Law of Mass Action Based Kinetic Approach for the Modelling of Parallel Mass Transfer Limited Reactions: Application to Metallurgical Systems

ISIJ International, Vol. 56 (2016),ISIJ International, No. 9 Vol. 56 (2016), No. 9, pp. 1543–1552

Mika JÄRVINEN,1)* Ville-Valtteri VISURI,2) Eetu-Pekka HEIKKINEN,2) Aki KÄRNÄ,2) Petri SULASALMI,2) Cataldo De BLASIO1) and Timo FABRITIUS2)

1) Department of Mechanical Engineering, Aalto University, P.O. Box 14400, 00076 Aalto, Finland. 2) Process Metallurgy Research Unit, P.O. Box 4300, FI-90014 University of Oulu, Finland. (Received on April 27, 2016; accepted on May 19, 2016; J-STAGE Advance published date: August 23, 2016)

The objective of this paper was to present a new law of mass action based rate expression for mass transfer limited reversible reactions. A simple reaction model was derived for parallel oxidation of silicon, chromium and carbon under conditions relevant to the argon-oxygen decarburization (AOD) process. Our hypothesis is that when the forward rate coefficients approach infinite values, the composition at the reaction surface approaches a constrained equilibrium. In numerical analysis, however, only finite numbers are allowed and therefore only finite values are accepted for rate coefficients. In order to circumvent this problem, additional residual affinity constraints were introduced. This assures that the affinities of all the reactions at the reaction front reach a pre-defined non-zero residual affinity and the rate coefficients remain finite. The calculated equilibrium composition is essentially the same as that obtained with the equilibrium coefficient method. In the case of effective gas side mass transfer, the component having the highest mole fraction or the highest mass transfer rate on the liquid side consumes most of the oxygen. When the gas side mass transfer rate is decreased, the mass transfer rate of oxygen begins to limit the overall rate and the partial pressure of oxygen at the reaction interface decreases. Then, the role of interfacial equilibrium becomes important as the species start to compete for the oxygen. The proposed method provides a transparent and direct solution of the mass transfer limited reaction rates and is thus suitable for process simulators and CFD software.

KEY WORDS: mass transfer limited parallel reactions; law of mass action; rate phenomena; residual affinity.

** kTB  ∆∆S   H  1. Introduction k = expe  xp −  ...... (2) In the primary and secondary metallurgy of steelmaking, h  R   RT  temperature of the metal bath is typically in the range of As seen from Eqs. (1) and (2), the reaction rate coef- 1 500–1 800°C, but local temperatures can be much higher. ficient becomes large at high temperatures, and thus the In basic oxygen furnace, for example, the temperature of mass transfer or the mixing becomes the rate limiting step. the gas jet impact area may exceed 2 000°C.1–3) Comparable It is thus reasonable to assume that the reaction front can circumstances are found also in other industrial applica- reach the chemical equilibrium composition that is subject tions: the temperature in the modern combustion furnaces to mass transfer constraints. However, there are examples of varies from 1 088–1 173 K (815–900°C) in fluidized boil- reactions having slow kinetic rates at high temperatures. In ers, 1 923 K (1 650°C) in pulverized coal boilers and up to particular, the presence of surface active elements can hin- 2 200–2 603 K (1 927–2 330°C) in premixed gaseous burn- der the interfacial reaction to the extent that the interfacial ers.4) In many of the aforementioned applications, the local reaction rate becomes the rate limiting step. A well-known kinetic reaction rate and the rate expression at the reaction example is the de-nitrification reaction of liquid steel.9) interface is unknown and virtually impossible to measure. The boundary condition for the rate expression of a The resistance of the interfacial reaction decreases as the single mass transfer limited reaction can be determined temperature increases. The temperature dependency of the directly with the equilibrium constant method. One of the reaction rate coefficient k can be described with the Arrhe- first qualitative descriptions of the boundary layer theory nius equation, Eq. (1), or the Eyring equation, Eq. (2).5) was proposed by Nernst10) in 1904. In the case of parallel reactions, however, the description of mass transfer in the  Ea  kA=−exp  ...... (1) boundary layer needs to be coupled with a description of the  RT  competitive thermodynamic equilibrium at the interface.7,11) The equilibrium constant method is not applicable as such, particularly if the rate limiting step is not known a priori or * Corresponding author: E-mail: [email protected] it varies during the process. DOI: http://dx.doi.org/10.2355/isijinternational.ISIJINT-2016-241 The mass transfer resistances of the considered phases

1543 © 2016 ISIJ ISIJ International, Vol. 56 (2016), No. 9 depend on the observed geometry. Typical geometries in the species that has been used to define the rate expression the primary and secondary steelmaking processes include and has molar stoichiometric coefficient of –1. The reaction gas jet in contact with surface of a steel bath (Fig. 1(a)), equations and equilibrium constants can be written either in a liquid metal droplet entrained in gas flow (Fig. 1(b)) as terms of the oxidized elements i, Eq. (5), or in terms of the well as slag droplet in liquid metal or liquid metal droplet in O2 as the key component, Eq. (6). slag (Fig. 1(c)). Small dots on the reaction surface present ii+=ν iOf22 O,νi or in12… ...... (5) existing small slag particles. The mass transfer rates of the 11 reagents and reaction products are interrelated through the ii+=Of22 O,νi or in12… ...... (6) reaction stoichiometry. ννii In this work, a novel approach based on the Law of Mass where νi is the molar stoichiometric coefficient of O2 in the

Action was studied and developed further in order to provide oxidation reaction of species i and iO2vi refers to the oxide of a thermodynamically consistent treatment of parallel mass species i. The equilibrium constants corresponding to Eqs. transfer controlled reactions in the context of steelmaking. (5) and (6) are obtained from Eqs. (7) and (8), respectively. The method can be used in metallurgical and chemical In this work, an atmospheric pressure of 1 atm was assumed. processes involving mass transfer controlled reactions. The   ∆Gi  aiO2νi method discussed in this paper has already been employed Ki =−exp  = ...... (7) RT apνi by the authors in different cases for defining the mass trans-   i O2 fer limited rates of reactions. The studied applications in the 1 AOD process include mathematical modelling of reactions  νi  ∆G  ai in a single gas bubble in liquid steel, during side-blowing G 1/νi i O2νi ...... (8) KKi ==i exp −  = 1 decarburization and during the reduction of slag. The RTν i   νi method has also been applied for mathematical modelling of api O2 chemical heating in the CAS-OB process. The objective of The source terms for all other species that participate in this paper is to re-derive, discuss and further develop the law the reaction are calculated simply by multiplying the rate of of mass action based method for multiple parallel reversible the key component by their stoichiometric coefficients. In reactions. A generalized approach for implementation into this work, the oxidation of Fe, i.e. the solvent, is neglegted. practice is also provided and discussed. FeO is often an intermediate product and acts effectively as oxygen storage. It is reasonable to expect that in an actual process, the supplied oxygen first oxidizes iron to form iron 2. Law of Mass Action Based Approach oxide, which is then reduced by species with higher oxygen The Law of Mass Action (LMA), proposed by Waage and affinity. In a transient case, the composition at the reaction Guldberg13,14) in 1864, is a mathematical expression of solu- surface approaches asymptotically the constrained chemical tions in dynamic equilibrium. More specifically, it relates equilibrium. The form of the chemical rate expression is the thermodynamic equilibrium to the reaction kinetics and irrelevant as long as the equilibrium composition and the implies that forward and backward reaction rates must be mass transfer control can be achieved. Therefore, it is con- equal at equilibrium. The reaction rate of an arbitrary reac- venient to employ the modified law of mass action, which tion can be expressed by: satisfies the equilibrium condition as a limit, to formulate 15) ν p the reaction rate expressions. The law of mass action  xp  ∏ p based approach has been shown to hold for elementary reac- νr ν p νr Rk′′ =−f xkr b xp =−kxf  r  ∏∏∏ K  ... (3) tions of radical gas species containing only a single reaction r p  r  step and to yield the correct equilibrium composition.15,16) Forwardreaction Backward reaction The surface mole balance for the elements i = 1...n diffusing where kf is the forward reaction rate coefficient, kb is the from the bulk liquid steel and reacting at the surface can be backward reaction rate coefficient x is the molar fraction, written as follows: r denotes reactants and p denotes reaction products. In cL ∞∞cL  ai   νi aiO2νi  ββL xxii− =−L xi =−kaf i p order to be generally applicable, it is necessary to replace ()    O2  ... (9) the concentrations with activities, which describe effective ML ML  γ i   Ki  concentration: where x denotes the mole fraction and γi is the Raoultian

ν p activity coefficient of species i. The rate expression is writ-  ap  νr ∏ p ten here using the law of mass action for Eq. (9). For the Rk′′ =−f  ar  ...... (4) ∏ sake of simplicity it is assumed that the transport rate or the  r K    amount of oxide species is not rate-controlling; the activities Here, the expression shown in Eq. (4) is referred to as the of the oxides were treated as constants. Here, a value of 0.5 modified Law of Mass Action. In the following, a case of n was employed. The mass balance for the oxygen diffus- parallel reactions between oxygen and dissolved elements in ing from the gas phase to reaction surface and distributing the liquid steel is studied. The term key component refers to between all the species can be defined as follows:

Fig. 1(a). Gas jet in contact with surface Fig. 1(b). Steel droplet entrained in gas Fig. 1(c). Gas bubble immersed in liquid of a steel bath. flow. steel.

 1  2.1. Numerical Implementation 1 r νi The implementation of the law of mass action based pG  pO2   aiO2ν  β x∞ − =−kaνi p i G  O2  ∑ fO i 2 G  .... (10) approach requires a finite value for the forward rate coef- RT γ O2 Ki   i=1   ficient kf. Therefore, it is necessary to develop a robust way   to define the values of kf such that the requirements for By using the mass balances shown in Eqs. (9) and (10), accuracy and numerical stability are fulfilled. An analytical the activities of dissolved elements and oxygen can be expression is not available due to non-linearity, and numeri- solved from Eqs. (11) and (12), respectively. cal methods need to be used. It should be noted that the β x∞ c a employed approach functions correctly also in the case that L i L + iO2νi the forward rate coefficient is known a priori. Therefore, kf ML Ki ai = for in=…12, ...... (11) the approach is applicable also for reactions, which are not β c L L + pνi limited by mass transfer. O2 kfγ i ML As an illustrative numerical example, a simple case of liquid stainless steel surface exposed to O2–CO gas is  1  studied; a schematic illustration is shown in Fig. 1(a). In ∞ νi β x p r  aiO  G O2 G + 2νi the simplified setting, the injected oxygen may oxidize ∑ i=1 G  kf RT Ki carbon, chromium and silicon according to Eqs. (15), (16)   17)   ...... (12) and (17), respectively. As pointed out by Wei and Zhu, pO2 = FeO is essentially an intermediate product and hence the  1  βG pG r ν oxidation of iron was not accounted for. The values of the +  a i  ∑ i==1 i  changes in standard Gibbs free energy were taken from kfOγ 2 RT   Rao.12) More specifically, the reference states of Si, Cr As seen from Eqs. (11) and (12), the system consists of and C were taken as pure liquid, pure liquid and graphite, n + 1 variables and n + 1 equations, and hence numerical respectively. The reference states of SiO2 and Cr2O3 were solution is possible. The main hypothesis of the law of mass taken as β-cristobalite and pure solid, respectively. Ideal gas action based approach is that as the rate of the surface reac- law was employed for the gaseous species. tion is infinitely fast and the reaction front is able to reach Si()lg+=OS22() iO ()l ∆GT0 −+938 913 193. 719 ....(15) equilibrium. Mathematically, this means that if kf → ∞, the following limits are obtained for the activities of species i 3 1 Cr()lg+ OC2() rO23()β −cristobalite and O2: 4 2 ...... (16) aiO2νi lim,ai ==aii,eq for =…12 n ∆GT1 =−566 934+ 128. 323 νi ...... (13) kf →∞ Kp i O2 1 1   CO()graphite +=2()gg CO () ∆GT2 −+119025 83. 482 ..(17) νi 2 r  aiO2ν   i  ∑ i=1 K G The corresponding equilibrium constants were defined  i  according to   ...... (14) lim pO = = pOOe, q 2 1 2 ν p kf →∞  ap r    ∆G  ∏ p  aνi  K =−exp  = ...... (18) ∑ i=1 i νr   RT ar     ∏ r It should be noted that Eq. (13) yields the equilibrium For the sake of simplicity, it was assumed that all spe- activities for the parallel reactions, while Eq. (14) defines cies behave ideally i.e. γi = 1 for all species i. Morever, the the equilibrium activity oxygen, which has to be same for Raoultian activities of SiO2 and Cr2O3 were assumed to be all reactions. The final surface composition can only be 0.5. In the case of transient applications also the bulk phase obtained after solving the system of non-linear equations compositions need to be accounted for. However, the calcu- defined by Eqs. (9) and (10), which include the mass trans- lations presented in this paper are related to a single moment fer rates. It is important to understand that the composition in time only, and hence the composition of the species in the solved at the reaction interface is not the full chemical bulk phases including the slag was assumed to be constant. equilibrium that can be determined with traditional methods, Employing the stationary medium approach, the conserva- such as the equilibrium constant method or Gibbs energy tion equations can be written as follows: minimization, but rather a constrained chemical equilibrium: ρL ∞  aSiO2  the equilibrium permitted by the mass transfer rates onto and βL ()xaSi− Si −−kafS,0  iOp 2  ==0 f0 ...... (19) from the reaction interface. In essence, the LMA method ML  K0  studied in this paper is a rate expression for mass transfer  a05.  ρL ∞ 07. 5 Cr23O limited dynamic equilibrium. βL xaCr− Cr −−kafC,1  r p  ==0 f1 ...... (20) () O2  Based on the analytical expressions for the activities of ML  K1  oxygen and oxidizing species at the reaction front it can be deduced that their values approach the equilibrium values ρL ∞  05. 1− pO2  βL xaCC− −−kafC,2 p ==0 f2 ...... (21) when the forward rate coefficient kf → ∞. This means that () O2  ML  K2  the results obtained from the law of mass action method p ρ approach the solution obtained with the traditional equi- G ∞∞L ββG ()xpOO22− −−L ()xaSi Si librium constant method. In comparison to the equilibrium RT ML constant method, the main advantage the LMA method is ... (22) ρL ∞ ρL ∞ that its starting point is the kinetic rate expression and the −−07..50βL ()xaCr Cr − 5ββL ()xaCC− ==0 f3 control volume approach. As discussed broadly by Rao,12) ML ML the equilibrium constant method is numerically very stiff In addition, the system was set subject to the following and the use of the standard Newton’s method might be constraints: numerically unstable. axSiOS22=iO = 05...... (23)

 ax=−1 ...... (24)  AG+ ∆ k  Cr23OSiO2 exp − *   In addition to these it is necessary to find sufficiently Kk RT  A  Error =−11 =−   =−1 exp − ...(29) large values for rate coefficients kf,k that satisfy the equi-    Kk  ∆Gk   RRT  librium requirement. At the equilibrium, the affinity i.e. exp−  the driving force of each reaction should be equal to zero,  RT  resulting in definitions of equilibrium constants. As dis- Assuming a temperature of T = 1 873 K (1 600°C), and cussed earlier, an accurate equilibrium can be solved only if assigning A with values of 100, 1, 0.01 and 0.0001 J/(mol∙K) the forward rate coefficient approaches infinity. This is not yields an relative error of 6.4×10,–3 6.4×10,–5 6.4×10–7 and computationally possible. Instead of having zero affinities at 6.4×10,–9 respectively. Thus it can be concluded that the the reaction front, a concept of residual affinity is introduced error in the equilibrium constant is negligible. The above and employed to move the reaction front thermodynamically 7×7 non-linear system can be solved effectively with to a small distance away from the equilibrium, making the numerical methods, for example the Newton’s method. solution numerically possible. In addition, the same value Mathematically, this can be expressed in matrix form as of residual affinity was employed for all reactions which follows: means that the thermodynamic driving forces of all reactions Jy×=−∆ f ...... (30) are balanced at the reaction front. The absolute value of the where J is the Jacobian matrix, Δy is the column vector of residual affinity is very small in comparison to the standard activities and rate coefficients and f is the column vector Gibbs free energy of reaction; for the studied reactions the of residuals. The Jacobian matrix consists of the first-order absolute values of ΔG° are in the order of 105–106 J/mol at differentiates of the residuals with respect to free variables 1 873 K. The residual affinity of a reaction can be expressed (see Table 1). The vectors Δy and f are defined as follows: as follows: T ∆∆y = []aaSi,,∆∆ Cr apCO,,∆∆2 kkff,,012,,∆∆kf, ... (31) ν p  ap  ∏ p −=AG∆  + RT ln  T νr ...... (25) f = []ffff01,,234,,ff,,56f ...... (32)  ar   ∏ r  Corrections to variables y are obtained by solving Eq. The residual affinities of reactions 0, 1 and 2 need to be (30) and updating them according to Eq. (33). re-written for implementation as residuals: yynn+1 =+∆y ...... (33) aSiO2 apSi O2 − ==0 f4 A small numerical constant, ε, was required for the  AG+ ∆   exp − 0 ...... (26) numerical stability in the case of negative exponents in the   −200  RT  Jacobian matrix. Here, a value ε = 10 was adopted. Ini- tial guess value of activities was 0.5 for all species and the a05. 07. 5 Cr23O iteration was continued until the root-mean square (RMS) apCr − ==0 f5 −16 O2  of the correction vector was less than 10 . This required  AG+ ∆ 1  ...... (27) exp −  typically only 5–20 iterations. The value of A was set equal  RT  to 0.001 J/mol and thus the driving residual force was equal for all the studied reactions. 05. 1− pO2 apC − ==0 f6 An additional and important aspect of the new method O2   AG+ ∆ 2  ...... (28) is the possibility to study reversible reactions. As discussed exp −  RT above, the interfacial composition is shifted a small distance   away from the equilibrium; this distance is measured using With the exception of the residual affinities added to the the residual affinity A. The value of A is always set to be change standard Gibbs free energy of reaction, Eqs. (26), negative which means that in principle, the direction of the (27) and (28) correspond to the traditional equilibrium con- reaction is always from left to right for a positive value stant method. The resulting relative error in the equilibrium of kf. Then, in the case of a reversed reaction, it would be constant can be calculated by comparing the restricted equi- necessary to change the sign of the residual affinity. But in * librium coefficient Kk to the true equilibrium coefficient,K k: practice, for this approach we would need to monitor the condition by a conditional if-statement, which would cause

∂fi Table 1. Elements of the Jacobian matrix, Jij, = . ∂J j i\j 0 1 2 3 4 5 6 ρ a L kp SiO2 0 −−βL fO,0 2 0 0 −kf,0asi −+apSi O2 0 0 ML K0 05. ρ 07. 5kafC,1 r L 07. 5 aCr O −−βL kpf,1 − 07. 5 23 1 0 O2 0 02. 5 0 −+apCr 0 M O2 L max()pO2 , K1

ρL 05. kafC,2 kf,2 05. 05. 1− pO2 −−βL kpf,2 − − 2 0 0 O2 05. 0 0 −+apC O M K2 2 L max,()pO2  K2 ρL ρL ρL pG 3 βL 075. βL 05. βL −βG 0 0 0 ML ML ML RT

4 pO2 0 0 aSi 0 0 0

07. 5aCr p075. 5 0 O2 0 02. 5 0 0 0 max()pO2 ,ε  A  exp 05.   p 05. aC  RT  6 0 0 O2 0 0 0 05. + K2 max()pO2 ,

© 2016 ISIJ 1546 ISIJ International, Vol. 56 (2016), No. 9 an undesirable discontinuity. Instead, we used the following on the rate limiting mechanism was studied at carbon mole simple approach to treat reversed reactions. fractions of 0.04 and 0.01. These correspond to the process As an example, the oxidation reaction of Cr is studied conditions during the early stage of carbon removal in the in the following. As discussed earlier, the forward rate AOD process. The model parameters were taken to be as −4 coefficient in the law of mass action rate expression should βL = 5×10 m/s, T = 1 873 K (1 600°C), xCr = 0.17, xSi = approach to ∞ in order to achieve equilibrium condition and 0.002 and A = 0.001 J/mol. The employed liquid phase fulfill the definition of equilibrium coefficient. To avoid the mass transfer coefficient corresponds roughly to the highest need to use infinite k values we defined a slightly falsified experimental value presented by Chatterjee et al.,48) who equilibrium, namely Eq. (27), having the residual affinity measured apparent liquid phase mass transfer coefficients (Gibbs free energy) of finite value on the negative side, if during top-blowing of oxygen on liquid silver. Figures 2 the reaction proceeds from left to right. We now solve the and 3 present the rate control of different mechanisms and “falsified” Cr2O3 activity from Eq. (27) and substitute it to the selectivity of oxygen, i.e. the share of oxygen consumed the rate of law of mass for Cr oxidation, resulting in Eq. (34) by each of the studied reactions. The values calculated by that then defines the rate of oxidation. the method of Wei and Zhu17) are also presented in Figs. 2 and 3, as defined later (and calculated) by Eqs. (41)–(43). 07. 5   A  Rk′′f =−fC,1apr 1 exp − O2    ...... (34) These are not dependent on the mass transfer coefficients.   RT  This example illustrates the flexibility of the proposed Following the definition of the residual affinity, Eq. (25), method as well as its sensitivity to boundary conditions. the value of this rate is always positive with positive values The simplified model was derived in such a way that of A and kf,1. It should also be noted that the value in the it solves automatically the mass transfer controlled rates, parenthesis is equal to the relative error in the equilibrium the surface composition, and also the values for rate coef- constant, see Eq. (29). Physically, the numerical values of k ficients required to reach a pre-defined residual affinity A. must to be always positive. But as a trick, being numerically The hypothesis was that if the values of rate coefficients are a free variable, we could let k have negative values too in increased, the rates become mass transfer limited and the the case of reversed reaction. Assuming the same values for surface will reach equilibrium. Consequently, the next task activities, but an opposite sign of A, results in a reversed is to determine the values of rate coefficients kf so that they reaction. More specifically, the driving force is moved to satisfy the requirement set for the residual affinity when the the other side of the equilibrium. gas side mass transfer coefficient is changed. Figures 4 and 5 show these rate coefficients obtained as a solution of the 07. 5   A  Rkb′′ =−fC,1apr 1 exp O2    ...... (35) system of equations.   RT  Three different regimes can be observed: a) the rate is This rate is always negative with positive values of A and controlled by the mass transfer of oxygen, b) the rate is kf,1. The condition on the surface can be such that the mass controlled by the mass transfer of the liquid phase and c) transfer wants to turn the reaction opposite but with the Eq. Cr2O3 is reduced and the rate is control by the mass trans- (34), this would not be possible and Eq. (35) should be used. fer of the liquid phase. As seen in Figs. 2 and 3 when the Our question is now: if we set the sign of A to be positive and let the sign of the kf change in Eq. (34), do we obtain the same result as by using Eq. (35)? The ratio of negative rate (Eq. (34)) and the accurate reversed reaction (Eq. (35)) can be written as follows:  A  exp − −1 −R′′  RT  exp()−X −1 f =   = Rb′′  A  1− exp()X 1− exp   RT  ... (36) X23X X23X 11−+X −…−1 −+X −… = 26 = 26≅ 1 X23X X23X 11−−X −−… −−X −… 26 26 A −Rf′′ When X = approaches zero, the value of Fig. 2. O2 selectivity and rate limiting mechanisms, xC= 0.04. RT Rb′′ approaches unity. Numerically, at 1 600°C, substituting A = 1 000 J/mol, 1 J/mol or 0.001 J/mol yields 0.937801, 0.999358 and 0.999994, respectively. Therefore, the use of negative values for kf is justified when the value of A is sufficiently small.

3. Results and Discussion By using the new LMA approach, we calculated the momentary mass transfer controlled rate of reactions between O2, Si, C and Cr and composition at the steel surface. The studied cases consider a single moment of time and hence bulk phase composition was assumed to be constant.

3.1. Effect of Mass Transfer Coefficients

The effect of the ratio of gas and liquid side mass transfer Fig. 3. O2 selectivity and rate controlling mechanisms, xC= 0.01.

gas side mass transfer is very effective typically βG > 1.3 kf are required. m/s, there is an excess of O2 available at the surface. All species on the liquid side are then driven by the maximum 3.2. Effect of Residual Affinity diffusion rates towards the surface and selectivity of O2 is The effect of residual affinity A on the reaction rates, defined by the molar composition of the bulk liquid and the was also studied. The following parameters were employed −4 stoichiometry. When the gas side mass transfer coefficient in the calculations:βL = 5×10 m/s, 1 873 K (1 600°C), decreases below this critical value, the amount of O2 at the xC = 0.02, xCr = 0.17 and xSi = 0.002. Figures 6 and 7 surface decreases and the mass transfer of O2 becomes the show the calculated rate coefficients required to reach pre- rate-determining step. In that case, the selectivity of O2 is defined affinities for parallel oxidation reactions of Si,Cr defined by the combination of the equilibrium and the liquid and C. The studied range of residual affinities is 10 −5 ≤ A ≤ side mass transfer rates and a different reaction mechanism 104 J/mol. It should be noted that the upper range of values is obtained. As the gas mass transfer coefficient decreases is considerably in excess of the general requirement sug- the selectivity of the carbon and the silicon increases. If gested for Gibbs free energy minimization methods, which –3 12) the mass transfer rate of O2 is close to or smaller than the is typically 10 J/mol. Figures 8 and 9 present the effect mass transfer rates of C and Si, Cr is no longer oxidized. of the residual affinity on the actual reaction rates. More As the oxygen partial pressure decreases to low values, specifically, it is illustrated which value of the residual affin- Cr2O3 starts to reduce and oxidizes Si and C. Because the ity required to obtain a sufficiently high accuracy in terms transport of oxides was not assumed to be limiting, the of the reaction rates. thermodynamic equilibrium defines the selectivity; this can As can be seen, the required residual affinityi.e. the accu- be seen as the constant selectivity levels at the left side of racy requirement has a considerable effect on the required the Figs. 2 and 3. Employing the same boundary conditions, rate coefficient values. The closer the solution is to the the approach presented by Wei and Zhu17) yields entirely equilibrium, the larger are the rate coefficient values. The different results, as seen in Figs. 2 and 3, selectivity values reverse, of course, is also true. Numerically, the higher the given in the text boxes. The calculated selectivity in the Wei value of kf is the more difficult it is to obtain convergence and Zhu17) method is not dependent on the mass transfer and numerical solution. Therefore, in order to have to have rate and only weakly on the composition too. It should be highest possible computational efficiency, the maximum noted that rate coefficients are not the same for all reactions values of the affinity that give correct results should be used. for reaching the same affinity level, as shown in Figs. 4 and The results shown in Figs. 6–9 leave room for some 5. The sharp drop in the curve of the Cr oxidation reaction interesting observations. In the case of the highest gas side results from the onset of the reduction and changing the mass transfer rate, Fig. 8, the value of residual affinity has direction of the reaction reverse when delivery rate of O2 is no effect on the reaction rates within the studied residual limited. In the analytical expressions (13) and (14), only the affinity range of 10–5…104 J/mol. The overall rate is limited value kf → ∞ resulted in the equilibrium for all reactions. entirely by the liquid side mass transfer and consequently, It is important to understand that large, but finite values of the concentrations of oxidizing species are very low and

Fig. 4. Rate coefficients required for 0.001 J/mol residual affini- Fig. 6. Rate coefficients required for achieving different residual ties, xC= 0.04. affinities, βG=2 m/s.

Fig. 5. Rate coefficients required for 0.001 J/mol residual affini- Fig. 7. Rate coefficients required for achieving different residual ties, xC= 0.01. affinities, βG= 0.02 m/s.

The studied applications in the AOD process include mathematical modelling of reactions in a single gas bubble in liquid steel,19) during side-blowing decarburization20,21) and during the reduction of slag.22,23) The method has also been applied for mathematical modelling of chemical heating in the CAS-OB process.24) The traditional equilibrium coefficient method would be to insert these activities as the boundary condition at the reaction surface and then solve the non-linear system of equations. In practice, this becomes very challenging as there is no rate expression in the system that could be differentiated and used as a method to stabilize the convergence of the solution. For CFD applications, the equilibrium coefficient method is not acceptable as the system on the surface should be somehow interlinked to flow field Fig. 8. The effect of residual affinity on reaction rates,β G=2 m/s. composition solution and the rates should be properly differentiated. We will discuss below, how to implement the LMA method to a practical problem and how to avoid infinite values for rate coefficients that would result in numerical overflow. The most common approach to consider mass transfer controlled chemical reactions in metallurgical applications, and especially in the steel converters, is to simply derive the mass transfer limited local rate for some selected geometry, see Figs. 1(a)–1(c), and then use the equilibrium composition, solved from equilibrium constant, at the reaction front as a boundary condition. A typical example could be the oxidation of carbon particles, by the reaction 2C + O2 ⇌ 2CO.4) The local mass transfer controlled oxidation rate of 2 carbon particles RC′′ (in mol/(m ∙s)) can be obtained simply as follows.

ββGGp ∞∞seGGp q Fig. 9. The effect of residual affinity on reaction rates,β = 0.02 m/s. RCO′′ =−22xxO2 =−xx2 G RT ()O2 RT ()O2 2 ...(37) βGGp  ∞ pCOOG βG p ∞ =−2  xO2  ≅ 2 xO2 there is plenty of O2 available on the surface. As noted ear- RT K RT lier, the rate mechanism is not controlled by the equilibrium   and the liquid side mass transfer defines the reaction rates. where x is the mole fraction. Overall rate can be obtained The component having the highest mole fraction or the high- just by multiplying Eq. (37) by the specific surface area Sa est rate of mass transfer gets most of the oxygen. When the [m2/m3] of carbon particles within the control volume. This gas side mass transfer begins to limit the rate, see Fig. 9, is very useful approach for single reactions or simple reac- this means that the maximum O2 delivery rate approaches to tion systems and can be easily implemented into process the mass transfer rate of minor species, here C and Si. These simulators or even to CFD.25,26) However, if the system simple examples given above show that in order to have a involves multiple and parallel reactions and the rate limit- proper solution for the case of parallel gas-liquid oxidation ing step is not known beforehand or it varies during the reactions, it is necessary to consider both mass transfer and process, this approach is not viable as the overall model restricted chemical equilibrium. Because the mass transfer would become a vast combination conditional of expres- conditions may vary during different process stages in steel- sions. Solving the equilibrium composition directly from an making, it is more preferable to use a more general model equilibrium constant based non-linear system of equations that incorporates all the relevant mass transfer and reaction is also complicated, because the system is stiff and the con- mechanisms instead of predefining the rate-determining vergence is difficult to obtain.12) step, as suggested also by Oeters.18) To overcome the challenges in heavy computational CFD problems, Lu and Pope27) and Ren and Goldin28) developed 3.3. Applications and Comparison to Existing Approaches the in situ adaptive tabulation (ISAT) method to speed up The main advantage of the employed approach is that extremely time consuming computational calculations of the local reaction rates and the equilibrium composition are reactive flows. As an example, they studied the possibil- solved simultaneously. Moreover, it provides a convenient ity to speed up the solution of the chemical equilibrium in approach to define the source term of chemical reactions connection to the CFD simulation. They used a relaxation in relation to volumetric unit. This is advantageous for model wherein the rate is assumed to be proportional to the mathematical and computational fluid dynamics (CFD) difference between the actual local and equilibrium con- based models, which employ the control volume method to centrations, i.e. Ri′′ ~(ci – ci,eq). In general, it is necessary to solve the conservation equations for momentum, mass and solve the equilibrium composition using restricted equilib- energy. In comparison to traditional Gibbs energy solvers, rium Gibbs free energy solvers, which is time consuming. an additional benefit is that the source terms of species can However, the equilibrium composition at the surface is not be differentiated with respect to the mass transfer rates and the true equilibrium composition, but instead a restricted the composition of the fluid flow, increasing the efficiency equilibrium for which, the traditional methods are not suit- of the Newton’s method. able. They were able to use the ISAT method successively The method discussed also in this paper has already been and showed that the calculation time was reduced by a fac- employed in different cases for defining the mass transfer tor of hundreds. limited rates of reactions based on the law of mass action. A short review of other methods comparable to the LMA

1549 © 2016 ISIJ ISIJ International, Vol. 56 (2016), No. 9 method is provided in the following. in AOD converter during combined top- and side-blowing and assumed that the O2 is distributed to different dissolved 3.3.1. Constrained Gibbs Free Energy Minimization elements, such as Si, Cr and C in linearly to chemical affini- 29,30)  Koukkari et al. proposed a method for calculat- ties, Ak = −∆Gk of oxidation reactions. Positive sign refers ing the constrained chemical equilibrium by means of to reactions going from left to right. The main assumption introducing rate constraints to traditional Gibbs energy in this method is that all the species on the liquid side have minimization routine. One possibility is to use constraints equal availability and the selectivity is linearly dependent on for rate-controlled (slow) reactions, which then leads to an the affinity of the reactions,i.e. their thermodynamic driving effective combined thermodynamic-kinetic algorithm for force. According to this method, O2 is distributed to three the calculation of the chemical state of the system. This parallel oxidation reactions of Si, Cr and C with fractions Γk: method is useful in process simulations, where the input and A0 Si +=OS22 iO Γ0 ... (38) the output flow rates can be described without diffusion or AA++10/.75 10 /.5A a more complex mass transfer effects. In the applications 012 studied in this study, the starting point is the species con- 10/.75A1 Cr +=07..50OC22 5 rO31Γ ..(39) servation equations such as Eq. (37) having a complex inter- AA01++10/.75 10 /.5A2 linked equations for chemical rate, mass transfer, diffusion, 10/.5A2 convection, turbulence effects and all these often in three CO+=05. 22 CO Γ ... (40) dimensions. For this approach, explicit and differentiable AA01++10/.75 10 /.5A2 rate expressions are required. where the affinities A0...Ak are calculated by the Eqs. (41)– (43) according to the bulk composition. 3.3.2. Affinity Based Selectivity Method Wei and Zhu17) calculated parallel competing reactions

Table 2. Comparison of methods to calculate parallel oxidation reactions.

Method References Applications Advantages Drawbacks Effective equilibrium constant method The concentration changes of ele- Parallel oxidation and Transparent, easy to implement, ments are determined based on reduction reactions in the suitable for on-line applications. 32–38) effective equilibrium constant and Hot Metal Dephosphori- electrical neutrality equation. zation and BOF processes Constrained Gibbs free energy minimization method Kinetic control considered Numerous applications, Commercial software available, Difficult implementation to CFD or through additional constraints in including combustion, combined with commercial complex mass transfer limited sys- the Gibbs free energy minimiza- chemical engineering and library on thermodynamic prop- tems with diffusion, convection and 29,30) tion routine. metallurgy. erties, derived from physical turbulence effects. Unsuitable for principles, well-suited for pro- on-line applications. cess simulators. Affinity based selectivity method Selectivity of oxygen corresponds Parallel oxidation and Transparent, easy to implement, Cannot be derived from physical to the oxygen affinity of species, reduction reactions in the suitable for on-line applications. principles. 17,43–47) additional constraints for mass AOD process. transfer limitation. Constrained solubility method Dissolved oxygen content is Parallel oxidation and Transparent, easy to implement, Cannot be derived from physical determined based on its solubility reduction reactions in the suitable for on-line applications. principles. Solution is sensitive to limit. Thereafter, the remaining VOD process. model parameters, such as volumet- oxygen is distributed so that the 31) ric flow rates into and from the reac- reaction with the largest oxygen tion zones. affinity consumes all gaseous oxygen during a sub-time step. Coupled Gibbs free energy minimization and volume element method Volume cells with multiple phases Parallel oxidation and State of the art combination of Treatment of the interface in unclear, are set into equilibrium with tra- reaction reactions in the CFD and equilibrium calcula- grid size dependency of the solution, 39,40) ditional Gibbs free energy mini- AOD and BOF processes. tions, uses commercial software large computational expense pre- mization routine. in CFD and equilibrium solution. vents on-line applications. Incremental step method Time steps are divided into sub- Parallel oxidation and Transparent, easy to implement, Solution is sensitive for model steps, during which the reaction reduction reactions in the possibly suitable for online appli- parameters and cannot be derived with the largest affinity consumes 41,42) BOF process. cations. from physical principles. The imple- all available oxygen mentation includes non-physical tun- ing parameters. Law of Mass Action based method Rate expressions are calculated Parallel oxidation and Derived from physical principles, No experience on suitability for CFD from the law of mass action, pre- reduction reactions in the non-sensitive to model parame- applications. defined residual affinity or equi- AOD and CAS-OB pro- ters, easy to implement, simulta- librium number determines 19–24) cesses. neous solution of the mass trans- unknown forward rate coeffi- fer limited rate and equilibrium cients in an iterative procedure. composition, suitable for online applications.

ematical model for the basic oxygen furnace (BOF) process.  a   SiO2 ...... (41) The principal approach in their model is that the supplied −=AG00∆ + RT ln ∞   apSi O2  oxygen is used momentarily only by a single reaction that has the highest affinity, as defined by Eqs. (41)–(43). Gas-  a05.  eous oxygen is supplied to the reaction surface by small  Cr23O −=AG11∆ + RT ln 07. 5  ...... (42) step amounts and the composition of the reaction surface  apCr   O2  is constantly updated. When one species is rapidly depleted from the surface, its affinity decreases and in the new situ-    aCO ation, another species that has then the highest affinity will −=AG22∆ + RT ln 05.  ...... (43)  apC  continue to consume O2 and so forth. The model has been  O2  reported to be very sensitive to the employed parameters, Generally, the term in the parenthesis is referred to as especially to the amount and size of the O2 steps supplied to 42) the reaction quotient, Qk. When the reaction reaches equi- the system. In general, a numerical model should always 16) librium, Qk = Kk, and the affinity is equal to zero. The be well posed and should converge to same final result inde- drawback of this method is that it does not define the reac- pendently on the numerical model parameters.25,26) Table 2 tion rate directly. For this reason, it is not suitable to be used summarizes the main features of all models. with the control volume method as such. The advantage of this approach is its simplicity and that it does not require a separate solver for the equilibrium composition. Unfor- 4. Conclusions tunately, no experimental validation was given for most of The objective of this paper was to present a novel law the components, and the detailed derivation of the model of mass action law based rate method for modeling parallel was not presented. reversible mass transfer limited reactions in metallurgical systems. At first, we discussed the available methods from 3.3.3. Constrained Solubility Method open literature for comparison. Then, we re-derived the new Ding et al.31) proposed a mathematical model for the method for n parallel reactions in general form and discuss vacuum oxygen decarburization (VOD) process. Reactions its practical implementation. were assumed to take place in two reaction zones (metal-gas As a numerical example, a simple reaction model was zone and metal-slag zone), while the volumetric flow rate of derived for the case of a liquid steel surface exposed to species from and to the reaction zones was estimated based O2–CO gas mixture. The model was employed for studying on the plant data. Two different oxygen distribution models the parallel oxidation of Si, Cr and C under conditions corre- were investigated for the metal-gas zone. In both models, sponding to those of the AOD process. To be transparent, the every time step Δt is divided into n smaller time steps Δts simple model developed here included only the conservation and the calculation proceeds so that only the reaction with equations of all species wherein the rate expressions were the largest affinity takes place and consumes all the oxygen based on a modified law of mass action. As an additional available during the time step Δts. The procedure is repeated requirement, the affinities of the reactions at the reaction in an iterative manner until the Gibbs free energy change of front were formulated such that they reach a pre-defined all reactions is zero or positive. value, referred to as the residual affinity. This parameter assures that all the reactions are at the same distance from 3.3.4. Effective Equilibrium Constant Method the constrained equilibrium. Our simulations showed that The effective equilibrium constant method, also known when the residual affinity is smaller than 10 J/mol, a suffi- as coupled reaction model, has been applied extensively cient degree of accuracy for the equilibrium is obtained. As for mathematical modelling of hot metal dephosphoriza- a practical matter, a residual affinity of A = 0.001 J/mol was tion,32–34) desiliconization35) and decarburization.36–38) In employed for all reactions in the final calculations. this approach, the equilibrium constants are modified to Despite the relative simplicity of the employed model, the effective equilibrium constants, which in combination with obtained results provide an in-depth analysis on the oxida- electro-neutrality condition, are employed to solve a system tion kinetics of molten stainless steel. In the case of the of parallel mass transfer limited reactions. highest gas side mass transfer the value of residual affinity has virtually no effect of the reaction rates, within the range 3.3.5. Coupled Gibbs Free Energy Minimization and Con- of 10–5…104 J/mol. This case is always limited by the liquid trol Volume Method side mass transfer and there is plenty of residual O2 avail- Ersson et al.39) and Andersson et al.40) coupled CFD able on the surface. The rate mechanism is not controlled models of BOF and AOD converters, respectively, with by the equilibrium and the liquid side mass transfer defines computational thermodynamics. The local control volume the reaction mechanisms. The component having the high- equilibrium composition is achieved from the present est mole fraction or the highest mass transfer rate consumes chemical species after each time step. The obtained rate most of the oxygen. of phase mass change from this is then used as a constant When the gas side mass transfer begins to control the rate for solving the rate at the new time step. Therefore, in rates, partial pressure of O2 at the reaction surface starts to addition to solution of the flow field, equilibrium calcula- decrease. Then, the role of the surface equilibrium becomes tions have to be carried at every time step. In order to save important and the species start to compete for the oxygen. computational effort, equilibrium calculations were carried Under such conditions the oxides of species with lower out only in the cells that have more than one phase present. oxygen affinity will be reduced by species with higher One major advantage of this approach is that it does not oxygen affinity. require information on the interfacial surface area as the The results obtained in this work show that in order to whole cell is set to equilibrium if an interphase exists. On have a proper solution for the case of parallel gas liquid the other hand, a drawback of the method is that the size oxidation reactions, it is an absolute requirement to consider of the computational cells at the reaction front should be both the mass transfer and the restricted chemical equilib- infinitesimal to describe the mass transfer correctly. rium. As the conditions vary during different process stages in steelmaking, it is preferable to employ a model that has 3.3.6. Incremental Step Method all the relevant reaction steps included. The proposed new Jalkanen41) and Virrankoski et al.42) presented a math- method is a transparent and direct solution of the reaction

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