474
Ind. Eng. Chem. Res.
Schnell, H.; Bottenbruch, L.; Krimm, H. Thermoplastic Aromatic Polycarbonates and their Manufacture, US. Patent 3,028,366, Apr 3, 1962. Silva, J. M. Manuscript in preparation, 1991. Vernaleken, H. Polycarbonates. In Interfacial Synthesis; Millich, F., Carraher, C. E., Jr., Eds.; Marcel Dekker: New York, 1977; Vol. 11, Chapter 13, pp 65-121.
1991, 30, 474-482 Wielgosz, 2.; Dobkowki, Z.; Krajeweki, B. Studies on Polycarbonate Preparation by the Interfacial Polycondensation Method. Eur. Polym. J. 1972,8, 1113-1119.
Received for review February 20,1990 Revised manuscript received September 7, 1990 Accepted September 23,1990
PROCESS ENGINEERING AND DESIGN Direct Reduction of Hematite in a Moving-Bed Reactor. Analysis of the Water Gas Shift Reaction Effects on the Reactor Behavior Enrique D. Negri, Orlando M. Alfano, and Mario G. Chiovetta* INTEC,’ U.N.L.-CONICET, Giiemes 3450,3000 Santa Fe, Argentina
A mathematical model of a moving-bed reactor for the direct reduction of iron oxides using mixtures of reducing gases is presented. The reactor model is based on a one-dimensional, heterogeneous, nonisothermal, steady-state scheme. Pellet modeling is performed by using three moving reduction fronts, with the water gas shift reaction (WGSR) taking place in the spongeiron layer. This reaction, which precludes the dropping of the hydrogen t o carbon monoxide concentration ratio, is of great importance for the performance of the shaft furnace. Given the high temperatures and the catalytic effects of the iron in the outer layer upon the WGSR, the latter is considered in equilibrium within this region. Model predictions are compared with those of a three-interface model with no side reactions and with the experimental data available in the literature for a pilot-plant-scale reactor. It is concluded that, although both models predict the variations observed in the experimental information, the rigorous (with WGSR) model more closely represents the behavior of the reducing furnace for the industrial operating range. 1. Introduction Different levels of simplification can be found when modeling the direct reduction of iron ores at the reactor scale (Negri et al., 1985). Simplified schemes range from those containing only mass balances (Hara et al., 1976; Yanagiya et al., 1979) to those considering a single reduction agent (Yagiand Szekely, 1979; Yu and Gillis, 1981) or to those taking into account a reducing mixture of hydrogen and carbon monoxide but neglecting the effects of side reactions in the gas phase (Tsay et al., 197613; Kam and Hughes, 1981; Negri et al., 1988). Among feasible secondary reactions in a reduction furnace fed with a mixture of gases coming out of a natural gas reformer, the most typical are (i) the water gas shift reaction (WGSR), (ii) the carbon deposition reaction, and (iii) the methane synthesis reaction. For the conditions typically found in an industrial reactor, only the first reaction is significant (Hara et al., 1976), because of usually high temperatures and the obvious presence of iron and its oxides, which are well-known catalysts for the reaction. This reaction among the gaseous species of the reacting system has the major effect of lowering the carbon monoxide contents while increasing the concentration of hydrogen, which is a better reducing agent. In addition, this water gas shift exothermic reaction provides energy to the reacting system. These Instituto de Desarrollo Tecnol6gico para la Industria Qdmica (INTEC). Universidad Nacional del Litoral (UNL) and Coneejo Nacional de Investigaciones Cientificas y TBcnicas (CONICET).
facts, in turn, increase the degree of ore reduction attained in the shaft furnace. For all these reasons, the inclusion of the WGSR in a furnace model should drastically improve the overall representation of the reactor performance. In spite of the evidence mentioned above, the effect of the water gas shift reaction on the behavior of an industrial reactor has received little attention in previous model efforts. Some simplified treatments can be found in the work by Hara et al. (1976), Hughes and Kam (1982), and Takenaka et al. (1986). Hara et al. and Takenaka et al. considered the WGSR in the reduction process via the inclusion of a source term in the reactor balances. Although the WGSR is, thus, included in the reduction process, no pellet model is presented in these papers in order to account for the evaluation of the source term introduced by this reaction. Hughes and Kam proposed a model including the effect of the WGSR a t the pellet level, and the calculation of the flux of chemical species entering the gas phase through the solid-gas interface. A single moving reaction front (hematite-iron) was considered in their work, with the WGSR taking place in the external layer of the pellet with the sponge-iron as the catalyst. In what follows, a study of the influence of the WGSR upon the behavior of the reduction furnace is presented. The pellet is modeled by using a heterogeneous scheme with three reaction fronts and with the WGSR taking place in the external, reduced-iron layer. This scheme is part of the mathematical representation of a one-dimensional,
0888-5885f 91f 26~0-0414$O2.50/0Q 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991 475 heterogeneous, nonisothermal, steady-state, moving-bed reactor used for the analysis of the direct reduction process. Axial profiles for the degree of reduction, gas temperature, reducing agent compositions, and degree of approximation to WGSR equilibrium conditions predicted by the model with and without the WGSR are compared with experimental information for a pilot-plant reactor published elsewhere (Takenaka et al., 1986). An analysis of the influence of the gas-to-solid feed ratio on the degree of reduction and the reducing gas utilization at the reactor outlet is also presented. +co,
2. Pellet Modeling 2.1. Model without the WGSR. Given the operating
conditions in most of the industrial processes for direct reduction of iron ores using a moving-bed reactor, the representation of the phenomena at the pellet scale using a heterogeneous model with moving reaction fronts has proved appropriate. In previous work (Negri et al., 1987; Negri, 1987), it was concluded that a model with a single moving front, usually hematite-iron, is not able to adequately represent the reaction and diffusion processes that occur along the reactor. On the basis of this fact, a mathematical scheme including the formation of intermediate oxides following the sequence hematite magnetite wustite iron via a three-moving-front representation is used in this analysis. The three reduction steps considered are 3FezO3+ H2 (or CO) e 2Fe304+ HzO (or COz) Fe304+ Hz (or CO) * 3"FeO" + HzO (or COz) "FeO" + Hz (or CO) F! Fe + HzO (or COz) The symbol "FeO" stands for the compound Fe0.94,0,the nonstoichiometric representation usually adopted for wustite, used for calculations throughout this paper. According to the solution for this case presented by Negri et al. (1987, 1988), the dimensionless rate of the reaction when no WGSR is considered is given by
- -
_ c
C
-
where the equation includes an "effectiveness factor" matrix and a vector of driving forces expressed as a function of the concentrations in the bulk gas phase. This effectiveness factor matrix combines all kinetic and transport resistances included in the model: magnetite, wustite, and iron layers and the gas film. The definition of the kinetic and transport resistance matrix (A) for the case with no WGSR can be found in the references cited above. The final matrix form does not introduce additional complications into the numerical scheme with respect to the single-front case (the model works with either a 3 X 3 or "1 X 1" matrix, with no substantial differences in computational requirements). 2.2. Model Including t h e WGSR. The side reaction taking place among the species in the gaseous mixture HzO + CO HZ + COS is added to the set of reactions representing the reduction processes above. A distinct characteristic of the pellet representation is the assumption of the presence of the WGSR only in the outermost, sponge-iron layer. It has been shown elsewhere that the local reaction rate is much higher with sponge-iron as the catalyst, when compared with the case where the reaction is catalyzed with iron
rl
rz
r3
rP
-
Figure 1. Schematic representation of the three-interface pellet model with the WGSR inside the iron layer.
oxides (Wagner, 1970; von Bogdandy and Engell, 1971). An additional assumption is introduced into the model: the WGSR reaches an equilibrium condition very rapidly, in accordance with the high temperatures in virtually all of the reactor and the catalytic effect of the sponge-iron layer. A schematic representation of the physical picture being modeled is presented in Figure 1. The analysis of the importance of the WGSR effects on the reactor performance and, thus, on its modeling is included in section 4.1. For the interior of the pellet, the transport of chemical species is envisioned in a manner similar to that developed for the case of three moving reduction fronts without secondary reactions (Negri et al., 1987). Figure 1 shows typical concentration profiles for a couple of species in the gas phase. A reducing agent "A" (either H2 or CO) moves through the sponge-iron layer, in equilibrium with the other chemical species, and reacts a t the first reduction front (wustite-iron). The balance between the flux reaching the wustite-iron interface and the chemical reaction consumption at this front diffuses through the wustite layer toward the magnetite-wustite front. Here, a similar sequence takes place: chemical reaction consume8 some reactant at the front, and the remainder flows across the magnetite layer toward the hematite-magnetite interface, where the last reduction step takes place. A reduction gaseous product "B" (either H20 or C02) moves through the pellet following a pattern similar to that of A but in the opposite direction, as shown in Figure 1. The mathematical model representing the reaction process in the pellet is based upon the following additional hypothesis: (1)quasi steady state; (2) equimolar counterdiffusion; (3) constant physical and transport properties; (4) isothermal pellet; ( 5 ) spherical pellet geometry; (6) constant temperature and composition for the bulk gas phase; (7) ideal gas mixture; and (8) negligible viscous flow contributions. The system of equations resulting from the mass balance is included in the Appendix.
476 Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991
After the linear portion of the system of equations (mass balance equations for the magnetite and wustite layers) was solved, an expression for the rates of reduction for the three reaction fronts considered is obtained:
[ 1;;]
= {[AA + &-'AB][)-'
[
wI,3
CA,3
-s,/6,1]
CA,3
- CB,3/KI,2
CA,3
- CB,3/KI,3
3
+ d#/COCHz,3
+
cs?(wH, j
j=l
+ @CO,j ) =
cCO,p CC0.3CH20,3
+ d&COCH~,p
= KwCHz,3CC0z,3
(3) (4)
3
cCO,p
+ *H2/COCH2,p + FCOcs?(wHz,
j
+ OCO, j ) =
1'1
CCO,b CCO,pCH~O,p
+ \kH2/COCHz.b ( 5 )
= K~CH~,pCCO~,p
(6)
There exists a linear relationship between the pairs of concentrations (H, - HzO) and (CO - COz)for any given position within the pellet and, in particular, at the pellet surface and at the wustite-iron interface. The following set of mathematical expressions fully determines the set of algebraic equations necessary to solve for the balances in the external layer of the pellet: CB,3
+ 4f)BCA,3 = CB,p + $ f ) B c A , p
GAS OUTLET
(2)
The matrix expression for the rate of reduction at each front is conceptually similar to that obtained for the case of no side reaction. It is built by using a matrix of kinetic and transport resistances located in the interior of the pellet (magnetite and wustite layers) and a vector of driving forces. For the case in this section, including the WGSR, the driving forces for each reaction are based upon the concentrations of reducing agents inside the spongeiron layer. Thus, it is of no convenience to use an effective factor in matrix form to compute the reaction rates as a function of the properties of the bulk gas phase, as was the case when no secondary reactions were included (Negri et al., 1987,1988). Concentrations in the reduced iron layer, namely CA,3and CB,3 in Figure 1, for A = Hz, CO and B = HzO, COP,are obtained by solving the following system of nonlinear algebraic equations (Appendix): cC0,3
HEMATITE .PELLETS
(7)
(8) + *A/BcA,p = CB,b + *A/BCA,b The definition of a dimensionless form for the reaction rate at front j renders CB,p
The, the rates of reaction for the case of the model including the WGSR are given by the dimensionless expression
DIRECT REDUCTION SHAFT FURNACE
z=OlnR S T. SOLID OUTLET
REDUCING GAS
lq
Figure 2. Schematic representation of the direct reduction shaft furnace.
For this particular case, it is not possible to obtain explicit expressions for the reaction rate at each reduction front, due to the nonlinearity introduced by the equilibrium condition into the system of equations for the mass balances (eqs 4 and 6). The numerical solution of the system of eqs 3-8 requires an iterative scheme to compute the concentration of the reduction agents in the iron layer. Brown's (1973) method is used to solve for these concentrations. Simultaneously, the rates of reaction at each front are computed by using eq 2. As was the case for the solution of the three-front model without side reactions, the mathematical representation obtained in this work includes as particular cases the set of processes where only one or two reaction fronts remain after the iron oxides with higher oxygen content have been depleted. Immediately after hematite has been consumed completely (t1= 0), the order of the mathematical system reduces by one. The mathematical expressions presented here are rendered valid for this situation by eliminating the first row and the first column in the corresponding matrices (Appendix). In turn, when the magnetite layer disappears due to reduction (4, = 01, the subsequent reduction by one renders a first-order system, with an expression valid for a single reaction front. At the early stages of the process and as long as there is no sponge-iron layer (13= 11, the model is equivalent to the mathematical scheme where no WGSR is considered (section 2.1). 3. Reactor Modeling The behavior of the reduction furnace is modeled by using a one-dimensional, heterogeneous scheme. The mathematical representation considers independent mass and energy balances for both the gaseous and solid phases and takes each pellet into account through the three moving-front reduction model analyzed in section 2. Figure 2 shows a simplified scheme of the reactor. 3.1. Mass and Energy Balances. The basic assumptions in the model are (1)steady-state reactor, (2) plug flow for both the solid and gaseous phases, (3) negligible radial and axial dispersion of mass and energy, (4) uniform bed of spherical particles, (6) ideal gas mixture, (6) constant physical properties, and (7) negligible side reactions other than the WGSR. With these hypotheses, the mass and energy balances for the solid and gaseous phases produce
Ind. Eng. Chem. Res., Vol. 30,No. 3, 1991 477 Table I. Values of Model Parameters parameter reactor operating conditions and dimensions
heat- and mass-transfer parameters
value employed Go = 26.1 mol/(mZs) So = 0.40 kg/(m2 s) = 298 K II" = 1113 K x i 2 = 0.547 x$20 = 0.038 .& = 0.388 xco2 = 0.025 r p = 0.007 m w& = 0.29 L = 2.7 m d = 0.25 m cp, = 840 J/(kg K) Cp, = 33.5 J/(mol K) DeH2 = 1.50 x 10"' m2/S &JH2/H20 = 1.7 Deco = 1.17 X IO-' mz/s &JCO/CO, = 2.2 kgH2 = 0.52 m/S *H2/H20 = kgco = 0.44 m/s *co/co* = 1.7 c = 0.4 (adopted) h = 165 W/(m2 K) U = 1.3 W/(m2 K)
the following set of dimensionless ordinary differential equations:
--
@awl (14)
Here, N, is the molar flux due to the WGSR, given by the following expression:
The dimensionless boundary conditions are yi(T=O) = yp (i = H2, CO) 0,({=0) = 0; Fj({=l)
= 1 ( j = 1, 2, 3)
e,({=l)
=e:
(16) (17)
The mathematical modeling of the reactor produces a set of nonlinear, first-order, ordinary differential equations with a set of split boundary conditions. Two dimensionless numbers are relevant to the representation of the process at the reactor level: the gas-solid feed ratio (a)and the heat capacity gas-solid feed ratio (P). When the WGSR is taken into account, two additional terms are introduced in the system corresponding to the case without side reactions: (i) in the gas-phase mass balance, the terms corresponding to the rate of reduction are replaced by the corresponding molar interface flux; and (ii) in the solid-phase energy balance, a new source term appears to model the energy flux due to the heat of reaction evolved by the water gas shift reaction. The replacement introduced in (i) is necessary because now the reduction process is no longer the only consumer (or producer) of gaseous chemical species. The addition in (ii) is performed in order to account for the energy addition associated with the water gas shift process, through the heat of reaction and the molar flux generated hy the WGSR (N,). The direction of the molar interface flux f o r t w h winpound (right-hand side of eq 11) is given by the t vd l I illICe between consumption (or production) of 1 h c spcv*ies through the reduction process and the consumption (or
ref Kaneko et al. (1982)
Perry and Chilton (1973) Reid et al. (1977) Sen Gupta and Thodos (1962) Kaneko et al. (1982)
production) via the WGSR. For those furnace regions where the reactor conditions are such that the WGSR equilibrium is severely disturbed, such as both ends of the vessel, it is possible that the pellet could act as a source of one of the reducing agents (either Hz or CO). When the WGSR is not considered,Nw= 0. In this case, the molar interface flux of either H2 or CO (first term in the right-hand side of eq 15) is determined by the rate of reduction of all heterogeneous process stages (second term in the right-hand side of eq 15), and the system of eqs 11-14 becomes coincident with that presented in previous work (Negri et al., 1987, 1988). 3.2. Numerical Solution. The resulting ordinary differential equation system is solved numerically by using a method based on a "shooting" strategy. Due to numerical stability problems usually found in these types of methods, the reactor is divided into sections, via a sequential-iterative scheme (Negri, 1987). Each of these sections (or modules) represents a split boundary condition, ODE system problem, and is solved independently by using the shooting technique. This method transforms the original boundary value scheme into an initial value problem and requires the assignment of trial values to the unknown variables at one end of the module. Please note that the shooting approach is necessary due to the countercurrent nature of the reactor (Figure 2). The fractioning of the reactor into modules creates intermediate pseudostreams that must also be assigned initial trial values in order to start the calculation sequence. The overall reactor scheme is solved sequentially; firstly, a direct-substitution scheme is applied to the convergence sequence of the pseudostream variables; secondly, Wegstein's convergence acceleration method is applied for the numerical solution of the system. 4. Comparison with Experimental Results Literature data concerning the behavior of direct reduction reactors operating under conditions similar to those found in industrial furnaces are scarce. Two relatively recent papers, by Kaneko et al. (1982) and Takenaka et al. (1986), presented studies of the direct reduction process at a pilot-plant level operating in a fashion similar to that of an industrial-scalefurnace. These pieces of work were performed by the same research and development group and could be considered complementary. In this section, a comparison between the predictions of both mathematical models (with and without the inclusion of the WGSR) and the pilot-plant data in the references above is performed. Dimensions and operating conditions of the pilot-plant-scale reactor in Kaneko et al. (1982) werc used for t,he calculations (Table I). Transport parametcrs are considered constant, and they are calculated at the reactor gas inlet conditions. In particular, the effective diffusivities for each component in the mixture are evaluated by using Wilke's equation at the bottom of
478 Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991 Table 11. Parameters for the Reaction Rates reaction
H2 reduction
CO reduction
WGSR
AI = 160 m/s A2 = 23 m/s A, = 30 m/s KP = 67672
KO2 = 0.965 Ki = 0.410 AI = 2700 m/s A2 = 25 m/s A, = 17 m/s KT = 103637 Ki = 1.283 K: = 0.546 Kk = 1.317
value employed E,= 92100 J/mol E; = 71200 Jjmol E, = 63600 J/mol AH1 = -12899 J/mol AH2 = 58443 J/mol AH3 = 15557 J/mol El= 113900 J/mol E2 = 73700 J/mol E , = 69500 J/mol AH1 = -47509 J/mol m2= 23831 J/mol AH3 = -19054 J/mol AHH, = -34611 J/mol
ref Tsar et al. (1976a,b) Weast (1976) Tsay et al. (1976a,b) Weast (1976) Weast (1976)
Table 111. Relative ConsumDtion of the Reducing Cases consumption, 5% results (at reactor outlet) H, co 26.1 37.3 exptl measurements 27.3 33.7 model including WGSR model without WGSR 30.3 28.9 ~
~
~~
~~~
Figure 3. Reactor axial profiles for a = 3.4 (a) gas temperature and degree of reduction and (b)hydrogen and carbon monoxide compositions. Key: (-) model with WGSR, (---) model without WGSR, and (A)experimental data (after Takenaka et al. (1986)).
the shaft furnace. The use of different concentration conditions (such as those at the center or top of the reactor) introduces no significant changes in the predictions. Kinetic data were obtained from the work by Tsay et al. (1976a) and is displayed in Table 11. This set of parameters is one of the most complete available in the open literature but was obtained for a narrow range of high temperatures. The remainder of the parameters used were obtain elsewhere. Firstly, axial profiles for the degree of reduction, gas temperature, and reducing agent concentrations are compared. Then, the degree of approximation to WGSR equilibrium conditions is studied. Finally, the effect of the gas-olid feed ratio on the degree of reduction and the gas utilization at reactor outlet are analyzed. The comparisons include the predictions previously obtained by Negri et al. (1987,1988) using the model without the WGSR. 4.1. Effects of the WGSR on the Reactor Behavior. The axial profiles predicted by the models with and without the WGSR are compared with the available pilot-plant data, and the results are shdwn in Figure 3 for the following case: gas-solid feed ratio a = 3.4, hydrogen to carbon monoxide feed ratio p = 1.4, and reducing to oxidizing gas feed ratio 6 = 14.8. For the set of conditions used, there exists an excess of gas. Consequently, as can be observed in the graph, the temperature level is high in practically the whole length of the furnace. Reduction takes place in the upper half of the vessel but without reaching full conversion at the bottom of the reactor [R({=O) = 0.9751. The pellets used in the pilot-plant experiments contained a small fraction of nonreducible compounds, associated, presumably, with impurities and alien chemical substances present in the original ore and not removed by the pelletizing process or added during it. This fact was taken into account in the modeling procedure: an amount of oxide equivalent to the nonreducible
fraction in the pilot-plant feeding material was substraded from the solid mass fraction of oxygen available in the ore composition entering the reactor model. This correction process does not affect in any way composition calculations in the gas phase and, consequently does not introduce artificial effects in the computation of the gas utilization. In Figure 3a, profiles for the gas temperature and the degree of reduction are displayed. The predictions of the model with no side reactions follow the same general trend observed in the experimental plots. However, the complete model (including the WGSR) is in much better agreement with the pilot-plant data. It can be observed that the temperature level and degree of reduction are higher in the predictions of the complete model, and thus in accordance with the experimental points, for most of the reaction zone within the furnace. The curves predicted by both models are coincident only for the region in the lower section of the furnace, where the solid reactant is depleted almost completely. Profiles corresponding to the predicted compositions for the reducing gases obtained by using the model including the WGSR match the available experimental points with better agreement (Figure 3b). It can be observed that the complete model predicts that CO relative consumption at the reactor outlet is higher than hydrogen relative consumption, while results from the model without the WGSR show the opposite behavior. With regard to this particular matter, only the predictions of the model including the WGSR are in good agreement with the experimental information. A summary of the comparison is presented in Table 111. Figure 4 displays the axial profiles corresponding to a = 2.5, p = 1.4, and 8 = 14.8. For this condition, because of a relatively lower reducing gas feed rate, the temperature level is lower than that in Figure 3, in particular, in the upper section of the reactor. Because of this fact, temperatures observed in the pilot-plant reactor are out of the range where the available kinetic data are valid for a reactor length of about 80% of the total furnace height. For this reason, model results have less good agreement with pilot-plant data. Nonetheless, the model including the WGSR produces the results with the values closest to the available experimental information. The curves for the degree of reduction and the gas-phase temperature as functions of the axial coordinate are presented in Figure 4a. The degree of reduction of the solid at the reactor outlet predicted by the complete model
Ind, Eng, Chem, Res., Vol. 30, No. 3,1991 479 12,
06
Table IV. Effect of the WGSR on the Bulk Gas-Phase Concentrations operating
conditions a = 3.4 p = 1.4 8 = 14.8
Q
-
0
0
-
00
05 (01
c
0
00
05
t
lo
(0)
Figure 4, Reactor axial profiles for (Y = 2.5 (a) gas temperature and degree of reduction and (b) hydrogen and carbon monoxide compositions. Key: (-) model with WGSR, (- - -) model without WGSR, and (A)experimental data (after Takenaka et al. (1986)).
closely matches the experimental value. However, the predicted profile along the reactor for the same variable is less smooth than the experimental curve. WGSR model predictions for temperature closely follow the corresponding experimental curve in the lower section of the reactor. For the upper section, predictions fall below the experimental curve, in a way that surely affects the predicted rate of growth of the degree of reduction, rendering the rate slower in this portion of the furnace. As was the case for a = 3.4,values for Og and R predicted by the model without the WGSR are even lower and thus less coincident with the experimental pilot-plant data. The axial profiles for the composition of H2 and CO are displayed in Figure 4b. The general shape of the curves predicted by both models is similar to that observed for the degree of reduction, with a region of sharply changing values of yi located in the lower section of the reactor (f < 0.4). Coincidentally, the predicted degree of reduction changes very rapidly in this zone, and the temperature level of the reactor is maximal in this portion. The model with the WGSR has a better matching of the trends, and a close agreement with the experimental points near the reactor solid outlet, with certain discrepancies, however, in the upper portion of the furnace. The importance of the inclusion of the WGSR into the reactor model is considered here; model predictions are compared with pilot-plant experimental data. In order to perform the analysis, the ratio Q / K , is used, where Q at any given temperature is
with all mole fractions measured in the bulk gas phase, and K, is the WGSR equilibrium constant expressed in terms of mole fractions at the same temperature. Although the WGSR takes place within the pellet, and mostly in the reduced iron layer, following the Q / K , ratio in the bulk gas phase gives an indication of the degree of approximation to the equilibrium. This, in turn, indicates how important the inclusion of the effect into a rigorous reactor model is. Because both Q and K, are formally equal, the only difference being that K, is evaluated at equilibrium conditions, values of the ratio Q / K , near to one will indicate that the concentrations of the species in the gaseous phase are close to equilibrium. Table IV displays Q / K , values from pilot-plant data and model predictions, both with and without the inclusion of the WGSR. The results are shown for two different operating situations of the pilot-plant reactor. They correspond to the reactor feed conditions previously presented
2.5
= 1.4 6 = 14.8
p
0
axial position (l) 1.00 0.84 0.66 0.30 0.00 1.00 0.84 0.66 0.30 0.00
exptl 0.106 1.158 0.925 0.963 1.001 0.025 0.973 0.964 0.955 1.001
(QlK,) with WGSR 0.172 0.979 0.984 0.998 1.078 0.000 0.637 0.575 0.982 1.070
without WGSR 0.133 0.522 0.688 0.977 1.002 0.067 0.550 0.490 0.625 1.002
in Figures 3 and 4. In both cases,the reducing gas mixture enters the reactor ({ = 0) at a state close to equilibrium. The ratio Q/K, obtained experimentally from pilot-plant data keeps close to one while the reducing gas travels the reactor. At a point near the top end of the furnace, a drastic drop in the Q / K , ratio is observed. This effect is due to the sharp decrease in temperature (and thus to a sharp increase in K,) typical of this vessel zone where the solid is entering the reactor at a much lower temperature. The behavior of Q / K , observed from the experimental data for most of the reactor length is a consequence of the existence of the WGSR. This reaction is able to sustain the equilibrium conditions in the gas bulk phase in spite of the fact that reduction kinetics favors a consumption of hydrogen larger than that of carbon monoxide. With regard to model calculations, the scheme with WGSR predicts exactly the same behavior for the case of a = 3.4. For CY = 2.5, the trend is the same. However, an early drop of the Q/K, ratio is observed at 1 = 0.66. (The reasons for this discrepancy were presented previously, when discussing the results in Figure 4.) Conversely, the model without the WGSR fails to predict the behavior observed following the pilot-plant data. The ratio Q / K , decreases when f increases, even for reactor zones close to the gas inlet, because of a faster hydrogen consumption. The drop in Q/K, in the predictions with this model is not balanced by the gaseous species interconvenion process associated with the inclusion of the WGSR. 4.2. Influence of the Gas-Solid Feed Ratio on Reactor Conversion. In this section, the predictions of both models are analyzed when one of the most important parameters of the reacting system is changed: the gas-solid feed ratio. The goal of this analysis is the comparison of predictions and experiments for a wide range of values of a. In Figure 5, the results are shown for the comparison associated with the two variables directly related to the conversion of the process: (i) the degree of reduction [R({=O)]and (ii) the gas utilization [s(f=l)]. For the set of parameters adopted ($ = 1.11;p = 1.4; 6 = 14.8),a region of total conversion is reached, with a maximum limiting value for R+O) = 0.975, according to the correction introduced to account for the presence of a nonreducible fraction mentioned in section 4.1. For values of a higher than 2.9,predictions of both models are coincident and in good agreement with the available experimental data. For decreasing values of a,a point is reached where the degree of reduction starts to decrease, thus resulting in a less pronounced increase in the gas utilization value. The latter starts decreasing after reaching a maximum value that represents a relatively optimal gas-solid feed ratio. The maximum in d{=l) predicted by the model without WGSR is observed at higher values of CY. These results are in good agreement
480
Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991
07 -
I
I
25
30
I a
(0)
0.3
p = 1.4
02
I
I
I
25
30
J a
(b)
Figure 5. Effect of the gas-solid feed ratio on (a) the degree of reduction and (b) the gas utilization. Key: (-) model with WGSR, (- - -) model without WGSR, and (A) experimental data (after Takenaka et al. (1986)).
with the trend observed previously, when equal or higher degrees of reduction are predicted by the model with WGSR with respect to the model with no side reactions. This trend has been discussed already, for the case of the axial profiles in Figures 3 and 4. The results in Figure 5 show a substantial agreement between predictions and experiments for high values of a,as wm discussed in section 4.1. The accordance is poorer for lower gas-solid feed ratios. When comparing the behavior of both models against the experimental data, predictions show better agreement across the whole range of a values in the case of the model including the WGSR. Differences between the predictions of the model including the WGSR and the experimental data at the reactor outlets are lower than 6% for the range of values 2.4 < a < 3.4. This range includes virtually all of the operating values of practical industrial interest. If one takes into account the fact that the model developed has no adjustable parameters and that the available kinetic data are strictly valid for a limited range of the reactor operating variables, then the overall agreement between the complete model predictions and the experimental data is good, in particular for the industrial operating conditions. 5. Conclusions A previous model for the mathematical representation
of a moving-bed reactor for the reduction of iron oxides using a mixture of reducing gases is generalized, in order to include the water gas shift reaction. The resulting model is based on a heterogeneous, onedimensional, nonisothermal, steady-state representation for the furnace and a three-moving-front reduction scheme for the pellet. The water gas shift reaction is assumed to take place inside the sponge-iron layer, once the latter is
formed by the reduction process. The WGSR is considered in equilibrium, because of high temperatures and reduced-iron catalytic effects. A nonlinear system of algebraic equations is obtained in order to compute both the reduction rates at the moving interfaces and the concentrations of reactants and producta within the pellet. No explicit solution scheme, as in the case for the model without WGSR, is obtained. Thus, an iterative method is employed in order to obtain a numerical solution for the system. For the integration of the reactor model and because of numerical stability problems of the shooting method when solving the system of ODE’S with split boundary conditions, a modification of the shooting technique is developed. The reactor is divided in sections, and pseudostreams linking them are introduced and solved by using a trial-and-error method. A Wegstein acceleration scheme is used for the convergence of the iterative procedure. A comparison is performed between model predictions with and without the WGSR and experimental results in the literature for a pilot-plant-scale reactor. The behavior of model predictions is in reasonably good agreement with the available pilot-plant data. The model including the WGSR is the representation that more closely matches the experiments. The WGSR model predicts axial profiles for the degree of reduction and the gas-phase temperature in good agreement with the experimental data, for gas-solid feed ratios larger than 2.9. Less good accordance is observed in the predictions of values for the variables along the reactor when a < 2.9. In spite of this fact, WGSR model predictions at the reactor outlets closely match the experimental data in the range of the industrial operating conditions.
Acknowledgment We are grateful to CONICET and to UNL for their financial support. Portions of this work were funded by CONICET Grant P.I.D. 3/906001/85.
Nomenclature a = 3(1 - c)/rp, specific surface area, l/m A = preexponential factor in reaction kinetic expression, m/s C = molar concentration, moi/m3 C, = molar specific heat, J/(mol K) = specific heat, J/(kg K) = reactor diameter, m D k =) diffusion resistance of the kth layer, s/m Da = hkRCR(1 + KR-~)/CO, Damkohler number, dimensionless De = effective diffusion coefficient, mz/s E = activation energy in the reaction kinetic expression,J/mol f, = relative content of oxygen removed in the jth step of reduction, dimensionless F = gas-film resistance, s/m G = molar specific flow rate for the gas phase, mol/(m2 s) h = gas-to-solid heat-transfer coefficient, W/(m2 K) AH, = enthalpy change of the ith chemical reaction at the jth interface, J/mol k = reaction rate constant, m/s kg = gas-to-solid mass-transfer coefficient, m / s K = equilibrium constant, dimensionless L = length of reactor, m M = molecular weight, kg/mol N = molar flux, dimensionless Q = concentration quotient, as defined in eq 18,dimensionless r = radius, m R = fractional reduction, dimensionless S = mass-specific flow rate for the solid phase, kg/(m* s)
2
Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991 481
T = temperature, K
U = overall heat-transfer coefficient through reactor wall, W / b ZK) w = mws fraction in the solid phase, dimensionless xi = mole fraction for gas component i, dimensionless yi = xi/xR, normalized mole fraction for gas component i, dimensionless z = axial coordinate, m Greek Letters a = G o ~ R M o x / S o ~ gas-solid O,,, feed ratio, dimensionless i3 = G0Cp,/Socp,,heat capacity gas-solid feed ratio, dimensionless y = LkgaCR/Go,gas-to-solid mass-transfer coefficient, dimensionless S = (yoH2+ yEo)/(y&o + yEo,), reducing gas to oxidizing gas feed ratio, dimensionless e = fractional void volume in the reactor, dimensionless t = z/L, axial coordinate, dimensionless 9 = gas utilization, dimensionless 0 = T/TR, temperature, dimensionless I ( . = chemical reaction resistance at the jth interface, s/m = kinetic and transport resistance matrix (for the model including the water gas shift reaction, as defined in eq A,12), s/m Y = stoichiometric coefficient, dimensionless f = r/rp, pellet radius, dimensionless p = yb2/y&-,, hydrogen to carbon monoxide feed ratio, dimensionless TI = ahL/ G°Cp,, gas-to-solid heat-transfer coefficient, dimensionless Tz = 4UL/dG°Cp , overall heat-transfer coefficient through the reactor wan, dimensionless 4 = ratio of reactant to product diffusion coefficients, dimensionless CP = (-AH)xR/Cp TR,heat of reaction, dimensionless = ratio of reaclant to product mass-transfer coefficients, dimensionless w = reaction rate per unit surface area, mol/(m2.s) 6 = reaction rate per unit volume, mol/(m3 s) Q = o / C R x R k R ( l + K R 9 ,reaction rate per unit surface area, dimensionless
with the boundary conditions
1
where wI,j = k ~j (,C ,j
Appendix Mass and energy balances, when applied to each chemical species i in each layer of solid within the pellet, with i = H2, HzO, CO, COz,render the following set of ordinary differential equations:
i = 1, 29 3
(A.9)
In eqs A.3-A.9 above, subindex I stands for the reduction reaction either with Hz(when I = Hz, i = Hzor H20)or with CO (when I = CO, i = CO or COP). The stoichiometric coefficients of the reacting system are shown in the table below:
*
Subscripts a = at atmospheric conditions A = for gaseous reactant A (H2 or CO) B = for gaseous product B (H20 or C02) b = in the bulk phase - in the gas phase - for the ith gas species j = for the jth reaction at the external surface of hematite 0' = l), magnetite 0' = 2), and wustite 0' = 3) ox = for the oxygen p = at the external surface of the particle R = at the reference state: TR = lo00 K, and xR = x i 2 + xeo s = in the solid phase w = for the water gas shift reaction Superscripts (k)= in the kth solid product layer: magnetite (k = l),wuetite (k = 2), and iron (k = 3) o = at the inlet condition
- Cg,j/KI,j )
i YiHp
hco yiw
Hz -1 0 1
HzO
co
-1
-1 -1
1 0
0
co, 0 1 1
When the equilibrium condition for the WGSR is applied, the following equation must be satisfied in the sponge-iron layer:
Kw = cfi;cf!!A,/cfl;ocf!!A
(A.lO)
With this assumption, the equations in (A.2) become homogeneous; the system is not any longer formed by independent equations, and one of them must be substituted by eq A.10. The expression for the dimensionless reaction rate obtained by solving the system of equations above can be written in a compact form as follows:
[ where
=
Ind. Eng, Chem. Rea. 1991,30, 482-490
482
with i = A (H2 or CO), B (H20 or COP),and KI, j = [k1,](1 + 1/KI, j)ttl-'
(A.13)
(A.15)
(A.18) Registry No. Fe, 7439-89-6; hematite, 1317-60-8.
Literature Cited Brown, K. M. Computer Oriented Algorithms for Solving Systems of Simultaneous Nonlinear Algebraic Equations. In Numerical Solution of Systems of Nonlinear Algebraic Equations; Byme, G. D., Hall, Ch. A., Eds.; Academic Press Inc.: New York, 1973;pp 281-348. Hara, Y.; Sakawa, M.; Kondo, S. Mathematical Model of the Shaft Furnace for Reduction of Iron-Ore Pellet. Tetsu to Hagane 1976, 62,315-323. Hughes, R.;Kam,E. K.T. Direct Reduction of Iron Ore in a Moving-Bed Reactor: Analysed by Using the Water Cas Shift Reaction. In Chemical Reaction Engineering-Boston; Wei, J., Georgakii, Ch., Eds.;ACS Symposium Series; American Chemical Society: Washington, DC, 1982;pp 29-38. Kam, E.K. T.; Hughes, R. A Model for the Direct Reduction of Iron Ore by Mixtures of Hydrogen and Carbon Monoxide in a Moving Bed. Trans. Inst. Chem. Eng. 1981,59,196-206. Kaneko, D.; Takenaka, Y.;Kimura, Y.; Narita, K. Production of Reduced Iron by Model Plant of Shaft Furnace. Trans. ISIJ 1982,22,88-97.
Negri, E. D. Doctoral Thesis, Universidad Nacional del Litoral, Santa Fe, Argentina, 1987. Negri, E. D.; Alfano, 0. M.; Chiovetta, M. G. Heat and Mass Transfer in the Modeling of Noncatalytic Moving Bed Reactors and its Application to Direct Radudion of Iron Oxides: A Review. Lat. Am. J. Heat Mass Transfer 1985,9,86129. Negri, E.D.;Alfano, 0. M.; Chiovetta, M. G. Direct Reduction of Hematite in a Moving Bed. Comparison between One- and Three-Interface Pellet Model. Chem. Eng. Sci. 1987, 42, 2472-2475. Negri, E. D.; Alfano, 0. M.; Chiovetta, M. G. Optimal Operating Conditions of the Direct Reduction of Hematite in a Shaft Furnace. Lat, Am, Appl, Res. 1988,18, 93-104. Perry, R,H., Chilton, C. H., Eds.Chemical Engineer's Handbook, 5th ed.; McGraw-Hill: Tokyo, 1973. Reid, R, C,; Prautnitz, J. M,; Shemood, T. K,The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Sen Gupta, A.; Thodos, G. Mass and Heat Transfer in the Flow of Fluids Through Fixed and Fluidized Beds of Spherical Particles. AIChE J. 1962,8,608-610. Takenaka, Y.; Kimura, Y.; Narita, K.; Kaneko, D. Mathematical Model of Direct Reduction Shaft Furnace and Its Application to Actual Operations of a Model Plant. Comp. Chem. Eng. 1986,10, 67-75. Tsay, Q. T.; Ray, W. H.; Szekely, J. The Modeling of Hematite Reduction with Hydrogen Plus Carbon Monoxide Mirtures. Part I-The Behavior of Single Pellets. AZChE J. 1976a, 22, 1064-1072, Tsay, Q. T.; Ray, W. H.; Szekely, J. The Modeling of Hematite Reduction with Hydrogen Plus Carbon Monoxide Mixtures. Part 11-The Direct Reduction Process in a Shaft Furnace Arrangement. AIChE J , 1976b,22, 1072-1079. von Bogdandy, L.; Engell, H. J. The Reduction of Iron Ores; Springer Verlag: Berlin, 1971. Wagner, C. Adsorbed Atomic Species as Intermediates in Heterogeneous Catalysis. Adu. Catal. 1970,21,323-381. Weast, R. C., Ed. Handbook of Chemistry and Physics, 57th ed.; CRC Press: Cleveland, 1976. Yagi, J.; Szekely, J. The Effect of Cas and Solids Maldistribution on the Performance of Moving-bed Reactors: The Reduction of Iron Oxide Pellets with Hydrogen. MChE J. 1979,26,800-810. Yanagiya, T.; Yagi, J.; Omori, Y.Reduction of Iron Oxide Pellets in Moving Bed. Ironmaking Steelmaking 1979,3,93-100. Yu, K. 0.; Gillis, P. P. Mathematical Simulation of Direct Reduction. Met. Trans. B 1981,12B,111-120.
Received for review November 21, 1989 Reuised manuscript receiued July 23, 1990 Accepted August 1,1990
Development of a Multivariable Forward Modeling Controller Kelvin T.Erickson* Department of Electrical Engineering, University of Missouri-Rolla, Rolla, Missouri 65401
Robert E. Otto Monsanto Company, St. Louis, Missouri 63198
The Forward Modeling Controller (FMC), a recently developed model-based predictive digital controller for single-input, single-output processes, is extended to multiinput, multioutput processes. The multivariable FMC is a promising approach to the control of complex industrial processes with many inputs and outputs. The theory presented in this paper includes stability analysis plus other features necessary for robustness in industrial control. The controller has only two types of adjustments: a robustness/performance setting for each controller variable and the controller sample interval. The performance of the multivariable FMC is demonstrated on a distillation column simulation and is compared with the performance of the Dynamic Matrix Controller. Introduction complex multiinput, multioutput ( ~ 1 ~chemical 0 ) we industrial processes such distillation *Author to whom correspondence should be addressed.
difficult to control. Manv of these Drocesses have large dead times and unusual djllamics and are often affeccd by persistent disturbances. Automatic control of these Drocesses is usually troublesome due to the interaction inherent in the process, requiring highly skilled operators to maintain acceptable product quality.
0888-6886/91/2630-0482$02.60/0Q 1991 American Chemical Society
