Ind. Eng. Chem. Res. 1999, 38, 2607-2614
2607
Oxidation Kinetics of Ferrous Sulfate over Active Carbon M. R. Ro1 nnholm,† J. Wa1 rnå,† T. Salmi,*,† I. Turunen,§ and M. Luoma‡ Laboratory of Industrial Chemistry, Process Chemistry Group, Faculty of Chemical Engineering, Åbo Akademi, FIN-20500 Åbo, Finland, Kemira Oy, Oulu Research Centre, P.O. Box 171, FIN-90101 Oulu, Finland, and Department of Chemical Engineering, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finland
Catalyzed oxidation kinetics of dissolved Fe2+ ions to Fe3+ over active carbon in concentrated H2SO4-FeSO4 solutions was studied with isothermal and isobaric experiments carried out in a laboratory-scale pressurized autoclave. The experiments were performed at temperatures between 60 and 130 °C, and the pressure of oxygen (O2) was varied between 4 and 10 bar. The kinetic results revealed that the oxidation rate was enhanced by increasing the temperature and pressure and that the catalytic and noncatalytic oxidations proceed as parallel processes. A rate equation was obtained for the catalytic oxidation process, based on the assumption that the oxidation of Fe2+ with adsorbed oxygen is rate determining. The rate equation for the catalytic oxidation has the simplified form r ) k′′cFecO0.5/(KO′′cO0.5 + 1) where cFe and cO are the concentrations of Fe2+ ions and dissolved oxygen, respectively, and k′′ and KO′′ are experimentally determined kinetic parameters. The total oxidation rate was simulated by including a previously determined rate equation for the noncatalytic oxidation into the global model, from which the kinetic parameters of the catalytic oxidation rate were determined. A comparison of the model fit with the experimental data revealed that the proposed rate equation is applicable for the prediction of the Fe2+ oxidation kinetics in acidic ferrous sulfate solutions. Introduction Ferrous sulfate is used in wastewater purification whereas ferric sulfate can be used in the production of potable water. The coagulation effect of trivalent cations is about 11 times that of a bivalent cation. Ferrous sulfate is oxidized to ferric sulfate in acidic solutions in the presence and also in the absence of a heterogeneous catalyst, e.g., active carbon. The overall reaction is written as follows:
Fe2+ + 1/4O2 + H3O+ f Fe3+ + 3/2H2O
(1)
The noncatalytic oxidation in the aqueous phase has been studied by many investigators (Cher and Davidson, 1955; King and Davidson, 1958; Chmielewski and Charewicz, 1984; Matthews and Robins, 1972; Ro¨nnholm et al., 1999) whereas practically nothing has been published concerning the oxidation mechanism and kinetics over heterogeneous catalysts. In a previous paper of our group (Ro¨nnholm et al., 1999), we considered the oxidation kinetics of ferrous sulfate to ferric sulfate with molecular oxygen in the absence of catalysts. A plausible reaction mechanism was considered, and rate equations were derived and fitted to kinetic data obtained from a batch reactor. It is, however, expected that the oxidation velocity can be considerably enhanced by the addition of a heterogeneous catalyst. The present publication is devoted to the oxidation of ferrous sulfate in the presence of an active carbon catalyst. In the presence of a heterogeneous catalyst, the noncatalytic oxidation in the liquid phase proceeds as a side process. Thus, a quantitative treatment of the * To whom correspondence should be addressed. † A ° bo Akademi. ‡ Kemira Oy, Oulu Research Centre. § Lappeenranta University of Technology.
heterogeneously catalyzed kinetics requires the consideration of the noncatalytic gas-liquid oxidation also. The endeavor of the current work is to present kinetic data of ferrous sulfate oxidation over active carbon and to model the data with a rate expression based on plausible surface reaction mechanisms. Experimental Section Isothermal kinetic experiments were carried out in a 600 mL Parr autoclave at temperatures of 60-130 °C and oxygen (AGA 99.5%) pressures of 4-10 bar. The total liquid volume was approximately 300 mL in the experiments. The initial concentrations of ferrous sulfate (FeSO4, technical quality) and sulfuric acid (H2SO4) were within the ranges of 2-2.5 and 1.3 M, respectively. The oxygen pressure was maintained constant during the experiment, and on-line pressure and temperature data were collected with a data acquisition system. The experiment was commenced by heating the vigorously stirred (>1000 rpm with a gas entrainment impeller) aqueous solution of FeSO4 and H2SO4 in the presence of dispersed active carbon particles. The reactor was coupled to a vacuum pump during the heating period. After the reaction temperature was attained, the oxygen pressure was applied and samples (2-10 mL) were withdrawn for chemical analysis. The amount of active carbon catalyst was 24 g. The catalyst was analyzed by nitrogen adsorption at -196 °C using a Carlo Erba Sorptomatic 1900 instrument. The BET specific surface area was found to be 770 m2/ g, and the pore specific volume was found to be 0.64 cm3/g. The contents of Fe2+ in the samples were determined by redox titration with a 0.1 M CeSO4 solution. The inflection point of the titration was recorded with Pt- and Ag-Cl electrodes connected to a potentiometer.
10.1021/ie980500p CCC: $18.00 © 1999 American Chemical Society Published on Web 05/25/1999
2608 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999
however, probable that both steps I and II contribute to the overall oxidation rate. For the sake of simplicity, we assume here that step II alone is rate-determining, whereas steps I and III-V are rapid enough so that the quasi-equilibrium hypothesis can be applied. Thus, the overall oxidation rate is given by
r ) r2
(2)
r2 ) k2cFeIIθO - k-2cFeIIIθO-
(3)
where r2 is written as
where θ0 and θ0- denote the coverages of the corresponding species. The application of the quasi-equilibrium approximation on steps I and III-V implies that
K1 ) θO2/cOθv2 K3 )
θOHcW θO-cW+
K4 ) K5 ) Figure 1. Surface reaction steps in the oxidation of Fe2+ to Fe3+ over active carbon.
Catalyzed Reaction Mechanism and Rate Equations To obtain a rate equation for the catalytic oxidation, it is necessary to assume a surface reaction mechanism over the active carbon catalyst. The heterogeneous oxidation mechanism was assumed to take place on the catalyst surface, where oxygen is dissociatively adsorbed. Fe2+ ions donate an electron to adsorbed oxygen, which reacts with a hydronium ion, forming a surface hydroxyl and releasing water. Two surface hydroxyls form water, leaving an oxygen-covered site and a vacant site on the catalyst surface. The molecular processes are illustrated in Figure 1, and the elementary steps on the catalyst surface are summarized below.
(4) (5)
θWθO
(6)
θOH2 cWθv θW
(7)
where subscripts v, W, and W+ refer to vacant sites, water (H2O), and hydronium ions (H3O+), respectively. From eqs 4-7, the surface coverages can be solved as a function of the fraction of vacant sites (θv). The following expressions are obtained:
θO ) K11/2cO1/2θv
(8)
θW ) K5-1cWθv
(9)
θOH ) K11/4(K4K5)-1/2cO1/4cW1/2θv
(10)
θO- ) K11/4(K4K5)-1/2K3-1cO1/4cW3/2cW+-1θv (11) After the expressions for θO and θO- are inserted in the rate equation (3) and it is recalled that the equilibrium constant of the overall reaction is given by
KC ) K11/4K2K3K41/2K51/2 the rate equation is transformed to
r)
(
k2K11/2cO1/4cW+-1
cFeIIcO1/4cW+
(12)
)
cFeIIIcW3/2 θv KC
(13)
The fraction of vacant sites is solved from the site balance
θO + θW + θOH + θO- + θv ) 1 Steps IIIb and IIIc are discarded in the subsequent treatment, because H2SO4 is a strong acid, whose contribution to the formation of surface hydroxyls is exclusively attributed to the hydronium ion. The ferrous (Fe2+) and ferric (Fe3+) ions are denoted by FeII and FeIII, respectively. The derivation of a rate equation is usually based on the assumption of a rate-determining step (rds). It is,
(14)
The expressions (8)-(11) are inserted in eq 14, after which θv is solved explicitly as
θv ) 1/(K11/2cO1/2 + K5-1cW + K11/4K3-1K4-1/2K5-1/2cO1/4cW3/2cW+-1 × (K3cW-1cW+ + 1) + 1) (15) The following simplifying notations are introduced:
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2609
KO ) K1
(16)
KW ) K5-1
(17)
K′ ) K11/4K3-1K4-1/2K5-1/2
(18)
K′′ ) K11/4K4-1/2K5-1/2
(19)
The fraction of vacant sites becomes now
-1
+
K′′)cO1/4cW1/2
+1 (20)
Furthermore, the product k2K11/2 is written as a lumped constant
k′ ) k2K11/2
(21)
After eqs 16-21 are inserted in eq 13, the final form of the rate equation is obtained:
r) + KWcW + (K′cWcW+
-1
+
K′′)cO1/4cW1/2
(
k′cO1/4
+1 (22)
The oxidation rate is dependent on the pH of the solution, because cW+ ) [H3O+] appears in the rate equation. The pH of the solution is calculated from the protolysis equilibria of sulfuric acid.
r)
r)
The rate and mechanism of the noncatalyzed gasliquid reaction and the liquid-phase processes between Fe2+ ions and dissolved oxygen are treated in detail in our previous paper (Ro¨nnholm et al., 1999). Here we just summarize the essential results. The noncatalyzed mechanism was assumed to proceed through the formation of an intermediate complex between dissolved oxygen and Fe ions and the cleavage of the O-O bond of the complex. The addition of two Fe2+ ions to dissolved oxygen leads to a peroxide-type complex, which is rapidly decomposed. The successive addition of Fe2+ ions to oxygen was assumed to be ratedetermining. The mechanism is summarized as follows:
Fe2+ + O2 f Fe-O-O2+ Fe
2+
+ Fe-O-O
(I) 4+
f Fe-O-O-Fe
(II)
The remaining steps are lumped to a quasi-equilibrium:
2Fe2+ + 4H3O+ + Fe-O-O-Fe4+ f 4Fe3+ + 6H2O (III) The application of the steady-state hypothesis on steps I and II, i.e., r1 ) r2, and the quasi-equilibrium
cFeIIcO1/4cW+
(KO1/2cO1/2
)
cFeIIIcW3/2 KC
+ 1)cW+ +
K′cO1/4cW3/2
(24)
Alternatively, if oxygen and water are the dominating surface compounds, the rate equation is simplified to
Noncatalytic Reaction Mechanism and Rate Equation
2+
(23)
The general form of the rate model is rather complicated; in particular, it contains several adjustable parameters. Thus, some simplifications are worth considering. These simplifications can be tested before progressing to the complete model, to achieve a suitable level of sophistication in the final model which is going to be used in practice. Some of the possible simplifications are discussed here. Very probably some of the surface intermediates are more abundant than the other ones. If the concentrations of the surface hydroxyls and water are negligible compared to those of O* and O*- (KW ≈ 0 and K′′ ≈ 0), the rate equation (22) is simplified to
k′cO1/4cW+-1(cFeIIcO1/4cW+ - cFeIIIcW3/2/KC) KO1/2cO1/2
knc2Knc1cFe2cO (1 - f) Knc1 1 + knc2 c knc1 Fe
Simplified Rate Equations
1 + KWcW + (K′cWcW+
r′ )
where f is a factor accounting for the reversibility of the process (see the Notation section).
θv ) KO1/2cO1/2
hypothesis on step III gives the rate expression for the noncatalytic process
( )(
)
cFeIIIcW3/2 k′cO1/4 cFeIIcO1/4cW+ cW+ KC KO1/2cO1/2 + KWcW
(25)
In practice, the rate equation (22) would be simplified even more, because the concentration of water is virtually constant, i.e., KWcW ≈ K′W, and eq 25 can be rewritten to
r)
( )(
)
k′′cO1/4 cFeIIIcW3/2 cFeIIcO1/4cW+ cW+ KC KO′′cO1/2 + 1
) k′′cO1/2cFeII
KO′′cO1/2 + 1
(26)
where k′′ ) k′/KWcW and K0′′ ) K01/2/KWcW. It is known that Fe2+ can be completely oxidized to Fe3+ in the presence of the catalyst; thus, a possible simplification of the rate equation is the irreversible case (KC f ∞), where the backward reaction terms in eqs 22 and 23-25 are discarded. These simplifications leave us with two parameters to fit (eq 26). The factor k′′ is estimated as a preexponential factor including an exponential factor E/R, which is estimated by transposing the temperature variables to be related to a reference temperature (1/x ) 1/T - 1/Tref) as proposed by Rose (1985). The concentration of H+ is considered to
2610 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 Table 1. Henry’s Coefficients for Oxygen Solubility in Water (Fogg and Gerard, 1991) A
B
C
temperature interval/K
-171.2542 -139.485
8391.24 6889.6
23.243 23 18.554
273-333 273-617
Table 2. Gas Solubility Parameters (Weisenberger and Schumpe, 1996) hG,O2/(m3 kmol-1)
103hT/(m3 kmol-1 K-1)
temperature interval/K
0
-0.334
273-353
hH+/(m3
The rate equations presented above include the liquidphase concentration of oxygen (cO). The concentration level of dissolved oxygen is determined by the masstransfer rates and the gas-liquid equilibrium in the oxygen-electrolyte solution. The gas-liquid equilibrium of oxygen is described with Henry’s law, which is suitable for sparingly soluble gases. The temperature dependence of Henry’s constant for oxygen in water is given by Fogg and Gerard (1991). For the actual case, however, the numerical value of Henry’s constant is influenced by the presence of the electrolytes in the solution. The estimation of the oxygen solubility in the electrolyte solution is considered in the sequel. The solubility of O2 in pure water is described by a temperature-dependent equation (Fogg and Gerard, 1991)
ln(atm/H0) ) A +
B + C ln T T
(27)
where H0 is Henry’s constant. The coefficients A, B, and C are listed in Table 1. In the present case, the solution is considered to contain the following electrolytes: Fe2+SO42-, Fe3+(SO42-)3/2, and H+HSO4-. For a mixture of electrolytes, Danckwerts (1970) has proposed the following correction of Henry’s constant:
log(H/H0) )
∑hiIi
(28)
applied for mixed electrolyte solutions according to Schumpe (1993), where hi is an ion-specific parameter
log
∑cizi
2
(29)
and the salting-out coefficients are calculated from
h ) hg + h+ + h-
(30)
where ci and zi denote the concentration and the valency of ion i, respectively. The salting-out coefficient is calculated from the contributions of the gas (hg), the cation (h+), and the anion (h-) (Charpentier, 1986). However, recent advances by Weisenberger and Schumpe (1996) convincingly showed that the Sechenov relation can be used
log
cG,0 ) KcS cG
(31)
where the parameter K (“Sechenov constant”) is specific to the gas as well as the salt and shows a moderate temperature dependency. The equation can also be
cG,0 cG
)
∑i (hi + hG)ci
(32)
Weisenberger and Schumpe (1996) extended the model of Schumpe to the temperature range of 273353 K by assuming hG, the gas-specific constant, to be a linear function of the temperature:
hG ) hG,0 + hT(T - 298.15 K)
(33)
The values used in this work are listed in Table 2. The mole fraction of oxygen at the gas-liquid interface is obtained from Henry’s law:
xO2* ) pO2/H
(34)
The interfacial liquid-phase concentration cO2* is obtained from
cO* ) xO2*cL
(35)
where cL is the total concentration of the liquid. In practice, cL is close to the water concentration: cL ≈ cW. Because the solvent has a notable vapor pressure at the highest experimental pressures, the measured total pressure (P) was corrected with the vapor pressure of the H2O-H2SO4-FeSO4 solution to obtain a reliable value of the partial pressure of oxygen (pO):
where hi is the salting-out coefficient and Ii is the contribution of electrolyte i to the ionic strength. The ionic strength is defined as
1 Ii ) 2
0 0.117 0.1523 0.448
hFe2+/(dm3 mol-1) hFe3+/(dm3 mol-1) hSO42-/(dm3 mol-1)
be constant at the present level of the acid concentration (Ro¨nnholm et al., 1999). Gas-Liquid Equilibria
kmol-1)
vp pO ) P - δPH 2O
(36)
vp where PH denotes the vapor pressure of water and δ 2O is a correction factor which depends on the solvent composition. The vapor pressure of water was calculated from the equation given by Reid et al. (1988)
ln
( )
(VPA)x + (VPB)x1.5 + (VPC)x3 + (VPD)x6 Pvp ) Pc 1-x
where
x ) 1 - T/Tc
(37)
The parameters VP A, VP B, VP C, VP D, Pc, and Tc are listed in Table 6. The correction factor (δ) is related to the weight fraction of (dihydrogen)sulfate (wH2SO4) in the solution. Based on data which are presented by Sippola (1992), the following empirical correlation was developed:
δ ) 1 + b1wH2SO4 + b2wH2SO42
(38)
The values of the coefficients are listed in Table 6. Calculations revealed that the vapor pressure of the solvent (wH2SO4 ) 0.3) varies typically from 0.1 bar at
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2611
of oxygen mass-transfer limitation. In the absence of mass-transfer limitation, the concentrations cO* and cO are equal, and cO is obtained from
Table 3. Results from Fitting the Parameters in Equation 26 (Rate Constant k′′ at Tref ) 80 °C) total SS (corrected for means) residual SS std. error of estimate explained R2 (%) fixed parameters estd parameters
estd std error
362 25 0.41 93 very large
KC
cO ) pO2cL/H
estd relative parameter/ std error (%) std error units
k′′ 0.614 × 10-3 0.33 × 10-3 KO′′ 0.09 0.44 E/R 1560 357
5.4 492 22.8
18.4 0.2 4.4
1/xM 1/M K
a
cW+, constant r, (22) HO, (27) h, (33) cG, (31) cO*, (36)-(39) dcFeII/dt, (51) dcFeIII/dt, (52) dcO/dt, (53)
Table 5. Results from Fitting the Parameters in Equation (23) (Rates at Tref ) 80 °C) 21.6 36.8 4176
M-2 min-1 M/min K
55 °C to 0.6 bar at 100 °C and 9 bar at 200 °C. This means that the solvent vapor pressure has to be included in the model, particularly at the lowest oxygen pressures and highest temperatures of the experimental domain. Because pure oxygen is present in the gas phase, the gas-phase mass-transfer resistance of oxygen is negligible, and the surface pressure pO2* equals the bulk phase pressure of oxygen, pO2. Consequently, the surface concentration of oxygen can be calculated from
cO* ) pO2cL/H
NO ) kLO(cO* - cO)
(41)
Liquid-Phase Mass Balances
The parentheses refer to equations in this paper.
knc2Knc1 knc1 E/R
In the presence of mass-transfer limitation, the surface and bulk phase concentrations are determined by the diffusional flux of oxygen (NO):
Equation 44 presupposes that the rate of the liquidphase oxidation is negligible in the liquid film. NO is incorporated in the oxygen mass balance for the reactor which is considered in the next section.
Table 4. Calculation Procedure for the Derivativesa 0. 1. 2. 3. 4. 5. 6. 7. 8.
(40)
(39)
Two different cases are of interest in relating the interfacial concentration (cO*) to the oxygen concentration in the bulk phase, i.e., the concentration (cO) used in the rate equation (22): the absence and the presence
The kinetic experiments were performed in a batch reactor. In the derivation of the reactor model, some fundamental assumptions were introduced. The catalytic and the noncatalytic oxidations were presumed to take place simultaneously. The reactor contents were assumed to be completely backmixed, and the diffusional resistance in the liquid film around the catalyst particles was neglected because of vigorous stirring. The kinetic and diffusional effects inside the catalyst particles were lumped together, to obtain a model for the global rate over the actual particles. Consequently, the internal diffusion effects were not considered in an explicit manner. On the basis of these hypotheses, the liquid-phase mass balances of the nonvolatile compounds (Fe2+ and Fe3+) in the reactor are given by
dni/dt ) rimcat + ni′VL
(42)
where ni is the amount of substance, mcat is the mass of catalyst, and ri and ri′ denote the noncatalytic and catalytic generation rates, respectively. The amount of substance is related to the concentration (ci) and the liquid volume (VL)
ni ) ciVL
(43)
If the liquid volume can be regarded to be virtually constant and the bulk density (FB) of the catalyst is
Table 6. Physical Parameters (Reid et al., 1988) M φ VO µ T
ln(µ/cP) ) A +
molecular weight of the solvent, 18 g/mol association factor of the solvent, 2.6 molar volume of O2 at 10 bar, 2.56 cm3/mol (Sherwood et al., 1975) viscosity (dynamic) of the solvent (water) temperature, 403 K
B + CT + DT2 T
Protolysis equilibrium constants: K1a ) 100/cH2O K2a ) 10(-1.5 - 4.5/295(T/K - 273.15))/cH2O (Sippola, 1992) A ) -24.71 B ) 4209 C ) 0.045 27 D ) -3.376 × 10-5 F ) 1333 kg/m3 ln(µ/cP) ) -1.2371 µ ) 0.2218 cP ) 0.22 × 10-3 Ns/m2 DO ) 5.2 × 10-8 m2/s kLO ) 0.0028 m/s
VP A ) -7.764 51 VP B ) 1.458 38 VP C ) -2.775 80 VP D ) -1.233 03 Tc ) 647.14 K (Lide, 1994) Pc ) 22.06 MPa (Lide, 1994) b1 ) -0.2542 (Sippola, 1992) b2 ) 3.91 (Sippola, 1992)
2612 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999
defined as
FB ) mcat/VL
(23) and (26) are inserted in the mass balance (45) of FeII. The result becomes
(
(44)
dci/dt ) FBri + ri′
(45)
FeII
(46)
-
ln (48)
The surface concentration (cO*) is obtained from eq 39. The generation rates (consumption and production rates) of the components are obtained by using the stoichiometric coefficients
ri ) νir
(49)
ri′ ) νir′
(50)
Taking into account the effect of the homogeneous reaction rate (Ro¨nnholm et al., 1999) that is of the same magnitude as the heterogeneous, we include the rates in the mass balances. After the stoichiometric coefficients of Fe2+, Fe3+, and O2 (νFe2+ ) -1, νFe3+ ) 1, and νO ) -1/4 in eq 1) are recalled and eqs 49 and 50 are inserted in eqs 45 and 48, the mass balances become
dcFeII/dt ) -FBri - ri′
(51)
dcFeIII/dt ) FBri + ri′
(52)
1 dcO/dt ) kLOa(cO* - cO) - (FBri + ri′) 4
(53)
where the rates r and r′ are obtained from eqs 22 and 23. For the case that oxygen mass-transfer resistance is negligible, eq 53 is omitted and cO is obtained directly from eq 39. The assumption that the role of the gasliquid reaction in the liquid film is negligible was confirmed by calculating the Hatta number and the enhancement number for a pseudo-first-order model r′ ) k′cO where k′ ) k1cFe. The role of the film reaction would be highest in the beginning of the reaction at the highest oxygen pressure. Calculations presented in our previous paper (Ro¨nnholm et al., 1999), however, revealed that the order of magnitude of the Hatta number is 7 × 10-2, which gives the enhancement factor ≈ 1. Consequently, the film reaction is negligible. An approximative analytical solution for the concentration of FeII can be obtained provided that the concentration of dissolved oxygen (cO) and the volume is assumed to be virtually constant and by neglecting the vapor pressure of the solvent. The rate equations
(
dcFeII
FBk′′xcOcFeII KO′′xcO + 1
+
a2cOcFeII
2
1 + (a2/a1)cFeII
)
)
∫0t dt (55)
After development of the integrand of eq 55 to partial fractions, integration, and rearrangement, the final result becomes
(47)
is introduced, the oxygen balance becomes
dcO/dt ) kLOa(cO* - cO) + FBrO + rO′
∫cc
Fe,O
where NO, the flux to the liquid phase, is given by eq 41, and A denotes the gas-liquid interfacial area. After the balance equation is divided by the liquid volume and the notation
a ) A/VL
(54)
which can be integrated by separation of variables
For the volatile compound, i.e., oxygen, the mass balance obtains the form
dnO/dt ) NOA + rOmcat + rO′VL
)
FBk′′xcOcFeII dcFeII a2cOcFeII2 + )dt KO′′xcO + 1 1 + (a2/a1)cFeII
the mass balance (42) can be rewritten to
[ [ c0,FeII cFeII
( )
] ]
cFeII +β c0,FeII R+β+1
(R + 1)
R/(R+1)
)
FBk′′xcOt KO′′xcO + 1
(56)
where the dimensionless quantities are R ) a1cO0.5/[FBk′′/ (KO′′cO0.5 + 1)] and β ) a1/a2c0,Fe. Computational Procedure The mathematical model is summarized as follows: mass balances of FeII, FeIII, and O2 hydronium ion concentration (cW+) water concentration (cW) gas-liquid equilibrium constant for O2
eqs 51-53 ∼constant ∼constant cW ≈ cW,O eqs 27 and 31-33
In the estimation of kinetic parameters from experimental data, the mass balance equations (51)-(53) were solved as a subproblem. The computational procedure for obtaining the values of dci/dt is summarized in Table 4. This procedure was implemented in a subroutine working in cooperation with a parameter estimation program, e.g., MODEST (Haario, 1994). The differential equations were solved with a stiff ODE-solver (Hindmarsh, 1983) during the parameter estimation. The approximative analytical model (53) was solved with a one-dimensional Newton-Raphson method using analytical derivatives. The following objective function (Q) was minimized in the parameter estimation:
Q)
∑(wFe ,exp - wFe )2 II
II
(57)
where wFeII,exp and wFeII denote the experimentally observed and from the model predicted amounts (in weight percent) of Fe2+ ion in the solution. The parameters were estimated with a consecutive simplex and a Levenberg-Marquardt algorithm. Parameter Estimation and Simulation Results Results from the kinetic experiments are displayed in Figure 2. From the figure we can observe that the kinetic behavior at different temperatures and pressures is consistent within the experimentally covered domain; the oxidation rate is enhanced by increasing the temperature and pressure.
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2613
Figure 2. Experiments at 4 bar: 60 °C (‚), 80 °C (O), 100 °C (×). Experiment at 10 bar: 100 °C (+). Lines by simulation.
Figure 4. Oxygen mass transfer: kLOa g 10 min-1, kLOa ) 1 min-1, and kLOa ) 0.1 min-1 as the slowest.
Figure 5. Approximate analytical (- - -) and experimental (‚‚‚) models compared to the numerical solution (s). Figure 3. Scaled maximum likelihood surface plot of parameters k′′, KO′′, and E/R.
The numerical values of the kinetic parameters obtained for the noncatalytic process (Ro¨nnholm et al., 1999) are listed in Table 5. These values were fixed when we fitted the catalytic rate equation (26) to experimental data. The results of the parameter fitting for the catalytic rate are summarized in Table 3; the degree of explanation is 93%. The presented model predicts not only the reaction in the presence of an active carbon catalyst but also the oxidation of ferrous sulfate proceeding in the absence of a catalyst. We conclude that there does not exist any competing reactions within the covered experimental domain. A sensitivity analysis is presented as a scaled maximum likelihood surface in Figure 3, where the same model with different parameter values is viewed as different models and the response surface is presented as the response divided by the optimum. The preexponential factor of k′′ was estimated to 0.0061 M-0.5 min-1sat Tref ) 80 °Csand the exponential factor to 1560 K, which gives an apparent activation energy of 13 kJ/mol. The parameter KO′′ was estimated to 0.09 M-1. The parameter KO′′ in the model is not allowed to be negative because being negative neither is easy to motivate scientifically nor would improve the model significantly. The parameters k′′ and KO′′ are strongly correlated and that is why it is recommended that all
three variables be used together in the modelsnot separately. For the sake of comparison, the concentration of FeII was simulated using the approximative analytical model (eq 56) at different temperatures and pressures. An average concentration of dissolved oxygen was used in simulations. A comparison of the approximative solution and the exact numerical solution is provided in Figure 5, which reveals that the approximative analytical solution gives a reasonable prediction of the progress of the oxidation process. The comparison is made at 4 bar and 100 °C. The sensitivity of the kinetic model with respect to the gas-liquid mass-transfer parameter (kLOa) was investigated by simulating the reactor model with different numerical values of the parameter. The results are displayed in Figure 4. The simulation reveals that the role of oxygen mass transfer in the prediction of the concentration of FeII is negligible for kLOa, values which are greater than 10 min-1. Acknowledgment The project was financed by the Technology Development Centre of Finland (TEKES). Notation A ) gas-liquid mass-transfer area A, B, C ) parameters for Henry’s constant
2614 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 a ) gas-liquid mass-transfer area-to-liquid volume a1, a2 ) rate parameters for noncatalytic oxidation bn ) parameters for estimating solvent vapor pressure correction ci ) concentration of ion i f ) reverse reaction rate ) f ) cFeIII4cW4/KCcFe4cOcW+4 h ) salting-out coefficient hG,0 ) gas-specific parameter hi ) ion-specific parameter hT ) gas-specific parameter for the temperature effect H, HO ) Henry’s constant I ) ionic strength K ) Sechenov constant (eq 31) K ) equilibrium constant K′, K′′ ) lumped equilibrium constants k ) rate constant k′, k′′ ) lumped rate constants kL ) mass-transfer coefficient in liquid phase m ) mass N ) molar flux n ) amount of substance p ) partial pressure Pc ) critical pressure Q ) objective function R2 ) degree of explanation; multiple correlation coefficient r ) reaction rate ri ) generation rate of compound i t ) time T ) absolute temperature Tc ) critical temperature V ) volume VP n ) parameter w ) weight fraction x ) mole fraction z ) valency Greek Letters R,β ) dimensionless parameters of the approximative analytical model δ ) vapor pressure correction ν ) stoichiometric coefficient ν′ ) stoichiometric number FB ) catalyst bulk density θ ) surface coverage Subscripts and Superscripts B ) bulk, in bulk density (FB) c ) concentration-based quantity cat ) catalyst g ) gas i ) component index j ) reaction index L ) liquid phase nc ) noncatalytic T ) thermodynamic quantity v ) vacant site + ) anion - ) cation * ) interfacial quantity
Abbreviations FeII ) ferrous ion (Fe2+) FeIII ) ferric ion (Fe3+ ) HSO4- ) hydrogen sulfate ion H2SO4 ) sulfuric acid 0 ) total (of sulfate) O ) oxygen (O2 or surface oxygen O*) O- ) charged oxygen ion (O*-) OH ) surface hydroxyl (OH*) SO4 ) sulfate ion (SO4-) W ) water (H2O) W+ ) hydronium ion (H3O+) * ) surface site
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Received for review July 30, 1998 Revised manuscript received April 6, 1999 Accepted April 6, 1999 IE980500P
