Indicadores de línea base

This chapter has provided a novel transformation to extents for generic distributed re- action systems that include tubular reactors and reactive separation columns. As a con- sequence, the well-known concept of extents for batch reactors, which has recently been extended to open homogeneous and heterogeneous reactors [18, 19], as shown in Chapter 74

3.7. Conclusion 0 50 100 150 200 250 0 2 4 6 8 nl, n ,A ( t) t n= 1 n= 2 n= 3 n= 4 0 50 100 150 200 250 0 5 10 15 nl, n ,B ( t) t n= 1 n= 2 n= 3 n= 4 0 50 100 150 200 250 0 2 4 6 8 10 12 nl, n ,C ( t) t n= 1 n= 2 n= 3 n= 4 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 nl, n ,D ( t) t n= 1 n= 2 n= 3 n= 4

Figure 3.10 – Numbers of moles nl,n(t)in kmol over time (in minutes) on each of the 4

trays of a reactive distillation column.

0 50 100 150 200 250 0 2 4 6 8 10 12 xr ,l ,n ,1 ( t) t n= 1 n= 2 n= 3 n= 4 0 50 100 150 200 250 0 0.1 0.2 0.3 0.4 0.5 xr ,l ,n ,2 ( t) t n= 1 n= 2 n= 3 n= 4 0 50 100 150 200 250 0 5 10 15 20 xm ,l ,n ,A ( t) t n= 1 n= 2 n= 3 n= 4 0 50 100 150 200 250 −15 −10 −5 0 5 xm ,l ,n ,D ( t) t n= 1 n= 2 n= 3 n= 4

Figure 3.11 – Extents of reaction xr,l,n(t) and of mass transfer xm,l,n(t)in kmol over time

Chapter 3. Concept of Extents for Distributed Reaction Systems

2, has been generalized to distributed reaction systems.

In agreement with the definition of a vessel extent for lumped reaction systems, each extent in a distributed reaction system describes uniquely and completely a particular rate process, taking into account the amount that has been transported by advection to a farther position and that has been removed by an outlet. The original concentration variables can always be expressed as the linear transformation of the extents. In many cases, these extents can, in turn, be obtained via the (inverse) linear transformation of the original concentrations. This linear transformation uses structural information about the reaction system, in particular its stoichiometry, the knowledge of the species that transfer between phases and diffuse, as well as information about the initial and boundary conditions. For distributed reaction systems, the initial and boundary conditions replace the initial and inlet conditions required for the transformation of lumped reaction systems. The applicability of these transformations for distributed reaction systems has been demonstrated via simulated examples.

Possible extensions of the concept of extents presented in this chapter include alterna- tive definitions of extents that require less strict rank conditions for the existence of the linear transformation to extents and the connection between extents and analytical or semi- analytical solutions to the PDEs that describe certain distributed reaction systems [91].

The generalization of the concept of extents to distributed reaction systems opens up new perspectives for industrially relevant applications in terms of design, modeling, model identification, model reduction, state reconstruction, data reconciliation, state estimation, monitoring, fault diagnosis, control and optimization of distributed reaction systems. These systems include one- and two-dimensional tubular reactors, three-dimensional reaction sys- tems, micro-reactors, and reactive separation systems, such as reactive absorption or reac- tive distillation columns. These perspectives for applications of the concept of extents in distributed reaction systems are justified by the fact that some of these applications have been investigated for lumped reaction systems, while others have already been mentioned for the case of distributed reaction systems, namely design, modeling, monitoring and fault diagnosis. For example, the concept of extents has been used for incremental model identi- fication of plug-flow reactors [92].

In summary, a clear understanding of the concept of extents in distributed reaction systems, which has been the main goal of this chapter, will certainly be helpful for the future development of useful applications in chemical engineering.

4

Estimation of Kinetic Parameters via

the Incremental Approach

This chapter is adapted from the postprints of the following articles [93, 94]:

D. Rodrigues, J. Billeter, and D. Bonvin. Global identification of kinetic parameters via the extent-based incremental approach. Comput. Aided Chem. Eng., 40:2119–2124, 2017.

Link: http://doi.org/10.1016/B978-0-444-63965-3.50355-X. Copyright © 2017 Elsevier Ltd.

D. Rodrigues, J. Billeter, and D. Bonvin. Maximum-likelihood estimation of kinetic pa- rameters via the extent-based incremental approach. Comput. Chem. Eng., in press, 2018.

Link: http://doi.org/10.1016/j.compchemeng.2018.05.024. Copyright © 2018 Elsevier Ltd.

The author of this thesis contributed to those articles by developing the main novel ideas, implementing the simulations, and writing a significant part of the text. Hence, the author retains the right to include the articles in this thesis since they are not published commercially and the journal is referenced as the original source.

4.1

Introduction

The identification of reaction kinetics represents one of the main challenges in building first-principles models for reaction systems. Although the literature on this topic is extensive and includes several well-established textbooks [95, 96, 97], there remain significant chal- lenges. This chapter addresses some of these challenges for lumped homogeneous reaction systems. Typically, the identification task consists in confronting a set of candidate rate laws to experimental data and identifying the rate laws and the kinetic parameters that provide the best fit. This identification task can be performed via a simultaneous or an incremental approach as discussed next.

• Simultaneous identification is performed by postulating a rate law for each reaction in the model and estimating all kinetic parameters simultaneously. The modeled rates are integrated numerically, and all the parameters are estimated together so as to minimize

Chapter 4. Estimation of Kinetic Parameters via the Incremental Approach

the deviations between model predictions and measurements. The procedure is re- peated for all combinations of rate candidates, and the best combination is selected via appropriate model discrimination techniques. The main advantage of the simultaneous approach is that it leads to statistically optimal parameter estimates in the maximum- likelihood sense. Although these parameter estimates are generally not unbiased, they are consistent in the sense that the estimates converge to the true parameter values as the number of data points tends to infinity [98]. However, simultaneous identification can be computationally costly when there are many combinations of rate candidates to be tested. Moreover, enforcing convergence to global optimality is often slow, diffi- cult and dependent on the initial guesses due to the large number of parameters that need to be estimated simultaneously. Finally, since structural mismatch in one part of the model typically results in errors in all parameters, it is difficult to attribute the mismatch to a particular part of the model.

• Incremental identification is performed in several steps by decomposing the identifica- tion task into a set of smaller subproblems, with each subproblem corresponding to a single reaction. Hence, since each reaction is investigated individually, only the rate candidates for one reaction at a time need to be compared, and there are fewer param- eters to be estimated simultaneously [59]. These parameters are estimated such that the model predictions fit the experimental data of the corresponding reaction. Then, the rate candidate with the best fit is selected.

Note that the estimation of kinetic parameters can be done using either the differential or the integral method. In the differential method, the kinetic parameters are estimated by fitting the candidate rate laws to the experimental rates that are generated through dif- ferentiation of measured concentrations [99, 60, 61]. In the integral method, the kinetic parameters are estimated by fitting the integral of the candidate rate laws to either experi- mental numbers of moles (with the simultaneous approach) or experimental extents (with the incremental approach) [64].

In the context of the incremental approach, the aforementioned differential and integral methods are also known as rate-based and extent-based approaches. Hence, in extent-based incremental identification, the measured numbers of moles are first transformed to exper- imental extents via linear transformation as shown in Chapter 2, and then the rate laws are identified individually by comparing the experimental extents to the modeled extents that result from integration of the candidate rate laws. Compared to the rate-based ap- proach, the extent-based incremental approach provides parameter estimates with less bias, tighter confidence intervals and increased ability to discriminate among rate law candidates. However, it also requires more computational effort due to the need to integrate the rates numerically [58].

The main advantage of the incremental methods is that, thanks to the decoupling of the estimation problems, the number of rate candidates can be kept low for each subproblem, and convergence is faster. The main drawback is the fact that the candidate rate laws must be evaluated using measured concentrations because each rate is simulated individually 78

4.1. Introduction

and there is no information about the other rates. Since the rates that are computed from measured concentrations are typically biased, this bias is propagated to the identification problem, and the parameter estimates are not statistically optimal. Hence, simultaneous identification is typically performed in a final step, using the model structure identified via the incremental approach. This procedure results in statistically optimal parameter esti- mates and less computational effort than a purely simultaneous approach, because the rate laws are fixed and good initial parameter values are available. An alternative is to use a sequential approach, whereby the rate laws for the various reactions are identified sequen- tially [100]. This approach shares the advantages of the simultaneous approach, provided that the correct model structure is chosen at each step, while it lies in between the simulta- neous and the incremental approaches in terms of computational effort.

A drawback of these kinetic identification approaches is that, since they typically rely on local search methods, they can only converge to locally optimal parameters. This means that, if the solution to the identification problem does not correspond to the global opti- mum, the identified model may yield an incorrect description of the reaction system. Note that the incremental approach is better suited to global optimization since each estimation subproblem involves a small number of parameters. Taking these considerations into ac- count, an incremental parameter estimation method has been proposed to obtain globally optimal parameter estimates [101]. However, this estimation method is based on (i) the rate-based incremental approach, which can have several disadvantages compared to the extent-based approach as mentioned above, and (ii) branch-and-bound techniques, which are known to exhibit a worst-case complexity that is exponential in the number of decision variables.

This chapter presents a novel methodology for the identification of kinetic parameters for lumped homogeneous reaction systems. The novelty consists in (i) adopting the extent- based incremental approach, (ii) leading to statistically optimal parameter estimates in the maximum-likelihood sense, and (iii) guaranteeing convergence to global optimality. The global solution to this incremental approach has a quality similar to the global solution to the simultaneous approach but with a much smaller computational effort. The proposed approach does not rely on branch-and-bound techniques but rather on the reformulation of the nonconvex optimization problem as a convex problem, thereby taking advantage of the developments in polynomial optimization using sum-of-squares polynomials and semidefi- nite programming [102].

The structure of the chapter is as follows. Section 4.2 reviews the concept of extents as well as the procedure for model identification using the extent-based incremental approach. Section 4.3 presents two important steps necessary to guarantee maximum-likelihood pa- rameter estimates with the extent-based incremental approach. Section 4.4 discusses the reformulation of the original identification problem as a convex optimization problem and its solution. A simulated case study is presented in Section 4.5, while Section 4.6 concludes the chapter.

Chapter 4. Estimation of Kinetic Parameters via the Incremental Approach