# (642g) Model-Based Design of Experiments to Achieve Kinetic Validation for Chemical-Looping Systems

- Conference: AIChE Annual Meeting
- Year: 2014
- Proceeding: 2014 AIChE Annual Meeting
- Group: Catalysis and Reaction Engineering Division
- Session:
- Time: Thursday, November 20, 2014 - 10:35am-10:55am

**
font-family:"Times New Roman","serif"'>Model-based design of experiments to
achieve kinetic validation for chemical-looping systems**

**
line-height:115%;font-family:"Times New Roman","serif"'>Lu Han**,

Zhiquan Zhou, George M. Bollas

**
line-height:115%;font-family:"Times New Roman","serif"'>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

H_{2} 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]:

line-height:115%;font-family:"Times New Roman","serif"'>where
position:relative;top:4.0pt'>**Â **is

the vector of best available estimates of the model parameters,
position:relative;top:4.0pt'>Â is

the design vector,
position:relative;top:4.0pt'>**Â **is

the output trajectories,
position:relative;top:4.0pt'>Â is

the weighting vector, and
position:relative;top:4.0pt'>Â 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.Â The objective

function is written as:
position:relative;top:5.5pt'>

.25in">Â Â Â Â Â Â Â Subject

to:** ****f**[**x**,**y**,**p**,**u**,**t**]

= **0**, **x**(**t**_{0}) = **x**_{0}

81.0pt;line-height:normal">**h**[**x**,**y**,**p**,**u**,**t**]

= **0**

81.0pt">**g**[**x**,**y**,**p**,**u**,**t**]

≤ **0**, **x**^{L }≤ **x** ≤ **x**^{U},

**u**^{L} ≤ **u** ≤ **u**^{U}

line-height:115%;font-family:"Times New Roman","serif"'>where
position:relative;top:4.0pt'>Â 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].

"Times New Roman","serif"'>Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â The cases studies under

investigation are performed in a bench-scale fixed-bed unit focusing on the

reduction of NiO/γ-Al_{2}O_{3}-SiO_{2} oxygen

carrier by CH_{4}. 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 CH_{4}

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.

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0in;margin-left:0in;margin-bottom:.0001pt;line-height:normal">CH |
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0in;margin-left:0in;margin-bottom:.0001pt;line-height:normal">H |
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0in;margin-left:0in;margin-bottom:.0001pt;line-height:normal">CO+NiOÃ Ni+CO |
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0in;margin-left:0in;margin-bottom:.0001pt;line-height:normal">CH |
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**
line-height:115%;font-family:"Times New Roman","serif"'>Acknowledgement**:

This material is based upon work supported by the National Science Foundation

under Grant No. 1054718.

**
line-height:115%;font-family:"Times New Roman","serif"'>References**

[1]Â Â Â Â Â Â Â Â Â Â 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]Â Â Â Â Â Â Â Â Â Â S.P. Asprey, S. Macchietto,

Statistical tools for optimal dynamic model building, Comput. Chem. Eng. 24

(2000) 1261?1267.

[3]Â Â Â Â Â Â Â Â Â Â G.E. Box, K.B. Wilson, On the

Experimental Attainment of Optimum Conditions, J. R. Stat. Soc. Ser. B. 13

(1951) 1?45.

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