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Abstract
The identification of kinetic models is an essential step for the monitoring, control and optimization of industrial processes. This is particularly true for the chemical and pharmaceutical industries, where the current trend of strong competition calls for a reduction in process development costs [1]. This trend goes in line with the recent initiative in favor of Process Analytical Technology (PAT) launched by the US Food and Drug Administration, which advocates a better understanding and control of manufacturing processes with the goal of ensuring final product quality.
Reaction systems can be represented by first-principles models that describe the evolution of the states (typically concentrations, volume and temperature) by means of conservation equations of differential nature and constitutive equations of algebraic nature. These models include information regarding the reactions (stoichiometry and reaction kinetics), the transfer of species between phases (mass-transfer rates), and the operation of the reactor (initial conditions, inlet and outlet flows, operational constraints). The identification of reaction and mass-transfer rates represents the main challenge in building these first-principles models. Note that first-principles models can include redundant states because the modeling step considers balance equations for more quantities than are necessary to represent the true variability of the process. For example, when modeling a closed homogeneous reaction system with R independent reactions, one typically writes a mole balance equation for each of the S species, whereas there are only R < S independent equations, that is S - R equations are redundant. The situation is a bit more complicated in open and/or heterogeneous reaction systems.
The identification of reaction systems can be performed in one step via a simultaneous approach, in which a kinetic model that comprises all reactions and mass transfers is postulated and the corresponding rate parameters are estimated by comparing predicted and measured concentrations [2]. The procedure is repeated for all combinations of model candidates and the combination with the best fit is typically selected. This approach is termed 'simultaneous identification' since all reactions and mass transfers are identified simultaneously. The advantages of this approach lie in the capability to handle complex reaction rates and in the fact that it leads to optimal parameters in the maximum-likelihood sense. However, the simultaneous approach can be computationally costly when several candidates are available for each reaction, and convergence problems can arise for poor initial guesses. Furthermore, structural mismatch in one part of the model may result in errors in all estimated parameters.
As an alternative to simultaneous identification, the incremental approach decomposes the identification task into a set of sub-problems of lower complexity [3]. With the differential method [2], reaction rates are first estimated by differentiation of transient concentrations measurements. Then, each estimated rate profile is used to discriminate between several model candidates, and the candidate with the best fit is selected. This approach is termed 'rate-based incremental identification' since each reaction rate and each mass-transfer rate is dealt with individually. However, because of the bias introduced in the differentiation step, the estimated rate parameters are not statistically optimal. With the integral method [4-5], extents are first computed from measured concentrations. Subsequently, postulated rate expressions are integrated individually for each reaction, and rate parameters can be estimated by comparing predicted and computed extents. Since each extent of reaction and mass transfer can be investigated individually, this approach is termed 'extent-based incremental identification' [6, 7].
The context of this work is the extent-based incremental identification of rate laws for fluid-fluid (F-F) reaction systems on the basis of process measurements. Process measurements are available for some of the species only, as it is difficult to measure the concentrations of all species due to limitations in the current state of sensor technology. Hence, it is necessary to reconstruct the unmeasured concentrations that appear in the rate laws from the available measurements. If a process model were available, this reconstruction could be done via state estimation using observers or Kalman filters. In the absence of such a reaction model, the idea is to perform instantaneous reconstruction by having as many measured quantities as there are non-redundant states. Hence the key question: How many measurements are needed to be able to reconstruct the full state? R measurements suffice in the case of a homogeneous batch reactor, whereas R + 2 pm + pl + pg + 2 measurements are needed in the case of an open gas-liquid reaction system without reaction/accumulation in the film [8], where pm is the number of mass transfers, pl the number of liquid inlets and pg the number of gas inlets, and there is one outlet in each phase.
After a review of the extent-based incremental identification, this contribution will extend the results on the minimal number of measured species required for reconstructing all states to the cases of F-F reaction systems with reactions taking place in one or two bulks, without and with accumulation/reactions in the film. For the case where the number of measured species is insufficient to compute all the states, this presentation will address the possibility of using additional measurements, such as calorimetry and gas consumption, to augment the number of measured quantities [9]. These theoretical results will be illustrated through simulated examples of F-F reaction systems.
[1] J. Workman et al, Anal. Chem. 83, 4557 (2011)
[2] A. Bardow et al, Chem. Eng. Sci. 59, 2673 (2004)
[3] M. Brendel et al, Chem. Eng. Sci. 61, 5404 (2006)
[4] M. Amrhein et al, AIChE Journal 56, 2873 (2010)
[5] N. Bhatt et al, Ind. Eng. Chem. Res. 49, 7704 (2010)
[6] N. Bhatt et al, Ind. Eng. Chem. Res. 50, 12960 (2011)
[7] N. Bhatt et al, Chem. Eng. Sci. 83, 24 (2012)
[8] N. Bhatt et al, ACC, Montreal (Canada), 3496 (2012)
[9] S. Srinivasan et al, Chem. Eng. J. 208, 785 (2012)
Reaction systems can be represented by first-principles models that describe the evolution of the states (typically concentrations, volume and temperature) by means of conservation equations of differential nature and constitutive equations of algebraic nature. These models include information regarding the reactions (stoichiometry and reaction kinetics), the transfer of species between phases (mass-transfer rates), and the operation of the reactor (initial conditions, inlet and outlet flows, operational constraints). The identification of reaction and mass-transfer rates represents the main challenge in building these first-principles models. Note that first-principles models can include redundant states because the modeling step considers balance equations for more quantities than are necessary to represent the true variability of the process. For example, when modeling a closed homogeneous reaction system with R independent reactions, one typically writes a mole balance equation for each of the S species, whereas there are only R < S independent equations, that is S - R equations are redundant. The situation is a bit more complicated in open and/or heterogeneous reaction systems.
The identification of reaction systems can be performed in one step via a simultaneous approach, in which a kinetic model that comprises all reactions and mass transfers is postulated and the corresponding rate parameters are estimated by comparing predicted and measured concentrations [2]. The procedure is repeated for all combinations of model candidates and the combination with the best fit is typically selected. This approach is termed 'simultaneous identification' since all reactions and mass transfers are identified simultaneously. The advantages of this approach lie in the capability to handle complex reaction rates and in the fact that it leads to optimal parameters in the maximum-likelihood sense. However, the simultaneous approach can be computationally costly when several candidates are available for each reaction, and convergence problems can arise for poor initial guesses. Furthermore, structural mismatch in one part of the model may result in errors in all estimated parameters.
As an alternative to simultaneous identification, the incremental approach decomposes the identification task into a set of sub-problems of lower complexity [3]. With the differential method [2], reaction rates are first estimated by differentiation of transient concentrations measurements. Then, each estimated rate profile is used to discriminate between several model candidates, and the candidate with the best fit is selected. This approach is termed 'rate-based incremental identification' since each reaction rate and each mass-transfer rate is dealt with individually. However, because of the bias introduced in the differentiation step, the estimated rate parameters are not statistically optimal. With the integral method [4-5], extents are first computed from measured concentrations. Subsequently, postulated rate expressions are integrated individually for each reaction, and rate parameters can be estimated by comparing predicted and computed extents. Since each extent of reaction and mass transfer can be investigated individually, this approach is termed 'extent-based incremental identification' [6, 7].
The context of this work is the extent-based incremental identification of rate laws for fluid-fluid (F-F) reaction systems on the basis of process measurements. Process measurements are available for some of the species only, as it is difficult to measure the concentrations of all species due to limitations in the current state of sensor technology. Hence, it is necessary to reconstruct the unmeasured concentrations that appear in the rate laws from the available measurements. If a process model were available, this reconstruction could be done via state estimation using observers or Kalman filters. In the absence of such a reaction model, the idea is to perform instantaneous reconstruction by having as many measured quantities as there are non-redundant states. Hence the key question: How many measurements are needed to be able to reconstruct the full state? R measurements suffice in the case of a homogeneous batch reactor, whereas R + 2 pm + pl + pg + 2 measurements are needed in the case of an open gas-liquid reaction system without reaction/accumulation in the film [8], where pm is the number of mass transfers, pl the number of liquid inlets and pg the number of gas inlets, and there is one outlet in each phase.
After a review of the extent-based incremental identification, this contribution will extend the results on the minimal number of measured species required for reconstructing all states to the cases of F-F reaction systems with reactions taking place in one or two bulks, without and with accumulation/reactions in the film. For the case where the number of measured species is insufficient to compute all the states, this presentation will address the possibility of using additional measurements, such as calorimetry and gas consumption, to augment the number of measured quantities [9]. These theoretical results will be illustrated through simulated examples of F-F reaction systems.
[1] J. Workman et al, Anal. Chem. 83, 4557 (2011)
[2] A. Bardow et al, Chem. Eng. Sci. 59, 2673 (2004)
[3] M. Brendel et al, Chem. Eng. Sci. 61, 5404 (2006)
[4] M. Amrhein et al, AIChE Journal 56, 2873 (2010)
[5] N. Bhatt et al, Ind. Eng. Chem. Res. 49, 7704 (2010)
[6] N. Bhatt et al, Ind. Eng. Chem. Res. 50, 12960 (2011)
[7] N. Bhatt et al, Chem. Eng. Sci. 83, 24 (2012)
[8] N. Bhatt et al, ACC, Montreal (Canada), 3496 (2012)
[9] S. Srinivasan et al, Chem. Eng. J. 208, 785 (2012)