- Email: [email protected]
Direct reduction of hematite in a moving bed. Comparison between one- and three-interface pellet models
2472
Shorter
Communications
HCl and HBr media, an improved method for the calculation of the X-functions and Ho scales. Can. J. Chem. 59, 21162121. Fokin, A. V., Kolomiets, A. F., Studnev, Yu. N. and Kuznetsova L. D., 1970, Acylation of phenol with carboxylic acids. Iriv. Sib. Otd. Akad. Nauk. SSSR, Ser. Khim. Nauk. 1, 87-90. Ghtzman, M. Kh., Dashevskaya, B. I. and Fridman, G. M., 1968, Esteritication of stearic acid with polyethylene glycols. Zh. Prikl. Khim. 41, 183-7. Hammett, L. P., 1970, Physical Organic Chemistry. Reaction Rates, Equilibria and Mechanisms, 2nd ed., pp. 315347. McGraw-Hill, Kogakusha. Hirsch, J. A. and Schwartzkopf G., 1974, Esterification of 7carboxy-2,3-dihydro-4-quinoline. Synth. Commun. 4, 215-27. Hussain, S. Z. and Kamath N. R., 1967, Esterification of salicylic acid with methanol: semibatch reactor design. Indian J. Technol. 5, 92-6. Jungers, J. C. and Sajus, L., 1967, L’Analyse Cinetique de la Transformation Chimique, pp. 322-325. Technip. Parts. Kapkin, V. D., Bashkirov, A. N. and Grozhan, M. M., 1967, Synthesis ofphtalatcs from secondary alcohols obtained by direct oxidation of hydrocarbons. Neftekhimiya 7,97-103. Larraiiaga, J. M., Ortiz I. and Irabien, J. A., 1986, Analysis
Chemical Engkring
Scierree, Vd.
42, No.
10, pp. 2472-2475.
and modelling of segregative reactions. 1-Butyl alcohol esterification with hydrobromic acid. Chem. Engng Sci. 41, 3031-36. Luddy, F. E., Barford, R. A., Herb, S. F. and Magidman P., 1968, A rapid and quantitative procedure for the preparation of methyl esters of butter oil and other fats. J. Am. Oil Chem. Sot. 45, 549-42. Manziuca. A.. 1975, Esters of dicarboxylic acids with lower a&holk. Ram. Patent 58, 661. _ Rev. I.. Ortiz. I. and Irabien A., 1987, Kinetic analysis of the &nogenc&s acid catalysis-in the chloromethylation of toluene. J. Molec. Catal. 39, 105-113. Roje, R. and Trombic, S., 1966, Synthesis of bis(2ethylhexyl)phtalate II. Effect of the catalyst H,SO,. Kern. Ind. 15,686-92. Rubinstein, B. I., Leont’ev, Ya. A., Morozov, L. A. and Ustavshchikov: B. F., 1972, Kinetics of the esterification of methacrylic acid by methanol in the presence of sulfuric acid. Neftekhimiya 12, 589-92. Saldao, M., 1977, Riboflavine tetrabutyric acid ester. Japan Patent 77 95, 698. Trambouze, P. J. and Piret, E. L., 1960, Chemical reaction processes in two phases systems. A.I.Ch.E. J. 6, 39-l. Zehner, L. R., 1977, Oxalic acid esters. Ger. Patent 2, 721, 734.
1987.
Printed in Great Britain.
0
Direct reduction of hematite in a moving bed. Comparison three-interface pellet models (Received
2 November
1986; accepted
Direct reduction without iron melting in a moving bed reactor, is a relatively new, alternative process in the production of iron for steelmaking. A recent review paper (Negri et al., 1985) analyzes the state of the art in the area of modeling of the reduction process at pellet and reactor scales. In the present work a generalized, heterogeneous pellet representation (one- and three-interface) is included in a onedimensional, steady state, non isothermal reactor model for the study of the reduction process. The objective of using these pellet models is to analyze the reliability of the oneinterface model to predict the behavior of the particle along the reactor path. Fractional reduction profiles of a reactor model using both pellet representations are compared and discussed.
PELLET
MODELING
The reduction process at pellet scale is modeled by mean& of an unreacted, shrinking core scheme. Generalized expressions for the reduction rates are reached both for one- and three-interface pellet models. The former implies a modeling
in revised form
OCC9-2509jS7 1987 Pergamon
S3.00+0.00 Journals Ltd.
between one- and
17 March
1987)
approach with a global kinetics for the reduction of the initial iron oxide directly to iron. The latter takes into account the intermediate reduction steps (hematite + magnetite + wustite + iron) found in the actual process. An explicit representation of the reaction rate is obtained. It shows a new and more rational approach to the problem of systematically representing the particle behavior. The statement of both pellet models requires similar hypotheses such as: (1) qua&-steady stite; (2) equal molar countercurrent diffusion fluxes; (3) constant gas diffusivity; (4) isothermal pellet; (5) spherical geometry; (6)constant bulk conditions (gas temperature and composition); (7) ideal gas mixture; (8) all reduction reactions at each interface proceeding independently for each reducing agent; (9) side reactions not considered; (10) negligible viscous-flow contribution to the fluxes and (11) reversible first order kinetics. For the case of a one-interface pellet model, each reactant gas species in the bulk phase is transported through the gas-film surrounding the pellet, and through the layer of porous solid-product to the iron-oxide interface where it reacts. Each product gas species follows an exactly inverse path from the reacting surface to the bulk fluid phase. From
Shorter
Communications
the solution of the mass balances in the porous solid-product layer for gaseous reactant (A)and product (B), the dimensionless reaction rate can be written as:
n, = rl$(l+l/KmJA
- Y%
(1)
II
where
i = A or B.
Fi’il/kg,;
(3b)
An overall effectiveness factor (q) can be identified as a function of the different resistances involved in the process: the chemical reaction resistance (JcJ, the diffusion resistance of the iron layer for both reactant (DA) and product (Ds), and the gas film resistance for both reactant (F4) and product (FB). For the case of the three-interf?xe pellet model, the reactant species must diffuse from the bulk phase to the pellet surface through the gas film surrounding the pellet, and through the porous iron layer to the iron-wustite interface. The balance between the flux arriving at the first interface, and the rate of reactant consumption due to the heterogeneous reaction, gives the flux continuing through the porous wustite layer to the wustite-magnetite interface. The remainder of the reactant species after chemical reaction at this interface is transported through the porous magnetite layer to the magnetite-hematite interface. In a similar fashion, the product gas species follows an inverse, outward directed path within the pellet. By solving the mass balance equations for the gas species for each solid-product porous layer, an explicit matrix representation of the interfacial reaction rates is obtained (Negri et al., 1987): 1 n=B([h,+K-lh,]<~$}-‘[I+K-‘](y,-yy*,) 9
2473
This generalized representation includes as a special case the set of processes where only one or two iron oxides remain after the complete consumption of oxides of higher oxygen content. After hematite is consumed (cl = 0), the order of the system is reduced by one, and the expressions above hold after the elimination of the frrst row and column of all the matrices involved. The second interface will also collapse (& = 0) at the moment when the magnetite layer disappears. The system is reduced to a first order one, eliminating the first two rows and columns in the original expressions above. The latter is equivalent to the representation discussed previously [eq. (l)]. However, here the core is a wustite sphere instead of the original oxide, and the parameters involved are different. With the assumption of equal mutual diffusivities and mass transfer coefficients for reactants and products, the expressions presented by Hara et al. (1976), Yagi and Szekely (1979) and Yanagiya et al. (1979) can be obtained as a special case of eq. (4). For the case of a binary gas mixture the matrix representation becomes equivalent to the solution of Spitzer et nl. (1966) and Tsay et al. (1976). REACTOR
MODELING
A heterogeneous, one-dimensional scheme is used to describe the behavior of the reduction shaft furnace. The model can handle a mixture of reducing agents (carbon monoxide/ hydrogen), independent energy balances for each phase (solid and gaseous), and one of the two different pellet models. The hypotheses considered in the model can be summarized as follows (Arce er al., 1982): (1) steady state reactor; (2) plug flow for both phases; (3) negligible axial and radial dispersion of mass and energy; (4) uniform bed composed of spherical pellets; (5) ideal gas mixture; (6) homogeneous reactions not considered; and (7) constant physical properties. The material and energy balances define the following set of dimensionless equations for the case of one (N = 1) or three (N = 3) interface pellet models (Negri et al., 1987):
dT= -jzl5 dyi
$’ = T(8.
(4)
Daij
i = HZ, CO
- 9,)
where
K* + Dj” + ez’ + Dj” + Fi D!2) 1 + Di3’ + F.1
hi =
Dj” + Dj”’ + Fi
tit” + Fi Df” + Ft
Q+~~‘+@~)+F~
~~ + Di3’ + F. ’ K, = [k,(l D!k’ , =
Klj= KR,,li’
+ l/K,)
$,(&&); 1
Kj; 0;
0;
+ l/KR,j)]-l;
(5)
J
j = 1, 2, 3
k=
1,2,X
l=j l+ j,
[kR.j(l
1
(64 Fi = l/kg,; 5
1= j l#
j,
I
u
u
=
(3
5f;l=j 1 0;
=
i=AorB
(64
l#j
l;l=j (W
0; l#j
Here again, it is possible to identify an “overall effectiveness factor” in matrix notation. This effectiveness matrix displays the different resistances to the species transport considered by the model: chemical reaction resistances (xi), product layer diffusion resistances (of”) and external gas film resistances (F’J. It can be seen that each chemical reaction equation is a function of the concentration differences in all the interfaces. Moreover, the matrices Ai admit the visualization of the links between each reaction and the array of pseudo-composition differences in a linear and simple expression.
dcj 1 ~ = ~ x [DaH,,jn~,,i+Daco,,nco,ll; j = 3fi dT de, 5
1 + N (9)
j=l
+ @co.+co,j
QCOJI .
(10)
2474
Shorter
The dimensionless
boundary
conditions
Communications
are:
Y,K = 0) = yP; i = HZ, CO,
O,(C = 0) = e
(lla)
&(C=l)=l;
@,(C = 1) = 0:.
(lib)
j=l+iv,
The extent of the reaction in the solid phase is represented by the position of the reaction interfams (cj) according to: R3F=
RIF=(l-r:), RESULTS
AND
i
j=
fj(l-<;).
(12)
1
DISCUSSION
Figure 1 represents typical temperature and concentration profiles for gas and solid phases predicted by the reactor model, when using the three-interface scheme to describe pellet behavior. A standard set of operating conditions as well as available transport and kinetic parameters, is employed (Negri et al., 1987). A high degree of reduction is predicted since, at the bottom of the bed, solid conversion is nearly 100 %. From the axial profiles of the three reaction fronts, it can be observed that hematite is totally consumed in the reactor upper half portion, magnetite is found completely reduced at about 70% of the reactor length and wustite is almost completely transformed into porous iron by the bottom outlet of the reactor. Model predictions for the axial variations of the fractional reduction are similar to the experimental data obtained by other authors at pilot plant (Hara et aZ., 1976, Yanagiya et al., 1979) and commercial (Takenaka et al., 1986) scales. Hydrogen consumption is larger than that of carbon monoxide; this is consistent with a higher reduction rate for the first reactant predicted by the available kinetic data. Bath gas and solid temperature profiles coincide, with the exception of a narrow zone in the top portion of the reactor. Inthis region, the incoming solid phase rapidly reaches the much higher temperature of the gas phase. (This behavior has also been observed by Tsay et al., 1976.)
The axial variations of the fractional reduction, computed for the one- and three-interface pellet models, are compared in Fig. 2. In order to perform a fair comparison of results, the kinetic information used for each of the two pellet models is taken from data produced by a single research group (Yapi et al., 1971; Yagi and Szekely, 1979). The figure shows that the fractional reduction predicted by the three-interface pellet model (R”) is always larger than that obtained with the oneinterface model (R lF). It is worth noting that the latter does not distinguish reduction zones along the axial coordinate, presenting a curve with a smooth variation along the reactor length. On the other hand, R” shows the effect of the three reaction steps that take place during the total reduction process. As a conclusion it can be stated that predictions obtained with the one-interface model are significantly different from those obtained with the complete, three-interface model. Thus, the former do not accurately represent the chemical transformations suffered by the pellet along the reactor length. Acknowledgments-The authors are grateful to CONICET and UNL for their financial support, and to E. Grimaldi for her assistance with the English version. E. D. NEGRI 0. M. ALFANO M. G. CHIOVETTA Instiruto de Desarrollo Tecnoldgico pro la Industria Quimica (INTEC) Universidad National de1 Litoral (UNL) and Consejo National de investigaciones Cient$cas~~y T.&r&as (CONICET) Casilla de Correo No. 91, 3000 Santa Fe, Argentina NOTATION D’k’
diffusion resistance of the kth layer, s/m Damkiihler number, dimensionless effective diffusion coefficient, m2/s relative content of oxygen removed in the jth reaction, dimensionless gas film resistance, s/m reaction rate constant, m/s gas to solid mass transfer coefficient, m/s
Da Solid 1.0
0.5
; 0.8 0.0
0.4
0.0
c=1.0
F k kg
i.0
R
~=o.s
‘\ RiF 0.5
\ \ \ \
\, ! \
FF
‘1\ ‘\ \ ‘\ \
Rz\ \
\
\ \\R.
1
i
\
\\ 1.2
1.0
- c-0.0 0.8
ply
0.0
Fig. 1. Typical temperature and composition and solid phases.
profiles for gas
R,
\
‘,,
c
Fig. 2. Axial profiles for the fractional reduction and three-interface pellet models.
R for one-
Shorter K rp
R W,, Yi
REFERENCES
equilibrium constant, dimensionless radius at the external surface of the particle, m fractional reduction, dimensionless mass fraction of oxygen in the solid phase, dhnensionless normalized mole fraction for gas component i, dimensionless
0. M. and Arri, L. Arce, P. E., Alfano. Heterogeneous model of a moving-bed reactor.
;3 K;
5,
T
rp R
9
Tetsu Hagane 62, 315-323. Neti. E. D.. Alfano. 0. M. and Chiovetta.
M. G.. 1985. Heat a% mass’transfe; in the modeling of n&catalytic moving bed reactors and its application to direct reduction of iron oxides: a review. Lat. Am. J. Heat Mass Transfer 9,85-129. Negri, E. D., Alfano, 0. M. and Chiovetta, M. G., 1987, Optimal operating conditions for the direct reduction of iron oxide; in a s&aft furnace. To be published. Soitzer, R. H., Manning. F. S. and Philbrook, W. O., 1966, ~Gendralized model f& the gaseous, topochemical ;educi tion of oorous hematite soheres. Trans. Met. Sot. AIME 236,17i>l724. Takenaka. Y.. Kimura. Y.. Narita. K. and Kaneko. D.. 1986. Mathe&a&al modei ofdirect reduction shaft furnace and its application to actual operations of a model plant.
gas-solid feed ratio, dimensionless heat capacity gas-solid feed ratio, dimensionless axial coordinate, dimensionless overall effectiveness factor, dimensionless temperature, dimensionless chemical reaction resistance at the jth interface, s/m _ pellet radius, dimensionless solid-fluid heat transfer coefficient, dimensionless heat of reaction, dimensionless specific reaction rate, dimensionless
i
R S
at the core surface in the gas phase at the external surface of: hematite and wustite (3) at the reference state in the solid phase
Comput. Chem. Engng 10, 67-75. (l), magnetite
Tsay, Q. T., Ray, W. H. and Szekely, J., 1976, The modeling of hematite reduction with hydrogen plus carbon monoxide mixtures. A.Z.Ch.E. J. 22, iO64LiOf9. Yaei. J.. Takahashi. R. and Omori, Y.. 1971. Study on the r&&on pro& of iron oxide ielle& in isothermal fixed bed. Sci. Rev. RITU A 23. 3147. Yagi, J. and Siekely, J., 197<, The effect of gas and solids maldistribution on the performance of moving-bed reactors: the reduction of iron oxide pellets with hydrogen.
(2)
Superscripts in the kth solid-product (k) 1F 3F 0 *
layer: magnetite (l), wustite (2) and iron (3) relative to the one-interface pellet model relative to the three-interface pellet mode1 inlet condition pseudo-equilibrium condition
Chemical Engineering Science, Vol. 42, No.
1982,
Heat Mass Transfer 6, 99-112.
Subscripts C
E.,
Lat. Am. J.
Hara, Y., Sakawa, M. and Kondo, S., 1976, Mathematical model of the shaft furnace for reduction of iron-ore pellet.
Greek letters !
2475
Communications
10, pp. 2475-2478,
A.1.Ch.E. J. 25, 800-810. Yanagiya, T., Yagi, J. and Omori, Y., 1979, Reduction of iron oxide pellets in moving bed. Ironmaking Steefmaking 6, 93-100.
0009-2509187 1987 Pergamon
1987. 0
Printed in Great Britain.
Some
comments
(Received 7 November
on flotation
S3.00f0.00 Journals Ltd.
kinetics
1986; accepted 11 March 1987)
The process of forth flotation is widely used in the mining, mineral, metallurgical and chemical industries for component separation of the processed raw materials. Process kinetics is an important factor describing the flotation results, and has received much attention hap&s by Jameson et al. (1977) and by Mori et al. (1986), contain exhaustive lists of relevant rdferenccs]. This-note is concerned with the etfects of some of the physical properties of the flotation system, in particular of air bubbles, on the rate of flotation. Effects of the chemical conditions of flotation are not considered. Hydrophobicity of particles sufficient for their adhesion to air bubbles is assumed. DlSCUSSION A differential equation describing the kinetics of the flotation process is usually written in the form analogous to that for a first order chemical reaction:
dC, = - k’C&
dt.
(1)
It is then assumed (e.g. Arbiter and Harris, 1976) that for a given flotation system, C, is constant, so eq. (1) reduces to dC, = - kc, dt,
(2)
A plot of the integrated form of eq. (2) is a straight line of slope -k, as shown in Fig. 1 by line a. This form is generally accepted as descriptive of most of the flotation processes. However, many of the published studies of flotation kinetics show that test results often do not follow the linear form (Mori et al., 1986). Figure 1, line b presents common deviations of exmrimental results from the theoretical characteristic. As &n be seen, the actual fiotation process can be divided into three parts distinguished by dilferent values of flotation rate (line c), and denoted A, B and C. Part A observed at the beginning of the process is
Shorter
Communications
HCl and HBr media, an improved method for the calculation of the X-functions and Ho scales. Can. J. Chem. 59, 21162121. Fokin, A. V., Kolomiets, A. F., Studnev, Yu. N. and Kuznetsova L. D., 1970, Acylation of phenol with carboxylic acids. Iriv. Sib. Otd. Akad. Nauk. SSSR, Ser. Khim. Nauk. 1, 87-90. Ghtzman, M. Kh., Dashevskaya, B. I. and Fridman, G. M., 1968, Esteritication of stearic acid with polyethylene glycols. Zh. Prikl. Khim. 41, 183-7. Hammett, L. P., 1970, Physical Organic Chemistry. Reaction Rates, Equilibria and Mechanisms, 2nd ed., pp. 315347. McGraw-Hill, Kogakusha. Hirsch, J. A. and Schwartzkopf G., 1974, Esterification of 7carboxy-2,3-dihydro-4-quinoline. Synth. Commun. 4, 215-27. Hussain, S. Z. and Kamath N. R., 1967, Esterification of salicylic acid with methanol: semibatch reactor design. Indian J. Technol. 5, 92-6. Jungers, J. C. and Sajus, L., 1967, L’Analyse Cinetique de la Transformation Chimique, pp. 322-325. Technip. Parts. Kapkin, V. D., Bashkirov, A. N. and Grozhan, M. M., 1967, Synthesis ofphtalatcs from secondary alcohols obtained by direct oxidation of hydrocarbons. Neftekhimiya 7,97-103. Larraiiaga, J. M., Ortiz I. and Irabien, J. A., 1986, Analysis
Chemical Engkring
Scierree, Vd.
42, No.
10, pp. 2472-2475.
and modelling of segregative reactions. 1-Butyl alcohol esterification with hydrobromic acid. Chem. Engng Sci. 41, 3031-36. Luddy, F. E., Barford, R. A., Herb, S. F. and Magidman P., 1968, A rapid and quantitative procedure for the preparation of methyl esters of butter oil and other fats. J. Am. Oil Chem. Sot. 45, 549-42. Manziuca. A.. 1975, Esters of dicarboxylic acids with lower a&holk. Ram. Patent 58, 661. _ Rev. I.. Ortiz. I. and Irabien A., 1987, Kinetic analysis of the &nogenc&s acid catalysis-in the chloromethylation of toluene. J. Molec. Catal. 39, 105-113. Roje, R. and Trombic, S., 1966, Synthesis of bis(2ethylhexyl)phtalate II. Effect of the catalyst H,SO,. Kern. Ind. 15,686-92. Rubinstein, B. I., Leont’ev, Ya. A., Morozov, L. A. and Ustavshchikov: B. F., 1972, Kinetics of the esterification of methacrylic acid by methanol in the presence of sulfuric acid. Neftekhimiya 12, 589-92. Saldao, M., 1977, Riboflavine tetrabutyric acid ester. Japan Patent 77 95, 698. Trambouze, P. J. and Piret, E. L., 1960, Chemical reaction processes in two phases systems. A.I.Ch.E. J. 6, 39-l. Zehner, L. R., 1977, Oxalic acid esters. Ger. Patent 2, 721, 734.
1987.
Printed in Great Britain.
0
Direct reduction of hematite in a moving bed. Comparison three-interface pellet models (Received
2 November
1986; accepted
Direct reduction without iron melting in a moving bed reactor, is a relatively new, alternative process in the production of iron for steelmaking. A recent review paper (Negri et al., 1985) analyzes the state of the art in the area of modeling of the reduction process at pellet and reactor scales. In the present work a generalized, heterogeneous pellet representation (one- and three-interface) is included in a onedimensional, steady state, non isothermal reactor model for the study of the reduction process. The objective of using these pellet models is to analyze the reliability of the oneinterface model to predict the behavior of the particle along the reactor path. Fractional reduction profiles of a reactor model using both pellet representations are compared and discussed.
PELLET
MODELING
The reduction process at pellet scale is modeled by mean& of an unreacted, shrinking core scheme. Generalized expressions for the reduction rates are reached both for one- and three-interface pellet models. The former implies a modeling
in revised form
OCC9-2509jS7 1987 Pergamon
S3.00+0.00 Journals Ltd.
between one- and
17 March
1987)
approach with a global kinetics for the reduction of the initial iron oxide directly to iron. The latter takes into account the intermediate reduction steps (hematite + magnetite + wustite + iron) found in the actual process. An explicit representation of the reaction rate is obtained. It shows a new and more rational approach to the problem of systematically representing the particle behavior. The statement of both pellet models requires similar hypotheses such as: (1) qua&-steady stite; (2) equal molar countercurrent diffusion fluxes; (3) constant gas diffusivity; (4) isothermal pellet; (5) spherical geometry; (6)constant bulk conditions (gas temperature and composition); (7) ideal gas mixture; (8) all reduction reactions at each interface proceeding independently for each reducing agent; (9) side reactions not considered; (10) negligible viscous-flow contribution to the fluxes and (11) reversible first order kinetics. For the case of a one-interface pellet model, each reactant gas species in the bulk phase is transported through the gas-film surrounding the pellet, and through the layer of porous solid-product to the iron-oxide interface where it reacts. Each product gas species follows an exactly inverse path from the reacting surface to the bulk fluid phase. From
Shorter
Communications
the solution of the mass balances in the porous solid-product layer for gaseous reactant (A)and product (B), the dimensionless reaction rate can be written as:
n, = rl$(l+l/KmJA
- Y%
(1)
II
where
i = A or B.
Fi’il/kg,;
(3b)
An overall effectiveness factor (q) can be identified as a function of the different resistances involved in the process: the chemical reaction resistance (JcJ, the diffusion resistance of the iron layer for both reactant (DA) and product (Ds), and the gas film resistance for both reactant (F4) and product (FB). For the case of the three-interf?xe pellet model, the reactant species must diffuse from the bulk phase to the pellet surface through the gas film surrounding the pellet, and through the porous iron layer to the iron-wustite interface. The balance between the flux arriving at the first interface, and the rate of reactant consumption due to the heterogeneous reaction, gives the flux continuing through the porous wustite layer to the wustite-magnetite interface. The remainder of the reactant species after chemical reaction at this interface is transported through the porous magnetite layer to the magnetite-hematite interface. In a similar fashion, the product gas species follows an inverse, outward directed path within the pellet. By solving the mass balance equations for the gas species for each solid-product porous layer, an explicit matrix representation of the interfacial reaction rates is obtained (Negri et al., 1987): 1 n=B([h,+K-lh,]<~$}-‘[I+K-‘](y,-yy*,) 9
2473
This generalized representation includes as a special case the set of processes where only one or two iron oxides remain after the complete consumption of oxides of higher oxygen content. After hematite is consumed (cl = 0), the order of the system is reduced by one, and the expressions above hold after the elimination of the frrst row and column of all the matrices involved. The second interface will also collapse (& = 0) at the moment when the magnetite layer disappears. The system is reduced to a first order one, eliminating the first two rows and columns in the original expressions above. The latter is equivalent to the representation discussed previously [eq. (l)]. However, here the core is a wustite sphere instead of the original oxide, and the parameters involved are different. With the assumption of equal mutual diffusivities and mass transfer coefficients for reactants and products, the expressions presented by Hara et al. (1976), Yagi and Szekely (1979) and Yanagiya et al. (1979) can be obtained as a special case of eq. (4). For the case of a binary gas mixture the matrix representation becomes equivalent to the solution of Spitzer et nl. (1966) and Tsay et al. (1976). REACTOR
MODELING
A heterogeneous, one-dimensional scheme is used to describe the behavior of the reduction shaft furnace. The model can handle a mixture of reducing agents (carbon monoxide/ hydrogen), independent energy balances for each phase (solid and gaseous), and one of the two different pellet models. The hypotheses considered in the model can be summarized as follows (Arce er al., 1982): (1) steady state reactor; (2) plug flow for both phases; (3) negligible axial and radial dispersion of mass and energy; (4) uniform bed composed of spherical pellets; (5) ideal gas mixture; (6) homogeneous reactions not considered; and (7) constant physical properties. The material and energy balances define the following set of dimensionless equations for the case of one (N = 1) or three (N = 3) interface pellet models (Negri et al., 1987):
dT= -jzl5 dyi
$’ = T(8.
(4)
Daij
i = HZ, CO
- 9,)
where
K* + Dj” + ez’ + Dj” + Fi D!2) 1 + Di3’ + F.1
hi =
Dj” + Dj”’ + Fi
tit” + Fi Df” + Ft
Q+~~‘+@~)+F~
~~ + Di3’ + F. ’ K, = [k,(l D!k’ , =
Klj= KR,,li’
+ l/K,)
$,(&&); 1
Kj; 0;
0;
+ l/KR,j)]-l;
(5)
J
j = 1, 2, 3
k=
1,2,X
l=j l+ j,
[kR.j(l
1
(64 Fi = l/kg,; 5
1= j l#
j,
I
u
u
=
(3
5f;l=j 1 0;
=
i=AorB
(64
l#j
l;l=j (W
0; l#j
Here again, it is possible to identify an “overall effectiveness factor” in matrix notation. This effectiveness matrix displays the different resistances to the species transport considered by the model: chemical reaction resistances (xi), product layer diffusion resistances (of”) and external gas film resistances (F’J. It can be seen that each chemical reaction equation is a function of the concentration differences in all the interfaces. Moreover, the matrices Ai admit the visualization of the links between each reaction and the array of pseudo-composition differences in a linear and simple expression.
dcj 1 ~ = ~ x [DaH,,jn~,,i+Daco,,nco,ll; j = 3fi dT de, 5
1 + N (9)
j=l
+ @co.+co,j
QCOJI .
(10)
2474
Shorter
The dimensionless
boundary
conditions
Communications
are:
Y,K = 0) = yP; i = HZ, CO,
O,(C = 0) = e
(lla)
&(C=l)=l;
@,(C = 1) = 0:.
(lib)
j=l+iv,
The extent of the reaction in the solid phase is represented by the position of the reaction interfams (cj) according to: R3F=
RIF=(l-r:), RESULTS
AND
i
j=
fj(l-<;).
(12)
1
DISCUSSION
Figure 1 represents typical temperature and concentration profiles for gas and solid phases predicted by the reactor model, when using the three-interface scheme to describe pellet behavior. A standard set of operating conditions as well as available transport and kinetic parameters, is employed (Negri et al., 1987). A high degree of reduction is predicted since, at the bottom of the bed, solid conversion is nearly 100 %. From the axial profiles of the three reaction fronts, it can be observed that hematite is totally consumed in the reactor upper half portion, magnetite is found completely reduced at about 70% of the reactor length and wustite is almost completely transformed into porous iron by the bottom outlet of the reactor. Model predictions for the axial variations of the fractional reduction are similar to the experimental data obtained by other authors at pilot plant (Hara et aZ., 1976, Yanagiya et al., 1979) and commercial (Takenaka et al., 1986) scales. Hydrogen consumption is larger than that of carbon monoxide; this is consistent with a higher reduction rate for the first reactant predicted by the available kinetic data. Bath gas and solid temperature profiles coincide, with the exception of a narrow zone in the top portion of the reactor. Inthis region, the incoming solid phase rapidly reaches the much higher temperature of the gas phase. (This behavior has also been observed by Tsay et al., 1976.)
The axial variations of the fractional reduction, computed for the one- and three-interface pellet models, are compared in Fig. 2. In order to perform a fair comparison of results, the kinetic information used for each of the two pellet models is taken from data produced by a single research group (Yapi et al., 1971; Yagi and Szekely, 1979). The figure shows that the fractional reduction predicted by the three-interface pellet model (R”) is always larger than that obtained with the oneinterface model (R lF). It is worth noting that the latter does not distinguish reduction zones along the axial coordinate, presenting a curve with a smooth variation along the reactor length. On the other hand, R” shows the effect of the three reaction steps that take place during the total reduction process. As a conclusion it can be stated that predictions obtained with the one-interface model are significantly different from those obtained with the complete, three-interface model. Thus, the former do not accurately represent the chemical transformations suffered by the pellet along the reactor length. Acknowledgments-The authors are grateful to CONICET and UNL for their financial support, and to E. Grimaldi for her assistance with the English version. E. D. NEGRI 0. M. ALFANO M. G. CHIOVETTA Instiruto de Desarrollo Tecnoldgico pro la Industria Quimica (INTEC) Universidad National de1 Litoral (UNL) and Consejo National de investigaciones Cient$cas~~y T.&r&as (CONICET) Casilla de Correo No. 91, 3000 Santa Fe, Argentina NOTATION D’k’
diffusion resistance of the kth layer, s/m Damkiihler number, dimensionless effective diffusion coefficient, m2/s relative content of oxygen removed in the jth reaction, dimensionless gas film resistance, s/m reaction rate constant, m/s gas to solid mass transfer coefficient, m/s
Da Solid 1.0
0.5
; 0.8 0.0
0.4
0.0
c=1.0
F k kg
i.0
R
~=o.s
‘\ RiF 0.5
\ \ \ \
\, ! \
FF
‘1\ ‘\ \ ‘\ \
Rz\ \
\
\ \\R.
1
i
\
\\ 1.2
1.0
- c-0.0 0.8
ply
0.0
Fig. 1. Typical temperature and composition and solid phases.
profiles for gas
R,
\
‘,,
c
Fig. 2. Axial profiles for the fractional reduction and three-interface pellet models.
R for one-
Shorter K rp
R W,, Yi
REFERENCES
equilibrium constant, dimensionless radius at the external surface of the particle, m fractional reduction, dimensionless mass fraction of oxygen in the solid phase, dhnensionless normalized mole fraction for gas component i, dimensionless
0. M. and Arri, L. Arce, P. E., Alfano. Heterogeneous model of a moving-bed reactor.
;3 K;
5,
T
rp R
9
Tetsu Hagane 62, 315-323. Neti. E. D.. Alfano. 0. M. and Chiovetta.
M. G.. 1985. Heat a% mass’transfe; in the modeling of n&catalytic moving bed reactors and its application to direct reduction of iron oxides: a review. Lat. Am. J. Heat Mass Transfer 9,85-129. Negri, E. D., Alfano, 0. M. and Chiovetta, M. G., 1987, Optimal operating conditions for the direct reduction of iron oxide; in a s&aft furnace. To be published. Soitzer, R. H., Manning. F. S. and Philbrook, W. O., 1966, ~Gendralized model f& the gaseous, topochemical ;educi tion of oorous hematite soheres. Trans. Met. Sot. AIME 236,17i>l724. Takenaka. Y.. Kimura. Y.. Narita. K. and Kaneko. D.. 1986. Mathe&a&al modei ofdirect reduction shaft furnace and its application to actual operations of a model plant.
gas-solid feed ratio, dimensionless heat capacity gas-solid feed ratio, dimensionless axial coordinate, dimensionless overall effectiveness factor, dimensionless temperature, dimensionless chemical reaction resistance at the jth interface, s/m _ pellet radius, dimensionless solid-fluid heat transfer coefficient, dimensionless heat of reaction, dimensionless specific reaction rate, dimensionless
i
R S
at the core surface in the gas phase at the external surface of: hematite and wustite (3) at the reference state in the solid phase
Comput. Chem. Engng 10, 67-75. (l), magnetite
Tsay, Q. T., Ray, W. H. and Szekely, J., 1976, The modeling of hematite reduction with hydrogen plus carbon monoxide mixtures. A.Z.Ch.E. J. 22, iO64LiOf9. Yaei. J.. Takahashi. R. and Omori, Y.. 1971. Study on the r&&on pro& of iron oxide ielle& in isothermal fixed bed. Sci. Rev. RITU A 23. 3147. Yagi, J. and Siekely, J., 197<, The effect of gas and solids maldistribution on the performance of moving-bed reactors: the reduction of iron oxide pellets with hydrogen.
(2)
Superscripts in the kth solid-product (k) 1F 3F 0 *
layer: magnetite (l), wustite (2) and iron (3) relative to the one-interface pellet model relative to the three-interface pellet mode1 inlet condition pseudo-equilibrium condition
Chemical Engineering Science, Vol. 42, No.
1982,
Heat Mass Transfer 6, 99-112.
Subscripts C
E.,
Lat. Am. J.
Hara, Y., Sakawa, M. and Kondo, S., 1976, Mathematical model of the shaft furnace for reduction of iron-ore pellet.
Greek letters !
2475
Communications
10, pp. 2475-2478,
A.1.Ch.E. J. 25, 800-810. Yanagiya, T., Yagi, J. and Omori, Y., 1979, Reduction of iron oxide pellets in moving bed. Ironmaking Steefmaking 6, 93-100.
0009-2509187 1987 Pergamon
1987. 0
Printed in Great Britain.
Some
comments
(Received 7 November
on flotation
S3.00f0.00 Journals Ltd.
kinetics
1986; accepted 11 March 1987)
The process of forth flotation is widely used in the mining, mineral, metallurgical and chemical industries for component separation of the processed raw materials. Process kinetics is an important factor describing the flotation results, and has received much attention hap&s by Jameson et al. (1977) and by Mori et al. (1986), contain exhaustive lists of relevant rdferenccs]. This-note is concerned with the etfects of some of the physical properties of the flotation system, in particular of air bubbles, on the rate of flotation. Effects of the chemical conditions of flotation are not considered. Hydrophobicity of particles sufficient for their adhesion to air bubbles is assumed. DlSCUSSION A differential equation describing the kinetics of the flotation process is usually written in the form analogous to that for a first order chemical reaction:
dC, = - k’C&
dt.
(1)
It is then assumed (e.g. Arbiter and Harris, 1976) that for a given flotation system, C, is constant, so eq. (1) reduces to dC, = - kc, dt,
(2)
A plot of the integrated form of eq. (2) is a straight line of slope -k, as shown in Fig. 1 by line a. This form is generally accepted as descriptive of most of the flotation processes. However, many of the published studies of flotation kinetics show that test results often do not follow the linear form (Mori et al., 1986). Figure 1, line b presents common deviations of exmrimental results from the theoretical characteristic. As &n be seen, the actual fiotation process can be divided into three parts distinguished by dilferent values of flotation rate (line c), and denoted A, B and C. Part A observed at the beginning of the process is