Ind. Eng. Chem. Res. 2003, 42, 5091-5098
5091
Parameter Estimation and Sensitivity Analysis of Lumped Kinetic Models for Wet Oxidation of Concentrated Wastewaters S. Verenich,* A. Laari, and J. Kallas Department of Chemical Technology, Lappeenranta University of Technology (LUT), P.O. Box 20, Lappeenranta FIN-53851, Finland
Three different lumped kinetic models for a wet oxidation process were used in this study to describe an originally complicated wet oxidation reaction system. The model parameters, the reaction orders for oxygen, frequency factors, and activation energies for the model reactions, were estimated by comparing measured data to predictions made using the model. Sensitivity analysis, a study on the cross-correlation and identification of the parameters, was carried out systematically during the parameter estimation. The models, which are presented together with the estimated parameters, explained the measured data to a good degree of agreement. This study shows that with proper parametrization the obtained parameters are well-identified and do not significantly correlate with each other. This is important in order to obtain meaningful parameters that are as reliable as possible. The parameter estimation and sensitivity analysis showed that the more complicated models, which include the biodegradability changes, were statistically more reliable and their prediction properties were better. It was also demonstrated that the wet oxidation reaction rates are dependent on the oxygen concentration in water. Introduction Wet oxidation (WO) is a well-established method for the purification of industrial wastewaters that cannot be treated with traditional purification processes because of their high concentration of organic matter and/ or toxicity.1,2 The WO process involves the use of oxygen as an oxidizing agent for the removal of pollutants in the liquid phase under elevated conditions (0.5-20 MPa and 125-300 °C).3 WO is, however, a complex process, especially if it involves the oxidation of industrial streams that contain large numbers of organic compounds. The oxidative process proceeds along complex reaction pathways and leads to the formation of different intermediates. It is impossible to trace all of the chemical transformations that take place in industrial wastewaters, which are enriched with different types of organic pollutants. However, merging different compounds, which have similar properties or structures, into certain lumps allows the oxidation process of multicomponent systems to be described. The generalized lumped kinetic model (GLKM) proposed by Li et al.4 is perhaps the model that has been applied the most extensively and has been employed for the different types of wastewaters.5-7 The complex reaction system has been divided into three categories. The first group represents the unstable parent organic substances that convert to both partially oxidized compounds, often represented by acetic acid (the second lump in the model), and the end products of oxidation. Zhang and Chuang,8 Belkacemi et al.,9 Donladgicˇ and Levec,2 and Lopez Bernal et al.10 have elaborated other reaction schemes that illustrate the process of heterogeneous catalysis and homogeneous oxidation. Verenich and Kallas11,12 have merged the compounds in wastewater according to their resistance toward bio-oxidation in order to be able to predict the biodegradability of the WO-treated effluent. * To whom correspondence should be addressed. Fax: +1 919 515 4556. E-mail:
[email protected].
The aforementioned models are able to predict the time dependence of the concentrations of the organic compounds in batch experiments. However, only a few studies have considered a detailed statistical analysis for the validation of the proposed model. Zhang and Chuang8 have used the variance inflation factor to determine if the model is overparametrized. Belkacemi et al.9 have validated their six-parameter model with a goodness-of-fit probability function, a correlation matrix, and an unweighted least-squares criterion. The aim of this work was to apply a detailed statistical analysis to the lumped models used for the prediction of time-dependent concentrations during the WO process. Sometimes, statistical parameters, such as the standard error, data-model plots, maximum likelihood, and so forth, when used alone can be misleading.13 This is an actual problem for nonlinear models and for cases of strongly correlated parameters such as the parameters in the Arrhenius law equation. However, the information from the sensitivity analysis provides a global picture of the identifiability of the estimated parameters. Kinetic Models For the present study, three lumped kinetic models have been chosen for the parameter estimation and sensitivity analysis. They are the lumped models by Zhang and Chuang8, Verenich and Kallas11 and a modified version of the latter model with the implementation of an additional lumped group.12 The first model (LKM)8 consists of two groups of organic substances that are present and formed during the WO process. In the reaction scheme, the parent organic material, A, which is present in the wastewater stream, transforms to a partially oxidized compound, B, and the reaction end products, E, during the course of the oxidative reaction. Thus, the reaction routes of this model can be shown as
10.1021/ie030134w CCC: $25.00 © 2003 American Chemical Society Published on Web 09/12/2003
5092 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
characterized by means of immediately available BOD (IA BOD), whereas the second subgroup, formed by the subtraction of IA BOD values from BOD5, consists of large molecular biodegradable organic compounds. In this way, the new reaction network can be presented as follows:
The oxidation process for such a reaction network can been described by the following system of ordinary differential equations (ODEs):
dcA ) -(k1 + k2)cAcO20 dt
(2)
dcB ) k2cAcO20 dt
(3)
where k1 and k2 are expressed via the Arrhenius law equation. The concentration of oxygen in the liquid phase is assumed to be in excess and, therefore, the reaction order with respect to oxygen was set to “zero”. The initial condition states that the concentration of group A equals some concentration [A0], the initial total organic carbon (TOC0) or chemical oxygen demand (COD0) of wastewater, whereas the initial concentration of B is equal to “zero”. Verenich and Kallas11 have divided the reaction system into lumps with respect to the biodegradability of the organic substances. The first lump includes the compounds that are difficult to oxidize biologically, Aref, and was obtained by subtracting the biochemical oxygen demand (BOD5) from the COD. As can be seen from scheme (4), under the action of oxygen, Aref converts to a biodegradable compound, Bbio, and to the reaction end products. The Bbio lump is then characterized via BOD analysis. The latter also undergoes further oxidation into carbon dioxide and water.
This reaction system was expressed as follows:
dcA,ref ) -k1cA,refcO2n1 - k2cA,refcO2n2 dt
(5)
dcB,bio ) k2cA,refcO2n2 - k3cB,biocO2n3 dt
(6)
where k1, k2, and k3 were well-described using the Arrhenius law. At “zero” time, the concentrations of lumps Aref and Bbio are the concentrations of (COD BOD5)0 and BOD5,0, respectively. Later in this text, this model will be abbreviated as BLKM1, where “B” refers to the lumping approach with respect to the bioresistance of the organic compounds. The modified version of the above model12 (BLKM2) splits the biodegradable compounds into two different subgroups. One of these subgroups, Bbio,S, corresponds to small molecular biodegradable compounds and is
The newly formed reaction system includes ODEs, where the last equation corresponds to the formation of end products, E, as a result of oxidative reaction:
dcA,ref ) -k1cA,refcO2n1 - k2cA,refcO2n2 dt
(8)
dcB,bioL ) k2cA,refcO2n2 - k3cB,bioLcO2n3 dt
(9)
dcB,bioS ) k3cB,bioLcO2n3 - k4cB,bioScO2n4 dt
(10)
dcE ) k1cA,refcO2n1 + k4cB,bioScO2n4 dt
(11)
By analogy with the aforementioned models, k1, k2, k3, and k4 were expressed using the Arrhenius law and, at initial “zero” time, Bbio,L, Bbio,S, and E are [BOD5 - IA BOD]0, IA BOD0, and “zero”, respectively. The activation energies, frequency factors, and oxygen reaction orders for the above briefly described models were obtained using a parameter estimation procedure. The computations were performed using the MODEST software package,14 which is designed for various tasks of model building such as simulation, parameter estimation, sensitivity analysis, and optimization. The software consists of a FORTRAN 95/90 library of objective functions, solvers, and optimizers, which are linked to model problem-dependent routines and the objective function. The parameters were estimated from the systems of differential equations (2-3), (5-6), (8-11) using the least-squares method. The objection function (SSQ) to be minimized was nsetsnobsnydata
SSQ )
∑∑ ∑ k)1 j)1 i)1
(yijk - ypijk)2wijk
(12)
where the values yp denote the concentrations (COD, BOD, and IA BOD) predicted by the model and the y values are the concentrations observed experimentally. The weight factors ωijk, used to improve the identifiability of parameters in the system, were set to 1 by default because all response components were of comparable magnitude. The differential equations were solved by means of linear multistep methods implemented in ODESSA, which is based on the LSODE software.15
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5093
Along with the common indicators, such as the regression coefficient, R2, standard error, and correlation matrix, Ω, a graphical sensitivity analysis was performed. The two-dimensional sensitivity contour plots produce a “landscape” of the objective functions for the identification of the probable region of the optimum values. The more centered the contours are, the better the parameters are identified. Experimental Section Materials. For the present study, model thermomechanical pulp (TMP) process water was prepared using TMP pulp obtained from a pulp and paper mill after the second stage of refining. The obtained concentrate had the following characteristics: COD of 6.0-8.0 g/L, 1.2-1.9 g/L of dissolved organic carbon (DOC), biodegradability of 20-35%, and pH of 5.2-6.0. The IA BOD values were measured at 0-200 mg/L. Experimental Procedure. The kinetic experiments were conducted in a 0.3-L stainless steel autoclave (Parr Instrument Co., Moline, IL) equipped with a magneticdriven stirrer. The rotation speed was kept at 900 min-1 because no mass-transfer limitation was encountered under these conditions. The operating temperature was maintained at the set level by means of a thermal sensor, a cooling loop implemented inside the reactor, and an external heating element. First of all, the sample of the TMP concentrate with a volume of 0.175 L was loaded into the cold reactor, which was then tightly sealed. After the preheating period, which lasted for about 30 min, the oxidation process was initiated by introducing pure oxygen into the preheated solution, and the desired oxygen partial pressure was maintained throughout the experimental run. Six liquid samples were extracted at designated time intervals during the 2-h reaction period. The reaction was stopped by rapidly cooling the reactor with chilled water. The COD, BOD5, and IA BOD analyses were carried out for the samples withdrawn from the reactor during the oxidation process. Analytical Methods. The COD analysis was performed by means of the closed reflux dichromate method16 using a COD reactor (Hach Co., Loveland, CO) and a direct-reading spectrophotometer DR/2000. The BOD and IA BOD analyses were assessed with the help of a SensorBOD device developed by Dr. Lange & Co. (Germany), which allows for BOD values identical with those of BOD5 to be obtained. The more detailed procedure of the analysis can be obtained elsewhere.12 Results and Discussion The activation energies, Eai, and frequency factors, A0i, were estimated from the Arrhenius equation for each of the models presented above. However, the Arrhenius equation, in its traditional form (eq 13), has two strongly correlating parameters, Eai and A0i. By a
ki ) A0i exp(-Eai/RT)
(13)
suitable increase of the values of both A0i and Eai, ki could remain virtually unchanged. Therefore, a new parametrization of eq 13 should be performed, and this can be written in the form14
[ (
ki ) kmean,i exp -
Eai 1 1 R T Tmean
)]
(14)
Table 1. Estimated Kinetic Parameters, Such as Frequency Factors, Activation Energies, and Oxygen Reaction Orders, for Each Studied Lumped Kinetic Model
reaction pathway
A0, min-1 or (mol-1 m3)n Ea, std error, kJ/mol kJ/mol min-1
LKMS 1 5.48 × 1011 115.6 2 5.48 × 1016 167.3 regression coefficient 88.6 R2, %
(13.7 (103.0
BLKM1 1 6.51 × 1011 55.31 (28.8 45.44 (23.8 2 4.97 × 109 3 1.37 × 1010 67.41 (29.9 regression coefficient 98.3 R2, % 1 2 3 4 regression coefficient R2, %
BLKM2 7.76 × 1012 58.59 40.41 2.39 × 109 4.40 × 108 66.76 7.60 × 106 53.31 98.3
(13.5 (19.0 (19.9 (20.9
n
std error
0 0
1.88 (0.40 1.53 (0.42 0.86 (0.95 99.8
2.09 1.31 0.19 0.27 99.1
(0.25 (0.29 (0.25 (0.57
where
kmean,i ) A0i exp(-Eai/RTmean)
(15)
where Tmean is a “mean” temperature between the lowest and highest temperatures used in the estimation. Later on, kmean,i values have to be recalculated from eq 15 to receive the values of the original frequency factors, A0i. Additionally, in eqs (5-6) and (8-11), the concentration of the dissolved oxygen was divided by its “mean” value at the operating conditions in order to improve parameter identification in the estimation of the oxygen reaction order. Therefore, after all of the frequency factors, A0i, obtained from eq 15 also have to be divided ni , which was obtained with an equaby a factor cO 2,mean tion by Tromans17 at Tmean of 180 °C and pO2,mean of 1 MPa. Estimation of the Oxygen Reaction Order. The reaction orders for oxygen in the two models, BLKM1 and BKLM2, were estimated by using data sets at a constant temperature of 170 °C and a variable oxygen partial pressure of 0.5-1.5 MPa. The estimated reaction orders are shown in Table 1. The estimated reaction orders for each lump showed a decline of their value with an increase in the degree of oxidation of the organic substances. Let us consider in more detail the BLKM2 model. The estimation of the reaction order for direct oxidation to the end products resulted in a value of about 2, while oxidation via intermediate products led to a decrease in the order to values of 0.2-0.3 with an increase in the degree of oxidation. This may be considered natural because more oxygenated organic compounds exhibit more resistance toward chemical oxidation and, therefore, have a lower dependence on the oxygen partial pressure. Moreover, to ensure that the process is controlled by chemical reaction and not by mass transfer, the Hatta number (Ha)18,19 was computed for each lump in the models. The mass-transfer coefficient present in the formula for Ha (eq 16) was evaluated according to Calderbank and MooYoung20 for agitated vessels. As a result, a Ha number of less than 10-2 was attained, which indicates that the oxidative processes were conducted under the kinetic regime and that the reaction orders, with respect to
5094 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 Table 2. Some Kinetic Data, Such as Activation Energies and Oxygen Reaction Orders, for the WO Lumped Kinetic Models kinetic parameters compound
model
chlorophenol (TOC)
GLKM
chlorophenol (TOC)
LKM
petrochemical WW (COD)
LKM
pulp paper WW (TOC)
LKM
petrochemical WW (COD)
GLKM
petrochemical WW (COD)
GLKM
azo dye solution, Orange II (TOC) TMP concentrate (COD, BOD)
BLKM1
desizing WW (COD)
GLKM
fuel-oil WW (COD)
a
reaction pathway
Ea, kJ/mol
m
n
operating conditions
ref
1 2 3 1 2 1 2 1 2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
72.0 36.0 167.1 72.7 26.4 53.5 24.0 18.3 2.62 42.5 34.6 a 15.8 3.6 a 90b 104 57 95.7 15.7 74.8 23.9 20.5 1.45 34.6 35.1 44.4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1
0 0 0.37 0 0 0 0 0 0 0 0 0 0 0 0
204-260 °C, O2 3.9-7.1 MPa
4
204-260 °C, O2 3.9-7.1 MPa
8
175-240 °C, O2 3-5 MPa
8
120-170 °C, O2 ∼1.5 MPa
8
175-240 °C, O2 3-5 MPa
6
150-240 °C, air 2.5-5 MPa
6
200-240 °C, O2 1-3 MPa
2
1 1 0.5 0.3 0.5 0 0 0 0 0 0
170-200 °C, O2 0.4-1 MPa
11
150-270 °C, O2 ∼1.92 MPa
22
250-300 °C, O2 ∼20 MPa
10
Temperature independent. b Thermal decomposition.
oxygen, can be attributed to the nature of the reaction between the oxidant and the substrate. A sensitivity
x2D ∫ Ha )
cO2,SAT (-ri) O2 0
dcO2
kLcO2,SAT
(16)
analysis for the model BLKM2 was conducted by presenting a global picture of the identifiability of the parameters in the form of contour plot pairs n1-n2, n1n3, n1-n4, n2-n3, n2-n4, and n3-n4. Figure 1 illustrates the typical shape of the contour lines observed during the estimation of the oxygen reaction orders for BLKM2. As can be seen, the contour lines are well-centered around the most probable point. Furthermore, the correlation matrix, Ω1 (eq 17), confirms that the esti-
n1 Ω1 ) n2 n3 n4
n1 1 -0.57 -0.57 -0.84
n2
n3
n4
1 0.32 1 0.83 0.50 1
(17)
mated parameters are not strongly cross-correlated. Its elements do not exceed the values of -0.9 or 0.9, which would indicate a strong dependency between the two parameters.21 Nevertheless, the standard errors for the reaction orders were observed to be rather high. For example, the standard error for the fourth reaction pathway (BLKM2) reaches 212% (see Table 1). This indicates that statistical information, such as the standard error or maximum likelihood, computed for each estimated parameter, may not be an absolute factor for drawing conclusions about the identifiability of the kinetic parameters. A very low standard error does not
Figure 1. Sensitivity contour plot (least-squares objection function) for the parameter pair n1-n2 of the BLKM2 model.
necessarily indicate well-identified parameters that are in the center of the optimum region. The obtained reaction orders were compared with some data of kinetic modeling for different lumped models and wastewaters that are given in the literature (Table 2). Oxygen reaction orders were observed experimentally only in two cases,2,11 and in the present study, and they are not equal to “zero”. The rest of the models made the assumption of a pseudo-first reaction order. Estimation of Activation Energies. The activation energies for the three reaction schemes were estimated by using the data sets at temperatures of 170-200 °C and oxygen partial pressures of 0.5-1.5 MPa (see the Supporting Information). Their values can be seen in Table 1. The estimated parameters for each model studied in this work were found to be in the same magnitude as those given in the literature (Table 2). It is also worth noting that the regression coefficients, R2, for these estimations were high, up to 98%, except the LKM model.
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5095
The statistical analysis of each considered lumped model indicated that the relative standard errors for the activation energies vary within the range of 13-61%. For instance, the relative standard errors for the activation energies Ea1, Ea2, Ea3, and Ea4 (BLKM2) are 23, 47, 30, and 39%, respectively. Nevertheless, the predicted curves with the estimated kinetic constants closely follow the trend of the experimental data. Figure 2 shows the predicted concentrations at the operating temperature of 170 °C and oxygen partial pressures of 1.5 MPa for models BLKM1 and BLKM2. The sensitivity analyses for the parameter pairs Ea1kmean,1, Ea2-kmean,2, Ea3-kmean,3, and Ea4-kmean,4 (BLKM2) suggest that the parameters are well-identified (Figure 3) and only weakly correlated [Ω2 (eq 18)]. Although the Ω2 ) kmean,1 kmean,2 kmean,3 kmean,4 Ea1 Ea2 Ea3 Ea4
kmean,1 1 -0.44 -0.43 -0.75 0.26 -0.13 -0.13 -0.11
kmean,2 kmean,3 kmean,4 Ea1 1 0.00 0.78 -0.11 0.54 -0.09 -0.26
1 0.03 -0.13 -0.09 0.59 -0.14
1 -0.11 0.29 -0.16 0.17
Ea2
Ea3
Ea4
1 -0.45 1 -0.45 0.03 1 -0.77 0.77 0.09 1
(18)
Figure 2. Models’ performance for the experimental data at 170 °C and 1.5 MPa of oxygen partial pressure where 9 is COD data, 1 BOD, 2 IA BOD, and [ reaction end products. A and B correspond to the BLKM1 and BLKM2 models, respectively.
correlation matrix Ω3 (eq 19) indicates a strong dependency of the kinetic parameters Ea3 and kmean,3 on the parameters Ea1 and kmean,1, respectively, the sensitivity analysis (Figure 5, Appendix A) shows that the parameters in BLKM1 are well-identified. Also taking into consideration the fact that the reaction order n3 crosscorrelates with the other oxygen reaction orders in the model [Ω4 (eq 20)], the third lump Bbio has to be
Figure 3. Sensitivity contour plot (least-squares objection function) for parameter pairs Ea1-kmean,1, Ea2-kmean,2, Ea3-kmean,3, and Ea4kmean,4 (BLKM2).
5096 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
Figure 4. Sensitivity contour plot (least-squares objection function) for the parameters of Ea2-kmean,2 (LKM).
reconsidered. Splitting the Bbio lump into two different subgroups, such as Bbio,S and Bbio,L, led to the formation of the BLKM2 model. As shown above, the modified version of BLKM1 was validated as a model with welldefined kinetic parameters. Special attention should be
kmean,1 kmean,2 Ω3 ) kmean,3 Ea1 Ea2 Ea3
kmean,1 1 -0.71 -0.93 0.26 -0.20 -0.12
kmean,2 kmean,3 Ea1 1 0.82 -0.16 0.41 0.12
Ea2
Ea3
1 -0.13 1 0.17 -0.71 1 0.03 -0.93 0.81 1 (19) n2 n3
n1 n 1 Ω4 ) n1 2 -0.73 1 n3 -0.90 0.86 1
(20) Figure 5. Sensitivity contour plot (least-squares objection function) for the parameter pairs Ea1-kmean,1, Ea2-kmean,2, and Ea3kmean,3 (BLKM1).
paid to the LKM model. First of all, the relative standard error for the activation energy, Ea2, was found to be the largest observed in this study, 61%. Moreover, the sensitivity plot for the Ea2-kmean,2 pair (Figure 4) reveals a slight expansion of the contour lines along the Ea2 axis. The correlation matrix Ω5 (eq 21) confirms the
kmean,1 Ω5 ) kmean,2 Ea1 Ea2
kmean,1 1 0.77 0.45 -0.51
kmean,2 Ea1
Ea2
1 0.12 1 -0.90 0.21 1
(21)
bad identification of this parameter. The activation energy, Ea2, correlates rather strongly with the frequency factor, kmean,2. This can be partially explained by the assumption of the zero oxygen reaction order that this model makes. The simulation results indicated that this model cannot fit the results at different oxygen partial pressures. Conclusions This work studied three lumped kinetic models for WO of concentrated wastewater and estimated the mo-
del parameters. In the parameter estimation, the obtained regression coefficients, R2, are high for models BLKM1 and BLKM2 and the predicted concentrations are in good agreement with the measured ones. The estimated rate constants indicate that the process of WO is likely to proceed via the direct formation of the reaction end products for the present wastewater rather than via intermediates. The reaction orders with respect to oxygen were found to be of values larger than “zero”, whereas in the literature, these values are commonly assumed to be “zero”. A sensitivity analysis was carried out systematically during the parameter estimation to study the identification and cross-correlation of the parameters. It shows that, if proper parametrization is used, the model parameters are well-identified and do not correlate with each other significantly, even in a complex reaction system. This is especially true for the model BLKM2, where the correlation factor between the parameters never exceeds the value of 0.8. However, the standard errors in the estimated parameter values were found to be rather high. This suggests the necessity for more experiments than were available in the present study.
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5097
Acknowledgment The authors acknowledge the financial support provided for this research by the Graduate School in Chemical Engineering (GSCE), Finland. Appendix A The appendix contains Figure 5, which depicts twodimensional sensitivity plots for the pairs Ea1-kmean,1, Ea2-kmean,2, and Ea3-kmean,3 that were obtained during the estimation of the activation energy and frequency factors for the model BLKM1. Appendix B Here are some definitions of terms used for sensitivity analysis
Correlation matrix: r ) covij(coviicovjj)-1/2 where covariance matrix (cov) is defined as follows:
cov ) (XTX)-1XTcovEX(XTX)-1 with X being the sensitivity matrix and covE the covariance matrix of errors. The sensitivity matrix, in its turn, is characterized as X ) (∇βYT)T, where β are parameters to be optimized from function Y. Supporting Information Available: Tables S1 and S2 contain some experimental data from the wet oxidation experiments at different operating temperatures and oxygen partial pressures. This material is available free of charge via the Internet at http://pubs.acs.org. Nomenclature A0 ) frequency factor, (L mol-1)ni min-1 biodegradability ) (BOD/COD) × 100% BOD ) biochemical oxygen demand, (mg of O2) L-1 cA ) parent organic material present in wastewater cA,ref ) refractory toward bio-oxidation compounds cB ) partially oxidized compounds cB,bio ) partially oxidized biodegradable compounds cB,bioL ) partially oxidized biodegradable large compounds cB,bioS ) partially oxidized biodegradable small compounds cE ) reaction end products cO2 ) concentration of oxygen in solution, mol L-1 cO2,mean ) mean concentration of oxygen in solution at 170200 °C, mol L-1 COD ) chemical oxygen demand, (mg of O2) L-1 DO2 ) diffusivity of oxygen in the liquid phase, m2 s-1 Ea ) activation energies, kJ mol-1 Ha ) Hatta number IA BOD ) immediately available biochemical oxygen demand, (mg of O2) L-1 k ) rate constants, (L mol-1)ni min-1 kL ) mass-transfer coefficient for oxygen in the liquid phase, m s-1 kmean ) frequency factors defined by eq 11, (L mol-1)ni min-1 pO2,mean ) mean partial pressure of oxygen, MPa R ) gas constant, J K-1 mol-1 r ) reaction rate, mol L min-1 T ) temperature, K Tmean ) mean temperature defined by eq 10, K WW ) wastewater Ω ) correlation matrix
Superscripts and Subscripts i ) reaction pathway number, 1, 2, 3, or 4 L ) liquid phase m ) reaction orders with respect to substrate n ) reaction orders with respect to oxygen SAT ) saturation
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Received for review February 11, 2003 Revised manuscript received June 16, 2003 Accepted July 22, 2003 IE030134W
