(642g) 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.

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.