A critical study of the Miura–Maki integral method for the estimation of the kinetic parameters of the distributed activation energy model

Abstract

Using some theoretically simulated data constructed from known sets of the activation energy distribution f(E) (assumed to follow the Gaussian distribution $f(E)=\frac{1}{\sigma \sqrt{2\pi }}\exp [-\frac{(E-E_0{)}^2}{2{\sigma }^2}]$, where E is the activation energy, E₀ is the mean value of the activation energy distribution, and σ is the standard deviation of the activation energy distribution) and the frequency factor k₀, a critical study of the use of the Miura–Maki integral method for the estimation of the kinetic parameters of the distributed activation energy model has been performed from three cases. For all cases, the use of the Miura–Maki integral method leads to important errors in the estimation of k₀. There are some differences between the assumed and calculated activation energy distributions and the differences decrease with increasing the assumed k₀ values (for Case 1), with increasing the assumed σ values (for Case 2), and with decreasing the b values (for Case 3).

Introduction

The distributed activation energy model (DAEM) has been proven very successful in describing the kinetics of biomass pyrolysis (Cai and Liu, 2007, Cai and Liu, 2007, Cai and Ji, 2007, Cai and Liu, 2008a, Cai and Liu, 2008b, Mani et al., 2009, Sonobe and Worasuwannarak, 2008), while it has also been applied to the kinetic description of the pyrolysis of other materials, such as coal (Güneş and Güneş, 2005, Güneş and Güneş, 2008, Li et al., 2009, Liu et al., 2001, Miura et al., 2001, Sun et al., 2006), and oil shale (Aboulkas et al., 2008). The model assumes an infinite number of irreversible first order parallel reactions with different activation energies to occur simultaneously, where the difference in activation energies is represented by a distribution function f(E) (Dawood and Miura, 2001):

$$\frac{V}{V^{\ast }}=1-{\int }_0^{\infty }\exp \left(-k_0{\int }_0^t\exp \left(-\frac{E}{\mathit{RT}}\right)\text{d}t\right)f(E)\text{d}E$$

In the above equation, V is the total amount of volatile formed by time t, V^∗ is the ultimate total amount volatile, k₀ is the frequency factor, E is the activation energy, R is the universal gas constant and T is the absolute temperature.

The kinetic analysis of biomass pyrolysis can be performed under different types of temperature programs, such as isothermal (Jankovic, 2009), linear nonisothermal (Cai and Liu, 2007, Cai and Ji, 2007), and nonlinear nonisothermal (Koga et al., 2000) measurements. Most laboratory experiments involving biomass pyrolysis are conducted at temperatures linearly varying with time. Under linear nonisothermal conditions, Eq. (1) can be rewritten as follows:

$$\frac{V}{V^{\ast }}=1-{\int }_0^{\infty }\exp \left(-\frac{k_0}{\beta }{\int }_0^T\exp \left(-\frac{E}{\mathit{RT}}\right)\text{d}T\right)f(E)\text{d}E=1-{\int }_0^{\infty }\mathrm{\Phi }(E\text{,}T)f(E)\text{d}E$$

where β = dT/dt is the heating rate, and $\mathrm{\Phi }(E\text{,}T)=\exp \left(-\frac{k_0}{\beta }{\int }_0^T\exp \left(-\frac{E}{\mathit{RT}}\right)\text{d}T\right)$

In the conventional DAEM, f(E) is generally assumed to follow a Gaussian distribution with a mean activation energy, E₀, and a standard deviation, σ:

$$f(E)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left[-\frac{(E-E_0{)}^2}{2{\sigma }^2}\right]$$

Some other specific mathematical forms of f(E) appearing in the literature are the Weibull (Burnham et al., 2004, Lakshmanan and White, 1994), Gamma (Burnham and Braun, 1999, Ding et al., 2005), Maxwell–Boltzmann (Skrdla, 2009), and logistic (Cai et al., 2010a, Cai et al., 2010b) distributions.

In the conventional DAEM, k₀ is assumed to be a constant for all reactions. However, in some existed papers (Miura, 1995, Miura and Maki, 1998b), the following compensation effect between k₀ and E was assumed.

$$k_0=a\exp (\mathit{bE})$$

where a and b are constants.

Considerable researches have focused on the estimation of f(E) and k₀. In the conventional methods for the estimation of the kinetic parameters of the DAEM, there are the following calculation steps: (1) k₀ is assumed to be an unknown constant (Güneş and Güneş, 2005, Liu et al., 2001) or is directly assigned to a given constant (Güneş and Güneş, 2002); (2) f(E) is assumed to be a Gaussian distribution; (3) the following objective function is constructed: $O.F.=\sum _{\text{i}=1}^{n_{\text{d}}}{\left[{\left(\frac{V}{V^{\ast }}\right)}_{\text{i,cal}}-{\left(\frac{V}{V^{\ast }}\right)}_{\text{i,exp}}\right]}^2$, where ${\left(\frac{V}{V^{\ast }}\right)}_{\text{i,exp}}$ and ${\left(\frac{V}{V^{\ast }}\right)}_{\text{i,cal}}$ are experimental and calculated values of the degree of conversion, respectively, n_d is the data number; (4) to determine the kinetic parameters minimizing the objective function using some optimization methods such as the direct search method (Güneş and Güneş, 2002), the pattern search method (Cai and Ji, 2007), and the simulated annealing optimization method (Mani et al., 2009).

Miura and Maki (1998a) proposed an integral method to determine f(E) and k₀ without any previous assumptions for f(E) and k₀. In this method, only three sets of experimental kinetic data obtained at different heating rates are required. The method has been widely used for estimating f(E) and k₀ of the DAEM. Thus, according to SCOPUS website (http://www.scopus.com/), more than 70 citations can be found in the literature for the method. Despite the popularity of the method, the accuracies of the estimation of the kinetic parameters (f(E) and k₀) of the DAEM is still in doubt. This is the scope of the present paper. Before proceeding to the evaluation of the method, it is necessary to briefly introduce the calculating procedure of the method. In further discussion, the method will be referred to as the Miura–Maki integral method.