Effect of Mass Transfer and Bulk Flow on the Zinc Oxide Reduction by

Thus, the effect of mass transfer and bulk flow can be considered in on the TG scale. Then a mixed control model was developed which contains the effe...
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Ind. Eng. Chem. Res. 2002, 41, 2630-2636

Effect of Mass Transfer and Bulk Flow on the Zinc Oxide Reduction by Methane H. Ale Ebrahim* and E. Jamshidi Department of Chemical Engineering, Amir-Kabir University (Tehran Polytechnic), Tehran 15875-4413, Iran

Zinc oxide reduction with methane can introduce a new zinc-producing technology. The advantages of this method with respect to conventional carbothermic furnaces are lower operating temperatures, a simple indirect condenser, and copreparation of synthesis gas. The kinetics of ZnO reduction by CH4 at the chemical control regime has been studied on the thermogravimetric (TG) scale and at moderate temperatures extensively (Ale Ebrahim, H.; Jamshidi, E. Trans. Inst. Chem. Eng. 2001, 79A, 62-70). On the industrial scale with large ZnO pellets, mass transfer and bulk flow will be effective even at moderate temperatures. Therefore, a successful mathematical model for the design of such industrial plants is needed. In this work the ZnO + CH4 reaction at high temperature (1000 °C) was studied, where the kinetic resistance decreased considerably. Thus, the effect of mass transfer and bulk flow can be considered in on the TG scale. Then a mixed control model was developed which contains the effects of external mass transfer, bulk flow, and chemical reaction at the pellet surface. The obtained experimental data from thermogravimetry in this work were in good agreement with the mixed control model predictions. Introduction Coproduction of zinc and synthesis gas by the zinc oxide reduction with methane was considered in recent researches.2,3 This method can be the starting point for introducing a new zinc production technology in metallurgy. On the other hand, zinc can be used as a clean fuel for producing electricity in fuel cells or a hydrogen generator by water splitting reaction.4,5 In each case ZnO (a product of the above reactions) must be recycled to the system for reacting again with methane.4,5 The preferences of this new technique with respect to the conventional zinc production methods are predicted as follows: 1. The operating temperature can be decreased to about 850 °C6 (with respect to about 1200 °C of cokebased plants7) because of the high thermodynamic reactivity of methane. Therefore, this method seems to possess fewer problems in external heat transfer and refractory materials. On the other hand, high temperature requirements of coke-based plants are the reason for the introduction of new proposals using solar energy for zinc production.8,9 2. In the moderate temperatures (about 850 °C), this method produces zinc in the liquid form. However, because of the high volatility of molten zinc, the liquid zinc recovery was about 20% in a single pellet thermogravimeter,1 but in an industrial packed-bed reactor, one can expect the more liquid zinc recovery. Moreover, the vaporized zinc can be condensed in an indirect condenser simply, because the gaseous products of the reaction (CO + 2H2) are not oxidizing. However, a common coke-based method such as the imperial smelting process needs a quick and direct lead splash condenser (because of the reduction of ZnO with CO and the existence of CO2) for preventing zinc vapor reoxidation.7 The rate of molten lead circulation to these condensers is very high (400 tons of Pb/ton of Zn), which creates some technical problems.10 * Corresponding author. E-mail: [email protected].

3. The valuable byproduct of this method is synthesis gas (CO + 2H2) from interaction of methane with oxygen of ZnO.1 These gaseous products may be used in petrochemical industries such as methanol production plants.2 Therefore, it seems that the catalytic steam reformer units of such petrochemical plants can be omitted. Another possible application of these gaseous products is as reducing agents for steel production from direct reduction methods. 4. This method can be used as a clean technology for zinc production because the gaseous products of the reaction may be the raw material for preparing petrochemical products. On the other hand, common pyrometallurgical methods produce a large amount of greenhouse gas (CO2), which discharges directly to the atmosphere.10 The complete kinetic study of zinc oxide reduction with methane at moderate temperatures was presented elsewhere.1 The kinetics of this reaction was studied on a thermogravimetric (TG) scale by simultaneous thermogravimetry of the pellet and mass spectroscopy of the gaseous products.1 It was found that under operating conditions (840-930 °C, pellet diameter 5 mm, and gas flow 130 cm3/min) the resistance of mass transfer and bulk flow is negligible.6 However, on the industrial scale with large pellets, mass transfer and bulk flow become effective even at these moderate temperatures, because the mass-transfer coefficient is nearly proportional to the inverse of the pellet radius. Therefore, it is essential to develop a mathematical model for these effects and to check the model with some suitable experimental data. In this work, a high-temperature condition (1000 °C) was selected in order for the mass-transfer and bulk flow effects of the ZnO + CH4 reaction on the TG scale to appear. When the temperature is increased, the rate constant increases severely and therefore the reaction becomes mixed control (instead of pure kinetic control at moderate temperatures). Then a suitable mixed control mathematical model was proposed for this

10.1021/ie010604r CCC: $22.00 © 2002 American Chemical Society Published on Web 05/02/2002

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inert) is switched to the thermogravimeter. At this isothermal temperature, the weight of the pellet is plotted versus time. From simultaneous mass spectrometric (MS) analysis of TG outlet gases in the previous work,1 it was proved that the gaseous products of the ZnO + CH4 reaction are carbon monoxide and hydrogen. Therefore, the reaction at high temperatures is as follows:

ZnO + CH4 f Zn(g) + CO + 2H2

(1)

Now at high temperatures (above the boiling point of zinc), the experimental conversion-time profile can be obtained from TG results as follows:

X(t) ) Figure 1. Flow diagram of the system for ZnO + CH4 reaction experiments.

reaction. Finally, by comparison of experimental data with the model predictions, a good agreement was obtained. Therefore, this mixed control model can be used as a design basis for industrial plants (with large pellets) at moderate or high temperatures. Experimental Section The flow diagram of the system for experimental study of the ZnO + CH4 reaction is presented in Figure 1. The system, which consists of a thermogravimeter from Rheometric Scientific and a mass spectrometer from Leda Mass, has been used for complete kinetic study of the ZnO + CH4 reaction at moderate temperatures.1 In the present work, a ZnO pellet is placed in a tungsten wire basket in a thermogravimeter and the system is heated to the desired temperature under an inert gas stream (gas 1). Then the isothermal period begins, and the reducing gas (a mixture of CH4 and

W0 - W(t) W0

(2)

In this work, zinc oxide is a guaranteed reagent (Merck Art. 8849) and methane is from Air Products with 99.95% purity. The inert gases (helium and argon) are also high-purity grades. It is usual to use nonporous pellets for the kinetic study of gas-solid reactions.11 Therefore, a low final porosity in ZnO pellets is achieved by high-pressure hydraulic pressing (2000 bar) and complete sintering. Cylindrical pellets with heights equal to their diameters were produced, which are similar to spherical pellets.12 The porosity of such pellets from the apparent density method was about 50%. Then these pellets were sintered at 1000 °C for 5 h to increase their mechanical strength and reduce their porosity. The porosity of sintered pellets from a mercury porosimetry was about 1%. Now the results of the high-temperature TG experiments are described. First of all, the rate constant at 1000 °C must be determined. Therefore, it is necessary to design a kinetic control experiment even at this high temperature. This was accomplished by using helium instead of argon as

Figure 2. TG results for a kinetic control experiment of the ZnO + CH4 reaction at 1000 °C, 40% CH4/He, r0 ) 0.15 cm, and Q ) 226 cm3/min.

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Figure 3. TG results for a mixed control experiment of the ZnO + CH4 reaction at 1025 °C, 26.3% CH4/Ar, r0 ) 0.235 cm, and Q ) 125 cm3/min.

Figure 4. TG results for a mixed control experiment of the ZnO + CH4 reaction at 1000 °C, 40% CH4/Ar, r0 ) 0.272 cm, and Q ) 100 cm3/min.

the diluant gas, where molecular diffusion increased between 3 and 4 times. Also in the kinetic control experiment, the small pellet and high gas flow rate were used in order to increase the mass-transfer coefficient. A TG curve of this kinetic control experiment is presented in Figure 2. This figure is for a ZnO sintered pellet with a 0.15 cm radius at 1000 °C. Then the mixed control experiments were designed with argon diluant, larger pellets, and low gas flow rates. TG curves of these mixed control experiments are presented in Figures 3-5. Figure 3 is for 26.3% CH4/ Ar, r0 ) 0.235 cm, and Q ) 125 cm3/min at 1025 °C. In

Figures 4 and 5, the temperature, methane mole fraction, and inlet flow of gases were fixed on 1000 °C, 40% CH4/Ar, and Q ) 100 cm3/min. The initial radii of pellets in Figures 4 and 5 are 0.272 and 0.345 cm, respectively. Now the experimental conversion function profiles (refer to Ale Ebrahim and Jamshidi1) for spherical pellets and two kinetic control and mixed control experiments are shown in Figure 6. If a conversion function profile deviates upward with the progress of reaction, the mass transfer will be effective in the overall rate.13 This phenomenon is apparent in run 27 (argon test) in Figure 6, while run 24 (helium test) is a

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Figure 5. TG results for a mixed control experiment of the ZnO + CH4 reaction at 1000 °C, 40% CH4/Ar, r0 ) 0.345 cm, and Q ) 100 cm3/min.

1. The porosity of the sintered semispherical pellets is negligible.1 2. The reaction is irreversible and first-order with respect to methane because the Langmuir-Hinshelwood relation for the ZnO + CH4 reaction1 approaches a first-order dependency at high temperatures of this work. 3. The pseudo-steady-state approximation is valid. 4. On a TG scale, the CH4 stream is in excess with respect to the solid reactant. Thus, the bulk gas concentration of methane is constant. 5. The system is isothermal. Now the molar flux of methane is expressed as follows:14 Figure 6. Conversion function profile for two kinetic control (He, r0 ) 0.15 cm, and Q ) 226 cm3/min) and mixed control (Ar, r0 ) 0.272 cm, and Q ) 100 cm3/min) experiments for the ZnO + CH4 reaction at 1000 °C and 40% CH4.

nearly straight line. Therefore, the rate constant at 1000 °C was determined (about 150 cm/min) from the slope of this kinetic control line.6 More evidence for the kinetic control mechanism in run 24 (helium test) is obtained from a comparison of kinetic resistance (6.67 × 10-3 min/cm) with mass transfer and bulk flow resistances (1.13 × 10-3 min/cm).6 Mathematical Modeling For mathematical modeling of the system, the nonequimolar counterdiffusion nature of reaction (1) must be considered. This means that at high temperature 1 mol of gas (methane) moves toward the pellet and 4 mol of gas moves from the pellet surface to the gas stream. Thus, a net bulk flow (3 mol from the pellet surface) is created, which introduces additional resistance. Now the mixed control model with the effect of external mass transfer, bulk flow, and surface reaction is developed. The following assumptions are used in the mathematical modeling:

NCH4 ) kmC(yAb - yAs) + yAb(NCH4 + NZn(g) + NCO + NH2 + NAr) (3) The first term of the right-hand side of eq 3 is mass transfer due to the concentration gradient, and the second term is mass transfer by bulk flow. From the stoichiometric of reaction (1), it is apparent that

1 NCH4 ) -NZn(g) ) -NCO ) - NH2, NAr ) 0 (4) 2 Therefore, from eqs 3 and 4 the molar flux of methane can be determined as

NCH4 )

kmC(yAb - yAs) 1 + 3yAb

(5)

From the mole fraction of methane at the pellet surface, the rate of reaction is as follows:

Rrx ) kCyAs

(6)

Because the mole fraction of methane at the pellet surface is an unknown quantity, it is usual to relate yAs

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to other system parameters. From the equality of the molar flux of methane (eq 5) with the rate of reaction (eq 6), the mole fraction of methane at the pellet surface can be computed as

yAs )

kmyAb

(7)

k(1 + 3yAb) + km

Therefore, when eq 7 is inserted into eq 6, the rate of reaction is expressed as follows:

Rrx )

CyAb 3yAb 1 1 + + km km k

(8)

The denominator of eq 8 from left to right consists of mass-transfer resistance, bulk flow resistance, and surface reaction resistance. If the reaction accomplishes under the boiling point of zinc, it will be 2yAb for the bulk flow term in eq 8. Now the mass-transfer coefficient can be expressed from Ranz and Marshall correlation:11

km )

( )( )]

[

DA 2ruFg 2.0 + 0.6 2r µg

1/2

1/3

µg FgDA

)

a1 a2 (9) + r xr

When km from eq 9 is inserted into eq 8 and from the equality of the rate of reaction with the rate of pellet shrinkage, the following equation can be derived:

CyAb dr ) -FB 1 + 3yAb 1 dt + a 1 a2 k + r xr

(10)

By integration of eq 10 from r ) r0 at t ) 0, the following radius-time relation is obtained:

t)

[ (

2FB(1 + 3yAb) CyAba24

a13 ln

a1 + a2xr

a1 + a2xr0

)

+

a1a22 (r - r0) - a12a2(xr - xr0) 2 FB(r - r0) a23 (rxr - r0xr0) (11) 3 kCyAb

]

Equation 11 is similar to the final equation of Guger and Manning,15 which was derived by another method for the ZnO + CO reaction at high temperature with 1 mol of bulk flow effect. Also Guger and Manning predicted the experimental weight-time behavior of the ZnO + CO system by such an equation.15 The radius of a spherical pellet can be related to the solid conversion as follows:

r ) r0(1 - X)1/3

(12)

Therefore, an implicit relation for conversion-time behavior in the mixed control model is obtained by inserting eq 12 into eq 11. This mixed control model can also be extended for considering the internal reaction of a porous pellet. The modified grain model with a simple approximate solution for its complicated equations16 can be used as a

Table 1. Parameters Used for the ZnO + CH4 Reaction Simulation run temperature (°C) DA (cm2/min) yAb u (cm/min) Fg (g/cm3) µg (g/cm‚min) FB (gmol/cm3) C (gmol/cm3) k (cm/min) r0 (cm)

26

27

28

1000 39.5 0.400 95 2.62 × 10-4 3.34 × 10-2 6.72 × 10-2 8.62 × 10-6 150 0.345

1000 39.5 0.400 95 2.62 × 10-4 3.34 × 10-2 6.72 × 10-2 8.62 × 10-6 150 0.272

1025 41.0 0.263 119 2.90 × 10-4 3.55 × 10-2 6.72 × 10-2 8.46 × 10-6 490 0.235

framework for this extension. The effect of bulk flow on the partial differential equations of a porous pellet model has been expressed by Sohn and Bascur.17 Comparison of the Results The values of all of the parameters used in the simulation for the ZnO + CH4/Ar reaction are presented in Table 1. The Wilke-Lee equation18 was used for the estimation of molecular binary diffusivities. Moreover, because of the production of saturated zinc vapor from the reaction, the diffusion of CH4/Ar in the gas stream and CH4/Zn(g) near the pellet surface were considered as consecutive resistances for computing DA. For the diffusion properties of zinc vapor, some information presented by Rao19 was used. Because it is essential to design a gas-solid reaction experiment under excess reacting gas conditions,11 the mole fraction of methane in the bulk gas stream (yAb) was determined from methane and argon flowmeters. The velocity of the gas phase (u) was evaluated from flowmeters and an ideal gas assumption. The gas flow is proportional to the absolute temperature and the inverse of the absolute pressure. When high-temperature gas flow was divided by the cross-sectional area of the TG value (4.9 cm2), the gas velocity was determined. The density and viscosity of a binary gas mixture (CH4/Ar) were computed by relations presented by Bird et al.20 and using the handbook quantity for each component. The molar density of ZnO (FB) is equal to its handbook density (5.47) divided by its molecular weight (81.4). The total concentration of gases (C) was computed from the ideal gas law at the TG condition (0.9 atm and 1273 or 1298 K). The rate constant (k) at 1000 °C was determined from the previous kinetic control test (run 24). The rate constant at 1025 °C was estimated from two kinetic control tests with negligible liquid zinc at the pellet surface. These two tests are a reduction at 930 °C (from the previous paper1) and run 24 at 1000 °C. By assumption of Arrhenius dependency between 930 °C (k ) 4.2 cm/min) and 1000 °C (k ) 150 cm/min), the activation energy of the reaction was estimated (155 kcal/gmol). Therefore, by this activation energy and the result of run 24, the rate constant at 1025 °C was estimated (490 cm/min). Finally, the initial radius of the pellet was estimated from the equality of the volume of the cylindrical pellet (with measured height and diameter) with a hypothetical spherical one. Now when eqs 11 and 12 are applied, the conversiontime profiles for the ZnO + CH4 reaction are predicted.

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Figure 7. Prediction of the conversion-time profile of the ZnO + CH4 reaction from models and comparison with experimental data for 1025 °C, 26.3% CH4/Ar, r0 ) 0.235 cm, and Q ) 125 cm3/ min.

Figure 10. Prediction of the conversion-time profile of the ZnO + CO reaction from models and comparison with experimental data of Guger and Manning15 at 1200 °C and r0 ) 0.501 cm. Table 2. Value of Parameters Used for the ZnO + CO Reaction temperature (°C) DA (cm2/min) yAb u (cm/min) Fg (g/cm3) µg (g/cm‚min) FB (gmol/cm3) C (gmol/cm3) k (cm/min) r0 (cm)

Figure 8. Prediction of the conversion-time profile of the ZnO + CH4 reaction from models and comparison with experimental data for 1000 °C, 40% CH4/Ar, r0 ) 0.272 cm, and Q ) 100 cm3/ min.

Figure 9. Prediction of the conversion-time profile of the ZnO + CH4 reaction from models and comparison with experimental data for 1000 °C, 40% CH4/Ar, r0 ) 0.345 cm, and Q ) 100 cm3/ min.

The comparisons between these predictions and experimental data are shown in Figures 7-9. These figures are for zinc oxide pellets with 0.235, 0.272, and 0.345 cm radii, respectively. Moreover, the conversion-time profile from the kinetic control model1 is presented for comparison. As Figures 7-9 show, the mixed control model predicts the experimental data with a good accuracy and much better than the kinetic control model. The small overprediction of the mixed control

1200 73.8 1 3036 2.32 × 10-4 3.30 × 10-2 6.72 × 10-2 8.28 × 10-6 132 0.501

1300 83.4 1 6516 2.17 × 10-4 3.54 × 10-2 6.72 × 10-2 7.75 × 10-6 384 0.503

model is apparently due to the effect of the tungsten basket cell (about 20 mesh) on the gas flow. Also much better agreement between the mixed model and experimental data may be obtained by more accurate estimation of the diffusion coefficient of methane (DA) in the system. The ability of the model for predicting another reaction with a bulk flow effect was checked in the ZnO + CO system (Guger and Manning work15). For this system, the term 1 + 3yAb in eq 11 must be replaced by 1 + yAb. The values of parameters used in the simulation of the ZnO + CO reaction are presented in Table 2. The diffusivity of Zn(g) through CO was computed by the Wilke-Lee equation.18 Also, CO velocities in the furnace were estimated from initial experimental weight loss slopes, which are in the operating range of Guger and Manning experiments.15 All other parameters (except the handbook data) were extracted from Guger and Manning work.15 The comparisons between mixed model conversion predictions and experimental data of Guger and Manning15 are presented in Figures 10 and 11. Figures 10 and 11 are for the reaction of ZnO + CO at 1200 and 1300 °C, respectively. As Figures 10 and 11 show, the agreement of mixed model predictions with experimental data of Guger and Manning15 is very good. Conclusions Zinc oxide reduction by methane at moderate temperatures may be an important metallurgical reaction with several preferences with respect to common cokebased zinc production methods. This reaction is not an equimolar counterdiffusion system and has a large bulk flow effect. The effect of mass transfer and bulk flow appears in the industrial plants with large pellets even at moderate temperatures because of a relatively low mass-transfer coefficient.

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Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 X ) conversion of the solid reactant at each time yAb ) mole fraction of methane in the bulk gas stream yAs ) mole fraction of methane at the pellet surface µg ) viscosity of the gas mixture (g/cm‚min) FB ) true molar density of the solid reactant (gmol/cm3) Fg ) density of the gas mixture (g/cm3)

Literature Cited

Figure 11. Prediction of the conversion-time profile of the ZnO + CO reaction from models and comparison with experimental data of Guger and Manning15 at 1300 °C and r0 ) 0.503 cm.

In this work, the mass-transfer and bulk flow phenomena in the ZnO + CH4 reaction were studied. In the mathematical section of this work, a mixed control model was developed for such systems. The Experimental Section was directed to high-temperature TG tests for the appearance of mass-transfer and bulk flow effects. The resulting experimental data were predicted by the mixed control model with good accuracy. Therefore, this model with related kinetic parameters (see work by Ale Ebrahim and Jamshidi1) is very useful for the design of moderate- and also high-temperature industrial plants. Nomenclature a1 ) first constant in the mass-transfer equation (cm2/min) a2 ) second constant in the mass-transfer equation (cm1.5/ min) C ) P0/(RgT0) ) total concentration of gases (gmol/cm3) DA ) molecular diffusion coefficient of methane (cm2/min) k ) surface rate constant (cm/min) km ) mass-transfer coefficient (cm/min) NAr ) molar flux of argon (gmol/cm2‚min) NCH4 ) molar flux of methane consumption (gmol/cm2‚min) NCO ) molar flux of carbon monoxide production (gmol/ cm2‚min) NH2 ) molar flux of hydrogen production (gmol/cm2‚min) NZn(g) ) molar flux of gaseous zinc production (gmol/cm2‚ min) P0 ) total pressure in the reactor (atm) Q ) flow rate of inlet gases to the reactor (cm3/min) r ) radius of a spherical pellet at each time (cm) r0 ) initial radius of the spherical pellet (cm) Rg ) gas constant (cm3‚atm/gmol‚K) Rrx ) rate of reaction per unit area (gmol/cm2‚min) t ) time (min) T0 ) temperature of the bulk gas stream in the reactor (°K) u ) velocity of the gas stream in the reactor (cm/min) W ) weight of the pellet at each time in thermogravimetry (mg) W0 ) initial pellet weight (after gas switching) in thermogravimetry (mg)

(1) Ale Ebrahim, H.; Jamshidi, E. Kinetic Study of Zinc Oxide Reduction by Methane. Trans. Inst. Chem. Eng. 2001, 79A, 6270. (2) Steinfeld, A.; Brack, M.; Meier, A.; Weidenkaff, A.; Wuillemin, D. A Solar Chemical Reactor for Co-Production of Zinc and Synthesis Gas. Energy 1998, 23, 803-814. (3) Jamshidi, E.; Ale Ebrahim, H. Preparation of Synthesis Gas and Zinc by Zinc Oxide Reduction with Methane and Complete Kinetic Study of the Reaction. Iranian Patent No. 25994, 1998. (4) Steinfeld, A.; Kuhn, P.; Reller, A.; Palumbo, R.; Murray, J.; Tamaura, Y. Solar-Processed Metals as Clean Energy Carriers and Water Splitters. Int. J. Hydrogen Energy 1998, 23, 767-774. (5) Steinfeld, A.; Frel, A.; Kuhn, P.; Wuillemin, D. Solar Thermal Production of Zinc and Syngas via Combined ZnO Reduction and CH4 Reforming Processes. Int. J. Hydrogen Energy 1995, 20, 793-804. (6) Ale Ebrahim, H. Kinetic Study and Mathematical Modeling of ZnO-PbO Mixtures Reduction with Methane. Ph.D. Thesis, Amir-Kabir University, Tehran, Iran, 1998. (7) Moore, J. J. Chemical Metallurgy; Butterworth: London, 1981; pp 150-155. (8) Fletcher, E. A. Solarthermal and Solar Quasi Electrolytic Processing and Separations: Zinc from Zinc Oxide as an Example. Ind. Eng. Chem. Res. 1999, 38, 2275-2282. (9) Steinfeld, A. High-Temperature Solar Thermochemistry for CO2 Mitigation in the Extractive Metallurgical Industry. Energy 1997, 22, 311-316. (10) Kirk-Othmer Encyclopedia of Chemical Technology; Wiley: New York, 1984; Vol. 24, pp 807-862. (11) Szekely, J.; Evans, J. W.; Sohn, H. Y. Gas-Solid Reactions; Academic Press: New York, 1976. (12) Gioia, F.; Mura, G.; Viola, A. Experimental Study of the Direct Reduction of Sintered Zinc Oxide by Hydrogen. Chem. Eng. Sci. 1977, 32, 1401-1409. (13) Szekely, J.; Themelis, N. J. Rate Phenomena in Process Metallurgy; Wiley: New York, 1971; p 624. (14) Sohn, H. Y.; Sohn, H. J. The Effect of Bulk Flow due to Volume Change in the Gas Phase on Gas-Solid Reactions: Initially Nonporous Solids. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 237-242. (15) Guger, C. E.; Manning, F. S. Kinetics of Zinc Oxide Reduction with Carbon Monoxide. Met. Trans. 1971, 2, 3083-3090. (16) Jamshidi, E.; Ale Ebrahim, H. A New Solution Technique for Gas-Solid Reactions with Structural Changes. Chem. Eng. Sci. 1999, 54, 859-864. (17) Sohn, H. Y.; Bascur, O. A. Effect of Bulk Flow due to Volume Change in the Gas Phase on Gas-Solid Reactions: Initially Porous Solids. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 658-663. (18) Treybal, R. E. Mass Transfer Operations; McGraw-Hill: New York, 1968; p 25. (19) Rao, Y. K. Diffusion-Limited Heterogeneous Processes. Can. Metall. Q. 1979, 18, 379-381. (20) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960.

Resubmitted for review December 30, 2001 Revised manuscript received December 30, 2001 Accepted February 4, 2002 IE010604R