Model-Based Design of Experiments to Achieve Kinetic Validation for Chemical-Looping Systems

Model-based design of experiments to achieve kinetic validation for chemical-looping systems

Lu Han, Zhiquan Zhou, George M. Bollas

Abstract

A model-based experimental design approach is applied to design experiments for model discrimination and parameter estimation of an ill-defined model structure. This concept is illustrated for a chemical-looping process, in which understanding of the intrinsic kinetics of the non-catalytic gas-solid reactions is crucial. Solid-state kinetic models are usually derived empirically and supported by advanced characterization techniques. These models can be classified according to their mechanistic basis, such as nucleation, geometrical contraction, diffusion, and reaction order. As these models vary in their degree of complexity, statistical approaches are used to discriminate the overall best-suited model that fits the experimental data. Recently, Zhou et al. used the corrected Akaike Information Criterion (AICc) and the F-test on twenty solid-state kinetic models for describing the reduction NiO by H₂ and oxidation of Ni by air [1]. All of the models were compared against experimental data from the literature and in-house experiments. However, due to subtle differences between the kinetic models, several cases presented by Zhou et al. appear to have multiple winning models. To address this issue, the approach employed in this work utilizes the rival models to design experiments that maximize the divergence of the model predictions. By inspection of the quality of fit, the choice of the winner model (i.e., model discrimination) can be made with improved statistical significance. The method for model discrimination is formulated as an optimal control problem [2]:

where  is the vector of best available estimates of the model parameters,  is the design vector,  is the output trajectories,  is the weighting vector, and  is the discrete sampling time. This problem can be applied for practically any number of N_M rival models and it is not certain which of the models is the best.

Furthermore, it becomes important to decrease the size of the confidence intervals of each of the parameters in the best-suited model. Optimal experimental design is cast as an optimization problem of the control variables that maximizes the sensitivity of the output variables with respect to the model parameters. This is reflected in the Fisher information matrix, F:

where y denotes the model outputs, p the set of unknown, parameters , and Q the inverse of the measurement error covariance matrix.  The objective function is written as:

        Subject to: f[x,y,p,u,t] = 0, x(t₀) = x₀

h[x,y,p,u,t] = 0

g[x,y,p,u,t] ≤ 0, x^L ≤ x ≤ x^U, u^L ≤ u ≤ u^U

where  is a metric of the selected design criterion, x is the vector of state variables, u the vector of manipulated variables, f is the system of ordinary differential equations, h and g are equality and inequality algebraic constraints, and U and L are the upper and lower bounds for x and u. In this work, D-optimality criterion, aimed at maximizing the determinant of the information matrix, is used, as it maximizes the overall information while at the same time decreasing the degree of correlation between parameters [3].

                The cases studies under investigation are performed in a bench-scale fixed-bed unit focusing on the reduction of NiO/γ-Al₂O₃-SiO₂ oxygen carrier by CH₄. In this configuration, over 20 redox cycles are typically conducted to analyze the stability of the oxygen carrier. Concerning only the reduction reactions with Ni, sets of parallel experiments are designed to maximize the statistical confidence in the Arrhenius rate constants. The set of manipulated variables include: reduction temperature, bed length, and CH₄ fraction. The experimental designs are evaluated in terms of the statistical quality of the model parameters fitted to the collected data. Figure 2 shows the experimental results for two different temperatures and corresponding model predictions. The quality of fit is excellent and the estimated kinetic parameters (Table 1) are in good agreement with studies. A decrease in statistical uncertainty is achieved with this selection of experiments in comparison to performing a single experiment.

Reaction	Frequency factor [m/s]	Activation energy [kJ/mol]
CH4+2NiOà2Ni+CO2+H2	1.7E-01	108
H2+NiOàNi+H2O	1.7E-03	56
CO+NiOàNi+CO2	2.4E-03	36
CH4+NiOàNi+2H2+CO	6.0E-03	20

 

 

 

 

 

 

Acknowledgement: This material is based upon work supported by the National Science Foundation under Grant No. 1054718.

References

[1]           Z. Zhou, L. Han, G.M. Bollas, Kinetics of NiO reduction and Ni oxidation at conditions relevant to chemical-looping combustion and reforming, Int. J. of Hydrogen Energy. 39 (2014) 8535–8556.

[2]           S.P. Asprey, S. Macchietto, Statistical tools for optimal dynamic model building, Comput. Chem. Eng. 24 (2000) 1261–1267.

[3]           G.E. Box, K.B. Wilson, On the Experimental Attainment of Optimum Conditions, J. R. Stat. Soc. Ser. B. 13 (1951) 1–45.