Journal of Chemistry

Volume 2018, Article ID 5127393, 8 pages

https://doi.org/10.1155/2018/5127393

## Global Sensitivity Analysis of Large Reaction Mechanisms Using Fourier Amplitude Sensitivity Test

^{1}School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China^{2}Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

Correspondence should be addressed to Shengqiang Lin; moc.liamg@qslnaijuf

Received 5 May 2018; Accepted 5 July 2018; Published 1 August 2018

Academic Editor: Kokhwa Lim

Copyright © 2018 Shengqiang Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Global sensitivity analysis (GSA) of large chemical reaction mechanisms remains a challenge since the model with uncertainties in the large number of input parameters provides large dimension of input parameter space and tends to be difficult to evaluate the effect of input parameters on model outputs. In this paper, a criterion for frequency selection to input parameter is proposed so that Fourier amplitude sensitivity test (FAST) method can evaluate the complex model with a low sample size. This developed FAST method can establish the relationship between the number of input parameters and sample size needed to measure sensitivity indices with high accuracy. The performance of this FAST method which can allow both the qualitative and quantitative analysis of complex systems is validated by a H_{2}/air combustion model and a CH_{4}/air combustion model. This FAST method is also compared with other GSA methods to illustrate the features of this FAST method. The results show that FAST method can evaluate the reaction systems with low sample size, and the sensitivity indices obtained from the FAST method can provide more important information which the variance-based GSA methods cannot obtain. FAST method can be a remarkably effective tool for the modelling and diagnosis of large chemical reaction.

#### 1. Introduction

Combustion in engines with a practical fuel is very complex and is commonly described by a detailed mechanism which may involve dozens or even hundreds of reactions. However, the detailed mechanism is not well understood especially for reaction rate coefficients which may be uncertain by one or more orders of magnitude [1]. For many gas-phase elementary reactions, reaction rates have uncertainty of 10% to 30% [2, 3]. The uncertainties in the rate coefficients will significantly affect the production rate of species and ultimately affect the combustion behaviors such as temperature profile, ignition delay time, and laminar flame speed. In order to investigate the detailed mechanism with large uncertainties in the reaction rate coefficients, sensitivity analysis is employed to study how the uncertainties in the outputs can be apportioned to different sources of uncertainties in its inputs. This investigation can help researchers to determine the important reactions and reaction pathways for practical fuels and model detailed mechanism. There are two classes of sensitivity analysis techniques: local sensitivity analysis (LSA) and global sensitivity analysis (GSA) [3–5]. LSA is only applicable to the best-known model and not applicable to the model with large uncertainties in the input parameters, and it may actually produce inaccuracies in evaluating detailed mechanism [6–11]. Hence, many GSA methods have been developed to analyze complex models such as FAST [12–14], Sobol’ [15], HDMR [6, 16–19], and the ordinary least squares approach [20]. These methods have been successfully applied to complex reaction models such as reaction mechanism construction [21–24] and chemical reaction mechanism analysis [6, 19, 25]. However, these GSA methods are rarely employed to evaluate large numbers of input numbers.

Sobol’ method is a standard variance-based sensitivity analysis tool to evaluate the complex systems, and it can obtain accurate variance-based sensitivity indices with a large sample size. However, it cannot be applied to the models with large numbers of input parameters [8]. Since Sobol’ method does not employ the dimensionality reduction techniques like FAST and HDMR methods, it needs very large sample size to evaluate the reaction systems. Thus, the computational cost for Sobol’ method will be very high. However, Sobol’ method with large sample size is widely applied to validate other GSA methods through a model with small input parameters since sensitivity indices for complex reaction systems could not be analytically evaluated. In this paper, the variance-based sensitivity indices from the FAST method can be validated by comparing it with sensitivity indices from Sobol’ method. Another version of the FAST method, which can identify the positive or negative effect of input parameter on products, is also validated by using other measures. When sensitivity indices obtained from the FAST method that employs the dimensionality reduction techniques is validated to be reliable, the FAST method is applied to GSA of larger numbers of input parameters.

The performance of the FAST method is validated by using it to diagnose two test cases: a H_{2}/air combustion model [26] and the well-known GRI3.0 (CH_{4}/air) combustion model [27]. A H_{2}/air combustion mechanism which has been defined the uncertainty factor of each reaction is commonly chosen to validate the performance of the GSA methods, such as Li et al. [8] and Davis et al. [19]. In this paper, the FAST method has been compared with other GSA methods in order to evaluate the performance of the FAST method. A GRI3.0 combustion mechanism which involves 53 species and 325 reactions is commonly applied to investigate the turbulent combustion regime [28, 29]. However, only few references of GSA methods are validated by using them to large combustion mechanisms with a small sample size and a low computational cost.

#### 2. Methodologies

Since the reaction rate coefficients are not known with high accuracy, an uncertainty factor (UF) of a reaction rate is employed to evaluate the degree of uncertainty in the reaction rate coefficients and is defined aswhere is the nominal value and and are the upper and lower bounds of the reaction rate, respectively. A classic FAST method was developed by Cukier et al. [12–14] to evaluate chemical reaction mechanisms with large uncertainties in rate coefficients, and the effect of the reaction rate coefficients on the output parameters can be written in the following form:where is the th reaction rate and is the th product, and can be the concentration of species, temperature, and ignition delay time. In this paper, perfectly stirred reactor (PSR) is selected as a reactor model to investigate the effect of uncertainties in the rate coefficients on the products. This reactor model is commonly used to predict profiles of species and ignition delay time, validate prediction ability of detailed reaction mechanisms, and evaluate the performance of the GSA methods [8, 19].

The FAST method can estimate sensitivity indices with a small sample size and low computational cost since it employs dimensionality reduction techniques. This method introduces different frequencies to represent different input parameters so that it can evaluate every input parameter with same equally spaced points [12]. When the importance of input parameters is evaluated by the FAST method, we must first choose a set of *r* integer frequencies, , for the input factors, where *r* is the number of independent reaction rate coefficients. Frequency selection is crucial for the FAST method to obtain sensitivity indices with high accuracy since frequency selection should avoid interferences up to a given order *M*, where *M* is the interference factor and commonly set as 4∼6 [30]. If interferences occur on the input parameters, the sensitivity indices can be overestimated. Note that the larger the *M*, the more the sample size is needed. Thus, *M* is selected to be 4 and a criterion for frequency selection is presented as follows:where is an integer and it is determined by , and the inequalities . can be set as 7, 9, and 11 when the model involves 3 input parameters. This formula for frequency selection can greatly avoid interferences from input parameters. When *M* and are given, the total sample size for the FAST method can be determined as follows:

Therefore, the FAST method can establish the relationship between the number of input parameters and total sample size needed. This is a good advantage for the GSA methods to evaluate the model with large input parameters since it is difficult to determine the total sample size for the GSA methods in order to measure the sensitivity indices with high accuracy.

FAST method generates sampling points in the *k*th dimensional input space by the space-filling curve. In this section, a space-filling curve introduced by Saltelli and Chan [30] is adopted since it can provide a uniformly distributed sample and is presented as follows:where is a probability density function (PDF) of the th input parameter, is the parametric variable varying in , which is sampled over its range using *N* points, and is the integer frequency determined by (3). After sample size is determined, in the Fourier series can be expanded aswhere Fourier amplitudes, and , can be written as follows:

The sensitivity indices measured in the classic FAST method [12] for input factor are written in the following form:

A classic FAST method only considers the Fourier amplitude as sensitivity index of an input parameter . Such sensitivity index can be positive or negative; the positive sign indicates increase in products with increasing and the negative sign indicates decrease in products with increasing since is proportional to . Due to the symmetry properties of space-filling curve and reduction in the number of model evaluations, (7) was upgraded by Koda et al. [11] and written as follows:

The sensitivity indices measured in such a variance-based FAST method can be written in the following form:where is referred to as the first-order sensitivity index, is the total variance of product , and is a part of total variance and indicates the contribution of input factor to the product since frequency is associated with . The variances and can be written respectively as follows:

Note that is a normalized sensitivity index in the variance-based FAST method so that may be ordered with respect to . Such sensitivity index is different when it is calculated by (10), and it cannot identify how an input parameter plays a positive or negative role in products.

When (3) is applied to select frequencies for input parameters, is always odd. Thus, = 0, and a brief formula can be obtained to calculate sensitivity indices by combining the method of Cukier et al. and Koda et al. Therefore, the formula can be written as follows:

In the present study, sensitivity index obtained from (10) and sensitivity index obtained from (12) will be compared. In other words, sensitivity indices with or without normalization will be compared.

#### 3. Results and Discussion

##### 3.1. H_{2}/Air Reaction Mechanisms

The aim of this section is to validate the formula of frequency selection proposed in (3) and evaluate the performance of the FAST method. The H_{2}/O_{2} combustion model developed by Konnov [26] is selected as test case to validate the performance of the FAST method since this combustion model has defined the uncertainty factor for each reaction. In addition, this model is employed to study the effect of uncertainty in the reaction rate coefficients on autoignition process, which were calculated with the Senkin code [31] for a stoichiometric H_{2}/air mixture at a temperature of 1000 K and pressure of 0.1 atm.

For a reaction system, products are always functions of time. Hence, the FAST and Sobol’ methods are extended to obtain the profiles of sensitivity indices with time in this section. Besides that, there are a few GSA researches assessing the entire autoignition process that plays important role in the validation of a detailed mechanism. In this case, we generated logarithmically 1000 points in the range of 10^{−5} to 10^{−2} s as time points for the sensitivity indices measured by the GSA methods, and the ordinary differential equation (ODE) solver should be modified for obtaining the calculation results at fixed points in time. Figure 1 shows the probability distribution of temperature prediction due to the uncertainties in the rate coefficients. It can be seen that the uncertainties in the rate coefficients can greatly affect the autoignition process. When reactions are selected as chemical kinetics mechanisms for hydrogen combustion, the uncertainty exists in the profile of temperature prediction and other products. GSA methods provide diagnostic tools to measure the importance of reactions in order to obtain reaction kinetics mechanisms which can accurately predict combustion behaviors, and identify important reactions and reaction pathways.