A RATE EQUATION FOR GAS-SOLID REACTIONS ACCOUNTING FOR THE EFFECT OF SOLID STRUCTURE AND ITS APPLICATION

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1 bv dt A RATE EQUATION FOR GAS-SOLID REACTIONS ACCOUNTING FOR THE EFFECT OF SOLID STRUCTURE AND ITS APPLICATION Shoichi KIMURA, Yoshikazu TAKAGI, Setsuji TONE and Tsutao OTAKE Department of Chemical Engineering, Osaka University, Toyonaka, Osaka 560 An integrated form of the rate equation for gas-solid reactions is proposed, taking into account the effect of solid structure on the reactivity of the solid. The rate equation is verified by the oxidation of iron sulfide particles, which follows the volume reaction model, with solid structure being varied due to different extent of sintering. The rate constant based on unit particle volume takes distinct value depending on the structure of the solid phase, yielding identical apparent activation energy. The surface rate constant is then evaluated on the basis of the effective surface area for the reaction being separated from the solid structure and is correlated in an Arrhenius equation. The proposed rate equation is approved to embrace a variety of kinetic equations used not only in the volume reaction model but in the grain model, in shrinking core kinetics, and in the gasification of a porous particle. Introduction It is knownthat the solid structure plays an important role in reactions between a gas and a solid. Since the solid participates directly in the reaction, the structure of the solid phase progressively changes in the process of reaction, making the system inherently transient. A number of models have been proposed to represent the overall behavior of this transient system, with emphasis being placed on the effect of solid structure on the reaction mechanism1'7'8'9'11}. The volume reaction model with the consideration of solid reactant depletion4'12) and the grain model9'11} maybe noted as representative examples, applicable with simplicity in practical situations. The former model, in which the reaction is considered to take place uniformly throughout the interior of the solid phase, was derived from a macroscopic point of view. The latter, derived from a microscopic viewpoint, is based on the assumption that reaction proceeds at the interface formed in each individual grain constituting the solid phase. Therefore distinct kinetic equations are applied in the two models in accounting for solid depletion. In this work a rate equation for general use is proposed in due consideration of the effect of solid structure on the reactivity of the solid. Experiments were carried out to verify the validity of the rate expression. Received May 8, Correspondence concerning this article should be addressed to S. Kimura. 456 In addition, it is elucidated that the proposed rate equation is general enough to embrace a variety of reaction kinetics used in many situations, including those described by the volume reaction model and the grain model. 1. Rate Equation Accounting for Solid Structure The gas-solid reaction taking place in a porous particle is represented by the general stoichiometry A(gas)+Z>B(solid) -åº products (1) Froma microscopic viewpoint, the reaction is considered to proceed on the reacting interface, which progressively changes with reaction. Taking a single particle as the basis, the conversion rate of A or B at any time is proportional to this reacting interfacial area formed in the particle. Hence, the rate of reaction maybe represented on the basis of unit particle volume by -ra=- V dt =k'sca (2) where first-order dependence on the gaseous reactant concentration is postulated4). S is defined as the reacting inter facial area per unit particle volume and k' is the rate constant based on the unit area of this reacting interface. In general, the solid particle is considered to consist of a mixture of the reactant and inerts. Product solids are also regarded as inerts since they do not take part in the reaction. In such a case, the area of JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

2 reacting interface is face area constituted not always equivalent to the sur- by pores in the particle, as is schematically shown in Fig. 1. To interpret the area of the actual reacting interface in relation to the measurable pore surface area, eter <j) which represents one mayintroduce a param- the ratio of the two. Accounting for the contribution of inerts to the pore surface area, only a fraction of the pore surface area is effective for the reaction even at the onset of reaction. Hence, taking this initial effective surface area as the basis, actual reacting inter facial area at any time ^~~ effective surface area of pores initially present in the solid structure -h (3) In Eq. (3), (p represents the fractional ratio of the effective surface area for the reaction to the total pore surface area So per unit particle volume. The value of (j> always reduces to unity at the onset of reaction. Assumingthat the effectiveness fraction <p of the pore surface is equivalent to the volumefraction of the reactant solid in the original mixture with inerts, (p may be given by * - $% <4> NB0 identifies moles of B originally contained in the particle. V is the particle volume, e0 is the porosity, and vb is the molar volume of B, which is a universal constant. Substituting Eqs. (3) and (4) into Eq. (2) gives ~ra~~b dt ^ (1-So) *Ca ^ where CB is defined by CB=NB/V, which designates the apparent concentration of solid reactant giving moles ofb per unit particle volume. CB0is its initial value. Using the average extent of conversion XB, CB is given by CB=CBo(l-XB) and then Eq. (5) reduces to dxb dt = bk fvb (^^yca (6) This rate equation maybe rewritten as with d^= bk<j) CA (7) k=k 'vj>-^s- (8) This rate constant is based on the unit particle volume. The rate constant k is conditioned by the original pore structure by Eq. (8). On the other hand, the parameter (j> maybe represented as a function of XB. Assuming simply that 0=1-XB, Eq. (7) is equivalent in form to the volume reaction model4'12). Thus VOL. 14 NO Fig. 1 Schematic illustration of reacting interface in relation to solid structure The rate equation given in this form often applies in cases where the solid is considered as an ensemble of fine lumps of reactant distributed uniformly throughout the solid phase10). This work experimentally reveals the effect of solid structure on the reactivity of solid using Eqs. (8) and (9) as the basis. 2. Experiments 2. 1 Materials Oxidation of iron sulfide particles is dealt with in the verification of the rate equation. Sample solids of iron sulfide were prepared by contacting sintered iron oxide particles with a hydrogen stream containing 0.5% hydrogen sulfide in a quartz tube reactor at 650 C. The original solid is a mixture of iron oxide and inert silica. Powdered reagent-grade ferric oxide (Fe2O3) was mixed with the powdered silica (SiO2) of 50wt%, blended with a proper amount of water and dried at room temperature. The solids obtained were then sintered at various temperatures in the range C to attain different solid structures due to different extent of sintering. The composition of sulfurized solids is FeSi.O4 on average, independently of solid structure5>. The solids were pulverized and screened for use as sample particles. To eliminate resistance to intraparticle diffusion, small particles passing through a 400-mesh screen were used in the oxidation runs Measurement of physical properties Physical properties concerning solid structure accounted for in the rate equation are the pore surface area So per unit particle volume and the porosity eq as the pore volume per unit particle volume. The pore surface area was measured by BETwith nitrogen as adsorption gas. The pore volume was measured by a mercury porosimeter capable of registering a minimumpore radius of 50 A. The contribution of external surface area is important when small particles are used. Hence, the BETsurface area was measured using a part of actual sample particles. Otherwise, particles of mm diameter were used, on confirming that the other 457

3 Table 1 Physical properties of iron sulfide solids Sintering Net BET Particle Porosity Pore temp. pore surface density surface vol. area area [ C] [cm3/g] [m2/g] [g/cm3] s0[-] So[1/m] 1) ) ) ) ) xl0r " " " " physical properties are independent of particle size in the range investigated Experimental apparatus and procedures A quartz tube reactor of 10 mminternal diameter was used. Sample particles were sandwiched in a thin layer between quartz wool so that differential reactor conditions were satisfied. The sample was spread evenly within the quartz wool for dilution and dusting and then sandwiched. The reactant gas is a mixture of air and nitrogen gas. The fractional content of oxygen was adjusted by controlling flow rates of air and nitrogen so that the intraparticle temperature rise due to heat of reaction was negligible. The flow velocity of the reactant gas in the reactor tube was maintained at levels where gas-film resistance at the exterior of the particle was negligibly small. The progress of reaction was followed by detecting the concentration of gaseous product SO2 in the effluent gas stream from the reactor. IR stream line SO2analyzer was used to detect SO2 concentration. 3. Results and Analyses 3. 1 Physical properties of solids All the relevant physical properties of the solids are summarized in Table 1. The pore volume and the pore surface area observed per unit mass of solid are converted into values per unit particle volume, using particle density. The difference of sintering conditions results in considerable difference of solid structure. It is indicated from the decrease in s0 and So that sintering occurs to sufficient extent at a temperature above 1100 C. The structural variation of the solid may be significant in testing the rate equation Interpretation of rate data It was confirmed by X-ray diffraction of product solids that the reaction ends up with hematite (Fe2O3). Hence, according to the average chemical composition of the iron sulfide, the oxidation of this sulfide may be represented by the following single-stage reaction : O2+0.56FeSx.04 >0.28 Fe2O SO2 (10) where one mole of oxygen is taken as the basis. The fractional conversion XBof a particle was then cal- 458 culated on the basis of the cumulative moles of SO2 produced, or XR= integrated moles of product SO2 total moles of S contained in the sample Both numerator and denominator of the equation above were evaluated by graphical integration of the production rate-time curve of SO2from the onset of the reaction to any time and to the end, respectively. The completion of the reaction was confirmed by detecting no SO2. The total moles of SO2 obtained by this graphical integration agreed well with the sulfur content by chemical analysis. Integrating Eq. (9) with XB=0 at /=0 yields an equation to describe the progress of reaction. Thus -ln (l-xb)=bkcj (ll) Hence, the conversion against time data are plotted according to Eq. (ll) on semi-log scale in Fig. 2, Fig. 3, and Fig. 4. These figures display data obtained at three different reaction temperatures using particles of distinct solid structures. The agreement of data with the linear relationships is satisfactory up to almost complete conversion. One may conclude from these figures that the oxidation of the present iron sulfide particles follows the volume reaction model represented by Eq. (ll). Figure 5 illustrates the temperature dependence of the rate constants k determined from the slope of these straight lines by Eq. (ll). All the data result in a group of parallel lines. Each line corresponds to those data obtained using solid of a kind. It is indicated from this figure that the rate constants have an identical apparent activation energy of 21 kcal/mol independently of solid structure. Accounting for the effect of solid structure on the reactivity of the solid, the volumetric rate constant k is correlated in terms of structural physical properties by Eq. (8). Then, the values of k corresponding to each solid of distinct structure, observed at any fixed reaction temperature, are plotted against S0/l-e0 in Fig. 6. Goodagreement of the data with the linear relationships, representing the proportionality of k to So/1-s0, is obtained. Hence, the rate constant can be evaluated on the basis of effective surface area by Eq. (8) as k'-^ik (12) Figure 7 shows the Arrhenius plot of k' determined by each value of k displayed in Fig. 5 with the aid of Eq. (12), assuming ^=18.2cm3/mol of FeS. As is obvious from the figure, all the data fall on a single line showing that k' is independent of solid structure. The rate constant k' based on the effective surface area is then correlated in the form of an Arrhenius JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

4 Fig. 2 Plot of conversion-time data based on volume reaction model; oxidation of iron sulfide particles at T=631 C Fig. 5 Arrhenius plot of volumetric reaction rate constant k, showing different magnitude depending on solid structure Fig. 3 Plot of conversion-time data based on volume reaction model; oxidation of iron sulfide particles at T=600 C Fig. 6 Correlation between volumetric reaction rate constant and solid structure Fig. 4 Plot of conversion-time data based on volume reaction model; oxidation of iron sulfide particles at J=555 C Fig. 7 Arrhenius plot of rate constant k' based on unit effective surface area, showing independence of solid structure 4. Discussion Confirming that the oxidation of iron sulfide particles follows the volume reaction model, the effect of solid structure on the rate constant was examined. It was verified that the rate constant becomes in- VOL. 14 NO

5 Table 2 Correlation of the proposed model represented by Eq. (7) with some other models k 4> Volume reaction model k'vs(so/l - e0) 1 -XB Grain model 3k'vB/rg (1 -XB)m Shrinking core model 3k'vB/R (1 -XB)2/S Gasification* k 'SJ CBQ S!S0 To represent <j) in terms of XBin the gasification the random pore model is usable1>2). dependent of solid structure if it is evaluated on the basis of effective surface area. One may expect from this the soundness of the rate equation represented by Eq. (6). This section deals with further inquiry into this rate equation in the light of its applicability in other situations. Reaction of a porous particle consisting of grains Assumethat a porous particle consists of spherical grains of dense solid, all of the same size, and that each grain reacts with a gas forming an unreacted core with reaction control. In this case, SQ is given as the sumof the external surface area of grains contained in the unit particle volume. Hence, So 4nrl 1-eo (4/3>rrJ r9 where rg is the grain radius. On the other hand, <j) is given as the surface area ratio of the unreacted core to the grain. Thus, - " (15) where rc is the radius of the unreacted core in the grain. In terms of conversion, Eq. (15) is expressed by <f>= (1-XB)2/3. Hence, with insertion ofeq. (14) and Eq. (15), Eq. (6) yields dxb dt 3bk'vBCA (16) (i -xey Equation (16) corresponds to the rate equation derived from the grain model in the absence of intraparticle diffusion resistance9' 11}. 2) Reaction of a densely packed particle Suppose a situation where the reaction proceeds, forming an unreacted core in the particle. This is often encountered when the porosity of the solid is originally so low as to be impervious to the reactant gas and product solid becomes sufficiently porous3}. Thus, eo^o in this case. In the absence of diffusion resistance through the product solid, the overall rate is controlled by the reaction on the unreacted core. Hence, So is given by So=47rR2/(4/3)7rRd=3/R. Using the radius rc of the unreacted core related to the fractional conversion XB, <f) is represented by 460 (14) ^=(^y=a-^)2/3 Equation (6) for this case ends up with dxb 3bk'vBCAn YW3 (17) (18) Equation (18) corresponds to the rate equation obtained by the shrinking core model6}. 3) Gasification of a porous particle When a particle consists of a pure solid reactant, (p=l and Eq. (6) reduces to dxbk' SCA (19) Cmin this case is given by CB0=(l-e0)lvB as the apparent molar density of the particle consisting of B. With no product solid as in gasification, S becomes measurablepore surface area per unit particle volume. Equation (19) has been verified in the steam gasification of activated carbon2k Thus, it was experimentally substantiated that the conversion rate at any time is proportional to the pore surface area per unit particle volume at that time during the course of gasification. The variation of the pore surface area S was well described in terms of XBby the random pore model1'2*. Onemayunderstand from these considerations that Eq. (6) is general enough to apply in a variety of situations. Consequently, the effect of solid structure on the reactivity of the solid can be represented by Eq. (7), the variation of which is summarized in Table 2. C onclusion Anintegrated form of rate equation is proposed to account for the effect of solid structure on the reactivity of solid in gas-solid reactions. The validity of this rate equation has been verified experimentally by the oxidation of iron sulfide particles, which follows the volume reaction model. Evaluated on the basis of the unit effective surface area, the rate constant becomes independent of solid structure and is correlated by an Arrhenius equation represented by Eq. (13). It was confirmed that the proposed rate equation also embraces a variety of kinetic equations, such as those used in the grain model, in the shrinking core model, and in the gasification of porous solid. Nomenclature ca cb Cbq E k stoichiometric coefficient in Eq. (1) [-] concentration of gaseous reactant A [mol/m3] apparent concentration of solid reactant B [mol/m3] initial concentration of B [mol/m3] apparent activation energy [kcal/mol] rate constant based on unit particle volume [m3/mol - min] JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

6 k' = rate constant based on unit effective surface area moles of gaseous reactant A moles of solid reactant B initial moles of B contained in the particle radius of a particle ideal gas law constant reaction volume rate based on unit particle So T t V VB Xb [m/min] [mol] [mol] [mol] [m] [cal/mol - K] [mol/m3 à" min] [m] = radius of an unreacted core radius of grains constituting a particle [m] area of reacting interface per unit particle volume [1 /fli] pore surface area per unit particle reaction temperature reaction time particle volume molar volume of solid reactant B fractional conversion of B in the p article = a parameter defined by Eq. (3) = fraction of pore surface area effective for the reaction volume [1/m] o = porosity as pore volume per unit particle volume [-] Literature Cited 1) Bhatia, S. K. and D.D. Perlmutter: AIChE /., 26, 379 (1980). 2) Chin, G., S. Kimura, S. Tone, and T. Otake: Petrol Inst., 24, 305 (1981): zm/., 24, 344 (1981). /. Japan 3) Froment, G. F. and K. B. Bischoff: "Chemical Reactor Analysis and Design", John Wiley, N. Y., ) Kimura, S., J. Nakagawa, S. Tone and T. Otake: /. Chem. Eng. Japan, 14, 190 (1981). 5) Kimura, S., T. Morita, S. Tone and T. Otake: ibid., 14, 389 (1981). 6) Levenspiel, O.: "Chemical Reaction Engineering", 2nd ed., John Wiley, N. Y., ) Park, J. Y. and O. Levenspiel: Chem. Eng. Sci., 30, 1207 (1975). 8) Petersen, E. E.: AIChEJ., 3, 443 (1957). 9) Szekely, J. and J.W. Evans: Chem. Eng. Sci., 25, 1091 (1970): ibid., 26, 1901 (1971). 10) Wen, C.Y.: Ind. Eng. Chem., 60(9), 34 (1968). ll) Wen, C. Y. and M. Ishida: Environ. Sci. Technol., 7, 703 (1973); Chem. Eng. Sci., 26, 1031 (1971). 12) Wen, C.Y. and N.T. Wu: AIChEJ., 22, 1012 (1976). VOL. 14 NO