- Email: [email protected]
Identification of multicomponent mass transfer by means of an incremental approach
European Symposium on Computer Aided Process Engineering - 13 A. Kraslawski and I. Turunen (Editors) © 2003 Elsevier Science B.V. All rights reserved.
563
IdentiHcation of Multicomponent Mass Transfer by Means of an Incremental Approach Andre Bardow, Wolfgang Marquardt* Lehrstuhl fiir Prozesstechnik, RWTH Aachen, D-52056 Aachen, Germany
Abstract In this work, an incremental approach to model identification is introduced. The new method follows the steps of model development in the identification procedure. This reduces uncertainty in the estimation problems. Furthermore, it allows the efficient exploitation of model structure and thereby reduces the computational expense substantially. The proposed method is applied to examples from binary and ternary diffusive mass transfer and its performance is compared to current approaches.
1. Introduction The identification of models for kinetic phenomena in multiphase reactive systems is still an open question. In the common approach to model identification model candidates based on prior knowledge or physical insight are proposed and inserted in the balance equations. Measured data is then used to estimate unknown parameters in the fully specified models. We therefore call this approach simultaneous identification. Model discrimination techniques are finally employed to identify the most suitable model (see e.g. Asprey and Macchietto, 2000). The method suffers from several drawbacks: introducing model candidates into the balance equations may bias the estimation problem (e.g. Walter and Pronzato, 1997); computational cost is high since the whole estimation problem is solved for every possible candidate (e.g. Asprey and Macchietto, 2000); and the methods are usually tailored for point measurements with low-resolution (e.g. Schittkowski, 2002) whereas modem experimental techniques offer data with high spatial and temporal resolution. In this work a general incremental approach to model identification is proposed which minimizes model uncertainty and reduces computational cost while efficiently utilizing data from high-resolution measurement techniques. The approach tries to reflect the inherent model structure in the identification procedure itself The method will be illustrated with examples from diffusive mass transfer.
2. Incremental Modeling and Model Identification 2.1. Model development The key idea of incremental model identification is to mimic model development in model identification. For conciseness we will present the approach for isothermal binary diffusion. The extension to multicomponent mixtures is given in the examples below. * Correspondence should be addressed to W. Marquardt, [email protected]
564 Modeling of diffusion processes usually starts by formulation of the balance equations for each component
Model B:
dci _ dt
3(qw^)
dJi
dz
dz
(1)
where u^ is the volume average velocity, cj the concentration and Ji^ the diffusive flux of species 1. Next, constitutive equations have to be given for the convective and the diffusive flux. In diffusion experiments volume effects can usually be neglected, leading to w^ = 0 (Tyrell and Harris, 1984). Still, the modeler has to specify the diffusive flux. For binary diffusion, all models can be related to Pick's law (Taylor and Krishna, 1993) M o d e l T : y r =-D- 12 ^^^ dz
(2)
where DJ^ is the binary Pick diffusion coefficient. It is a function of composition in concentrated liquid mixtures. Therefore, a further constitutive equation has to be employed. Today, there is still uncertainty about a suitable model, especially for multicomponent mixtures (Taylor and Krishna, 1993). We therefore state the generic relationship Model D: Dl^^
f{x,e)
(3)
where x represents the mole fraction and the vector 0 collects all constant coefficients. If the function/and its parameters are known the model can be solved. 2.2. Model identiHcation The simultaneous approach to identification is computationally expensive and may lead to biased estimates. It neglects the inherent model structure with its sequence of models, each containing further assumptions about the process. In contrast, the incremental approach follows the steps of model development for model identification (Pig. 1). measured data structure of system
I states Xj(z,t) fluxes J(z,t)
model B
balances
model BT
balances
transport law
model BID
balances
transport law
structure for fluxes
structure for coefficients!
~1
coefficients D(z,t)
diffusion coefficient
model structure and parameters for diffusion process
parameters
n
Figure 1: Incremental approach to identification of diffusion models.
565 2.2.7. Model B: Balances The balance equation (1) contains the least uncertainty. Without introducing potentially uncertain constitutive equations, the diffusive flux itself is computed from this equation as a function of space and time
Model B: J^zJ)
=- f ^ ^ ^ ^ z ' J dt
(4)
assuming the boundary {z=0) impermeable. The main difficulty in the evaluation of Eq.(4) is the estimation of the time derivative of the measured concentration. This is an ill-posed problem, i.e. small errors in the data will be amplified. Regularization techniques have to be employed. A smoothing spline is used here (Reinsch, 1967). The diffusive flux estimation requires only the solution of the linear Eq. (4) independent of the number of candidate models. This decoupling of the problem carries fully over to the multicomponent case and reduces the computational expense substantially. 2.2.2. Model BT: Transport laws Different candidate models for the description of the diffusive flux may be introduced. The unknown diffusion coefficients are then computed as function of space and time. In the binary case, the diffusion coefficients are calculated from Eq. (2) as
ModelT: D\^{z,t) = J)^^'']^ . dc^{z,t)ldz
(5)
The spatial derivative is also calculated by the smoothing spline approach. Since transport coefficients have a physical interpretation which results in certain restrictions (e.g. positivity), those models violating any restriction could be discarded already at this stage. 2.2.3. Model BTD: Diffusion coefficients Several m o d e l s / ^ (Eq. (3)) for the diffusion coefficient are assumed. The parameters are estimated using the diffusion coefficients from Eq. (5) and the measured mole fractions. Since both quantities contain errors this is an error-in-variables problem:
^f^[^e\D^2iZi,0-f''{x{z,,U)
+ S^,e'')] + wsSl^
(6)
where J represents the error in the mole fraction and w weights for both types of errors. Here, the efficient solution method by Boggs et al. (1992) is used. Finally, the adequacy of the model candidates is quantified using the a posteriori probability for each model M according to the data Y (Stewart et al., 1998). In the case of unknown variance it can be calculated from p(M|y)-p(M)2-''«'%/2r'"
(7)
566 where the sum SM is the residual sum of squares and PM is the number of independent parameters in model M, To rank the models, probability shares n are calculated as TT^ = p(M I y ) / \ p(M'| Y). Based upon that value the most suitable model is chosen. It should be noted that this criterion does not carry fully over to the error-in-variables case. It is used here based on the heuristic argument that the errors in the measured mole fractions are expected to be much smaller than the values in the estimated diffusion coefficient. The corrected mole fractions are therefore assumed to be errorfree in the model discrimination step.
3. Numerical Example 3.1. Binary diffusion In this simulation study the diffusion coefficient in the mixture toluene-cyclohexane as given by Sanni et al. (1971) is estimated from mole fraction data. The assumed experimental setup was presented by Bardow et al. (2003). A typical experiment gives 60 concentration profiles with a spatial resolution of 400 points, i.e. a total of 24,000 data points. The simulated concentration profiles were corrupted by Gaussian noise with variance o^=70"^ which corresponds to very unfavorable experimental conditions and shows the robustness of the approach. The analysis is carried out as described above. Fig. 2 compares estimated and true flux values. There is a substantial difference for very small times. The estimation is excellent for larger times. This is due to the steep gradients at the beginning of the run which cannot be easily distinguished from measurement noise. Different polynomials were proposed for the diffusion coefficient. Fig. 3 shows the resulting fit from Eq. (6) and gives the probability shares TIM- Model discrimination favors the linear model due to the difficulty of estimating the full concentration dependence from a single experiment. Commonly, more than 10 experiments are used (e.g. Sanni et al, 1971). Therefore, only the linear trend can safely be deduced and the probability shares indicate the need for further experiments since no model gains high values. 1.2 x10"
true —
0.9
1
estimated |
t = 2 min-^c
—— true constant (71 =13.2%) ^ CO
'
linear (7t|j=39.3%)
1r It
M
•^0 6 t = 6 min
0.3
0
2
4.^. . 6 , position [mm]
Figure 2: Estimated diffusive flux.
8
10
0.2
0.4 ^Toluol
iV
0.8
Figure 3: Estimated diffusion coefficients.
567 Table 1: Ternary diffusion coefficient from a single experiment (noise level o^=10'^). coefficient D,/ Djj Djj D27 D22 value in IQ-^m^/s 4.35 ±0.006 1.69 ±0.012 3.56 ±0.006 6.15 ±0.012 error in % -2.0 -7.8 -2.3 -2.3
In order to compare the proposed method to the simultaneous approach we performed a classical parameter estimation for the linear model. It lead to a very similar solution and the residual sums of squares differ only by 0.11%, even though this objective is not directly employed in the incremental approach. Furthermore, the computational time for the simultaneous approach lies in the order of hours due to the distributed nature of the problem and the high measurement resolution whereas the incremental procedure takes only minutes including the fit of all models and the model discrimination. 3.2. Ternary diffusion In ternary mixtures, the diffusive flux of one component is influenced by the concentration gradients of the other components. This requires the use of Generalized Pick's Law leading to a matrix of four Fick diffusion coefficients (Taylor and Krishna, 1993). Current measurement procedures in ternary mixtures require at least two experiments to estimate these coefficient and the cross coefficients are still ill-determined (e.g. van de Ven-Lucassen et al, 1995). In this section the full diffusion coefficient matrix is estimated from a single experiment using the incremental approach. Sample diffusion coefficients are taken from Arnold and Toor (1967) who studied gas systems for which the diffusion coefficients (Eq. 3) are constant. The estimated values are compared to the true solution in Tab. 1. The diffusion matrix can be estimated from a single experiment with good precision using the incremental approach. It should be noted that the four diffusion coefficients are not identifiable from Eq. (5). But the insertion of the constitutive law for the diffusion coefficient (Eq. (3)) into the flux expression allows to overcome this situation. Furthermore, it should be stressed that this estimation problem is very difficult to solve by the simultaneous approach. The Fick matrix is positive definite which is enforced by three inequality constraints (Taylor and Krishna, 1993). In parameter estimation, a sequential approach with an infeasible path optimization routine is often used. This may not be possible since the model cannot be integrated if the matrix is not positive definite. This limitation does not apply to the incremental approach since no solution of the direct problem is required. It could therefore also be used to initialize the simultaneous procedure.
4. Conclusions and Future Work In this work a new incremental approach to model identification was presented which makes use of the inherent model structure. Thereby, uncertainty in each step of the calculation is reduced. It allows furthermore a decoupling of the problems which reduces the computational cost to several minutes even for distributed systems. The approach is especially suited for high-resolution measurements. The data is used to solve an infinite dimensional problem, the estimation of the diffusive flux. This step
568 may be error-prone with low-resolution data. But it could be shown that the method compares well with results from simultaneous identification strategies which may fail completely for ternary mixtures whereas the incremental approach is robust. A stepwise procedure as proposed here may be even more advantageous when it is difficult to formulate suitable candidate models. This was recognized by Tholudur and Ramirez (1999) for the estimation of reaction rates in a protein production model. Based on the estimation of the generalized fluxes data mining techniques may be employed to discover constitutive relationships. This possibility is currently investigated.
5. References Arnold, J.H. and Toor, H.L., 1967, Unsteady diffusion in ternary gas mixtures, AIChEJ. 13,909-914. Asprey, S.P. and Macchietto, S., 2000, Statistical tools in optimal model building, Comput. Chem. Eng., 24, 831-834. Bardow, A., Marquardt, W., Goke, V., Ko6, H.-J. and Lucas, K., 2003, Model-based measurement of diffusion using Raman spectroscopy, AIChE J., in press. Boggs, P.T., Byrd, R.H., Rogers, J.E. and Schnabel, R.B., 1992, User's reference guide for ODRPACK version 2.01 software for weighted orthogonal distance regression. National Institute of Standards and Technology, NISTIR 92-4834. Reinsch, C.H., 1967, Smoothing by spline functions. Num. Math., 10, 177-183. Sanni, S.A., Felland, C.J.D. and Hutchison, H.P., 1971, Diffusion coefficients and densities for binary organic liquid mixtures. J. Chem. Eng. Data, 16,424-427. Schittkowski, K., 2002, EASY-FIT: A software system for data fitting in dynamic systems. Struct. Multidiscip. O., 23(2), 153-169. Stewart, W.E., Shon, Y. and Box, G.E.P., 1998, Discrimination and goodness of fit of multiresponse mechanistic models, AIChE J., 44, 1404-1412. Taylor, R. and Krishna, R., 1993, Multicomponent Mass Transfer, Wiley, New York. Tholudur, A. and Ramirez, W.F., 1999, Neural-network modeling and optimization of induced foreign protein production, AIChE J., 8, 1660-1670. Tyrell, H.J.V. and Harris, K.R., 1984, Diffusion in Liquids, Butterworths, London. van de Ven-Lucassen, I.M.J.J., Kieviet, E.G. and Kerkhof, P.J.A.M., 1995, Fast and convenient implementation of the Taylor dispersion method, J. Chem. Eng. Data, 40,407-411. Walter, E. and Pronzato, L., 1997, Identification of Parametric Models: from Experimental Data, Springer, Berlin.
6. Acknowledgements The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center (SFB) 540 "Model-based Experimental Analysis of Kinetic Phenomena in Fluid Multi-phase Reactive Systems".
563
IdentiHcation of Multicomponent Mass Transfer by Means of an Incremental Approach Andre Bardow, Wolfgang Marquardt* Lehrstuhl fiir Prozesstechnik, RWTH Aachen, D-52056 Aachen, Germany
Abstract In this work, an incremental approach to model identification is introduced. The new method follows the steps of model development in the identification procedure. This reduces uncertainty in the estimation problems. Furthermore, it allows the efficient exploitation of model structure and thereby reduces the computational expense substantially. The proposed method is applied to examples from binary and ternary diffusive mass transfer and its performance is compared to current approaches.
1. Introduction The identification of models for kinetic phenomena in multiphase reactive systems is still an open question. In the common approach to model identification model candidates based on prior knowledge or physical insight are proposed and inserted in the balance equations. Measured data is then used to estimate unknown parameters in the fully specified models. We therefore call this approach simultaneous identification. Model discrimination techniques are finally employed to identify the most suitable model (see e.g. Asprey and Macchietto, 2000). The method suffers from several drawbacks: introducing model candidates into the balance equations may bias the estimation problem (e.g. Walter and Pronzato, 1997); computational cost is high since the whole estimation problem is solved for every possible candidate (e.g. Asprey and Macchietto, 2000); and the methods are usually tailored for point measurements with low-resolution (e.g. Schittkowski, 2002) whereas modem experimental techniques offer data with high spatial and temporal resolution. In this work a general incremental approach to model identification is proposed which minimizes model uncertainty and reduces computational cost while efficiently utilizing data from high-resolution measurement techniques. The approach tries to reflect the inherent model structure in the identification procedure itself The method will be illustrated with examples from diffusive mass transfer.
2. Incremental Modeling and Model Identification 2.1. Model development The key idea of incremental model identification is to mimic model development in model identification. For conciseness we will present the approach for isothermal binary diffusion. The extension to multicomponent mixtures is given in the examples below. * Correspondence should be addressed to W. Marquardt, [email protected]
564 Modeling of diffusion processes usually starts by formulation of the balance equations for each component
Model B:
dci _ dt
3(qw^)
dJi
dz
dz
(1)
where u^ is the volume average velocity, cj the concentration and Ji^ the diffusive flux of species 1. Next, constitutive equations have to be given for the convective and the diffusive flux. In diffusion experiments volume effects can usually be neglected, leading to w^ = 0 (Tyrell and Harris, 1984). Still, the modeler has to specify the diffusive flux. For binary diffusion, all models can be related to Pick's law (Taylor and Krishna, 1993) M o d e l T : y r =-D- 12 ^^^ dz
(2)
where DJ^ is the binary Pick diffusion coefficient. It is a function of composition in concentrated liquid mixtures. Therefore, a further constitutive equation has to be employed. Today, there is still uncertainty about a suitable model, especially for multicomponent mixtures (Taylor and Krishna, 1993). We therefore state the generic relationship Model D: Dl^^
f{x,e)
(3)
where x represents the mole fraction and the vector 0 collects all constant coefficients. If the function/and its parameters are known the model can be solved. 2.2. Model identiHcation The simultaneous approach to identification is computationally expensive and may lead to biased estimates. It neglects the inherent model structure with its sequence of models, each containing further assumptions about the process. In contrast, the incremental approach follows the steps of model development for model identification (Pig. 1). measured data structure of system
I states Xj(z,t) fluxes J(z,t)
model B
balances
model BT
balances
transport law
model BID
balances
transport law
structure for fluxes
structure for coefficients!
~1
coefficients D(z,t)
diffusion coefficient
model structure and parameters for diffusion process
parameters
n
Figure 1: Incremental approach to identification of diffusion models.
565 2.2.7. Model B: Balances The balance equation (1) contains the least uncertainty. Without introducing potentially uncertain constitutive equations, the diffusive flux itself is computed from this equation as a function of space and time
Model B: J^zJ)
=- f ^ ^ ^ ^ z ' J dt
(4)
assuming the boundary {z=0) impermeable. The main difficulty in the evaluation of Eq.(4) is the estimation of the time derivative of the measured concentration. This is an ill-posed problem, i.e. small errors in the data will be amplified. Regularization techniques have to be employed. A smoothing spline is used here (Reinsch, 1967). The diffusive flux estimation requires only the solution of the linear Eq. (4) independent of the number of candidate models. This decoupling of the problem carries fully over to the multicomponent case and reduces the computational expense substantially. 2.2.2. Model BT: Transport laws Different candidate models for the description of the diffusive flux may be introduced. The unknown diffusion coefficients are then computed as function of space and time. In the binary case, the diffusion coefficients are calculated from Eq. (2) as
ModelT: D\^{z,t) = J)^^'']^ . dc^{z,t)ldz
(5)
The spatial derivative is also calculated by the smoothing spline approach. Since transport coefficients have a physical interpretation which results in certain restrictions (e.g. positivity), those models violating any restriction could be discarded already at this stage. 2.2.3. Model BTD: Diffusion coefficients Several m o d e l s / ^ (Eq. (3)) for the diffusion coefficient are assumed. The parameters are estimated using the diffusion coefficients from Eq. (5) and the measured mole fractions. Since both quantities contain errors this is an error-in-variables problem:
^f^[^e\D^2iZi,0-f''{x{z,,U)
+ S^,e'')] + wsSl^
(6)
where J represents the error in the mole fraction and w weights for both types of errors. Here, the efficient solution method by Boggs et al. (1992) is used. Finally, the adequacy of the model candidates is quantified using the a posteriori probability for each model M according to the data Y (Stewart et al., 1998). In the case of unknown variance it can be calculated from p(M|y)-p(M)2-''«'%/2r'"
(7)
566 where the sum SM is the residual sum of squares and PM is the number of independent parameters in model M, To rank the models, probability shares n are calculated as TT^ = p(M I y ) / \ p(M'| Y). Based upon that value the most suitable model is chosen. It should be noted that this criterion does not carry fully over to the error-in-variables case. It is used here based on the heuristic argument that the errors in the measured mole fractions are expected to be much smaller than the values in the estimated diffusion coefficient. The corrected mole fractions are therefore assumed to be errorfree in the model discrimination step.
3. Numerical Example 3.1. Binary diffusion In this simulation study the diffusion coefficient in the mixture toluene-cyclohexane as given by Sanni et al. (1971) is estimated from mole fraction data. The assumed experimental setup was presented by Bardow et al. (2003). A typical experiment gives 60 concentration profiles with a spatial resolution of 400 points, i.e. a total of 24,000 data points. The simulated concentration profiles were corrupted by Gaussian noise with variance o^=70"^ which corresponds to very unfavorable experimental conditions and shows the robustness of the approach. The analysis is carried out as described above. Fig. 2 compares estimated and true flux values. There is a substantial difference for very small times. The estimation is excellent for larger times. This is due to the steep gradients at the beginning of the run which cannot be easily distinguished from measurement noise. Different polynomials were proposed for the diffusion coefficient. Fig. 3 shows the resulting fit from Eq. (6) and gives the probability shares TIM- Model discrimination favors the linear model due to the difficulty of estimating the full concentration dependence from a single experiment. Commonly, more than 10 experiments are used (e.g. Sanni et al, 1971). Therefore, only the linear trend can safely be deduced and the probability shares indicate the need for further experiments since no model gains high values. 1.2 x10"
true —
0.9
1
estimated |
t = 2 min-^c
—— true constant (71 =13.2%) ^ CO
'
linear (7t|j=39.3%)
1r It
M
•^0 6 t = 6 min
0.3
0
2
4.^. . 6 , position [mm]
Figure 2: Estimated diffusive flux.
8
10
0.2
0.4 ^Toluol
iV
0.8
Figure 3: Estimated diffusion coefficients.
567 Table 1: Ternary diffusion coefficient from a single experiment (noise level o^=10'^). coefficient D,/ Djj Djj D27 D22 value in IQ-^m^/s 4.35 ±0.006 1.69 ±0.012 3.56 ±0.006 6.15 ±0.012 error in % -2.0 -7.8 -2.3 -2.3
In order to compare the proposed method to the simultaneous approach we performed a classical parameter estimation for the linear model. It lead to a very similar solution and the residual sums of squares differ only by 0.11%, even though this objective is not directly employed in the incremental approach. Furthermore, the computational time for the simultaneous approach lies in the order of hours due to the distributed nature of the problem and the high measurement resolution whereas the incremental procedure takes only minutes including the fit of all models and the model discrimination. 3.2. Ternary diffusion In ternary mixtures, the diffusive flux of one component is influenced by the concentration gradients of the other components. This requires the use of Generalized Pick's Law leading to a matrix of four Fick diffusion coefficients (Taylor and Krishna, 1993). Current measurement procedures in ternary mixtures require at least two experiments to estimate these coefficient and the cross coefficients are still ill-determined (e.g. van de Ven-Lucassen et al, 1995). In this section the full diffusion coefficient matrix is estimated from a single experiment using the incremental approach. Sample diffusion coefficients are taken from Arnold and Toor (1967) who studied gas systems for which the diffusion coefficients (Eq. 3) are constant. The estimated values are compared to the true solution in Tab. 1. The diffusion matrix can be estimated from a single experiment with good precision using the incremental approach. It should be noted that the four diffusion coefficients are not identifiable from Eq. (5). But the insertion of the constitutive law for the diffusion coefficient (Eq. (3)) into the flux expression allows to overcome this situation. Furthermore, it should be stressed that this estimation problem is very difficult to solve by the simultaneous approach. The Fick matrix is positive definite which is enforced by three inequality constraints (Taylor and Krishna, 1993). In parameter estimation, a sequential approach with an infeasible path optimization routine is often used. This may not be possible since the model cannot be integrated if the matrix is not positive definite. This limitation does not apply to the incremental approach since no solution of the direct problem is required. It could therefore also be used to initialize the simultaneous procedure.
4. Conclusions and Future Work In this work a new incremental approach to model identification was presented which makes use of the inherent model structure. Thereby, uncertainty in each step of the calculation is reduced. It allows furthermore a decoupling of the problems which reduces the computational cost to several minutes even for distributed systems. The approach is especially suited for high-resolution measurements. The data is used to solve an infinite dimensional problem, the estimation of the diffusive flux. This step
568 may be error-prone with low-resolution data. But it could be shown that the method compares well with results from simultaneous identification strategies which may fail completely for ternary mixtures whereas the incremental approach is robust. A stepwise procedure as proposed here may be even more advantageous when it is difficult to formulate suitable candidate models. This was recognized by Tholudur and Ramirez (1999) for the estimation of reaction rates in a protein production model. Based on the estimation of the generalized fluxes data mining techniques may be employed to discover constitutive relationships. This possibility is currently investigated.
5. References Arnold, J.H. and Toor, H.L., 1967, Unsteady diffusion in ternary gas mixtures, AIChEJ. 13,909-914. Asprey, S.P. and Macchietto, S., 2000, Statistical tools in optimal model building, Comput. Chem. Eng., 24, 831-834. Bardow, A., Marquardt, W., Goke, V., Ko6, H.-J. and Lucas, K., 2003, Model-based measurement of diffusion using Raman spectroscopy, AIChE J., in press. Boggs, P.T., Byrd, R.H., Rogers, J.E. and Schnabel, R.B., 1992, User's reference guide for ODRPACK version 2.01 software for weighted orthogonal distance regression. National Institute of Standards and Technology, NISTIR 92-4834. Reinsch, C.H., 1967, Smoothing by spline functions. Num. Math., 10, 177-183. Sanni, S.A., Felland, C.J.D. and Hutchison, H.P., 1971, Diffusion coefficients and densities for binary organic liquid mixtures. J. Chem. Eng. Data, 16,424-427. Schittkowski, K., 2002, EASY-FIT: A software system for data fitting in dynamic systems. Struct. Multidiscip. O., 23(2), 153-169. Stewart, W.E., Shon, Y. and Box, G.E.P., 1998, Discrimination and goodness of fit of multiresponse mechanistic models, AIChE J., 44, 1404-1412. Taylor, R. and Krishna, R., 1993, Multicomponent Mass Transfer, Wiley, New York. Tholudur, A. and Ramirez, W.F., 1999, Neural-network modeling and optimization of induced foreign protein production, AIChE J., 8, 1660-1670. Tyrell, H.J.V. and Harris, K.R., 1984, Diffusion in Liquids, Butterworths, London. van de Ven-Lucassen, I.M.J.J., Kieviet, E.G. and Kerkhof, P.J.A.M., 1995, Fast and convenient implementation of the Taylor dispersion method, J. Chem. Eng. Data, 40,407-411. Walter, E. and Pronzato, L., 1997, Identification of Parametric Models: from Experimental Data, Springer, Berlin.
6. Acknowledgements The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center (SFB) 540 "Model-based Experimental Analysis of Kinetic Phenomena in Fluid Multi-phase Reactive Systems".