pure hydrogen to 4 pct moist hydrogen), the effect on the mass-transfer coefficient (km) being too small. The finding that the dry hydrogen reduction runs (Reaction [1]) of Lewis[2] did not follow a mass-transfer-controlled mechanism can be further ascertained with the help of the intrinsic activation energy for diffusion (ED) extracted from the measured[2] apparent activation energy (Eapp) of Reaction [1] for those runs. Following Eq. [18], which is the mass-transfercontrolled rate equation for dry hydrogen experiments,[2] the following relationship is obtained, noting  (D1/3 v1/6), i.e.,  (D1/3 T1/12), as mentioned previously: Eapp 

2 13 1 (ED)  (RT)  H° 3 12 2

[19]

From Eq. [19], for which Eapp  339 kJ/mol, in the temperature range 1558 to 1723 K,[2] ED is obtained as 128 kJ/mol, at T  1700 K. Now, the characteristic value for ED (for gas film diffusion) is ED  2 RT (for low temperatures) to 1.65 RT (for high temperatures),[4] as previously mentioned, i.e., ED  17 to 27 kJ/mol, in the temperature range of 1000 to 2000 K. Because the experimentally obtained ED value (128 kJ/mol) is far greater than the characteristic ED value for gas film diffusion (17 to 27 kJ/mol),[4] the reduction of SiO2 in dry (pure) hydrogen, given by Reaction [1] (having a small equilibrium constant), evidently did not follow mass-transfercontrolled kinetics, violating the author’s theory.[1] To conclude the discussion, one small yet important point may be noted in passing. The author,[1] in order to defend the prospect of mass-transfer-controlled kinetics (which requires that the interfacial reaction be under equilibrium) of reactions with small equilibrium constants, concludes that “the presence of even a small concentration of the fluid product near the reaction interface brings the condition there close to equilibrium.” However, this argument is not clear, inasmuch as it is only the relative values of the kinetic parameters associated with the different kinetic steps that dictate whether chemical equilibrium (even for a small equilibrium constant) exists at the fluid/solid interface, and, hence, dictate the interfacial concentrations. In other words, the equilibrium interfacial concentrations are a consequence, rather than the cause, of attainment of interfacial equilibrium. REFERENCES 1. H.Y. Sohn: Metall. Mater. Trans. B, 2004, vol. 35B, pp. 121-31. 2. S.D. Lewis: Master’s Thesis, University of Utah, Salt Lake City, UT, 1979. 3. L.B. Pankratz, J.M. Stuve, and N.A. Gokcen: Thermodynamic Data for Mineral Technology, United States Bureau of Mines Bulletin 677, U.S. Department of the Interior, Washington, DC, 1984. 4. R.B. Bird, W.E. Stewart, and E.N. Lightfoot: Transport Phenomena, John Wiley & Sons, New York, NY, 1960, p. 511.

Author’s Reply H.Y. SOHN Essentially, all of Ghosh’s disagreements are based on an apparent lack of experimental experience in the field of fluidsolid reaction kinetics, not reading carefully the contents of METALLURGICAL AND MATERIALS TRANSACTIONS B

my article (or, if he did, choosing to ignore certain aspects of my statements), careless examination of mathematical results, or erroneous understanding of the theory of chemical kinetics. I have established this assertion in this Reply. Additional comments and elaborations, which I hope will provide increased understanding of some of the related finer points in fluid-solid reaction analysis, are also presented. Let us start with Ghosh’s analysis of the experimental data of Lewis [1] for the case of wet hydrogen. (Comments on the results with dry hydrogen are also given subsequently.). Ghosh has “developed” what is tantamount to a new “theory,” which, if correct, would establish a result that the intrinsic activation energy values (completely unaffected by mass-transfer effects) of all chemical reactions with small KC values are equal to the standard enthalpy change of the reaction. This would indeed be a very useful, ground-breaking result. Alas, unfortunately, Ghosh’s “theoretical derivation” is based on fatal errors. The first error made by Ghosh is the use of an incomplete mathematical argument that leads to his incorrect “conclusion.” He thought the derivation of his Eq. [13] by the use of his Eqs. [11] and [12] gave an independent relationship between Ef and Eb, or an independent measure of Eb/Ef, leading to a negligibly small value of Eb. The following closer examination will show that this is wrong. When we divide his Eq. [4] by his Eq. [5] and incorporate his Eqs. [7] and [9], we get Af Ef  Eb K  a b # exp a b RT Ab RT

[A]

Since K  exp a

G ° H° # S ° b  exp a b exp a b [B] RT RT R

Equation [A] reduces to exp a

Af Ef  Eb S° H° # b exp a b  a b # (RT ) # exp a b RT R Ab RT [C]

When we substitute the first part of Ghosh’s Eq. [13] (although the validity of even this equation is highly questionable, as indicated subsequently) into Eq. [C], we get exp a

Ef  Eb H ° b  (RT ) # exp a b RT RT

[D]

This just re-establishes Ghosh’s Eq. [6], but not any additional independent relationship between Ef and Eb, or an independent measure of Eb /Ef. Thus, Ghosh’s “intriguing” conclusion (his Eq. [16]) that in such a case Ef  H° is false. Even the expressions of Af and Ab given by Ghosh’s Eqs. [11] and [12], respectively, are incorrect. In the application of the activated complex theory of reaction rates, even to an elementary, homogeneous reaction, the pre-exponential factors must contain a combination of partition functions for the species involved in the reaction, including the activated complex.[2,3,4] These partition functions have temperature dependences that can add up to be large enough to significantly affect the activation energy values of the forward and reverse reactions to different degrees. Furthermore, the Af and Ab terms as written in Ghosh’s Eqs. [11] and [12], purported to be based on the activated complex theory, should be VOLUME 36B, DECEMBER 2005—897

multiplied by exponential functions of temperature containing the enthalpies of formation of the activated complex, which are different from Ef and Eb.[3] Especially, for complex heterogeneous reactions such as the hydrogen reduction of silica, which may involve several adsorbed complexes,[5,6] a loose application of the activated complex theory as attempted by Ghosh is not justified. If the preceding discussion is not enough to show Ghosh’s errors, the units of the terms Af and Ab cannot be the same, as written in his Eqs. [11] and [12]. This provides additional evidence that his Eq. [13], which gives Af /Ab as a unitless quantity, is incorrect. This can be shown, as follows, considering the fact that the net forward intrinsic rate, per unit area, of this reaction must satisfy the thermodynamic relationship at equilibrium (when the net rate R, as follows, is zero): R  kf CHn2  kb(CSiO # CH2O)n

[E]

in which n can be, and often is, different from unity. From this equation, even when n is unity, kf and kb, and thus Af and Ab, must have different units, further proving Ghosh’s Eq. [13] incorrect. Even just the fact that the net rate of reaction can follow Eq. [E] with n different from unity shows the falsehood of Ghosh’s Eqs. [9] and [6], because in this case kf /kb  (Kc)n

[F]

E f  E b  n (H °)

[G]

and

Ghosh’s statement that the experimental results obtained by Lewis[1] with wet hydrogen could represent a case of rate control by chemical reaction, solely based on the aforementioned erroneous interpretation of mathematical relationships, is thus proved to be erroneous. The fact that the apparent activation energy value is essentially the same as the enthalpy of reaction is one of the strongest indications for rate control by mass transfer. However, one should always look for other corroborating evidence. In this case, Lewis[1] had performed a rate analysis and found the actual magnitude of the rate to be consistent with that based on mass transfer (for a detailed reanalysis of Lewis’ data, the reader is referred to Han and Sohn[7]), not just the apparent activation energy, and even more convincingly, Lewis[1] obtained gas-flow-rate dependence consistent with mass-transfer control—a fact that Ghosh either had not read (which I strongly suspect) or ignored. Earlier, Schwerdtfeger[8] had also analyzed his experimental results for the same reaction based on mass transfer. With the large apparent temperature effect, it is highly unlikely that the rate just happens to be close to the rate determined by mass transfer at several different temperatures, if chemical reaction were rate controlling. Furthermore, the chemically controlled reaction rate does not vary with the flow rate of fluid phase, as observed by Lewis. Only when different corroborating pieces of evidence are combined, can one suggest the rate-controlling mechanism with a reasonable degree of confidence. In a general case, other such corroborating evidence may include the effects of solid particle shape/size and fluid reactant concentration, considering what a correctly derived rate equation would suggest. As to Lewis’ experimental results with dry hydrogen, I myself stated in my original article that “the reaction rate 898—VOLUME 36B, DECEMBER 2005

was . . . substantially affected by intrinsic chemical kinetics.” My original premise was that “reactions with small K values tend to be rate controlled by mass transfer,” rather than “all such reactions are rate controlled by mass transfer.” One should carefully read and try to understand others’ work better before rushing into erroneous criticism. Even in this case of dry hydrogen, the apparent activation energy is not overly distant from the value calculated for masstransfer control. Comparing the apparent activation energy values of wet and dry hydrogen, together with the similar temperature ranges for the two cases, these results also provide an excellent piece of evidence supporting my statement that the presence of the fluid product (at least one of the two in this case) makes it easier to establish equilibrium at the reaction interface, leading to rate control by mass transfer. (This is related to another of Ghosh’s comments discussed later.) Further examination of the difference between the results with and without water vapor in the bulk hydrogen gas shows the fallacy of Ghosh’s comment that, “It is reasonable not to have a shift in controlling mechanism with a small change in composition of the inlet gas mixtures (from pure hydrogen to 4 pct moist hydrogen), the effect on the mass-transfer coefficient (km) being too small.” On the contrary, this example clearly illustrates the importance of even a small change in composition for reactions with small K values, if it involves gaseous product species. For such reactions the presence of even a small amount of a gaseous product, in this case H2O, can exert an important influence on the equilibrium by reducing the already small equilibrium concentration of the other product gas, SiO, the rate of transfer of which must equal the overall rate of reaction. This can be shown by considering the expressions of the equilibrium SiO gas concentrations in the two cases, as follows: For wet hydrogen, (pSiO)e  K # (pH2)b /(pH2O)b

[H]

and for dry hydrogen, (pSiO)e  [K # (pH 2)b]1/2

[I]

Thus, the ratio of the two values is K1/2/(pH2O)b, assuming pH2  1 for a small water vapor content. With K  2.5

107 at 1700 K, this ratio is far from unity even for “a small change in composition of the inlet gas mixtures (from pure hydrogen to 4 pct moist hydrogen).” In fact, the ratio of the equilibrium SiO partial pressures  K1/2/(pH2O)b represents the ratio of mass-transfer rates, and thus overall reaction rates, at the same temperature and bulk gas flow rate for the two cases. With (pH2O)b  0.04, this ratio is about 0.13. This is close to the ratio of rates obtained experimentally by Lewis.[1] This presents another piece of convincing evidence for mass-transfer control, because such a small amount of water vapor would not have such a large effect on the rate if chemical reaction were rate controlling and its effect agrees quantitatively with the analysis based on masstransfer control—another fact that Ghosh missed. Ghosh must realize that whether a quantity is “small” is always relative. In this case, the apparently small concentration of 4 pct is large compared with its equilibrium concentration generated from dry hydrogen. The decrease in the equilibrium concentration of SiO gas, by a large ratio because of the very METALLURGICAL AND MATERIALS TRANSACTIONS B

small value of K as shown earlier, enhances the importance of its mass transfer. Thus, the “small change in composition of the inlet gas mixtures” pushed the system from incomplete mass-transfer control with dry hydrogen to dominant masstransfer control with wet hydrogen. Let us now turn to Ghosh’s last comment. Ghosh contends that only chemical reaction can bring the attainment of equilibrium condition at the interface. This is a narrow view that is true if the bulk fluid contains no fluid product. The presence of fluid product added to the bulk fluid phase can cause equilibrium or bring the system close to equilibrium, and for a reaction with a small KC value, its concentration only needs to be rather small. Even without considering such a case, the first statement in my “Concluding Remarks” section was meant to present a physical interpretation of the reason why reactions with small KC values tend to be rate controlled by mass transfer. For reactions with large KC values, a fluid product concentration of as high as say 5 pct of the fluid reactant concentration represents 95 pct toward chemical control, and thus, more rapid removal (mass transfer) of the fluid product is not critical. On the other hand, for a reaction with say KC  108, a very small fluid product concentration of 108 times the fluid reactant concentration brings the condition to equilibrium (when one mole of product gas is produced from one mole of reactant gas). If the mass-transfer coefficient is extremely large, the possibility exists that there is little or no gaseous product species present at the interface (or no product that is not added to the reactant gas, for reactions with more than one gaseous product species), and the overall rate would be controlled by chemical kinetics. As the mass-transfer coefficient decreases, the concentration of the product gas species that is generated by the reaction starts to appear. For reactions with small KC values, equilibrium is approached even when this concentration is small. Even a relatively slow chemical reaction could readily generate such a small fluid product concentration. When equilibrium is approached at the reaction interface, the overall rate is controlled by mass transfer. This is another way of explaining why reactions with small KC values tend to be rate controlled by mass transfer, and why such reactions are slow, i.e., because the concentration difference for mass transfer is small. To elaborate on my statement that reactions with small values of equilibrium constant tend to be rate controlled by mass transfer and slow, let us examine Eqs. [7], [8], and [11] of my original article, reproduced here: (NA) 

kf (CAb  CCb /KC) 1  s o2

[J]

where s o2 

kf d DA

(1  1/KC)

[K]

and for a small KC value, s o2  a

kf d

1 b#a b DA KC

[L]

While a small value of KC tends to make s 2o large, and thus the rate tends to be mass-transfer controlled, it is obvious that s 2o could be made small even for a small KC value, if kf is METALLURGICAL AND MATERIALS TRANSACTIONS B

sufficiently small. In this case, the overall reaction rate would be controlled by chemical kinetics. This is why I stated that reactions with small KC values tend to be, rather than are, rate controlled by mass transfer. Having said this, let us further examine the case of a small kf value. To be small enough to make s 2o much smaller than unity despite the small KC value, the kf value must be very small. Thus, the rate of such a reaction would accordingly be very slow and usually of little interest in real situations. This is another reason why reactions with small KC values are of interest in most cases at temperatures high enough to make the system rate controlled by mass transfer. On a related subject regarding fluid-solid reactions, it is often attempted to verify the rate-controlling mechanism by testing for linearity between time and a conversion function based on either chemical or mass-transfer control, such as 1  (1  X)1/3  k

[M]

1  (1  X)1/2  kt

[N]

 kt

[O]

1  (1  X)

2/3

in which X represents the fractional conversion of the solid reactant, t is time, and k is an apparent rate constant, which may contain an intrinsic chemical rate constant or masstransfer-related parameters. I wish to point out that this is a very risky method and should not be relied upon as a sole means of determining the mechanism. Equations such as these should be used in the context that the conversion-vs-time relationship is expected to follow such an equation, if the reaction mechanism is known, but not the reverse. Experience with real fluid-solid reaction systems reported in the literature as well as those studied in my laboratory over the years has shown that it is almost impossible, and often leads to an erroneous conclusion, to determine the rate-controlling mechanism based solely on a plot of a conversion function. There are many reasons for this. (1) Several different mechanisms can give rise to the same or an almost indistinguishable form of conversion function. (2) Real experimental data always have uncertainties that make the distinction between the conversion functions difficult. Especially for fluid-solid reactions, the conversion behavior very often deviates from the expected relationship, especially at high conversion values. This typically causes a long tail in the conversion-vs-time curve. Some of the factors causing this include changes in the overall shape and sintering of the solid as it is consumed. The deviation from the expected behavior also often occurs at the beginning of a reaction, a frequent reason for this being the need for nucleation to start at selected surface sites. (3) The solid reactants almost always contain, or develop during the reaction, different crystal/grain structures and morphologies with different reactivities. (4) The shape of the solid is often imperfect and its size is seldom uniform when the sample is particulate. The effect of irregularity in solid shape on the deviation from the expected conversion function increases with decreasing particle size. (5) A slow reaction such as the hydrogen reduction of silica is seldom continued to high conversion and typically carried out only to a low overall conversion. Within such VOLUME 36B, DECEMBER 2005—899

small conversion ranges, any rate equation appears as a straight line for X vs t. Thus, this equation for small values of X becomes, upon linearization of the overall rate equation, X  (const) kt,

for X  1

[P]

This does not allow testing of the assumed conversion function. One must still examine the nature of k, which could contain any or all the temperature-dependent kinetic, masstransfer, and thermodynamic parameters. Factors 3 and 4, as well as the presence of minor impurities contained in the solid, also contribute to the built-in uncertainties in measured activation energy values. I have also been presented with an argument that the apparent activation energy value could be falsified (lowered) for a chemically controlled reaction with a large KC value, even to the extent that the rate is controlled by mass transfer, if some fluid product species is present in the reactant fluid. I will show that this would almost never happen in practice. Let us consider a simple reaction, whose net forward rate under chemical control is given by Eq. [5] in my original article, as follows: NA  NC  kf (CAb  CCb /KC)

[Q]

The apparent activation energy is obtained by taking the natural logarithm of both sides and differentiating the resulting terms with respect to 1/T. Assuming that CCb/(KCCAb)  1 and KC  K and considering only the usually temperature-dependent terms, one obtains E app  E f  [CCb /(K C CAb)] # (H°) [R] Since a large KC value usually means a large negative value of H°, it is easy to assume that Ef, which is related to (H°) as in Ghosh’s Eq. [6], can be reduced substantially. Upon more careful examination of the terms, it will be shown that this is an erroneous assumption. Let us select a rather modest Ef value of 80 kJ/mol as the activation energy of the intrinsic chemical reaction. For the appearance of rate control by mass transfer, Eapp needs to be less than about 30 kJ/mol (usually smaller). This means that [CCb /(K C CAb)] (H °)  50 kJ, and CCb /CAb  50 K C /(H°) or CCb /CAb  50 exp (S °/R) # exp (H°/RT )/(H°) [S] The last relationship is obtained when Eq. [B] is substituted into the second relationship. The calculated values of CCb /CAb at 2000 K (a temperature much more favorable for the righthand side of Eq. [S] to be small than temperatures at which most fluid-solid reactions of interest take place) are shown below for several values of (H°), ranging from a large to a small value encountered in most fluid solid reactions: (Ho), kJ

400 200 100

CCb /CAb 10 exp (S°/R) 105 exp (S°/R) 102.6 exp (S°/R) 9

For most gas-solid reactions producing another solid and gas, 40 J  S°  40 J. (For a liquid-solid reaction, the range 900—VOLUME 36B, DECEMBER 2005

is expected to be smaller.) Thus, 102  exp (S°/R)  102. For the type of fluid-solid reaction under discussion, in which an interfacial reaction and fluid-phase mass transfer are important, the product solid must be porous. This means some of the constituent atoms of the reactant solid are removed into the fluid phase, as in the gaseous reduction of metal oxides. For such reactions, S° is positive, which makes the CCb /CAb value larger than the numbers multiplying the exp (S°/R) term. The value of S° is expected to be negative if some atoms of the fluid reactant are incorporated into the reactant solid. In such a case, the solid volume increases and thus the product solid tends to form a dense layer, and the slow solidstate diffusion becomes important. This type of fluid-solid reaction is not covered in this discussion. To create the situation in which Eapp is substantially different from Ef, the range of the calculated CCb /CAb values indicates that the bulk fluid phase must contain a very high concentration of the fluid product and in most cases must consist essentially entirely of the fluid product. It is noted that even in the extreme case of a low value of (H°) (100 kJ) combined with a negative S° value, CCb /CAb needs to be much greater than unity. It is further noted that I made the preceding calculations at a temperature much higher than almost all fluid-solid reactions of the type covered in this discussion. At lower temperatures, the necessary value of CCb /CAb only becomes larger, because the exp (H °/RT) term increases rapidly with decreasing temperature. Including the small value of Ef, I chose an evaluation of a case of small (H°); I used all the parameter values that are favorable for the falsification of the activation energy of a reaction with a large KC value to be possible. The results still prove that this situation practically never arises. It is further noted that the last term in Eq. [R], which determines the extent of possible “falsification,” is more strongly affected by the large KC rather than by the “large” value of (H°). In other words, as (H°) becomes larger, the possibility of activation energy falsification diminishes rather than increases as a casual examination of Eq. [R] would indicate. Having done all this exercise, it is noted that it would be pointless in the first place to maintain any presence of the fluid product in the bulk when one is interested in determining the activation energy of the intrinsic chemical reaction. The preceding comments regarding the reaction rate analysis show us that the determination of the reaction mechanism should be based on a sufficiently comprehensive understanding of the implications of mathematical relationships and, especially, having applied them to real reactions studied in the laboratory or other situations. This is particularly appropriate in this age of “virtual” realities, although I recognize their often significant usefulness when they are developed properly and their limitations are fully understood. Having spent time to present my counterarguments to Ghosh’s erroneous disagreements, the one positive outcome of this exercise is to have another opportunity to point out that a fluid-solid reaction with a small KC value tends to be rate controlled by mass transfer and the apparent activation energy value for such a reaction is large with a value close to the standard enthalpy of the reaction divided by the number of moles of fluid product(s) produced from one mole of fluid reactant. This is in contrast to a mass-transfer-controlled reaction with a large equilibrium constant, which has a small apparent activation energy value. This exercise has also preMETALLURGICAL AND MATERIALS TRANSACTIONS B

sented an opportunity to elaborate on some of the often misunderstood finer aspects of fluid-solid reaction analysis, which has long been a subject of my research and teaching activities. REFERENCES 1. S.D. Lewis: Master’s Thesis, University of Utah, Salt Lake City, UT, 1979. 2. S. Glasstone, K.J. Laidler, and H. Eyring: The Theory of Rate Processes, McGraw-Hill, New York, NY, 1941. 3. A.A. Frost and R.G. Pearson: Kinetics and Mechanism, 2nd ed., Wiley, New York, NY, 1961, pp. 88-100. 4. J.H. Espenson: Chemical Kinetics and Reaction Mechanisms, McGrawHill, New York, NY, 1981, pp. 153-55. 5. E.E. Petersen: Chemical Reaction Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1965, pp. 28-37. 6. J. Szekely, J.W. Evans, and H.Y. Sohn: Gas-Solid Reactions, Academic Press, New York, NY, 1976, pp. 41-46. 7. G. Han and H.Y. Sohn, J. Am. Ceram. Soc., 2005, vol. 88, pp. 882-88. 8. K. Schwerdtfeger: Trans. TMS-AIME, 1966, vol. 236, pp. 1152-56.

Application of the Genetic Algorithm to Estimate the Parameters Related to the Kinetics of the Reduction of the Iron Ore, Coal Mixture AMIT KUMAR and G.G. ROY A novel methodology has been developed to calculate the kinetic parameters associated with reduction of ore-coal composite mixtures and to describe the time course of reduction of hematite to iron. The empirical parameters, namely, the three sets of activation energies and frequency factors, have been estimated by employing an evolutionary optimization tool, the genetic algorithm (GA). The model prediction matches well with the experimental literature data. The estimated activation energies are higher than the corresponding intrinsic values, indicating the role of heat transfer in the process.

The direct reduction of iron ore, coal composites is a cokefree and environmentally friendly process. Use of noncoking coal, unused iron-rich ore fines, coal fines, process integration (ore and coke preparation in one unit), less emission, and economic production are major driving forces toward studies evaluating prereduced ore-coal composites as a feed to the blast furnace and even in the steelmaking unit. Several fundamental studies have also been conducted on reduction kinetics of ore-coal composite pellets.[1,2] Several semiempirical models have also been put forward to determine the time course of the reduction of ore-coal composite pellets.[3,4] The validity of these models depends critically on accurate estimation of the parameters related to kinetics of the iron oxide reduction process. The present model

AMIT KUMAR, Undergraduate Student, and G.G. ROY, Associate Professor, are with the Department of Metallurgical & Materials Engineering, Indian Institute of Technology, Kharagpur 721 302, India. Contact e-mail: [email protected] Manuscript submitted March 18, 2005. METALLURGICAL AND MATERIALS TRANSACTIONS B

applies an evolutionary optimization tool, the genetic algorithm, to estimate these parameters for the first time. The reactions for the coal-based direct reduction actually take place in three steps by gaseous intermediates (CO, CO2), namely, hematite to magnetite, magnetite to wustite, and wustite to iron. Hematite to magnetite: 3Fe2O3  CO  2Fe3O4  CO2

[1]

Magnetite to wustite: Fe3O4  CO  3FeO  CO2

[2]

Wustite to iron: FeO  CO  Fe  CO2

[3]

The CO gas in turn is produced by gasification of carbon by CO2, as follows: C  CO2  2CO

[4]

Assuming that the transition of hematite to magnetite, magnetite to wustite, and wustite to iron are first-order reactions, the production of these species may be given by the following mass balance equations:[3] EH dH  Hk H expa b dt RT

[5]

EH EM dM  xHk H expa b  Mk M expa b dt RT RT

[6]

EW EM dW  yMk M expa b  Wk W expa b dt RT RT

[7]

EW dF  zWk W expa b dt RT

[8]

Here, x, y, and z are the stoichiometric factors; kH, kM, and kW are the frequency factors; and EH, EM, and EW are the activation energies for the reduction of hematite to magnetite, magnetite to wustite, and wustite to iron, respectively. Note that the preceding four first-order ordinary differential equations contain six extra unknowns besides the concentration of hematite, magnetite, wustite, and iron, namely, the three pairs of frequency factors and activation energies for Eqs. [1] through [3]. It may be further noted that activation energies indicated in Eqs. [5] through [8] will represent the intrinsic activation energies as long as there is no internal thermal gradients in the reaction zone; otherwise, those will represent apparent activation energies, which will also include the heat-transfer resistance in series. Several previous authors[5,6] have indicated that the gasification reaction, given by Eq. [4], can be the rate controlling step. Assuming gasification as the rate controlling step indicates that the oxide reduction reactions are in equilibrium. Under this condition, the CO/CO2 ratio of the gas mixture for the respective reduction may be given by its equilibrium CO/CO2 ratio. Let us consider the Eq. [1], i.e., hematite to magnetite reduction. If Keq[1] represents the equilibrium constant for this reaction, the equilibrium weight ratio of carbon to oxygen (WC/WO) in the VOLUME 36B, DECEMBER 2005—901