficiency by more effective management of furnace operation. A full account of this type of approach is given by Peacey and Davenport.’ The next step was to include chemical reactions in the model formulation; the work of Muchi* and Yagi and Muchi in particular included both chem-

Address reprint requests to Dr. Bradley at the Department of Aerospace Engineering, University of Glasgow, Glasgow, G12 8QQ, UK.

Received 14 June 1990; accepted 26 April 1991

ical kinetics and heat transfer between gases and solids.

Despite these considerable advances, the development of modelling techniques was impeded by the lack of information about the internal structure of the furnace until, in the early 197Os, Japanese investigators quenched and excavated four operating furnaces. Their findings4 brought to an end the idea that simple gas flows could adequately represent the aerodynamics of the blast furnace. Figure I shows schematically the internal structure of a blast furnace as revealed by the Japanese investigations. The region between the stockline and the 1200” K solids isotherm is seen to be com- posed of a descending burden of distinct, alternate layers of ore and coke. This region is termed the granular zone; it ends at the isotherm of the softening temperature of 1200” K, where the ore becomes a cohesive mass with low permeability to the flow of gas. The cohesive zone, as it is named, extends from the 1200” K to the 1400” K isotherm, and in this zone, gas seeks out the low-resistance layers of coke and uses them to move radially, convecting heat from the center of the furnace toward the side. It is clear that the alternate charging of layers of coke is an important feature that has a dramatic effect on the flow of gas and hence the distribution of heat internally. Below the cohesive zone the temperature is sufficiently high to melt the ore, and one can identify two regions: the active zone, where iron and slag flow through the loosely packed coke, and a central region called the deadman or stagnant

506 Appl. Math. Modelling, 1991, Vol. 15, October 0 1991 Butterworth-Heinemann

coke ‘slits’

I slag+coke I ;, hearth -z metd+coke

I

Figure 1. Schematic diagram showing internal structure of the iron blast furnace

zone, which is a region of dense, unburnt coke. Ad- jacent to the tuyeres and extending into the active zone is the raceway, a small region generated by the incom- ing blast, where the combustion of the coke takes place. Molten iron percolates through the active zone and raceway to collect in the hearth, from where it is tapped.

These discoveries showed that the hitherto ignored complexities of blast furnace aerodynamics were an important factor in the distribution of heat through the internal structure. Consequently, the aerodynamic properties became a target of modelling activity, with recent research including the work of Yagi et al.,5 Cross et al.,h and Fenech et al.’

The present paper describes a strategy for modelling the aerodynamics and the heat transfer process of an inert blast furnace and for implementing a computationally efficient simulation. The three main ingredi- ents in the modelling strategy are models for the gas flow, the solids flow, and the heat transfer. They must be allied to an appropriate treatment of the raceway and the deadman regions. Computationally, the strategy includes devising a formulation of the models that is amenable to discretization, followed by the adoption of an efficient solution algorithm. These aspects of the strategy are developed in the sections that follow, and they conclude with an analysis of a set of results produced by this strategy.

Bla st furnace simulation: D. A. Bennett and R. Bradley

the slowly descending packed bed structure with typical gas and solids residence times of 1 second and 5 hours, respectively. As far as the gas dynamics are concerned, the solid material can be considered sta- tionary. Further, since the gas is forced through a relatively dense medium, the interphase friction terms dominate, so the convective accelerations, body force terms, and macroscopic viscous effects can be safely discarded. (These terms were retained by Fenech et al.’ as part of the PHOENICS* package and suffered a corresponding computational burden.) The consequence of this simplification is that with a knowledge of the physical properties, such as porosity E and particle size 6, of the various regions of the furnace, the gas flow may be described by the Ergun’ equation:

AplAy = (puls*)aq + (p/6)bq2 (1)

This fundamental equation represents the gas flow through a uniform porous medium, relating the pressure drop Ap across a uniform porous layer of thickness Ay to the volumetric flow rate per unit area, q. The dimensionless constants a and b are given by

a = 150( 1 - E)WIEW

b = 1.75(1 - e)SIe3d

where d is the effective solid particle diameter, p is the gas viscosity, and p is the gas density. The Ergun formulation has been extensively validated experi- mentally, in particular by the work of Dybbs and Ed- wards. lo

The incorporation of the Ergun equation into the flow through distinct layers of dissimilar material can be achieved simply by employing a sufficient density of cells or elements in the computational scheme to locate a number of nodes across the width of each layer. This technique of representing each layer individually was successfully implemented by Yagi et a1.5 using the Radestock and Jeschar” pressure formulation to model the cross-flow induced by the layered structure. The disadvantage of this approach is the computational load that is generated by a dense distribution of nodes. With the aim of capturing the cross- flow effects, but with a more economical distribution of nodes, Cross and Gibson’* employed the model

-grad p = (pIS)q B q (2) _

of a uniform anisotropic porous medium that approximates the averaged, isotropic, layered structure. In fact, the representation is directly equivalent in direc- tions that are aligned with, and perpendicular to, the layers of material. For the zones of the furnace where the gas has a relatively high flow rate this model has been validated by Cross and co-workers.” A linear model

Gas flow

- grad p = (p/6*) q A q -

The pioneering work of the Japanese investigators in revealing the multilayered porous structure of the blast furnace interior cleared the way for a realistic study of the internal gas flow. The gas moves upward through

(3)

developed by Bear and Bachmat13 for Darcy flow through an isotropic material, may be similarly employed to model low-speed flow. A definitive study of applying the Ergun equation to a layered structure and deriving a fully equivalent anistropic representation

Appl. Math. Modeiling, 1991, Vol. 15, October 507

Blast furnace simulation: D. A. Bennett and R. Bradley

may be found in the work of Bennett et aLI4 The re- lationship between q and grad p at any point is an implicit one and requires, in general, to be solved nu- merically, but an important outcome of the study is that a simple form derived via a hybrid combination of asymptotic results for large and small q could adequately represent the flow in all regions of the furnace interior. This result is

-gradp = [(cl.lS*)A + (plG)qBlq (4) -

where the matrices A and B include the effects of material porosity and particle size of the layers together with their inclination and relative thickness. The model described in equation (4) is the choice for the gas flow component of our strategy. It satisfies the criteria for accuracy and, looking forward to the computational requirements, economy. The influence of the layered structure on the gas flow is represented without the necessity of representing each layer individually. For the present application it is convenient to use the equation of state of a perfect gas and introduce the gas mass flux vector:

pg = q -

so that equation (4) becomes

-gradp* = 2R,T,[(p/6*)A + (1/6)g,B]g, (5)

where R, is the gas constant and T,: is the gas temperature. The resistance matrices A and B depend in- directly on the solids temperature T,, so for known distributions of Tg and T, the gas mass flux gg and pressure p can be found from (5) and the inert gas mass balance equation

divg, = 0 (6)

In the current work, appropriate boundary conditions for the gas flow are that the pressure is imposed at the stockline and raceway levels.

Solids flow

The relatively slow movement of the packed bed structure is determined by the rate at which molten iron and slag are produced by the chemical reactions of the furnace interior and subsequently tapped from the hearth. In equilibrium this is matched by an equivalent rate of charging of the furnace at the top of the stack. There is no evidence at present to suggest that the details of the solids flow are crucial to an acceptable overall model. It has little direct effect on the much faster flow of gas and is important to the blast furnace operation only in the quantity of new material intro- duced into the heat transfer and reduction processes. For these reasons, despite the complex physical structure, simple flow patterns are used. Cross et a1.6 used a simple axial flow together with conservation of mass to model the solids flow. Yagi et a1.,5 on the basis of blast furnace dissections, asserted that the flow was essentially irrotational. The latter assumption is made

508 Appl. Math. Modelling, 1991, Vol. 15, October

in the present study, and a potential & satisfies

-grad & = 1, (7)

where !S is the volumetric solids flow. In terms of the mass flux vector,

& = PSVS

where pS is the bulk solids density, we have

-grad & = gJpS (8) -

and conservation of mass requires that

divg, = 0 (9) _

Boundary conditions appropriate to the problem are to set $Q to zero at the upper boundary and prescribe the component of g, perpendicular to a horizontal surface at the raceway level. On the remainder of the surface the boundary is considered to be impermeable to solids flow.

Solids and gas heat flow

A prime effect of the flow of gas in the furnace interior is to carry heat generated by the burning of coke up into the stack to soften, then melt, the iron ore. It is essential therefore to represent the convection of heat and the exchange of heat between gas and solid. The standard energy equations for steady inert conditions are

div (g,H,) = - div qS + A(T, - T,) (10) - -

div (gx H,) = - div CJ, - A(T, - T,) (11)

for the solid and gas, respectively, where the final term in each equation represents heat exchange and the term in gg HR represents the convection by gas flow. The heat flux vectors q3 and 5, are related to the respective temperatures T, and T, by the Fourier laws. For the gas the isotropic form

- grad T, = (l/~,)q, (12)

is used, but for the solid the layered structure of the burden, by the same reasoning as that employed for the flow of gas, an anisotropic relation

-grad T, = (l/~)Rg~ (13)

is appropriate, where R is a matrix depending on the characteristics of the layered structure. The cohesive zone lies between the isotherms T, = T,(1200” K) and T, = T2( 1400” K), so the final internal temperature distribution will predict the cohesive region, in contrast to the work of Yagi et al.,5 who specify the position in advance. An appropriate solids enthalpy-temperature equation is the final part of the model. Following Cross et a1.,6 we write

(c, + c,) T, T, 5 T,

(c,+c,)T,--c,T, TI < T.7 < T2

Cc, + cm,) Ts + moLo Ts > T2

(14)

where the melting function c, is defined by

I

%U(T2 - T,) T, < T, < Tz c, =

0 T,>Ts, T,>T,

For the gas,

H, = c, T,

where c, is a constant. As a final step in this section it is convenient to

express the field equations in terms of the enthalpies. Inverting the equations above gives the following con- stitutive equations:

-grad H., = [(c, + c,Y~lRq, (1%

-grad H, = [ cW/KX]q, (16) -

Similarly, the energy balance equations become

div (gs H, + q.J = PIH, - P2K + P3 (17) - -

div (gg Hg + qx) = -PI H, + P2Hs - P3 (18)

where

p, = A/c,

p2 = hl(c, + c,)

1

0 T, 5 T,

P3 = Ac,T,I(c,~ + c,) T, < T, -=c T2

-Arnf&/(c, + c,) T,s 2 T2

The boundary conditions for the heat flow problem are for an ambient specified temperature at the top of the stack and a specified flame temperature at the raceway level. To model the conditions at the furnace wall, provision for allowing a prescribed flow of heat to es- cape is included.

The stagnant and raceway zones

Since there is relatively little flow of gas in the central region of the furnace, it may be argued that a natural stagnant zone (the deadman) is created. Accordingly, Cross et a1.6 take this view and do not make any special provision for the location and physical state of this zone. Other authors, for example, Yagi et aL5 and Fe- nech et al.,’ use values for material data that distin- guish between the active and stagnant coke zones. The strategy in this current work is to model the deadman as a centrally positioned porous cone. It is, in fact, straightforward to show I4 that such a model is ac- ceptably consistent with the formula derived by My- asaka et aLiS on the basis of dimensional analysis and associated experiments. One of the aims of the present work was to assess the effect of the inclusion and re- moval of a specified deadman.

At the tuyere level the region between the base of the cone defining the deadman and the furnace wall is the raceway region into which the blast of gas is in- jected. The blast creates a loosely packed combustion zone that is generated by the burning of coke. Fenech


et al.‘s investigation’ showed little dependence of blast furnace performance on the raceway size, provided that the mass flow of the gas was maintained. It would appear that an elaborate raceway model is not required, and the approach taken here is to identify the raceway with the boundary conditions imposed at the tuyere level. The raceway boundary is stipulated as having a constant gas pressure and prescribed solids flow, and the base of the deadman is considered to be impermeable to the gas.

The generic C-AD problem

The set of equations (5), (6), (8), (9), and (15-18) describe the furnace process flows of mass and energy. Each of the four problems-solid mass flow, gas mass flow, solids heat flow, and gas heat flow-may be identified as a particular example of the generic convection-anisotropic diffusion (C-AD) problem:

div (G@ + Q) = E0 - E,@ + E2@’ (19) -

-grad 0 = HQ (20) -

with

divG = 0 (21)

The generic scalar field (3 is appropriate to the phase under consideration, and where a two-phase approach is relevant, @’ is the generic scalar field relating to the alternate phase. The furnace process flows may then be summarized as in Table I. The common structure of the equations can be exploited to advantage, since a solution procedure can be developed on a modular basis

Transformation

With the exclusion of the hearth and raceway regions the flow domain is that shown in Figure 2. The natural coordinates for the problem are cylindrical polar coordinates (R, Z), and in terms of these the domain is

0 5 R < Bf (Z/H) O<Z<H

where the function f(ZIH) describes the profile of the furnace wall. There is some advantage to be gained by simplifying the domain to achieve a rectangular region that can be discretized by a regular grid. This step might not be essential if the solution method is to em- ploy finite elements, but it facilitates formulating the conservation of flux in the control volume approach adopted in this work.

Table 1. Process flows as instances of the generic C-AD problem

Flow + G Q W - -

Gas mass P2 0 sg -

Solids mass 4s 0 ss -

Gas energy 47 Solids energy H,

&I 5s H,

ss 4s 4

Appl. Math. Modelling, 1991, Vol. 15, October 509


Similarly, Q is obtained from q. The application of the transformaxon in equations (22)-(24) gives the following model equations for the computational C-AD problem:

Figure 2. Flow domain

An alternative formulation, used for example by Fe- nech et al7 is to adopt a cylindrical geometry and shape the furnace to the required profile by a blocking procedure whereby certain cells are made to have very high resistance to gas flow. The current authors have not found this method to be acceptable in the context of the present work, which is to obtain accurate representation with a modest grid pattern. The blocking procedure produces an irregular boundary that inter- feres in an unacceptable way with the flow of gas when the cell size is relatively large. A better approach is to transform the independent variables (R, Z) to the computational equivalents (x, y) in the unit square by

R = Bxf(y) Z = Hy (22)

The transformation of the dependent variables needs careful consideration. While the scalar field @ is simply transformed by

@(R, Z) = @‘rer 4(x, Y> (23)

the flux vector G = [G,(R, Z), G,(R, Z)] is transformed to the computational equivalent [g,(x, y), g,(x, y)] by the equations

[z] = G,ef [ (BiH;‘2xf ;;;$I [ ;] (24)

J-(g,+ + qx) + fj(gl$ + qJ = e0 - ed + e24’

(25)

(26)

a4 -ax = h,q, + h,q, (27)

-x = h,xqx + h,,q, (28)

where L = (H G,,,/@,,,)](B/H)2 (~~~x)HRRI h, = h,, = W G,,‘/~~,f)[(BIH)(1/2xf )HRZ

+ WH)2U-‘/2f )HmI h,, = W G,,f/~,,,f)[(1/2xf2)H,, +

WfKf’lf2P&z + WH)*W)(f’~f V~RRI

[e 0, Ed, el = WG,f) (2xf*)[E0, 6, &I

The role of the furnace aspect ratio, (B/H), can be clearly seen from these equations. It is also clear that the resistance matrix retains its symmetry and that the physical source coefficients Eo, E,, E2 are directly pro- portional to the computational coefficients eo, e,, e2. This is an important property of the transformation. Alternative transformations may generate the left-hand side of equation (25) but, in the authors’ experience, are invariably accompanied by the presence of additional, fictitious source terms on the right-hand side. Such terms require careful treatment in a computational scheme, since they often have a catastrophic effect on the convergence of the solution procedure.

Discretization

The model equations are cast into discrete form by integration over a control volume surrounding each node. The approach is similar to that used by Spalding and co-workers.i6 The grids for the scalar field @ and the flux components g, and g, are staggered to allow values to be located at the most appropriate points for the control volume integration. The scheme is shown in Figure 3. The flux components qx and qy are treated the same as g, and g,. Integrating equation (27) over the relevant control volume leads to the discrete equation

9c = -&(4E - 4P) - ce where

(2%

d, = 2/h([h,l, + [hxxld

c, = ([h,yqylP + [h,qyldl([hxxl~ + [hxxld (30)

with a similar result for equation (28).



erative scheme that operates on two levels: an outer iteration and an inner iteration. The outer iteration updates the temperature-dependent coefficents, leav- ing an inner problem that is linear and can be solved either directly or, as is the choice here, by a sequence of iteratively solved subproblems that are either single- phase, as in the case of solids and gas flow or double- phase as in the solids and gas heat flow shown in Figure 4.

0 $I _ grid points 0 qx _ grid points 0 q,, - grid points

Figure 3. Schematic discretization of the computational domain

The integration of equation (25) over the appropriate control volume for 4 together with the use of equation (26) leads, after some manipulation, to the algebraic equation

ap& - WFP

= a& + ahi& + aw+w + a& + b (31)

where

aE = HIlO, -sell + &I, aN = NO, -hII + 4) aw = ~(1[0,g,ll + dJ, as = ~(lKhg.Jl + 4) up = hk[ e& + aE + aN + as + aw ap = hk[ ezlp

b = hk[e,] + k(c, - c,) + h(c, - c,) (32)

In this derivation the convective flux terms are represented by standard upwinding.

Boundary conditions on scalar fields or normal flux components are easily accommodated so that equations (29) and (31) retain their form but with modifications to equations (30) and (32) to reflect the new coefficients.

The model equations representing the four blast furnace process flows of mass and energy are nonlinear and interlinked, the principal coupling mechanism being the gas mass-flow resistances and, to a lesser extent, the heat exchange terms. Each problem is a particular case of the generic C-AD problem and is transformed and discretized accordingly. The solution of the whole may then be approached by applying a similar solution technique to each problem. This modular approach to the solution has advantages in software development. The computational procedure is constructed to solve the resulting nonlinear algebraic equations by an it-

The C-AD single-phase solver

The inner stage of the iterative procedure involving the solids flow potential Q, and the gas pressure p2 are formulated as single-phase problems cast as equation (31) with (Ye = 0:

ap@+l) = a&+‘) + aN&(n+l)

+ aw$w(n+‘) + a&~+” + b (33)

together with equations such as

all coefficients being calculated at the nth stage of the outer iteration. Equation (33) is recast in terms of a 4

0

PROBLEM 3: SOLIDS HEAT FLOW PROBLEM 4: GAS HEAT FLOW TWO PHASE SOLVER

SOLUTION: I@‘+‘), g,(“+‘)

convergence of outer iteration

Figure 4. System solution procedure

Appt. Math. Modelling, 1991, Vol. 15, October 511


increment S4 by writing

4’” + 1) = #n) + 84 (35)

and obtaining

+ a+#$+ as&) + hap@) (37)

The inner iteration procedure is now to calculate S4 from equation (36) and then update 4 and the flux! components using equations (34) and (35), respectively. The preferred method of solution is line suc-

cessive over relaxation (LSOR).” In the present application, LSOR is preferred because it is more efficient than point relaxation, it is relatively simple to implement, and it is particularly effective when the line sweeps are taken along the direction of the prevailing flow so that advantage can be taken of the physical characteristics of the problem.

The C-AD two-phase solver

Where exchange terms between the phases are present as in the gas and solids heat flow, it has been found advantageous to solve the relevant equations for each phase simultaneously. The inner iteration for a two- phase model consists of equations (33) and (34) for each phase. In terms of the scalar field increments S+, S@, equation (3 1) becomes

b (38)

The inner iteration procedure is to solve equations (38) and (39) simultaneously, using LSOR to obtain S+, S+‘, and then update the dependent variables by equations of the form (35) and (34) for each phase.

The system solution procedure

The solution procedure for the complete system of computational equations is shown in Figure 4. In the current model of the blast furnace the solids mass flow is decoupled form the other flow problems, and the solution is obtained by using the single-phase solver. The gas mass flow equations are also solved by using the single-phase solver, but this problem is strongly coupled to the heat flows via the gas flow resistances. The solids and gas heat flow equations are solved simultaneously by using the two-phase solver. Nonlin- earities arising from variable dependent coefficients are handled by the outer iteration. Convergence is not guaranteed but, in the authors’ experience, can be achieved by underrelaxation.

Results and discussion

The model equations and solution procedure described in the previous sections have been applied to a model problem that approximates a large operational blast furnace. The data used are consistent with those used by Fenech et al.’ Consequently, the results obtained by these authors provide a basis for comparison.

Model results

Cohesive zone. The results from the current model highlight a number of interesting features. The most

512 Appt. Math. Modelling, 1991, Vol. 15, October

significant is the way in which the gas mass flow is dominated by the shape and layered structure of the cohesive zone. The existence of a corridor of low resistance, parallel to the layers, is observed, along which the gas must flow to pass from the active zone to the granular zone. Further evidence is manifested by the way in which the gas pressure contours (Figure 5) in that region are aligned with the layers and, being closely grouped, signify a considerable pressure gradient in the direction of high flow resistance, that is, across the layers. The observation of the corridor effect is particularly revealing, since it demonstrates clearly the ability of the new anisotropic Ergun equation to represent the transfer of gas through the cohesive zone by way of the coke slits. The sparsity of the isobars (Figure 5) in the active zone is consistent with the relatively low resistance there. In the granular zone the pressure contours are fairly evenly spaced and aligned with the layers. These findings are consistent with the observations made by other researchers.’ Fi- nally, an inspection of the solids isotherms (Figure 6) shows clearly the shape of the cohesive zone to be a slightly flattened, inverted U.

The results shown are those obtained from a coarse grid simulation. A qualitative comparison with results obtained from a finer grid shows that the essential features are retained. It is interesting to compare the results with those published by Fenech et al.’ using as principal features the gas mass flow distribution and the shape and location of the cohesive zone. In the Fenech model, the streamline distribution indicates that the gas moves through the central region of the active zone and is then marginally directed toward the furnace

Figure 5. Gas pressure contours

3.

L3 6,

1, \ >: : . . . 8.: '1: '6

7.

0.

Figure 6. Solids temperature contours

- cohesive zcme


wall by the cohesive zone before continuing up through the granular zone. An inspection of the pressure contours reveals an almost uniform pressure gradient. In- deed, across the cohesive zone a surprisingly small increase in the magnitude of the pressure gradient is observed. In the granular zone the isobars are ap- proximately aligned along the layers. An examination of the solids isotherms shows that the cohesive zone is thin in the central region of the furnace, thickening slightly toward the wall. Its general shape is in the form of an inverted V with the apex located high in the stack. The principal features of the results exhibited by Fe- nech et al.’ and those presented in the current work are significantly different. A possible explanation lies in the shape of the cohesive zone.

An examination of the cohesive zone of the Fenech et al. model reveals a wide corridor of low resistance consistent with the observed small pressure drop across the layers and lower furnace flow resistance. A higher gas flow rate is the result, and this contributes to the high apex location. There are, of course, differences in the model equations used. In the current work the gas momentum is represented by the new anisotropic Ergun equation, whereas Fenech et al.’ include the convective acceleration terms. The work of other researchers on gas flow through packed beds1’,‘2,‘8-20 has led to the conclusion that the convective acceleration terms may be neglected, so the omission of these terms in the current work is not thought to be significant. Possibly more significant is that Fenech et a1.7 calculate the ore mass fraction ma from a formula that takes into account variation of the charge pattern. In the current work a fixed value for ma is used. A variable ma may have considerable influence on the shape of the cohesive zone and could explain the differences in the predicted shape of the cohesive zone.

It is encouraging to observe the manner in which the application of the current strategy has highlighted the importance of the corridors of low resistance, through their dependence on the shape and layered structure of the cohesive zone, to the blast furnace operation.

Stagnant zone. The simulation model is a useful tool for investigating the effect of blast furnace parameters and modelling concepts. The effect of the deadman on the simulation was evaluated by including a high-flow- resistance (stagnant) zone in the shape of a cone. An examination of the results reveals that such an inclusion has no observable effect in and above the low- resistance corridor. This is important in view of the future development of the model as the predominant chemical activity occurs in this region. Elsewhere, the influence of the deadman is observable but small. The observations above tend to support the case argued by Cross et al.” for a natural stagnant zone without any special provision for location or physical state.

Gas viscosity. An increase in the gas viscosity for the model problem was observed to result in a narrower low-resistance corridor, signalling a more severe pressure drop across the layers, a higher furnace flow re-



sistance, and a lower gas flow rate. The position lower in the furnace of the cohesive zone and isotherms is also consistent with a lower gas flow rate. Such results are to be expected, since the effect of higher viscosity is to increase the gas flow resistance and hence reduce the overall gas flow rate.

Wall losses. Finally, the effect of wall heat loss has been evaluated. In comparison with the adiabatic case an examination of the isotherms reveals evidence of the expected cooling in the vicinity of the wall. This cooling alters the shape and location of the cohesive zone. There is a broader low-resistance corridor over which the pressure drop across the layers is less severe, signifying a lower furnace resistance and con- sequent higher gas flow rate. The cohesive zone and isotherms are higher in the furnace; this feature too is indicative of a higher gas flow rate. The higher gas flow rate related to higher wall heat loss is plausible. Al- though the wall heat loss amounts to only -8% of the total heat loss, it is sufficient to lower the flow resistance and thus increase the gas flow rate.

Performance

For a typical blast furnace simulation the outer iteration was terminated on a tolerance of 10 p3. Several computations were performed with finer tolerances, but no significant changes in the solution values were observed. For a coarse grid of 8 x 24 a converged solution required -50 outer iterations and a computer run (CPU) time of -5 minutes. For a finer grid of 16 x 48 a converged solution required ~75 iterations and a CPU time of -45 minutes. All computations were carried out on a Harris 800 minicomputer. The procedure compares favorably with the procedure used by Fenech et al.,’ which requires some 600 sweeps to convergence and several hours of CPU time.

The superior performance of the current procedure may be attributed to a number of features. The recently developed anisotropic Ergun equation is able to cap- ture the physical behavior of the gas flow with an economy of grid points and is computationally efficient. The use of the transformation results in a procedure that is considerably faster than the cell-blocking equivalent. For appropriate test problems using a grid of 16 x 8 a 20% reduction in CPU time has been observed.2’ Finally, for the heat flow equations the use of the two- phase solver is demonstrably superior to the sequential use of the single-phase solver.

The authors wish to thank Professor M. Cross for the advice that he has given during the course of this research.

Nomenclature

a, b aE7 . . . , aP, b, aP

ak, . . . , a;, b’, cub

A, B

B

Conclusion

A strategy has been developed for an efficient simulation of the process flows of an iron blast furnace.

The first aspect of the strategy was to establish suit- able models for the process flows: a recently developed anisotropic Ergun equation has been applied to the gas flow problem, while the solids flow has been assumed to be irrotational. Finally, a standard representation of the heat flow completed the model formulation.


Results from the model have identified important features relating the gas flow distribution to the shape and location of the cohesive zone. In particular, the influence of a corridor of low gas flow resistance in- terconnecting the active and granular zones has been highlighted. Results relating to the effect of the deadman on the process flows support the concept of a natural stagnant zone. Modelling the effects of changes to gas viscosity and wall heat loss has produced phys- ically plausible results, providing further encouraging evidence of the credibility of the model equations.

The second aspect of the strategy was to develop and implement an efficient solution technique. This was achieved by using a novel transformation to sim- plify the computational domain and applying a control volume scheme to discretize the models. The resulting algebraic system was solved by a doubly iterative scheme, certain elements of which have been analyzed for stability.2’ The consequence of this strategy is a procedure that compares favorably with other published work and whose modular development lends itself to future enhancements, including, for example, provision for the chemical dynamics of the furnace.

Acknowledgments

C, CS C, C c, . . . , C,; 6, . . .

d

e0, el, e2

Eo, 6, E2

f, f’

dimensionless constants coefficents in discretized

equations coefficients in discretized

equations gas flow resistance matri-

ces belly radius of blast fur-

nace gas specific heat solids specific heat melting function

d, coefficients in discretized equations

effective solid particle diameter

computational equivalents of Eo, E,, E2

source coefficients (generic C-AD problem)

wall shape function and derivative

discrete values of g,, g, magnitude of gg components of the compu-

tational equivalent of G gas and solids mass flux-

vectors


EL gas viscosity P gas density

h, BII solids bulk density max (A, B)

References

components of G mass flux vector (generic

C-AD problem) representative flux magni-

tude control volume dimensions elements of the computa-

tional equivalent to H

height from tuyere level to stockline

gas and solids enthalpies elements of H

flow resistance matrix (generic C-AD problem)

latent heat of ore ore mass ratio superscript, index of outer

iteration gas pressure magnitude of q volumetric flow vector discrete values of qx, qv

components of the computational equivalent to Q

gas and solids heat flux vectors

diffusion flux vector (generic C-AD problem)

equation residuals cylindrical polar coordi-

nates gas constant solids thermal resistance

matrix solids isotherms defining

the cohesive zone gas and solids tempera-

tures solids velocity vector computational (Cartesian)

coordinates

K

Kg

physical coefficients reference particle diameter scalar field increments porosity pressure drop across por-

ous layer thickness of porous layer computational equivalents

of @, @’ discrete values of 4 discrete values of 4’ solids velocity potential scalar fields (generic C-AD

problem) representative scalar field

magnitude representative solids ther-

mal conductivity bulk thermal conductivity

of the gas

4

5

6

7

8

9

10

I1

12

13

14

15

16

17

18

19

20

21

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