Numerical investigation of nonisothermal reduction

IOP PUBLISHING

MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 15 (2007) 487–507

doi:10.1088/0965-0393/15/5/008

Numerical investigation of nonisothermal reduction of hematite using Syngas: the shaft scale study Mohammad Sadegh Valipour and Yadollah Saboohi School of Mechanical Engineering, Sharif University of Technology, Azadi St., PO Box 11365-9567, Tehran, Iran

Received 3 March 2007, in final form 24 May 2007 Published 11 July 2007 Online at stacks.iop.org/MSMSE/15/487 Abstract A mathematical model is developed to investigate the reduction process of hematite in the reduction zone of the Midrex shaft furnace as a moving bed reactor. It is described as a counter-current moving bed cylindrical reactor in which hematite pellets are reduced by a gaseous mixture of hydrogen, water vapour, carbon dioxide and carbon monoxide, namely Syngas. It is laterally injected into the bed at the lower part close to the bottom. Governing equations, including continuity, momentum, energy and mass equations, are derived based on the conceptual model for both gas and solid phases in the cylindrical coordinate system. A three interface unreacted shrinking core model (USCM) is applied to describe the reduction process in the pellet scale at the hematite–magnetite, magnetite–wustite and wustite–iron interfaces. The concluded equations are solved numerically based on the finite volume method. The model predictions are validated by a comparison with the operating data of the Gilmore Midrex plant. It was seen that the model can reproduce the operating data satisfactorily. Finally, the distribution of process variables in the bed is exhaustively explained. (Some figures in this article are in colour only in the electronic version)

Nomenclature Variables aP , aW , aE , aS , aN ags Cij Cie Ctg O Ci,0 , CiO

Description coefficients of general discrete equation (28) special reaction surface (1/m) concentration of species i in phase j (mole m−3 ) concentration of species i in equilibrium state (mole m−3 ) total concentration of gas phase (mole m−3 ) initial atomic oxygen concentration and atomic oxygen concentration of iron oxides in solid phase, respectively (g-atom O m−3 )

0965-0393/07/050487+21$30.00

© 2007 IOP Publishing Ltd

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487

488

M S Valipour and Y Saboohi

CPj

specific heat of phase j (J kg−1 K−1 )

∗ De,i , De,i

effective diffusivity of species i and its product after reaction, respectively, in the pellet (m2 s−1 )

Dj,eff

effective diffusivity in j phase in the bed (m2 s−1 )

DP

Pellet diameter (m)

Db

Bed diameter (m)

f1 f2

150(1 − εb )2 µg Coefficient in equations (23) and (24) = ρg (ϕDp )2 εb3 1.75(1 − εb ) Coefficient in equations (23) and (24) = ρg (ϕDp )εb3

fl , fh , fm , fw

local fractional reduction of iron oxides, haematite to magnetite,

F

overall fractional reduction (—)

(−Hk )

heat of reaction k (J mole−1 )

hgs

convective heat transfer coefficient between gas and solid phases

magnetite to wustite and wustite to iron, respectively (—)

(W m−2 K−1 ) kr,il

reaction rate constant for species i at interface l (m s−1 )

K

permeability of the bed (m)

∗ km,i , km,i

mass transfer coefficient of species i and its product in reaction (m s−1 )

Keil

equilibrium constant of reaction with species i at interface l (—)

L

bed height (m)

M

molecular weight (g mole−1 )

Nj,d

diffusive molar flux in phase j (mole m−2 S−1 )

P

absolute pressure (Pa)

Pe

Peclet number (—)

Qg

volumetric flow rate of Syngas (Nm3 h−1 )

qj R˙ k

conductive heat flux in phase j (W m−2 )

Rb

bed radius (m)

Rp

pellet radius (m)

RG

universal gas constant (J mole−1 K−1 )

reaction rate (mole m−3 s−1 )

r

radial coordinate in cylindrical system (m)

Sφ

source term in general discrete equation (27)

Tj Vj

temperature of the phase j (K)

Xl

fractional unreduced iron oxides at each step of reduction (—)

Ygi

mole fraction of gaseous components (—)

z

axial coordinate in cylindrical system (m)

velocity vector for phase j

Numerical investigation of nonisothermal reduction of haematite using the Syngas

489

Greek symbols αi β γ ρp ρb ρg εb εp µg λj,eff ψj φ ϕ νik

amount of reducible oxygen in each iron oxide (—) gas ratio of Syngas (= H2 /CO) gas utility of Syngas (= H2 + CO/CO2 + H2 O) pellet density (kg m−3 ) bed apparent density (kg m−3 ) gas density (kg m−3 ) porosity of the bed (—) porosity of the pellet (—) gas phase viscosity (N s m−2 ) effective thermal conductivity of phase j (W m−1 K−1 ) stream function in phase j (kg s−1 ) general property in equation (26) shape factor of pellets stoichiometric coefficient for species i in reaction k (—)

Subscripts j l s g in out

related to gas or solid phase related to interface of solid conversion(h: haematite to magnetite, m: magnetite to wustite and w: wustite to iron) related to solid flow related to gas flow inlet condition outlet condition

1. Introduction Haematite pellets are generally reduced to sponge iron in a shaft furnace as a moving bed reactor. The most common industrial application consists of the reduction of iron oxides with gaseous reductants in a moving bed shaft furnace. A number of such industrial processes can be mentioned: Midrex, Armco, Purofer and Hyl III [1, 2]. The subject of reduction of iron oxides has been considered in the open literature using two different approaches. In the first one, the process is analysed at the single pellet scale which is comprehensively discussed in the previous paper [3], whilst in the second one the reduction of pellets is taken into account in the shaft scale. The state of the art with regard to the iron ore reduction process in the shaft scale is summarized in table 1. They may be classified into two major groups: experimental study and mathematical modelling (see table 1). 1.1. Experimental study Generally haematite reduction has been experimentally investigated in a laboratory scale of shaft furnace. As can be seen in table 1 DelCorso et al [4], Yagi et al [5], Hara et al [6], Yanagiya et al [7], Takahashi et al [8] and Takenaka et al [9] are the pioneers in this field. DelCorso et al have used a mixture of nitrogen and carbon monoxide as reducing gas for nonisothermal

490

Table 1. Summary of studies on the modelling of iron ore reduction in the shaft scale. Pellet model

WGSR

ISO/NON

Reactor model(steady state) 1D/2D Gas Solid

Three interface (h–m, m–w, w–f), USCM One interface (h–f) USCM

No

Iso

1D(Axial) Plug flow

Plug flow MATH.

No

1D(Axial) Ergun equation

Plug flow EXP/MATH

1D(Axial) Plug flow 1D(Axial) Plug flow

— EXP/MATH Plug flow EXP/MATH

Yes

Non (Tg = Ts ) experimentally induced Iso (Tg = Ts ) Non (Tg = Ts ) experimentally induced Non (Tg = Ts )

Plug flow MATH

No

Non (Tg = Ts )

1D(Axial) Ergun equation 1D(Axial) Plug flow

Plug flow MATH

No

Non (Tg = Ts )

2D

Plug flow MATH

No

No

NON (Tg = Ts ) experimentally induced NON (Tg = Ts )

No Yes

Reactants Solid

Spitzer et al [11]

Fe2 O3 , Fe3 O4 , FeO H2

Del Corso et al [4]

Fe2 O3

Yagi et al [5] Hara et al [6]

Fe2 O3 H2 and CO One interface (h–f) USCM Fe2 O3 , Fe3 O4 , FeO H2 , and CO Three interface (h–m, m–w, w–f), USCM

No Yes

Hara et al [14]

Fe2 O3 , Fe3 O4 , FeO H2 and CO Three interface (h–m, m–w, w–f) USCM Fe2 O3 , Fe3 O4 , FeO H2 and CO Three interface (h–m, m–w, w–f) USCM Fe2 O3 , Fe3 O4 FeO H2 Three interface (h–m, m–w, w–f) USCM Fe2 O3 , Fe3 O4 , FeO H2 Three interface (h–m, m–w, w–f) USCM

Yu and Gillis [10]

Fe2 O3 , Fe3 O4 , FeO H2 or CO

Kam and Hughes [17] Takahashi et al [8]

Fe2 O3 H2 and CO Fe2 O3 , Fe3 O4 , FeO H2 and CO

Takenaka et al [9]

Fe2 O3 , Fe3 O4 , FeO H2 and CO

Tsay et al [15] Yagi and Szekely [12, 13] Yanagiya et al [7]

Gas

CO

Rao and Pichestapong [18] Fe2 O3 , Fe3 O4 , FeO H2 and CO Negri et al [16]

Fe2 O3 , Fe3 O4 , FeO H2 and CO

Aguilar et al [19] Parisi and Laborde [20]

Fe2 O3 , Fe3 O4 , FeO H2 and CO H2 and CO Fe2 O3

Three consecutive reactions as homogeneous One interface (h–f) USCM Three interface (h–m, m–w, w–f) USCM Three interface (h–m, m–w, w–f) USCM Three interface (h–m, m–w, w–f) USCM Three interface (h–m, m–w, w–f) USCM Three consecutive stage One interface (h–f) USCM

EXP./MATH

1D

Ergun equation Plug Flow

Plug flow EXP/MATH

1D

Plug flow

Plug flow MATH

NON (Tg = Ts ) NON (Tg = Ts )

1D 1D

Plug flow MATH Plug flow EXP/MATH

Yes

NON (Tg = Ts )

1D

Plug flow Ergun equation Plug flow

No

NON (Tg = Ts )

1D

Plug flow

Plug flow MATH

Yes

NON (Tg = Ts )

1D

Plug flow

Plug flow MATH

No Directly No

NON (Tg = Ts ) NON (Tg = Ts )

1D 1D

Plug flow Plug flow

— MATH Plug flow MATH

Plug flow EXP/MATH


Authors


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reduction of haematite pellets in which the temperature profile is induced from the outside [4]. Yagi et al have employed a mixture of hydrogen and carbon monoxide as a reducing gas for isothermal reduction of haematite in a fixed bed reactor [5]. Hara et al have applied a mixture of hydrogen and carbon monoxide as a reducing gas for nonisothemal reduction in which the temperature profile is encouraged from the outside longitudinally [6]. Yanagiya et al have carried out two series of experiments in the reduction of haematite pellet by hydrogen: (a) temperature distribution is controlled by external heating and (b) pellets are heated by reducing gas only [7]. Takahashi et al have reported an experimental study on the nonisothermal reduction of haematite in a pressurized laboratory shaft furnace using a gaseous mixture of hydrogen, carbon monoxide, carbon dioxide, water vapour and methane [8]. Takanaka et al have also applied a mixture containing hydrogen and carbon monoxide as a reducing gas in the semi-isolated reactor tube in which the pellets are heated by the reducing gas [9]. It should be noted that all the experimental studies mentioned above are usually followed by a mathematical formulation, but the authors have commonly emphasized experimental study rather than mathematical modelling. 1.2. Mathematical modelling Mathematical models are usually applied to explain the reduction process in detail at low cost compared with experimental studies. As shown denoted in table 1 nearly all the mathematical models presented in the shaft scale study are one dimensional to explain the longitudinal distribution of process variables. Some authors have developed a mathematical model in which carbon monoxide has been used as a reducing agent [4, 10], some have developed their models based on reduction with hydrogen [7, 10–13] and others have used both hydrogen and carbon monoxide as reducing agents in mathematical modelling [6, 8, 9, 15–20]. However, some differences exist among these models due to the assumptions made . Table 1 indicates some differences between these models in detail. Virtually all these models suffer from two shortcomings with respect to one-dimensional conduct: (i) the possible effect of flow maldistributions is neglected. As a result of the work done by Stanek and Szekely [21] and Kuwabara and Muchi [22] it has been shown that quite significant flow maldistribution may occur in a packed bed if the distribution of the packing material is non uniform. Such flow maldistribution may have a marked effect on the performance of the packed bed reactors. (ii) In the actual shaft furnace reducing gas is injected laterally at the lower part of the reduction zone, so the gas flow in the vicinity of the inlet section may not definitely simulate as a one-dimensional flow. However, Yagi and Szekely have presented a simple two-dimensional model for a counter current moving bed reactor to investigate the effect of gas and solid flow maldistribution on the performance of the reactor [13], but it is restricted by a single gaseous reductant and by some simplifications in mass and energy equations. In this paper to overcome the aforementioned shortcomings a general two-dimensional mathematical model is presented to describe the reduction of haematite using Syngas (a gaseous mixture mainly comprising hydrogen, water vapour, carbon dioxide and carbon monoxide) in a shaft furnace in which the reducing gas is laterally blown close to the bottom of the bed. 2. Mathematical formulation Let us consider a cylindrical moving bed reactor (figure 1) in which haematite pellets are reduced with Syngas enriched by hydrogen and carbon monoxide. The solid pellets are charged at the top of the bed and descend to the bottom of the bed subsequently. The Syngas is laterally injected close to the bottom of the bed and it ascends in a counter flow arrangement with the

492


Figure 1. Schematic configuration of a moving bed reactor with lateral gas injection.

solid pellets. The haematite pellets are reduced according to the following reactions in three interfaces simultaneously. 3Fe2 O3 + H2 → 2Fe3 O4 + H2 O,

(1)

w 3 Fe3 O4 + H2 = Few O + H2 O, 4w − 3 4w − 3

(2)

Few O + H2 → wFe + H2 O,

(3)

3Fe2 O3 + CO → 2Fe3 O4 + CO2 ,

(4)

w 3 Fe3 O4 + CO = Few O + CO2 , 4w − 3 4w − 3

(5)

Few O + CO → wFe + CO2 .

(6)


493

While considering a two-dimensional flow within a cylindrical coordinate system for the gas and solid phases, the following assumptions are made: • • • • • •

steady state, cylindrical symmetry, heat of reaction is released to the solid phase, in the gas momentum equation inertia and diffusion terms may be neglected, potential flow model is considered for solid phase motion in the bed and any side reactions are neglected.

Therefore the governing equations are expressed as overall continuity, momentum and energy equations for the gas and solid phases together with continuity of species for solid and gas reactants and products. Furthermore, some auxiliary relations are needed to close the set of governing equations. 2.1. Overall continuity The equation of overall continuity for a reactive incompressible gas and solid flow can be written as the following: j = δj βk R˙ k , j = g, s (7) ∇ ·G k

where βk = i (νik Mi )g denotes the rate of mass generation in which νik is the stoichiometric j coefficient for species i appearing in the kth reaction and Mi denotes the molecular weight; G is the mass velocity vector for gas, and solid flow is expressed as Gj = ρj εj Vj ; δj is indicated for gas flow δg = 1 and for solid flow δs = −1. 2.2. Momentum equation By volume averaging of the Navier–Stokes equation on the representative elementary volume (REV) of the bed, a general momentum equation of fluid flow in porous media have been developed by Vafai and Tien [23]. Using the scale analysis of this equation it was found that the inertia and viscous diffusion terms can be neglected in the gas flow in the moving bed reactor [24]. So the gas momentum equation becomes as follows: µg g. g − c√F G g G (8) G ∇P = − ρg K ρg K This is an extended form of Forchheimer’s equation [25], where Kand cF are known as the permeability of the bed and a dimensionless drag constant, respectively. K and cF for a packed bed of spherical particles have been correlated empirically by Ergun as the following [26]. K=

DP2 εb3 , 150(1 − εb )2

1.75 cF = . 150εb3

(9)

The flow of solid phase may be simulated as a potential flow theory in the following form [23]: 1 ∇ × Vs = 0, 2 where ω is the rotation vector. ω =

(10)

494


2.3. Energy equation The energy equation in porous media, in principle, should include conductive, radiant and convective terms. The relative importance of these mechanisms varies, depending on solid properties, pore structure, temperature range and gas flow in each case. The common approach in treating radiant heat transfer in porous media at high temperatures is to include it in conductive heat transfer formulation by the definition of effective heat conductivity [2]. Thus the differential form of energy equation may be expressed as follows: j Tj ) + ∇ · ( ∇ · (CPj G R˙ k (−Hk ) = 0, qj ) + δj hgs ags (Tg − Ts ) + αj j = g, s (11) k

where δg = 1, δs = −1, αg = 0, αs = 1, Cpj is specific heat, Tj is temperature, qj is conductive heat flux, hgs is the convective heat transfer coefficient between gas and solid particle and ags is also specific surface area. 2.4. Continuity of species For the species iin the j th multi component phase flow, the molar conservation equation may be written as follows: ∇ · (N j υ ) + ∇ · (N j d ) − νj k R˙ k = 0 (j = g, s), (12) k

where Nj υ = Cj i εj Vj signifies the convective molar flux and Nj d is the diffusive molar flux. 2.5. Reaction rates To simulate the reaction rate of a pellet in the shaft scale we should use the unreacted shrinking core model (USCM) at each step of the reduction; haematite to magnetite, magnetite to wustite and wustite to iron as the following: ags (1 − fl )2/3 R˙ k = (Ci − Cie ), Ril∗

i = CO, H2

(13)

where, R˙ kl is the reaction rate of the kth reaction and Ril∗ is expressed as the following with regard to the chemical kinetics of the reduction in each step [24]. (1 − fl )2/3 (1 − fl )2 − (1 − fl )2/3 (1 − fl )2 − (1 − fl )2/3 1 ∗ Ril = + + ∗ km,i De,i /rp kr,il (1 + (1/Keil )) De,i /rp (1 − fl )2/3 (14) + ∗ Keil km,i 2.6. Auxiliary relations Some other equation will be needed to solve the set of governing equations. These relations may be written as follows: Cgt = Pgt /RG Tg , Cgt = Cgi ,

(15) (16)

i

Ygi =

Cgi . Cgt

(17)


495

Table 2. Boundary conditions. T Gas flow Inlet r = Rb , Z = Zin

Tg = Tg,in

Yi

Yi = Yi,in ∂Yi =0 Outlet 0 < r < Rb , Z = L — ∂z ∂Tg ∂Yi Outlet 0 < r < Rb , Z = 0 =0 =0 ∂z ∂z ∂Tg ∂Yi Wall r = Rb , 0 < z < L = Ueff (T − T∞ ) = 0.0 −λg ∂r ∂r ∂Tg ∂Yi Axis r = 0, 0 < z < L =0 =0 ∂r ∂r Solid flow Inlet 0 < r < Rb , Z = L Ts = Ts,in Xi = 0 ∂Xi ∂Ts Outlet 0 < r < Rb , Z = 0 =0 =0 ∂z ∂z ∂Xi ∂Ts −λs Wall r = Rb , 0 < z < L = Ueff (Ts − T∞ ) =0 ∂r ∂r ∂Xi ∂Ts =0 =0 Axis r = 0, 0 < Z < L ∂r ∂r

P

Gr

Gz

—

Ggr = Gg,in

Ggz = 0.0

Pg = Pg,out

Ggr = 0

—

—

Ggz = 0

Ggr = 0.0

—

Ggr = 0

—

—

Gsr = 0

Gsz = Gs,in

—

Gsr = 0

—

—

Gsr = 0

—

—

Gsr = 0

—

∂Pg =0 ∂z ∂Pg =0 ∂r ∂Pg =0 ∂r

In the solid state the fractional reduction is determined based upon eliminated atomic oxygen in three interfaces: hematite–magnetite, magnetite–wustite and wustite–iron, so the following relation can be written between the fractional reductions and the atomic oxygen concentration in each interface: O O O O O O CFe = CFe (1 − fh ), CFe = CFe (1 − fm ), CFeO = CFeO,0 (1 − fw ), (18) 2 O3 2 O3 ,0 3 O4 3 O4 ,0 Xl + fl = 1.

(19)

The overall fractional reduction, F , is calculated depending on the amount of reducible oxygen at each stage of reduction as the following relationship [3]. αl fl , (20) F = l=h,m,w

where αl is the amount of reducible oxygen at each step of the reduction. 2.7. Boundary conditions The required boundary conditions are shown in table 2 for the geometry which is shown in figure 1. 3. Numerical solution Equations (7) to (20) together with the boundary conditions indicated in table 2 represent the complete statement of the problem. Considering the fact that nearly all the equations are nonlinear and coupled, the analytical solution will be complicated. Therefore, the set of equations is solved numerically based upon computational fluid dynamics (CFD). In order to solve the pressure–velocity linked equations, the momentum equation for gas flow, it is convenient to recast them in terms of a stream function [22]. With regard to the continuity equation, the stream function for reactive flow in the cylindrical coordinate system is defined as the following. 1 ∂ψj SR,j Gj r = + r ∂z r j = g, s (21) 1 ∂ψj Gj z = − r ∂r

496

M S Valipour and Y Saboohi Table 3. Specification of the general property φ and the other variables of general equation (26). Equation

φ

φ

1 1

0 0

Stream function

g

1 g |) − (f1 + f2 |G r

Pressure

P

−1

Stream function

s

−

Pressure

Overall continuity (gas flow) Overall continuity (solid flow) Momentum (gas flow)

Sφ ˙ k βk R k − k βk R˙ k

ζ 1 1

S ∂ g |) R,g (f1 + f2 |G ∂z r g |) G g −∇ · (f1 + f2 |G −

Momentum (solid flow)

−

—

Energy (gas flow) Energy (solid flow)

Tg Ts

λg,eff λs,eff

Mass equation (gas flow) Mass equation (solid flow)

Ci Xl

Deff Db,eff

−hgs ags (Tg − TS ) hgs ags (Tg − TS ) + k R˙ k (−H )k ν R˙ k j k k νj k R˙ k

where SR,j is defined as the following:

r δj βk R˙ k r dr. SR,j = 0

k

SR,s rρs

0

1 rρs —

∂ ∂z —

0

0 — CP ,g CP ,s 1 1

(22)

k

Using the definition of stream function, the momentum equation (8) is converted into the following two equations. Equation (23) is applied to calculate the stream function and equation (24) is applied to estimate the pressure distribution in the gas flow.

SR,g 1 ∂ψg 1 ∂ψg ∂ ∂ ∂ (f1 + f2 |Gg |) + (f1 + f2 |Gg |) =− (f1 + f2 |Gg |) , (23) ∂r r ∂r ∂z r ∂z ∂z r (24) ∇ 2 P = −∇ · [(f1 + f2 G g )Gg ]. Also equation (10) will be converted as the following using the definition of stream function for solid flow:

∂ ∂ ∂ 1 ∂ψs 1 ∂ψs 1 SR,s . (25) + =− ∂r rρs ∂r ∂z rρs ∂z ∂z rρs 3.1. Discretization There are significant commonalities between the aforementioned equations. When we introduce a general variable φ the conservative form of all governing equations can usefully be written in the following general form:

1 ∂ ∂ ∂φ ∂φ r ζ Gr φ − φ + ζ Gz φ − φ = Sφ , (26) r ∂r ∂r ∂z ∂z where the property φ and the other variables for each equation are specified in table 3. The general equation, equation (26), has been discretized based upon the finite volume method (FVM) [28]. This method is more reliable than other CFD techniques and ensures numerical stability. The first step in FVM is to divide the domain into discrete control volumes. As shown in figure 2, the geometry which is shown in figure 1 is divided into n + 1 grid in the Z-direction and m + 1 grid in the r-direction.


497

Figure 2. Schematic representation of the grids and the control volumes.

The boundaries (or faces) of control volumes are positioned midway between the adjacent nodes. Thus each node is surrounded by a control volume or cell. It is common practice to set up control volumes near the edge of the domain in such a way that the physical boundaries coincide with the control volume boundaries. Figure 3 shows a control volume is considered around the central grid point P with the control volumes around the neighbouring grid points: N, S, W and E in the north, south, west and east of the grid, respectively. The midpoint of the control volume interfaces are n, s, w and e in the north, south, west and east, respectively. The general governing equation is integrated over the control volumes which are shown in figure 3 with regard to r from the western face w to the eastern face e and with regard to z from the southern face s to the northern face n as follows:

e n ∂φ ∂φ 1 ∂ ∂ (27) r ζ Gr φ − φ + ζ Gz φ − φ = Sφ dr dz. r ∂r ∂r ∂z ∂z w s Afterwards the discrete equation is rearranged in the following general form: aP φP = aW φW + aE φE + aS φS + aN φN + Su ,

(28)

498


Figure 3. Schematic representation of a grid and its adjacent grids. Table 4. The cell face values of the variables F , D and Pe. Cell faces

Diffusion conductance

Mass flux

Face area

Peclet number

East(e)

De =

(φ A)e δ re (φ A)w Dw = δrw (φ A)s Ds = δzs (φ A)n Dn = δzn

Fe = (Aζ Gr )e

Ae = re ze

Fw = (Aζ Gr )w

Aw = rw zw

Fs = (Aζ Gz )s

As =

Fe De Fw P ew = Dw Fs P es = Ds Fn P en = Dn

West(w) South(s) North(n)

Fn = (Aζ Gz )n

(re2 − rw2 )s 2 (r 2 − rw2 )n An = e 2

P ee =

where aE , aW ,aN , aS and aP are specified as follows: aE = De A(|P ee |) + max[−Fe , 0], aW = Dw A(|P ew |) + max[Fw , 0], aN = Dn A(|P en |) + max[−Fn , 0],

(29)

aS = Ds A(|P es |) + max[Fs , 0], aP = aW + aE + aS + aN − SP .

(30)

F , D and Pe are, respectively, defined as convective mass flux, diffusion conductance and nondimensional cell Peclet number as a measure of the relative strengths of convection and diffusion. The cell face value of these variables can be described as in table 4. A(|P |) is a function which depends on the scheme that is usually applied to identify the value of properties on the cell faces. It is indicated in table 5 for different schemes which are usually applied in this model.


499

Table 5. Definition of A(|P |) in different schemes [28]. Schemes Upwind Power law

A(|P |) 1 max 0, (1 − 0.1|P |)5

3.2. Solution methodology Formerly the governing equations were rendered to a set of algebraic equations by discretization based upon FVM. These algebraic equations are solved by an iterative method as tri-diagonal matrix algorithm (TDMA) [28]. TDMA is used as it saves computing time and memory space to a great extent. The solution procedure is schematically shown in figure 4. The convergence condition is used as equation (31) for each of the variables. anb φnb + b ∗ −4 − φ (31) P < O(10 ), aP where φP∗ is the value of φP in the previous iteration. 4. Physico-chemical properties The essential physico-chemical properties which are applied in this model such as effective heat conductivity, heat capacity, effective diffusivity, heat of reactions, rate of reactions, equilibrium constant and so on are estimated, the same as those calculated in previous published papers of the authors [3, 27, 29]. 5. Results and discussion This model was applied to simulate the shaft furnace of the Midrex plant of Gilmore Steel Corporation in USA. Table 6 contains the essential operating data of this plant which are needed to execute the model [18, 20]. In order to decrease the effect of grid size on the results of the model and to reduce the computing time, a grid independence study was done to specify the most efficient grid, so the model was run using the grid sizes G121×41, G121×81, G242×41 and G341×41. Then it was recognized that the efficient grid for a shaft scale geometry is about G242×41. In table 7 the model prediction of the Gilmore plant data are compared with the operating data. It can be seen that the model reproduces the operating data satisfactorily. However a slight deviation has been introduced as a result of ignoring the side reactions. Figure 5 shows the distribution of fractional reductions in a longitudinal cross section area of the bed. Figure 5(a) shows that the haematite is fully converted to magnetite very soon in the upper part of the bed. This is verified by the experimental observations reported by Rao and Pichestapong [18] together with Takahashi et al [8]. Then magnetite is reduced to wustite as shown in figure 5(b). Also wustite slowly begins to reduce to iron after the appearance of the magnetite in the pellet. However, the reduction of wustite to iron has the slowest rate of reduction among the iron oxides and usually a fresh reducing gas as well as a high temperature environment is needed. Therefore as depicted in figure 5(c), it is usually reduced to iron close to the entrance of the reducing gas. Figure 6 shows the velocity field inside the bed close to the reducing gas entrance. This figure shows the direction of the reducing gas flow inside the bed. However in the micro

500


Figure 4. Schematic representation of solution algorithm.

scale, there may occur some fluctuations due to mass transfer or reactions, but they cannot change the direction of the bulk flow. Figure 7(a) shows the distribution of the stream function in the bed. It is clear that the stream functions are tangential to the velocity vectors of figure 6. Pressure distribution on the longitudinal cross sectional area of the bed is illustrated


501

Table 6. Operating condition of Gilmore plant. Reactor Length of reaction zone Diameter of reactor Distance of gas inlet from bottom of reactor a

L Db Zin

9.1 m 4.27 m 1.71 m

Solid Production (Fe)

˙ s,out m

26.4 ton h−1

Bed porosity Diameter of the pellets Pellet initial porosity Pellet density Sponge iron density

εb dP εp ρp ρFe

0.5624 11 mm 0.13 4.7 g cm−3 3.2 g cm−3

Gas Discharge pressure Gas flow rate Inlet temperature

Pg,out Qin Tg,in

1.3 bar 53863 Nm3 h−1 1100 K

β γ

29.97 4.82 52.58 4.65 8.26 1.755 8.717

Inlet compositions (vol %) CO CO2 H2 H2 O N2 + CH4 H2 /CO (H2 + CO)/(H2 O + CO2 ) a

This value is not reported clearly so it is estimated similar to the other Midrex plants.

Table 7. Comparison of Gilmore data with model predictions. Outlet gas compositions

Gilmore data [20]

Model data (radial average)

CO CO2 H2 H2 O N2 + CH4 Reduction degree

18.9 ± 1 14.3 ± 1 37. ± 2 21.2 ± 1 8.7 ± 3 94 ± 1

18.58 15.94 34.17 23.05 8.26 97

in figure 7(b). It shows that the pressure is decreases linearly alongside the bed from the inlet of the reducing gas to its outlet. Figure 8 shows the temperature distribution for the gas and solid phases. It is depicted in figure 8(a) and (b) that the gas phase and the solid phase temperature are very close to each other, except in the upper part of the bed close to the solid flow entrance. This has already been reported using the 1D mathematical models which are explained in section 2.1 [10, 20]. Figure 9 illustrates the mole fraction distribution of carbon monoxide and carbon dioxide on the longitudinal cross section area of the bed. As shown in figure 9(a) reducing gas containing a relatively high concentration of carbon monoxide is injected into the bed. It is converted into carbon dioxide by developing the reduction reactions in the bed, so its concentration will be decreased along the bed. Conversely, the concentration of carbon dioxide is increased as shown in figure 9(b). Figure 10 illustrates the mole fraction distribution of hydrogen and water vapour on the longitudinal cross section area of the bed. As shown in figure 10(a) reducing gas with a

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Figure 5. Distribution of fractional reductions: (a) reduction of haematite to magnetite, Fh , (b) reduction of magnetite to wustite, Fm , (c) reduction of wustite to iron, Fw and (d) overall reduction, F .

high concentration of hydrogen is injected into the bed. It is converted into water vapour by developing the reduction reactions, so its concentration will be decreased in the bed. Conversely, the concentration of water vapour is increased as shown in figure 10(b). It can be seen in this figure that the concentration of hydrogen and water vapour does not change considerably in the middle of the bed (it is much clearer in figure 11). The reduction of magnetite by hydrogen is more endothermic than the reduction of magnetite by carbon


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Figure 6. Velocity vectors of reducing gas flow inside the bed close to the gas entrance.

monoxide, so in the middle of the bed reduction by hydrogen progresses very slowly compared with carbon monoxide, therefore the distribution of hydrogen seems a unique distribution in the middle of the bed. Figure 11 shows the longitudinal variation of the gaseous compositions together with pressure. It is clear that the average pressure decreases linearly from the inlet to the outlet. However, it is not changed too much at the bottom of the bed (underneath the gas entrance). It is also indicated that the concentration of carbon monoxide and hydrogen are decreased smoothly along the bed from the inlet to the outlet. Conversely, the concentration of carbon dioxide and water vapour are raised smoothly along the bed. Longitudinal variations of the average fractional reduction are illustrated in figure 12 for haematite, magnetite, wustite and overall reductions in the bed. It indicates that haematite and then magnetite are fully converted in the upper part of the bed. However wustite is reduced in the lower part of the bed. Also the average value of gas temperature is plotted along the bed in figure 12. As shown in this figure, gas temperature is decreased alongside the bed because of the endothermic reactions. As we know, magnetite and wustite are reduced by hydrogen as an endothermic reaction; however haematite is reduced by a weak exothermic reaction. Also haematite and wustite are reduced by carbon monoxide as an exothermic reaction, while magnetite is reduced as an endothermic reaction [3]. In the reducing gas the gas ratio (β) is greater than one, so the endothermic reactions (particularly during the magnetite reduction) are overcome in the bed, so the temperature is decreased alongside the bed.

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Figure 7. Distribution of (a) stream function and (b) absolute pressure.

(a)

(b) Figure 8. Temperature distribution (a) in gas phase (b) in solid phase.

6. Conclusions A two-dimensional mathematical model was presented to investigate the reduction of haematite in a moving packed bed using Syngas. The governing equation was solved numerically based on the FVM which is a powerful technique in CFD.


(a)

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(b) Figure 9. Concentration distribution (a) for carbon monoxide and (b) for carbon dioxide.

(a)

(b) Figure 10. Concentration distribution (a) for hydrogen and (b) for water vapour.

The model predictions for the Gilmore plant condition were compared with the operating data. It was seen that the model reproduced the operating data satisfactorily. During the reduction of the haematite pellets in a shaft furnace haematite was fully reduced to magnetite in the upper part of the bed, whereas wustite was converted to iron

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Figure 11. Profile of average reducing gas composition and pressure along the bed.

Figure 12. Profile of average fractional reduction and temperature along the bed.

in the lower part of the bed. Also magnetite was reduced to wustite in the middle of the bed. The concentration of hydrogen was greater than that of carbon monoxide in Syngas (β = 1.755) so the endothermic reactions were overcome, and therefore temperature was decreased alongside the bed. Acknowledgments One of the authors, Mohammad Sadegh Valipour, wishes to thank Professor Mamoru Kuwabara at Nagoya University in Japan for his great help and kindly discussions.


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